Electrostatic and electromagnetic linear perturbation analyses of electron shear-flow in a planar, crossed-field diode produce a second-order differential eigenvalue equation. The growth rate and spatial structure of unstable modes are determined by the roots of the associated dispersion relation, which depends on the equilibrium plasma profile and boundary conditions on the perturbation profile at the two ends of the diode. Conventional approaches address this boundary value problem using the so-called shooting method, in which the eigenvalue problem is recast as an initial value problem and solved iteratively until the boundary conditions are satisfied. While effective, the shooting method requires locating the solutions of the dispersion relation one at a time in the complex plane, making it computationally intensive and restrictive for exploring the full instability spectrum. In the present work, the eigenvalue equation is reformulated as a global eigenvalue problem based on the discretized differential equation. The global approach captures the roots obtained from the shooting method, which correspond to the Kelvin–Helmholtz type instability, as well as additional unstable modes not captured using the shooting method. These newly identified solutions are found to be associated with the detrapping of electrons due to large electric field gradients.
Castillo et al. (Sun,) studied this question.