Magnetohydrodynamics

Magnetohydrodynamics (MHD) is a theory that regards a plasma as a single electrically conducting fluid and treats the forces that the flow and the magnetic field exert on each other. It is the foundation for studying the equilibrium and stability of fusion plasmas.

Simple Definition (High School)

A plasma is a collection of countless charged particles, but tracking every single one of them is overwhelming. So magnetohydrodynamics takes the view that we can treat the whole thing as “a single flowing body,” like water or air. Unlike ordinary water, however, this fluid conducts electricity, so it is strongly affected by magnetic fields.

Within a magnetic field, the fluid behaves like beads threaded onto the magnetic field lines. It helps to think of the field lines as having two kinds of forces: a “pushing-back” force and a “pulling-back” force. The sideways force that crowded field lines exert on one another is the magnetic pressure, and the force by which taut field lines try to return to their original shape is the magnetic tension. When these two forces balance with the plasma’s own pressure, the plasma is confined stably.

Precise Definition (Undergraduate and Above)

Magnetohydrodynamics incorporates the Lorentz force into the fluid equations of motion and couples them with Maxwell’s equations. The momentum equation can be written as follows.

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

This reads as “a fluid of density $\rho$ accelerates due to the sum of the pressure gradient $-\nabla p$ and the force $\mathbf{J} \times \mathbf{B}$ produced by the current density $\mathbf{J}$ and the magnetic flux density $\mathbf{B}$.”

This force term can be decomposed into magnetic pressure and magnetic tension.

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

The first term is the gradient of the magnetic pressure $B^2/(2\mu_0)$, representing the effect of field lines pushing against one another sideways. The second term is the magnetic tension, representing the effect of bent field lines trying to straighten back out. Here $\mu_0$ is the permeability of vacuum. The ratio of the plasma pressure to the magnetic pressure is called the plasma beta ($\beta$), and it serves as a measure of confinement efficiency.

Role in Fusion Research

In fusion, the starting point of design is to find a state in which the plasma can rest stably, that is, the MHD equilibrium. The magnetic field configurations of tokamaks and stellarators are determined by solving for the shape of this equilibrium. In addition, MHD stability analysis, which asks whether that equilibrium survives against small perturbations, is indispensable. When stability is lost, the plasma deforms drastically, leading in a tokamak to a sudden loss of confinement such as a disruption. Although it is a simplified approximation, magnetohydrodynamics is a central tool that underpins device design and the estimation of operational limits.

Learn More

• Magnetohydrodynamics
• Instabilities

Related Terms

• Plasma - The subject that magnetohydrodynamics treats
• Plasma Beta - The ratio of plasma pressure to magnetic pressure
• Safety Factor - An indicator that governs MHD stability
• Disruption - A sudden loss of confinement due to loss of stability