The instability which arises in thin ionization layers in the ionospheric E region is inherently nonlocal at mid-latitudes because of the high conductivity along the inclined magnetic field lines. We attack the gradient-drift stability problem using a numerical matrix method, with a finite Fourier basis for the density and potential perturbations, and find complex eigenfunctions and eigenvalues. A thin density layer with steep edges, superimposed on a constant background, is unstable at modes that are centered on the locally unstable side but extend into the locally stable region. The finite background density is essential, but of course there always is such a background in the ionosphere. An interesting aspect of the solutions is that the electric field perturbations peak at distances farther away from the layer than do the density perturbations. The growth rate of the nonlocal modes is significantly smaller than that predicted by local theory (which omits the damping effect of coupling along the magnetic field lines), as expected, and so the waves must be strongly driven. But for ambient electric field strengths of a few mV/m, or an equivalent wind velocity, the growth rates are sufficient to explain sounding rocket and radar observations of plasma waves associated with sporadic-E layers. We also find that the unstable plasma waves propagate in directions that differ from normal to the magnetic field by less than a degree, probably less than half a degree, in agreement with radar observations.
