Multimode theory of electron hole instability

We analyze 3-D Vlasov-Poisson instabilities, which limit final shape

and lifetime, of an initially planar electron hole (BGK) structure.

The $f_1(v)$ is found by integration along unperturbed orbits in

a non-sinusoidal potential perturbation shape (parallel to $B$)

expanded in eigenfunctions of the adiabatic Poisson operator. This

generalizes the prior assumption of a rigid shift of the equilibrium.

The shiftmode is then modified by a second discrete mode plus an

integral over a continuum of wave-like modes. A rigorous treatment

shows that the continuum can be approximated effectively by a single

mode that satisfies the external wave dispersion relation, thus making

the perturbation a weighted sum of three modes. We find numerically

the solution for the complex instability frequency, and for the

corresponding 3 mode amplitudes determining the perturbation

eigenmode. This multimode analysis refines the accuracy of the prior

shiftmode results, giving slightly higher growth rates at most

parameters, as expected from the extra mode shape freedom. Oscillating

modes near stability boundaries have larger eigenmode distortions

which helps explain PIC simulations that observe spatial narrowing of

the perturbation, and instability at up to $\sim20$\% beyond the prior

shiftmode thresholds.