Magnetohydrodynamics (MHD)

Magnetohydrodynamics, or MHD for short, is a theory that regards a plasma as a single “electrically conducting fluid” and describes how that fluid pushes against a magnetic field. Whether a plasma inside a fusion device can float stably or instead collapses and slams into the wall is determined by MHD. On this page we first build intuition through everyday analogies, then work our way up through the basic equations, magnetic pressure and magnetic tension, the frozen-in theorem, plasma beta, equilibrium and stability, and finally to research frontiers such as magnetic reconnection and simulation.

Start with Intuition (High School)

A plasma is a gas in which atoms have been broken apart into electrons and ions. Think of it as a gas that conducts electricity. The starting point of MHD is a simplification: we view this conducting gas as “a single fluid,” something continuous like water or air. In reality, ions and electrons are flying around separately, but on a large scale they move like a single jelly.

What makes this jelly special is that it becomes strongly entangled with the magnetic field. A magnetic field is usually pictured as a bundle of invisible “field lines.” They are the streaked pattern that appears when you scatter iron filings around a bar magnet. In the world of MHD, these field lines pierce the plasma and cannot come loose. If the plasma moves, the field lines are dragged along with it; if the field lines move, the plasma is carried along too. It is as though the field lines and the plasma are stitched together with thread. This is called being frozen-in.

Magnetic field lines have two “personalities.” One is that neighboring field lines dislike being crowded together. This acts like the pressure of an air-filled balloon trying to expand outward. This is magnetic pressure. The other is that a bent field line tries to straighten back out. Picture a taut rubber band or a string. When you bend it, it tries to return to its original shape. This is magnetic tension.

What a fusion device wants to do is confine an ultra-hot plasma exceeding a hundred million degrees, floating it in midair without letting it touch the wall. If it touches the wall, the plasma cools down and the wall is damaged. So we build a “cage” made of magnetic field lines and use this magnetic pressure and magnetic tension to hold the hot plasma inside. The plasma tries to expand outward, and the magnetic field pushes back inward. The state in which this tug-of-war is exactly balanced is called equilibrium. And whether, when nudged slightly, the plasma returns to its original state or the disturbance grows and collapses is the problem of stability. MHD is the theory that handles this tug-of-war of confinement and the ways it can collapse.

Understand the Physics (Undergraduate)

Let us check the conditions under which MHD is valid in the first place. A plasma can be regarded as a fluid when the spatial scale of the phenomenon of interest is much larger than the ion gyroradius (ion Larmor radius) and the time scale is much longer than the ion gyration period (ion cyclotron period). The circular motion of each individual particle (see Motion of Charged Particles for details) is averaged out and blurred, leaving only the collective fluid-like behavior. This “coarse-grained” picture is MHD. Furthermore, since the phenomena are slow compared to the speed of light, we may neglect the displacement current, which is another underlying assumption.

Within this picture, a plasma is described by combining the fluid equations with Maxwell’s equations. The basic system of equations is as follows. First, the continuity equation, which expresses conservation of mass.

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

Here $\rho$ is the mass density and $\mathbf{v}$ is the fluid velocity. This equation states the obvious fact that “the mass in a region increases by the amount that flows in and decreases by the amount that flows out.”

Next is the equation of motion, which expresses the balance of forces.

$$\rho \left( \frac{\partial \mathbf{v}}{\partial t} + \mathbf{v} \cdot \nabla \mathbf{v} \right) = -\nabla p + \mathbf{J} \times \mathbf{B}$$

The left-hand side is the mass times the acceleration of the plasma fluid, that is, the “mass times acceleration” of Newton’s equation of motion. On the right-hand side, the first term $-\nabla p$ is the force that pushes from high pressure toward low pressure (the pressure-gradient force), and the second term $\mathbf{J} \times \mathbf{B}$ is the Lorentz force arising from the current $\mathbf{J}$ and the magnetic field $\mathbf{B}$. The tug-of-war between these two forces determines the motion of the plasma.

The third is the energy equation, which in many cases is closed by the adiabatic relation between pressure and density, $p \rho^{-\gamma} = \text{constant}$ (where $\gamma$ is the specific-heat ratio). The fourth is Ohm’s law.

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

$\mathbf{E}$ is the electric field and $\eta$ is the electrical resistivity. The case in which the resistance on the right-hand side can be taken as zero is called ideal MHD, and the case in which it cannot be neglected is called resistive MHD. Finally, of Maxwell’s equations, Faraday’s law $\partial \mathbf{B} / \partial t = -\nabla \times \mathbf{E}$ and Ampère’s law $\nabla \times \mathbf{B} = \mu_0 \mathbf{J}$ are added, and the system of equations is closed.

Here, if we rewrite the Lorentz force $\mathbf{J} \times \mathbf{B}$ using Ampère’s law so that it is expressed in terms of the magnetic field alone, it separates neatly into the two forces described in the intuition section.

$$\mathbf{J} \times \mathbf{B} = -\nabla \left( \frac{B^2}{2\mu_0} \right) + \frac{1}{\mu_0} (\mathbf{B} \cdot \nabla) \mathbf{B}$$

The first term is the magnetic pressure, whose magnitude $p_B = B^2 / (2\mu_0)$ is proportional to the square of the magnetic field strength and pushes isotropically in all directions. The second term is the magnetic tension, which acts only when a field line is bent and tries to straighten out the bend. $\mu_0$ is the permeability of vacuum.

The ratio of the plasma pressure to the magnetic pressure is an important indicator of confinement efficiency, called the plasma beta.

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

$\beta$ indicates “how much hot plasma can be confined with a given magnetic field.” The larger it is, the more economical, but if it is too large the plasma becomes unstable. In tokamaks it is typically a few percent.

Deepen the Theory (Graduate)

Let us start with equilibrium. In a steady state ($\partial / \partial t = 0$) with no flow ($\mathbf{v} = 0$), the equation of motion reduces to just the balance of forces.

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

Taking the inner product of this equation with $\mathbf{B}$ gives $\mathbf{B} \cdot \nabla p = 0$, which tells us that the pressure does not change along a field line. Similarly, we can derive $\mathbf{J} \cdot \nabla p = 0$, so both field lines and current lines lie on the same constant-pressure surface. This surface is called a flux surface, and we obtain a picture in which the plasma is confined in layers as nested surfaces of constant pressure.

In an axisymmetric system like a tokamak, this equilibrium reduces to a single two-dimensional partial differential equation called the Grad-Shafranov equation.

$$\Delta^* \psi = -\mu_0 R^2 \frac{dp}{d\psi} - F \frac{dF}{d\psi}$$

Here $\psi$ is the poloidal magnetic flux, $R$ is the coordinate in the major-radius direction, $F$ is a quantity related to the toroidal magnetic field, and $\Delta^*$ is an elliptic differential operator. Once the two free functions on the right-hand side, the pressure profile $p(\psi)$ and $F(\psi)$ (which corresponds to the current profile), are given, the shape of the flux surfaces is determined. In tokamak design, solving this equation numerically to design the equilibrium is the starting point.

Even when an equilibrium is found, it is not necessarily stable. The energy principle judges stability in a unified way. Consider displacing the plasma from equilibrium by a small displacement $\boldsymbol{\xi}$, and examine the change in the potential energy of the system, $\delta W[\boldsymbol{\xi}]$. If $\delta W > 0$ for every possible displacement, then energy increases no matter how you displace it (you are at the bottom of a valley), so the state is stable. Conversely, if even one displacement exists for which $\delta W < 0$, the state is unstable and collapses in that direction while spontaneously lowering its energy. $\delta W$ can be written as the sum of terms that bend the field lines and compress the magnetic field (these act to stabilize) and terms driven by the pressure gradient and the current (these act to destabilize), and which terms win determines the stability.

Let us list some representative instabilities. The kink instability is a deformation in which the plasma column twists into a helix, and it tends to occur when the safety factor $q$ (an indicator of how much the field lines twist during one trip around the torus) drops below 1. The ballooning instability is driven by the pressure gradient and bulges locally on the outer side of the torus, where the curvature of the field lines is unfavorable. The tearing mode is a resistive instability in which the flux surfaces are torn apart by the reconnection described later, creating a break in the nested structure known as a magnetic island. The detailed classification of these is covered in Plasma Instabilities. Such instabilities set the beta limit (Troyon limit) and the density limit (Greenwald limit), and trigger disruptions in which confinement is suddenly lost.

Here the difference between ideal MHD and resistive MHD becomes essential. In ideal MHD ($\eta = 0$), the frozen-in theorem holds rigorously. Combining Ohm’s law and Faraday’s law, it can be shown that the magnetic flux through any closed curve does not change with time, so the field lines move as one with the plasma and never reconnect. But if there is even a slight resistivity $\eta$, this constraint comes loose. In thin layers where the flux surfaces bunch together and the current concentrates (current sheets), the resistive term becomes effective even if $\eta$ is small, and the field lines can reconnect. This is magnetic reconnection. When reconnection occurs, field lines that were previously separate reconnect, and the magnetic energy that had been stored is converted all at once into kinetic energy and heat. The sawtooth oscillation in tokamaks and the explosive energy release of solar flares are also explained by this reconnection. Whereas ideal MHD tells us “whether it collapses or not,” resistive MHD tells us “how, and how fast, it reconnects and collapses.”

Research Frontier (PhD)

Modern MHD research has shifted its center of gravity beyond linear stability analysis to the nonlinear domain, following what happens after an instability has fully grown. At the center of this is nonlinear MHD simulation. The resistive MHD equations, or the extended MHD equations obtained by adding two-fluid effects (corrections that treat electrons and ions separately), are solved numerically as three-dimensional, time-evolving problems on a torus geometry. Codes such as JOREK, M3D-C1, and NIMROD are used internationally, and they attempt to reproduce the violent phenomena observed in experiments, such as the entire course of a disruption, the eruption of edge-localized modes (ELMs), and the growth of magnetic islands.

Let us list some active research topics. The prediction and mitigation of disruptions is a problem that becomes more serious as devices grow larger. In large devices like ITER, a single disruption imposes a large thermal and mechanical load on the wall, so the generation mechanism of runaway electrons and mitigation methods that inject impurity gas to dissipate the energy safely are being vigorously researched. The neoclassical tearing mode (NTM) is a resistive mode that grows spontaneously as the pressure gradient is lost inside a magnetic island, and active control that injects electron cyclotron current drive (ECCD) aimed at the location of the island is being researched and demonstrated. The resistive wall mode (RWM) is a mode in which the stabilizing effect of the wall is lost in a finite time because of the resistance of the conducting wall, and the interplay of control by feedback coils and stabilization by plasma rotation is being studied.

The methodology of theory and computation itself is also advancing. MHD is originally a framework that averages out and discards the kinetic effects of particles, but in reality fast ions and resonant particles couple strongly with MHD modes. So gyrokinetic MHD, which connects fluid MHD with kinetic theory, and hybrid simulations (methods that superimpose kinetic fast particles on a fluid background) have been developed, and topics such as the interaction between Alfvén eigenmodes and fusion-produced alpha particles are being studied. For reconnection as well, how to model the bridge between the MHD scale and the tiny scale at which reconnection actually occurs remains a theoretical focus. When you read papers, keywords such as ideal/resistive MHD, Grad-Shafranov, energy principle, Newcomb’s equation, $\delta W$, tearing mode, reconnection rate, and extended/two-fluid MHD appear frequently. These numerical and theoretical results are ultimately fed back into the design of operating scenarios for real machines such as the tokamak.

Check Your Understanding

Q1. In MHD, what state does the phrase 'the field lines are frozen into the plasma' describe by analogy?

A A state in which the field lines melt and vanish from the heat of the plasma B A state in which the field lines and the plasma move as one, as if stitched together with thread, so that if one moves the other is carried along too C A state in which the field lines avoid the plasma and are distributed only on the outside D A state in which the plasma cools and freezes solid

Q2. What kinds of forces are magnetic pressure and magnetic tension? Explain them with everyday analogies.

A Magnetic pressure is a contracting force like a rubber band, and magnetic tension is an expanding force like a balloon B Both are the same kind of force that pushes the field lines in one direction C Magnetic pressure is a balloon-like force by which crowded field lines try to spread outward, and magnetic tension is a rubber-band-like force by which bent field lines try to straighten out D Magnetic pressure is a force proportional to gravity, and magnetic tension is a force proportional to the electric field

Q3. Plasma beta beta is the ratio of what to what, and what are the advantages and disadvantages of it being large?

A The ratio of electric field to magnetic field; the larger it is, the higher the heating efficiency, but the more expensive the device B The ratio of plasma pressure to magnetic pressure; the larger it is, the more economical, but if it is too large it invites MHD instabilities C The ratio of ion temperature to electron temperature; the larger it is, the higher the reaction rate, but the harder it is to measure D The ratio of plasma density to neutral-gas density; the larger it is, the better the confinement, but the harder it is to heat

Q4. What physical balance does the Grad-Shafranov equation, which describes tokamak equilibrium, represent?

A The balance of the energy budget between plasma heating and radiative cooling B The balance of the charge-neutrality condition between ions and electrons C The relativistic balance between particles near the speed of light and the magnetic field D The balance of forces in a flowless steady state (the balance between the pressure-gradient force and the Lorentz force), rewritten for an axisymmetric torus

Q5. What is the essential difference between ideal MHD and resistive MHD, and what is a representative phenomenon that arises from that difference?

A Ideal MHD has zero resistivity so field lines do not reconnect, whereas in resistive MHD the field lines reconnect and magnetic reconnection occurs B Ideal MHD applies only to hot plasma and resistive MHD only to cold plasma; the difference is just the temperature range C Ideal MHD can handle only two-dimensional phenomena and resistive MHD only three-dimensional ones D Ideal MHD ignores the magnetic field, and only resistive MHD accounts for it

Related Topics

• Plasma Physics Overview
• Motion of Charged Particles
• Plasma Instabilities
• Tokamak