Communications in Mathematical Sciences
Volume 14 (2016)
Lagrangian averaged gyrokinetic-waterbag continuum
Pages: 593 – 626
In this paper, we first present the derivation of the anisotropic Lagrangian averaged gyrowaterbag continuum (LAGWBC-$\alpha$) equations. The gyrowaterbag (short for gyrokinetic-waterbag) continuum can be viewed as a special class of exact weak solution of the gyrokinetic-Vlasov equation, allowing us to reduce the latter into an infinite-dimensional set of hydrodynamic equations while keeping its kinetic features, such as Landau damping. In order to obtain the LAGWBC-$\alpha$ equations from the gyrowaterbag continuum we use an Eulerian variational principle and Lagrangian averaging techniques introduced by Holm, Marsden, and Ratiu [27, 28], Marsden and Shkoller [32, 33] for the mean motion of ideal incompressible flows, extended to barotropic compressible flows by Bhat et al.  and some supplementary approximations for the electrical potential fluctuations. Regarding the original gyrowaterbag continuum, the LAGWBC-$\alpha$ equations show some additional properties and several advantages from the mathematical and physical viewpoints, which make this model a good candidate for accurately describing gyrokinetic turbulence in magnetically confined plasma. In the second part of this paper, we prove local-in-time well-posedness of an approximate version of the anisotropic LAGWBC-$\alpha$ equations, which we call the isotropic LAGWBC-$\alpha$ equations, by using quasilinear PDE type methods and elliptic regularity estimates for several operators.
gyrokinetic-waterbag model, gyrowaterbag model, well-posed problem, gyrokinetic turbulence, Lagrangian averaged models, Eulerian and Lagrangian variational principles, gyrokinetic-Vlasov equations, multi-fluids systems, infinite-dimensional hyperbolic system of conservation laws in several space dimension, magnetically confined fusion plasmas
2010 Mathematics Subject Classification
35F55, 35L65, 35Q35, 35Q83, 76B03, 76F02, 76N10