A nonlocal theory of an electrostatic sinusoidal density drift instability

The stability of space plasmas in which the macroscopic variables vary in the direction x perpendicular to the magnetic field is usually studied by means of the local approximation, in which the fluctuating potential is assumed to be independent of x. To remove this approximation, a nonlocal problem is studied in which the background density, and hence the fluctuating potentials, are periodic in x. From the linear Vlasov/Poisson equations, a set of coupled, linear, homogeneous, algebraic equations relating the Fourier amplitudes of the eigensolution is derived. The equations are solved numerically and a hierarchy of exact eigenmodes characterized by different growth rates and spatial structures is found. At points where the density gradient is locally zero, the mode amplitude is generally several orders of magnitude lower than the peak amplitude. The dependence of these solutions on the parameters which define the background density perturbation is studied and a correspondence between these nonlocal results and those obtained with the local approximation is demonstrated.