The plasma making up the Sun can be modeled as an MHD system

Simulation of the Orszag–Tang MHD vortex problem, a well-known model problem for testing the transition to supersonic 2D MHD turbulence^[1]

Magnetohydrodynamics (MHD; also called magneto-fluid dynamics or hydro­magnetics) is a model of electrically conducting fluids that treats all interpenetrating particle species together as a single continuous medium. It is primarily concerned with the low-frequency, large-scale, magnetic behavior in plasmas and liquid metals and has applications in numerous fields including geophysics, astrophysics, and engineering.

The word magneto­hydro­dynamics is derived from magneto- meaning magnetic field, hydro- meaning water, and dynamics meaning movement. The field of MHD was initiated by Hannes Alfvén, for which he received the Nobel Prize in Physics in 1970.

History

The MHD description of electrically conducting fluids was first developed by Hannes Alfvén in a 1942 paper published in Nature titled "Existence of Electromagnetic–Hydrodynamic Waves" which outlined his discovery of what are now referred to as Alfvén waves.^[2][3] Alfvén initially referred to these waves as "electromagnetic–hydrodynamic waves"; however, in a later paper he noted, "As the term 'electromagnetic–hydrodynamic waves' is somewhat complicated, it may be convenient to call this phenomenon 'magneto–hydrodynamic' waves."^[4]

Equations

In MHD, motion in the fluid is described using linear combinations of the mean motions of the individual species: the current density ${\displaystyle \mathbf {J} }$ and the center of mass velocity ${\displaystyle \mathbf {v} }$. In a given fluid, each species ${\displaystyle \sigma }$ has a number density ${\displaystyle n_{\sigma }}$, mass ${\displaystyle m_{\sigma }}$, electric charge ${\displaystyle q_{\sigma }}$, and a mean velocity ${\displaystyle \mathbf {u} _{\sigma }}$. The fluid's total mass density is then ${\textstyle \rho =\sum _{\sigma }m_{\sigma }n_{\sigma }}$, and the motion of the fluid can be described by the current density expressed as

${\displaystyle \mathbf {J} =\sum _{\sigma }n_{\sigma }q_{\sigma }\mathbf {u} _{\sigma }}$

and the center of mass velocity expressed as:

${\displaystyle \mathbf {v} ={\frac {1}{\rho }}\sum _{\sigma }m_{\sigma }n_{\sigma }\mathbf {u} _{\sigma }.}$

MHD can be described by a set of equations consisting of a continuity equation, an equation of motion, an equation of state, Ampère's Law, Faraday's law, and Ohm's law. As with any fluid description to a kinetic system, a closure approximation must be applied to highest moment of the particle distribution equation. This is often accomplished with approximations to the heat flux through a condition of adiabaticity or isothermality.

In the adiabatic limit, that is, the assumption of an isotropic pressure ${\displaystyle p}$ and isotropic temperature, a fluid with an adiabatic index ${\displaystyle \gamma }$, electrical resistivity ${\displaystyle \eta }$, magnetic field ${\displaystyle \mathbf {B} }$, and electric field ${\displaystyle \mathbf {E} }$ can be described by the continuity equation

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

the equation of state

${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}\left({\frac {p}{\rho ^{\gamma }}}\right)=0,}$

the equation of motion

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

the low-frequency Ampère's law

${\displaystyle \mu _{0}\mathbf {J} =\nabla \times \mathbf {B} ,}$

Faraday's law

${\displaystyle {\frac {\partial \mathbf {B} }{\partial t}}=-\nabla \times \mathbf {E} ,}$

and Ohm's law

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

Taking the curl of this equation and using Ampère's law and Faraday's law results in the induction equation,

${\displaystyle {\frac {\partial \mathbf {B} }{\partial t}}=\nabla \times (\mathbf {v} \times \mathbf {B} )+{\frac {\eta }{\mu _{0}}}\nabla ^{2}\mathbf {B} ,}$

where ${\displaystyle \eta /\mu _{0}}$ is the magnetic diffusivity.

In the equation of motion, the Lorentz force term ${\displaystyle \mathbf {J} \times \mathbf {B} }$ can be expanded using Ampère's law and a vector calculus identity to give

${\displaystyle \mathbf {J} \times \mathbf {B} ={\frac {\left(\mathbf {B} \cdot \nabla \right)\mathbf {B} }{\mu _{0}}}-\nabla \left({\frac {B^{2}}{2\mu _{0}}}\right),}$

where the first term on the right hand side is the magnetic tension force and the second term is the magnetic pressure force.^[5]

Ideal MHD

The simplest form of MHD, ideal MHD, assumes that the resistive term ${\displaystyle \eta \mathbf {J} }$ in Ohm's law is small relative to the other terms such that it can be taken to be equal to zero. This occurs in the limit of large magnetic Reynolds numbers during which magnetic induction dominates over magnetic diffusion at the velocity and length scales under consideration.^[5] Consequently, processes in ideal MHD that convert magnetic energy into kinetic energy, referred to as ideal processes, cannot generate heat and raise entropy.^[7]: 6

A fundamental concept underlying ideal MHD is the frozen-in flux theorem which states that the bulk fluid and embedded magnetic field are constrained to move together such that one can be said to be "tied" or "frozen" to the other. Therefore, any two points that move with the bulk fluid velocity and lie on the same magnetic field line will continue to lie on the same field line even as the points are advected by fluid flows in the system.^[8][7]: 25 The connection between the fluid and magnetic field fixes the topology of the magnetic field in the fluid—for example, if a set of magnetic field lines are tied into a knot, then they will remain so as long as the fluid has negligible resistivity. This difficulty in reconnecting magnetic field lines makes it possible to store energy by moving the fluid or the source of the magnetic field. The energy can then become available if the conditions for ideal MHD break down, allowing magnetic reconnection that releases the stored energy from the magnetic field.

Structures in MHD systems

Schematic view of the different current systems which shape the Earth's magnetosphere

In many MHD systems most of the electric current is compressed into thin nearly-two-dimensional ribbons termed current sheets.^[10] These can divide the fluid into magnetic domains, inside of which the currents are relatively weak. Current sheets in the solar corona are thought to be between a few meters and a few kilometers in thickness, which is quite thin compared to the magnetic domains (which are thousands to hundreds of thousands of kilometers across).^[11] Another example is in the Earth's magnetosphere, where current sheets separate topologically distinct domains, isolating most of the Earth's ionosphere from the solar wind.

Waves

The wave modes derived using the MHD equations are called magnetohydrodynamic waves or MHD waves. There are three MHD wave modes that can be derived from the linearized ideal-MHD equations for a fluid with a uniform and constant magnetic field:

• Alfvén waves
• Slow magnetosonic waves
• Fast magnetosonic waves

Phase velocity plotted with respect to θ

v_A > v_s

v_A < v_s

These modes have phase velocities that are independent of the magnitude of the wavevector, so they experience no dispersion. The phase velocity depends on the angle between the wave vector k and the magnetic field B. An MHD wave propagating at an arbitrary angle θ with respect to the time independent or bulk field B₀ will satisfy the dispersion relation

${\displaystyle {\frac {\omega }{k}}=v_{A}\cos \theta }$

where

${\displaystyle v_{A}={\frac {B_{0}}{\sqrt {\mu _{0}\rho }}}}$

is the Alfvén speed. This branch corresponds to the shear Alfvén mode. Additionally the dispersion equation gives

${\displaystyle {\frac {\omega }{k}}=\left({\tfrac {1}{2}}\left(v_{A}^{2}+v_{s}^{2}\right)\pm {\tfrac {1}{2}}{\sqrt {\left(v_{A}^{2}+v_{s}^{2}\right)^{2}-4v_{s}^{2}v_{A}^{2}\cos ^{2}\theta }}\right)^{\frac {1}{2}}}$

where

${\displaystyle v_{s}={\sqrt {\frac {\gamma p}{\rho }}}}$

is the ideal gas speed of sound. The plus branch corresponds to the fast-MHD wave mode and the minus branch corresponds to the slow-MHD wave mode. A summary of the properties of these waves is provided:

Mode	Type	Limiting phase speeds		Group velocity	Direction of energy flow
		${\displaystyle \mathbf {k} \parallel \mathbf {B} }$	${\displaystyle \mathbf {k} \perp \mathbf {B} }$
Alfvén wave	transversal; incompressible	${\displaystyle v_{A}}$	${\displaystyle 0}$	${\displaystyle {\frac {\mathbf {B} }{\sqrt {\mu _{0}\rho }}}}$	${\displaystyle \mathbf {Q} \parallel \mathbf {B} }$
Fast magnetosonic wave	neither transversal nor longitudinal; compressional	${\displaystyle \max(v_{A},v_{s})}$	${\displaystyle {\sqrt {v_{A}^{2}+v_{s}^{2}}}}$	equal to phase velocity	approx. ${\displaystyle \mathbf {Q} \parallel \mathbf {k} }$
Slow magnetosonic wave		${\displaystyle \min(v_{A},v_{s})}$	${\displaystyle 0}$		approx. ${\displaystyle \mathbf {Q} \parallel \mathbf {B} }$

The MHD oscillations will be damped if the fluid is not perfectly conducting but has a finite conductivity, or if viscous effects are present.

MHD waves and oscillations are a popular tool for the remote diagnostics of laboratory and astrophysical plasmas, for example, the corona of the Sun (Coronal seismology).

Extensions

Resistive

Resistive MHD describes magnetized fluids with finite electron diffusivity (η ≠ 0). This diffusivity leads to a breaking in the magnetic topology; magnetic field lines can 'reconnect' when they collide. Usually this term is small and reconnections can be handled by thinking of them as not dissimilar to shocks; this process has been shown to be important in the Earth-Solar magnetic interactions.

Extended

Extended MHD describes a class of phenomena in plasmas that are higher order than resistive MHD, but which can adequately be treated with a single fluid description. These include the effects of Hall physics, electron pressure gradients, finite Larmor Radii in the particle gyromotion, and electron inertia.

Two-fluid

Two-fluid MHD describes plasmas that include a non-negligible Hall electric field. As a result, the electron and ion momenta must be treated separately. This description is more closely tied to Maxwell's equations as an evolution equation for the electric field exists.

Hall

In 1960, M. J. Lighthill criticized the applicability of ideal or resistive MHD theory for plasmas.^[12] It concerned the neglect of the "Hall current term" in Ohm's law, a frequent simplification made in magnetic fusion theory. Hall-magnetohydrodynamics (HMHD) takes into account this electric field description of magnetohydrodynamics, and Ohm's law takes the form

${\displaystyle \mathbf {E} +\mathbf {v} \times \mathbf {B} \underbrace {-{\frac {1}{n_{e}e}}(\mathbf {J} \times \mathbf {B} )} _{\text{Hall current term}}=\eta \mathbf {J} ,}$

where ${\displaystyle n_{e}}$ is the electron number density and ${\displaystyle e}$ is the elementary charge. The most important difference is that in the absence of field line breaking, the magnetic field is tied to the electrons and not to the bulk fluid.^[13]

Electron MHD

Electron Magnetohydrodynamics (EMHD) describes small scales plasmas when electron motion is much faster than the ion one. The main effects are changes in conservation laws, additional resistivity, importance of electron inertia. Many effects of Electron MHD are similar to effects of the Two fluid MHD and the Hall MHD. EMHD is especially important for z-pinch, magnetic reconnection, ion thrusters, neutron stars, and plasma switches.

Collisionless

MHD is also often used for collisionless plasmas. In that case the MHD equations are derived from the Vlasov equation.^[14]

Reduced

By using a multiscale analysis the (resistive) MHD equations can be reduced to a set of four closed scalar equations. This allows for, amongst other things, more efficient numerical calculations.^[15]

Limitations

Applications

Geophysics

Beneath the Earth's mantle lies the core, which is made up of two parts: the solid inner core and liquid outer core.^[16][17] Both have significant quantities of iron. The liquid outer core moves in the presence of the magnetic field and eddies are set up into the same due to the Coriolis effect.^[18] These eddies develop a magnetic field which boosts Earth's original magnetic field—a process which is self-sustaining and is called the geomagnetic dynamo.^[19]

Reversals of Earth's magnetic field

Based on the MHD equations, Glatzmaier and Paul Roberts have made a supercomputer model of the Earth's interior. After running the simulations for thousands of years in virtual time, the changes in Earth's magnetic field can be studied. The simulation results are in good agreement with the observations as the simulations have correctly predicted that the Earth's magnetic field flips every few hundred thousand years. During the flips, the magnetic field does not vanish altogether—it just gets more complex.^[20]

Earthquakes

Some monitoring stations have reported that earthquakes are sometimes preceded by a spike in ultra low frequency (ULF) activity. A remarkable example of this occurred before the 1989 Loma Prieta earthquake in California,^[21] although a subsequent study indicates that this was little more than a sensor malfunction.^[22] On December 9, 2010, geoscientists announced that the DEMETER satellite observed a dramatic increase in ULF radio waves over Haiti in the month before the magnitude 7.0 M_w 2010 earthquake.^[23] Researchers are attempting to learn more about this correlation to find out whether this method can be used as part of an early warning system for earthquakes.

See also

References