MHD equations

The time evolution of the fluid velocity $\mathbf{u}$ is governed by the momentum equation

$$\rho \left( \frac{\partial \mathbf{u}}{\partial t} +\mathbf{u} \c... ...\mathbf{u} \right) = \rho_q \mathbf{E}- \nabla p +\mathbf{J} \times \mathbf{B},$$ (1)

where $\rho$, $\rho_q$, $p$, $\mathbf{J}$, $\mathbf{E}$, and $\mathbf{B}$ are mass density, charge density, thermal pressure, current density, electric field, and magnetic field, respectively. The time evolution of the mass density $\rho$ is governed by the mass continuity equation

$$\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{u}) = 0.$$ (2)

The time evolution equation for pressure $p$ is given by the equation of state

$$\frac{d}{d t} (p \rho^{- \gamma}) = 0,$$ (3)

where $\gamma$ is the ratio of specific heats. The time evolution of $\mathbf{B}$ is given by Faraday's law

$$\frac{\partial \mathbf{B}}{\partial t} = - \nabla \times \mathbf{E}.$$ (4)

The current density $\mathbf{J}$ can be considered as a derived quantity, which is defined through Ampere's law (the displacement current being ignored)

$$\mathbf{J}= \frac{1}{{\textmu}_0} \nabla \times \mathbf{B}$$ (5)

The electric field $\mathbf{E}$ is considered to be a derived quantity, which is defined through Ohm's law

$$\mathbf{E}= -\mathbf{u} \times \mathbf{B}+ \eta \mathbf{J}.$$ (6)

The charge density $\rho_q$ can be considered to be a derived quantity, which is defined through Poisson's equation,

$$\rho_q = \varepsilon_0 \nabla \cdot \mathbf{E}.$$ (7)

The above equations constitute a closed set of equations for the time evolution of four quantities, namely, $\mathbf{B}$, $\mathbf{u}$, $\rho$, and $p$ (the electric field $\mathbf{E}$, current density $\mathbf{J}$, and charge density $\rho_q$ are eliminated by using Eqs. (5), (6), and (6)). In addition, there is an equation governing the spatial structure of the magnetic field, namely

$$\nabla \cdot \mathbf{B}= 0.$$ (8)

In summary, the MHD equations can be categorized into three groups of equations, namely[1],

1. Evolution equations for base quantities $\mathbf{B}$, $\mathbf{u}$, $\rho$, and $p$:

   $\frac{\partial \mathbf{B}}{\partial t} = - \nabla \times (-\mathbf{u} \times ... ... \times (\mathbf{u} \times \mathbf{B}) + \eta \nabla^2 \mathbf{B}/{\textmu}_0$

   $\rho \left( \frac{\partial \mathbf{u}}{\partial t} +\mathbf{u} \cdot \nabla \... ...athbf{E}- \nabla p + (\nabla \times \mathbf{B}) /{\textmu}_0 \times \mathbf{B}$

   $\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{u}) = 0$,

   $\frac{d}{d t} (p \rho^{- \gamma}) = 0$,

2. Equation of constraint: $\nabla \cdot \mathbf{B}= 0$.

3. Definitions: (i.e., they are considered to be derived quantities.)

   $\mathbf{E}= \eta (\nabla \times \mathbf{B}) /{\textmu}_0 -\mathbf{u} \times \mathbf{B}$,

   $\mathbf{J}= (\nabla \times \mathbf{B}) /{\textmu}_0$,

   $\rho_q = \varepsilon_0 \nabla \cdot \mathbf{E}$.

The electrical field term $\rho_q \mathbf{E}$ in the momentum equation (1) is usually neglected because this term is usually much smaller than other terms for low-frequency phenomena in tokamak plasmas.