Magnetohydrodynamics (MHD)

Magnetohydrodynamics (MHD) is the study of the dynamics of electrically conducting fluids in the presence of magnetic fields. In fusion research, MHD provides the framework for understanding plasma equilibrium, stability, and large-scale dynamics.

Fundamental Concept

MHD treats the plasma as a single conducting fluid rather than tracking individual particles. This approximation is valid when:

1. Length scales are much larger than the ion Larmor radius
2. Time scales are much longer than the ion cyclotron period
3. Collisions are frequent enough to maintain local thermal equilibrium

Under these conditions, the plasma can be described by fluid equations coupled with Maxwell’s equations.

Basic MHD Equations

Continuity Equation

Mass conservation:

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

where $\rho$ is mass density and $\mathbf{v}$ is fluid velocity.

Momentum Equation

The equation of motion including the magnetic force:

$$\rho \frac{d\mathbf{v}}{dt} = -\nabla p + \mathbf{j} \times \mathbf{B}$$

where $p$ is pressure, $\mathbf{j}$ is current density, and $\mathbf{B}$ is magnetic field. The term $\mathbf{j} \times \mathbf{B}$ is the Lorentz force per unit volume.

Ohm’s Law

In ideal MHD (infinite conductivity):

$$\mathbf{E} + \mathbf{v} \times \mathbf{B} = 0$$

In resistive MHD:

$$\mathbf{E} + \mathbf{v} \times \mathbf{B} = \eta \mathbf{j}$$

where $\eta$ is the plasma resistivity.

Energy Equation

For an adiabatic process:

$$\frac{d}{dt}\left(\frac{p}{\rho^\gamma}\right) = 0$$

where $\gamma = 5/3$ is the adiabatic index for a monatomic ideal gas.

Maxwell’s Equations

The relevant Maxwell’s equations (in the MHD approximation where displacement current is neglected):

$$\nabla \times \mathbf{B} = \mu_0 \mathbf{j}$$ $$\nabla \cdot \mathbf{B} = 0$$ $$\frac{\partial \mathbf{B}}{\partial t} = -\nabla \times \mathbf{E}$$

Magnetic Pressure and Tension

The $\mathbf{j} \times \mathbf{B}$ force can be decomposed into:

$$\mathbf{j} \times \mathbf{B} = -\nabla\left(\frac{B^2}{2\mu_0}\right) + \frac{(\mathbf{B} \cdot \nabla)\mathbf{B}}{\mu_0}$$

This reveals two physical effects:

Magnetic Pressure

The term $B^2/(2\mu_0)$ acts as a magnetic pressure. The plasma pushes back against increasing magnetic field strength.

Magnetic Tension

The term $(\mathbf{B} \cdot \nabla)\mathbf{B}/\mu_0$ represents tension along the field lines, like a stretched rubber band. This restores bent field lines toward straightness.

Plasma Beta

The plasma beta $\beta$ is the ratio of plasma pressure to magnetic pressure:

$$\beta = \frac{p}{B^2/(2\mu_0)} = \frac{2\mu_0 p}{B^2}$$

This is a crucial parameter in fusion:

• Low $\beta$ ($\ll 1$): Magnetic pressure dominates; easier to confine but less efficient use of magnetic field
• High $\beta$ ($\sim 1$): More plasma per unit magnetic field; economically desirable but harder to stabilize

Typical values in tokamaks are $\beta \sim 0.01-0.10$ (1-10%).

MHD Equilibrium

Force Balance

In equilibrium, the plasma is stationary ($\mathbf{v} = 0$), so:

$$\nabla p = \mathbf{j} \times \mathbf{B}$$

This fundamental equation states that pressure gradients are balanced by magnetic forces.

Grad-Shafranov Equation

For axisymmetric equilibria (like in tokamaks), the equilibrium condition reduces to the Grad-Shafranov equation:

$$R^2 \nabla \cdot \left(\frac{\nabla \psi}{R^2}\right) = -\mu_0 R^2 \frac{dp}{d\psi} - F\frac{dF}{d\psi}$$

where $\psi$ is the poloidal flux function, $R$ is the major radius, $p(\psi)$ is the pressure profile, and $F(\psi) = RB_\phi$ relates to the toroidal field.

Solving this equation determines the shape and properties of plasma equilibrium in a tokamak.

MHD Stability

Stability Concept

A plasma equilibrium is stable if small perturbations decay or oscillate with bounded amplitude. It is unstable if perturbations grow exponentially.

Energy Principle

Stability can be analyzed using the energy principle. The change in potential energy $\delta W$ for a displacement $\boldsymbol{\xi}$ determines stability:

$$\delta W = -\frac{1}{2}\int \boldsymbol{\xi}^* \cdot \mathbf{F}(\boldsymbol{\xi}) \, d^3r$$

where $\mathbf{F}$ is the force operator. If $\delta W > 0$ for all $\boldsymbol{\xi}$, the equilibrium is stable.

Key Instabilities

Several instabilities are important in fusion plasmas:

Kink Instability

A helical distortion of the plasma column. The safety factor $q$ must satisfy:

$$q = \frac{r B_\phi}{R B_\theta} > 1$$

(Kruskal-Shafranov limit) to avoid the most dangerous external kink mode.

Sausage Instability

Axisymmetric pinching of the plasma column, stabilized by sufficient magnetic shear.

Ballooning Instability

Pressure-driven instability that bulges on the outboard (low-field) side of the torus.

Tearing Mode

Resistive instability that can break and reconnect magnetic field lines, forming magnetic islands.

Ideal vs Resistive MHD

Ideal MHD

In ideal MHD ($\eta = 0$), magnetic field lines are “frozen” into the plasma:

$$\frac{\partial \mathbf{B}}{\partial t} = \nabla \times (\mathbf{v} \times \mathbf{B})$$

The magnetic topology is preserved; field lines move with the plasma.

Resistive MHD

Finite resistivity allows magnetic field diffusion:

$$\frac{\partial \mathbf{B}}{\partial t} = \nabla \times (\mathbf{v} \times \mathbf{B}) + \frac{\eta}{\mu_0}\nabla^2\mathbf{B}$$

This enables magnetic reconnection, which can change field topology and release stored magnetic energy.

Importance for Fusion

Plasma Shaping

MHD equilibrium calculations guide the design of magnetic coils to achieve optimal plasma shapes for confinement and stability.

Operational Limits

MHD stability considerations set operational limits on:

• Maximum plasma pressure (beta limit)
• Minimum safety factor (q-limit)
• Maximum plasma current (disruption avoidance)

Disruptions

Violent MHD instabilities can terminate the plasma discharge in tokamaks, potentially damaging the device. Understanding and avoiding disruptions is critical for fusion reactor design.

Active Control

Real-time MHD control systems use external coils to stabilize certain modes and extend operational boundaries.

Related Topics

• Plasma Physics Overview - Introduction to plasma parameters
• Debye Shielding - Microscopic plasma properties
• Charged Particle Motion - Single-particle dynamics
• Tokamak - Primary application of MHD equilibrium
• Confinement - How MHD enables confinement