# Magnetohydrodynamics: Overview

# Magnetohydrodynamics: Overview

- E.R. PriestE.R. PriestSt Andrews University, Mathematics Institute

### Summary

Magnetohydrodynamics is sometimes called magneto-fluid dynamics or hydromagnetics and is referred to as MHD for short. It is the unification of two fields that were completely independent in the 19th, and first half of the 20th, century, namely, electromagnetism and fluid mechanics. It describes the subtle and complex nonlinear interaction between magnetic fields and electrically conducting fluids, which include liquid metals as well as the ionized gases or plasmas that comprise most of the universe.

In places such as the Earth’s magnetosphere or the Sun’s outer atmosphere (the corona) where the magnetic field provides an important component of the free energy, MHD effects are responsible for much of the observed dynamic behavior, such as geomagnetic substorms, solar flares and huge eruptions from the Sun that dominate the Earth’s space weather. However, MHD is also of great importance in astrophysics, since many of the MHD processes that are observed in the laboratory or in the Sun and the magnetosphere also take place under different parameter regimes in more exotic cosmical objects such as active stars, accretion discs, and black holes.

The different aspects of MHD include determining the nature of: magnetic equilibria under a balance between magnetic forces, pressure gradients and gravity; MHD wave motions; magnetic instabilities; and the important process of magnetic reconnection for converting magnetic energy into other forms. In turn, these aspects play key roles in the fundamental astrophysical processes of magnetoconvection, magnetic flux emergence, star spots, plasma heating, stellar wind acceleration, stellar flares and eruptions, and the generation of magnetic fields by dynamo action.

### Keywords

### Subjects

- Fluid Dynamics

### 1. Introduction

Plasmas are ubiqitous in the universe and are referred to as the *fourth state of matter*. On Earth humanity sits in a bubble consisting of the first three states, namely, solids, liquids, and gases, such that one can go from solid to liquid to gas by progressively raising the temperature. However, if the temperature of a gas is raised sufficiently it becomes ionized, with the outer electrons of the previously neutral atoms being stripped off and moving around independently of their parent atoms, which remain behind as positive ions. The flows of these charged particles represent electric currents. Indeed, an ionized gas behaves so differently from a neutral gas that it is given a new name, *plasma*.

The main way that plasmas differ from normal gases is that they interact in complex ways with magnetic fields, which may be modeled either by *kinetic plasma physics* (when the particle motions are followed) or, as here, by *magnetohydrodynamics* (when the plasma is treated as a continuous medium), which is referred to as *MHD* throughout this article. The condition that a collection of particles be regarded as a plasma is that it is electrically neutral.

As one goes up in the Earth’s atmosphere, eventually the ionosphere is reached, and that is where the plasma universe starts. Indeed, over 99.9% of the observable matter in the universe is in the plasma state. Surprisingly, even gases that are only 1% ionized may behave as a plasma. Thus, the Earth’s magnetosphere is plasma, as is the tenuous interplanetary medium between the Sun and planets. The whole of the Sun is plasma and so is the interstellar medium, even though it includes regions of neutral hydrogen gas and dust. Further afield, galaxies and the intergalactic medium are also comprised mainly of plasma. The main exceptions are solid bodies, such as asteroids and rocky planets.

The magnetic field influences a plasma in several ways which are described by the MHD equations:

exerting a force, which can create nonuniform structure or accelerate the plasma;

storing energy, which can heat plasma such as the solar corona or be converted into other forms in a solar flare (e.g., Longcope, 2020);

acting as a heat blanket, which isolates hot or cool structures from the surrounding medium, as in a cool solar prominence (e.g., Mackay, 2021) situated in the hot corona;

channeling plasma, fast particles, and heat that are liberated in, for example, a solar flare;

providing stability or driving instabilities in a nonuniform medium;

supporting several different types of MHD wave.

This article gives an overview of MHD, which is developed further in other articles and in Priest (2014). It starts by summarizing the historical development of the field and discussing the definition of plasma and the validity of MHD. It also introduces the MHD equations together with the physical implications of the induction equation and equation of motion. Then it develops the different aspects of MHD, namely, magnetic equilibria (Wiegelmann, 2020), MHD waves (Nakariakov, 2020), MHD instabilities, and magnetic reconnection (Pontin, 2020).

#### 1.1 Historical Development

One of the main stimuli for the development of MHD was the realisation of the importance of magnetic fields on the Sun (e.g., Priest, 2014). Sunspots (Rempel and Borrero, 2021) had been observed in ancient Greece and China and were rediscovered around 1610 with newly invented telescopes by Thomas Harriot in England and later by Johannes Fabricius, Christoph Scheiner, and Galileo Galilei. In the 19th century, prominences (Mackay, 2021) and the solar corona were photographed during solar eclipses, and solar flares (Longcope, 2020) were first observed. Then, in 1908, George Ellery Hale discovered that sunspots possess strong magnetic fields.

The bases of electromagnetism and of fluid mechanics had been laid in the previous century, and their unification to form MHD involved several different topics and actors. First of all, in solar and stellar physics, a key role was played by Tom Cowling (1906–1990), who worked on gas dynamics and the structure and stability of stars between 1930 and 1937 with Sydney Chapman (1888–1970) and, in reaction to work by Joseph Larmor, produced his famous anti-dynamo theorem (Cameron, 2020) for the generation of magnetic fields in stars (Cowling, 1933). He also had discussions with Ludwig Biermann (1907–1986) for a study of the structure of sunspots (Cowling, 1935, 1946), and calculated the possible non-radial oscillations of stars (Cowling, 1941). Later, he brought together a mature understanding of MHD in the first monograph on the subject (Cowling, 1957).

A second strong contribution to the development of MHD came from magnetospheric physics and in particular a debate about the nature of geomagnetic substorms and aurorae between Chapman and Vicenzo Ferraro (1907–1974), on the one side, and Hannes Alfvén (1908–1995) and Jim Dungey (1923–2015), on the other. Dungey, in particular, developed the current understanding for solar–terrestrial coupling and other magnetospheric phenomena. In addition, important contributions by Alfvén include his theory for Alfvén waves (Alfvén, 1942), where he also coined the term “magnetohydrodynamic,” and his physical comments about the nature of the solar corona (Alfvén, 1941) following the discovery that it is hot by Edlén (1941).

A third contribution was from two types of laboratory study. The first concerns liquid metals (Davidson, 2001), where magnetic fields are used in engineering applications to heat, pump, stir, and levitate liquid metals. The second is in the behavior of fusion plasmas, where, in particular, the nature of complex equilibria, instabilities, and wave modes has been developed for cylindrical and toroidal geometries (Goedbloed & Poedts, 2004, 2010; Manheimer & Lashmore-Davies, 1989; Wesson, 1997).

The condition that a large collection of positively and negatively charged particles of densities ${n}_{+}$ and ${n}_{-}$, respectively, can be regarded as a plasma is that they are electrically neutral to a high degree of approximation, in other words, that

where

is the total number density. Before presenting the equations of magnetohydrodynamics, it is therefore worth using Maxwell’s equations to explore this condition in more detail and also to summarise the other conditions for the validity of MHD.

#### 1.2 Definition of a Plasma

Maxwell’s equations may be written in *SI* units as

in terms of the *electric field*$E$ measured in volts per metre $(\text{V}\phantom{\rule{0.2em}{0ex}}{\text{m}}^{-1})$, the magnetic induction (loosely called the *magnetic field*) $B$ in tesla (T), the *electric current density*$j$ in amps per square metre $(\text{A}\phantom{\rule{0.2em}{0ex}}{m}^{-2})$ and the *charge density*$({\rho}^{*})$. Later, the *plasma velocity*$(v)$ will be introduced, measured in $\text{m}\phantom{\rule{0.2em}{0ex}}{\text{s}}^{-1}$. Here the usual relations $(H=B/\mu ,D=\u03f5E$ have been used to eliminate the magnetic field $(H)$ and electric displacement $(D)$, where for low-density plasma such as exists on the Sun, the *magnetic permeability*$(\mu )$ and *permittivity of free space*$(\u03f5)$ are approximated by their vacuum values, ${\mu}_{0}$ and ${\u03f5}_{0}$, respectively, but the 0 subscripts are removed for simplicity. Note that the *speed of light* in a vacuum is

where

and ${\u03f5}_{0}=8.854\times {10}^{-12}$ farad ${\text{m}}^{-1}$

Although MKS units are being used, magnetic field and magnetic flux values are often quoted in gauss (where $1\phantom{\rule{0.2em}{0ex}}\text{G}={10}^{-4}\text{T}$) and in maxwells, since these are more commonly used in practice.

The fundamental variables in MHD are the plasma velocity and magnetic field, with typical values of, say, ${V}_{0}$ and ${B}_{0}$, and so, if they are varying on a typical length scale ${l}_{0}$ and time scale ${t}_{0}={l}_{0}\text{/}{V}_{0}$, it is possible to rewrite the condition for neutrality (Equation 2) in terms of these values. First of all, the difference in charge density may be written

where $e\approx 1.6\times {10}^{-19}\cdot \text{C}$ and ${\rho}^{*}$ is in order of magnitude from Equation (6) given by

However, Faraday’s law (Equation 5) implies that

and so the condition for neutrality (Equation 1) may be rewritten

where ${B}_{0}$ is in tesla. This condition is very well satisfied in practice for many solar system and astrophysical applications.

Another way of considering charge neutrality is that a local imbalance in charge gives rise to an electrical field, whose spatial range is the *Debye length* given by

where $T$ is in K and $n$ in ${\text{m}}^{-3}$. It is a measure of the distance over which ${n}_{-}$ and ${n}_{+}$ can deviate appreciably from each other. Thus, a *plasma* may be defined as an ionized gas for which

so that the number of particles in a Debye sphere (a sphere of radius ${\lambda}_{D}$) is very large.

#### 1.3 MHD Equations

Maxwell’s Equations (3)–(6) represent four equations for the curl and divergence of $\left(B\right)$ and $\left(E\right)$, where $E=j\text{/}\sigma $ is Ohm’s law, and $\sigma $ is the electrical conductivity. They describe the behavior of electric and magnetic fields, but in a fluid (a gas or a liquid), another independent set of equations of motion, mass continuity, energy, and a perfect gas law (i.e., the equations of *fluid mechanics*) determine the behavior of the fluid density $\left(\rho \right)$, velocity $\left(v\right)$, pressure $\left(p\right)$, and temperature $\left(T\right)$.

However, in an ionized gas (plasma) or liquid metal, electric currents flow and produce new forces, which leads to a remarkable coupling and unification of electromagnetism and fluid mechanics to give the equations of MHD. This occurs by making three assumptions:

in a sub-relativistic plasma, $v\ll c$, so that the displacement current $(\partial D\text{/}\partial t)$ in Ampère’s law (Equation 3) is negligible;

in a moving plasma, an extra term $(v\times B)$ is added to the electric field $E$ acting on plasma at rest;

an extra (Lorentz) force $(j\times B)$ is present in the equation of motion.

Thus, it is the new terms ($v\times B$ and $j\times B$) that provide the coupling. By eliminating $j$ and $E$, Maxwell’s equations and the new form of Ohm’s law can be reduced to the so-called *induction equation* (9) for $v$ and $B$. The MHD equations for a plasma are completed by adding the new form of the equation of motion (11), together with three other equations for $\rho $, $p$ and $T$, namely, equations for the conservation of mass (10) and energy (13) and the perfect gas law (12). The result is as follows:

where $\eta \equiv {(\mu \sigma )}^{-1}$ is the *magnetic diffusivity* (here assumed constant), $g$ is the gravitational acceleration, $F$ includes other terms such as viscous or Coriolis forces, $\mathcal{R}$ is the specific gas constant (the gas constant per molar mass), $d\text{/}dt=\partial \text{/}\partial t+v\cdot \nabla $ is the total time derivative in a frame of reference moving with an element of plasma, and $\mathcal{L}$ is the *energy loss function*, which includes both losses and gains.

There are several alternative forms of the energy equation (Priest, 2014). For example, at constant pressure

where ${c}_{p}$ is the specific heat at constant pressure. However, when the fluid is incompressible, with a density that does not change following the motion (as in a liquid), $\nabla \cdot v=0$ replaces the continuity equation (10), and the gas law (12) is removed, while the energy equation is replaced by

where ${c}_{v}$ is the specific heat at constant volume.

The energy loss function $(\mathcal{L})$ depends on the application. Thus, if thermal conduction and Ohmic heating are the only losses,

where $q=\kappa \nabla T$ is the heat flux vector, whereas in the solar corona (where there is an extra optically thin radiation loss ${L}_{r}$ and extra heat sources ${F}_{H}$),

Equations (9)–(13) are coupled and determine the primary variables in MHD (namely, $v$, $B$, $p$, $\rho $, and $T$), in which the basic physics of MHD resides. Here $j$ and $E$ are secondary variables and, once $v$ and $B$ have been determined, they may be calculated if required from

but they do not affect the fundamental MHD physics of the process being studied. This represents a profound change of philosophy from electromagnetism, where the primary variables are instead $E$ and $j$. Furthermore, $B$ has to obey

which plays the role of an initial condition for the time-dependent equations (9)–(13). Indeed, the divergence of (9) shows that, if $\nabla \cdot B$ vanishes initially, then it continues to vanish for all time.

#### 1.4 Assumptions and Validity of MHD Equations for a Plasma

Two assumptions have been mentioned, namely, that a charge-neutral plasma is being assumed and that relativistic effects are neglected (so that the flow speed and other characteristic speeds such as the sound speed and Alfvén speed are much smaller than the speed of light). Other assumptions include:

The plasma is regarded as a *continuous medium*, which is valid for a *collisional* plasma, namely, one for which the length scale $({L}_{0})$ for variations greatly exceeds the mean-free path for collisions $({\lambda}_{mfp})$. For a *collisionless* plasma (i.e., one for which ${L}_{0}<{\lambda}_{mfp}$), the plasma may still be treated as continuous provided ${L}_{0}$ exceeds internal plasma lengths such as the ion gyroradius.

The plasma is in thermodynamic equilibrium, so that the distribution functions are close to Maxwellian, which holds for time scales much larger than the collision times and for length scales much longer than the mean free paths.

The transport coefficients such as $\eta $ are uniform, and most of the plasma properties are *isotropic*. A more complex theory using tensor transport coefficients has also been developed (Braginsky, 1965).

The equations are written in an *inertial frame*. The extra terms such as Coriolis forces that arise for a frame rotating with the Sun are important for large-scale processes.

The simple form of *Ohm’s law* (15) is adopted for many applications, rather than a generalized version.

The plasma is a *single fluid*, although two- or three-fluid models (Khomenko, 2020) may be more relevant for the solar photosphere or low chromosphere, where the plasma is mostly neutral with low ionization or for the high corona where the plasma is collisionless.

In MHD, the plasma is treated as a continuous medium rather than a collection of individual particles, which is valid for a collisional plasma, where typical length scales $({L}_{0})$ are larger than the collisional mean-free path between particles (typically 3 cm in the solar chromosphere and 30 km in the corona). But MHD often works very well also in a collisionless plasma, where this condition is not satisfied, such as the high corona, the solar wind or the magnetosphere.

In a two-fluid or three-fluid plasma (where the electrons, ions, and neutrals may be regarded as separate fluids that interact with one another), the normal equations of MHD are modified in two important ways. Firstly, the pressure is not isotropic and so one needs a pressure tensor, and secondly, Ohm’s law generalizes with new terms representing electron inertia, a Hall term, and electron stress (Khomenko, 2020). The generalized Ohm’s law is derived from a three-fluid model for electrons, protons, and neutral atoms (with number densities ${n}_{e}$, ${n}_{i}={n}_{e}$ and ${n}_{a}$, respectively) and may be written when ${m}_{e}/{m}_{p}\ll 1$

where ${\text{\Omega}}_{e}=eB/{m}_{e}$ and ${\text{\Omega}}_{i}=eB/{m}_{p}$ are the electron and ion (proton) gyration frequencies, $f={n}_{a}{({n}_{a}+{n}_{e})}^{-1}$ is the fraction of atoms not ionized, ${\tau}_{ei}$ is the electron-ion collision time, while ${\tau}_{en}$ and ${\tau}_{in}$ are collision times for neutrals with electrons and ions.

In a collisionless plasma, why does MHD work so well, where Vlasov-kinetic theory is the most complete description? There are several reasons:

ideal MHD describes large-scale dynamics both with and without collisions since it embodies conservation of mass, momentum, and energy, which are universal relations in both collisional and collisionless plasmas;

for particle motion perpendicular to the magnetic field, gyro-motion impedes the motion, and so the net effect may be described by MHD-like equations;

for particle motion along the magnetic field, often wave–particle interactions impede the motion;

when the $E\times B$-drift is dominant in a collisionless plasma, the drift velocity is ${u}_{drift}\equiv E\times B/{B}^{2}$, which leads to an ideal Ohm’s law

the Lorentz force acts on individual particles and so, when they are summed over many particles, a net force results of

where $qE\ll j\times B$ when ${u}_{drift}\ll c$.

#### 1.5 Implications of the Induction Equation

The two key equations in MHD are the induction equation (9) and the equation of motion (11), which embody the basic physics of how an MHD plasma behaves, and together they determine the magnetic field $(B)$) and plasma velocity $(v)$ provided the density, pressure, and temperature are known. It is therefore important to describe the physical significance of these two equations for the interaction between a plasma and the magnetic field that threads it.

According to the induction equation, the magnetic field at a fixed point in space changes in time due to two physical effects. The first term on the right implies that the magnetic field has a tendency to be transported with the plasma, while the second term represents the diffusion of magnetic field through the plasma. The ratio of these two terms is in order of magnitude

where ${V}_{0}$ and ${l}_{0}$ are typical velocity and length scale. This key dimensionless parameter is called the *magnetic Reynolds number*.

##### 1.5.1 Limits ${R}_{m}\ll 1$ and ${R}_{m}\gg 1$

In industrial MHD, usually ${R}_{m}<1$. However, in heliophysical or astrophysical MHD, usually ${R}_{m}\gg 1$, so that magnetic diffusion is negligible and Equation (16) reduces to $E=-v\times B$. This implies that, in most of the cosmos, the magnetic field moves with the plasma and does not dissipate its energy by Ohmic diffusion. In addition, the electrical field is just $-v\times B$ and so bears no direct relationship to the current at all. As an example, in the solar corona where typically $\eta \approx 1\phantom{\rule{0.2em}{0ex}}{\text{m}}^{2\phantom{\rule{0.2em}{0ex}}}{\text{s}}^{-1}$, ${l}_{0}\approx {10}^{6}\text{m}$ and ${V}_{0}\approx {10}^{3}\text{m}\phantom{\rule{0.2em}{0ex}}{\text{s}}^{-1}$, the magnetic Reynolds number becomes ${R}_{m}\approx {10}^{8}$. An exception is in current singularities or sheets, where electric currents and magnetic field gradients are very much larger than normal. Current sheets form at null points or along separators and are locations where magnetic energy is converted to other forms by magnetic reconnection.

The second term on the right of Equation (9) becomes significant only in regions where the length scale ${l}_{0}$ is small enough and the electric currents

are large enough that ${R}_{m}$ reduces to a value less than or of order unity. The resulting intense sheets of current are precisely the locations where magnetic energy conversion takes place.

Consider first a region where ${R}_{m}\ll 1$, so that Equation (9) reduces to

$B$ is therefore governed by a diffusion equation, and so magnetic field variations on a scale ${l}_{0}$ diffuse away on a *diffusion time*${\tau}_{d}\equiv {l}_{0}^{2}/\eta $ and with a *diffusion speed*${v}_{d}\equiv {l}_{0}/{\tau}_{d}=\eta /{l}_{0}$. The decay time for a sunspot, for instance, where $\eta =1\phantom{\rule{0.2em}{0ex}}{\text{m}}^{2}{\text{s}}^{-1}$ and ${l}_{0}{=10}^{6}\text{m}$ becomes ${\text{10}}^{12}$ secs or 30,000 years. See Priest (2014) for a discussion of diffusion of magnetic fields in one, two and three dimensions according to Equation (17).

Consider next a region where ${R}_{m}\gg 1$, so that Equation (9) reduces instead to

while Ohm’s law becomes $E+v\times B=0$. One consequence of this equation is *magnetic flux conservation*, namely, that the magnetic flux through a curve C that moves with the plasma does not change in time. Another consequence is *magnetic field line conservation*, namely, that field lines are “frozen to the plasma” so that two elements of plasma that are initially joined by a magnetic field line will continue to be joined by a field line at later times. Motion of plasma along magnetic field lines is freely allowed, but motion perpendicular to the field lines carries the field lines with the plasma, while motion of the field lines drags the plasma in a direction perpendicular to the field. See Priest (2014) for proofs of these results and examples.

#### 1.6 Implications of the Equation of Motion

The equation of motion

describes how plasma is accelerated by a variety of forces, such as a pressure gradient, a magnetic force and the force of gravity, and so the way the plasma velocity $(v)$ varies in time. Usually, in the solar atmosphere these are the main forces acting, but when there are strong velocity shears it is important to add viscous forces, and over large scales the effect of solar rotation produces Coriolis forces. In most of the corona, especially active regions, in a direction normal to the magnetic field, the magnetic force dominates the structure and behavior. In a direction along the magnetic field, however, the situation is different, since $j\times B$ vanishes, which makes the pressure gradient and gravity important.

In order of magnitude, the ratio of the sizes of the pressure gradient and gravity is just $p/({l}_{0}\rho g)$. Thus, the pressure gradient dominates gravity when the length scale is ${l}_{0}<H$, where $H\equiv p/(\rho g)=RT/g$ is the so-called *pressure scale height*, which is proportional to the temperature. An implication is that, within an active region, the shape and position of a coronal loop is determined by the magnetic field, whereas the structure of plasma along the loop is governed by pressure and gravity, such that, if the height of a loop is much less than $H$, gravity is unimportant. Typical values of $H$ are 150 km in the photosphere and 100 Mm in a 2MK-degree corona, so that, starting in the photosphere and moving upwards in the atmosphere, the plasma pressure decreases very rapidly at first and then much more slowly as the temperature reaches coronal values.

In gas dynamics, a plasma pressure gradient $(-\nabla p)$ acts from regions of high pressure to regions of low pressure in a direction perpendicular to the curves of constant pressure (the isobars). However, the magnetic (or Lorentz) force may be rewritten, after using $j=\nabla \times B\text{/}\mu $ and $\nabla \cdot B=0$ as

Three important deductions can be made from this equation. Firstly, the Lorentz force is always perpendicular to the magnetic field. Secondly, the first term on the right may be regarded as a *magnetic tension force*. It is non-vanishing whenever the magnetic field lines are curved and points in a direction toward the centre of curvature. The field lines behave in a similar way to elastic bands with a tension ${B}^{2}/\mu $. Thirdly, the second term on the right has the same form as $-\nabla p$, and so the magnetic field behaves as if it possesses a *magnetic pressure force*, acting from regions of high magnetic pressure $({B}^{2}/(2\mu ))$ to those of low magnetic pressure.

A key parameter in MHD is the *plasma beta*, which measures the ratio of the pressure gradient to the magnetic force, namely,

Thus, its value determines whether the pressure gradient or the Lorentz force is most important, such that, when $\beta \ll 1$, the magnetic force dominates. In the corona of an active region, typically $\beta \approx {10}^{-4}$, whereas in a sunspot or a photospheric flux tube $\beta \approx 1$.

The speed to which the magnetic field tends to accelerate plasma when it dominates other forces is called *Alfvén speed*. Its value may be found by equating the acceleration term on the left of Equation (11) to the magnetic force, namely, in order magnitude $v\approx {v}_{A}\equiv B/\sqrt{\mu \rho}$. Typical values are $1,000\phantom{\rule{0.2em}{0ex}}\text{km}\phantom{\rule{0.2em}{0ex}}{\text{s}}^{-1}$ in the solar corona and $5\phantom{\rule{0.2em}{0ex}}\text{km}\phantom{\rule{0.2em}{0ex}}{\text{s}}^{-1}$ in a photospheric flux tube. Whereas the Mach number of a flow is the ratio $M\equiv v/{c}_{s}$, of the flow speed $v$ to the sound speed $({c}_{s}=\sqrt{\gamma p/\rho})$, the *Alfvén Mach number* is defined by analogy to be

Note also that the plasma beta (Equation 20) can be written in terms of the sound and Alfvén speeds as

### 2. Aspects of MHD

Bodies of theory have been developed for several basic aspects of MHD, and they are described in separate entries in this encyclopedia, but it is worth summarising the main features here. First of all, there is the nature of the equilibrium structures that are possible, either under a balance between pressure gradients, gravity and magnetic fields, or, when the magnetic field dominates, under the influence of the Lorentz force alone (see Wiegelmann, 2020).

When magnetic fields are perturbed away from equilibrium, if the equilibrium is stable the injected energy propagates away in a series of wave modes (see Nakariakov, 2020), which are simple enough in a uniform medium but have a complex nature in nonuniform media. Alternatively, the perturbation may grow in time as an instability, whose nature depends on the structure of the equilibrium.

A fundamental process in a magnetic field with high magnetic Reynolds number is *magnetic reconnection*, whereby in extremely thin sheets of current the magnetic energy is converted into kinetic energy, heat and fast particle energy (see Pontin, 2020). The classical theory in two dimensions is highly developed and well understood, but in three dimensions there are many new features whose properties are just being developed.

#### 2.1 Magnetic Equilibria

If plasma velocities are much smaller than the Alfvén speed $(v\ll {v}_{A})$, then the inertial term on the left of the equation of motion (11) is much smaller than the Lorentz force and so Equation (11) reduces to

which represents *magnetohydrostatic equilibrium* for a balance between a pressure gradient, a magnetic force and gravity (Wiegelmann, 2020).

If, furthermore, the pressure and magnetic forces are comparable and the height $(h)$ of a structure such as a coronal loop is much smaller than the scale height, i.e., $h\ll H\equiv p/(\rho g)=RT/g$, then the force of gravity is negligible. If also the *plasma beta*$\beta \equiv 2\mu p/{B}^{2}\ll 1$, then Equation (21) approximates to

for a so-called *force-free field*.

A simple case is when the current $(j=\nabla \times B/\mu )$ vanishes. This may be satisfied identically by a magnetic field of the form $B=\nabla \psi $, where the potential $\psi $ satisfies Laplace’s equation, ${\nabla}^{2}\psi =0$, which gives a *potential field*. Calculating potential fields for the corona using the normal component of the observed magnetic field in the photosphere is extremely valuable in determining the structure and topology of the coronal magnetic field. But, in order to model highly sheared or twisted fields typical of solar prominences and solar flares, it is essential to solve the force-free field equation instead (e.g., Mackay & Yeates, 2012).

Equation (22), where $j=\nabla \times B/\mu $ and $\nabla \cdot B=0$, looks disarmingly simple, but very little is known in general about its properties. It implies that the electrical current $(j)$ is parallel to the magnetic field $(B)$, so that

where $\alpha $ is a function of position that is constant along each magnetic field line.

If $\alpha $ is uniform $(={\alpha}_{0})$, the force-free field is called “constant-$\alpha $” or “linear.” The curl of Equation (23) in this case gives $({\nabla}^{2}+{\alpha}_{0}^{2})B=0$, as a generalization of Laplace’s equation. Unfortunately, often $\alpha $ is far from uniform, and so linear force-free solutions are not as useful as first thought. Thus, one often resorts to numerical nonlinear force-free solutions, which are far from trivial. A detailed account of force-free solutions can be found in Priest (2014).

#### 2.2 MHD Waves

Sound waves propagate in a uniform compressible fluid (i.e., a gas). In terms of its pressure ${p}_{0}$, density ${\rho}_{0}$, and *sound speed*${c}_{s}=(\gamma {p}_{0}/{\rho}_{0}{)}^{1/2}$, their dispersion relation is

where $\omega $ is the frequency and $k=k\phantom{\rule{0.2em}{0ex}}\widehat{k}$ the wavenumber vector, such that the wave propagates in a direction $\widehat{k}$ with wavenumber $k$, and the perturbed gas properties (density ${\rho}_{1}$, pressure ${p}_{1}$, velocity ${v}_{1}$, and temperature ${T}_{1}$) have the form

The speed of propagation of the wave is the phase speed

so that the wave propagates at a uniform speed, independent of $k$ and the same in all directions (i.e., it is *isotropic*). The plus sign represents propagation in the same direction as $\widehat{k}$, while the negative sign represents propagation in the opposite direction.

The dispersion relation is derived by linearizing the basic equations of fluid mechanics about a uniform equilibrium with density ${\rho}_{0}$, pressure ${p}_{0}$, velocity ${v}_{0}=0$, and temperature ${T}_{0}$ and then simply eliminating the variables ${\rho}_{1},\phantom{\rule{0.2em}{0ex}}{p}_{1},\phantom{\rule{0.2em}{0ex}}{v}_{1},\phantom{\rule{0.2em}{0ex}}{T}_{1}$. The physical cause for the existence of the wave is the fact that, when the gas is perturbed from equilibrium, the direction of the resulting force is such as to move the gas back toward its equilibrium state. The restoring force for sound waves is the plasma pressure gradient, and the wave is *longitudinal* in the sense that the perturbed velocity $({v}_{1})$ is parallel to the direction of propagation $(k)$.

Whereas in fluid mechanics there is one type of wave when the medium is uniform (the sound wave), in MHD there are three independent wave modes, created by the three restoring forces, a pressure gradient, a magnetic pressure gradient and a magnetic tension force (assuming gravity is not important)(Nakariakov, 2020). In addition, the modes are no longer isotropic but depend on the inclination $(\theta )$ of the direction of propagation $(\widehat{k})$ to the background uniform magnetic field $({B}_{0})$.

The first type of wave is the *Alfvén wave* with dispersion relation

where ${v}_{A}^{2}={B}_{0}^{2}/(\mu {\rho}_{0})$. The speed of propagation $(\omega /k={v}_{A}\mathrm{cos}\theta )$ has a maximum magnitude of ${v}_{A0}$ when it propagates along the magnetic field $(\theta =0)$, and it falls to zero for propagation normal to the field $\left(\theta =\frac{1}{2}\pi \right)$. The wave is incompressible (since it produces no change in density) and it is also *transverse* since the perturbed velocity is normal to the direction of propagation $(k\cdot v=0)$. The restoring force is magnetic tension.

The two other wave modes are the *fast* and *slow magnetoacoustic (or magnetosonic) waves*, sometimes called the *fast mode* and the *slow mode*. They propagate at phase speeds larger and smaller, respectively, than the Alfvén speed $({v}_{A})$, and so the Alfvén mode is sometimes known as the *intermediate mode*. Their restoring forces are a combination of the plasma and magnetic pressure gradients, and the dispersion relations are

For propagation along the magnetic field $(\theta =0)$, the phase speed is either ${c}_{s}$ or ${v}_{A}$, since one magnetoacoustic mode degenerates into an acoustic mode and the other into an Alfvén mode; in terms of slow and fast modes, when ${c}_{s}<{v}_{A}$ the slow-mode propagates at the sound speed, but when ${c}_{s}>{v}_{A}$ it does so at the Alfvén speed. On the other hand, propagation across the field $\left(\theta =\frac{1}{2}\pi \right)$. makes the slow mode vanish and the fast mode propagate at ${({c}_{s}^{2}+{v}_{A}^{2})}^{1/2}$.

##### 2.2.1 Magnetic Instabilities

Understanding stability and instability is often important for astrophysical and laboratory phenomena. Sometimes one needs to explain how a structure can remain stable, and at other times, the quest is to fathom why magnetic structures on, say, the Sun suddenly become unstable and erupt to produce events such as erupting prominences, coronal mass ejections, or solar flares. Stability methods and the main instabilities will be briefly described, and further details and references can be found in Goedbloed & Poedts (2010), and Priest (2014).

The methods used to study the linear stability of an MHD system are natural generalizations of those for studying a particle in one-dimensional motion along the $x$-axis under the action of a force. One starts in equilibrium at, say, $x=0$, and then linearizes the equation for 1D motion. One method is to seek *normal-mode solutions* of the form $x={x}_{0}{e}^{i\omega t}$ and to determine the value of $\omega $. If ${\omega}^{2}>0$, the particle oscillates about $x=0$ and the initial equilibrium is stable. If instead ${\omega}^{2}<0$, the displacement $(|x|)$ increases in time from the equilibrium position, which is now *unstable*.

An alternative approach is to consider the change $(\delta W)$ in potential energy due to the displacement $(x)$ from equilibrium. To first order in $x$, $\delta W=x{(dW/dx)}_{0}$, which vanishes since the initial state is in equilibrium. To second order,

The particle is therefore in stable equilibrium if $\delta W>0$ for *all* small displacements from $x=0$, with both $x>0$ and $x<0$. It is unstable if $\delta W<0$ for at least one small displacement, with either $x>0$ or $x<0$.

The stability of an MHD system may be analyzed in just the same way. First, linearize the equations and then seek either normal modes or the variation of the energy. The advantage of the normal-mode method is that a *dispersion relation* can be found, linking the frequency $(\omega )$ to the wavenumber $(k)$ of the disturbance. However, the variational or energy method may be applied to more complex equilibrium states.

As the parameters of an equilibrium configuration are varied, a transfer from stability to instability may occur in two possible ways via a state of *marginal* (or *neutral*) stability. If ${\omega}^{2}$ is real and it decreases through zero, the marginal state is then stationary $(\omega =0)$, and a monotonic growth in the perturbation occurs. If, instead, the frequency $(\omega )$ is complex and its imaginary part decreases from positive to negative values, the marginal state has oscillatory motions, and a state of growing oscillations appears, called *overstability*.

The behavior of an ideal (dissipationless) MHD system is governed by Equations (9)–(13) in the limit as the magnetic diffusivity (and viscosity) approach zero and as the non-adiabiatic terms on the right of the energy equation vanish, so that

Suppose the initial state is in equilibrium with a vanishing plasma velocity $({v}_{0}\equiv 0)$, together with a magnetic field $({B}_{0})$, plasma pressure $({p}_{0})$, density $({\rho}_{0})$, and electric current $({j}_{0}=\nabla \times {B}_{0})$ that are all independent of time. Next, perturb the equilibrium by setting

where ${j}_{1}=\nabla \times {B}_{1}/\mu $ and ignore squares and products of the perturbation quantities (denoted by subscript 1).

The displacement $[\xi ({r}_{0},t)]$ of a plasma element from equilibrium is just $\xi =r-{r}_{0}$, in terms of the position vector $(r)$ and its initial (equilibrium) position $({r}_{0})$. By changing from *Eulerian coordinates*$(r,t)$ to *Lagrangian coordinates*$({r}_{0},t)$, the perturbed velocity may be written in terms of $\xi $ as

and, after integrating the induction and mass continuity equations, the perturbed magnetic field and density become

and

Also, the set of linearized MHD equations may be reduced to a single equation, namely,

where

If the displacement is written in the form

the equation of motion reduces to

which is used as a basis for both the normal-mode and energy methods.

The main MHD instabilities are as follows.

##### 2.2.2 Interchange Instability or Flute Instability

This is a general class of instability that occurs at an interface between two different plasmas and magnetic fields. It is fed when equal volumes of plasma are interchanged while the magnetic energy decreases. The perturbed interface has ripples, such that the magnetic field lies parallel to the crests and troughs of the ripples.

A particular case occurs when a region of plasma is free of magnetic field and is confined by a magnetic field that wraps around the plasma with a radius of curvature ${R}_{c}$ in a concave manner. For a perturbation of wavenumber $k$ and time dependence ${e}^{i\omega t}$, the growth rate $(i\omega )$ is determined by

Examples of interchange instabilities include the *Rayleigh–Taylor instability*, the *Kelvin–Helmholtz instability* and the *sausage instability*. Indeed, for a plane interface, an interchange of magnetic field lines with wavenumber vector normal to the field $(k\cdot B=0)$ is usually the most unstable, since the field lines remain straight during such a perturbation, with magnetic tension forces doing no work.

##### 2.2.3 Rayleigh–Taylor Instability

Consider first a horizontal boundary separating two incompressible, inviscid plasmas with uniform density ${\rho}_{0}^{(-)}$ and ${\rho}_{0}^{(+)}$, and gravity acting down across the interface. When perturbations behave like ${e}^{i\omega t}$ and the denser plasma lies above the rarer plasma $({\rho}_{0}^{(+)}>{\rho}_{0}^{(-)})$, the boundary is unstable to a rippling (or fluting) with a growth rate $(i\omega )$ determined by

An extra uniform vertical magnetic field $({B}_{0}\widehat{z})$ changes the growth rate but not the stability. However, a uniform horizontal field $({B}_{0}\widehat{x})$ has no effect for wavenumbers normal to the field $({k}_{x}=0)$, but tends to inhibit the instability for ripples along the field $(k={k}_{x})$.

The *hydromagnetic Rayleigh–Taylor instability* occurs when a plasma of density ${\rho}_{0}^{(+)}$ is supported against gravity by a magnetic field $({B}_{0}^{(-)}\widehat{x})$ (Figure 1). Its dispersion relation is

for which the most unstable mode has wavenumber ${k}_{x}=0$ and a growth rate

##### 2.2.4 Sausage and Kink Instability of a Flux Tube

In the laboratory a *linear pinch* is a cylindrical plasma tube (of radius *a*, pressure ${p}_{0}$ and density ${\rho}_{0}$), confined by the azimuthal magnetic field created by a current ($J\phantom{\rule{0.2em}{0ex}}\widehat{z}$) that flows inside the tube or near its surface (Figure 2a). A Lorentz force is directed inwards and balanced by a pressure gradient that acts outward. When the plasma column has no magnetic field, it is unstable to the *sausage instability* (Figure 2b) with a growth rate $i\omega =[2{p}_{0}k/({\rho}_{0}a{)]}^{1/2}$. Modes with a large enough wavelength to destroy the whole column ($k\approx {a}^{-1}$) have growth rate

A linear pinch with a purely azimuthal external field ${B}_{\varphi}$ and a helical kink perturbation $\left\{\xi =\xi (R)\mathrm{exp}[i(\varphi +kz)+i\omega t]\right\}$ is unstable to a *kink instability* for all axial wavenumbers. The same is true for a *lateral kink* (Figure 2c) with a perturbation proportional to ${e}^{i\varphi}\mathrm{cos}kz$, which may be obtained by superposing two oppositely twisted helical perturbations like ${e}^{i(\varphi +kz)}$ and ${e}^{i(\varphi -kz)}$. Unlike the sausage instability, the kink instability cannot be stabilised by an axial field outside the plasma tube.

For a simple laboratory torus of major radius ${R}_{0}$, helical kink instability is present when the twist ($\Phi $) is large enough ($\Phi \ge 2\pi $). An interesting solar example is a coronal loop of uniform twist, for which, in the presence of photospheric line-tying, the loop becomes kink unstable when its twist exceeds a value that depends on the magnetic profile but is typically $2.5\pi $.

##### 2.2.5 Hydrodynamic Instability

There are many types of hydrodynamic instability. Viscous flows in channels, boundary layers, jets, and shear layers become unstable and can develop into turbulent states. If ${L}_{0}$ is a typical length, ${V}_{0}$ a typical fluid speed, and $\nu $ the kinematic viscosity, instability ensues when the Reynolds number $(Re\equiv {L}_{0}{V}_{0}/\nu )$ exceeds a critical value $(R{e}^{*})$, typically 40,000.

The following flows have been studied in detail, including the effects of adding magnetic fields:

plane Couette flow, with a linear profile [${v}_{z}(x)={V}_{0}x/{L}_{0}$];

Couette flow, namely, a circular flow with angular speed $\text{\Omega}(R)$ between two coaxial rotating cylinders;

plane Poiseuille flow, with a parabolic profile $[{v}_{2}(x)={V}_{0}(1-{x}^{2}/{L}_{0}^{2})]$;

Poiseuille flow, namely, the corresponding flow in a circular pipe;

Kelvin–Helmholtz instability, which occurs when one fluid rests on top of another and the two are in relative motion.

##### 2.2.6 Resistive Instability

This includes three types, namely, gravitational, rippling, and tearing modes, which occur in a current sheet or in a sheared plasma due to the presence of magnetic diffusion.

##### 2.2.7 Convective Instability

A horizontal slab of fluid of thickness $d$ that is heated from below can become unstable when the temperature difference $\text{\Delta}T$ between the upper and lower boundaries is too large. The temperature difference is measured by the Rayleigh number

where $\alpha ,\kappa ,\nu $ are the coefficients of volume expansion, thermometric conductivity, and kinematic viscosity. The onset of instability occurs when $Ra$ exceeds a critical value, which for a Boussinesq fluid (see, e.g., Priest, 2014) with free boundaries is $27{\pi}^{4}/4\approx 658$. The effects of rotation and magnetic field on the onset of convection and development of instability have been studied in detail.

##### 2.2.8 Radiatively-Driven Thermal Instability

In an optically thin plasma such as the solar corona, the radiative loss term in the energy equation naturally produces a thermal or radiative instability unless thermal conduction along field lines is strong enough to stop it. Consider, for example, a plasma of temperature ${T}_{0}$ and density ${\rho}_{0}$, in thermal balance per unit mass

between mechanical heating ($h=constant$) due to some unknown mechanism and optically thin radiation ($\chi \rho {T}^{\alpha}$), where $\chi $ and $\alpha $ are constants. Now consider a constant-pressure $({p}_{0})$ perturbation satisfying an energy equation

After substituting for $h$ from the equilibrium equation and for $\rho $ from the perfect gas law $\left[\rho =m{p}_{0}\text{/}({k}_{B}{T}_{0})\right]$, this becomes

Hence, when $\alpha <1$, a small decrease in temperature $(T<{T}_{0})$ makes $\partial T/\partial t<0$ and so the perturbation continues on a time scale

In practice, the radiative loss is such that $\alpha <1$ for temperatures larger than about ${10}^{5}\text{K}$. The effect of magnetic fields on the instability, as well as its nonlinear development, have been studied.

#### 2.3 Magnetic Reconnection

Often in astrophysics, such as the solar corona, the magnetic field is the main source of free energy, but since the magnetic Reynolds number is invariably enormous in value the energy is not easy to extract. To circumvent this problem, extremely thin sheets of current are formed where the magnetic gradients are much larger than normal and where the magnetic field can slip through the plasma rather than being frozen to the plasma. Then the process of magnetic reconnection converts magnetic energy to other forms and is responsible for many dynamic processes in astrophysics and space physics, such as solar flares (Longcope, 2020) and geomagnetic substorms. It also may well be responsible for heating at least part of the solar corona. In two dimensions, the theory is well developed, but in three dimensions it is in its infancy (Pontin, 2020).

##### 2.3.1 Two-Dimensional Reconnection

Magnetic field changes due to transport and diffusion of field lines are described by the induction equation (9). The advection term is much larger than the diffusion term in most of the Sun, and so the magnetic field is attached to the plasma and cannot use its energy to heat the plasma ohmically. The exception is in extremely thin current sheets, where the gradient of the magnetic field is so large that the magnetic field lines can diffuse through the plasma, break and reconnect.

These current sheets can form in 2D at neutral points where the magnetic field vanishes, and which therefore act as weak spots in the magnetic field. The current sheets look like singularities when viewed from afar, but they are resolved by magnetic diffusion (Figure 3). The effects of this local process are:

to alter the magnetic connectivity (and so the topology) of magnetic field lines, and this in turn changes the paths of fast particles and heat along the field;

to change magnetic energy into heat and kinetic energy;

to produce enormous electric currents, electric fields, shock waves and turbulence, each of which may play a role in accelerating fast particles to high energies.

##### 2.3.2 2D Reconnection: Sweet-Parker Mechanism

Suppose the magnetic field in a current sheet of width $l$ is modeled by a one-dimensional field ${B}_{y}(x,t)$ with a current density ${j}_{z}={\mu}^{-1}d{B}_{y}/dx$. If the field slips through the plasma at rest according to Equation (17), the field lines diffuse inwards at a speed ${v}_{d}=\eta /l$ and cancel or “annihilate” at $x=0$, while the width of the current sheet diffuses outward at the same speed, and the magnetic energy is transformed into heat by Ohmic dissipation $({j}^{2}/\sigma )$.

The Sweet–Parker model (Parker, 1957; Sweet, 1958) considers a steady state balance between such diffusion and the bringing in from large distances of magnetic flux and plasma at the same speed ${v}_{d}$ (Figure 4). It is an order of magnitude model for a simple diffusion region of length $2L$ and width $2l$ between oppositely directed fields of magnitude ${B}_{i}$.

Such a balance between outward diffusion and inwards advection means that the speed $({v}_{i})$ at which the field lines are brought in is

A steady-state conservation of mass implies that the rate $(4\rho L{v}_{i})$ at which mass is entering both sides of the sheet must equal the rate $(4\rho l{v}_{o})$ at which it is leaving both ends at speed ${v}_{o}$, so that, if the density is uniform ($\rho =constant$),

The sheet width $(l)$ may then be eliminated between Equations (26) and (27) to give the reconnection rate (i.e., the inflow speed) $\left({v}_{i}\right)$ as

In dimensionless variables this may be rewritten ${M}_{i}=\sqrt{{v}_{o}/{v}_{Ai}}/\sqrt{{R}_{mi}},$ in terms of the *inflow Alfvén Mach number*$({M}_{i}\equiv {v}_{i}/{v}_{Ai})$ and *magnetic Reynolds number*$({R}_{mi}\equiv L\phantom{\rule{0.2em}{0ex}}{v}_{Ai}/\eta )$ based on the inflow Alfvén speed $({v}_{Ai}={B}_{i}/\sqrt{\mu \rho})$).

But, what is the outflow speed ${v}_{o}$? This is determined by the equation of motion. Suppose that $x$- and $y$-axes are set up along and normal to the sheet, respectively, with the origin situated at the centre of the sheet. Suppose also that the sheet is long and thin, and that the inflow is much slower than the Alfvén speed. Then inertial and magnetic tension forces are negligible in the transverse direction (i.e., $y$-direction), and so the $y$-component of Equation (11) reduces to $0=-\partial /\partial y[p+{B}^{2}/(2\mu )]$ By integrating from the inflow point $(0,l)$ to the neutral point $(0,0)$, the neutral point pressure is then ${p}_{N}={p}_{i}+{B}_{i}^{2}/(2\mu ).$ However, the longitudinal (i.e., $x$) component of Equation (11) is $\rho {v}_{x}\partial {v}_{x}/\partial x=j{B}_{y}-\partial p/\partial x,$ which may be evaluated at $({\scriptscriptstyle \frac{1}{2}}L,0)$ to give

Here $j\approx \partial {B}_{x}/\partial y\approx {B}_{i}/l$, $\partial {v}_{x}/\partial x\approx {v}_{o}/L$, ${v}_{x}\approx \frac{1}{2}{v}_{o}$ and ${B}_{y}\approx \frac{1}{2}{B}_{o}$. After substituting for ${B}_{o}/l\approx {B}_{i}/L$ and ${p}_{N}$, it becomes

The outflow speed therefore depends on the imposed values of the Alfvén speed $({v}_{Ai})$ and the inflow $({p}_{i})$ and outflow $({p}_{o})$ pressures. In particular, when there is no pressure gradient along the sheet $({p}_{o}={p}_{N}={p}_{i}+{B}_{i}^{2}/(2\mu ))$, so that acceleration is, by the Lorentz force alone, the classical Sweet–Parker result $({v}_{o}={v}_{Ai})$, so that plasma is expelled from the sheet at the inflow Alfvén speed ${v}_{Ai}={B}_{i}/\sqrt{(}\mu {\rho}_{i})$.

The fields in that case reconnect at a speed given by Equation (28) as

or, in dimensionless form, as

where ${M}_{i}\equiv {v}_{i}/{v}_{Ai}$ is the *Alfvén Mach number* and ${R}_{mi}=L{v}_{Ai}/\eta $.

Now that the outflow speed ${v}_{o}$ is known, Equation (27) in turn determines the sheet width as $l=L\phantom{\rule{0.6em}{0ex}}{v}_{i}/{v}_{o}=L/{R}_{mi}{\phantom{\rule{0.6em}{0ex}}}^{1/2}$, while the outflow magnetic field strength $({B}_{o})$ from flux conservation $({v}_{i}\phantom{\rule{0.6em}{0ex}}{B}_{i}={v}_{o}\phantom{\rule{0.6em}{0ex}}{B}_{o})$ becomes ${B}_{o}={B}_{i}\phantom{\rule{0.6em}{0ex}}{v}_{i}/{v}_{o}={B}_{i}/{R}_{mi}{\phantom{\rule{0.6em}{0ex}}}^{1/2}$. Since ${R}_{m}\gg 1$, these imply ${v}_{1}\ll {v}_{Ai,}{B}_{o}\ll {B}_{i}$ and $l\ll L$.

However, if there is a pressure gradient along the sheet, the outflow speed and reconnection rate can be different. For example, if there is a strong outflow pressure ${p}_{o}>{p}_{N}$, the outflow will be smaller than the Alfvén speed and the reconnection rate slower than the classical rate.

Furthermore, the previous analysis does not include compressibility, since it assumes the density is uniform. Suppose instead that the input and output densities are ${\rho}_{i}$ and ${\rho}_{o}$, respectively, and consider acceleration by the magnetic force alone. Then the equation for continuity of mass (27) is altered to

and the equation of motion along the sheet is modified to ${\rho}_{o}{v}_{o}^{2}/L={B}_{i}{B}_{o}/(\mu l).$ The result is that the inflow (28) and outflow speeds become

Thus, it can be seen that, when ${\rho}_{o}>{\rho}_{i}$, compressibility slows down the outflow by a factor ${({\rho}_{i}/{\rho}_{o})}^{\frac{1}{2}}$, while the reconnection rate [${v}_{i}=({\rho}_{o}/{\rho}_{i}{)}^{\frac{1}{4}}\phantom{\rule{0.2em}{0ex}}{(\eta {v}_{Ai}/L)}^{\frac{1}{2}}$] is enhanced by a factor ${({\rho}_{o}/{\rho}_{i})}^{\frac{1}{4}}$, since the width of the sheet is decreased by compression. The density ratio ${\rho}_{o}/{\rho}_{i}\equiv ({p}_{o}/{p}_{i})({T}_{i}/{T}_{o})$ depends on the temperature ratio, which in turn depends on the details of the energy equation.

##### 2.3.3 2D Reconnection: Tearing-Mode Instability

Furth, Killeen and Rosenbluth (1963) discovered that, in the presence of magnetic diffusion, Lorentz forces can drive three kinds of *resistive instability* in a current sheet or a sheared magnetic field. The effect is to make the magnetic field reconnect in a time-dependent way. In a current sheet of width $2l$ there are two time scales of interest, namely, the time ${\tau}_{d}={l}^{2}/\eta $ at which the magnetic fields diffuse through the plasma and the time ${\tau}_{A}=l/{v}_{A}$ at which magnetic waves propagate across the sheet.

Since the magnetic Reynolds number is generally very much larger than unity, diffusion is very much slower than magnetic disturbances (i.e., ${\tau}_{d}\gg {\tau}_{A}$). ). Each of the three resistive instabilities grow on a time scale ${\tau}_{d}^{(1-c)}{\tau}_{A}^{c}$, which is intermediate between ${\tau}_{d}$ and ${\tau}_{A}$, since $c$ is a constant lying between 0 and 1. They have the effect of creating in 2D many small-scale magnetic loops (or magnetic islands) or in 3D magnetic flux ropes (or so-called plasmoids or current filaments), which subsequently diffuse away. These instabilities may play an important role in filamentation of magnetic fields and current sheets.

The *gravitational* and *rippling modes* are driven by gradients in density $[{\rho}_{0}(x)]$ and resistivity $[\eta (x)]$, respectively, in a direction $\left(x\right)$ across a sheet. Their wavelengths are short, similar to the width of the sheet $(kl\approx 1)$. The fine-scale filamentary structure that they create may lead to turbulent diffusion of plasma across the magnetic field, with accompanying heating. The growth rates (${\omega}_{g}$ and ${\omega}_{r}$, respectively) are given by

where ${\tau}_{G}=(-g/{\rho}_{0}\phantom{\rule{0.6em}{0ex}}d{\rho}_{0}/dx{)}^{-1/2}$ is the gravitational time scale. In the solar corona, the magnetic diffusivity depends on temperature, and so variations in temperature from one field line to another naturally give rise to spatial variations $({\eta}_{0}(x))$ of diffusivity.

In contrast, the *tearing-mode instability* has a longer wavelength than the sheet width $(kl<1)$. Its growth rate is

for wavenumbers $(k)$ in the range ${({\tau}_{A}/{\tau}_{d})}^{1/4}<kl<1$. The smallest allowable wavelengths $(l)$ grow on a time scale ${\tau}_{d}^{3/5}{\tau}_{A}^{2/5}$. Diffusion, which allows the reconnection, is significant in a very narrow layer of width ${(kl)}^{13/5}{({\tau}_{A}/{\tau}_{d})}^{-2/5}l$. The fastest growing mode occurs at the longest wavelength and has a growth rate

The physical cause of the instability is that, if one starts with a one-dimensional sheet with straight field lines and perturbs the field in the way shown in Figure 5, then the resulting force imbalance is in such a direction as to make the perturbation grow. Magnetic tension pulls the new loops of magnetic field to left and right away from the X-points (see Figure 5), while a magnetic pressure gradient pushes plasma toward the X-points from above and below. The restoring magnetic tension force with a large-scale field line curvature is minimised for long wavelengths.

Resistive modes can also take place in a sheared magnetic field, because the addition of a uniform field normal to the plane of Figure 5 has no effect on the stability analysis. Sheared fields tend to be resistively unstable at many thin sheaths throughout a structure. At any particular location, the instability has a vector wavenumber $(k)$ that is perpendicular to the equilibrium field $({B}_{0})$, i.e., $k\cdot {B}_{0}=0.$

##### 2.3.4 2D Fast Reconnection:

The rate of reconnection by the Sweet–Parker model was found to be much too slow to explain the energy release in solar flares, but Petschek (1964) discovered the first model for much faster reconnection. Later, Priest and Forbes (1986) discovered a whole family of such models, of which Petschek reconnection is one member. The reason why these models give a faster rate of reconnection is that the diffusion region is very much smaller (Figure 6), and so the magnetic field can slip through the plasma at a much faster speed. Surrounding the diffusion region there are two pairs of slow-mode shock waves, which stand in the flow and convert the inflowing magnetic energy into heat and kinetic energy of two outflowing hot streams of high-speed plasma.

Whereas for the Sweet–Parker diffusion region there is equipartition, with half of the incoming magnetic energy converted into heat and half into kinetic energy, Petschek’s mechanism converts 2/5 into heat and 3/5 into kinetic energy. However, the kinetic energy of the hot jets of plasma can later be dissipated viscously and ohmically in the surrounding medium.

Consider a plasma with very large magnetic Reynolds number so that in most of the region of consideration the plasma is frozen to the magnetic field and moves with it. For driven reconnection, it is the plasma that carries the magnetic field along. The reconnection rate then depends on the speed with which the plasma is carrying the magnetic field into the reconnection region, where the magnetic field can slip through the plasma. This process will occur at any rate between the Sweet-Parker rate and a maximum value given by

where ${M}_{e}$ and ${R}_{me}$ are evaluated at some large external distance from the origin. Whereas for typical solar values (say, ${R}_{me}{=10}^{8}$) the Sweet–Parker rate equation (28) is about ${10}^{-4}$, for Petschek reconnection, the maximum rate equation (31) is typically about ${10}^{-2}$ and so is fast enough for a solar flare. For free reconnection, where the reconnection has free boundary conditions and so is allowed to proceed at its natural rate, it is found that it does in fact tend to operate at the maximum rate.

Priest & Forbes’s (1986) additional regimes of fast reconnection have different boundary conditions and their inflow regions possess pressure gradients, whereas for Petschek’s model the inflow region has no pressure gradients and so consists of a potential magnetic field. They have been reproduced in numerical experiments provided the diffusion region diffusivity is enhanced, which is often expected in practice due to current-driven micro-turbulence.

One aspect of the simulations was for long a puzzle. When $\eta $ is uniform, usually the steady-state solutions can no longer be sustained. However, Baty et al. (2014) used time-dependent numerical simulations and analytical modeling to clarify the conditions under which fast reconnection with an enhanced resistivity is stable. Baty, Forbes, and Priest (2014), Baty, Priest, and Forbes (2006), and Forbes, Priest, Seaton, and Litvinenko (2013) found that, for uniform resistivity, Petschek reconnection is structurally unstable, and so reconnection can only occur at the slow, Sweet–Parker rate. For nonuniform resistivity, however, reconnection can occur at the much faster rate provided the resistivity profile has a sufficiently strong maximum near the X-point. If this condition is satisfied, then the scale length of the nonuniformity determines the reconnection rate.

Three possible modes for fast reconnection have been studied, which have different microscopic plasma physics at work, but which all lead to roughly the same maximum mean rate of reconnection. The first is the Petschek or almost-uniform reconnection that has just been outlined. The second mode occurs when reconnection is collisionless rather than collisional. In this case, the resistive diffusion region is replaced by an ion diffusion region (governed by the Hall effect) together with a much smaller electron diffusion region (e.g., Birn & Priest, 2007; Shay & Drake, 1998). The third mode takes place when the central sheet is so long that it goes unstable to secondary tearing mode instability and a regime of impulsive bursty reconnection results (e.g., Loureiro, Schekochihin, & Cowley, 2007; Priest, 1986), but the mean reconnection rate is similar to the other two modes.

##### 2.3.5 Three-Dimensional (3D) Reconnection

Whereas the theory for 2D reconnection is well established, the way reconnection operates in 3D is only just beginning to be investigated (e.g., Hesse, Schindler, & Birn, 1991; Hornig, 2001; Parnell, Haynes, & Galsgaard, 2010; Priest & Forbes, 2000). Surprisingly, several key differences from 2D have been discovered, as follows.

First of all, magnetic neutral points, at which the magnetic field vanishes, have a completely different structure in 3D. A neutral point in 2D possesses either an X-type or an O-type structure, where the nearby magnetic field lines have a hyperbolic or elliptical structure, respectively. In 3D the simplest field that vanishes at the origin and satisfies $\nabla \cdot B=0$ has components

whose field lines are sketched in Figure 7. Two types of field line link to the null point: a *spine curve*, which is isolated and approaches the null along the positive and negative $z$-axes; and a *fan surface* consisting of field lines that exit the null in the $xy$-plane. The fan surface separates the field lines above the $xy$-plane from those below it, and it is referred to as a *separatrix surface*.

Second, the topology is much more complex in 3D. An example is shown in Figure 8a, which shows the field in the solar corona produced by four magnetic sources on the base (the photosphere). In 2D, they are situated in the order $+,-,+,-$, and suppose they are called ${\text{P}}_{+},{\text{P}}_{-},{\text{Q}}_{+},{\text{Q}}_{-}$. They produce an X-type null point in the corona, from which four field lines radiate called *separatrix curves*. They separate the corona into four topologically distinct regions, since field lines immediately to the left of the null all start at ${\text{P}}_{+}$ and end at ${\text{P}}_{-}$, whereas those to the right of the null link the other two sources. Furthermore, all the field lines below the null join ${\text{Q}}_{+}$ to ${\text{P}}_{-}$, while all the field lines above the null join ${\text{P}}_{+}$ to ${\text{Q}}_{-}$.

In 3D (Figure 8b), two 3D null points are located on the base at ${\text{N}}_{1}$ and ${\text{N}}_{2}$, say, shown by large dots, and the fan surfaces from ${\text{N}}_{1}$ and ${\text{N}}_{2}$ form dome-shaped surfaces that separate field lines below and above the dome. The domes intersect in a *separator*, a special field line that joins ${\text{N}}_{1}$ to ${\text{N}}_{2}$.

The third difference between 2D and 3D concerns reconnection. In 2D, reconnection occurs at an X-point, and it transfers magnetic flux from two regions to two other regions (Figure 8a). In 3D, there is one kind of reconnection, namely, *separator reconnection*, which is similar to 2D reconnection, since it transfers magnetic flux from two 3D regions to two other regions (Figure 8b). It occurs after a strong current has accumulated along a separator.

However, in 3D, reconnection take place in other ways when current has built up at other locations, such as (see Pontin, 2020):

at *null points* by torsional spine reconnection, torsional fan reconnection or, usually by *spine-fan reconnection*;

or at a so-called *quasi-separator* by *quasi-separator reconnection*.

Suppose the footpoints of magnetic field lines are mapped from one part of the photosphere to another. When there is a 3D null point in the overlying atmosphere, then a separatrix surface exists, across which the mapping possesses a sudden jump or discontinuity. When there is no null point or separatrix surface, there can still exist *quasi-separatrix surfaces* or *QSLs* for short, across which the mapping is quite continuous but it does possess a steep gradient. The result is that large currents can again build up along the quasi-separator and can give rise to magnetic reconnection.

Another new feature that turns up in three dimensions is a topological invariant known as *magnetic helicity*. This arises in two forms called *self-helicity* and *mutual helicity* (Berger, 1984). Self-helicity measures twisting and kinking of a magnetic flux tube. Mutual helicity represents the linkage between different flux tubes. The interesting feature is that, during 3D reconnection, the total magnetic helicity (i.e., self-helicity + mutual helicity) is conserved. However, magnetic helicity can be converted from one form to the other. For example, the conversion of mutual helicity to self-helicity has been suggested as the means whereby erupting flux ropes in the core of solar coronal mass ejections are created or enhanced (Priest & Longcope, 2017).

### 3. Conclusion

This article has briefly set the scene for several other articles on the different aspects of MHD and their application in solar and space physics. The physics that underlies the equations of MHD and the way they may be set up has been emphasized, including the assumptions and conditions for validity. In particular, the implications of the induction equation for the time-behavior of the magnetic field, and of the equation of motion have been laid out.

Different aspects of MHD have then been summarised, namely, magnetic equilibria, MHD waves, magnetic instabilities, and magnetic reconnection. One extension to the basic theory for single-fluid resistive MHD concerns the addition of two-fluid effects, which is treated in Khomenko (2020).

These aspects play important roles in many different phenomena in the Sun and its atmosphere and in the solar wind, which are described in the following articles in this encyclopedia. First of all in the *solar interior*, its dynamics are being observed with the help of helioseismology and modeled with numerical experiments. Great advances have been made here in understanding how the solar magnetic field may be generated by *dynamo action*, but it is not clear whether there is just one dynamo mechanism or several dynamos operating at different scales (Cameron, 2020).

Turning to the different parts of the solar atmosphere, the solar surface layer or photosphere outside sunspots at about 6,000 K used to be regarded as devoid of magnetic fields, but now ubiquitous small patches of strong magnetic field have been detected at the edges of both supergranules and granules (Chitta, Smitha, & Solanki, 2020). The warmer chromosphere is known to be extremely dynamic and highly filamentary, while the million-degree corona is dominated by the magnetic field, both for its structure and its heating. The corona consists of coronal loops, coronal holes, X-ray bright points, and a background diffuse component. It is thought to be heated somehow by magnetic reconnection or by MHD waves, but the detailed mechanisms and their relative contribution in different parts of the chromosphere and corona are unclear.

Traditionally, the photosphere, chromosphere and corona were classified as part of the quiet Sun, whose overall properties do not vary much with the 11-year cycle in sunspot numbers and therefore of solar activity. However, it is now known that they all vary, especially the intensity and structure of the corona with the solar cycle (Gestelyi & Owens, 2020). Sunspots (Rempel and Borrero, 2021) are one component of solar activity and are surrounded by regions of enhanced magnetic flux called active regions. Another is huge magnetic flux ropes called prominences that are located up in the corona, but containing cool plasma at 10,000 K (Mackay, 2021), but the causes of their fine-scale structure and dynamics and their eruption have not been identified. Other components include solar flares (Longcope, 2020) and coronal mass ejections (or CMEs). When large quiescent prominences erupt from outside active regions they produce slow CMEs, but when prominences erupt from active regions, where the magnetic fields and therefore the Lorentz forces are much larger, they tend to be accompanied by rapid CMEs occur and eruptive two-ribbon flares.

The solar corona extends into the heliosphere out to the Earth and beyond, but, apart from the corona that is contained within closed magnetic fields, it is flowing outward as the *solar wind* (Cranmer, 2019). The fast solar wind emerges from coronal holes in a steady manner and may possibly be accelerated by MHD waves, while the more dynamic slow solar wind escapes from regions around coronal streamers and may be produced by magnetic reconnection (Owens, 2020). When the solar wind reaches the neighbourhood of the Earth or other planets, there is a complex interaction, whose details vary considerably from one planet to another, depending partly on the strength of their magnetic field (Arridge, 2020).

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