Ambipolar damping

Ambipolar damping is attenuation of magnetic disturbances by friction between the magnetically coupled charged fluid and a neutral fluid. In a strongly collisional mixture, the fluids nearly move together, but their small relative velocity dissipates energy. With drag coefficient ${\rho }_i{\mu }_{in}={\rho }_n{\mu }_{ni}$, the frictional energy loss is $-{\rho }_i{\mu }_{in}\int |{\mathbf{u}}_i-{\mathbf{u}}_n{|}^{2}dV$. Neglecting other dissipative processes does not remove this collisional loss.

For equal ion and neutral mass densities, let $\mu ={\mu }_{in}={\mu }_{ni}$, $a=k_{\parallel }{B}_0/\sqrt{4\pi {\rho }_i}$ and use Fourier time dependence $e^{-i\omega t}$. Solenoidal displacements obey ${\ddot{\xi }}_i=-a^2{\xi }_i-\mu ({\dot{\xi }}_i-{\dot{\xi }}_n)$ and ${\ddot{\xi }}_n=-\mu ({\dot{\xi }}_n-{\dot{\xi }}_i)$. For nonzero frequency, the neutral equation gives ${\xi }_i=(1-i\omega /\mu ){\xi }_n$. Substitution in the ion equation givesEquivalently, the growth rate $s=-i\omega$ obeys $s^3+2\mu s^2+a^{2}s+\mu a^2=0$. A time-independent neutral displacement with zero velocity is a relabeling of a uniform fluid, not an additional wave root of the velocity/magnetic-field system.

For $|a|\ll \mu$, the equal-density ion-neutral cubic has a rapidly damped mode ${\omega }_d=-2i\mu +i{a}^2/(4\mu )+O(a^4/{\mu }^3)$ and two propagating modes ${\omega }_{\pm }=\pm a/\sqrt{2}-i{a}^2/(8\mu )+O(a^3/{\mu }^2)$. The reduced phase speed uses the total density ${\rho }_i+{\rho }_n=2{\rho }_i$. The exact displacement ratio is ${\xi }_n/{\xi }_i=\mu /(\mu -i\omega )$. It is nearly $-1$ for the rapidly damped counterflow and nearly $1\pm ia/(\sqrt{2}\mu )$ for the propagating modes. Thus the latter have small ion-neutral slippage of relative size $|a|/\mu$, responsible for ambipolar damping.