The Liley form for the stress tensor in magnetized plasma

Abstract:

“Transport Processes in a Plasma” (Braginskii, 1965) is one of the most cited articles in plasma physics. Clarity throughout its English transla- tion provided the Western research community with an early insight into the basis of macroscopic plasma models beyond the ideal or even resistive formulations previously widely used. This talk will describe a significant result derived by Bruce Liley at the Australian National University, but not yet so widely known in the plasma research community. More details are provided in Chapter 2 of Hosking & Dewar, “Fundamental Fluid Me- chanics and Magnetohydrodynamics” (Springer, 2016).

Macroscopic field equations may be obtained by taking moments of a Bolzmann-like kinetic equation, producing a countably infinite hierarchy that must somehow be truncated to render a closed set. Thus familiar field quantities related to the lower moments for each microscopic particle species appear in the basic equations for the mass (continuity), momen- tum, thermal energy, the traceless part of the stress tensor the heat con- duction vector, etc. etc.. An asymptotic closure like Chapman-Enskog for neutral gas transport or one of various parameter ordering procedures for plasmas may be considered, but there is another “careful and elegant” ap- proach (Hazeltine & Meiss, Plasma Confinement, Dover, 2003) – notably the Grad “thirteen moment” approximation, corresponding to a trunca- tion of the hierarchy of moment equations up to the equation for the heat flux.

In particular, the moment equation for the traceless part of the stress tensor can be written in a form that isolates the term proportional to the magnetic field – from which Bruce Liley derived an explicit expression for this important part of the stress tensor which is invariant (i.e. valid in any coordinate system) and applies for any magnetic field strength, including in the neighbourhood of a magnetic null point where his result reduces to the familiar Navier-Stokes form of fluid mechanics. Moreover, in a typical magnetised plasma where the magnetic field is characteristically large, ex- cept near a magnetic null the closed Liley form may be expanded to yield leading terms well known in the literature as the parallel, cross or gyrovis- cous (or FLR) and perpendicular components of the plasma stress tensor. Further expansion of these three components obtained directly from the Liley form produce epressions obtained independently by Jim Callen and co-workers, which in turn exhibit all of the terms in the traceless part of the stress tensor given by Braginskii – i.e. where apart from his collisional coefficients the W0 expression corresponds to the parallel component, the sum of his W1 and W2 to the perpendicular component, and the sum of his W3 and W4 to the gyroviscous component.