UNIVERSITY OF OKLAHOMA
GRADUATE COLLEGE
CHARACTERIZATION OF THE CONVECTIVE BOUNDARY LAYER
THROUGH A COMBINATION OF LARGE-EDDY SIMULATIONS AND A
RADAR SIMULATOR
A DISSERTATION
SUBMITTED TO THE GRADUATE FACULTY
in partial fulfillment of the requirements for the
Degree of
DOCTOR OF PHILOSOPHY
By
DANNY SCIPIO´N
Norman, Oklahoma
2011

c￿Copyright by DANNY SCIPIO´N 2011
All Rights Reserved.
Acknowledgments
This work was primarily supported by the National Science Foundation (NSF) for
their support via grant ATM-0553345. The observational data for this study were
obtained from the Atmospheric Radiation Measurement (ARM) Program sponsored
by the U.S. Department of Energy, Office of Science, Office of Biological and Envi-
ronmental Research, Environmental Science Division. Special thanks to committee
co-chairs Dr. Robert Palmer and Dr. Phillip Chilson for their guidance, comments,
and suggestions during the course of this project. Additional thanks to Dr. Evgeni
Fedorovich for his time and advice, especially in giving a meteorlogical perspective to
the project. I would also like to acknowledge my committee members, Dr. Tian-You
Yu and Dr. Richard Doviak for their efforts put forth in reviewing my work and for
offering their comments, suggestions, and corrections.
In addition, I would like to thank my parents, Eddy Scipio´n and Juany Castillo
de Scipio´n for being my motivation, and showing me the value of pursuing higher
education in order to succeed in a competitive world. Finally, to Dustin Baum for his
support, encouragement, understanding, patience, and especially his time throughout
this project.
iv
Contents
Acknowledgments iv
List Of Tables viii
List Of Figures ix
Abstract xv
1 Introduction 1
1.1 The Atmospheric Boundary Layer: A Brief Review . . . . . . . . . . 1
1.2 Radars for Sensing the Atmosphere . . . . . . . . . . . . . . . . . . . 6
1.2.1 Weather Radars: Particle Scattering From Precipitation . . . 7
1.2.2 Profiling Radars: Clear-Air Scattering from Refractive Index
Variations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.3 Large-Eddy Simulations to Characterize the Atmospheric Boundary
Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.4 Motivation for Dissertation . . . . . . . . . . . . . . . . . . . . . . . . 18
1.5 Outline of Dissertation . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2 Large-Eddy Simulations and Doppler Radars: Theoretical Back-
ground 20
2.1 Large-Eddy Simulations . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.2 Pulsed Doppler Radar . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.2.1 Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.2.2 The Radar Equation . . . . . . . . . . . . . . . . . . . . . . . 33
2.2.3 Scattering Mechanisms . . . . . . . . . . . . . . . . . . . . . . 33
2.2.3.1 Weather Radar - Rayleigh Scatter . . . . . . . . . . . 33
2.2.3.2 Clear-Air Radar - Bragg Scatter . . . . . . . . . . . 35
2.3 Signal Processing for Radars that Observe the Atmosphere . . . . . . 37
2.3.1 Spectral Moments . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.3.1.1 Moment Estimation in the Time Domain . . . . . . . 40
2.3.1.2 Moment Estimation in the Frequency Domain . . . . 42
2.4 Techniques for the Estimation of the Three-Dimensional Wind Field . 43
2.4.1 Doppler Beam Swinging . . . . . . . . . . . . . . . . . . . . . 43
2.4.2 Spaced Antenna . . . . . . . . . . . . . . . . . . . . . . . . . . 45
v
3 Radar Simulator Design Based on Large-Eddy Simulations 49
3.1 Time Series Radar Simulator . . . . . . . . . . . . . . . . . . . . . . . 49
3.2 Possible Configurations of the Virtual Radar . . . . . . . . . . . . . . 56
3.2.1 Monostatic Configuration . . . . . . . . . . . . . . . . . . . . 58
3.2.1.1 Doppler Beam Swinging . . . . . . . . . . . . . . . . 59
3.2.1.2 Multiple Radar Configuration . . . . . . . . . . . . . 61
3.2.2 Spaced Antenna . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4 Characterization of the Boundary Layer: Validation with Realistic
Data 64
4.1 Experimental Configuration . . . . . . . . . . . . . . . . . . . . . . . 64
4.2 Structure Function Parameter of Refractive
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.3 Wind Field Comparisons . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.4 Vertical Velocity Variance . . . . . . . . . . . . . . . . . . . . . . . . 73
4.5 Vertical Velocity Skewness . . . . . . . . . . . . . . . . . . . . . . . . 75
5 Comparison between LES and Virtual Boundary Layer Radar 78
5.1 Effect of Horizontal Shear of Vertical Velocity on Horizontal Wind
Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
5.1.1 Theoretical Formulation . . . . . . . . . . . . . . . . . . . . . 78
5.1.1.1 Doppler Beam Swinging . . . . . . . . . . . . . . . . 78
5.1.1.2 Spaced Antenna . . . . . . . . . . . . . . . . . . . . 79
5.1.2 Test Configurations . . . . . . . . . . . . . . . . . . . . . . . . 80
5.1.2.1 DBS Configuration . . . . . . . . . . . . . . . . . . . 81
5.1.2.2 Spaced Antenna Configuration . . . . . . . . . . . . 83
5.1.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
5.2 Turbulence Kinetic Energy Based on Radar Wind Estimates . . . . . 97
5.2.1 Theoretical Formulation . . . . . . . . . . . . . . . . . . . . . 97
5.2.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
5.3 Turbulence-Eddy Dissipation Rate Based on Radar Spectrum Width 99
5.3.1 Theoretical Formulation . . . . . . . . . . . . . . . . . . . . . 99
5.3.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
6 Conclusions and Future Work 110
6.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
6.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
Reference List 116
Appendix A - List Of Symbols 127
Appendix B - List Of Acronyms and Abbreviations 135
Appendix C
Calculations of Structure Function Parameter of Refractive Index . . . . . 138
vi
Index 139
vii
List Of Tables
1.1 Fractional concentration by volume of the major constituents of the
Earth’s atmosphere up to 105 km, with respect to dry air (fromWallace
and Hobbs (2006)). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Example of typical VCPs utilized by the WSR-88D based on the type
of weather occurring (from http://www.wdtb.noaa.gov/tools/RPS/). 10
1.3 Classification of radars according to their altitude coverage . . . . . . 15
2.1 Description of the LES settings . . . . . . . . . . . . . . . . . . . . . 24
2.2 Examples of the values of Z . . . . . . . . . . . . . . . . . . . . . . . 35
2.3 Examples of the magnitude and intensity of C2n . . . . . . . . . . . . 36
3.1 Description of the LES settings . . . . . . . . . . . . . . . . . . . . . 51
5.1 Description of the field domain used by the radar simulators . . . . . 81
5.2 Specifications of DBS configuration . . . . . . . . . . . . . . . . . . . 82
5.3 Specifications of SA configuration . . . . . . . . . . . . . . . . . . . . 84
viii
List Of Figures
1.1 The structure of the Earth’s atmosphere (modified from Ahrens (1991)).
The structure reflects how temperature, density, and composition of the
atmosphere change with height, and how it is determined in large part by
how it interacts with the sun’s light. . . . . . . . . . . . . . . . . . . . . 3
1.2 Depiction of the diurnal development of the atmospheric boundary layer
(adapted from Stull (1988)). . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 Location of radars present at the OPERA radar database updated in June
2010 (from http://www.knmi.nl/opera/). The color dots represent the
frequency bands: C-band = green, S-band = blue, X-band = cyan. The
center color represent the type of radar: dual polarization radar = red
center, Doppler radar= yellow center. . . . . . . . . . . . . . . . . . . . 8
1.4 Location of the operational WSR-88D radars (National Oceanic and At-
mospheric Administration (NOAA) 2007 (from http://radar.weather.
gov/). The color coding represent the US agency in charge of the correct
operation of the radar. . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.5 (a) NEXRAD Radar at the WSR-88D Radar Operations Center in Nor-
man, OK (from http://en.wikipedia.org/wiki/NEXRAD). (b) Phased-
Array Radar (PAR) in Norman, OK (from http://www.thunderchaser.
com/june2005/spc_tour.html). . . . . . . . . . . . . . . . . . . . . . . 11
1.6 (a) Jicamarca Incoherent Scatter Radar at Lima, Peru (from http://jro.
igp.gob.pe). (b) The SOUSY (SOUnding SYstem) Radar at Lindau,
Germany (from http://radars.uit.no/sousy/index.html). (c) NARL
MST Radar at Gadanki, India (from http://www.narl.gov.in/. (d) MU
(Middle and Upper atmosphere) Radar at Shigaraki, Japan (from http:
//www-lab26.kuee.kyoto-u.ac.jp/study/mu/mu_e.html). . . . . . . . 14
1.7 Schematic of the evolution of the inertial subrange with height. The hori-
zontal axis represent the Bragg scale. In the troposphere, the Bragg scale ex-
tends safety from 10−1 to 10 m, which corresponds to frequencies of 1.5 GHz
(UHF) - 15 MHz (VHF), respectively (from Hocking (1985)). . . . . . . . 15
2.1 Schematic representation of idealized (a) and realistic (b) initial data (from
Botnick and Fedorovich (2008)). . . . . . . . . . . . . . . . . . . . . . . 23
2.2 Examples of LES fields in the virtual radar sub-domain. Top-left: zonal
wind. Top-right: meridional wind. Middle-left: potential temperature.
Middle-right: specific humidity. Bottom-left: vertical wind. Bottom-right:
sub-grid kinetic energy. All data presented refer to the same single realiza-
tion in time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
ix
2.3 Three-dimensional wind fields obtained at the center of the LES sub-domain
for the ∼11.5 hours of simulation. Top: zonal wind. Middle: meridional
wind. Bottom: vertical wind. . . . . . . . . . . . . . . . . . . . . . . . . 26
2.4 Simple schematic of the basic principles of a radar. (a) A pulse is transmitted
into space. (b) The pulse reaches the target. (c) The target scatters in all
directions. (d) Some of the energy is reradiated towards the antenna and
delivered to the receiver. . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.5 Block Diagram of a homodyne Doppler Radar. The RF signal generated
by the STALO is modulated, amplified, and transmitted into space by the
antenna. The received signal is “mixed” with the same RF from STALO,
filtered, processed, and finally, displayed for the user. . . . . . . . . . . . 28
2.6 Block Diagram of a heterodyne Doppler Radar. A difference from the homo-
dyne system presented in Figure 2.5 is the generation of the RF frequency
by combining the STALO and COHO signals. The received signal is mixed
with the STALO to obtained an IF signal. This IF signal is sampled by
a digital receiver, where matched filtering, baseband conversion and phase
referencing for coherent-on-receive operation are realized by digital signal
processing hardware and software. . . . . . . . . . . . . . . . . . . . . . 29
2.7 Idealized traces for the radar signal (adapted from (Doviak and Zrnic´ 1993)).
Each trace represent a composite of echoes for each transmitted pulse. In-
stantaneous samples are taken at Tr1, Tr2, etc. The red stars represent a
probable time dependence of the sample at Tr. Samples at fixed Tr taken at
Ts intervals are used to construct the Doppler spectrum from targets located
at approximately the range cTr/2. . . . . . . . . . . . . . . . . . . . . . 31
2.8 Simple diagram of the Kolmogorov turbulence spectrum (adapted from Pope
(2000)). The energy spectrum of turbulence E(κ) and wavenumber κ axes
are in logarithmic scale. In the inertial subrange, the large eddies from the
energy-containing range cascades into the small eddies in the dissipation
range following a slope of -5/3. . . . . . . . . . . . . . . . . . . . . . . . 36
2.9 Schematic of the Doppler spectrum. S represents the signal power, N is the
noise power (usually assumed “white”), vr is the mean radial velocity, and
σv represents the Doppler spectral width. . . . . . . . . . . . . . . . . . 40
2.10 Schematic of the autocorrelation function. Top: magnitude of the ACF.
Signal power is represented by S, N is the noise power. Bottom: phase of
the ACF. The radial velocity vr is calculated based on the phase of the ACF
at lag 1 (τ = 1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.11 Schematic of the calculations of radial velocity vr as a function of the three
wind fields components. Left: the radial velocity is expressed as a function
of the horizontal velocity and the vertical wind field. Right: the horizontal
velocity is expressed as a function of the zonal and meridional wind fields. 44
2.12 Schematic of the Doppler Beam Swinging Technique. DBS uses at least
three non-coplanar beams to estimate the three components of the wind
field under the assumptions of homogeneity and stationarity during the time
of the scan. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
x
2.13 Normalized autocorrelation (c11(τ)) and cross-correlation (c12(τ)) functions.￿η represents the decorrelation paramenter, ￿τp is the peak time of CCF, ￿τc
is the square root of the second central moment of CCF, and τx is the lag
such that |c11(τx)| = |c12(0)|. . . . . . . . . . . . . . . . . . . . . . . . . 47
3.1 Left: scheme that represents an off-vertical pointing beam. The dotted
lines represent the radial from the location of the radar to the grid points
of the LES within the resolution volume. Along these dotted lines, C2n is
calculated, and later weighted in range (Wr) and beam width (Wb). Right:
the dotted line represents the axis along which C2n needs to be estimated.
First, the value of n at level z is obtained from the LES matrix; however,
at levels z + ∆z and z − ∆z, the dotted line does not match any of the
LES grid points. In order to obtain an estimate at those heights, a linear
interpolation is made using the four closest points. Finally, an average of
the two estimates of structure function parameter of n (from z and z+∆z,
and z and z −∆z) with an oblique separation ∆ is computed. . . . . . . 52
3.2 Left: vertical profile of specific humidity characteristics for the CBL. Center:
vertical profile of refractivity calculated from the LES fields. Right: vertical
profile of the horizontally averaged structure function parameter of refractive
index C2n. All data were calculated from a single realization in time. . . . 53
3.3 Time-series data from the simulated BLR data V (t) obtained at a range of
475 m. The blue line represents the real part (in-phase) of V (t), and the
red line is the imaginary part (quadrature) of V (t). Top: without AWGN.
Bottom: with AWGN (minimum SNR = -3 dB). . . . . . . . . . . . . . . 57
3.4 Schematic of the monostatic radar configuration. The simulated radar has
the capability to point at any desired azimuth and zenith angle provided
the resolution volume is located within the LES sub-domain. . . . . . . . 58
3.5 Doppler beam swinging configuration. The configuration is based in a sin-
gle monostatic radar that points in five different, non-coplanar directions.
Usually, the oblique beams point to the four cardinal directions. . . . . . 60
3.6 Multiple monostatic radar setup. The configuration uses the assumption
that five simultaneous monostatic radars point towards the same resolution
volume inside the LES. . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.7 Spaced Antenna Setup. This experiment uses one antenna for transmission
and three or more for reception. The distance between the transmitter (Tx)
and receivers (RxA and/or RxB) is small, so it allows the beams to intercept.
The different transmitter and receiver beamwidths have been incorporated
in the radar simulator. Here, the beamwidth is exaggerated for diagram
purposes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.1 Picture of the Radial Wind Profiler (RWP) located at the SGP ACRF. In
addition to the wind profiles, the system can obtain virtual temperatures
through the Radio Acoustic Sounding System (RASS) co-located with the
radar (Courtesy of Dr. P. Chilson). . . . . . . . . . . . . . . . . . . . . 65
xi
4.2 Structure function parameter of the refractive index estimates on June 8,
2007. Top: estimates from the vertical beam of the radar simulator. Middle:
LES-profile at center of domain. Bottom: estimates from the vertical beam
of the SGP ACRF radar. . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.3 Zonal wind estimates averaged every 12 min on June 8, 2007. Top: LES-
DBS. Middle: LES-Profile at center of domain. Bottom: DBS estimates
from the SGP ACRF radar. . . . . . . . . . . . . . . . . . . . . . . . . 69
4.4 Meridional wind estimates averaged every 12 min on June 8, 2007. Top:
LES-DBS. Middle: LES-Profile at center of domain. Bottom: DBS esti-
mates from the SGP ACRF radar. . . . . . . . . . . . . . . . . . . . . . 70
4.5 Vertical wind estimates averaged every 12 min on June 8, 2007. Top: LES-
DBS. Middle: LES-Profile at center of domain. Bottom: DBS estimates
from the SGP ACRF radar. . . . . . . . . . . . . . . . . . . . . . . . . 71
4.6 Wind comparison from the RadSimDBS , LES, RadarARM , and a sounding
(launched at three different times: 12:00, 18:00, and 00:00 UTC) at ∼530 m
(left) and ∼1030 m (right) on June 8, 2007. Top: zonal wind. Middle:
meridional wind. Bottom: vertical velocity. . . . . . . . . . . . . . . . . 72
4.7 Vertical velocity variance as a function of height for different periods of
time from left to right (14:00-16:00; 16:00-18:00; 18:00-20:00; and 20:00-
22:00 UTC). Red line: estimates from the virtual BLR. Blue line: estimates
from the LES. Black line: estimates from the BLR located at the SGP
ACRF site. The horizontal dashed line represents the CBL top obtained
from the LES C2n maximum averaged over the period under study. . . . . 74
4.8 Skewness of the vertical velocity as a function of height for different periods
of time from left to right (14:00-16:00; 16:00-18:00; 18:00-20:00; and 20:00-
22:00 UTC). Red line: estimates from the virtual BLR. Blue line: estimates
from the LES. Black line: estimates from the BLR located at the SGP
ACRF site. The horizontal dashed line represents the CBL top obtained
from the LES C2n maximum averaged over the period under study. . . . . 76
5.1 Zonal wind estimates obtained using DBS for different shear conditions.
Left: sfrac = 0.00 (no shear). Middle: sfrac = 0.50. Right: sfrac = 1.00. . 82
5.2 Zonal wind bias. Line: Ideal bias. Dots: estimated from the DBS simulation
for different values of sfrac. . . . . . . . . . . . . . . . . . . . . . . . . . 83
5.3 Diagram showing the location of the SA transmitter (Tx) and receivers
(RxA, RxB, RxC, and RxD). The black dots represent the location of the
four receivers in the cross-configuration. As noted, they are located sym-
metrically, with respect to the Tx, along the zonal and meridional axes. . . 84
5.4 Zonal wind estimates obtained using SA for different shear conditions. Left:
sfrac = 0.00 (no shear). Middle: sfrac = 0.50. Right: sfrac = 1.00. . . . . 85
5.5 Zonal wind bias. Line: Ideal bias. Dots: estimated from the SA simulation
for different values of sfrac. . . . . . . . . . . . . . . . . . . . . . . . . . 86
5.6 Top: Signal-to-Noise Ratio as a function of time. The asterisks represent
estimates of the BL top. Bottom: censored SNR plot. Values less than
-3 dB and above the BL top have been removed. . . . . . . . . . . . . . . 87
xii
5.7 Zonal wind estimates as a function of time for sfrac = 1.00. Top: LES
estimates. Middle: DBS estimates. Bottom: SA estimates obtained from
the FCA-Gaussian method. The first height is removed for DBS, and the
first three heights for SA. . . . . . . . . . . . . . . . . . . . . . . . . . 89
5.8 Same as Figure 5.7, except for the meriodional wind. . . . . . . . . . . . 90
5.9 Comparison between the DBS estimates versus LES fields. Top: zonal wind
for different sfrac values. Bottom: meridional wind for the same sfrac values.
Left: sfrac = 0.00. Middle: sfrac = 0.50. Right : sfrac = 1.00. . . . . . . 91
5.10 Error analysis as a function of time for sfrac = 1. Top: zonal wind error
from DBS estimates. Middle-top: range corrected zonal shear from LES.
Middle-bottom: meridional wind error from DBS estimates. Bottom: range
corrected meridional shear from LES. . . . . . . . . . . . . . . . . . . . 92
5.11 Error from the DBS estimates vs. range corrected shear from LES for sfrac
= 1.00. Left: zonal wind. Right: meridional wind. . . . . . . . . . . . . 93
5.12 Wind bias for DBS. Left: zonal wind. Right : meridional wind. . . . . . . 94
5.13 Comparison between the SA estimates vs. LES fields. Top: zonal wind for
different sfrac values. Bottom: meridional wind for the same sfrac values.
Left: sfrac = 0.00. Middle: sfrac = 0.50. Right : sfrac = 1.00. . . . . . . 94
5.14 Error analysis as a function of time for sfrac = 1. Top: zonal wind error
from SA estimates. Middle-top: range corrected zonal shear from LES.
Middle-bottom: meridional wind error from DBS estimates. Bottom: range
corrected meridional shear from LES. . . . . . . . . . . . . . . . . . . . 95
5.15 Error from the SA estimates vs. range corrected shear from LES for sfrac
= 1.00. Left: zonal wind. Right: meridional wind. . . . . . . . . . . . . . 96
5.16 Wind bias for SA. Left: zonal wind. Right: meridional wind. . . . . . . . 97
5.17 Horizontal velocity variances. Gray lines: time spread of plane statistic esti-
mates from the LES over a one-hour period. Black: statistics obtained from
LES data over the entire BLR domain by plane averaging and additional
time averaging over one hour. Red: velocity variances calculated by time
averaging from LES data over the DBS volume scan. Blue: velocity variance
estimated from an ideal DBS setting (all beams available at dwell time of
30 s). Green: velocity variance estimated from a realistic DBS setting (all
beams available after the scan time of 150 s). Statistics in the two upper
plots are evaluated with original vertical velocities (sfrac = 1). Statistics in
the two lower plots are obtained with zero vertical velocity (sfrac = 0). . . 100
5.18 Horizontal velocity variances. For notation see Figure 5.17. The averaging
time for the DBS estimates is 1200 s instead of 150 s. . . . . . . . . . . . 101
5.19 Left: vertical velocity variance. Right: turbulence kinetic energy. For nota-
tion see Figure 5.17. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
xiii
5.20 Kolmogorov turbulence spectra obtained from the velocity fields. The sub-
script represents the longitudinal l or the transverse t component, and the
superscript represents the velocity component. All five components are pre-
sented as a time average obtained between 18:00-19:00 UTC at approxi-
mately 700 m. The black dashed line represents the -5/3 law expected
from isotropic turbulence (see Section 2.2.3.2). All spectra behave similarly,
which confirms the isotropy in the LES fields. Also, the transverse spec-
trum is larger than the longitudinal spectrum, which is consistent with the
structure parameter in the inertial subrange. The beginning of the dissipa-
tion range can be identified as ≈157 m, which is caused by dumping in the
sub-grid closure of the LES model. . . . . . . . . . . . . . . . . . . . . . 104
5.21 Turbulence eddy dissipation rate as a function of time and height. Top: es-
timates from LES. Middle-top: estimates from the Doppler spectrum width
of the signal from the vertical beam. Middle-bottom: estimates from the
Doppler spectrum width of the signal from the oblique beam pointing to the
north. Bottom: estimates from the Doppler spectrum width of the signal
from the oblique beam pointing to the east. . . . . . . . . . . . . . . . . 106
5.22 Hourly averages of turbulence eddy dissipation rate profiles from 16:00 -
17:00. Left: estimates from the virtual BLR with different directed beams
(red: vertical beam; blue: east beam; green: north beam; cyan: west beam;
magenta: south beam). Right: ￿ profile from the LES (black) compared to
the profile from BLR averages among different beam directions (red). . . . 107
5.23 Same as Figure 5.22, except with time averaging over period 17:00 - 18:00
UTC. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
5.24 Same as Figure 5.22, except with time averaging over period 18:00 - 19:00
UTC. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
5.25 Same as Figure 5.22, except with time averaging over period 19:00 - 20:00
UTC. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
C.1 Scheme that describes the calculations of C2n at each LES cell. The difference
in refractive index ∆nk is calculated at edge of the cube considering the
refractive index at each vertex. The length of each of the twelve edges of
the cube corresponds by the LES grid spacing ∆. . . . . . . . . . . . . . 138
xiv
Abstract
The boundary layer (BL) is the lowest part of the atmosphere where the flow field
is directly influenced by interactions among air, heat, and the Earths surface. The
structure of flow in the BL is dynamic due to the fact that the atmospheric flow is
turbulent. Turbulence in the daytime convective boundary layer (CBL) is primarily
caused by buoyancy forced from the heated underlying surface. Wind profilers are one
of the many instruments used to study and characterize the atmosphere. In addition
to in-situ observations, numerical large-eddy simulations (LES) have been probed to
adequately reproduce the CBL under different conditions. The focus of this work is
to bring the advantages of LES techniques to assist in the interpretation of data from
wind profilers.
The present study focuses on an example of flow structure of the CBL as observed
in the U.S. Southern Great Plains Atmospheric Radiation Measurement Climate Re-
search Facility in Lamont, Oklahoma on June 8, 2007. The considered CBL flow has
been reproduced using LES, sampled with a LES-based virtual boundary layer radar
(BLR), and probed with an actual operational wind profiler. The LES-generated CBL
flow data are then ingested by the virtual BLR and treated as a proxy for prevailing
atmospheric conditions. The virtual BLR has been used to simulate radar signals
obtained from wind profilers through the synthesis of Doppler beam swinging (DBS)
and spaced antenna (SA) techniques and to retrieve the three-dimensional wind fields.
Comparisons of the estimates of the structure parameter of refractive index C2n, wind
fields, vertical velocity variance, and vertical velocity skewness have been presented
for the LES, virtual BLR, and actual radar.
It has been observed that during the presence of strong horizontal shear of ver-
tical velocity, the estimates of the horizontal wind fields are biased. This study will
quantify the effect of this shear for both wind estimation techniques under different
conditions. Additionally, it has been noticed that this shear also biases the estimates
of turbulence kinetic energy (TKE) calculated from the variances of the wind fields.
Finally, the TKE (eddy) dissipation rate ￿ can be obtained from radar estimates of
xv
Doppler spectral width. Values of ￿ are obtained from the different oblique and verti-
cal beams and contrasted with the LES estimates obtained through a parameterized
expression that relates the dissipation rate to sub-grid TKE and turbulence length
scale.
xvi
Chapter 1
Introduction
1.1 The Atmospheric Boundary Layer: A Brief
Review
The Earth’s atmosphere is a layer of gases surrounding the planet that is retained
by its gravity (Ahrens 1991; Stull 2000; Wallace and Hobbs 2006). The atmosphere
protects life on Earth by absorbing ultraviolet solar radiation which warms the surface
through heat retention (greenhouse effect) and reduces temperature extremes between
day and night. The atmosphere is a complex fluid system (Stull 2000) which generates
chaotic motions known as weather. In 340 B.C., Aristotle definedMeteorologica as the
study of the weather and climate, as well as astronomy, geography, and chemistry.
Today, Meteorology is the study of the atmosphere and its phenomena. Some of
the most intensely studied topics include all parts of severe weather: thunderstorm,
clouds, rain, snow, wind, hail, lighting, and hurricanes (Ahrens 1991). In the absence
of precipitation, Meteorology studies changes in temperature, winds, air turbulence,
etc.
The atmosphere is composed of a mixture of gases in proportions shown in Ta-
ble 1.1. Nitrogen and Oxygen are the main constituents of the Earth’s atmosphere,
and their percentage remains constant from the surface up to approximately 80 km
(Ahrens 1991; Wallace and Hobbs 2006). Consequently, there is a balance between
production and destruction of these gases near the surface. Nitrogen is removed from
the atmosphere by biological processes that involve soil bacteria. It is returned to
the atmosphere mainly through the decay of plant and animal matter. Oxygen, on
the other hand, is removed from the atmosphere when organic matter decays and
when oxygen combines with other substances, producing oxides. The oxygen returns
to the atmosphere through photosynthesis (Ahrens 1991). Argon is present in higher
1
concentrations than other noble gases like neon, helium, krypton, and xenon. Water
vapor accounts for approximately 0.25% of the mass of the atmosphere, and its con-
centration varies from 10 ppmv (parts per million per volume) in the coldest regions
of the Earth’s atmosphere up to 5% by volume (5×10−4 ppmv) in hot and humid air
masses (Wallace and Hobbs 2006). Ozone concentrations also have large variabili-
ties, and a concentration of greater than 0.1 ppmv is considered dangerous to human
health.
Table 1.1: Fractional concentration by volume of the major constituents of the Earth’s
atmosphere up to 105 km, with respect to dry air (from Wallace and Hobbs (2006)).
Constituent Molecular weight Fractional concentration by volume
Nitrogen (N2) 28.013 78.08%
Oxygen (O2) 32.000 20.95%
Argon (Ar) 39.95 0.93%
Water vapor (H2O) 18.02 0-5%
Carbon dioxide (CO2) 44.01 380 ppm
Neon (Ne) 20.18 18 ppm
Helium (He) 4.00 5 ppm
Methane (CH2) 16.04 1.75 ppm
Krypton (Kr) 83.80 1 ppm
Hydrogen (H2) 2.02 0.5 ppm
Nitrous oxide (N2O) 53.02 0.3 ppm
Ozone (O3) 48.00 0 - 0.1 ppm
The structure of the atmosphere indicates changes in temperature, density, and
composition with height and is depicted in Figure 1.1. The lowest layer of the at-
mosphere is the troposphere, which extends from the surface up to approximately
7 km at the poles and 17 km at the equator. The troposphere is mostly heated by
transfer of energy from the surface, so on average the lowest part of the troposphere
is warmest and the temperature decreases with altitude. The rate at which temper-
ature decreases with height is called adiabatic lapse rate, and its value can be found
by combining the first law of thermodynamics, the equation of state for ideal gas, and
the hydrostatic balance equation (Γd ∼9.8◦C km−1) (Holton 2004; Wallace and Hobbs
2006). If the atmospheric lapse rate Γ is lower than the adiabatic lapse rate (Γ < Γd),
2
http://burro.cwru.edu/Academics/Astr201/Atmosphere/atmosphere1.html
Figure 1.1: The structure of the Earth’s atmosphere (modified from Ahrens (1991)). The
structure reflects how temperature, density, and composition of the atmosphere change with
height, and how it is determined in large part by how it interacts with the sun’s light.
3
an air parcel that undergoes an adiabatic displacement from its equilibrium level
will be positively buoyant when displaced downward and negatively buoyant when
displaced upward so that it will tend to return to its equilibrium level, and the at-
mosphere is said to be statically stable. Static stability considers only buoyancy to
estimate the flow stability, and ignores shears in the mean wind. The air parcel is
considered unstable, if the buoyant force is in the same direction as the displacement.
This instability causes the air parcel to continue accelerating in the initial direction
of movement. This results in convective circulations and convective clouds (Stull
2000). This promotes the vertical mixing of gases. The troposphere also contains all
weather that we are familiar with on Earth. The boundary between the troposphere
and stratosphere is called the tropopause.
The stratosphere extends from the tropopause up to approximately 50 km. Tem-
perature increases with height, which restricts turbulence and mixing. Stratospheric
air is extremely dry and ozone rich. The absorption of solar radiation in the ul-
traviolet region of the spectrum by this stratospheric ozone layer is critical to the
inhabitability of the Earth (Wallace and Hobbs 2006). The stratopause, which is
the boundary between the stratosphere and mesosphere, is typically at 50 - 55 km.
The mesosphere extends from the stratopause up to 80 - 85 km. This is the layer
where most meteors ablate upon entering the atmosphere. Temperature decreases
with height in the mesosphere. A phenomenon due, in part, to the fact that there
is little ozone in the air to absorb solar radiation. Hence, the molecules are able to
lose more energy than they absorb, which results in an energy deficit and cooling
(Ahrens 1991). The mesopause, the temperature minimum that marks the top of
the mesosphere, is the coldest region of the Earth and has an average temperature
around -85◦C. Due to the cold temperature of the mesosphere, water vapor can be
frozen, which might allow for the formation of ice clouds (or Noctilucent clouds).
Temperature increases with height in the thermosphere from the mesopause up to
the thermopause due to the absorption of solar radiation in association with the dis-
sociation of diatomic nitrogen and oxygen molecules and the stripping of electrons
from atoms (Wallace and Hobbs 2006). These process is known as photodissociation
and photoionization. Temperatures in the Earth’s outer thermosphere can change
over a wide range due to variations in the emission of ultraviolet and x-ray radiation
from the sun’s outer atmosphere.
4
The Atmospheric Boundary Layer (ABL) is the lowest part of the troposphere
in which the flow field is directly influenced by interactions with the Earth’s sur-
face (Holton 2004). For average midlatitude conditions, the atmospheric Convective
Boundary Layer (CBL) extends through the lowest kilometer and thus contains about
10% of the mass of the atmosphere. The structure of the flow in the boundary layer is
dynamic due to the fact that the atmospheric flow is turbulent. The CBL depth pro-
duced by turbulent transport may range from as little as 30 m in conditions of large
static stability to more than 3 km in highly convective conditions. Vertical shears
can be extremely intense within a few millimeters of the surface, which continuously
leads to the development of turbulent eddies.
In the daytime CBL, turbulence is primarily forced by surface heating, radiational
cooling from clouds at the CBL top, or by both mechanisms. A pure buoyancy-
driven CBL rarely exists, and there are many situations in which surface heating is
relatively weak while the turbulence production by wind shears is relatively strong.
In these cases, the shear effects on the CBL turbulence dynamics cannot be ignored
(Conzemius and Fedorovich 2006).
The CBL is considered clear when no clouds are present (Holtslag and Duynkerke
1998). A clear CBL is the focus of this study. The bottom of the CBL is called the
surface layer and extends upwards from 20 to 200 m. In this layer, frictional drag, heat
conduction, and evaporation from the surface cause substantial changes in wind speed,
temperature, and humidity with height (Stull 2000). This layer is also known as the
constant flux layer because the turbulent fluxes are relative uniform with height. In
this case, the CBL’s main forcing mechanism is surface heating. Turbulent convective
motions in the CBL transport heat upward in the form of convective plumes or
thermals. These rising motions and associated downdrafts effectively mix momentum
and potential temperature fields in the middle portion of the CBL (Zilitinkevich 1991).
The resulting mixed layer is typically the CBL’s thickest sublayer. The CBL is topped
by the entrainment zone, which has relatively large vertical gradients of averaged (in
time or over horizontal planes) meteorological fields. Above the entrainment zone is
the rest of the troposphere, also called the free atmosphere (see Figure 1.2).
The entrainment zone is often called the interfacial or capping inversion layer
because it is co-located with the region of maximum gradients in the potential tem-
perature profile. This stable layer traps turbulence, pollutants, and moisture below it
and prevents most of the surface friction from being felt by the free atmosphere. The
height of the capping inversion is usually denoted by zi and is frequently a measure
5
Surface Layer
Entrainment ZoneCapping Inversion
Residual 
Layer
Mixed Layer
Nocturnal
Inversion
NBL
He
ig
ht
Sunset Sunrise
Surface 
Layer
Nocturnal
Inversion
NBL
Residual 
Layer
Sunset
Time
Capping 
Inversion
He
ig
ht
1 - 2 km
100 -  300 m
Free Atmosphere
Figure 1.2: Depiction of the diurnal development of the atmospheric boundary layer
(adapted from Stull (1988)).
of the CBL depth. Many quantities within the CBL can be scaled with this value
(Deardorff 1970). These include mean flow parameters (mean wind speed, potential
temperature, and specific humidity), turbulent fluxes of momentum, heat, and mois-
ture, and variances of the velocity components and passive scalars. As a consequence,
accurate estimates of zi are extremely important in any study of CBL structure. A
detailed understanding of the CBL is important for numerical weather prediction and
climate modeling applications. The turbulence contained in the CBL is an important
transport mechanism of heat, pollutants, and momentum.
1.2 Radars for Sensing the Atmosphere
One of the many instruments that is used to study the atmosphere is radar (RAdio
Detection And Ranging) (Doviak and Zrnic´ 1993). There are many other uses for
radars such as detection of military targets, radar-assisted ground-controlled approach
(GCA) systems in airports, radar guns used by police to monitor vehicle speeds on the
roads, etc. Radars use electromagnetic waves to irradiate and measure the observed
fields. The returned signal is used to infer the properties of the atmosphere. Doppler
6
radars can detect tracers of wind and measure their radial velocities, both in the
clear-air and/or within regions of precipitation. Depending on the type of scattering
mechanism, radars can be separated as weather or clear-air radars. There have been
many studies that summarize their characteristics (Doviak and Zrnic´ 1993; Balsley
and Gage 1982; Rinehart 2004). Nevertheless, a short review is provided here.
1.2.1 Weather Radars: Particle Scattering From Precipitation
Precipitation refers to all liquid or solid phase aqueous particles that originate in
the atmosphere and fall to the Earth’s surface (AMS Glossary). They are usually
identified as rain, sleet, hail, snow, and any other form of water falling from the sky.
The scattering mechanism of electromagnetic radiation by spherical particles is called
Mie scattering. Hydrometeors correspond to this type of scattering. However, when
their size is much smaller than the radar wavelength, they correspond to a sub-set of
the Mie theory called Rayleigh scattering .
Radar was not developed by one person, it was the product of the effort of many.
Hertz, Marconi, Tesla, Thompson, Edison, Braun, and many others can all be credited
with contributing to the invention of radar (Dunlap Jr. 1946). It is difficult to trace
the first radar detection of precipitation, but it is likely to have been in the latter half
of the 1940s (Atlas 1990; Doviak and Zrnic´ 1993), during World War II when Britain
invested more than $41.5 million in radar technology, which corresponds to more 500
million in today’s dollars (Dunlap Jr. 1946). This new technology allowed Britain’s
Royal Air Force to intercept the German Luftwaffe at night through fog and clouds.
The use of radar definitely aided the Allies in their victory over Germany and Japan,
and it also saved countless lives.
Today, Doppler radars are commonly used for weather observations. Weather
radars come in a variety of designs and use different frequencies such as X-band
(8-12 GHz), C-band (4-8 GHz), and S-band (2-4 GHz), depending on the applica-
tion. Mobile radars usually operate at X-band due to its low cost and relatively
good resolution. However, they are susceptible to suffer from severe attenuation
and range-velocity ambiguity. Range-velocity ambiguity refers to the inability of a
pulsed Doppler radar to resolve range and velocity without aliasing (Rinehart 2004).
Most stationary radars operate at S-band due to their low attenuation and less se-
vere range-velocity ambiguity. Nonetheless, they are expensive and less sensitive (e.g.
WSR-88D (Weather Surveillance Radar - 1998 Doppler) sensitivity is -7.5 dBZ at
50 km). Radars that operate at C-band are mobile and stationary systems alike due
7
to their low cost and higher sensitivity (e.g. OU PRIME (Polarimetric Radar for
Innovations in Meteorology and Engineering) sensitivity is -18 dBZ at 50 km).
There are many weather radar networks around the world. The European weather
radar network (Operational Program for the Exchange of weather RAdar information
– OPERA) standard is a C-band Doppler radar, although the Spanish network is
composed of C- and S-band Doppler radars. Italy operates several polarimetric C-
band Doppler radars for local applications (Meischner 2004). A map of the location
of the current radars operated by OPERA is presented in Figure 1.3.
Figure 1.3: Location of radars present at the OPERA radar database updated in June
2010 (from http://www.knmi.nl/opera/). The color dots represent the frequency bands:
C-band = green, S-band = blue, X-band = cyan. The center color represent the type of
radar: dual polarization radar = red center, Doppler radar= yellow center.
The United States weather radar network is composed of 154 WSR-88D (Next-
Generation Radar - NEXRAD) S-band radars (see Figure 1.4) and is operated by
the National Weather Service, the Federal Aviation Administration (FAA), and the
8
Department of Defense (DOD) (Rinehart 2004). The WSR-88D (see left panel of
Figure 1.4: Location of the operational WSR-88D radars (National Oceanic and Atmo-
spheric Administration (NOAA) 2007 (from http://radar.weather.gov/). The color cod-
ing represent the US agency in charge of the correct operation of the radar.
Figure 1.5) began as the Joint Doppler Operational Project (JDOP) in 1976 at the
National Severe Storms Laboratory (NSSL) in Norman, Oklahoma (OK) (Crum and
Alberty 1993). The goal was to create a national network of Doppler radars to im-
prove the detection of hazardous storms, tornados, and squall lines. The program
was an effort put forth by the three U.S. agencies sharing common interests in the ca-
pabilities of the radar: Department of Commerce (DOC), DOD, and the Department
of Transportation (DOT). Installation of the network began with the first WSR-88D
constructed near Oklahoma City, OK in 1990.
The WSR-88D systems collect, process, and display high-resolution and high ac-
curacy reflectivity, radial velocity, and spectrum width continuously. The antenna
scans its environment in predefined sequences of 360◦ azimuthal sweeps at various
angles or volume coverage patterns (VCP). A complete multiple elevation sequence is
9
a volume scan (Crum and Alberty 1993). Typical VCPs used by the WSR-88D are
presented in Table 1.2.
Table 1.2: Example of typical VCPs utilized by the WSR-88D based on the type of
weather occurring (from http://www.wdtb.noaa.gov/tools/RPS/).
Tscan Elev.
VCP
(min) scans
Elevation angles (◦) Usage
0.5, 1.5, 2.4, 3.4, 4.3, 5.3, 6.2, Convection, especially
11 5 14
7.5, 8.7, 10, 12, 14, 16.7, 19.5 when close to the radar
0.5, 0.9, 1.3, 1.8, 2.4, 3.1, 4, Convection, especially
12 4.5 14
5.1, 6.4, 8, 10, 12.5, 15.6, 19.5 activity at longer ranges
0.5, 1.5, 2.4, 3.4, 4.3, 6, 9.9,
21 6 9
14.6, 19.5
Shallow precipitation
Default clear-air mode,
32 10 5 0.5, 1.5, 2.5, 3.5, 4.5
reduces wear on antenna
The WSR-88D network will be start to be modified for dual polarization opera-
tions in 2011 (Istok et al. 2009). This new technology will allow the WSR-88D radars
to simultaneously transmit and receive in horizontal and vertical polarization. Dual
polarization provides a unique tool for classification of radar echoes. Having dual
polarization capability is essential for weather discrimination and classification of dif-
ferent hydrometeor types like rain, hail, graupel, and snow of different types. The
dual polarization technology has been the subject of research since the 1970’s. How-
ever, it was not until the Joint Polarization Experiment (JPOLE) that the technology
was demonstrated to provide significant benefits to the forecaster. During JPOLE,
using a prototype dual polarimetric WSR-88D, hydrometeor classification was tested
using fuzzy logic algorithms (Ryzhkov et al. 2005b). The version of hydrometeor
classification algorithm (HCA) which will be implemented on the WSR-88D radars
is described in Park et al. (2009). This algorithm includes the estimation of a data
quality index that is an indicator of many possible sources of error in the radar
measurements, the assignment of a matrix of weights that sets values to the radar
retrieval according to its importance to the variable to retrieve, and implementation
of class designation based on the distance from the radar and the parameters of the
melting layer which are determined as a function of azimuth with polarimetric radar
10
measurements (Istok et al. 2009). The Melting Layer Detection Algorithm (MLDA)
is part of the HCA, and was tested using S-band radar in OK (Giangrande et al.
2008) and C-band in Ontario, Canada (Boodoo et al. 2008). The performance and
robustness of the MLDA was proven in both conditions (Istok et al. 2009). Finally,
the quantitative precipitation estimation (QPE) algorithm (Ryzhkov et al. 2005a)
was optimized by combining QPE and HCA in the second version of the polarimetric
QPE algorithm (QPE v2) which was validated using the Oklahoma Mesonet for 43
events (Giangrande and Ryzhkov 2008).
(a) (b)
Figure 1.5: (a) NEXRAD Radar at the WSR-88D Radar Operations Center in Norman,
OK (from http://en.wikipedia.org/wiki/NEXRAD). (b) Phased-Array Radar (PAR) in
Norman, OK (from http://www.thunderchaser.com/june2005/spc_tour.html).
As mentioned previously, the WSR-88D scans the volume in sequences of 360◦,
and depending on the VCP and the current weather conditions, the total time for a
volume scan is approximately 4.5 minutes. However, it is convenient to revisit regions
of interest more frequently and other regions less frequently maintaing adequate data
quality. These constitute a limitation for conventional weather radars due to the
mechanic inertial of the rotating antennas. The Phased-Array Radar (PAR) or the
National Weather Radar Testbed (NWRT) is presented in the right panel of Figure 1.5
and has a unique antenna that collects the same information as a conventional radar in
about one-forth the time. The S-band radar has a 10-cm wavelength and is being used
in studying and developing faster and more accurate severe and hazardous weather
warnings (Forsyth et al. 2005; Heinselman et al. 2008; Zrnic´ et al. 2007). The PAR is
able to electronically steer in both elevation and azimuth within a 45◦ cone-shaped
volume. An advantage of the system is its capability to change the beam pointing
direction from pulse to pulse, which allows the system to be used in weather forecast
11
and primary aircraft surveillance. This could be an alternative to current surveillance
radars (e.g. WSR-88D) (Weadon et al. 2009). The system was developed by an
interactive group from government and industry organizations (Forsyth et al. 2002)
consisting of NSSL and Radar Operations Center (ROC), the United States Navy’s
Office of Naval Research, Lockheed Martin Corporation, the University of Oklahoma,
the Oklahoma State Regents for Higher Education, and the FAA’s Technical Center.
The PAR was installed successfully in September 2003, and started full operational
data collection in May 2005.
It has demonstrated in Heinselman et al. (2008) that the NWRT PAR can achieve
high temporal volumetric resolution. Its time improvement over a conventional WSR-
88D scan is at least four times. This high temporal resolution is necessary for rapid
identification and confirmation of weather phenomena. Also, the beam multiplexing
(BMX) technique has been tested and implemented on the NWRT (Yu et al. 2007;
Zrnic´ et al. 2007). This approach improves the scan time by changing the pointing di-
rection, which reduces the time to achieve independent samples. In both approaches,
the time to obtain a volume scan was significantly reduced in comparison with a
WSR-88D, which opens more possible uses for the PAR.
The Multifunction Phased-Array Radar (MPAR) system is a specialized applica-
tion of general phased array radar technology that is designed to simultaneously fulfill
the multiple functions of national air and weather surveillance (Weadon et al. 2009).
The MPAR system will be able to improve meteorological forecasts and warnings and
reduce the number of existing radars by combining meteorological and air surveillance
functions into a single system. This also will reduce the cost of maintaing the current
systems, and in time MPAR will become more affordable. Due to the capability of
instantaneously and dynamically controlling beam position on a pulse-by-pulse basis,
the MPAR system can perform both tasks with different update times to achieve
the goal of better characterizing and forecasting the storms of interest. In order to
achieve both tasks at the same time, an algorithm based on the concept of time
balance (TB) was developed for adaptive weather sensing (Reinoso-Rondinel et al.
2010). A simulated experiment was conducted to test the feasibility of the technique
and to test the performance of the TB scheduling algorithm. As a result, the storm of
interest was scanned more frequently within a relatively short period time compared
to conventional scanning while air surveillance was done “simultaneously”.
12
1.2.2 Profiling Radars: Clear-Air Scattering from Refractive
Index Variations
Wind profilers are pulsed Doppler radars that transmit radio waves vertically or
nearly vertically. They receive backscattered signals from refractive index fluctua-
tions in clear-air (Bragg scattering). These fluctuations are caused by variations in
temperature, humidity, and electron density of the atmosphere. The contribution of
the latter is only observed in the upper atmosphere.
The history of clear-air radars started during the same period of weather radars
in which “angel echoes” were recorded when no precipitation was present (Baldwin
1948; Friend 1949). There were discrepancies regarding the origin of the echoes;
insects and birds were one possibility, and sharp gradients in refractive index were
another (Atlas 1990). Since 1950 there has been significant exploration of the clear-air
echoes with different radar wavelengths and for long periods of time. From 1963 to
1972 there were notable achievements in detection of clear-air turbulence in ranges up
to 10-20 km with 10 cm or longer wavelength radars (UHF). Atlas and Hardy (1966)
conducted an experiment at Wallops Islands with 3.2-, 10.7-, and 71.5-cm radars to
study the atmosphere. Through the use of range-height-intensity (RHI) displays,
they discovered that some of the “angel echoes” that appeared between 1 and 3 km
were seen most prominently with the 3.2-cm radar and not with the 71.5-cm radar,
which indicates that these targets were small compared to the radar wavelength, and
led to the conclusion that the return signals were from insects and/or birds. Atlas
and Hardy (1966) also discovered a layer around 1 km that was clearly observed with
the 10.7- and 71.5-cm radars, but it was more intensely observed with the 10.7-cm
radar. This layer was clearly caused by variations in the refractive index (Hardy
et al. 1966). Also, using low elevation plan-position-indicator (PPI) plots Hardy
and Ottersten (1969); Konrad (1970); Harrold and Browning (1971), with the 10.7-
cm radar, were able to study convective processes in the boundary layer. The first
Frequency-Modulated Continuous-Wave (FM-CW) radar (Richter 1969; Gossard and
Richter 1970) was built in the late 1960s, pointed vertically, and had a resolution of
1 m. The main contribution from the use of FM-CW radar is the understanding of
Kelvin-Helmholtz instabilities and the interaction of instabilities on a variety of scales
(Atlas 1990).
Woodman and Guillen (1974) obtained Doppler wind velocities both in the lower
atmosphere and mesosphere with the 50 MHz (VHF) radar located at the Jicamarca
13
Radio Observatory near Lima, Peru. The Jicamarca radar was previously used pri-
marily for ionospheric research. After this discovery at Jicamarca, many researchers
recognized the potential of VHF radars for exploring the clear atmosphere, from the
lower troposphere up to the mesosphere (∼100 km) (Atlas 1990). Consequently, the
radar wavelengths to study the clear-air atmosphere were switched from the UHF to
VHF. These new VHF radars were typically phase-array antennas which often op-
erated unattended at low power. Many radars were built as a direct result of the
Jicamarca research such as the 53 MHz SOUSY (SOUnding SYstem) radar in the
Harz, Germany, the 50 MHz Poker Flat radar in Alaska (no longer in operation),
the 46 MHz MU (Middle and Upper atmosphere) radar in Shigaraki, Japan; and the
53 MHz Gadanki radar in Tirupati, India. Some of these VHF radars are presented in
Figure 1.6. Today, this type of radar usually operates in frequencies between 50 MHz
(a) (b)
(c) (d)
Figure 1.6: (a) Jicamarca Incoherent Scatter Radar at Lima, Peru (from http://jro.igp.
gob.pe). (b) The SOUSY (SOUnding SYstem) Radar at Lindau, Germany (from http:
//radars.uit.no/sousy/index.html). (c) NARL MST Radar at Gadanki, India (from
http://www.narl.gov.in/. (d) MU (Middle and Upper atmosphere) Radar at Shigaraki,
Japan (from http://www-lab26.kuee.kyoto-u.ac.jp/study/mu/mu_e.html).
and 1 GHz (VHF and UHF frequencies - see Figure 1.7), and they can be classified
depending on their altitude coverage (see Table 1.3):
14
Table 1.3: Classification of radars according to their altitude coverage
Type of radar Approximate range
MST (Mesosphere, Stratosphere, and Troposphere) 4 km - 85 km
ST (Stratosphere and Troposphere) 2 km - 22 km
BLT (Boundary Layer and Troposphere) 300 m - 8 km
BL (Boundary Layer) 200 m - 2 km
1410 HOCKING: MEASUREMENT OF TURBULENT DISSIPATION RATES 
80' 
70' 
•60 
vscous 
'• 30 - 
20 
inertial range turbulence up to altitudes 
of about 65-70 km. 
5 . RADIO WAVE BACKS CATTER 
104 10-' 10-' 104 I 10 10 • 10' l•f 
SCALE f•-TRES/ 
Fig. 1. Typical inner and inertial 
range/buoyancy range transition scales 
([0 and L B respectively) for turbulence in unit solid angle, per unit incident power density, and per unit volume (i.e. the the atmosphere. The formulae used are given in the text. The profile of the cross section of backscatter) is
Having illustrated some appropriate 
formulae which relate refractive index 
fluctuations to turbulence parameters, it 
is no• necessary to determine how these 
formulae relate to radio wave backscatter. 
Assume that the refractive index fluct- 
uations within a turbulent patch are 
Fourier analysed into various scales 
A, and that the mean square refractive 
index fluctuation associated2with wave numbers k (magnitude -- k = -• ) to 
k + dk is •_2(k)dk. If the patch b-•cks-•atter• ra--di• waves of wavelength 
%, the effective scale producing the 
scatter is A = %/2. Booker [1956] has 
shown that the power backscattered per 
less than L R and larger than [0- solid angle, per unit incident power 
Approximate values of L_ and [0 are density, per unit volume of scatterer. It 
shown in Figure 1. For a radar wavelength is often denoted by •, and although this 
%, backscatter is from scales of %/2. Thus can at times be confused with the Kolmo- 
if a 50 MHz radar is used (% = 6 m), then goroff microscale, this usage will be 
scatter should be possible from isotropic maintained here. Then 
strictly only applies for scales somewhat 
assumed Brunt-Vaisala period is also 
shown. It wa• assum_•d that the mean value 
of c was 10 W kg _2at 901km , decreasing 
exponentially to 10 W kg at 80 km. 
Betwee• 80 an_• 60 km, this mean was taken 
as 10 W kg . In the region 60-90 km, 
the bounds of the dotted areas correspond 
to turbulent energy dissipation rates of 
1/3 rd and 3 times these mean values. 
Below 40 km, and down to the tropopause, 
the upper and lo•er lim.•ts of c were taken as 10 • and 10 --2 W kg-- . L_ and [0 were assumed to vary smoothly between 40 and 60 
km. (The region between 30 and 80 km is 
the most uncertain part of the graph.) 
Below the tropopause, • •as assumed to be imited. between 10- and 10 -•W kg -•. Larger • values correspond 
to smaller [0 and larger L_ values (i.e. 
the inertial range widens at both ends 
when • increases). 
•n2(k).is similar to •n(k) in equation (3) ([ = n), but Booker [1956] used the norm- 
alization 
• 2 
so that 
•n2(k_) = (2•)3 •n(k_) (26) 
For radar backscatter at wavelength 
•, k -- 4• •-1, and so using (26) in (25), 
and using (3) for •n(k)' 
4/3 1/3 • -- 0.00655 • C 2 •- (27) 
n 
Sometimes an alternative definition of 
backscatter cross section is used. •is 
is the total power which would be scat- 
tered if power were scattered isotro- 
pically with an intensity equal to that of 
the backscattered radiation, per unit 
Figure 1.7: Schematic of the evolution of the inertial subrange with height. The horizontal
axis represent the Bragg scale. In the troposphere, the Bragg scale extends safety from 10−1
to 10 m, which corresponds to frequencies of 1.5 GHz (UHF) - 15 MHz (VHF), respectively
(from Hocking (1985)).
15
To study the ABL, boundary layer radars (BLR) have been developed in the
UHF band near 1 GHz and in the S-band with frequencies near 3 GHz (Van Zandt
2000). The S-band radars are more susceptible to hydrometeor, and they often can-
not measure the vertical wind and the estimation of horizontal winds is difficult.
Consequently, the UHF frequency is more common in clear-air studies. The UHF
radar detects irregularities in the refractive index when the Bragg condition is met.
This condition refers to scattering waves when continuous turbulent interfaces are in
phase, and constructive interference provides a maximum in the intensity of the re-
sulting wave. Therefore, the Bragg scale is such that BLRs are sensitive to turbulent
structures with spatial scales near 15 cm (λ/2, where λ is the radar wavelength).
Enhanced refractive index variations are often associated with the entrainment
zone just above the CBL, which can be detected by clear-air radar. There have been
many studies in which BLRs are used to estimate the CBL depth and thickness of the
entrainment layer (e.g., Angevine et al. 1994b; Angevine 1999; Cohn and Angevine
2000; Grimsdell and Angevine 2002). A good estimate of the CBL depth is important
to understand the evolution and also for simulations of the CBL (Angevine et al.
1994b). Continuous profiles of the wind vector directly above the instrument can be
obtained using the Doppler beam swinging (DBS) method (Balsley and Gage 1982).
Due to its frequency of operation, as mentioned before, BLRs are also sensitive to
Rayleigh scatter from hydrometeors and are used to study clouds and precipitation
(e.g., Atlas and Hardy 1966; Hardy and Ottersten 1969; Konrad 1970; Harrold and
Browning 1971; Gage et al. 1994; Ecklund et al. 1995). Thus, the BLR can be used
to study the boundary layer under a wide variety of meteorological conditions and
has been proven invaluable for such investigations (e.g., Rogers et al. 1993; Angevine
et al. 1994b; Wilczak et al. 1996; Dabberdt et al. 2004). However, for cloud-free CBL
studies, precipitation might be seen as signal contamination.
Another radar used in the ABL studies is the previously mentioned FM-CW. This
radar differs from the conventional pulsed radars in that utilizes the Doppler frequency
shift to detect moving targets (Skolnik 2001), and it requires a large bandwidth (a
wide-frequency spread). Calculations of the radial velocity are very complicated in
FM-CW radars. Strauch (1976) implemented a Doppler processing technique that
uses a data acquisition system developed for pulsed Doppler radar for the extrac-
tion of the Doppler spectrum. Nevertheless, this radar has been used to show that
the thermodynamic fields within the CBL can exhibit a high degree of complexity
and that organized fine-scale structures with spatial scales of roughly one meter are
16
common (Richter 1969; Gossard and Richter 1970; Eaton et al. 1995). In contrast
to FW-CW radars, BLRs must operate within stringent frequency management con-
straints, which limits their range resolution. The typical range resolution for BLR
measurements of the ABL is approximately 100 m. This resolution is too coarse to
adequately reproduce the entrainment zone’s embedded spatial structure. Several
multiple frequency techniques have been introduced in the past as a means of im-
proving the range resolution (e.g., Kudeki and Stitt 1987; Palmer et al. 1990, 1999;
Yu and Palmer 2001; Luce et al. 2001). These techniques use several close frequencies
for transmission and reception, and after post-processing, the coarse original range
resolution was improved. The minimum layer thickness obtained by Chilson et al.
(2003) was 16 m. Multiple-frequency techniques have also been successfully used to
study the ABL at UHF (Chilson et al. 2003; Chilson 2004; Yu and Brown 2004).
1.3 Large-Eddy Simulations to Characterize the
Atmospheric Boundary Layer
In addition to field observations of the CBL by in-situ and remote sensing measure-
ment methods, numerical simulation approaches – specifically, the large-eddy sim-
ulation (LES) technique – are widely employed to study physical processes in the
atmospheric CBL. Large-eddy simulations of CBL-type flows have developed into a
routine scientific exercise over the last three decades, see Deardorff (1972); Moeng
(1984); Mason (1989); Schmidt and Schumann (1989); Moeng and Sullivan (1994);
Sorbjan (1996, 2004); Khanna and Brasseur (1998); Sullivan et al. (1998); Van Zan-
ten et al. (1999); Fedorovich et al. (2001, 2004a); Conzemius and Fedorovich (2006).
These cited works are indicative of how the LES has gradually evolved into an applied
research technique in CBL studies. However, the relation of LES to observations of
the CBL, as well as to conceptual CBL models or theories, needs further examination
and quantitative evaluation (Wyngaard 1998; Stevens and Lenschow 2001).
The LES method is based on the numerical integration of filtered equations of
flow dynamics and thermodynamics, which resolves most of the energy-containing
scales of turbulent transport. Any unresolvable motions are assumed to carry only
a small fraction of the flow’s total energy and are parameterized with a subgrid (or
subfilter) closure scheme. In the LES of the atmospheric CBL, environmental pa-
rameters such as surface heating, stratification, and shear can be precisely controlled.
Retrieval of spatial turbulence statistics in LES do not necessarily rely on additional
17
assumptions like the Taylor (1938) frozen-turbulence hypothesis, which states that
thermodynamic and kinematic properties of the simulated flow are known at all points
of the numerical grid simultaneously. In this manner, LES has been a helpful tool in
studying the statistics of (resolved) CBL turbulence and in visualizing its turbulent
structure. However, the applicability of LES in atmospheric boundary layer studies
is limited by the subgrid model’s ability to adequately describe the subgrid motions’
effect on the filtered (resolved) fields. This is the compromise to simulate turbulence
in larger domains with heterogeneous turbulence properties. Also, like all numerical
approaches based on temporal and spatial discretization of the governing flow equa-
tions, LES is subject to various numerical artifacts including: phase speed errors,
artificial viscosity, and dispersion errors.
1.4 Motivation for Dissertation
A thorough understanding of the CBL is of significant importance in meteorological
studies. One of the most important parameters used in scaling the CBL parameters
is the CBL depth zi or inversion layer height. However, there are discrepancies when
this height is obtained from radar measurements (Botnick and Fedorovich 2008). A
close comparison of this parameter can be obtained through the combination of LES
and a LES-based radar simulator. Under these conditions, the gap between numerical
simulations and remote sensing methods will be addressed.
Turbulence parameters needing quantification are the turbulence kinetic energy
(TKE) and energy eddy dissipation rate (￿). Studies of ￿ have been made from radar
measurements of the return radar backscatter (Hocking 1983, 1985, 1992; Hocking
and Mu 1997) and Doppler spectral width (Hocking 1983, 1985; White 1997; White
et al. 1999; Jacoby-Koaly et al. 2002). However, these studies have been developed
for the troposphere and stratosphere and have never been applied to the CBL. Also,
derivations from the radar backscattering were made under stable conditions which
consider gradients of potential refractive index profiles, and due to the high level of
turbulence present in CBL, only the estimates obtained from the Doppler spectral
width are suitable in CBL studies.
Horizontal wind components obtained from DBS or Spaced Antenna (SA) tech-
niques have been found to be biased by the horizontal shear of vertical velocity, and
depending on the level of the shear present in the CBL, estimates can be significantly
18
biased (Flowers et al. 1994; Zhang and Doviak 2007). Three-dimensional wind es-
timates have been obtained from both DBS and SA techniques for different levels
of shear and compared with the theoretical formulation. Later, with the the combi-
nation of LES and LES-based radar simulator, the bias was estimated for realistic
conditions.
By definition, estimates of TKE have to be obtained from the variance of the
three-dimensional winds. TKE estimates from LES will be compared and contrasted
with the ones obtained from radar simulator measurements to find to what extent
the estimates from radar are comparable with those commonly used in fluid dynamics
(Stull 1988). Also, the influence of the horizontal shear of vertical velocity in the TKE
estimates will be studied for different time averaging schemes.
1.5 Outline of Dissertation
Following the introduction, a more in-depth review of Doppler radars and LES is
presented in Chapter 2, including scattering mechanisms and signal processing. In
Chapter 3, a radar simulator based on LES is described, along with the different exper-
imental setups and techniques that will be used to characterize the CBL. Estimations
of turbulence parameters based on the virtual radar such as structure function pa-
rameter of refractive index C2n, three-dimensional wind fields (u, v, and w), vertical
velocity variance σ2w, and the vertical velocity skewness are presented in Chapter 4.
Here, these parameters are compared with estimates obtained from an operational
wind profiler located in Lamont, OK. In Chapter 5, the effect of horizontal shear of
vertical velocity on horizontal wind estimations will be studied, along with turbulence
kinetic energy and turbulence-eddy dissipation rate for different conditions. The dis-
sertation concludes in Chapter 6 with an assessment of the suitability of the radar
estimates for the characterization of the CBL and future work recommendations.
19
Chapter 2
Large-Eddy Simulations and Doppler Radars:
Theoretical Background
2.1 Large-Eddy Simulations
The LES code used in this work was developed following the methodology described
in Nieuwstadt (1990), Fedorovich et al. (2004b,a), and Conzemius and Fedorovich
(2006). This code was extensively tested in comparison with several other represen-
tative LES codes and with experimental data for clear CBLs with and without wind
shear (Fedorovich et al. 2004b; Fedorovich and Conzemius 2008). The LES was found
to confidently reproduce turbulence structure for a broad variety of flow regimes ob-
served in the clear CBL. In its last version, the code was revised, optimized, and
modified to incorporate realistic atmospheric sounding data retrieved from observa-
tions or larger-scale atmospheric model/analysis outputs (Botnick and Fedorovich
2008).
Following the established LES methodology adopted in atmospheric and engineer-
ing turbulence studies, motions in the simulated turbulent CBL flow are subdivided
into larger-scale resolved motions. These motions are directly reproduced on the
computational grid, and the smaller-scale so-called sub-grid motions are modeled
through additional stress/flux terms in the discretized governing equations for the
resolved quantities (Botnick and Fedorovich 2008). Here, the Einstein summation
convention is used.
20
∂ ￿ui
∂t
= −∂ ￿ui ￿uj
∂xj
+ g
￿Θv −Θv0
Θv0
δi3 − ∂￿π
∂xi
+ f
￿￿uj − ugj￿ εij3
+
∂
∂xj
￿
ν
￿
∂ ￿ui
∂xj
+
∂ ￿uj
∂xi
￿
− (￿uiuj − ￿ui ￿uj)￿ , (2.1)
∂ ￿ui
∂xi
= 0 , (2.2)
∂￿Θv
∂t
= −∂ ￿ui￿Θv
∂xi
+
∂
∂xi
￿
µ
￿
∂￿Θv
∂xi
￿
−
￿
￿Θvui − ￿Θv ￿ui￿￿ , (2.3)
where the tildes denote volume averaged (resolved) quantities, i, j = {1, 2, 3}; t is
time, xi = (x, y, z) are the right-hand Cartesian coordinates, ￿ui = (￿u, ￿v, ￿w) represent
resolved velocity components, ugj=1,2 = (ug, vg) are the geostrophic wind components,
ν is kinematic viscosity, µ is molecular thermal diffusivity, and ￿Θv is resolved virtual
potential temperature of dry-air that would have the same density as moist-air and is
used as a convenient surrogate for density in buoyancy calculations (AMS Glossary).
Components of subgrid kinematic momentum and subgrid heat fluxes are represented
by ￿uiuj − ￿ui ￿uj and￿Θvui − ￿Θv ￿ui, respectively. Normalized pressure, ￿π, is defined as￿π = (￿p− p0) /￿0, where ￿p is resolved pressure, p0 is hydrostatic atmospheric pressure,
￿0 is constant reference density, and Θv0 is constant reference potential temperature.
The sub-grid momentum and buoyancy flux are modeled in terms of the sub-grid
eddy viscosity (Km) and the sub-grid eddy diffusivity (Kh) (Deardorff 1980):
￿uiuj − ￿ui ￿uj = 2
3
Eδij − 2Km￿sij, (2.4)
￿Θvui − ￿Θv ￿ui = −Kh∂￿Θv
∂xi
, (2.5)
where ￿sij = (∂ ￿ui/∂xj + ∂ ￿uj/∂xi) /2 represents the deformation tensor for filtered
velocity, and subgrid turbulence kinetic energy E is determined from the balance
equation,
∂E
∂t
+
∂ ￿uiE
∂xi
= 2Km￿sij ∂ ￿ui
∂xj
−Kh∂
￿Θv
∂x3
+
∂
∂xi
￿
2Km
∂E
∂xi
￿
− ε, (2.6)
where ε is the sub-grid TKE viscous dissipation rate. Eddy viscosity and diffusivity
are expressed as
Km = 0.12lE
0.5, (2.7)
Kh =
￿
1 +
2l
∆
￿
Km, (2.8)
21
where ∆ = (∆x∆y∆z)1/3 is the effective grid-cell size, and l is the sub-grid mixing
length. Finally, ε and l are evaluated as
ε = fc
￿
0.19 + 0.51
l
∆
￿
E3/2
l
, (2.9)
l =

∆ , ∂
￿Θv
∂z ≤ 0;
min
∆,
0.5
√
E￿
∂￿Θv
∂z
 ,
∂￿Θv
∂z > 0,
(2.10)
fc = 1 +
2￿
zw
∆zw
+ 1.5
￿2
− 3.3
, (2.11)
where zw is the distance from the ground, and ∆zw is the vertical dimension of the
lowest grid cell.
A Poisson equation for ￿π is constructed by combining the continuity and momen-
tum balance equations as in Nieuwstadt (1990). The equation is solved numerically
by the fast Fourier transform technique over horizontal planes and by tri-diagonal ma-
trix decomposition in the vertical planes. Periodic boundary conditions for prognostic
variables (velocity components, potential temperature, specific humidity, and E) are
prescribed on the simulation domain side-walls. In the upper 20% of the domain, a
sponge layer is introduced in order to dampen vertical motions close to the domain top
and to ensure steady-state conditions at the upper boundary. To minimize the sponge
layer’s inevitable spurious influence on the numerical solution, the time advancement
is only maintained as long as the CBL depth is less than 70% of the domain height.
The no-slip boundary condition for velocity and the zero-gradient condition for sub-
grid energy are prescribed at the heated bottom surface. Monin-Obukhov similarity
relationships are used point-by-point to couple local buoyancy and mechanical tur-
bulence forcing within the lowest layer of grid cells. The surface roughness length,
temperature, and moisture fluxes are prescribed as external parameters.
The LES was applied to reproduce a daytime CBL case observed at the Southern
Great Plains (SGP) Atmospheric Radiation Measurement (ARM) Climate Research
Facility (ACRF) in Lamont, OK (LMN), on June 8, 2007. The LES was initialized
with realistic environmental profiles of horizontal velocity components u and v, poten-
tial temperature Θ, and turbulent kinematic virtual heat flux w￿Θ￿v. As schematically
22
indicated in Botnick and Fedorovich (2008), realistic environment settings should in-
clude multiple inversions (areas of high stability) and sharp wind changes with height
(wind shears) typically observed in the atmosphere. The surface heat flux (see bottom
panel of Figure 2.1b) reflects the solar heating evolution during the day. Represented
in the top panel of Figure 2.1a, are the initialization profiles of wind and potential
temperature. These profiles are kept constant above the CBL depth for the whole sim-
ulation. In the bottom panel of this figure is the surface heat flux for the simulation.
Θ
Θ
Θ Θ
w￿Θ￿v w￿Θ￿v
Figure 2.1: Schematic representation of idealized (a) and realistic (b) initial data (from
Botnick and Fedorovich (2008)).
Surface fluxes of temperature and moisture were calculated from measured sensible
H and latent LH heat fluxes using (Botnick and Fedorovich 2008):
w￿Θ￿ =
H
￿0 cp
, (2.12)
w￿q￿ =
LH
￿0 Lv
, (2.13)
w￿Θ￿v = w￿Θ￿ + 0.61Θv0w￿q￿, (2.14)
where w￿Θ￿ is surface kinematic heat flux and w￿q￿ is surface kinematic moisture
flux. cp is specific heat for air, and Lv is latent heat of vaporization. Geostrophic
wind components were evaluated from the Rapid Update Cycle (RUC) objective
analysis data (Benjamin et al. 2004), which are available hourly. The geostrophic wind
23
is assumed to be height-constant. The geostrophic wind components are obtained
through the calculation of horizontal gradients of pressure, which use four RUC grid
points surrounding the LMN site.
ug =
−1
￿0f
∂p
∂y
, (2.15)
vg =
1
￿0f
∂p
∂x
, (2.16)
where f = 2Ω sinΦ is the Coriolis parameter, Ω is the Earth’s angular velocity, p
is the surface pressure from RUC, and Φ is the latitude. Wind profiles (obtained
either from RUC and/or from the local sounding at LMN) are interpolated to the
LES domain’s vertical nodes. Additionally, near-surface portions of these profiles are
adjusted in order to match the no-slip condition at the surface, under the assumption
of a logarithmic wind profile throughout the lowest LES cell layer. The simulation
for the present study has been performed in a rectangular domain composed of 20 m
grid cells. The LES settings are summarized in Table 2.1.
Table 2.1: Description of the LES settings
Parameter Setting
Domain size 5.12 × 5.12 × 4.0 km3
Number of grid points 256 × 256 × 200
Spatial resolution 20 m
Time step 1 s
Output variables Resolved u, v, w, q, and Θ and E
The LES output included instantaneous fields of the filtered flow velocity compo-
nents, potential temperature, specific humidity, and the sub-grid TKE are available
with one-second time increments in the BLR sub-domain. Flow statistics were ob-
tained locally by temporal averaging. Spatial statistics, obtained through horizontal
planes averaging, were also included in the output. Evaluated statistics included mean
flow variables, velocity, temperature, and humidity variances (both resolved and sub-
grid), vertical fluxes of momentum, heat and moisture, and third-order moments of
the resolved vertical velocity and temperature fields.
A sub-set of the LES output was employed as an input data-set for the virtual
Boundary Layer Radar - BLR (see Chapter 3). This sub-domain had spatial limits
24
of 1860 m≤X≤3260 m, 1860 m≤Y≤3260 m, and 0 m≤Z≤2000 m. The virtual BLR
used three-dimensional fields of potential temperature Θ, specific humidity q, sub-grid
TKE E, and flow velocity components u, v, and w along the coordinate directions
x, y, and z, respectively. A snapshot of the simulated CBL flow structure is given in
Figure 2.2.
y(m)
z (
m
)
Zonal Wind (m s−1)
 
 
−600 −400 −200 0 200 400 600
0
500
1000
1500
2000
−20
−10
0
10
20
y(m)
z (
m
)
Meridional Wind (m s−1)
 
 
−600 −400 −200 0 200 400 600
0
500
1000
1500
2000
−20
−10
0
10
20
y(m)
z (
m
)
Potential Temperature (K)
 
 
−600 −400 −200 0 200 400 600
0
500
1000
1500
2000
285
290
295
300
305
310
y(m)
z (
m
)
Specific Humidity (g/g)
 
 
−600 −400 −200 0 200 400 600
0
500
1000
1500
2000
0
2
4
6
x 10−3
y(m)
z (
m
)
Vertical Wind (m s−1)
 
 
−600 −400 −200 0 200 400 600
0
500
1000
1500
2000
−5
0
5
y(m)
z (
m
)
Subgrid Kinetic Energy (m2 s−2)
 
 
−600 −400 −200 0 200 400 600
0
500
1000
1500
2000
0
0.2
0.4
0.6
0.8
1
Cross Section at X = 0 m − Time: 17:30:00
Figure 2.2: Examples of LES fields in the virtual radar sub-domain. Top-left: zonal wind.
Top-right: meridional wind. Middle-left: potential temperature. Middle-right: specific
humidity. Bottom-left: vertical wind. Bottom-right: sub-grid kinetic energy. All data
presented refer to the same single realization in time.
The presented instantaneous flow fields illustrate the main features of the CBL
turbulent flow structure. By its nature, the clear atmospheric CBL is a turbulent
boundary layer whose turbulence is primarily forced by heating from the underlying
surface. One may see an intensive turbulent mixing of momentum and scalar fields in
the main portion of the CBL, beneath the capping inversion (seen in the plots as a zone
25
of sharp vertical gradients of potential temperature and specific humidity). Typically
collocated with the capping inversion layer is the entrainment zone, through which
the strongly turbulent CBL flow interacts with the relatively inactive free-atmospheric
flow aloft. Another characteristic feature of the CBL flow is its coherent structure
on larger scales that is represented by convective updrafts (thermals) and downdrafts
which are clearly observed in the vertical velocity field pattern (see bottom panel of
Figure 2.3). Rare, but more intense in comparison with downward motions, updrafts
observed in the plot are indicative of positive skewness of vertical velocity in the
atmospheric CBL.
He
igh
t (
m
)
Zonal Wind (m sï1)
UTC Time
 
 
12:00 13:00 14:00 15:00 16:00 17:00 18:00 19:00 20:00 21:00 22:00 23:00 00:00
0
500
1000
1500
2000
ï20
ï10
0
10
20
He
igh
t (
m
)
Meridional Wind (m sï1)
UTC Time
 
 
12:00 13:00 14:00 15:00 16:00 17:00 18:00 19:00 20:00 21:00 22:00 23:00 00:00
0
500
1000
1500
2000
ï20
ï10
0
10
20
He
igh
t (
m
)
Vertical Wind (m sï1)
UTC Time
 
 
12:00 13:00 14:00 15:00 16:00 17:00 18:00 19:00 20:00 21:00 22:00 23:00 00:00
0
500
1000
1500
2000
ï5
0
5
Figure 2.3: Three-dimensional wind fields obtained at the center of the LES sub-domain
for the ∼11.5 hours of simulation. Top: zonal wind. Middle: meridional wind. Bottom:
vertical wind.
26
2.2 Pulsed Doppler Radar
2.2.1 Basic Concepts
A radar is an electromagnetic system for detecting, locating, and characterizing re-
flective objects. These objects can be aircrafts, ships, spacecrafts, vehicles, people,
and the natural environment (Skolnik 2001). Radars operate by radiating energy into
space and detecting the echo signal reflected from an object or target. The returned
echo indicates the presence of a target, as well as other valuable information, such
as its distance from the radar, and information related to the target’s size, and in
some cases, its velocity relative to the radar. In most cases, a radar can operate both
automatically and under extreme weather conditions.
The basic principle of operation is depicted in Figure 2.4. For illustrative purposes,
only the antenna is depicted. A more detailed explanation of the design of a radar
system is presented later in this section. An electromagnetic pulse is generated in
the transmitter and radiated into space by the antenna (a). The pulse travels in
space until it reaches the target (b). The transmitted energy is then intercepted by
the target and reradiated in all directions (c). Some of this energy moves toward the
antenna (d) and is delivered to the receiver. In a monostatic system, the transmitting
and receiving antenna are the same. The system is considered bi-static if the antennas
are physically separated.
a) b)
c) d)
Adapted from http://weather.noaa.gov/radar/radinfo/radinfo.html
Figure 2.4: Simple schematic of the basic principles of a radar. (a) A pulse is transmitted
into space. (b) The pulse reaches the target. (c) The target scatters in all directions. (d)
Some of the energy is reradiated towards the antenna and delivered to the receiver.
27
A simplified block diagram of a homodyne Doppler radar is presented in Figure 2.5.
The stabilized local oscillator (STALO) generates a continuous wave (CW) signal
that represents an almost perfect sinusoid (carrier frequency). This stable RF (radio
frequency) signal is converted into a pulsed signal (on/off modulator). Once it has
been modulated, the RF signal is amplified to produce intense microwave power
(Doviak and Zrnic´ 1993). The amplified pulse is radiated into space by the antenna.
The T/R switch has the important function of separating the transmitted pulse from
the received signal. The received signal is “mixed” with the same RF signal which
was used for the transmitter pulse obtained from the STALO. Through this coherent
quadrature detection process, the so-called in-phase I(t, r0) and quadrature Q(t, r0)
signals are obtained.
Pulse Modulator
Amplifier
T/R Switch
Mixer
Signal Processing
Antenna
Q(t,r0)I(t,r0)
STALO
Display
LPF
-90º
Figure 2.5: Block Diagram of a homodyne Doppler Radar. The RF signal generated by the
STALO is modulated, amplified, and transmitted into space by the antenna. The received
signal is “mixed” with the same RF from STALO, filtered, processed, and finally, displayed
for the user.
An heterodyne Doppler radar (see Figure 2.6) generates the RF signal by com-
bining the STALO with a coherent oscillator (COHO) signal. The modulated and
28
amplified pulse is transmitted through the antenna. Here, the received signal is mixed
with the STALO generating an intermediate frequency (IF) signal. In an analog re-
ceiver, this IF signal is mixed with the COHO frequency to generate the I(t, r0) and
Q(t, r0) signals. Today, a digital receiver samples the intermediate frequency (IF) sig-
nal of a radar receiver. Matched filtering, baseband conversion and phase referencing
for coherent-on-receive operation are realized by digital signal processing hardware
and software. Usually the filtering and demodulation is an integrated decimation
process. The advantages of using digital receivers include higher dynamic range, per-
fect coherent quadrature detection, programable low pass filters (LPF), and matching
filter to the transmitted pulse.
Pulse 
Modulator Amplifier
T/R Switch
-90º
Signal 
Processing
Antenna
Q(t,r0)
I(t,r0)
LPF Display
BPF
STALO
COHO
IF Signal
Mixer
Digital Receiver
Figure 2.6: Block Diagram of a heterodyne Doppler Radar. A difference from the homo-
dyne system presented in Figure 2.5 is the generation of the RF frequency by combining the
STALO and COHO signals. The received signal is mixed with the STALO to obtained an
IF signal. This IF signal is sampled by a digital receiver, where matched filtering, baseband
conversion and phase referencing for coherent-on-receive operation are realized by digital
signal processing hardware and software.
The distance, or the range, to a target is obtained by measuring the time it takes
the radar signal to travel from the radar to the target and back. If the target is in
motion, I(t, r0) and Q(t, r0) signals will be used to measure a return signal frequency
29
shift from the Doppler effect. This frequency shift is proportional to the radial velocity
(velocity of the target relative to the radar). The Doppler frequency shift is used to
separate the target’s desired movement from fixed “clutter” echoes originating from
mountains, sea, buildings, etc. A recent upgrade to digital receivers of the WSR-
88D was completed in 2006 as part of the NEXRAD Product Improvement Program
Open Systems Radar Data Acquisition (ORDA) project (Crum et al. 1998). One of
its main features was the incorporation of the ground clutter suppression algorithm
known as Gaussian Model Adaptive Processing (GMAP) (Siggia and Passarelli Jr.
2004). GMAP identifies and removes 20 clutter points. Then, it replaces them with
an estimated Doppler spectrum, thereby improving the quality of the post-filtering
spectral estimates. Also, the algorithm only applies the filter to spectra that follow
the known shape of stationary clutter. The problem with this assumption is that
unwanted signals like cars, sea, and/or wind turbines are not removed because they
appear as desired targets for the algorithm.
As mentioned previously, the target’s range is calculated by the time, Tr, it takes
the radar signal to travel to the target and back (range-time) (see horizontal axis on
Figure 2.7). The electromagnetic energy in free space travels at the speed of light,
which is c ≈ 3× 108 m s−1, so that the expression for the range r to the target can
be expressed as:
r =
cTr
2
. (2.17)
Typically, multiple transmitted pulses are sent in a sequence at a rate determined by
the longest expected target’s range. The time between pulses is also known as “pulse
repetition time (PRT)” or “inter-pulse period (IPP)” and is represented by Ts. The
PRT or sample-time (see oblique axis on Figure 2.7) also represents the frequency at
which the radar signals are sampled. If the PRT is too short, a backscattered signal
from a long-range target could arrive after the transmission of the next pulse, and this
can be mistakenly associated with the latter pulse rather than the previous, which
could lead to an incorrect or ambiguous range estimation (Skolnik 2001). These kinds
of echos are called “second-trip echoes”. The maximum range after which targets
appear as second-trip echoes is the “maximum unambiguous range” ra and can be
represented as:
ra =
cTs
2
=
c
2fs
, (2.18)
where fs = 1/Ts represents the “pulse repetition frequency” (PRF).
30
Range-Time
Sa
mp
le-
Tim
e
Ts
Tr
Tr1
Tr2
Figure 2.7: Idealized traces for the radar signal (adapted from (Doviak and Zrnic´ 1993)).
Each trace represent a composite of echoes for each transmitted pulse. Instantaneous sam-
ples are taken at Tr1, Tr2, etc. The red stars represent a probable time dependence of the
sample at Tr. Samples at fixed Tr taken at Ts intervals are used to construct the Doppler
spectrum from targets located at approximately the range cTr/2.
31
The target’s velocity measurements are obtained from the received signal phase.
The echo phase is sampled at intervals Ts, and its change over the interval Ts is
a measure of the Doppler frequency fd = −2vr/λ, where vr is the radial velocity,
λ = c/f is the radar wavelength, and f is the frequency of the radar. According to
the “Nyquist Sampling theorem”, the sampling frequency has to be greater than twice
the highest frequency of interest in order to calculate the Doppler frequency without
any ambiguities fs > 2fN . By combining both expressions, the “aliasing velocity”
can be expressed as:
va = ± λ
4Ts
. (2.19)
Unfortunately, both ambiguity conditions (Equations (2.18) and (2.19)) cannot be
improved simultaneously. A large Ts is needed to avoid range ambiguities, and a short
Ts is needed to avoid velocity aliasing. Both requirements must compromise to achieve
good range and velocity estimates (“Doppler dilemma”). In practice (especially in
weather radars), the volume scan is designed with two distinct PRTs to analyze range
and velocity independently, as well as to possibly correct second-trip echos in post-
processing. This method is called batch PRT, and it uses a sequence of two short
PRTs to estimate velocity and another sequence of long PRTs to estimate power.
Other methods to mitigate the range velocity ambiguity include staggered PRT
(SPRT) (Sirmans et al. 1976), dual PRF, and phase coding. SPRT alternates between
a short and long PRT, with the preferred ratio of 2/3. The maximum unambiguous
range is determined by the long PRT, while the maximum unambiguous velocity is
obtained by SPRT velocity de-aliasing algorithm (Doviak and Zrnic´ 1993; Torres et al.
2004). The dual PRF (Dazhang et al. 1984) technique alternates two “batches” of
PRTs, which improves the ground clutter filtering. The technique uses the SPRT
dealiasing velocity algorithm. However, it fails when there is strong shear present.
The systematic phase coding technique uses deterministic codes to reduce the effects
of range overlaid echoes (Sachidananda and Zrnic´ 1986). This code is usually part
of the SZ-2 algorithm developed by NSSL and The National Center for Atmospheric
Research (NCAR) and has been operational on the NEXRAD network since 2007.
The algorithm transmits a long PRT without coding and short PRT with a SZ(8/64)
code to recover velocities from overlaid echoes.
32
2.2.2 The Radar Equation
Radars transmit pulses at a high power to maintain the required sensitivity to detect
reflections from targets at a desired maximum range. The radar equation is used to
determine the returned power Pr based on multiple parameters from a single target
with backscattered cross-section σb (Skolnik 1990; Doviak and Zrnic´ 1993).
Pr =
PtG2λ2σbf 4(θ,φ)
(4π)3r4l2
, (2.20)
where Pt is the transmitted power, and G2 represents the gain of the transmitting and
receiving antennas for the monostatic case. The range to the target is represented by
r, l represents the attenuation losses, and f 4(θ,φ) is the two-way normalized power
density pattern. The radar equation is important for characterizing a radar system.
There are many forms of the radar equation, and depending on the application,
different parameters are considered. For atmospheric scatter, however, it is almost
impossible to have a single target within the resolution volume. Consequently, it is
necessary to modify the equation to consider volume scattering. Hence, the the radar
equation for a volume filled with targets (Doviak and Zrnic´ 1993; Cohn 1995; Hocking
1985, 1996; White 1997) can be presented as:
Pr =
PtG2λ2
(4π)3r20l
2
× cτw
2
× πθ
2
1
8 ln 2
× η (2.21)
where r0 is the range to the center of the resolution volume. The radar resolution
volume refers to the minimum separation between two volume targets that permits
them to be distinguished by a radar. The second term on the right-hand-side of
the equation represents the range resolution ∆r = cτw/2, where τw is the transmit-
ting pulse width. The third term on the right-hand-side represents the transmitting
beamwidth, where θ1 is the 3-dB width (in radians) of the one-way pattern (Doviak
and Zrnic´ 1993). Finally, the radar reflectivity η, or average backscatter cross-section
per unit volume (White 1997), is a measurement of the radar scattering intensity in
units of m−1.
2.2.3 Scattering Mechanisms
2.2.3.1 Weather Radar - Rayleigh Scatter
For weather radars, the radar reflectivity is related to the amount and type of pre-
cipitation in the resolution volume. If the diameter D of a spherical drop is small
33
compared to λ (D ≤ λ/16), then the process is approximated by Rayleigh scatter-
ing (Doviak and Zrnic´ 1993). Under the Rayleigh approximation, the backscattered
cross-section σb is proportional to D6. The radar reflectivity can be expressed as a
function of the drop size distribution N(D) and σb:
η =
￿ ∞
0
σb(D)N(D)dD, (2.22)
σb ≈ π
5
λ4
|Km|2D6. (2.23)
The term Km is called the complex dielectric factor and is defined as (Doviak and
Zrnic´ 1993):
Km =
m2 − 1
m2 + 2
, (2.24)
where m = n− jnκ is the complex refractive index for water. The refractive index is
n, and κ is the attenuation index. The typical value of |Km|2 for liquid water is 0.92
for wavelengths between 0.01 and 0.10 m. Ice spheres have a |Km|2 of approximately
0.18, which is independent of temperature and is valid for radar wavelengths in the
microwave region (Doviak and Zrnic´ 1993).
The reflectivity can also be expressed as a function of Z, the so called “reflec-
tivity factor”. This is a very common term which can be related to the intensity of
precipitation.
η =
π5
λ4
|Km|2Z, (2.25)
Z =
￿ ∞
0
N(D)D6dD. (2.26)
Whenever the Rayleigh approximation does not apply (usually for short-wavelength
(λ < 10 cm) weather radars), it is acceptable to express the reflectivity as (Doviak
and Zrnic´ 1993):
η =
π5
λ4
|Km|2Ze, (2.27)
where Ze is the “equivalent reflectivity factor”. The units of Z are mm6 m−3, but it
is more common to express Z on a logarithmic scale (dBZ or 10log10
Z
1mm6m−3
). An
example of the values of Z are taken from Doviak and Zrnic´ (1993) and presented in
Table 2.2.
34
Table 2.2: Examples of the values of Z
Z value Type of precipitation
0 dBZ cumulus clouds
20 dBZ light rain
60 dBZ heavy rainfall and hail
2.2.3.2 Clear-Air Radar - Bragg Scatter
The energy spectrum of turbulence E(κ) proposed by Kolmogorov can be classified
into three regions (see Figure 2.8): the energy-containing range, the inertial subrange,
and the dissipation range (Pope 2000). Under this hypothesis (White 1997), there
is no exchange of energy between the energy-containing range and the dissipation
range. However, the rate of energy transfer from large scales in the energy-containing
range is equivalent to the energy dissipation rate at small scales in the dissipation
range. In the inertial subrange, the bulk of energy transfers from larger to smaller
scales following a slope of -5/3, and it can be observed in Figure 2.8. This is also
known as “energy cascade” assumption or “Kolmogorov -5/3 turbulence spectrum”
for isotropic turbulence, where the turbulence has the same variance in all directions
(Stull 2000). Under these conditions the energy spectrum of turbulence per unit mass
E(κ) can be expressed as:
E(κ) = Ae￿
2/3κ−5/3, (2.28)
where Ae is a universal dimensionless constant between 1.53 and 1.68 (Gossard and
Strauch 1983), ￿ is the turbulence kinetic energy (eddy) dissipation rate, and κ rep-
resents the wavenumber. In the ABL inertial subrange, the large eddies usually have
sizes of 200 m and the small eddies sizes of 1 cm (Wallace and Hobbs 2006).
The radar reflectivity for clear-air radars is a measurement of the intensity caused
by refractive index fluctuations present in the radar resolution volume. If the radar
half-wavelength (Bragg scale) lies within the inertial subrange, the radar reflectivity
can be represented as:
η = 0.379C2nλ
−1/3, (2.29)
where C2n is the structure function parameter of refractive index (m
−2/3) (Tatarskii
1961; Ottersten 1969; Cohn 1995; Doviak and Zrnic´ 1993; White 1997).
35
slope = -5/3
Inertial subrange
Energy-
containing
range
Dissipation
range
κ
E(κ)
Figure 2.8: Simple diagram of the Kolmogorov turbulence spectrum (adapted from Pope
(2000)). The energy spectrum of turbulence E(κ) and wavenumber κ axes are in logarithmic
scale. In the inertial subrange, the large eddies from the energy-containing range cascades
into the small eddies in the dissipation range following a slope of -5/3.
Typical values of C2n are extracted from Doviak and Zrnic´ (1993) and presented in
Table 2.3. These values give an idea of the turbulence magnitude typically observed
in the atmosphere.
Table 2.3: Examples of the magnitude and intensity of C2n
C2n value Intensity of turbulence
6 × 10−17 m s−2/3 weak
2 × 10−15 m s−2/3 intermediate
3 × 10−13 m s−2/3 strong
36
2.3 Signal Processing for Radars that Observe the
Atmosphere
The received signal after the matched filter in Figure 2.5 has an in-phase and quadra-
ture component, and can be represented as:
I(t, r0) =
|A|
2
cosψeU(t− 2r0/c), (2.30)
Q(t, r0) = − |A|
2
sinψeU(t− 2r0/c), (2.31)
where A is the complex amplitude, and U(t) is the transmitted pulse shape defined
as:
U(t) =
￿
1 0 ≤ t ≤ τw
0 elsewhere
. (2.32)
The echo phase of the signal ψe is represented as (Doviak and Zrnic´ 1993):
ψe =
4πr0
λ
+
4πvrt
λ
− ψt − ψs, (2.33)
where vr is the average radial velocity within the resolution volume, ψt and ψs rep-
resent the transmitted phase and the scattering phase shift, respectively. Normally,
the received complex echo voltage is the combination of the in-phase and quadrature
components as:
V (t, r0) = I(t, r0) + jQ(t, r0), (2.34)
V (t, r0) =
|A|
2
exp(−jψe)U(t− 2r0/c). (2.35)
The components of the complex echo voltage (I, Q) are considered random vari-
ables because the motion of the scatters within the resolution volume is unpredictable
(Doviak and Zrnic´ 1993). Also, within this volume, the scatters are large in number,
and none are dominant. This condition implies, through the Central Limit Theorem,
that I and Q have a Gaussian distribution. Moreover, the atmosphere is constantly
changing, and the time over which the complex echo voltage is taken is small com-
pared to the time it takes for significant changes to occur. Under these conditions,
the complex echo voltage V is a wide-sense stationary (WSS) random process. Fi-
nally, the received signals are considered to be ergodic, which means that temporal
averaging will converge to ensemble averages and be used to investigate the statistical
parameters of the process.
37
The time-series data obtained from the radar can reveal a wealth of information
of the characteristics within the resolution volume. The most practical approach is
to analyze the Doppler spectrum, which is the power-weighted distribution of radial
velocities within the resolution volume. The Doppler spectrum is obtained from the
Fourier transform of the autocorrelation function (ACF):
S(f) = F{R(τ)}, (2.36)
where R(τ) is the ACF assuming a WSS random process
R(τ) = E[V ∗(t)V (t+ τ)], (2.37)
where ∗ represents complex conjugate, and E[·] represents the expected value.
Since the complex echo voltage is sampled through the pulsed radar design, the
Doppler spectrum or power spectral density (PSD) must be obtained through the
calculation of the discrete time Fourier transform (DTFT):
￿S(f) = Ts M−1￿
m=−(M−1)
￿R(k)e−j2πfTsk, (2.38)
where k represents the lag, and M represents the total number of samples. The ￿·
represents an estimated value. For a discrete WSS signal, the biased estimator of
ACF is given by:
￿R(k) = 1
M
M−|k|−1￿
m=0
V ∗(m)V (m+ k). (2.39)
In this equation, the term Ts is omitted in the finite complex echo voltage V (mTs).
The bias of a signal estimator α refers to the difference between the expected
value of the estimator and the true value Bias(￿α) = E[￿α]−α. The mean-square-error
(MSE) can be expressed in terms of the variance and the bias E[(￿α− α)2] = Var(￿α)
+ Bias2(￿α).
Another estimator of the Doppler spectrum can be obtained through the peri-
odogram as:
￿S(f) = Ts
M
|Z(f)|2, (2.40)
where Z(f) is the DTFT of V (mTs)(Doviak and Zrnic´ 1993; Shanmugan and Breipohl
1988) calculated as:
Z(f) =
M−1￿
m=0
V (m)e−j2πfmTs . (2.41)
38
Since ￿S(f) is estimated based on a finite number of pulses, there is a bias resulting
from the multiplication of the complex echo voltage signal and the rectangular window
of length M (data window), which is required for truncated data. The Doppler
spectrum estimate is the convolution between the true spectrum with the Fourier
transform of the corresponding lag window and results in a broadening of the true
spectrum and transfer of power from the peak into the sidelobes (Doviak and Zrnic´
1993). The lag window is obtained through the calculation of ACF of the data
window.
2.3.1 Spectral Moments
The Doppler spectrum is often assumed to have a Gaussian shape and is represented
as:
S(v) =
S√
2πσv
exp
￿−(v − vr)2
2σ2v
￿
+
2NTs
λ
, (2.42)
where S represents the signal power, N is the noise power which is assumed to be
white, and vr = −λfd/2 is the mean radial velocity within the resolution volume.
A schematic of the Doppler spectrum is depicted in Figure 2.9. Also, for direct
calculations of the radial velocity, the Doppler spectrum presented in Equation (2.42)
is now represented in terms of velocity instead of frequency (see Equation (2.38)).
The most important characteristics of the signals obtained from the Doppler spec-
trum are described below (Doviak and Zrnic´ 1993).
1. Total power or the zeroth moment of the Doppler spectrum: Depending on the
type of radar, this can be indicative of the turbulence level (clear-air radars),
or the liquid water content or precipitation (weather radars).
2. Mean Doppler velocity or the first moment of the power-normalized spectra: This
is indicative of the scatterers’ radial velocity.
3. Spectrum width σv, the square root of the second moment about the first of the
normalized spectrum: This a measure of shear or turbulence within the resolu-
tion volume.
The ACF (see Figure 2.10) is calculated from the inverse Fourier transform of
S(v) and is presented below:
R(mTs) = S exp
￿
−8
￿
πσvmTs
λ
￿2￿
e−j4πvrmTs/λ +Nδm, (2.43)
where δm is the Kroneker delta function.
39
σv
S
N
v
S(f)
vr
Figure 2.9: Schematic of the Doppler spectrum. S represents the signal power, N is the
noise power (usually assumed “white”), vr is the mean radial velocity, and σv represents
the Doppler spectral width.
2.3.1.1 Moment Estimation in the Time Domain
The total power P , or the zeroth moment, is estimated from the ACF at lag zero
(m = 0). In comparison to Equation (2.43), it can be observed that the estimated
power is the sum of the estimated power signal S and estimated noise power N :
￿P = ￿S + ￿N, (2.44)
￿P = ￿R(0) = 1
M
M−1￿
m=0
V ∗(m)V (m), (2.45)
￿S = ￿R(0)− ￿N. (2.46)
The noise power appears at the center of the ACF (see Figure 2.10). The signal power
is estimated by removing the center point and replacing it by a interpolated value
using the closest four points around the center. The noise power is then obtained by
subtracting the initial ACF at lag zero minus the estimated signal power.
An improvement in the accuracy of the power estimation is obtained when the
dwell time is long (i.e., number of samples M large). However, this improvement in
accuracy sacrifices the data update time (Doviak and Zrnic´ 1993).
40
m∠R(mTs)
S
N
m
|R(mTs)|
slope = −4π
λ
vrTs
Figure 2.10: Schematic of the autocorrelation function. Top: magnitude of the ACF.
Signal power is represented by S, N is the noise power. Bottom: phase of the ACF. The
radial velocity vr is calculated based on the phase of the ACF at lag 1 (τ = 1).
41
An estimate of the mean radial velocity can be obtained from the autocorrelation
function at lag one (m = 1) in Equation (2.43):
￿vr = −λ
4πTs
∠( ￿R(Ts)), (2.47)
where ∠ represents the phase of the ACF.
Finally, for the Doppler spectral width estimates, two lags are necessary. The
direct way is to use lag zero and lag one; however, the magnitude of the ACF at lag
zero is contaminated with noise power. Here, the estimate of σv will be obtained from
lag one ( ￿R(Ts)) and lag two ( ￿R(2Ts)) as:
￿σv = λ
2
√
6πTs
￿
ln
￿ ￿R(Ts)￿R(2Ts)
￿￿1/2
. (2.48)
Note that this procedure could lead to undetermined values of Doppler spectral width
due to calculations of fractional values of ln(·), and these values need to be discarded
as σv should always be positive.
Computationally, estimates of the moments are more efficient in the time domain
(Doviak and Zrnic´ 1993). Nevertheless, estimates in the frequency domain can help
to remove unwanted signals (i.e., interference) or to study other signal characteristics
(i.e., non-Gaussian shape (Yu et al. 2009)). Finally, Lei et al. (2011) introduced a
new technique that uses multi-lag correlation estimators to obtain polarimetric radar
measurements in the presence of low signal-to-noise ratio (SNR).
2.3.1.2 Moment Estimation in the Frequency Domain
An estimate of the total power in the frequency domain is obtained by summing the
contributions of all velocities of the periodogram as:
￿P = λ
2MTs
M−1￿
k=0
￿S(k). (2.49)
The signal power ￿S is obtained after subtracting the white noise contribution ￿N using
a noise level estimation scheme such as that proposed by Hildebrand and Sekhon
(1974).
42
The estimate of the mean radial velocity ￿vr and Doppler spectrum width ￿σv are
obtained from Doviak and Zrnic´ (1993):
￿vr = −λ
2MTs
km+M/2+1￿
k=km−M/2
k￿Sn(modM(k))− M
2
 , (2.50)
￿σv2 = ￿ λ
2MTs
￿2 km+M/2−1￿
k=km−M/2
(k − km)2￿Sn(modM(k)), (2.51)
where km is a rough estimate of the peak of the spectra, and mod(·) represents the
“modulo” operator. These calculations mitigate biases due to aliasing if spectra are
symmetric (Doviak and Zrnic´ 1993). An expression for the normalized spectrum￿Sn(k) is obtained from:
￿Sn(k) = ￿S(k)￿M−1
k=0
￿S(k) . (2.52)
The measured spectrum width σ2v is the contribution of many independent broad-
ening factors such as: shear σ2s , turbulence σ
2
11, and signal processing σ
2
x (Doviak
and Zrnic´ 1984; Doviak and Zrnic´ 1993). The spectrum width can be represented
as σ2v = σ
2
s + σ
2
11 + σ
2
x. Consequently, the Doppler spectrum due to turbulence is
contaminated by all the other contributions. These contributions can be eliminated
depending on their cause. For example, the shear contribution can be represented
as a function of the horizontal wind (Doviak and Zrnic´ 1993), and the signal pro-
cessing contribution is related to M number of points. Both effects can be removed
successfully. However, if the turbulence is very weak, there might be cases where the
resultant spectrum could be negative and should be discarded (Hocking 1988).
2.4 Techniques for the Estimation of the Three-
Dimensional Wind Field
2.4.1 Doppler Beam Swinging
Doppler Beam Swinging (DBS) is a technique used to estimate the three components
of the wind field in clear-air. This technique is based on measurements of mean radial
velocity using three or more non-coplanar beams. The radar usually points vertically,
or nearly vertically. DBS is based on the assumption of homogeneity and stationarity
during the time of the scan. A summary of the technique can be found in the work
of Gage and Balsley (1978), Balsley (1981), and Balsley and Gage (1982).
43
Radial velocity can be expressed as a function of the three wind field components:
the zonal wind uo positive to the East, the meridional wind vo positive to the North,
the vertical wind wo positive away from the radar (upward). The radial velocity,
presented in the left panel of Figure 2.11, can be expressed as a function of the
horizontal velocity vH and wo, and vH , presented in the right panel of Figure 2.11, as
a function of uo, vo, the azimuth angle θ, and zenith angle φ as:
vr = vH sinφ+ wo cosφ, (2.53)
vH = uo sinφ+ vo cos θ, (2.54)
vr(θ,φ) = uo sin θ sinφ+ vo cos θ sinφ+ wo cosφ. (2.55)
O E
N
vr
φ
vH
wo
N
E
vH
θ
uo
vo
Figure 2.11: Schematic of the calculations of radial velocity vr as a function of the three
wind fields components. Left: the radial velocity is expressed as a function of the horizontal
velocity and the vertical wind field. Right: the horizontal velocity is expressed as a function
of the zonal and meridional wind fields.
When combining the radial velocities from at least three beams of similar height
(see Figure 2.11), it is more convenient to express Equation (2.55) in matrix form,
vr = Au, (2.56)
44
where :
A =

sin θ1 sinφ1 cos θ1 sinφ1 cosφ1
...
...
...
sin θn sinφn cos θn sinφn cosφn
 , (2.57)
u =
￿
uo vo wo
￿T
, (2.58)
vr =
￿
vr1 · · · vrn
￿T
. (2.59)
N
EO
Beam 1Beam 3
Beam 2
Figure 2.12: Schematic of the Doppler Beam Swinging Technique. DBS uses at least
three non-coplanar beams to estimate the three components of the wind field under the
assumptions of homogeneity and stationarity during the time of the scan.
The minimum number of independent radial velocities needed to estimate the
three-dimensional velocity vector u is three (n = 3). If n < 3, the system becomes
undetermined. If n > 3, the solution to the set of equations can be estimated in the
least-squares sense using:
u =
￿
AAT
￿−1
ATvr. (2.60)
2.4.2 Spaced Antenna
Another technique used to estimate the three-dimensional wind is called the Spaced
Antenna (SA) method, which technique uses one transmitter and three or more closely
45
spaced receivers. By calculating the cross-correlation function (CCF) between the re-
ceived radar signals, it is possible to estimate the horizontal wind and wind variability
(turbulence). The SA technique was first presented by Briggs et al. (1950), and its
implementation using the Full Correlation Analysis (FCA) was described in Briggs
(1984).
The generalized theoretical expression for the CCF used by SA to measure hor-
izontal wind is presented in Doviak et al. (1996) and Holloway et al. (1997). The
equation is presented below:
c12(τ) = exp
￿
−2jkowoτ − 2(koσtτ)2 − β2h
￿
uoτ − ∆x
2
￿2
−β2h
￿
voτ − ∆y
2
￿2
− 1
8
￿
woτ
σr
￿2￿
, (2.61)
where τ represents the time lag, ko = 2π/λ is the radar wavenumber, and ∆x and
∆y represent the separation of the two receivers along the x and y direction, respec-
tively. The root-mean-square error (rms) wind variability of each wind component
for isotropic turbulence is represented by σt, σr is the second central moment of the
range weighting function, which is assumed to be equal for both receivers, βh is re-
lated to the scale length of the diffraction pattern (see Equations (3), (4), (5), and
(6) of Holloway et al. (1997)). The effect of the vertical wind wo is typically small
and will be omitted in the future (Holloway et al. 1997; Zhang and Doviak 2007).
By taking the magnitude of Equation (2.61) and rewriting in a more convenient
form, the normalized ACF (∆x,∆y=0) and CCF (∆y=0) are obtained:
|c11(τ)| = exp
￿−β2h(u2o + v2o)− 2k2oσ2t τ 2￿+ NS δ(τ), (2.62)
|c12(τ)| = exp
￿
−β2h
￿
uoτ − ∆x
2
￿2
− β2hv2oτ 2 − 2k2oσ2t τ 2
￿
. (2.63)
A new notation for ACF c11(τ) is included here to be consistent with CCF. However
both c11(τ) and R(τ) are equivalent to denote the ACF. Typically, the ACF and CCF
are assumed to be Gaussian functions. The wind estimates using FCA are obtained
based on their Gaussian fitted parameters using:
|￿c12(τ)| = exp ￿−￿η − (τ − ￿τp)2
2￿τ 2c
￿
+ ￿c0, (2.64)
where ￿η is the decorrelation parameter, ￿τp is the peak time delay of |￿c12(τ)|, ￿τc is the
square root of the second central moment of |￿c12(τ)|, and ￿c0 accounts for the observed
46
amplitude offset of ACF and CCF (Holloway et al. 1997). This offset is caused by a
DC component present in the received voltage signals. A schematic of the ACF and
CCF is presented in Figure 2.13.
!!" !# !$ !% !& " & % $ # !"
"
"'!
"'&
"'(
"'%
"')
"'$
"'*
"'#
"'+
!
!,-./0
1
2
3.
4
567
8
9
,:
2
33
8
54
;6
2
<
,=
4
>
<
6;
?
9
8
c12(τ)
c11(τ)
exp(−ηˆ)
τˆc
τˆp τˆx
Figure 2.13: Normalized autocorrelation (c11(τ)) and cross-correlation (c12(τ)) functions.￿η represents the decorrelation paramenter, ￿τp is the peak time of CCF, ￿τc is the square root
of the second central moment of CCF, and τx is the lag such that |c11(τx)| = |c12(0)|.
The process of calculating the winds from the Briggs’ FCA algorithm can be
complicated, but for horizontal isotropic scatters, the baseline wind in the zonal
direction can be written as (Briggs 1984; Holloway et al. 1997; Zhang et al. 2003;
Doviak et al. 2004):
￿u = ∆x￿τp
2￿τ 2x , (2.65)
where ￿τx is the lag such that |c11(τx)| = |c12(0)|. If both the ACF and CCF have
Gaussian shape, the technique method is called FCA-Gaussian (FCA-G), and the
expression for the baseline wind can be expressed in terms of the Gaussian parameters
as (Holloway et al. 1997; Zhang et al. 2003; Doviak et al. 2004):
￿τ 2x = 2￿η￿τ 2c + ￿τ 2p , (2.66)￿u = ∆x￿τp
2(2￿η￿τ 2c + ￿τ 2p ) . (2.67)
47
In this chapter, the basics of the LES have been presented along with fundamental
radar theory. The LES is used in the simulation of a realistic CBL case. This case will
be studied with the synthesis of a virtual BLR based on LES. Through radar theory,
estimates of power, radial velocity, and spectral width obtained from the virtual BLR
will be used to compare and contrast parameters from the CBL. Finally, calculations
of the ACF and CCF will be used to estimate the three-dimensional winds using SA,
compared with those obtained from the radial velocities through DBS, and contrasted
with the “true” winds from LES.
48
Chapter 3
Radar Simulator Design Based on Large-Eddy
Simulations
3.1 Time Series Radar Simulator
The radar equation (Equation (2.21)) reveals that the backscattered signal power
from a distributed target is directly proportional to the radar reflectivity η. The
value of η represents the average backscattering cross-section per unit volume. Clear-
air scatter will develop from turbulent variations in the refractive index n. The radar
reflectivity η is given as a function of the structure function parameter of refractive
index C2n and radar wavelength λ (see Equation (2.29)).
In addition to the assumptions that the turbulence is both isotropic and in the
inertial subrange, it is further assumed that the radar resolution volume is uniformly
filled with turbulence. In this case, the structure function parameter of refractive
index C2n is derived from the refractive index field using (Tatarskii 1971):
Dn(δ) =
￿
[n(r+ δ)− n(r)]2￿
r
, (3.1)
Dn(δ) = C
2
nδ
2/3, (3.2)
C2n =
￿
[n(r+ δ)− n(r)]2￿
r
|δ|2/3 , (3.3)
whereDn is the structure function for locally homogeneous perturbations, and< · >r
denotes the spatial average over a volume within which the n irregularities are as-
sumed to be statistically isotropic and homogeneous. Here, r represents the position
vector, δ denotes the spatial separation, and δ = |δ|. The refractive index is related
to the refractivity N through n = 1 + N × 10−6. If the background atmospheric
pressure profile is in hydrostatic balance, then the refractivity is found directly from
49
the simulation output parameters through the following equations (Bean and Dutton
1966; Holton 2004):
N =
77.6
T
￿
P + 4811
e
T
￿
, (3.4)
dlnP = − g
RT
dz, (3.5)
T = Θ
￿
P
P0
￿0.286
, (3.6)
e =
qP
0.622 + q
, (3.7)
where P is total atmospheric pressure (hPa), e(q, T ) is the partial pressure of water
vapor (hPa), P0 represents the pressure at z = 0 m (1000 hPa), g is the gravitational
acceleration (9.81 m s−1), R is the gas constant for dry air (287 J kg−1 K−1), and
T is the absolute temperature (K). The electron density term has been omitted in
Equation (3.4) due to its influence only in the ionosphere.
Various approaches are available in the literature that have been used to generate
time-seres data for radar simulations. One approach that was studied by Sheppard
and Larsen (1992); Holdsworth and Reid (1995); Yu (2000); and Cheong et al. (2004a)
uses the Lagrangian framework approach. This approach consists of creating a sam-
pling domain populated with scattering points. These points move within the domain
according to the instantaneous wind vector field.
Another method follows the Eulerian framework approach which considers the
grid cells of the model as a scattering center. The phase of the radar signal from the
scattering center is modulated by the local instantaneous velocity field (Muschinski
et al. 1999; Scipio´n et al. 2008). Therefore, by varying the phase without actually
moving the scatterer, the expected Doppler velocity can be generated. Some imple-
mentation advantages exist with this method such as simple calculations of oblique
C2n and fast updates of velocity at each grid point.
The LES sub-domain description and output variables used in the radar simulator
are presented in Table 3.1. To calculate C2n within each LES domain grid cell, Equa-
tion (3.3) is applied along a line (beam) which connects the position of the virtual
radar to the center of the grid cell where C2n is estimated (see left panel of Figure 3.1).
However, the radar is sensitive to the Bragg scale (δ = λ/2) to which the scale δ cor-
responds. This Bragg scale (∼16 cm for a frequency of 915 MHz) is much smaller
50
Table 3.1: Description of the LES settings
Parameter Setting
Domain size 1.4 × 1.4 × 2.0 km3
Number of grid points 70 × 70 × 100
Spatial resolution 20 m
Time step 1 s
Output variables u, v, w,E, q,Θ
than the LES grid cell size (∆ ∼20 m). Nevertheless, due isotropy assumed within
the LES fields, Equation (3.3) can be expressed as:
C2n =
(∆n￿)2
δ2/3
=
(∆n)2
∆2/3
, (3.8)
where ∆n￿ represents the refractive index n gradient at the Bragg scale (which is
considered isotropic on the radar scale), ∆n represents the refractive index gradient
at the grid scale, and ∆ represents the oblique distance between the layers where the
∆n is estimated.
To estimate ∆n, the refractive index n is calculated in the center of the grid cell
at level z with respect to the ground and at two levels displaced from z in height by
an increment ∆z (see right panel of Figure 3.1). That is, the points at z, z−∆z, and
z +∆z are considered. A bilinear interpolation is then applied in order to obtain an
estimate of n between the closest four points on the upper and lower planes, thereby
calculating values of n along the beam:
nxy = (1− dx)(1− dy)nx0y0 + (dx)(1− dy)nx1y0 + (3.9)
(1− dx)(dy)nx0y1 + (dx)(dy)nx1y1 ,
dx =
x− x0
x1 − x0 , dy =
y − y0
y1 − y0 ,
where nxy represents the refractive index n at point (x, y) (which does not match any
of the grid points), and nxiyj denotes refractive index values at xi,yj; i, j = 0, 1.
This modification along with the average over two consecutive layers constitute a
significant refinement from the work of Muschinski et al. (1999) and yields:
C2n(x, y, z, t) =
1
2
￿
(n1 − n)2
∆2/3
+
(n− n2)2
∆2/3
￿
, (3.10)
where n1, n, and n2 represent the refractive index at levels z + ∆z, z, and z − ∆z,
respectively, as depicted in right panel of Figure 3.1.
51
XY
Z
Oblique
Beam
LES Grid
Wb
Wr
z
∆
z - ∆z
dx
dx
dy
dy
n1
n
n2
z + ∆z
∆
Figure 3.1: Left: scheme that represents an off-vertical pointing beam. The dotted lines
represent the radial from the location of the radar to the grid points of the LES within the
resolution volume. Along these dotted lines, C2n is calculated, and later weighted in range
(Wr) and beam width (Wb). Right: the dotted line represents the axis along which C2n needs
to be estimated. First, the value of n at level z is obtained from the LES matrix; however,
at levels z+∆z and z−∆z, the dotted line does not match any of the LES grid points. In
order to obtain an estimate at those heights, a linear interpolation is made using the four
closest points. Finally, an average of the two estimates of structure function parameter of
n (from z and z +∆z, and z and z −∆z) with an oblique separation ∆ is computed.
52
Presented as a single realization in time, Figure 3.2 depicts an example of the spe-
cific humidity q vertical profile from the LES output at the center of the sub-domain,
as well as the corresponding calculated profiles of refractivity N , and the horizontal
averaged structure parameter of refractive index C2n. The observed turbulence level of
C2n is intermediate with an enhancement of at least one order of magnitude at approx-
imate 1000 m. This enhancement can be identified as the CBL top, and according
to Table 2.3, the intensity level of turbulence is strong. The vertical profile structure
of C2n are in excellent agreement with Figure 11.15 of Doviak and Zrnic´ (1993), and
radar measurements of marine air over OK gave a mean of C2n = 5×10−13 m2/3 and
peak values as high as 3×10−12 m2/3 for mid-afternoon skies (Doviak and Berger 1980;
Doviak and Zrnic´ 1993).
4.8 5 5.2 5.4 5.6 5.8
x 10−3
200
300
400
500
600
700
800
900
1000
1100
1200
Ra
ng
e(
m
.)
Specific Humidity (kg/kg)
270 275 280 285 290 295 300
200
300
400
500
600
700
800
900
1000
1100
1200
Ra
ng
e(
m
.)
Refractivity (N)
10-14 10-13 10-12
200
300
400
500
600
700
800
900
1000
1100
1200
Ra
ng
e (
m.
)
m2/3
C2n,φ = 0
o, θ = 0o
Figure 3.2: Left: vertical profile of specific humidity characteristics for the CBL. Center:
vertical profile of refractivity calculated from the LES fields. Right: vertical profile of the
horizontally averaged structure function parameter of refractive index C2n. All data were
calculated from a single realization in time.
53
The received voltage V from the Eulerian framework is expressed in its exponential
form (recall Equation (2.35)) at time t0 +∆t as:
V (t0 +∆t) =
M￿
p=1
A(p)(t0 +∆t) exp(−jψ(p)e (t0 +∆t)), (3.11)
A(p)(t0 +∆t) = A
￿
￿
C2n(t0 +∆t)
(p)W (p)r W
(p)
btxW
(p)
brx, (3.12)
A￿ =
√
GtxGrx
λrtx0rrx0
√
0.0330k−11/6B , (3.13)
ψ(p)e (t0 +∆t) = ψ
(p)
0 + k0(r
(p)
tx + r
(p)
rx )
+k0(eˆ
(p)
tx + eˆ
(p)
rx ) · v(p)(t0 +∆t)∆t, (3.14)
where the amplitude A(p) is proportional to the square root of C2n and inversely
proportional to the range of the center to the resolution volume from the transmitter
rtx0 and receiver rrx0. The LES time is represented as t0 and ∆t is the time increment
which is a function of the PRT. Each individual grid point of the M points contained
within the resolution volume is represented by p, and Gtx and Grx are constants
proportional to the gain of the transmitter and receiver, respectively. Random initial
phase is represented as ψ(p)0 , k0 is the radar wavenumber, eˆ
(p)
tx is the unit vector directed
from the transmitter antenna to the pth LES grid cell, eˆ(p)rx is the unit vector directed
from the receiver antenna to the pth LES grid cell, and kB is the Bragg wavenumber
(kB =
4π
λ ). The instantaneous three-dimensional wind field v
(p) is estimated from the
LES resolved wind fields (uo,vo, and wo) and the sub-grid TKE E as:
u = uo + u
￿, (3.15)
v = vo + v
￿, (3.16)
w = wo + w
￿, (3.17)
where u, v, and w represent the resultant wind fields, and u￿, v￿, and w￿ are the
sub-grid fluctuations of the velocity fields. Under the assumptions of isotropy, the
variance of the fluctuations is defined as a function of E as:
E =
1
2
(σ2u + σ
2
v + σ
2
w), (3.18)
σ2u = σ
2
v = σ
2
w =
2
3
E. (3.19)
Therefore, the fluctuation component of any of the wind fields can be represented as
a random Gaussian variable with zero mean and
￿
2
3
E standard deviation.
54
The range weighting function Wr represents the convolution between the impulse
response of the receiver with the transmitted pulse and is described by (Holdsworth
and Reid 1995; Scipio´n et al. 2008) as:
Wr(x, y, z) = exp
￿
(r − r0)2
2σ2r
￿
, (3.20)
where r represents the projection of each grid point range
￿
x2 + y2 + z2 over the
pointing direction, and the variance is σr = 0.35cτw/2, where τw is the pulse width
(Doviak and Zrnic´ 1993). Finally, Wbtx and Wbrx represent the one-way transmitter
and receiving beam pattern weighting function (Yu 2000; Cheong et al. 2004a; Scipio´n
et al. 2008).
Wbtx(x, y, z) = exp
￿
−(θx − θxtx)
2
4σ2x
− (θy − θytx)
2
4σ2y
￿
, (3.21)
Wbrx(x, y, z) = exp
￿
−(θx − θxrx)
2
4σ2x
− (θy − θyrx)
2
4σ2y
￿
, (3.22)
θx = tan
−1
￿x
z
￿
, θy = tan
−1
￿y
z
￿
,
where θxtx and θytx describe the transmitting antenna beam pointing; θxrx and θyrx
describe the receiving antenna beam pointing in degrees for both cases; and σx =
σy = θ1/2.36 is proportional to the beamwidth. The range weighting function and
beam weighting function are illustrated in the right panel of Figure 3.1.
Muschinski et al. (1999) introduced a radar simulator based on LES only with a
vertical beam and for one realization in time. Improvements over that work are:
• Values of C2n are calculated over oblique beams along the radial.
• C2n and velocity are interpolated at each time ∆t.
• The location of each grid point with respect to transmitter and receiver are now
part of the equation.
• Radial velocities at each LES are calculated as the resultant of both the trans-
mitter and receiver components.
• Radar beamwidth effect is also calculated independently for the transmitter and
receiver.
All these additions permit multiple frequency applications, as well as monostatic and
bi-static. This allows opportunities for many experiments based on LES fields.
55
Since the LES generates data for variables with 1 s temporal resolution, and the
radar inter-pulse period (IPP) is on the order of milliseconds, an estimation of all of
the variables X at t0 + ∆t is needed. In order to achieve this, the desired value is
calculated using the linear interpolation scheme presented by Cheong et al. (2004a):
X(t0 +∆t) = (1−∆t)X(t0) + (∆t)X(t1), (3.23)
where t0 and t1 are two consecutive time steps of the LES, and ∆t is the intermediate
value taken from 0 to 1. Also suppose that the V AR(X(t0)) = V AR(X(t1)) =
V AR(X), where V AR(·) denotes the variance. Thus the variance of the interpolated
random variable X(t0 +∆t) is:
V AR(X(t0 +∆t)) = [(1−∆t)2 + 2ρ(1−∆t)∆t+∆t2]V AR(X), (3.24)
which causes V AR(X(t0+∆t)) to be reduced in a quadratic form as a function of ∆t
as shown in the previous equation. In order to reverse this artifact, the interpolated
value can be scaled with the inverse-square-root function of the equation. Therefore,
the scaled interpolated value is (Cheong et al. 2004a):
f ￿i(x) =
1￿
(1−∆t)2 + 2ρ(1−∆t)∆t+∆t2fi(x), (3.25)
where ρ is the cross-correlation factor of X(t0) and X(t1).
In order to produce realistic results from the radar simulator, it is necessary to
introduce additive white Gaussian noise (AWGN) to our generated time-series data.
First, the minimum power of the signal is estimated from the time-series data. Second,
the variance of the background is computed based on a desired SNR. The complex
white noise is generated based on the assumption that it has a Gaussian distribution
described by the previously calculated variance. This procedure is independently
applied to the real and imaginary components of the time-series data. The complex
Gaussian noise is simply added to the original time-series data. An example of time-
series data corresponding to a vertically pointing beam is presented in Figure 3.3.
The complex AWGN shown was calculated for a minimum SNR of -3 dB.
3.2 Possible Configurations of the Virtual Radar
Some of the “virtual” experiments that will be used along with this work require
special radar configurations. These configurations might require one or more virtual
radars which are located at different positions with the LES sub-domain and are
56
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
2
1.5
1
0.5
0
0.5
1
1.5
2
x 104
Time (s)
V(
t)
Simulated wind profiler signal without noise (r = 475 m)
 
 
I
Q
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 22
1.5
1
0.5
0
0.5
1
1.5
2 x 10
4
Time (s)
V(
t)
Simulated wind profiler signal with noise (r = 475 m)
 
 
I
Q
Figure 3.3: Time-series data from the simulated BLR data V (t) obtained at a range
of 475 m. The blue line represents the real part (in-phase) of V (t), and the red line is
the imaginary part (quadrature) of V (t). Top: without AWGN. Bottom: with AWGN
(minimum SNR = -3 dB).
57
presented in the following sub-sections. Some of them may require a simplification
in the radar equation, while others just include a diagram of the radar location and
pointing direction to specify the experiment configuration.
3.2.1 Monostatic Configuration
The monostatic configuration uses the same antenna for both transmission and re-
ception. An illustration of the beam pattern is presented in Figure 3.4. As can be
observed, the resolution volume is located inside the LES sub-domain, which is rep-
resented by the rectangular volume. This configuration allows the radar to possess
both a vertical and an oblique pointing beam.
O
φ
Figure 3.4: Schematic of the monostatic radar configuration. The simulated radar has the
capability to point at any desired azimuth and zenith angle provided the resolution volume
is located within the LES sub-domain.
As a consequence of the co-located antennas, they share the same beam-pattern
weighting function. Also, the relative distance and unit vectors from each of the LES
58
particles to the transmitter and receiver are the same. The resulting simplified radar
voltage can be expressed as (Scipio´n et al. 2008):
V (t0 +∆t) =
M￿
p=1
A(p)(t0 +∆t) exp(−jψ(p)e (t0 +∆t)),
A(p)(t0 +∆t) = A
￿
￿
C2n(t0 +∆t)
(p)W (p)r W
(p)
b , (3.26)
A￿ =
G
λr20
√
0.0330k−11/6B , (3.27)
ψ(p)e (t0 +∆t) = ψ
(p)
0 + kBr
(p) + kB(eˆ
(p) · v(p))(t0 +∆t)∆t, (3.28)
where eˆ(p) represents the unit vector from the center of the antenna to the to pth LES
grid cell, G =
√
GtxGrx is proportional to the transmitted power and transmitting
and receiving antenna gain, and r20 = rtx0rrx0 is the distance from the radar to the
center of the resolution volume. Finally, Wb is the two-way beam weighting function
(Yu 2000; Cheong et al. 2004a; Scipio´n et al. 2008):
Wb(x, y, z) = exp
￿
−(θx − θx)
2
2σ2x
− (θy − θy)
2
2σ2y
￿
, (3.29)
θx = tan
−1
￿x
z
￿
, θy = tan
−1
￿y
z
￿
.
By combining three or more monostatic radar beams, DBS or multiple radar
configurations can be obtained, as explained next.
3.2.1.1 Doppler Beam Swinging
DBS is the most common configuration to estimate the three-dimensional wind fields
using a single monostatic radar. In this configuration, the radar beam stays in one
position for a specific amount of time (dwell time), then switches to the next position
until all the needed beams to estimate the three wind components have been scanned.
The time that takes the radar to go back to the starting beam position is called the
“scan time”. Even though the technique only requires three non-coplanar beams, it
usually uses five: one vertical beam and four oblique beams which point toward the
cardinal directions. The use of more than three beams provides a better estimate of
the horizontal wind while reducing the variance of the estimates. A representation of
DBS is illustrated in Figure 3.5.
To obtain the wind estimates, the radial velocity of the five beams are combined
after the scan time (see Section 2.4.1). The assumptions made in these calculations
are homogeneity over the volume containing the radar beams and stationarity during
the time of the scan.
59
NS E
W V
O
Figure 3.5: Doppler beam swinging configuration. The configuration is based in a single
monostatic radar that points in five different, non-coplanar directions. Usually, the oblique
beams point to the four cardinal directions.
60
3.2.1.2 Multiple Radar Configuration
The multiple radar configuration (MRC) consists of five different radars which are lo-
cated equidistant from one another within the LES sub-domain. A diagram indicates
the location of the radars in Figure 3.6. A vertical pointed radar is located in the
center of the LES sub-domain. The other four radar beams are directed off-vertically
but point towards the vertical beam of the first radar at different heights. The spacing
between the heights will be chosen depending on the resolution desired for the study.
A
B
D
C E
Figure 3.6: Multiple monostatic radar setup. The configuration uses the assumption that
five simultaneous monostatic radars point towards the same resolution volume inside the
LES.
The procedure to obtain the three-dimensional winds at each height is the same
presented for DBS. However, this configuration allows us to obtain more accurate
results than DBS when compared with the “true” LES fields because they do not
rely on spatial homogeneity or stationarity over the dwell time. This is possible
because the radar coverage volume is small in comparison with the effective DBS
sampling volume, and all radars are available simultaneously. That avoids the scan
time. Although this configuration obtained better results than DBS (Scipio´n et al.
2007), it is impractical due to expense.
61
3.2.2 Spaced Antenna
The SA technique (Briggs et al. 1950) consists a pseudo-monostatic radar where the
transmitting and receiving antennas point vertically and are not collocated in the
strict sense. However, the separation between the center of the antennas can be as
small as the diameter of the receiving antenna. The method uses one transmitter
and three or more closely spaced receivers. The common volume is reached as a
consequence of the close spacing between the transmitting and receiving radar beams.
Estimates of the horizontal wind are obtained using the interferometry pattern
of signals received along the baseline of pair of separate (spaced) antennas (see RxA
and RxB in Figure 3.7). The calculations of the interference or diffraction pattern
are based on the ACF and CCF of the received signals, and the detailed procedure
is described in Section 2.4.2.
A basic diagram of the transmitting and receiving beams is illustrated in Fig-
ure 3.7. It can be observed that the transmitting and receiving beamwidths are
not the same, which is certainly not a mandatory condition. In the illustration, the
transmitting beamwidth is half the receiving beamwidth. Also, at low heights the
transmitting and receiving beams do not intercept, which will result in erroneous
calculations of the refractive pattern and, consequently, in undetermined horizontal
wind estimates.
In this chapter, the radar simulator based on LES fields was presented along
with the modifications and/or simplifications for different experimental setups. In
the next chapter, the specific parameters for each radar simulator will be presented.
Also, results and comparisons with real data will be provided.
62
Tx RxBRxA
Figure 3.7: Spaced Antenna Setup. This experiment uses one antenna for transmission and
three or more for reception. The distance between the transmitter (Tx) and receivers (RxA
and/or RxB) is small, so it allows the beams to intercept. The different transmitter and
receiver beamwidths have been incorporated in the radar simulator. Here, the beamwidth
is exaggerated for diagram purposes.
63
Chapter 4
Characterization of the Boundary Layer:
Validation with Realistic Data
In this chapter, the simulated radar signals based on LES fields that were presented
in Chapter 3 are compared with the “true” LES fields and realistic data. The com-
pared parameters are the structure function parameter of refractive index, the three-
dimensional wind fields, the vertical velocity variance, and the vertical velocity skew-
ness. The realistic data are obtained from the 915 MHz radar wind profiler/radio
acoustic sounding system (RWP/RASS) which is located at the SGP ACRF site.
4.1 Experimental Configuration
The 915 MHz RWP/RASS provides wind profiles and backscattered signal strength
in the height range of 0.1 km to 5 km (nominally) and virtual temperature profiles
from 0.1 km to 2.5 km (see Figure 4.1). Typically, the radar operates in two modes
with different range resolutions. The low-mode has a range resolution of 62.5 m
and a maximum range of 2600 m, while the high-mode has a 212.55 m resolution
and a maximum range of 6670 m. The radar has a 9◦ beamwidth, and points in
the four cardinal directions with a zenith angle of 15.5◦. The dwell time in each
mode during the considered observation period is approximately 30 s, while the scan
time is between 6 and 7 min. The radar simulator has a range resolution of 60 m.
Consequently, only the low-mode is considered in this study. Three spectral moments
are calculated based on the Doppler wind spectra downloaded from the ARM website
(http://www.arm.gov/).
64
Figure 4.1: Picture of the Radial Wind Profiler (RWP) located at the SGP ACRF. In
addition to the wind profiles, the system can obtain virtual temperatures through the Radio
Acoustic Sounding System (RASS) co-located with the radar (Courtesy of Dr. P. Chilson).
4.2 Structure Function Parameter of Refractive
Index
As mentioned in Chapter 2, the received radar power Pr is proportional to the radar
reflectivity η, which is a measure of the radar scattering intensity. Also, η is propor-
tional to the structure function parameter of refractive index C2n (see Equation (2.29)).
If the radar is calibrated, profiles of C2n can be obtained for every radar dwell time.
Unfortunately, an accurate radar calibration can be complex, requiring detailed
measurements of the various radar parameters (see Equation (2.21)), such as the
transmitted power Pt, antenna gain G, radar beamwidth θ1, etc. After combining
Equations (2.21) and (2.29) and grouping all the known radar parameters, the esti-
mation of C2n can be expressed as:
logC2n = log(Pr × r2)− 2 log l + logC, (4.1)
where Pr × r2 represents the total range-corrected power, C is the radar calibration
constant, and l represents the one-way system losses.
65
Range-time-intensity (RTI) plots of C2n are presented in Figure 4.2. Here, the
middle panel represents the C2n that was obtained directly from the LES fields for a
vertically pointed radar beam. This represents the same estimate obtained from the
LES parameters following the procedure described in Scipio´n et al. (2008) and used
in the magnitude term of the radar simulator (see Equation (3.11)). The top panel
represents the C2n that was obtained from the range-corrected power of the virtual
BLR (RadSim). To calculate C2n from Equation (4.1), the loss term was omitted, and
the constant C was obtained empirically by comparing the level of turbulence around
the maximum of C2n from RadSim with the “true” C
2
n from LES. This represents
a rudimentary estimate of the CBL top. Finally, in the bottom panel, there is an
empirical calibration of the range-corrected SNR that was obtained from the vertical
beam of the radar located at the SGP ACRF (RadarARM) for the same date of the
simulation.
As studied in Section 2.1, the LES was setup to reproduce the cloud free CBL
case of June 8, 2007. Profiles of potential temperature and horizontal wind for the
LES initialization were obtained from the same location as the RWP (SGP ACRF).
The evolution of the LES is controlled by the turbulent kinematic heat flux for the
whole simulation period obtained from the same location. However, the LES code
is nudged with hourly profiles obtained from the RUC interpolated at every second.
This combination gives the LES the freedom to evolve accordingly to the heat flux
from LMN and RUC profiles. There is a good agreement between the estimates
from the RadSim and LES, except at the lowest height where there are not enough
points within the virtual resolution volume to characterize the C2n appropriately. Also,
there is an unrealistic increase in the intensity of C2n observed above 1000 m at the
RadSim RTI, which is caused by applying the range-corrected algorithm to regions
with low SNR. Additionally, the LES RTI presents some artificial layers that appear
as discrete bands at determinant heights, which are caused by residual layers present
at the initialization profiles of the LES run. Finally, the LES captures the evolution
of the CBL during the day. Even so, the code is apparently unable to reproduce
the evolution of external atmospheric forcing, which is observed at the heights of
maximum C2n between LES and RadarARM . Nevertheless, the LES produces C
2
n that
closely matches the trend in radar data (Botnick and Fedorovich 2008).
66
UTC Time
He
igh
t (
m
)
log10(Cn
2 (mï2/3)) ï RadSim (Vertical)
 
 
12:00 13:00 14:00 15:00 16:00 17:00 18:00 19:00 20:00 21:00 22:00 23:00 00:00
0
500
1000
1500
ï14
ï13.5
ï13
ï12.5
ï12
ï11.5
ï11
UTC Time
He
igh
t (
m
)
log10(Cn
2 (mï2/3)) ï LES
 
 
12:00 13:00 14:00 15:00 16:00 17:00 18:00 19:00 20:00 21:00 22:00 23:00 00:00
0
500
1000
1500
2000
ï14
ï13.5
ï13
ï12.5
ï12
ï11.5
ï11
UTC Time
He
igh
t (
m
)
log10(Cn
2 (mï2/3)) ï RadarARM (Vertical)
 
 
12:00 13:00 14:00 15:00 16:00 17:00 18:00 19:00 20:00 21:00 22:00 23:00 00:00
500
1000
1500
2000
ï14
ï13.5
ï13
ï12.5
ï12
ï11.5
ï11
Figure 4.2: Structure function parameter of the refractive index estimates on June 8, 2007.
Top: estimates from the vertical beam of the radar simulator. Middle: LES-profile at center
of domain. Bottom: estimates from the vertical beam of the SGP ACRF radar.
67
4.3 Wind Field Comparisons
The following three quantities are analyzed for each of the three wind components (u,
v, and w): DBS values calculated from the radial velocities of the virtual BLR which
points in the same directions of RWP (RadSimDBS), resolved LES values along a
vertical profile at the center of the simulation sub-domain (LES), and DBS estimates
from the RWP (RadarARM). Based on the revisit time of the RWP, wind estimates
are retrieved every 12 minutes. In order to provide a fair comparison between the
three quantities, the averaging time for RadSimDBS and LES velocities is also set to
12 min.
The following procedure is used to obtain the RadSimDBS values. First, radial
velocities from each of the five virtual BLR beams are estimated with a dwell time
of 30 s. Second, velocity samples are computed every 6 min corresponding to the
scan time of the RWP measurements. Third, the velocities are averaged over 12-min
periods based on the DBS revisit time. Finally, the DBS technique is used to retrieve
the three wind components. The LES data are averaged in time to obtain 12-min
estimates used for comparison. Radial velocities from RadarARM are averaged over
the same time period. Values with SNR≤-10 dB are censored from further analysis.
Values of the three wind components obtained by different techniques are pre-
sented in Figures 4.3 (zonal), 4.4 (meridional), and 4.5 (vertical). The RadSimDBS
estimates do not appear as smooth as those from LES. This is primarily due to the
realistic noise contamination and associated measurement error. In general, the wind
fields retrieved from LES of the fair CBL case, in terms of the classification applied
in Botnick and Fedorovich (2008), agree well with the RadarARM estimates. The ob-
served discrepancies (especially, in the zonal and vertical winds) seem to result from
the evolution of external atmospheric forcing, which is unaccounted for in this ver-
sion of the LES code. The LES is initialized using data from a rawinsonde launched
at 11:30 UTC at the SGP ACRF site. The background atmospheric state repre-
sented by this initial sounding is assumed to be steady throughout the simulation.
Therefore, evolution of turbulence fields within the CBL is determined entirely by
surface heating. It is reproduced realistically based on data from surface heat bal-
ance measurements at the SGP ACRF site and the variable geostrophic forcing. The
agreement observed between the LES and the RadarARM reflects the extent of the
LES code’s ability to reproduce the basic properties of the evolving CBL flow fields.
68
The relatioship between the RadSimDBS and the RadarARM demonstrates how the
virtual BLR is able to reproduce these fields similarly to the real radar.
UTC Time
He
igh
t (
m
)
Zonal Wind (ms−1) − RadSimDBS
 
 
12:00 13:00 14:00 15:00 16:00 17:00 18:00 19:00 20:00 21:00 22:00 23:00 00:00
500
1000
1500
−20
−10
0
10
20
UTC Time
He
igh
t (
m
)
Zonal Wind (ms−1) − LES
 
 
12:00 13:00 14:00 15:00 16:00 17:00 18:00 19:00 20:00 21:00 22:00 23:00 00:000
500
1000
1500
2000
−20
−10
0
10
20
UTC Time
He
igh
t (
m
)
Zonal Wind (ms−1) − RadarARM
 
 
12:00 13:00 14:00 15:00 16:00 17:00 18:00 19:00 20:00 21:00 22:00 23:00 00:00
500
1000
1500
2000
−20
−10
0
10
20
Figure 4.3: Zonal wind estimates averaged every 12 min on June 8, 2007. Top: LES-DBS.
Middle: LES-Profile at center of domain. Bottom: DBS estimates from the SGP ACRF
radar.
To quantify the wind parameters, two cross-sectional cuts are made in the wind
fields referring to altitudes ∼530 m and ∼1030 m. Wind component estimates for
all three sources are presented in Figure 4.6 along with corresponding sounding data
at 12:00, 18:00, and 00:00 UTC. There is a general agreement between the estimates
from RadSimDBS, LES, and the sounding data for both heights and at all times. The
agreement between the estimates from the LES and RadarARM is especially good
for low elevations. The reason for this agreement is that the wind estimates for all
quantities at that altitude correspond to the CBL mixed layer region.
There are some discrepancies at higher altitudes partially due to low SNR at
particular time intervals. An additional cause might be attributed to the horizontal
69
UTC Time
He
igh
t (
m
)
Meridional Wind (ms−1) − RadSimDBS
 
 
12:00 13:00 14:00 15:00 16:00 17:00 18:00 19:00 20:00 21:00 22:00 23:00 00:00
500
1000
1500
−20
−10
0
10
20
UTC Time
He
igh
t (
m
)
Meridional Wind (ms−1) − LES
 
 
12:00 13:00 14:00 15:00 16:00 17:00 18:00 19:00 20:00 21:00 22:00 23:00 00:000
500
1000
1500
2000
−20
−10
0
10
20
UTC Time
He
igh
t (
m
)
Meridional Wind (ms−1) − RadarARM
 
 
12:00 13:00 14:00 15:00 16:00 17:00 18:00 19:00 20:00 21:00 22:00 23:00 00:00
500
1000
1500
2000
−20
−10
0
10
20
Figure 4.4: Meridional wind estimates averaged every 12 min on June 8, 2007. Top: LES-
DBS. Middle: LES-Profile at center of domain. Bottom: DBS estimates from the SGP
ACRF radar.
70
UTC Time
He
igh
t (
m
)
Vertical Wind (ms−1) − RadSimDBS
 
 
12:00 13:00 14:00 15:00 16:00 17:00 18:00 19:00 20:00 21:00 22:00 23:00 00:00
500
1000
1500
−5
0
5
UTC Time
He
igh
t (
m
)
Vertical Wind (ms−1) − LES
 
 
12:00 13:00 14:00 15:00 16:00 17:00 18:00 19:00 20:00 21:00 22:00 23:00 00:000
500
1000
1500
2000
−5
0
5
UTC Time
He
igh
t (
m
)
Vertical Wind (ms−1) − RadarARM
 
 
12:00 13:00 14:00 15:00 16:00 17:00 18:00 19:00 20:00 21:00 22:00 23:00 00:00
500
1000
1500
2000
−5
0
5
Figure 4.5: Vertical wind estimates averaged every 12 min on June 8, 2007. Top: LES-
DBS. Middle: LES-Profile at center of domain. Bottom: DBS estimates from the SGP
ACRF radar.
71
12:00 14:00 16:00 18:00 20:00 22:00 00:00
0
10
20
UTC Time
Ve
loc
ity
 (m
 s-
1 )
Zonal Wind (~530 m)
RadSimDBS
LES
RadarARM
SoundingARM
12:00 14:00 16:00 18:00 20:00 22:00 00:00
0
10
20
UTC Time
Ve
loc
ity
 (m
 s-
1 )
Meridional Wind
RadSimDBS
LES
RadarARM
SoundingARM
12:00 14:00 16:00 18:00 20:00 22:00 00:00
0
1
2
UTC Time
Ve
loc
ity
 (m
 s-
1 )
Vertical Wind
RadSimDBS
LES
RadarARM
12:00 14:00 16:00 18:00 20:00 22:00 00:00
0
10
20
UTC Time
Ve
loc
ity
 (m
 s-
1 )
Zonal Wind (~1030 m)
RadSimDBS
LES
RadarARM
SoundingARM
12:00 14:00 16:00 18:00 20:00 22:00 00:00
0
10
20
UTC Time
Ve
loc
ity
 (m
 s-
1 )
Meridional Wind
RadSimDBS
LES
RadarARM
SoundingARM
12:00 14:00 16:00 18:00 20:00 22:00 00:00
0
1
2
UTC Time
Ve
loc
ity
 (m
 s-
1 )
Vertical Wind
RadSimDBS
LES
RadarARM
-10
-10-10
-10
-20
-20-20
-20
-1-1
-2-2
-3 -3
33
Figure 4.6: Wind comparison from the RadSimDBS , LES, RadarARM , and a sounding
(launched at three different times: 12:00, 18:00, and 00:00 UTC) at ∼530 m (left) and
∼1030 m (right) on June 8, 2007. Top: zonal wind. Middle: meridional wind. Bottom:
vertical velocity.
72
shear of vertical velocity (see Section 5.1), which can bias the horizontal wind field
estimates when the CBL is very active. Another cause for the observed discrepancies
is the apparent inability of the LES code in its current version to account for the
evolution of environmental (larger-scale) forcing, whose contribution at certain stages
of the CBL evolution is comparable to the effects of simulated surface and shear
forcing. This causes the pace of the CBL growth in the LES to be more gradual
than is observed at the SGP ACRF site and reproduced by the local radar. For
instance, wind field estimates at approximately 1030 m correspond to different CBL
regions. In the RadarARM , this elevation is within the mixed layer, while for the LES
and RadSimDBS, the estimated wind refers to an elevation with two different CBL
regions. Before 15:45 UTC, the free atmosphere above the CBL top, and after 15:45
UTC, the mixed layer inside the CBL. The estimations of the different regions for the
LES and RadSimDBS are obtained based on C2n, which were presented in Section 4.2.
4.4 Vertical Velocity Variance
The values of vertical velocity w obtained by all three techniques (LES, RadSimDBS,
and RadarARM) are used to estimate the vertical variance of w. With regard to
the LES, this procedure involves estimating the fluctuating component of the vertical
velocity (w￿) from the time series of the resolved w field retrieved from the LES output
for a particular location (center of the LES domain). To account for the effect of sub-
grid turbulence, a sub-grid component of variance is added to the resolved variance
(Equation (4.2)). This component is represented (assuming isotropy of the sub-grid
turbulence) by two thirds of the time-averaged values of the sub-grid turbulence
kinetic energy.
σ2w(LES) = w
￿w￿ +
2
3
E, (4.2)
σ2w(Radar) = w
￿w￿, (4.3)
where the over-bar represents a time average. Before calculating the deviation from
the mean, a linear de-trending procedure is performed for all the vertical wind es-
timates (Angevine et al. 1994a). Estimates of the vertical velocity variances from
RadSim, LES, and RadarARM for the CBL case of June 8, 2007 applying an averaging
time of two hours for different periods of time (14:00-16:00; 16:00-18:00; 18:00-20:00;
and 20:00-22:00 UTC) are presented in Figure 4.7. The CBL top is also presented
in the same figure as a horizontal dashed line at each of the averaged periods as a
73
reference. The CBL/inversion layer top estimated from the maximum of the LES C2n
from Section 4.2 is located at 850 m, 1050 m, 1100 m, and 1250 m for each averaged
period, respectively. The LES w field, considered as a reference data source, and
RadSim vertical velocity are sampled every 6 min. The averaging and sampling times
used clearly filter the high frequency components of the variance, especially for LES.
However, their use is justified for realistic comparisons with the RWP at SGP ACRF
since the vertical velocity estimates from the RadarARM are obtained after each re-
visit time. Later, in Section 5.2.2, a more appropriate study of the vertical velocity
variance obtained from LES and RadSim will reveal that the best agreement between
the estimates are obtained when the calculations are made with high-temporal res-
olution (at each dwell time ∼30 s), which avoids the filtering of the high frequency
components of the vertical velocity.
0.5 0 0.5 1 1.5 2 2.50
200
400
600
800
1000
1200
1400
m2 s-2
He
igh
t (
m
)
14:00 - 16:00
 
 
RadSim
LES
RadarARM
0.5 0 0.5 1 1.5 2 2.50
200
400
600
800
1000
1200
1400
m2 s-2
He
igh
t (
m
)
16:00 - 18:00
 
 
RadSim
LES
RadarARM
0.5 0 0.5 1 1.5 2 2.50
200
400
600
800
1000
1200
1400
m2 s-2
He
igh
t (
m
)
18:00 - 20:00
 
 
RadSim
LES
RadarARM
0.5 0 0.5 1 1.5 2 2.50
200
400
600
800
1000
1200
1400
m2 s-2
He
igh
t (
m
)
20:00 - 22:00
 
 
RadSim
LES
RadarARM
Figure 4.7: Vertical velocity variance as a function of height for different periods of time
from left to right (14:00-16:00; 16:00-18:00; 18:00-20:00; and 20:00-22:00 UTC). Red line:
estimates from the virtual BLR. Blue line: estimates from the LES. Black line: estimates
from the BLR located at the SGP ACRF site. The horizontal dashed line represents the
CBL top obtained from the LES C2n maximum averaged over the period under study.
The decrease in the vertical velocity variance observed (especially noticeable in the
RadSim profiles) is a clear indication of the CBL/inversion layer top. As can be seen in
the plot, the drop in variance corresponds roughly to the upper boundary of the CBL,
and the height of this feature increases over time as the CBL continues to develop. For
74
the four periods, the minimum is observed at approximately 850 m, 1050 m, 1100 m,
and 1250 m for the 14:00-16:00; 16:00-18:00; 18:00-20:00; and 20:00-22:00 UTC period,
respectively. These results are in excellent agreement with the estimates obtained as
reference from C2n. The discrepancy between the RadSim and the LES estimates are
≤1.4 m2 s−2 and may be in part to the differences in spatial and temporal extent
of the vertical velocity samples and to the noise added to the simulated time-series
data. The primary cause of discrepancies in the w variance profile shapes from the
LES and RadarARM estimates is attributed to the aforementioned inconsistency of
prediction of the CBL depth evolution associated with external forcings.
4.5 Vertical Velocity Skewness
The skewness of the vertical velocity in the CBL is a signature feature of the CBL
flow structure (Moeng and Rotunno 1990):
skewness =
w￿w￿w￿
(w￿w￿)
3
2
. (4.4)
Positive w skewness throughout the main portion of the CBL is indicative of localized
and intense upward vertical motions (updrafts) as compared to relatively widespread
and gentle downward motions (downdrafts) (see Figure 2.3). The height dependence
of the vertical velocity skewness in the CBL has been previously analyzed using LES
in conjunction with observations (Moeng and Rotunno 1990) and LES in conjunction
with wind tunnel measurements (Fedorovich et al. 1996, 2001). The chosen sampling
period filters the high frequency components of the spectrum and skewness as well, but
it is the most appropriate for comparisons with real data. Here, we calculate vertical
profiles of skewness using vertical velocity data from RadSim, LES, and RadarARM
for the CBL case of June 8, 2007 applying an averaging time of two hours for different
stages of the CBL evolution (14:00-16:00; 16:00-18:00; 18:00-20:00; and 20:00-22:00
UTC). The vertical velocity fields from LES and RadSim are sampled every 6 min to
reproduce the revisit time of the RadarARM . The results are presented in Figure 4.8
for comparison. Note that the estimates of skewness from all three sources exhibit
similar behavior. That is, within the main portion of the CBL, the values are found
to be positive. This is consistent with the large-scale pattern in the vertical velocity,
which is characteristic of the mixed layer. Skewness increases in magnitude and range
of positive values of w toward the CBL top; this growth can be clearly identified at
each time period.
75
2 0 2 4
0
200
400
600
800
1000
1200
1400
Skewness
He
igh
t (
m
)
14:00 - 16:00
 
 
RadSimDBS
LES
RadarARM
2 0 2 4
0
200
400
600
800
1000
1200
1400
Skewness
He
igh
t (
m
)
16:00 - 18:00
 
 
RadSimDBS
LES
RadarARM
2 0 2 4
0
200
400
600
800
1000
1200
1400
Skewness
He
igh
t (
m
)
18:00 - 20:00
 
 
RadSimDBS
LES
RadarARM
2 0 2 4
0
200
400
600
800
1000
1200
1400
Skewness
He
igh
t (
m
)
20:00 - 22:00
 
 
RadSimDBS
LES
RadarARM
Figure 4.8: Skewness of the vertical velocity as a function of height for different periods of
time from left to right (14:00-16:00; 16:00-18:00; 18:00-20:00; and 20:00-22:00 UTC). Red
line: estimates from the virtual BLR. Blue line: estimates from the LES. Black line: esti-
mates from the BLR located at the SGP ACRF site. The horizontal dashed line represents
the CBL top obtained from the LES C2n maximum averaged over the period under study.
76
In the simulated CBL, the skewness reaches a maximum near the base of the
inversion, and then drops to zero in the interior of the inversion layer, where the
upward component of turbulent motion is in approximate equilibrium with the down-
ward component (Fedorovich et al. 2001). From 14:00 to 16:00 UTC, the increase in
skewness is observed at approximately 600 m, reaching the top of inversion layer at
nearly 750 m. In this period, the skewness estimates are rather inconsistent and, as
a consequence, the estimate of the CBL top based on the skewness profile appears to
be unreliable. For the other time periods (16:00-18:00; 18:00-20:00; and 20:00-22:00
UTC), the maximum in skewness occurs at 900 m, 950 m, and 1100 m; and the
CBL/inversion layer top estimated form vertical velocity variance and the maximum
of C2n are located at 1050 m, 1100 m, and 1250 m, respectively. Coincidently, the dif-
ference between the estimates from the vertical velocity variance, and the maximum
in skewness is 150 m. Radar estimates of skewness agree well with the LES estimates
and are similar to those presented in Fedorovich et al. (1996, 2001). While analyzing
the obtained skewness profile, one should keep in mind that skewness is extremetely
sensitive to the sample size, which in the reported retrieval procedures was rather
small due to the necessity of bringing data from all sources to one sampling time
window that corresponds to the radar revisit time.
In this chapter, estimates of C2n, three-dimensional winds, vertical velocity vari-
ance, and vertical velocity skewness have been compared between LES and RadSimDBS.
These estimates have also been contrasted with “real” radar data obtained from the
RWP located in SGP ACRF. Even when the average time for comparison is set too
high (∼12 min for wind comparisons and ∼6 min for vertical velocity variance and
vertical velocity skewness), which corresponds to the RWP revisit time, it is accept-
able for realistic comparisons. The obtained results are satisfactory in all studied
cases. This confirms the ability of the LES to reproduce the real atmosphere.
In the next chapter, the horizontal shear of vertical velocity effect will be investi-
gated in depth using theoretical formulations and comparisons using LES fields and
virtual radar. Also, turbulence kinetic energy will be obtained from the DBS wind
estimates and analyzed for different averaging periods. Finally, the turbulence (eddy)
dissipation rate will be obtained from the Doppler spectrum width for the five DBS
beams and compared with the estimates obtained from LES.
77
Chapter 5
Comparison between LES and Virtual Boundary
Layer Radar
Virtual radar estimates of structure function parameter of refractive index, three-
dimensional winds, vertical velocity variance, and vertical velocity skewness have
been compared with their corresponding estimates obtained directly from LES, in
Chapter 4. These estimates have been contrasted with “real” data from the wind
profiler located in LMN, OK for the date of the simulation with good agreement.
This confirms the ability of the LES code to accurately reproduce the CBL. Here, the
effect of horizontal shear of vertical velocity on horizontal wind estimations will be
studied, along with turbulence kinetic energy and turbulence-eddy dissipation rate
for different conditions.
5.1 Effect of Horizontal Shear of Vertical Velocity
on Horizontal Wind Estimates
5.1.1 Theoretical Formulation
5.1.1.1 Doppler Beam Swinging
In order to estimate winds, DBS technique assumes that the three-dimensional wind
components are constant across the region defined by the beam locations (see Sec-
tion 2.4.1). However, if the vertical velocity assumption is not considered constant
78
within the DBS sampling volume and the horizontal components of the vertical ve-
locity shear are introduced, then the equations for the radial velocity for any oblique
beam can be written as:
vr(θ,φ) = uo sinφ sin θ + vo sinφ cos θ + w cosφ, (5.1)
w = wo + suzo tanφ sin θ + svzo tanφ cos θ, (5.2)
su =
wu − wo
zo tanφ
, θ = 90o, (5.3)
sv =
wv − wo
zo tanφ
, θ = 0o, (5.4)
where zo is the vertical range where the radial velocity is calculated; w is the true
vertical velocity at the beam position; wu and wv represent the true vertical velocity at
the zonal and meridional directions, respectively ; and finally, su and sv represent the
horizontal shear components of the vertical velocity caused by the horizontal gradient
in the true vertical velocity (w) and the assumed constant mean vertical wind (wo)
within the DBS sampling volume.
Regrouping terms leads to an expression similar to the one presented in Equa-
tion (2.55) for radial velocity:
vr(θ,φ) = (uo + suzo) sinφ sin θ + (vo + svzo) sinφ cos θ + wo cosφ,
vr(θ,φ) = ￿u sinφ sin θ + ￿v sinφ cos θ + wo cosφ, (5.5)
where ￿u = uo + suzo and ￿v = vo + svzo are the DBS-measured zonal and meridional
wind biased by the horizontal shear of the vertical wind in the x and y direction,
respectively. The solution to the set of equations of radial velocities for multiple
beams shows that the horizontal shear effect caused by the vertical velocity cannot
be separated from the horizontal wind estimates.
5.1.1.2 Spaced Antenna
The generalized theoretical expression that includes the shear terms for the CCF used
in spaced antenna interferometry is presented in Zhang and Doviak (2007). However,
the equations have been derived for a horizontal pointing weather radar. This theory
needs to be adapted for a vertically pointing wind profiler in the same fashion that
79
is presented in Doviak et al. (1996), and Holloway et al. (1997), and Equation (2.61).
The modified equation is presented below:
c12(τ) = exp
￿
−2jkowoτ − 2k2o(σ2rs2w − σ2t )τ 2 − β2h
￿
(uo + suzo)τ − ∆x
2
￿2
−β2h
￿
(vo + svzo)τ − ∆y
2
￿2
− 1
8
￿
woτ
σr
￿2￿
, (5.6)
where su, sv, and sw are the three components of the vertical velocity shear. The
effects of the vertical wind wo and vertical shear sw are typically small and will be
omitted in the future (Holloway et al. 1997; Zhang and Doviak 2007).
By re-writting the magnitude of Equation (5.6) in a more convenient form, the
expression for the normalized ACF and CCF are obtained. The expressions are similar
to Equations (2.62) and (2.63):
|c11(τ)| = exp
￿−β2h(￿u2 + ￿v2)− 2k2oσ2t τ 2￿+ NS δ(t), (5.7)
|c12(τ)| = exp
￿
−β2h
￿￿uτ − ∆x
2
￿2
− β2hv˜2τ 2 − 2k2oσ2t τ 2
￿
, (5.8)
where ￿u = uo+suzo and ￿v = vo+svzo are the SA measured wind components in the x
and y direction, respectively. They represent the baseline wind biased by the baseline
shear (Zhang and Doviak 2007). These equations are the same as the ones presented
in Holloway et al. (1997) and Section 2.4.2 and are used to estimate the measured
wind and turbulence. Once again, it is clear that it is impossible to separate the
horizontal shear component of the vertical velocity from the measured wind. If the
horizontal shear term is significantly large, the estimates of the horizontal wind will
be erroneous.
To calculate the magnitude of the effect of the shear in both simulations of DBS
and SA techniques, a constant amount of shear of vertical velocity will be applied
to the zonal wind. In this way, the amount of bias will be measured and compared
between the simulated and the theoretical described previously.
5.1.2 Test Configurations
A controlled experiment is setup in order to quantify the effects of the vertical ve-
locity shear in the horizontal wind estimates. Of all the LES fields used in the radar
simulator presented in Section 3.1, the only quantities that are variables here are the
80
Table 5.1: Description of the field domain used by the radar simulators
Property Specification
Number of grid points 70×70×100
Spatial resolution 20 m
Time step 1 s
Dimensions of the sub-domain 1400×1400×2000 m3
Variables u, v, w
Constants E, q,Θ
wind components. The amplitude of the signal is assumed to be constant and to have
high SNR. The parameters used in the radar simulators are presented in Table 5.1.
The meridional wind is set to be constant and equal to 5 m s−1. The zonal
wind is variable and changes from -15 m s−1 to 15 m s−1 in steps of 0.125 m s−1.
The vertical velocity fields are generated (using Equation (5.2)) with shear only in
the zonal direction (sv = 0). The magnitude of the vertical velocity is controlled
using an external parameter called “vertical velocity fraction” (sfrac) and denotes
the fractional contribution of the shear in the range of 0 to 1. That is, when sfrac is
0, the vertical velocity is set to zero; when it is one, the vertical velocity is set to the
maximun shear.
5.1.2.1 DBS Configuration
The main parameters used in the DBS simulation are presented in Table 5.2. The
experiment setup consists of generating 5 beams: one vertical and four steered at
15.5◦ off-vertical along 4 azimuth angles: 0◦, 90◦, 180◦, and 270◦. Radial velocities
are estimated using the spectral moments, which are later used to calculate the wind
components.
Presented in Figure 5.1 are the zonal wind estimates calculated using different
values of sfrac versus the ideal zonal wind. The x-axis in the scatter plot represents the
“ideal” zonal winds used by the radar simulator. The y-axis represents the measured
zonal wind (￿u) for the different shear conditions. The estimates have a correlation
of 0.99 for the three presented cases of Figure 5.1. However, as the shear increases
(controlled by sfrac), bias increases. This increase in bias is observed as a displacement
of the wind estimates toward larger values (see right panel of Figure 5.1).
81
Table 5.2: Specifications of DBS configuration
Quantity Value
Frequency 915 MHz
Wavelength 32.8 m
Full half-power beam width 9◦
Inter-Pulse Period 10 ms
Resolution 60 m
Range (zo) 1650 m
Beam inclination variable
Simulation Time 30000 s
ï20 ï10 0 10 20
ï20
ï10
0
10
20
Zonal Wind (msï1)
Zo
na
l D
BS
 W
ind
 (m
sï
1 )
6h = 60 m ï sfrac = 0.00
ï20 ï10 0 10 20
ï20
ï10
0
10
20
Zonal Wind (msï1)
Zo
na
l D
BS
 W
ind
 (m
sï
1 )
6h = 60 m ï sfrac = 0.50
ï20 ï10 0 10 20
ï20
ï10
0
10
20
Zonal Wind (msï1)
Zo
na
l D
BS
 W
ind
 (m
sï
1 )
6h = 60 m ï sfrac = 1.00
Figure 5.1: Zonal wind estimates obtained using DBS for different shear conditions. Left:
sfrac = 0.00 (no shear). Middle: sfrac = 0.50. Right: sfrac = 1.00.
82
In this unique situation, w at each DBS beam direction is recorded directly from
the input vertical wind field. The ideal bias for DBS is also referred to as “range
corrected shear (suzo)”, and it is plotted as a continuous line in Figure 5.2. The
cumulative bias calculated from the DBS zonal wind estimates plotted in Figure 5.1
for the different sfrac conditions are plotted as dots.
0 0.2 0.4 0.6 0.8 1ï0.5
0
0.5
1
1.5
2
sfrac
Bi
as
 (m
 sï
1 )
Bias (Zonal Wind ï DBS)
 
 
Ideal
Simulated
Figure 5.2: Zonal wind bias. Line: Ideal bias. Dots: estimated from the DBS simulation
for different values of sfrac.
The correlation between the ideal bias calculated from w using Equation (5.3)
and the cumulative error obtained from the data presented in Figure 5.1 for the
different values of sfrac is extremely high. The simulated values are well represented
by the theory. When sfrac is zero, the bias is zero, and when sfrac is one, the bias is
maximum. The mean square error (MSE) between the measured and theoretical bias
from Figure 5.2 is 1.55×10−4 m2 s−2.
5.1.2.2 Spaced Antenna Configuration
Four closely spaced receivers (see Figure 5.3) are used to estimate the zonal and
meridional wind using the SA technique. The transmitter is located at the center
of the sub-domain. Furthermore, the receivers are located in a cross-configuration:
receivers RxA and RxB are located in the zonal direction at -0.25 m and 0.25 m,
respectively. Receivers RxC and RxD are located in the meridional direction at -
0.25 m and 0.25 m, respectively. This configuration is designed to allow for direct
determination of the horizontal wind components along each baseline. The parameters
used in the SA simulation are described in Table 5.3.
83
NERxA(-.25,0) RxB(.25,0)
RxC(0,-.25)
RxD(0,.25)
Tx
Figure 5.3: Diagram showing the location of the SA transmitter (Tx) and receivers (RxA,
RxB, RxC, and RxD). The black dots represent the location of the four receivers in the
cross-configuration. As noted, they are located symmetrically, with respect to the Tx, along
the zonal and meridional axes.
Table 5.3: Specifications of SA configuration
Quantity Value
Frequency 915 MHz
Wavelength 32.8 m
Full half-power transmitter beam width 9◦
Full half-power receiver beam width 18◦
Number of receivers 4
Inter-Pulse Period 2 ms
Resolution 60 m
84
Zonal wind estimates for the different sfrac cases versus the ideal zonal wind
are presented in Figure 5.4. As in Figure 5.1, the y-axis corresponds to the zonal
wind estimates but the estimates are obtained using the FCA-Gaussian for the SA
technique (see Section 2.4.2). The correlation for the SA case for the different sfrac
values is 0.99. However, the estimates are not aligned as well as in the case of DBS.
The effect of the shear in the estimates can still be seen (i.e., the estimates shift up
in value as sfrac increases).
ï20 ï10 0 10 20
ï20
ï10
0
10
20
Zonal Wind (msï1)
Zo
na
l S
A 
(F
CA
ïG
) W
ind
 (m
sï
1 ) 6h = 60 m ï sfrac = 0.00
ï20 ï10 0 10 20
ï20
ï10
0
10
20
Zonal Wind (msï1)
Zo
na
l S
A 
(F
CA
ïG
) W
ind
 (m
sï
1 ) 6h = 60 m ï sfrac = 0.50
ï20 ï10 0 10 20
ï20
ï10
0
10
20
Zonal Wind (msï1)
Zo
na
l S
A 
(F
CA
ïG
) W
ind
 (m
sï
1 ) 6h = 60 m ï sfrac = 1.00
Figure 5.4: Zonal wind estimates obtained using SA for different shear conditions. Left:
sfrac = 0.00 (no shear). Middle: sfrac = 0.50. Right: sfrac = 1.00.
The shear applied to both SA and DBS techniques are the same. However, in
contrast to DBS, the ideal bias for the SA is calculated within the resolution volume
as the averages of the vertical velocities gradient along the x and y directions. The
ideal bias for the SA case is the same as the DBS bias, and it is plotted as a continuous
line in Figure 5.5. The cumulative measured bias for the different sfrac values obtained
from Figure 5.4 are presented as black dots. The MSE for the SA (FCA-G) obtained
from Figure 5.5 is 1.83×10−2 m2 s−2, and is larger than the one calculated from DBS.
The good agreement between the ideal bias and the cumulative bias obtained
from the zonal wind estimates for DBS and SA techniques supports the theoretical
formulation of the horizontal shear of vertical velocity effect in the horizontal wind
estimates.
5.1.3 Results
After the theory has been validated through a controlled experiment in the previous
sections, it is pertinent to study the effect of the horizontal shear of vertical velocity
85
0 0.2 0.4 0.6 0.8 1ï0.5
0
0.5
1
1.5
2
sfrac
Bi
as
 (m
 sï
1 )
Bias (Zonal Wind ï SA (FCAïG))
 
 
Ideal
Simulated
Figure 5.5: Zonal wind bias. Line: Ideal bias. Dots: estimated from the SA simulation for
different values of sfrac.
under realistic conditions. For this study, radar signals for both techniques (DBS
and SA) are simulated for approximately 8 hours of the LES run and within the
LES sub-domain. The simulated radar signals follow the same parameters presented
in Table 5.2 for DBS and Table 5.3 for SA. To produce realistic signals, AWGN has
been added to the radar signals obtained at each receiver (see top panel of Figure 5.6).
As a consequence, wind estimates at certain sectors are not reliable due to low SNR.
The SNR threshold used for censoring the radar signals is set to -3 dB. However,
due to initialization parameters of the LES, the returned signal from a residual layer
found at the beginning of the simulation is not removed. Usually, turbulence above
the BL top is weak, so the wind estimates are not reliable and have been removed.
The procedure used consists of estimating the maximum of the C2n, which matches
the maximum of the SNR (see asterisks at the top panel of Figure 5.6) corresponding
to an estimate of the BL top (see Section 4.2). Any estimates above the BL top are
censored (see bottom panel of Figure 5.6).
Estimates of horizontal wind (￿u and ￿v) are analyzed for three different quantities:
resolved LES values along a vertical profile at the center of the simulation sub-domain,
DBS estimates calculated from the radial velocities of the virtual BLR pointing in
the five directions, and SA estimates calculated from the four spaced receivers. The
averaging time chosen for all the estimates is 5 min.
To obtain the DBS values, the following procedure is used. First, radial velocities
from each of the five virtual BLR beams are estimated with a dwell time of 30 s.
Second, velocity samples are computed every 2.5 min representing the DBS scan time.
86
He
igh
t (
m
)
SignalïtoïNoise Ratio
 
 
12:00 13:00 14:00 15:00 16:00 17:00 18:00 19:00 20:000
500
1000
1500
ï40
ï30
ï20
ï10
0
10
20
30
UTC Time
He
igh
t (
m
)
SignalïtoïNoise Ratio ï SNRth = ï3 dB
 
 
12:00 13:00 14:00 15:00 16:00 17:00 18:00 19:00 20:000
500
1000
1500
ï40
ï30
ï20
ï10
0
10
20
30
Figure 5.6: Top: Signal-to-Noise Ratio as a function of time. The asterisks represent
estimates of the BL top. Bottom: censored SNR plot. Values less than -3 dB and above
the BL top have been removed.
87
Third, the DBS technique is used to retrieve the three wind components. Finally, the
wind components are averaged over 5 min. The LES data are averaged in time to
obtain the 5-min (revisit time) estimates used for comparison. Estimates from SA are
calculated every 150 s (DBS scan time) and are averaged over the same 5-min period.
Zonal and meridional wind estimates plots for sfrac= 1 are presented in Figures 5.7
and 5.8, respectively. In the top panel the “true” wind obtained from the LES is
presented. Estimates obtained from the DBS technique are presented in the middle
panel. Finally, the bottom panel displays the estimates obtained from SA (FCA-G).
The first height is removed for DBS due to corrupted wind estimates; meanwhile,
for SA, the three first heights are removed due to the lack of points in the common
volume used by the technique.
A comparison between the horizontal wind estimates (zonal and meridional) ob-
tained from DBS are presented versus their corresponding “ideal” values obtained
from the LES in Figure 5.9 for different sfrac values. The measured wind estimates
exhibit good agreement with their corresponding values from LES. However, as the
sfrac increases, the bias of the estimates increases. The effect of this increase in bias
is observed as a spread in the wind estimates for both horizontal components. The
maximum spread is observed when sfrac = 1, and its minimum when sfrac = 0. As
previously studied, the main cause of this bias error is the horizontal shear of vertical
velocity. In realistic measurements, it is impossible to separate this effect from the
real wind estimates and/or to evaluate this quantity.
The range corrected shear, or ideal bias, is calculated from the LES fields in the
same fashion as described in Section 5.1.1.1. Presented in the middle-top and bottom
panels of Figure 5.10 are the ideal biases for the case when sfrac = 1 for the zonal and
meridional components, respectively. The calculations have been made at each height
and along the whole simulation time. The ideal bias has low values at the beginning
of the simulation. The top and middle-bottom panels represent the measured bias
from the wind estimates obtained from the radar measurements. The plot shows a
good correlation between the bias error in the zonal and meridional estimates, and
the bias caused by the horizontal shear of vertical velocity calculated from the LES
wind fields. However, there is an overestimation of the wind in certain areas. This
overestimation might be caused by inhomogeneities within the volume that encloses
the five DBS beams (Cheong et al. 2008).
For better comparison, the ideal bias error is compared in a scatter plot with
the range corrected shear for each horizontal wind component in Figure 5.11. The
88
He
igh
t (
m
)
Zonal Wind (msï1) ï LES ï Avg. time = 300 s
 
 
12:00 13:00 14:00 15:00 16:00 17:00 18:00 19:00 20:000
500
1000
1500
ï20
ï10
0
10
20
He
igh
t (
m
)
Zonal Wind (msï1) ï DBS ï Dwell time = 30 s ï Avg. time = 300 s ï SNRth = ï3 dB
 
 
12:00 13:00 14:00 15:00 16:00 17:00 18:00 19:00 20:000
500
1000
1500
ï20
ï10
0
10
20
He
igh
t (
m
)
UTC Time
Zonal Wind (msï1) ï SA(FCAïG) ï Dwell time = 150 s ï Avg. time = 300 s ï SNRth = ï3 dB
 
 
12:00 13:00 14:00 15:00 16:00 17:00 18:00 19:00 20:000
500
1000
1500
ï20
ï10
0
10
20
Figure 5.7: Zonal wind estimates as a function of time for sfrac = 1.00. Top: LES estimates.
Middle: DBS estimates. Bottom: SA estimates obtained from the FCA-Gaussian method.
The first height is removed for DBS, and the first three heights for SA.
89
He
igh
t (
m
)
Meridional Wind (msï1) ï LES ï Avg. time = 300 s
 
 
12:00 13:00 14:00 15:00 16:00 17:00 18:00 19:00 20:000
500
1000
1500
ï20
ï10
0
10
20
He
igh
t (
m
)
Meridional Wind (msï1) ï DBS ï Dwell time = 30 s ï Avg. time = 300 s ï SNRth = ï3 dB
 
 
12:00 13:00 14:00 15:00 16:00 17:00 18:00 19:00 20:000
500
1000
1500
ï20
ï10
0
10
20
He
igh
t (
m
)
UTC Time
Meridional Wind (msï1) ï SA(FCAïG) ï Dwell time = 150 s ï Avg. time = 300 s ï SNRth = ï3 dB
 
 
12:00 13:00 14:00 15:00 16:00 17:00 18:00 19:00 20:000
500
1000
1500
ï20
ï10
0
10
20
Figure 5.8: Same as Figure 5.7, except for the meriodional wind.
90
ï20 ï10 0 10 20ï25
ï20
ï15
ï10
ï5
0
5
10
15
20
25
Zonal LES Wind (m sï1)
Zo
na
l D
BS
 W
ind
 (m
 sï
1 )
6h = 60 m ï sfrac = 0.0
ï20 ï10 0 10 20
ï20
ï10
0
10
20
Meridional LES Wind (m sï1)
M
er
idi
on
al 
DB
S 
W
ind
 (m
 sï
1 )
6h = 60 m ï sfrac = 0.0
ï20 ï10 0 10 20ï25
ï20
ï15
ï10
ï5
0
5
10
15
20
25
Zonal LES Wind (m sï1)
Zo
na
l D
BS
 W
ind
 (m
 sï
1 )
6h = 60 m ï sfrac = 0.5
ï20 ï10 0 10 20
ï20
ï10
0
10
20
Meridional LES Wind (m sï1)
M
er
idi
on
al 
DB
S 
W
ind
 (m
 sï
1 )
6h = 60 m ï sfrac = 0.5
ï20 ï10 0 10 20ï25
ï20
ï15
ï10
ï5
0
5
10
15
20
25
Zonal LES Wind (m sï1)
Zo
na
l D
BS
 W
ind
 (m
 sï
1 )
6h = 60 m ï sfrac = 1.0
ï20 ï10 0 10 20
ï20
ï10
0
10
20
Meridional LES Wind (m sï1)
M
er
idi
on
al 
DB
S 
W
ind
 (m
 sï
1 )
6h = 60 m ï sfrac = 1.0
Figure 5.9: Comparison between the DBS estimates versus LES fields. Top: zonal wind
for different sfrac values. Bottom: meridional wind for the same sfrac values. Left: sfrac =
0.00. Middle: sfrac = 0.50. Right : sfrac = 1.00.
91
He
igh
t (
m
)
Zonal Wind Error (uDBS ï uLES) (ms
ï1) ï SNRth = ï3 dB
 
 
12:00 13:00 14:00 15:00 16:00 17:00 18:00 19:00 20:00
500
1000
1500
ï5
0
5
He
igh
t (
m
)
Range Corrected Zonal Shear (su z0) ï LES (ms
ï1)
 
 
12:00 13:00 14:00 15:00 16:00 17:00 18:00 19:00 20:00
500
1000
1500
ï5
0
5
He
igh
t (
m
)
Meridional Wind Error (vDBS ï vLES) (ms
ï1) ï SNRth = ï3 dB
 
 
12:00 13:00 14:00 15:00 16:00 17:00 18:00 19:00 20:00
500
1000
1500
ï5
0
5
He
igh
t (
m
)
UTC Time
Range Corrected Meridional Shear (sv z0) ï LES (ms
ï1)
 
 
12:00 13:00 14:00 15:00 16:00 17:00 18:00 19:00 20:00
500
1000
1500
ï5
0
5
Figure 5.10: Error analysis as a function of time for sfrac = 1. Top: zonal wind error
from DBS estimates. Middle-top: range corrected zonal shear from LES. Middle-bottom:
meridional wind error from DBS estimates. Bottom: range corrected meridional shear from
LES.
92
region after the censoring corresponds to the CBL, where strong updrafts and mild
downdrafts are present. Consequently, the zone is more susceptible to the horizontal
shear of vertical velocity effect. The correlation for zonal wind is 0.36. Meanwhile,
for the meridional wind is 0.66, which indicates the presence of other factors that can
cause an error in the wind estimates that cannot be attributed to the vertical velocity
shear, such as the horizontal wind shear especially observed in the LES zonal wind.
ï5 0 5
ï6
ï4
ï2
0
2
4
6
Zonal Wind Error (m sï1)
Id
ea
l B
ias
 ï
 Z
on
al 
W
ind
DBS ï sfrac = 1.0
ï5 0 5
ï6
ï4
ï2
0
2
4
6
Meridional Wind Error (m sï1)
Id
ea
l B
ias
 ï
 M
er
idi
on
al 
W
ind
DBS ï sfrac = 1.0
Figure 5.11: Error from the DBS estimates vs. range corrected shear from LES for sfrac
= 1.00. Left: zonal wind. Right: meridional wind.
As in the theoretical analysis, the cumulative measured bias as a function of sfrac
is presented in Figure 5.12. Notice that when sfrac = 0, the measured bias is different
than zero (theoretical value). This is another indication that there are other factors
that contribute to the error in the wind estimates. However, the theoretical tendency
of the bias increases as the sfrac does, and it is observed in both wind components.
As in the case of DBS, the bias error is analyzed for the SA technique. The
comparison between the zonal and meridional wind estimates from SA and LES are
presented in Figure 5.13. Again, it can be observed that as the sfrac increases, the
bias in the measured wind estimates also increases. Here, the estimates are more
variable than for DBS, especially in the zonal wind.
The theoretical shear is presented in Figure 5.14 for the zonal (middle-top) and
meridional (bottom) wind components. Again, there is a correlation between the bias
error and the horizontal shear of vertical velocity. Nevertheless, there are discrepan-
cies that need further analysis.
93
0 0.5 1
0.03
0.035
0.04
0.045
0.05
0.055
0.06
sfrac
Bi
as
 (m
 sï
1 )
Zonal Wind ï DBS
0 0.5 1
0.2
0.22
0.24
0.26
0.28
0.3
0.32
sfrac
Bi
as
 (m
 sï
1 )
Meridional Wind ï DBS
Figure 5.12: Wind bias for DBS. Left: zonal wind. Right : meridional wind.
ï20 ï10 0 10 20ï25
ï20
ï15
ï10
ï5
0
5
10
15
20
25
Zonal LES Wind (m sï1)
Zo
na
l S
A(
FC
Aï
G)
 W
ind
 (m
 sï
1 )
6h = 60 m ï sfrac = 0.0
ï20 ï10 0 10 20
ï20
ï10
0
10
20
Meridional LES Wind (m sï1)
M
er
idi
on
al 
SA
(F
CA
ïG
) W
ind
 (m
 sï
1 ) 6h = 60 m ï sfrac = 0.0
ï20 ï10 0 10 20ï25
ï20
ï15
ï10
ï5
0
5
10
15
20
25
Zonal LES Wind (m sï1)
Zo
na
l S
A(
FC
Aï
G)
 W
ind
 (m
 sï
1 )
6h = 60 m ï sfrac = 0.5
ï20 ï10 0 10 20
ï20
ï10
0
10
20
Meridional LES Wind (m sï1)
M
er
idi
on
al 
SA
(F
CA
ïG
) W
ind
 (m
 sï
1 ) 6h = 60 m ï sfrac = 0.5
ï20 ï10 0 10 20ï25
ï20
ï15
ï10
ï5
0
5
10
15
20
25
Zonal LES Wind (m sï1)
Zo
na
l S
A(
FC
Aï
G)
 W
ind
 (m
 sï
1 )
6h = 60 m ï sfrac = 1.0
ï20 ï10 0 10 20
ï20
ï10
0
10
20
Meridional LES Wind (m sï1)
M
er
idi
on
al 
SA
(F
CA
ïG
) W
ind
 (m
 sï
1 ) 6h = 60 m ï sfrac = 1.0
Figure 5.13: Comparison between the SA estimates vs. LES fields. Top: zonal wind for
different sfrac values. Bottom: meridional wind for the same sfrac values. Left: sfrac =
0.00. Middle: sfrac = 0.50. Right : sfrac = 1.00.
94
He
igh
t (
m
)
Zonal Wind Error (uSA(FCAïG) ï uLES) (ms
ï1) ï SNRth = ï3 dB
 
 
12:00 13:00 14:00 15:00 16:00 17:00 18:00 19:00 20:00
500
1000
1500
ï5
0
5
He
igh
t (
m
)
Range Corrected Zonal Shear (su z0) ï LES (ms
ï1)
 
 
12:00 13:00 14:00 15:00 16:00 17:00 18:00 19:00 20:00
500
1000
1500
ï5
0
5
He
igh
t (
m
)
Meridional Wind Error (vSA(FCAïG) ï vLES) (ms
ï1) ï SNRth = ï3 dB
 
 
12:00 13:00 14:00 15:00 16:00 17:00 18:00 19:00 20:00
500
1000
1500
ï5
0
5
He
igh
t (
m
)
UTC Time
Range Corrected Meridional Shear (sv z0) ï LES (ms
ï1)
 
 
12:00 13:00 14:00 15:00 16:00 17:00 18:00 19:00 20:00
500
1000
1500
ï5
0
5
Figure 5.14: Error analysis as a function of time for sfrac = 1. Top: zonal wind error
from SA estimates. Middle-top: range corrected zonal shear from LES. Middle-bottom:
meridional wind error from DBS estimates. Bottom: range corrected meridional shear from
LES.
95
The ideal bias error and the measured range corrected shear are presented as a
scatter plot for sfrac = 1 in Figure 5.15. There is good correlation between both
quantities. However, in contrast to what was presented for DBS (see Figure 5.11),
the correlation for both horizontal wind components has improved. That is, the
correlation for zonal and meridional winds are 0.66 and 0.79, respectively. Probably,
because the scan volume for SA is smaller than for DBS.
ï5 0 5
ï6
ï4
ï2
0
2
4
6
Zonal Wind Error (m sï1)
Id
ea
l B
ias
 ï
 Z
on
al 
W
ind
SA(FCAïG) ï sfrac = 1.0
ï5 0 5
ï6
ï4
ï2
0
2
4
6
Meridional Wind Error (m sï1)
Id
ea
l B
ias
 ï
 M
er
idi
on
al 
W
ind
SA(FCAïG) ï sfrac = 1.0
Figure 5.15: Error from the SA estimates vs. range corrected shear from LES for sfrac =
1.00. Left: zonal wind. Right: meridional wind.
Finally, the cumulative measured bias error as function of sfrac is presented in
Figure 5.16. As observed for DBS (see Figure 5.12), there is a bias different than
zero for sfrac = 0, which indicates the presence of other factors contributing to the
measured bias. Also, their increase as a function of sfrac is also observed. However,
they are not as well aligned as in the case of DBS.
In realistic radar setups, the horizontal shear caused by vertical velocity cannot be
separated from the horizontal wind estimates, and depending on the level of activity
of the CBL, it can considerably bias their estimates (Flowers et al. 1994; Zhang and
Doviak 2007). From the results shown for both techniques when sfrac = 0, it can
inferred that this is not the only cause that generated bias in the wind estimates.
Other might include estimation errors, and/or horizontal wind shear, etc.
96
0 0.5 1
ï0.65
ï0.6
ï0.55
ï0.5
ï0.45
ï0.4
sfrac
Bi
as
 (m
 sï
1 )
Zonal Wind ï SA(FCAïG)
0 0.5 1
ï0.1
ï0.05
0
0.05
0.1
sfrac
Bi
as
 (m
 sï
1 )
Meridional Wind ï SA(FCAïG)
Figure 5.16: Wind bias for SA. Left: zonal wind. Right: meridional wind.
5.2 Turbulence Kinetic Energy Based on Radar
Wind Estimates
5.2.1 Theoretical Formulation
Turbulence kinetic energy (TKE) estimates have been calculated from both LES and
virtual BLR data using a variety of averaging approaches. The reference TKE values
have been obtained from LES data by horizontal plane averages over the entire sub-
domain as:
σ2u(LES) = u
￿u￿ +
2
3
E, (5.9)
σ2v(LES) = v
￿v￿ +
2
3
E, (5.10)
σ2w(LES) = w
￿w￿ +
2
3
E (5.11)
TKELES =
1
2
￿
σ2u + σ
2
v + σ
2
w
￿
, (5.12)
where u￿, v￿, and w￿, respectively, are the zonal-, meridional-, and vertical- velocity
fluctuations against the corresponding plane means, E is sub-grid TKE, and the
over-bars represent the horizontal plane averaging. To enable comparisons with the
BLR statistics, the TKE estimates from LES are additionally averaged in time over
a one-hour period.
Another TKE estimate is obtained from the LES velocity fields first averaged in
space over a cross-section that encloses five DBS beams for each LES time step. The
97
resulting time series of velocity is then used to estimate TKE by temporal averag-
ing over a period of one hour. The employed expressions for statistics in this case
are analogous to Equations (5.9) to (5.12), where the over-bars would now specify
temporal averaging and primes which denote a deviation from temporal means. The
optimal averaging period is investigated empirically.
Estimates of TKE obtained from the virtual BLR in the DBS mode are calculated
from velocity fluctuations as functions of time within the DBS volume sampled over
the same period of time (one hour):
σ2u(Radar) = u
￿u￿, (5.13)
σ2v(Radar) = v
￿v￿, (5.14)
σ2w(Radar) = w
￿w￿, (5.15)
TKERadar =
1
2
￿
σ2u + σ
2
v + σ
2
w
￿
, (5.16)
where the over-bars represent temporal averaging in this case.
5.2.2 Results
Estimates of TKE from the virtual BLR with two DBS settings are considered. In the
ideal setting (DBS ideal), the velocity readings from individual beams are assumed to
be available at the same time. In the realistic setting (DBS real), data from individual
beams are available sequentially every 30 s so that the wind estimates are calculated
after the scan time period (150 s) is completed.
All considered methods of the TKE evaluation have been analyzed accordingly.
The horizontal wind variances from LES and BLR which are obtained from different
averaging techniques are presented in Figure 5.17. The estimates over the DBS do-
main represent a smaller velocity field sample as compared to the hour-long sample of
the velocity field from the LES within the whole BLR sub-domain. Only the statistics
obtained by plane averaging show that they are variable in time. The spread of plane
statistics over one-hour time period is demonstrated by gray lines in Figure 5.17.
Both the estimates from the ideal and realistic DBS settings overestimate the hori-
zontal velocity variance. DBS estimates from the ideal configuration are consistently
smaller than the estimates from realistic settings.
After careful analysis, the overestimation of the TKE for horizontal components
by the DBS technique are attributed to the horizontal shear of vertical velocity effect
(see Section 5.1), which generates a bias in the horizontal wind estimates. The shear
98
effect in these estimates can be easily removed by setting the vertical velocity to zero
(sfrac = 0). Once it has been removed, the BLR estimates of TKE are in better
agreement with the LES horizontal velocity variance data.
A longer averaging time for the DBS signal (1200 s instead of 150 s) reduces the
effect of the vertical velocity variability on the retrieved horizontal wind variances.
The corresponding variance estimates are presented in Figure 5.18. However, the
number of points used in the calculations of the horizontal variance has been reduced
from 24 (for 150 s) to 3 (for 1200 s). This reduction in the number of points, although
they provide better agreement with the LES estimates, is not appropriate for variance
calculations.
Estimates of the vertical velocity variances are presented in the left panel of Fig-
ure 5.19 for the same cases that have been analyzed with respect to the horizontal
velocity variances. The discrepancies among the estimates are due to the differences
in the spatial and temporal dimensions of the vertical velocity samples. Same effect
was observed in Section 4.4, where the sampling time of the vertical velocities was set
to ∼6 min to replicate the radar revisit time. The velocity variances calculated from
the LES across the DBS volume scan over a one-hour time period agree reasonably
well with estimates from the DBS in ideal and realistic settings.
Finally, the estimated values of all three flow components (zonal, meridional, and
vertical) variances are brought together to obtain TKE estimates. The horizontal
velocity variances from the DBS are computed with 1200 s averaging applied. The
vertical component variances are evaluated with the maximum sampling time. The
results are presented in right panel of Figure 5.19. Remarkably, the DBS estimates
from the realistic setting are rather close to the LES estimates calculated over the
DBS volume with one-hour averaging.
5.3 Turbulence-Eddy Dissipation Rate Based on
Radar Spectrum Width
5.3.1 Theoretical Formulation
The TKE (eddy) dissipation rate ￿ is commonly estimated from the Doppler spec-
trum width σv of the radar signal when the outer scale in the inertial subrange is at
least three times greater than the scale of the radar’s resolution volume (Istok and
Doviak 1986; Doviak and Zrnic´ 1993). As indicated in White (1997); Jacoby-Koaly
99
Figure 5.17: Horizontal velocity variances. Gray lines: time spread of plane statistic
estimates from the LES over a one-hour period. Black: statistics obtained from LES data
over the entire BLR domain by plane averaging and additional time averaging over one hour.
Red: velocity variances calculated by time averaging from LES data over the DBS volume
scan. Blue: velocity variance estimated from an ideal DBS setting (all beams available at
dwell time of 30 s). Green: velocity variance estimated from a realistic DBS setting (all
beams available after the scan time of 150 s). Statistics in the two upper plots are evaluated
with original vertical velocities (sfrac = 1). Statistics in the two lower plots are obtained
with zero vertical velocity (sfrac = 0).
100
Figure 5.18: Horizontal velocity variances. For notation see Figure 5.17. The averaging
time for the DBS estimates is 1200 s instead of 150 s.
Figure 5.19: Left: vertical velocity variance. Right: turbulence kinetic energy. For notation
see Figure 5.17.
101
et al. (2002); Scipio´n et al. (2009), the spectral broadening is caused by spatial and
temporal variations of radial velocity within the resolution volume. As mentioned in
Section 2.3.1.2, the square of the spectrum width can be expressed as the sum of at
least three different contributions (Doviak and Zrnic´ 1984; Doviak and Zrnic´ 1993):
σ2v = σ
2
11 + σ
2
s + σ
2
x, (5.17)
where σ11 represents broadening due to turbulence, σs represents the shear broadening
due to large-scale (larger than the radar resolution volume) variations of the wind
field, and σx represents other contributions that can be attributed to signal processing
inaccuracies.
As discussed in White (1997), Jacoby-Koaly et al. (2002), and Scipio´n et al. (2009),
the shear broadening effect for a vertical pointing beam is primarily determined by
beam broadening. It may be represented as:
σ2s = σ
2
a × V 2H , (5.18)
σa =
θ1
4
√
ln 2
, (5.19)
where VH is the horizontal wind magnitude.
For a case corresponding to oblique beams, and under the assumptions of narrow
beams, constant wind shear, and separable beam illumination, Doviak and Zrnic´
(1984) and Jacoby-Koaly et al. (2002) recommend expressing the shear broadening
effect as:
σ2s = (a1kθ)
2 + (a2kφ)
2 + (bkr)
2, (5.20)
where kθ, kφ, and kr represent the components of the wind shear in the spherical
coordinates. Under the additional assumption of a circularly symmetric Gaussian
beam pattern, a1 and a2 have the same value a = σar, which is a function of range
and beamwidth. Finally,
akφ = σa
￿￿
∂u
∂z
sin θ +
∂v
∂z
cosθ
￿
r0 cos
2 φ− u sin θ sinφ (5.21)
− v cos θ sinφ+ ∂w
∂z
r0 cosφ sinφ+ w cosφ
￿
,
akθ = σa(u cos θ − v sin θ), (5.22)
bkr = b
￿￿
∂u
∂z
sin θ +
∂v
∂z
cos θ
￿
cosφ sinφ+
∂w
∂z
sin2 φ
￿
, (5.23)
where b is a related to the pulse width as b = 0.3∆r.
102
Following Scipio´n et al. (2009), the windowing effect is considered the only signal
processing effect that broadens the estimated spectrum. This contribution is generally
constant and is easily calculated from the Fourier transform of the window used in
the processing (3-dB spectrum width).
After removing all the external effects, the value of σ11 can be employed to esti-
mate the turbulence eddy dissipation rate ￿. Under the assumptions of homogeneity
and isotropy, and Gaussianity of the beam and range weighting functions, the turbu-
lence broadening contribution σ211 may be related to the dissipation rate ￿ by Equation
(5.32) from White (1997). After converting to spherical coordinates and approximat-
ing sinc2(x) with exp(−x2/3) (the approximation has a margin of error of 2%), White
et al. (1999) obtains a more manageable expression for ￿, namely:
￿Rad = σ
3
11(4π/Ae)
3/2J−3/2, (5.24)
J = 12Γ(2/3)
π/2￿￿
0
(sin3 ϕ)(b2 cos2 ϕ+ a2 sin2 ϕ (5.25)
+ (L2/12) sin2 ϕ cos2 φ)1/3dϕdφ,
L = VT tD, (5.26)
where Γ is the Gamma function, VT is the wind speed transversal to the radar beam,
and tD is the radar dwell-time (White 1997). The parameter Ae = 1.6 is related to
the empirical constant in the inertial subrange of the velocity spectrum. Note that ￿
is proportional to the cube of σ11. This implies that the TKE dissipation rate in the
considered formulation is quite sensitive to variations of σ11.
The best way to obtain ￿ from LES data is to use the parameterized expression
that enters the LES prognostic equation for the sub-grid kinetic energy (Deardorff
1980) as presented in Equations (2.9) to (2.11).
5.3.2 Results
The energy spectrum of turbulence E(κ) was estimated from the LES velocity fields
at approximately 700 m following the procedure described in Kaiser and Fedorovich
(1998). The average longitudinal and transverse spectra obtained between 18:00 -
19:00 UTC are presented in Figure 5.20. However, due to the limited number of points
of the LES sub-domain used in the radar simulator, it is impossible to determine the
outer scale of the inertial subrange Λ from the estimated spectrum. Nevertheless, its
103
scale is greater than 1400 m, which corresponds to the maximum scale resolvable by
the spectrum.
!"
!#
!"
!$
!"
!!
!"
"
!"
!!
!"
"
!"
!
!"
$
!"
#
!"
%
&'()*+,-)./0!1
2
0!
1
/
/
2
3
0+10!1
2
4
0+10!1
2
3
0(10!1
2
4
0(10!1
2
3
0&10!1
!56#/.)7
20π 2π200π2000π
scale (m)
Figure 5.20: Kolmogorov turbulence spectra obtained from the velocity fields. The sub-
script represents the longitudinal l or the transverse t component, and the superscript
represents the velocity component. All five components are presented as a time average ob-
tained between 18:00-19:00 UTC at approximately 700 m. The black dashed line represents
the -5/3 law expected from isotropic turbulence (see Section 2.2.3.2). All spectra behave
similarly, which confirms the isotropy in the LES fields. Also, the transverse spectrum is
larger than the longitudinal spectrum, which is consistent with the structure parameter in
the inertial subrange. The beginning of the dissipation range can be identified as ≈157 m,
which is caused by dumping in the sub-grid closure of the LES model.
The scale of the resolution volume is defined as (Istok and Doviak 1986):
Λ6 = rmaxθ1, (5.27)
104
where rmax = 2 km is the maximum vertical range available from the LES. Therefore,
for the virtual BLR with θ1 = 9◦, the resulting Λ6 ≈ 220 m is less than one third of
the outer scale of the inertial subrange. In the BLR simulation, the eddy dissipation
rate has been obtained for all five DBS beams (4 oblique and 1 vertical) without
censoring and for the whole run of the simulation (approximately 11.5 hrs). The
obtained estimates are compared with the ￿ estimates from LES. The estimates are
presented in Figure 5.21. Qualitatively, all estimates look similar; however, a more
in-depth analysis is needed to quantify the observed differences.
Hourly averaged profiles of ￿ from the BLR are presented in Figures 5.22 through 5.25
for four time periods: 16:00-17:00, 17:00-18:00, 18:00-19:00, and 19:00-20:00. In the
left-hand plots, the radar estimates for different beams are presented in different col-
ors. Different between the estimates for different beams are ≤4×10−2 m2 s−3. This
agreement confirms the previous findings of Jacoby-Koaly et al. (2002) and comple-
ments their results since their estimates of ￿ from the vertical beam were rather poor
due to ground-clutter contamination. In the right-hand plots of Figures 5.22 to 5.25,
averages from the five beams are compared with the LES estimates of ￿. In general,
the radar and LES estimates of dissipation rate agree well, with discrepancies in the
mixed layer ≤2×10−2 m2 s−3. The observed discrepancies correspond to heights below
∼100 m where the radar simulator does not reproduce the spectrum width accurately.
Another region of discrepancies is above the CBL top (free atmosphere) where the
estimates are not reliable because the turbulence levels there are extremely low.
In this chapter, the horizontal shear of vertical velocity effect has been studied
in depth. First, the theoretical approach using LES-like fields as an input for the
virtual radar simulator proves that it affects both DBS and SA techniques for wind
estimations. The theoretical measured bias for the simulated case is in excellent
agreement for both techniques. The second approach uses realistic LES fields for
8 hours of simulation. Here, the measured biases are also in good agreement with
the theoretical biases estimated at each time and height. However, from scatter plot
comparison between the two quantities, the horizontal shear of vertical velocity is
not the only factor that contributes to discrepancies observed in the horizontal wind
estimates. Other factors might include errors in the wind estimates inherent to each
technique, horizontal wind shear, etc. Nevertheless, these factors are not subject of
the study in this work.
Also, estimates of TKE have been contrasted and compared for different averag-
ing times. LES and radar horizontal estimates are in excellent agreement when the
105
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Figure 5.21: Turbulence eddy dissipation rate as a function of time and height. Top:
estimates from LES. Middle-top: estimates from the Doppler spectrum width of the signal
from the vertical beam. Middle-bottom: estimates from the Doppler spectrum width of the
signal from the oblique beam pointing to the north. Bottom: estimates from the Doppler
spectrum width of the signal from the oblique beam pointing to the east.
106
10ï6 10ï5 10ï4 10ï3 10ï2 10ï1
0
200
400
600
800
1000
1200
1400
1600
1800
2000
m2 sï3
He
igh
t (
m
)
Eddy Dissipation Rate
 
 
Vïbeam
Eïbeam
Nïbeam
Wïbeam
Sïbeam
10ï6 10ï5 10ï4 10ï3 10ï2 10ï1
0
200
400
600
800
1000
1200
1400
1600
1800
2000
m2 sï3
He
igh
t (
m
)
16:00 ï 17:00
 
 
LES
Radaravg
Figure 5.22: Hourly averages of turbulence eddy dissipation rate profiles from 16:00 - 17:00.
Left: estimates from the virtual BLR with different directed beams (red: vertical beam;
blue: east beam; green: north beam; cyan: west beam; magenta: south beam). Right: ￿
profile from the LES (black) compared to the profile from BLR averages among different
beam directions (red).
107
10ï6 10ï5 10ï4 10ï3 10ï2 10ï1
0
200
400
600
800
1000
1200
1400
1600
1800
2000
m2 sï3
He
igh
t (
m
)
Eddy Dissipation Rate
 
 
Vïbeam
Eïbeam
Nïbeam
Wïbeam
Sïbeam
10ï6 10ï5 10ï4 10ï3 10ï2 10ï1
0
200
400
600
800
1000
1200
1400
1600
1800
2000
m2 sï3
He
igh
t (
m
)
17:00 ï 18:00
 
 
LES
Radaravg
Figure 5.23: Same as Figure 5.22, except with time averaging over period 17:00 - 18:00
UTC.
10ï6 10ï5 10ï4 10ï3 10ï2 10ï1
0
200
400
600
800
1000
1200
1400
1600
1800
2000
m2 sï3
He
igh
t (
m
)
Eddy Dissipation Rate
 
 
Vïbeam
Eïbeam
Nïbeam
Wïbeam
Sïbeam
10ï6 10ï5 10ï4 10ï3 10ï2 10ï1
0
200
400
600
800
1000
1200
1400
1600
1800
2000
m2 sï3
He
igh
t (
m
)
18:00 ï 19:00
 
 
LES
Radaravg
Figure 5.24: Same as Figure 5.22, except with time averaging over period 18:00 - 19:00
UTC.
108
10ï6 10ï5 10ï4 10ï3 10ï2 10ï1
0
200
400
600
800
1000
1200
1400
1600
1800
2000
m2 sï3
He
igh
t (
m
)
Eddy Dissipation Rate
 
 
Vïbeam
Eïbeam
Nïbeam
Wïbeam
Sïbeam
10ï6 10ï5 10ï4 10ï3 10ï2 10ï1
0
200
400
600
800
1000
1200
1400
1600
1800
2000
m2 sï3
He
igh
t (
m
)
19:00 ï 20:00
 
 
LES
Radaravg
Figure 5.25: Same as Figure 5.22, except with time averaging over period 19:00 - 20:00
UTC.
horizontal shear of vertical velocity effect is not present. When the effect is present,
using a long time-average can reduce its effect. Vertical variances have to be cal-
culated using the best time resolution available to avoid filtering the estimates (see
Chapter 4).
Finally, TKE (eddy) dissipation rate estimates have been obtained from the radar
Doppler spectrum width for all DBS beams. These estimates are in excellent agree-
ment among each other and have been compared with the LES estimates obtained
using the Deardorff (1980) closure scheme. Both estimates show similar results, the
discrepancies observed at lower heights (<100 m) are due the lack of points in the
LES. Discrepancies above the BL top are caused by weak turbulence. It is important
to notice, that these estimates are not affected by the horizontal shear of vertical
velocity since they are not obtained from radial velocities but from spectrum width.
109
Chapter 6
Conclusions and Future Work
6.1 Conclusions
In this work, the combination of LES and an LES-based wind profiler was used to
study and characterize the lower atmosphere. The LES had succeeded in adequately
reproducing different ideal and realistic CBL cases. Meanwhile, the virtual BLR had
replicated complex voltage radar signals within the LES sub-domain used in this
study.
A LES-based radar simulator that uses an Eulerian framework approach was pre-
sented for studies of the CBL in Section 3.1. The virtual radar was based on the
work of Muschinski et al. (1999). However, it was refined to allow a wide range of
virtual radar experiments. The simulator was capable of producing realistic complex
time-series data for the case of a cloud-free CBL. For example, one could select the
beam direction, beam width, pulse width, and simultaneously deploy a large number
of virtual radars. The radar simulator was also designed to accommodate multiple-
frequency applications, as well as monostatic, bi-static, and SA configurations. The
primary differences between the virtual BLR presented here and the one developed
by Muschinski et al. (1999) can be summarized as follows:
• The virtual BLR, in its present form, could accommodate multiple beams (verti-
cal and off-vertical), which was made possible though the calculation of oblique
C2n at each LES grid point considering the radial from the transmitter to the
point.
• C2n and velocity values were interpolated for each time step of the virtual BLR.
• Noise was generated and injected into the time-series data.
110
• Location of each grid point with respect to the transmitter and receiver were
included as part of the equation.
• Radial velocities at each LES point were calculated as a resultant of the trans-
mitter and receiver components.
• Radar beamwidth was also calculated independently for the transmitter and
receiver.
The LES was initialized with the CBL case observed in the U.S. Southern Great
Plains Atmospheric Radiation Measurement Climate Research Facility in Lamont,
Oklahoma on June 8, 2007. As indicated in Botnick and Fedorovich (2008), this
particular case was one for which LES predictions compare rather favorably with
sounding data (see Figure 4.6). The LES output was also compared with actual
radar data from the same location and date.
Range-corrected power estimates were used to retrieve profiles of C2n at every
height for the entire simulation time using an empirical calibration method (see Sec-
tion 4.2). There was good agreement between the estimates obtained from the LES,
the virtual (RadSim) and real (RadarARM). Discrepancies were observed in the Rad-
Sim at regions with low SNR when the noise power was amplified through the range
correction algorithm. Also, the LES code was apparently unable to reproduce the
evolution of external atmospheric forcing, which caused a discrepancy in the heights
at which the C2n from LES and RadarARM attain maxima values. Comparisons of
horizontal winds were presented, in addition to the C2n in Section 4.3. The results
showed a reasonable agreement in the three wind fields estimates between the RadSim
and the LES, which reflects the ability of the employed LES code to reproduce basic
properties of the CBL mean flow for the studied case. Estimates of the mean wind
by virtual (RadSim) and real (RadarARM) radars were also in good agreement, which
indicates that the virtual BLR is capable of reliable characterization of atmospheric
flow in a similar way as the real radar.
The wind fields estimates obtained from the LES and the radar simulator were
in good agreement. The discrepancies observed especially on the mixed layer were
≤5 m s−1 for the horizontal wind fields and were attributed by the AWGN contam-
ination, associate error measurements, and the horizontal shear of vertical velocity.
Discrepancies observed between the LES and the real (RadarARM) radar, in particu-
lar at high elevations were ≤10 m s−1 and were due to the apparent inability of the
LES to account for the evolution of large-scale forcing.
111
It was shown that the LES was able to reproduce the evolution of a cloud free CBL
case. The virtual BLR replicated realistic radar signals under the same conditions.
The combination of both allows the characterization of different CBL parameters.
Additional comparisons between the three initial quantities were made to confirm
the agreement observed. Estimates of the vertical velocity variance (see Section 4.4)
from LES and RadSim were generally in good agreement. The small discrepancies
between these estimates were ≤1.4 m2 s−2 and apparently to the differences in the
spatial and temporal extent of the vertical velocity samples. The differences between
the LES and RadarARM variances might be attributed to the inability of LES to
sufficiently reproduce the actual evolution of the CBL depth, particularly during
the morning stages. The sharp decrease in the vertical velocity variance in the upper
portion of the CBL can be used as a rough indicator of the top of the capping inversion
layer and of the overall CBL top. This decrease was clearly observed in the variance
data retrieved from LES and virtual radar for subsequent stages of CBL growth (see
Figure 4.7).
The skewness of the vertical velocity, presented in Section 4.5, was analyzed as
a function of height. In the well-mixed portion of the CBL, skewnessVertical ve-
locity!skewness estimates are essentially positive and their magnitude increases with
height in agreement with Moeng and Rotunno (1990). The estimates from the three
sources agree well with an error of ≤1, especially within the well-mixed portion of the
CBL. The estimates of CBL depth from skewness profiles demonstrate fair agreement
with those retrieved from vertical velocity variance distributions.
The LES and LES-based radar simulator were also used to investigate other atmo-
spheric parameters, phenomena, or effects in the CBL. The effect of horizontal shear
of vertical velocity on horizontal wind estimates was studied for DBS and SA in Sec-
tion 5.1. The virtual BLR was setup during the theoretical studies to only change its
phase due to variations in the three wind fields. Its magnitude was kept constant for
simplicity. A known shear was introduced in the vertical velocity, and its magnitude
was controlled by an external parameter sfrac, which allowed the amount of vertical
shear within the simulator. The presence of the shear biased the zonal wind estimates.
The theoretical and the measured bias were in excellent agreement, and the MSE be-
tween both quantities is 1.55×10−4 m2 s−2 for DBS and 1.83×10−2 m2 s−2 for SA.
Under realistic conditions, the biases showed the same behavior as in the theoretical
studies; when the sfrac increased, the bias also increased. It is important to mention
that the theoretical bias was calculated at each time and height from the LES fields.
112
The correlation between the theoretical bias from the LES and the measured bias
from the virtual BLR indicated the presence of other factors that could contribute
to the measured bias. Other factors might include estimation errors, and/or hori-
zontal wind shear. Also, the effect of the horizontal shear of the vertical velocity is
observed mostly in the mixed layer, where the strong updrafts and mild downdrafts
were present.
The radar was employed to retrieve variances of the wind velocity components
from the simulated radar signals with different DBS settings in Section5.2. The ver-
tical profiles of TKE obtained with BLR were evaluated against the reference TKE
profiles directly computed from the LES data. Different averaging approaches for
evaluating velocity statistics were studied, and the effect of the horizontal shear of
vertical velocity on the accuracy of the evaluation of the horizontal wind component
variances was investigated. It was noticed that the horizontal variances were influ-
enced by the horizontal shear in the vertical velocity, and in some cases, the bias in
the variance was more than 600%. A proposed method to reduce the horizontal biases
was to average the wind estimates before the calculation of the variances. However,
this procedure might reduce the number of points used in the calculations making the
results unrealistic. On the other hand, calculations of the vertical variance required
high time resolution. Also, it is important to mention the lack of agreement observed
in the horizontal variances between the time- and the space/time- average from the
radar simulator and LES when the shear of vertical velocity was artificially removed.
A method of estimating turbulence intensity is through remotely sensed velocity
variations (e.g., Brewster and Zrnic 1986). The Doppler spectrum is an approxima-
tion of a power-weighted distribution of radial velocities within the resolution volume
of the radar. The standard deviation of the Doppler spectrum is termed the spectrum
width σv, and it can be related to the turbulence energy dissipation rate ￿ (Hocking
1985). The measurement of the width of the Doppler spectrum can be problematic
in that there are several phenomena that tend to bias it to larger values, such as
wind shear and finite beam-width effects (Nastrom 1997; White 1997; White et al.
1999). However, given a known beam pattern of the radar and standard DBS mea-
surements, it is possible to minimize these effects and estimate the velocity variation
due to turbulence (Cohn 1995; White 1997; White et al. 1999). The broadening of the
Doppler spectrum resulting from turbulence within the radar resolution volume was
estimated using the procedure outlined in White (1997); Jacoby-Koaly et al. (2002)
113
in Section 5.3. The TKE (eddy) dissipation rates have been estimated for alterna-
tively (vertically and obliquely) directed radar beams. The estimates of ￿ for the
differently directed beams agreed extremely well. In LES, the dissipation rate val-
ues were estimated using a parameterized expression from Deardorff (1980). Radar
and LES estimates of ￿ in the mixed layer were found to be in good agreement with
discrepancies ≤2×10−2 m2 s−3. Major discrepancies have been found at low heights
where the radar simulator does not reproduce the spectrum width with accuracy due
to the limited number of points within the resolution volume, and above the CBL top
(free atmosphere) where the turbulence was low and/or the presence of residual noise
was not removed.
6.2 Future Work
At the conclusion of this work, it is important to assess areas of potentially important
future research. Subsequent work in studies of the atmosphere with LES and radar
simulators should include the simulation of more LES cases to complement the realis-
tic case presented in this dissertation. An alternative to oblique calculations of C2n at
each LES grid cell presented in Section 3.1 can be made considering an average over
the twelve edges of the LES cube surrounding the desired grid cell (see Appendix C
for details).
Some of these cases will include the simulation of new realistic setups to help
study the following:
• Study and compare the evolution of CBL between LES and realistic radar data
from LMN. This study will include accurate measurements of the CBL depth
and quantification of the discrepancies.
• Quantify the discrepancies observed between the LES and LES-based virtual
radar estimates of the horizontal wind fields.
• Implement stable boundary layer (SBL) cases which are often observed at night.
These cases will complement the daytime CBL presented in this study and will
give a complete idea of the evolution of the BL.
Additional cases will incorporate the simulation of several idealized cases. Some
will include cases with non-horizontal shear, and/or constant wind velocity. These
cases will clarify the following:
114
• Identify and isolate cases where discrepancies that cannot be attributed to the
horizontal shear of the vertical velocity are observed in the horizontal winds.
• Quantify the contribution of each of the factors identified before.
These new cases will also help to improve the estimates of TKE, in particular the
discrepancies observed between the time and the space/time average in the horizontal
variances. Additionally, it is important to fully calculate the second-order statistics,
which will complement the calculations of vertical velocity variance and TKE, in
particular the calculations of the kinematic momentum fluxes (u￿w￿ and v￿w￿). A first
attempt to calculate these fluxes was made by Scipio´n et al. (2007), but the results
were inconclusive. This new set of cases will allow the study of these fluxes to be more
detailed and to incorporate different conditions. Complementary to the calculation
of the kinematic momentum fluxes, profiles of kinematic heat flux (Θ￿w￿) can also be
obtained from the same data set. Profiles of virtual potential temperature can be
obtained if the simulation of a RASS system is included within the radar simulator.
The mixed layer height in the SBL reaches its maximum at approximately 200 m
(e.g., Asimakopoulos et al. 2004) at night. Usually, radars have a range resolution of
100 m, which might be too coarse for SBL studies. A more appropriate instrument for
remote sensing of the SBL is a SODAR (SOund Detection And Ranging), which has
a range resolution as small as 5 m. In the literature, estimations of the SBL mixed
layer height are widely spread (e.g., Beyrich and Weill 1993; Beyrich 1995; Crescenti
1997; Asimakopoulos et al. 2004), as well as studies of the atmospheric fluxes based on
wind profiles retrieved from a Doppler sodar (e.g., Giannini et al. 1997). For studies
of the SBL, the LES will have a finer resolution (at least 2 m) than the LES used in
CBL studies (10 m and 20 m). A LES-based sodar can be simulated from the fine
LES fields to study the SBL. As in the case of CBL, true sodar retrievals can aid in
the validation of LES and virtual sodar.
Finally, the LES with fine resolution, especially designed for SBL studies, can be
extended to improve the previous daytime CBL studies. These new high-resolution
cases will give a better population in the radar resolution volume, especially at lower
elevations. It can also help with the determination of small layers using FDI or RIM.
Additionally, other techniques can be implemented and tested to study the CBL, like
the turbulent eddy profiler (TEP) (Cheong et al. 2004b, 2008).
115
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Appendix A - List Of Symbols
A Complex amplitude
Ae Empirical constant in the inertial subrange of ve-
locity spectrum
A(p) Amplitude at the pth LES grid cell
A Matrix for the solution of DBS
c Speed of Light
cm Centimeter, 10−2 meters
C Radar calibration constant
C2n Structure function parameter of refractive index
C2(p)n Structure function parameter of refractive index
at the pth LES grid cell
cp Specific heat for air
c11 Autocorrelation function￿c11 Estimate of autocorrelation function
c12 Cross-correlation function￿c12 Estimate of cross-correlation function
c0 Amplitude offset of the auto- or cross-correlation
function
D Diameter of a spherical drop
DC Direct Current
e Partial pressure of water vapor￿e(p)tx Unit vector from the transmitting antenna to the
pth LES grid cell￿e(p)rx Unit vector from the receiving antenna to the pth
LES grid cell
E Sub-grid TKE viscous dissipation rate
E(κ) Energy spectrum of turbulence
127
E(u)t (κ) Transverse energy spectrum of turbulence ob-
tained from LES zonal wind
E(u)l (κ) Longitudinal energy spectrum of turbulence ob-
tained from LES zonal wind
E(v)t (κ) Transverse energy spectrum of turbulence ob-
tained from LES meridional wind
E(v)l (κ) Longitudinal energy spectrum of turbulence ob-
tained from LES meridional wind
E(w)t (κ) Transverse energy spectrum of turbulence ob-
tained from LES vertical wind
f Frequency
fd Doppler frequency
fN Nyquist frequency
fs Pulse repetition frequency
f 2(θ,φ) (One-way) normalized power density pattern
g Gravitational acceleration
G Gain of the transmitting or receiving antennas
Gtx Gain of the transmitting antenna
Grx Gain of the receiving antenna
GHz Gigahertz, 109 Hz
H Sensible heat flux
hPa Hectopascal, 102 Pascals
Hz Hertz, unit increment of frequency
I(t, r0) In-phase component of the complex voltage signal
J Joules
k Wavenumber
kB Bragg wavenumber
kg Kilogram
km Estimate of the peak of the Doppler spectrum
k0 Radar wavenumber
K Kelvin, unit increment of temperature
Kh Sun-grid eddy diffusivity
Km Sub-grid eddy viscosity
Km Complex dielectric factor
Kw Complex dielectric factor for liquid water
128
Ki Complex dielectric factor for ice
km Kilometer, 103 meters
kW Kilowatts, 103 Watts
l Sub-grid mixing length
l Losses
LH Latent heat flux
Lv Latent heat of vaporization
m Meter, unit increment of length
m Complex refractive index for water
M Number of signal samples
M Number of LES grid points within the resolution
volume
mb Millibar, 10−3 bar
mm Millimeter, 10−3 meters
MHz Megahertz, 106 Hz
n Refractive index
n Refractive index at level z
n1 Refractive index at level z +∆z
n2 Refractive index at level z −∆z
nxy Refractive index at point (x,y)
nxiyi Refractive index at the fixed grid cells (xi,yi)
N Refractivity
N Noise power￿N Estimated noise power
N(D) Particle size distribution
p Surface pressure from RUC￿p Resolved pressure￿P Estimated total power
Pa Pascal
Pr Received power
Pt Transmitted power
P Atmospheric pressure
P0 Pressure at z=0 m
p0 Hydrostatic atmospheric pressure
q Specific humidity
129
Q(t, r0) Quadrature component of the complex voltage
signal
r Range
r Position vector
ra Range Aliasing
rtx0 Range from the transmitting antenna to the cen-
ter of the resolution volume
rrx0 Range from the receiving antenna to the center of
the resolution volume
rmax Maximum range available at the LES sub-domain
r0 Range from the antenna to the center of the res-
olution volume
R Gas constant for dry air
RxA Location of receiver A in the spaced antenna con-
figuration
RxB Location of receiver B in the spaced antenna con-
figuration
RxC Location of receiver C in the spaced antenna con-
figuration
RxD Location of receiver D in the spaced antenna con-
figuration
R(τ) Autocorrelation function￿R(k) Estimated autocorrelation function
sfrac Vertical velocity fraction
s Second
su Shear in the zonal direction
suz0 Range corrected shear in the zonal direction
sv Shear in the meridional direction
svz0 Range corrected shear in the meridional direction
sw Shear in the vertical direction
S Signal power￿S Estimated signal power￿sij Deformation tensor for filtered velocity
S(f) Doppler spectrum in frequency
130
￿S(f) Estimated Doppler spectrum
S(v) Doppler spectrum in velocity￿Sn(k) Normalized Doppler spectrum
SNR Signal-to-Noise Ratio
t Time
t0 Initial time for variable interpolation from LES
t1 Final time for variable interpolation from LES
tD Radar dwell time
T Absolute temperature
Tr Range time
Ts Pulse repetition time
Tx Location of transmitter in the spaced antenna
configuration
u Zonal wind field
u￿ Fluctuating component of the zonal velocity￿u Measured zonal wind field biased by the horizontal
shear of vertical velocity
u Three dimensional velocity vector
u0 Uniform zonal wind field￿ui Resolved velocity components - Einstein summa-
tion convention
ug Zonal geostrophic wind
U(t) Transmitted pulse shape
v Meridional wind field
v￿ Fluctuating component of the meridional velocity￿v Measured meridional wind field biased by the hor-
izontal shear of vertical velocity
v0 Uniform meridional wind field
vg Meridional geostrophic wind
V (t, r0) Received complex echo voltage
V (t0 +∆t) Received complex echo voltage at t0 +∆t
V (mTs) Baseband time series signal
va Aliasing velocity
vH Horizontal velocity
VH Horizontal wind magnitude
131
VT Wind speed transverse to the radar beam
vr Radial velocity￿vr Estimate of the mean radial velocity
vr Radial velocity vector
v(p) Instantaneous radial velocity at the pth LES grid
cell
w Vertical wind field
w￿ Fluctuating component of the vertical velocity
w0 Uniform vertical wind field
Wr Range weighting function
Wb Two-way beam weighting function
Wbtx One-way transmitter beam pattern weighting
function
Wbrx One-way receiver beam pattern weighting func-
tion
xi Right cartesian coordinates - Einstein summation
convention
z Level of the grid cell
z0 Vertical range of the radial velocity
Z Reflectivity factor
Z(f) Discrete time Fourier transform
Ze Equivalent reflectivity factor
zw Distance from the ground
βh Length of diffraction pattern
δ Separation scale
δ Bragg scale (δ = λ/2)
δm Kroneker delta function
∆ Effective grid-cell size
∆ Oblique distance between layers in calculations of
∆n
∆n Refractive index gradient at the grid scale
∆n￿ Refractive index gradient at the Bragg scale
∆r Range resolution
∆t Increment in time, usually a function of PRT
∆x Grid cell size in the x-direction
132
∆y Grid cell size in the y-direction
∆z Grid cell size in the z-direction
∆zw Vertical dimension of the lower grid cell
￿ Turbulence (eddy) dissipation rate
η Radar reflectivity - average cross-section per unit
volume￿η Decorrelation parameter of ￿c12
κ Attenuation index
κ Wavenumber
λ Radar wavelength
Λ Outer scale of the inertial subrange
Λ6 Scale of the resolution volume
ν Kinematic viscosity
Ω Earth’s angular velocity
φ Zenith angle
ψe Echo phase of the received signal
ψ(p)e Echo phase at the pth LES grid cell
ψt Transmitted phase
ψs Scattering phase shift
ψ(p)0 Random initial phase at the p
th LES grid cell
ρ0 Constant reference density
ρ Cross-correlation factor at t0 and t1
σ11 Doppler spectrum broadening due turbulence
σb Backscattered radar cross-section
σr Second central moment of the range weighting
function
σs Doppler spectrum broadening due shear
σx Doppler spectrum broadening due signal process-
ing
σt rms wind variability for isotropic turbulence
σv Doppler spectrum width￿σv Estimate of the Doppler spectrum width
σ2w Vertical velocity variance￿π Normalized pressure
θ Azimuth angle
133
θxtx Transmitting antenna beam pointing in x-
direction
θytx Transmitting antenna beam pointing in y-
direction
θxrx Receiving antenna beam pointing in x-direction
θyrx Receiving antenna beam pointing in y-direction
Θ Potential temperature
Θv0 Constant reference potential temperature
θ1 3-dB width of the one-way pattern (radians)￿τc Square root of the second moment of ￿c12￿τp Peak time delay of ￿c12￿τx Lag when |￿c11(τx)| = |￿c12(0)|
τw Transmitting pulse width
134
Appendix B - List Of Acronyms and Abbreviations
ABL Atmospheric Boundary Layer
ACF Autocorrelation Function
ACRF Atmospheric radiation measurement Climate Re-
search Facility
ARM Atmospheric Radiation Measurement
AWGN Additive White Gaussian Noise
BMX Beam MultiPlexing
BLR Boundary Layer Radar
CBL Convective Boundary Layer
CCF Cross-Correlation Function
COHO Coherent Oscillator
CW Continuous Wave
DBS Doppler Beam Swinging
DC Direct Current
DOC Department of Commerce
DOD Department of Defense
DOT Department of Transportation
DTFT Discrete Time Fourier Transform
FAA Federal Aviation Administration
FCA Full Correlation Analysis
FCA-G Full Correlation Analysis - Gaussian
FM-CW Frequency Modulated - Continuous Wave
GMAP Gaussian Model Adaptive Processing
HCA Hydrometeor Classification Algorithm
IF Intermediate Frequency
IPP Inter-Pulse Period
JDOP Joint Doppler Operational Project
JPOLE Joint Polarization Experiment
135
LES Large-Eddy Simulations
LMN Lamont, OK
MLDA Melting Layer Detection Algorithm
MPAR Multifunction Phased Array Radar
MRC Multiple Radar Configuration
MSE Mean Square Error
MST Mesosphere-Stratosphere-Troposphere
MU Middle and Upper
NASA National Aeronautics and Space Administration
NCAR National Center for Atmospheric Research
NEXRAD Next-Generation Radar
NOAA National Oceanic and Atmospheric Administra-
tion
NSSL National Severe Storms Laboratory
NPTS Number of Points
NWRT National Weather Radar Testbed
NWS National Weather Service
OPERA Operational Program for the Exchange of
weather RAdar information
ORDA Open Systems Radar Data Acquisition
OK Oklahoma
PAR Phased Array Radar
PPI Plan Position Indicator
PRT Pulse Repetition Time
PRF Pulse Repetition Frequency
PSD Power Spectral Density
QPE Quantitative Precipitation Estimation
RADAR Radio-Detection-And-Ranging
RASS Radio Acoustic Sounding System
RF Radio Frequency
RHI Range Height Intensity
RMS Root Mean Square Error
ROC Radar Operations Center
RTI Range Time Intensity
RUC Rapid Update Cycle
136
RWP Radar Wind Profiler
SA Spaced Antenna
SBL Stable Boundary Layer
SGP Southern Great Plains
SNR Signal-to-Noise Ratio
SODAR SOunding Detection And Ranging
SOUSY SOUnding SYstem
SPRT Staggered PRT
ST Stratosphere-Troposphere
STALO Stable Local Oscillator
TB Time Balance
TEP Turbulent Eddy Profiler
TKE Turbulence Kinetic Energy
UHF Ultra High Frequency
VCP Volume Coverage Pattern
VHF Very High Frequency
WSR-88D Weather Surveillance Radar - 1988 Doppler
WSS Wide Sense Stationary
137
Appendix C
Calculations of Structure Function Parameter of
Refractive Index
Calculations of C2n due to isotropy at each LES grid cell can be alternatively estimated
by computing the structure function as an average over the twelve edges of the cell
(see Figure C.1). The square of the difference in the refractive index is estimated at
each edge of the grid cell and divided by the the LES grid spacing ∆ as:
C2n =
1
12
12￿
k=1
(∆nk)
2
∆2/3
. (C.1)
n3
n4
n1 n2
n6
n7n8
n5
∆
∆nk = ni − nj
C2n
Figure C.1: Scheme that describes the calculations of C2n at each LES cell. The difference
in refractive index ∆nk is calculated at edge of the cube considering the refractive index
at each vertex. The length of each of the twelve edges of the cube corresponds by the LES
grid spacing ∆.
138
Index
Additive White Gaussian Noise, 56, 86,
110
Autocorrelation function, 38–42, 46–48,
62, 80
bias
ideal, 83, 85, 88, 93, 96
measured, 85, 88, 93, 96, 104, 112
Boundary Layer
Atmospheric, 5, 16, 17, 35
Convective, 5, 6, 16–20, 22, 25, 26, 48,
113
depth, 5, 6, 16, 18, 22, 23
top, 53, 66, 73–75, 77, 86, 104, 108,
111, 113
Stable, 113, 114
Bragg
scale, 15, 16, 35, 50, 51
scatter, 13
Central Limit Theorem, 37
COHO, 28, 29
Cross-correlation function, 46–48, 62, 79,
80
dissipation range, 35, 36
Doppler Beam Swinging, 16, 18, 19, 43,
45, 48, 59, 61, 68, 77–81, 83, 85,
86, 88, 93, 96–100, 104, 108, 111,
112
Doppler spectrum, 30, 31, 38–40, 112
width, 43, 77, 99, 102–105, 108, 113
dwell time, 40, 59, 61, 64, 65, 68, 74, 86,
100
Einstein summation, 20
energy-containing range, 35, 36
entrainment zone, 5, 16–18, 26, 74, 77,
111
ergodic, 37
Eulerian framework approach, 50, 54
free atmosphere, 5, 73, 104, 113
Full Correlation Analysis, 46, 47
Gaussian, 47, 85, 88
homogeneity, 43, 45, 59, 61, 103
homogeneous, 49
Horizontal shear of vertical velocity, 18,
19, 77, 78, 85, 88, 93, 98, 104, 108,
111, 112, 114
in-phase, 28, 29, 37
inertial subrange, 15, 35, 36, 49, 103
inversion layer, see entrainment zone
isotropy, 73, 103
Lagrangian framework approach, 50
Large-eddy simulations, 17–20, 22, 24, 48,
50–56, 58, 59, 61, 62, 64, 66, 68,
69, 73–75, 77, 78, 80, 86, 88, 93,
97–99, 103, 104, 108–112
sub-domain, 50, 56, 58, 61, 86
mean square error, 83, 85
mixed layer, 5, 69, 73, 75, 104, 112
Power
Noise, 39–42
Signal, 39, 40, 42
Total, 39, 40, 42
quadrature, 28, 29, 37
radar, 6, 7
bi-static, 27, 55, 109
Boundary Layer, 16, 17, 24, 25, 48,
66, 68, 69, 86, 97–99, 104
clear-air, 7, 35, 39
Doppler, 7–9, 13, 19
heterodyne, 28, 29
139
homodyne, 28, 29
monostatic, 27, 33, 55, 58–61, 109
PAR, 11, 12
pseudo-monostatic, 62
reflectivity, 9, 33–35, 49, 65
factor, 34
simulator, 18, 19, 50, 55, 56, 62–64,
66, 67, 80, 81, 104, 109, 111, 114
weather, 7, 32–34, 39
Wind Profilers, 13, 19, 64–66, 68, 74,
77–79
WSR-88D, 8–12
radial velocity, 9, 30, 32, 37, 39–44, 48,
55, 59, 79, 102
Radio Acoustic Sounding System, 64, 65
range corrected shear, see bias
RASS, see Radio Acoustic Sounding Sys-
tem
Rayleigh
scatter, 7, 34
refractive index, 13, 16, 34, 35, 49, 51
structure function parameter of, 19,
35, 36, 49, 50, 52–55, 64–66, 73,
75, 77, 78, 86, 109, 110
refractivity, 49, 53
revisit time, 68, 74, 75, 77, 99
scan time, 59, 61, 64, 68, 86, 88, 98, 100
sfrac, 81–83, 85, 86, 88, 89, 91–96, 99, 100,
111
sodar, 114
Spaced Antenna, 18, 19, 45, 46, 48, 80,
83, 85, 86, 88, 93, 96, 104, 109,
111
STALO, 28, 29
stationarity, 43, 45, 59, 61
surface layer, 5
Taylor frozen-turbulence hypothesis, 18
turbulence, 4–6, 13, 17–20, 22, 25, 36, 39,
46, 49, 53, 66, 68, 80, 86, 108
energy spectrum, 35
isotropic, 35, 46, 49, 51
Spectrum width due, 43, 102, 103
Turbulence kinetic energy, 18, 19, 97–99,
104, 112, 114
(eddy)dissipation rate, 18, 19, 77, 78,
99, 103, 104, 108, 113
Vertical velocity
skewness, 19, 26, 64, 75, 77, 111
variance, 19, 64, 73, 74, 77, 111, 114
vertical velocity fraction, see sfrac
weighting function
beam, 55, 58, 59
range, 55
Wide Sense Stationary, 37, 38
140