Part 3, Chapter 2
MHD Equations
Ideal and resistive magnetohydrodynamics
2.1 The MHD Approximation
MHD treats plasma as a single conducting fluid. Key assumptions:
- Quasi-neutrality: ne β Zni (charge separation negligible)
- Low frequency: Ο βͺ Ξ©ci (slower than ion cyclotron)
- Large scales: L β« Οi, Ξ»D (larger than ion gyroradius)
- Non-relativistic: u βͺ c
2.2 Ideal MHD Equations
Mass Conservation
$$\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{u}) = 0$$
Momentum Equation
$$\rho\left(\frac{\partial \mathbf{u}}{\partial t} + \mathbf{u} \cdot \nabla \mathbf{u}\right) = \mathbf{J} \times \mathbf{B} - \nabla p$$
Adiabatic Energy
$$\frac{d}{dt}\left(\frac{p}{\rho^\gamma}\right) = 0 \quad \Rightarrow \quad p \propto \rho^\gamma$$
Faraday's Law
$$\frac{\partial \mathbf{B}}{\partial t} = -\nabla \times \mathbf{E}$$
Ampère's Law (MHD limit)
$$\nabla \times \mathbf{B} = \mu_0 \mathbf{J}$$
Ideal Ohm's Law
$$\mathbf{E} + \mathbf{u} \times \mathbf{B} = 0$$
2.3 Magnetic Induction Equation
Combining Faraday's law with ideal Ohm's law gives the induction equation:
$$\frac{\partial \mathbf{B}}{\partial t} = \nabla \times (\mathbf{u} \times \mathbf{B})$$
Ideal MHD Induction Equation
Frozen-in Flux Theorem: In ideal MHD, magnetic field lines move with the fluid. Magnetic flux through any surface moving with the plasma is conserved.
2.4 Resistive MHD
Including resistivity Ξ· in Ohm's law:
$$\mathbf{E} + \mathbf{u} \times \mathbf{B} = \eta \mathbf{J}$$
Resistive Ohm's Law
The induction equation becomes:
$$\frac{\partial \mathbf{B}}{\partial t} = \nabla \times (\mathbf{u} \times \mathbf{B}) + \frac{\eta}{\mu_0}\nabla^2 \mathbf{B}$$
Magnetic Reynolds Number
$$R_m = \frac{\mu_0 u L}{\eta} = \frac{\text{advection}}{\text{diffusion}}$$
Rm β« 1: ideal MHD valid | Rm ~ 1: resistive effects important
2.5 MHD Force Balance
The Lorentz force can be written as:
$$\mathbf{J} \times \mathbf{B} = \frac{(\nabla \times \mathbf{B}) \times \mathbf{B}}{\mu_0} = -\nabla\left(\frac{B^2}{2\mu_0}\right) + \frac{(\mathbf{B} \cdot \nabla)\mathbf{B}}{\mu_0}$$
Magnetic Pressure
$$p_B = \frac{B^2}{2\mu_0}$$
Isotropic pressure from B-field
Magnetic Tension
$$\frac{(\mathbf{B} \cdot \nabla)\mathbf{B}}{\mu_0}$$
Tension along curved field lines
Key Takeaways
- β Ideal MHD: E + uΓB = 0 β frozen-in flux
- β Resistive MHD adds Ξ·βΒ²B diffusion term
- β Magnetic Reynolds number Rm determines regime
- β Lorentz force = magnetic pressure + tension