MHD(Magnetohydrodynamics) simulation

The complete description of plasma system is given by Vlasov equation (5) and Maxwell equations (6)-(9).

Vlasov Equation

$$\frac{\partial f}{\partial t}+\textbf{u}\cdot\nabla f+\frac{\... ...{F}}{m} \nabla_{\textbf{u}}f=(\frac{\partial f}{\partial t})_c$$ (5)

where f is the particle distribution function, u particle velocity, F is the force density exerted on the particles. The right hand side of the equation is the changing of particle distribution according to particle collisions.

Maxwell Equations

$$\nabla\times\textbf{E}=-\frac{\partial\textbf{B}}{\partial t}$$ (6)

$$\nabla\times\textbf{B}=\mu_0\textbf{J}+\mu_0\epsilon_0\frac{\partial\textbf{E}}{\partial t}$$ (7)

$$\nabla\cdot\textbf{E}=\frac{\rho}{\epsilon_0}$$ (8)

$$\nabla\cdot\textbf{B}=0$$ (9)

where B is the magnetic flux density, E the electric field, J the current density, the charge density.

In contrast to the description of each particle in a particle simulation, an MHD simulation describes the plasma as a fluid. When the time scale of interest is much longer than the longest plasma kinetic time scale(proton gyroperiod), and the space scale is much larger than the largest kinetic space scale(proton gyroradius), we can use integrated quantities to replace the parameters of each particle.

Mass density:

$$\rho(\textbf{x})=m\int f(\textbf{x},\textbf{v})d\textbf{v}$$ (10)

Charge density:

$$\rho_e(\textbf{x})=q\int f(\textbf{x},\textbf{v})d\textbf{v}$$ (11)

Velocity:

$$\textbf{u}=\frac{m}{\rho(\textbf{x})}\int \textbf{v}f(\textbf{x} ,\textbf{v})d\textbf{v}$$ (12)

Other parameters of the system can also be expressed in the similar form. Here we assume the velocity distributions of all the particles are not far away from a Maxwellian distribution. One thing we should notice here is that, since all the system parameters are integrated quantities, we are unable to determine the particle velocity distribution now.

After certain approximations, we can change the control equations(5)-(9) to the MHD form

$$\frac{\partial \rho}{\partial t}+\nabla\cdot(\rho\textbf{v})=0$$ (13)

$$\rho\frac{d\textbf{u}}{dt}=\textbf{j}\times\textbf{B}-\nabla p$$ (14)

$$\textbf{j}=\sigma(\textbf{E}+\textbf{u}\times\textbf{B})$$ (15)

$$\nabla\times\textbf{B}=\mu_0\textbf{j}+\mu_0\epsilon_0\frac{\partial\textbf{E}}{\partial t}$$ (16)

$$\nabla\times\textbf{E}=-\frac{\partial\textbf{B}}{\partial t}$$ (17)

$$\nabla\cdot\textbf{B}=0$$ (18)

To make the equations complete, we need to add another equation. It can be one of the following:

Incompressible fluid:

$$\nabla\cdot\textbf{u}=0$$ (19)

Adiabatic approximation:

$$\frac{d}{dt}(P\rho^{-\gamma})=0$$ (20)

where .

Isothermal approximation:

$$\frac{d}{dt}(P/\rho)=0$$ (21)