Fundamentals of Plasma Physics 2025, 32 - Warm, Magnetized, Electrostatic Dispersion
An alternate limit, allowing for k||, with k2⊥r2L << data-preserve-html-node="true" 1, thus retaining the lowest-order finite Larmor radius terms. k|| is assumed small enough so ω/k|| >> vTe, vTi. The lowest-order finite-temperature terms of the dispersion relation define a thermal coefficient.
For large S, the modes split into cold and hot plasma waves. The WKB approximation fails at mode coalescence, as dk⊥/dx = ∞. Linear mode conversion is analyzed by the Airy equation y" + xy = 0 split into the x < 0, and x > 0 region. It resolves to y(x) = ∫C ef(p, x)dp. Saddle-point solutions at large |x| obtained by d/dx → ik solves to k2 = x and the WKB mode corresponds to a WKB mode.
More realistic plasma have finite extent and pressure gradients along their volume. This case is inspected by way of 2-fluid analysis and Vlasov analysis. The 2-fluid analysis uncovers a diamagnetic drift velocity along the azimuthal directional, and an associated current. Said current happens to be associated with the MHD equilibrium equation, which can still be disassembled into the ion and electron portions. The diamagnetic velocity does not apply to individual particle, and is more a collective quantity. The Vlasov analysis extends this into the pressure gradient.
A simple two-fluid model of these drift waves assumes an exponential density gradient, defined by its length scale, and gives both an ion density perturbation in the continuity equation, and a most basic form of drift wave dispersion relations.
Including collisions modifies the equations of motion through a collision term, characterized through a collision frequency. It turns out that collisional drift waves are always unstable, and they have a diffusion coefficient of
At energies high enough that collision frequencies become insignificant, then by Vlasov analysis, take a perturbed distribution function evaluated along unperturbed orbits, which takes into account the density gradient.
The dispersion relation shows that the ions are adiabatic and electrons are isothermal, and the cyclotronic frequencies are much larger than the collision relations, and the Larnor orbit radius for electrons is negligible compared to the perpendicular wavelength. Further integration leads to expressions for the collisionless destabilization mechanisms, namely normal Landau damping, and the current.