Magnetohydrodynamic (MHD) waves represent one of the macroscopic processes responsible for the transfer of the energy and information in plasmas. The existence of MHD waves is due to the elastic and compressible nature of the plasma, and by the effect of the frozen-in magnetic field. Basic properties of MHD waves are examined in the ideal MHD approximation, including effects of plasma nonuniformity and nonlinearity. In a uniform medium, there are four types of MHD wave or mode: the incompressive Alfvén wave, compressive fast and slow magnetoacoustic waves, and non-propagating entropy waves. MHD waves are essentially anisotropic, with the properties highly dependent on the direction of the wave vector with respect to the equilibrium magnetic field. All of these waves are dispersionless. A nonuniformity of the plasma may act as an MHD waveguide, which is exemplified by a field-aligned plasma cylinder that has a number of dispersive MHD modes with different properties. In addition, a smooth nonuniformity of the Alfvén speed across the field leads to mode coupling, the appearance of the Alfvén continuum, and Alfvén wave phase mixing. Interaction and self-interaction of weakly nonlinear MHD waves are discussed in terms of evolutionary equations. Applications of MHD wave theory are illustrated by kink and longitudinal waves in the corona of the Sun.

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## Magnetohydrodynamic Waves

### V.M. Nakariakov

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## Magnetohydrodynamic Equilibria

### Thomas Wiegelmann

Magnetohydrodynamic equilibria are time-independent solutions of the full magnetohydrodynamic (MHD) equations. An important class are static equilibria without plasma flow. They are described by the magnetohydrostatic equations
j
×
B
=
∇
p
+
ρ
∇
Ψ
,
∇
×
B
=
μ
0
j
,
∇
·
B
=
0.
B
is the magnetic field,
j
the electric current density,
p
the plasma pressure,
ρ
the mass density,
Ψ
the gravitational potential, and
µ
0
the permeability of free space. Under equilibrium conditions, the Lorentz force
j
×
B
is compensated by the plasma pressure gradient force and the gravity force.
Despite the apparent simplicity of these equations, it is extremely difficult to find exact solutions due to their intrinsic nonlinearity. The problem is greatly simplified for effectively two-dimensional configurations with a translational or axial symmetry. The magnetohydrostatic (MHS) equations can then be transformed into a single nonlinear partial differential equation, the Grad–Shafranov equation. This approach is popular as a first approximation to model, for example, planetary magnetospheres, solar and stellar coronae, and astrophysical and fusion plasmas.
For systems without symmetry, one has to solve the full equations in three dimensions, which requires numerically expensive computer programs. Boundary conditions for these systems can often be deduced from measurements. In several astrophysical plasmas (e.g., the solar corona), the magnetic pressure is orders of magnitudes higher than the plasma pressure, which allows a neglect of the plasma pressure in lowest order. If gravity is also negligible, Equation 1 then implies a force-free equilibrium in which the Lorentz force vanishes.
Generalizations of MHS equilibria are stationary equilibria including a stationary plasma flow (e.g., stellar winds in astrophysics). It is also possible to compute MHD equilibria in rotating systems (e.g., rotating magnetospheres, rotating stellar coronae) by incorporating the centrifugal force. MHD equilibrium theory is useful for studying physical systems that slowly evolve in time. In this case, while one has an equilibrium at each time step, the configuration changes, often in response to temporal changes of the measured boundary conditions (e.g., the magnetic field of the Sun for modeling the corona) or of external sources (e.g., mass loading in planetary magnetospheres). Finally, MHD equilibria can be used as initial conditions for time-dependent MHD simulations. This article reviews the various analytical solutions and numerical techniques to compute MHD equilibria, as well as applications to the Sun, planetary magnetospheres, space, and laboratory plasmas.

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## Multi-Fluid Effects in Magnetohydrodynamics

### Elena Khomenko

Multi-fluid magnetohydrodynamics is an extension of classical magnetohydrodynamics that allows a simplified treatment plasmas with complex chemical mixtures. The types of plasma susceptible to multi-fluid effects are those containing particles with properties significantly different from those of the rest of the plasma in either mass, or electric charge, such as neutral particles, molecules, or dust grains. In astrophysics, multi-fluid magnetohydrodynamics is relevant for planetary ionospheres and magnetospheres, the interstellar medium, and the formation of stars and planets, as well as in the atmospheres of cool stars such as the Sun. Traditionally, magnetohydrodynamics has been a classical approximation in many astrophysical and physical applications. Magnetohydrodynamics works well in dense plasmas where the typical plasma scales (e.g., cyclotron frequencies, Larmor radius) are significantly smaller than the scales of the processes under study. Nevertheless, when plasma components are not well coupled by collisions it is necessary to replace single-fluid magnetohydrodynamics by multi-fluid theory. The present article provides a description of environments in which a multi-fluid treatment is necessary and describes modifications to the magnetohydrodynamic equations that are necessary to treat non-ideal plasmas. It also summarizes the physical consequences of major multi-fluid non-ideal magnetohydrodynamic effects including ambipolar diffusion, the Hall effect, the battery effect, and other intrinsically multi-fluid effects. Multi-fluid theory is an intermediate step between magnetohydrodynamics dealing with the collective behaviour of an ensemble of particles, and a kinetic approach where the statistics of particle distributions are studied. The main assumption of multi-fluid theory is that each individual ensemble of particles behaves like a fluid, interacting via collisions with other particle ensembles, such as those belonging to different chemical species or ionization states. Collisional interaction creates a relative macroscopic motion between different plasma components, which, on larger scales, results in the non-ideal behaviour of such plasmas. The non-ideal effects discussed here manifest themselves in plasmas at relatively low temperatures and low densities.

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## Magnetohydrodynamic Reconnection

### D. I. Pontin

Magnetic reconnection is a fundamental process that is important for the dynamical evolution of highly conducting plasmas throughout the Universe. In such highly conducting plasmas the magnetic topology is preserved as the plasma evolves, an idea encapsulated by Alfvén’s frozen flux theorem. In this context, “magnetic topology” is defined by the connectivity and linkage of magnetic field lines (streamlines of the magnetic induction) within the domain of interest, together with the connectivity of field lines between points on the domain boundary. The conservation of magnetic topology therefore implies that magnetic field lines cannot break or merge, but evolve only according to smooth deformations. In any real plasma the conductivity is finite, so that the magnetic topology is not preserved everywhere: magnetic reconnection is the process by which the field lines break and recombine, permitting a reconfiguration of the magnetic field. Due to the high conductivity, reconnection may occur only in small dissipation regions where the electric current density reaches extreme values. In many applications of interest, the change of magnetic topology facilitates a rapid conversion of stored magnetic energy into plasma thermal energy, bulk-kinetic energy, and energy of non-thermally accelerated particles. This energy conversion is associated with dynamic phenomena in plasmas throughout the Universe. Examples include flares and other energetic phenomena in the atmosphere of stars including the Sun, substorms in planetary magnetospheres, and disruptions that limit the magnetic confinement time of plasma in nuclear fusion devices. One of the major challenges in understanding reconnection is the extreme separation between the global system scale and the scale of the dissipation region within which the reconnection process itself takes place. Current understanding of reconnection has developed through mathematical and computational modeling as well as dedicated experiments in both the laboratory and space. Magnetohydrodynamic (MHD) reconnection is studied in the framework of magnetohydrodynamics, which is used to study plasmas (and liquid metals) in the continuum approximation.

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## Solar Wind: Interaction With Planets

### Chris Arridge

The interaction between the solar wind and planetary bodies in our solar system has been investigated since well before the space age. The study of the aurora borealis and australis was a feature of the Enlightenment and many of the biggest names in science during that period had studied the aurora. Many of the early scientific discoveries that emerged from the burgeoning space program in the 1950s and 1960s were related to the solar wind and its interaction with planets, starting with the discovery of the Van Allen radiation belts in 1958. With the advent of deep space missions, such as Venera 4, Pioneer 10, and the twin Voyager spacecraft, the interaction of the solar wind other planets was investigated and has evolved into a sub-field closely allied to planetary science. The variety in solar system objects, from rocky planets with thick atmospheres, to airless bodies, to comets, to giant planets, is reflected in the richness in the physics found in planetary magnetospheres and the solar wind interaction. Studies of the solar wind-planet interaction has become a consistent feature of more recent space missions such as Cassini-Huygens (Saturn), Juno (Jupiter), New Horizons (Pluto) and Rosetta (67/P Churyumov–Gerasimenko), as well more dedicated missions in near-Earth space, such as Cluster and Magnetosphere Multiscale. The field is now known by various terms, including space (plasma) physics and solar-terrestrial physics, but it is an interdisciplinary science involving plasma physics, electromagnetism, radiation physics, and fluid mechanics and has important links with other fields of space science, including solar physics, planetary aeronomy, and planetary geophysics. Increasingly, the field is relying on high-performance computing and methods from data science to answer important questions and to develop predictive capabilities. The article explores the origins of the field, examines discoveries made during the heyday of the space program to the late 1970s and 1980s, and other hot topics in the field.

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## Magnetohydrodynamics: Overview

### E.R. Priest

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.