Landau fluid equations for electromagnetic and electrostatic fluctuations

Abstract

Closure relations are developed to allow approximate treatment of Landau damping and growth using fluid equations for both electrostatic and electromagnetic modes. The coefficients in these closure relations are related to approximations of the plasma dispersion function by ratios of polynomials. Thirteen different numerical sets of coefficients are given and explicitly related to previous fits to the plasma dispersion function. The application of the techniques presented in this paper is illustrated with the specific example of resistive g modes. Comparisons of full kinetic and approximate results are made for the solutions to the dispersion relation, radially resolved modes in sheared magnetic geometry, and the plasma dispersion function itself.

Fluctuation-dissipation relations for a plasma-kinetic Langevin equation

“…This material and, indeed, most of the results described above, perhaps belong to elementary textbooks, but we are not aware of any where they are adequately explained-although implicitly they underlie the thinking behind both Landau-fluid closures (Hammett & Perkins 1990;Hammett et al 1992;Hedrick & Leboeuf 1992;Snyder et al 1997;Passot & Sulem 2004;Goswami et al 2005;Passot & Sulem 2007) and Hermitespace treatments for plasma kinetics (Armstrong 1967;Grant & Feix 1967;Hammett et al 1993;Parker & Carati 1995;Ng et al 1999;Watanabe & Sugama 2004;Zocco & Schekochihin 2011;Loureiro et al 2013;Hatch et al 2013;Plunk & Parker 2014). …”

“…The simplest Landau-fluid closure Simplistically described, the idea of Landau-fluid closures is to truncate the Hermite hierarchy of equations (4.3-4.5) at some finite m and to replace in the last retained equation 36) where R m+1 , which in general depends on the complex frequency ζ (equation (4.32)), is approximated by some suitable frequency-independent expression leading to the correct recovery of the linear physics from the truncated system. A considerable level of sophistication has been achieved in making these choices and we are not proposing to improve on the existing literature (Hammett & Perkins 1990;Hammett et al 1992;Hedrick & Leboeuf 1992;Snyder et al 1997;Passot & Sulem 2004;Goswami et al 2005;Passot & Sulem 2007). It is, however, useful, in the context of the result of section 3.1 that the "fluid" version of FDR is recovered in the limit of low frequency and weak damping, to show how the same conclusion can be arrived at via what is probably the simplest possible Landau-fluid closure.…”

Fluid models for kinetic effects on coherent nonlinear Alfvén waves. I. Fundamental theory

“…This effect depends on the angle between the wave propagation direction and the ambient magnetic field direction as sin 2 Θ, and vanishes for a strictly parallel-propagating wave. We studied this effect using a gyrofluid model constructed from gyrofluid equations [18,19]. This effect results in additional terms in the amplitude evolution equation.…”

“…In the gyrofluid equations [18,19] there are corrections from both E × B velocity and an effect of gyro-averaging of fields over the Larmor orbit.…”

“…Several authors have suggested that kinetic effects, such as Landau damping may be modelled in fluid equations by adding parallel dissipation terms [15,16]. Recently, a closure method for modelling of kinetic effects was developed [17][18][19][20][21] and applied to iontemperature gradient instabilities using gyrofluid models. This closure anzatz for fluid moment equations i) ensures particle, momentum and energy conservation, ii) takes a simple form in wave-number space, and iii) has a linear response function very close to that of a collisionless, Maxwellian plasma.…”