Kinetic Approximation of MHD Equations

The magnetohydrodynamic (MHD) model has found applications in many space and astrophysical plasma problems. Two major assumptions built in the MHD model are the isotropic pressure and the neglect of heat fluxes in the energy equation. In space plasmas the condition of isotropic pressure however is unlikely to be fulfilled due to the lack of sufficient collisions. In particular, in low $\beta$ plasmas the pressure tensor exhibits the gyrotropic form of $\mathsf{P} = p_\perp \vec{I} + \left( p_\parallel - p_\perp \right) \hat{b} \hat{b}$, where $p_\parallel$ and $p_\perp$ are the pressure component parallel and perpendicular to the magnetic field, respectively, and $\hat{b} = {\vec{B}}/{B}$ is a unit vector in the direction of the magnetic field.

In the limit that all fluctuations of interest are at scales larger than the proton Larmor radius and have frequencies much smaller than the proton cyclotron frequency, a collisionless plasma can be described by the ideal hydromagnetic equations with the so-called double-adiabatic of CGL laws, $p_\perp / \rho \vec{B} = C_\perp$ and $p_\parallel B^2/\rho^3 = C_\parallel$, as the energy closure chew56. In the CGL-MHD formulation, the plasma is assumed to be a perfect conductor and the heat flux is neglected. The breakdown of the CGL laws may be attributed to the nonideal MHD effects, including imperfect conducting effects and heat fluxes, etc. hau96.

The conservative form of CGL-MHD equations with the magnetic field normalization (e.g. $\frac{\vec{B}}{\sqrt{4 \pi}}\rightarrow \vec{B}$) can be written as follow:

$$\frac{\partial}{\partial t} \left[ \begin{array}{c} \rho ... ... \vec{V} \vec{B} - \vec{B} \vec{V} \end{array} \right] = 0,$$ (1)

where $\rho$ is the mass density, $\vec{V}$ is the fluid velocity, $\vec{B}$ is the magnetic field, $E = \frac{1}{2} \rho V^2 + \frac{1}{2} B^2 + \frac{1}{2} p_\parallel + p_\perp$ is the total energy, $S_\perp = p_\perp B^{1 - \gamma_\perp}$ is the magnetic moment, and $\hat{b} = \frac{\vec{B}}{B}$ is a unit vector in the direction of the magnetic field.

In addition to the MHD waves, two types of low-frequency hydromagnetic instabilities: fire-hose and mirror instabilities may arise in homogeneous anisotropic plasmas.