Magnetohydrodynamics

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, who received the Nobel Prize in Physics in 1970 for his work in the field.

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 known 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]

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 mean velocity ⁠${\displaystyle \mathbf {u} _{\sigma }}$⁠. The fluid's total mass density is then ⁠${\displaystyle \textstyle \rho =\sum _{\sigma }m_{\sigma }n_{\sigma }}$⁠, and the fluid's motion 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 (the Cauchy momentum equation), an equation of state, Ampère's law, Faraday's law, and Ohm's law. As with any fluid description of a kinetic system, a closure approximation must be applied to the highest moment of the particle distribution equation. This is often accomplished through approximations to the heat flux under conditions of adiabaticity or isothermality.

In the adiabatic limit, that is, under the assumption of an isotropic pressure ${\displaystyle p}$ and isotropic temperature, a fluid with 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 applying Ampère's law and Faraday's law yields the induction equation, $${\displaystyle {\frac {\partial \mathbf {B} }{\partial t}}=\nabla \times \left(\mathbf {v} \times \mathbf {B} \right)+{\frac {\eta }{\mu _{0}}}\nabla ^{2}\mathbf {B} ,}$$ 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 yield $${\displaystyle \mathbf {J} \times \mathbf {B} ={\frac {\left(\mathbf {B} \cdot \nabla \right)\mathbf {B} }{\mu _{0}}}-\nabla \left({\frac {B^{2}}{2\mu _{0}}}\right),}$$ where the first term on the right-hand side is the magnetic tension force and the second term is the magnetic pressure force.^[5]

The simplest form of MHD, ideal MHD, assumes that the resistive term ${\displaystyle \eta \mathbf {J} }$ in Ohm's law is sufficiently small relative to the other terms that it can be taken to be zero. This occurs in the limit of large magnetic Reynolds numbers, in which magnetic induction dominates 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 or increase entropy.^[7]: 6

A fundamental concept underlying ideal MHD is the frozen-in flux theorem, which states that the bulk fluid and the 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 that 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 is tied into a knot, then it 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 to release the stored energy from the magnetic field.

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 within which the currents are relatively weak. Current sheets in the solar corona are thought to be between a few meters and a few kilometers thick, which is quite thin compared to the magnetic domains, which are thousands to hundreds of thousands of kilometers across.^[11] Another example occurs in the Earth's magnetosphere, where current sheets separate topologically distinct domains, isolating most of the Earth's ionosphere from the solar wind.

The wave modes derived from 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 wave vector, so they experience no dispersion. The phase velocity depends on the angle between the wave vector ⁠${\displaystyle \mathbf {k} }$⁠ and the magnetic field ⁠${\displaystyle \mathbf {B} }$⁠. An MHD wave propagating at an arbitrary angle ⁠${\displaystyle \theta }$⁠ with respect to the time-independent, or bulk, field ⁠${\displaystyle \mathbf {B} _{0}}$⁠ satisfies the dispersion relation:$${\displaystyle {\frac {\omega }{k}}=v_{\text{A}}\cos \theta }$$ where $${\displaystyle v_{\text{A}}={\frac {B_{0}}{\sqrt {\mu _{0}\rho }}}}$$ is the Alfvén speed. This branch corresponds to the shear Alfvén mode. Additionally the dispersion equation yields:$${\displaystyle {\frac {\omega }{k}}=\left[{\frac {1}{2}}\left(v_{\text{A}}^{2}+v_{\text{s}}^{2}\pm {\sqrt {\left(v_{\text{A}}^{2}+v_{\text{s}}^{2}\right)^{2}-4v_{\text{s}}^{2}v_{\text{A}}^{2}\cos ^{2}\theta }}\right)\right]^{1/2}}$$ where $${\displaystyle v_{\text{s}}={\sqrt {\frac {\gamma p}{\rho }}}}$$ 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 below:

More information , ...

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	${\displaystyle v_{\text{A}}}$	${\displaystyle 0}$	${\displaystyle {\frac {\mathbf {B} }{\sqrt {\mu _{0}\rho }}}}$	${\displaystyle \mathbf {Q} \parallel \mathbf {B} }$
Fast magnetosonic wave	neither transversal nor longitudinal; compressional	${\displaystyle \max(v_{\text{A}},v_{\text{s}})}$	${\displaystyle {\sqrt {v_{\text{A}}^{2}+v_{\text{s}}^{2}}}}$	equal to phase velocity	approx. ${\displaystyle \mathbf {Q} \parallel \mathbf {k} }$
Slow magnetosonic wave		${\displaystyle \min(v_{\text{A}},v_{\text{s}})}$	${\displaystyle 0}$		approx. ${\displaystyle \mathbf {Q} \parallel \mathbf {B} }$

Close

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

MHD waves and oscillations are popular tools for the remote diagnostics of laboratory and astrophysical plasmas, for example, the Sun's corona (coronal seismology).

Resistive

Resistive MHD describes magnetized fluids with finite electron diffusivity (⁠${\displaystyle \eta \neq 0}$⁠). This diffusivity leads to a break in the magnetic topology; magnetic field lines can "reconnect" when they collide. Usually, this term is small, and reconnections can be treated as similar to shocks; this process has been shown to be important in Earth-solar magnetic interactions.

Extended

Extended MHD describes a class of phenomena in plasmas that are higher-order than resistive MHD, but 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 because 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] This 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: $${\displaystyle \mathbf {E} +\mathbf {v} \times \mathbf {B} -\underbrace {{\frac {1}{n_{\text{e}}e}}(\mathbf {J} \times \mathbf {B} )} _{\text{Hall current term}}=\eta \mathbf {J} ,}$$ where ${\displaystyle n_{\text{e}}}$ 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 rather than to the bulk fluid.^[13]

Electron MHD

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

Collisionless

MHD is also 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, among other things, more efficient numerical calculations.^[15]