# Magnetohydrodynamics

Post-publication activity

Curator: Søren Bertil F. Dorch

Magnetohydrodynamics (MHD) is the physical-mathematical framework that concerns the dynamics of magnetic fields in electrically conducting fluids, e.g. in plasmas and liquid metals. The word magnetohydrodynamics is comprised of the words magneto- meaning magnetic, hydro- meaning water (or liquid) and -dynamics referring to the movement of an object by forces. Synonyms of MHD that are less frequently used are the terms magnetofluiddynamics and hydromagnetics.

One of the most famous scholars associated with MHD was the Swedish physicist Hannes Alfvén (1908-1995), who received the Nobel Prize in Physics (1970) for fundamental work and discoveries in magnetohydrodynamics with fruitful applications in different parts of plasma physics.

The central point of MHD theory is that conductive fluids can support magnetic fields. The presence of magnetic fields leads to forces that in turn act on the fluid (typically a plasma), thereby potentially altering the geometry (or topology) and strength of the magnetic fields themselves. A key issue for a particular conducting fluid is the relative strength of the advecting motions in the fluid, compared to the diffusive effects caused by the electrical resistivity. Other topics belonging to the fundamental framework of magnetohydrodynamics include, e.g. MHD turbulence, MHD waves (Alfvén waves), magneto-convection, MHD reconnection, and hydromagnetic dynamo theory.

Figure 1: Three-dimensional visualization of the magnetic field resulting from a numerical MHD simulation of hydromagnetic dynamo action in the red supergiant star "Betelgeuse" (field lines in white and temperature surface in red). Adapted from [1]

## The MHD equations

MHD is a macroscopic theory. The partial differential equations of MHD can in principle be derived from Boltzmann's equation assuming space and time scales to be larger than all inherent scale-lengths such as the Debye length or the gyro-radii of the charged particles. It is, however, more convenient to obtain the MHD equations in a phenomenological way as the electromagnetic extension of the hydrodynamic equations of ordinary fluids, where the main approximation is to neglect the displacement current

In the standard nonrelativistic form the MHD equations consist of the basic conservation laws of mass, momentum and energy together with the induction equation for the magnetic field. The equations are, written in SI units, $\tag{1} \frac{\partial \rho}{\partial t} +\nabla \cdot \rho \vec u = 0,$

where $$\rho$$ is the mass density and $$\vec u$$ the fluid bulk velocity;

the equation of motion, $\tag{2} \frac{\partial (\rho \vec u)}{\partial t} + \nabla \cdot (\rho\vec u\vec u)= -\nabla p + \vec j \times \vec B + \nabla\cdot\sigma,$

where $$p$$ is the gas pressure, $$\vec B$$ the magnetic field (properly speaking, the magnetic flux density), $$\vec j= \nabla\times \vec B/ \mu_0$$ the current density, $$\mu_0$$ the vacuum permeability, and $$\sigma$$ is the viscous stress tensor;

the equation for the internal energy, which is usually written as an equation for the pressure $$p\ ,$$ $\tag{3} \frac{\partial p}{\partial t} + \vec u\cdot\nabla p +\gamma p \nabla \cdot \vec u = Q,$

where $$Q$$ comprises the effects of heating and cooling as well as thermal conduction and $$\gamma$$ is the adiabaticity coefficient. Equation (3) implies the equation of state of the ideal ionized gas $p = 2(\rho/ m_i)k_BT \ ,$ which is well satisfied for most dilute plasmas, $$T$$ is the temperature, $$m_i$$ the ion mass, $$k_B$$ the Boltzmann constant, and the factor 2 arises because ions and electrons contribute equally;

the induction equation, or Faraday's law, $\tag{4} \frac{\partial \vec B}{\partial t} = -\nabla \times \vec E = \nabla \times ( \vec u \times \vec B) + \eta \nabla^2 \vec B,$

inserting Ohm's law $$\vec E = -\vec u\times\vec B + \eta \vec j \ ,$$ where $$\eta$$ is the electrical resistivity (properly speaking, the magnetic diffusivity). The magnetic field is coupled to the fluid by the Lorentz force $$\vec j\times\vec B$$ in the equation of motion (2).

In total the MHD equations thus consist of two vector and two scalar partial diffenrential equations (or eight scalar equations) that are to be solved simultaneously, either analytically or numerically.

## Basic MHD parameters

When the resistivity is negligible, $$\eta=0\ ,$$ which is called ideal MHD, equation (4) implies that the magnetic field is tightly coupled to the fluid, it said to be frozen into the fluid. The relative strength of resistivity is measured by a dimensionless number, the magnetic Reynolds number, $$Rm\ :$$ $Rm = \frac{U \ell}{\eta},$ where $$U$$ is the characteristic amplitude of the fluid velocity and $$\ell$$ the dominating length scale. The magnetic Reynolds number can be thought of as a typical ratio of the advective and diffusive terms in the induction equation (4). In astrophysical systems $$Rm$$ is usually very large, but mostly because the scales are large, not because the resistivity is small. Only the case of superconductors is the resistivity identically zero. Additionally, even if the resistivity is very low, MHD turbulence often produces very small-scale structures and large magnetic gradients, which lead to a finite rate of magnetic diffusion through the last term in (4), though the magnetic Reynolds number may be huge.

Another important dimensionless number in MHD theory is the plasma beta $$\beta\ ,$$ defined as the ratio of gas pressure $$p\ ,$$ to the magnetic pressure, $\beta = \frac{p}{B^2/2\mu_0}.$

When the magnetic field dominates in the fluid, $$\beta \ll 1\ ,$$ the fluid is forced to move along with the field. In the opposite case, when the field is weak, $$\beta \gg 1\ ,$$ the field is swirled along along by the fluid.

In ideal MHD the topology of the magnetic field (the connectivity of the field lines) does not change. If a set of magnetic field lines is twisted into a knot, they will remain so. This means that it is very difficult to change the magnetic topology of some field configuration by MHD reconnection, if the resistivity is very low. It is thus possible to store large amounts of energy in the magnetic field by twisting it into a complex topology by the fluid motions. This energy can subsequently be released into heat and kinetic energy, if the scales of the magnetic field become very small leading to large magnetic gradients or if the resistivity increases. In this case the ideal MHD breaks down allowing magnetic reconnection to occur.

## Applications of MHD

Magnetism is found throughout the Universe. Magnetic fields are known to exist in planets, stars, accretion disks, the interstellar medium, galaxies and in active galactic nulcei (cf. the article on galactic magnetic fields). Often these magnetic fields are generated and maintained by magnetohydrodynamic dynamo action (e.g. the solar dynamo or planetary dynamos, which are described by hydromagnetic dynamo theory), and in many cases the magnetic field is dynamically dominant, determining the evolution of the object.

Topics studied within MHD include typical computational astrophysics topics, such as magneto-convection, MHD turbulence and hydromagnetic dynamo action. Typically, a multitude of intermittent magnetic structures are generated in such systems, e.g. magnetic flux tubes, loops, filaments and arcades. On the Sun and other stars, these magnetic structures may take the form of cool spots (sunspots) and magnetic bright points.

Magnetic structures can store enormous amounts of energy that may be released through reconnection of magnetic field lines (MHD reconnection) leading to solar flares and stellar or even galactic eruptions. It is in this way that the Sun causes storms in the heliospherical magnetic field and ultimately aurorae and space weather.

In engineering MHD is employed to study, e.g., the magnetic behavior of plasmas in fusion reactors, liquid-metal cooling of nuclear reactors and electromagnetic casting.

## References

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