MHD Astrophysics

Mathematical Models

1. MHD Continuity Equation: $\( \frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{v}) = 0 \)\( where \)\rho\( is the fluid density, and \)\mathbf{v}$ is the fluid velocity.

2. MHD Momentum Equation: $\( \rho \frac{D\mathbf{v}}{Dt} = -\nabla p + \rho \mathbf{g} + \nabla \cdot \boldsymbol{\tau} + \mathbf{J} \times \mathbf{B} + \nu \nabla^2 \mathbf{v} \)\( incorporating the viscous term (\)\nu \nabla^2 \mathbf{v}$) for fluid viscosity.

3. MHD Induction Equation: $\( \frac{\partial \mathbf{B}}{\partial t} = \nabla \times (\mathbf{v} \times \mathbf{B} - \eta \nabla \times \mathbf{B}) \)\( where \)\eta$ is the magnetic diffusivity.

4. MHD Ohm’s Law: $\( \mathbf{J} = \sigma (\mathbf{E} + \mathbf{v} \times \mathbf{B}) \)\( with \)\sigma\( as the electrical conductivity and \)\mathbf{E}$ as the electric field.

5. MHD Energy Equation: $\( \rho C_p \frac{DT}{Dt} = -p \nabla \cdot \mathbf{v} + \nabla \cdot (k \nabla T) + \frac{1}{\sigma}(\mathbf{J} \cdot \mathbf{E}) \)\( including heat conduction (\)k \nabla T\() and Joule heating (\)\frac{1}{\sigma}(\mathbf{J} \cdot \mathbf{E})$) terms.