Rotating waves in confined nonneutral plasmas elucidate criterion for negative energy

Determining the wave energy in plasma is a complicated calculation that often relies on an idealized version of the equilibrium and dynamics of the system. But, it is possible to determine the sign of the wave energy without directly calculating the energy of the wave, providing insight into equilibrium states of plasmas and how negative energy waves can become unstable and grow.

In a recent paper, O’Neil takes a pedagogical approach to modernizing historical methods for determining the sign of wave energy using ideas applicable to modern plasma theory. He highlights negative wave energy examples in which nonneutral plasmas are confined in Malmberg-Penning traps, and he examines the confinement, equilibrium and distribution qualities that lead to a negative wave energy in each example.

The symmetry properties of the plasma equilibrium in the Malmberg-Penning trap and the general property of collisionless kinetic theory for the waves, namely that the flow in phase space is incompressible, can be used to determine the criterion for negative wave energy. The work uses the Vlasov-Poisson equations, a description combining mechanics, electricity and magnetism, and it establishes the criterion that the wave rotates in the same direction of the plasma but more slowly.

Further, if the Malmberg-Penning apparatus is modified to include an energy sink, such as resistance on the conducting wall surrounding the plasma, the negative energy wave can dump enough energy to become unstable, and the wave will grow. One example is diocotron waves, which have been shown to destabilize and grow when resistance is added.

Source: “Criterion for the sign of wave energy,” by Thomas M. O’Neil, Physics of Plasmas (2019). The article can be accessed at https://doi.org/10.1063/1.5120401 .