Criterion for the sign of wave energy

Thomas M O'Neil

doi:10.1063/1.5120401

Abstract

A criterion for the sign of wave energy is developed by using the symmetry properties of the plasma equilibrium and the fact that Vlasov dynamics is an incompressible flow in phase space, rather than the usual and more difficult procedure of calculating the value of the wave energy directly. Applications are made to the case of waves excited on a non-neutral plasma in a Malmberg–Penning trap and to waves excited on an infinitely long non-neutral beam.

Highlights

A plasma wave is said to have positive energy if energy must be added to the plasma when the wave is excited
A criterion for the sign of wave energy is developed by using the symmetry properties of the plasma equilibrium and the fact that Vlasov dynamics is an incompressible flow in phase space, rather than the usual and more difficult procedure of calculating the value of the wave energy directly
Applications are made to the case of waves excited on a non-neutral plasma in a Malmberg–Penning trap and to waves excited on an infinitely long non-neutral beam

Summary

INTRODUCTION

A plasma wave is said to have positive energy if energy must be added to the plasma when the wave is excited. This paper considers weakly damped, electrostatic waves that propagate on a stable non-neutral plasma, and establishes criterion that the waves have negative energy as viewed in the laboratory reference frame. Scitation.org/journal/php see that the wave energy, as viewed in the laboratory reference frame, is negative if the ratio ðx À lxrÞ =x is negative This simple criterion applies to all of the weakly damped, electrostatic waves that can be excited on the plasma, including diocotron waves, Trivelpiece-Gould waves, cyclotron waves, etc.. Davidson and Krall showed that such a beam is stable under Vlasov–Poisson dynamics if the particle distribution is a monotonic decreasing function of the Hamiltonian in a frame that translates axially with velocity u and rotates with angular frequency xr: We will see that a weekly damped, electrostatic wave on the beam has negative energy, as seen in the laboratory frame, when the ratio ðx À lxr À kuÞ=x is negative. There are non-neutral plasma equilibria that do not have cylindrical symmetry but are stable under 2D E Â B drift dynamics, and we will see that all of the low frequency drift modes on these plasmas have negative energy

WAVES ON A PLASMA CONFINED IN A MALMBERG–PENNING TRAP

WAVES ON AN INFINITELY LONG BEAM

WAVES GOVERNED BY 2D E 3 B DRIFT DYNAMICS

ASYMMETRICAL EQUILIBRIA