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Dark matter is one of the most fundamental and perplexing issues of modern physics. Its presence is deduced from a straightforward application of Newton’s theory of gravity to astronomical systems whose dynamical motion should be simple to understand. The success of Newton’s theory in describing the behavior of the solar system was one of the greatest achievements of the 18th century. Its subsequent use to deduce the presence of a previously unknown planet, Neptune, discovered in 1846, was the first demonstration of how minor departures from its predictions indicated additional mass. The expectation in the early 20th century, as astronomical observations allowed more distance and larger celestial systems to be studied, was that galaxies and collections of galaxies should behave like larger solar systems, albeit more complicated. However, the reality was quite different. It is not a minor discrepancy, as led to the discovery of Neptune, but it is extreme. The stars at the edges of galaxies are not behaving at all like Pluto at the edge of the solar system. Instead of having a slower orbital speed, as expected and shown by Pluto, they have the same speed as those much further in. If Newton’s law is to be retained, there must be much more mass in the galaxy than can be seen, and it must be distributed out to large distances, beyond the visible extent of the galaxy. This unseen mass is called “dark matter,” and its presence was becoming widely accepted by the 1970s. Subsequently, many other types of astrophysical observations covering many other types of object were made that came to the same conclusions. The ultimate realization was that the universe itself requires dark matter to explain how it developed the structures within it observed today. The current consensus is that one-fourth of the universe is dark matter, whereas only 1/20th is normal matter. This leaves the majority in some other form, and therein lies another mystery—“dark energy.” The modern form of Newton’s laws is general relativity, due to Albert Einstein. This offers no help in solving the problem of dark matter because most of the systems involved are nonrelativistic and the solutions to the general theory of relativity (GR) reproduce Newtonian behavior. However, it would not be right to avoid mentioning the possibility of modifying Newton’s laws (and hence GR) in such a way as to change the nonrelativistic behavior to explain the way galaxies behave, but without changing the solar system dynamics. Although this is a minority concept, it is nonetheless surviving within the scientific community as an idea. Understanding the nature of dark matter is one of the most intensely competitive research areas, and the solution will be of profound importance to astrophysics, cosmology, and fundamental physics. There is thus a huge “industry” of direct detection experiments predicated on the premise that there is a new particle species yet to be found, and which pervades the universe. There are also experiments searching for evidence of the decay of the particles via their annihilation products, and, finally, there are intense searches for newly formed unknown particles in collider experiments.

The fluid–gravity correspondence establishes a detailed connection between solutions of relativistic dissipative hydrodynamics and black hole spacetimes that solve Einstein’s equations in a spacetime with negative cosmological constant. The correspondence can be seen as a natural corollary of the holographic anti–de Sitter (AdS)/conformal field theory (CFT) correspondence, which arises from string theory. The latter posits a quantum duality between gravitational dynamics in AdS spacetimes and that of a CFT in one dimension less. The fluid–gravity correspondence applies in the statistical thermodynamic limit of the CFT but can be viewed as an independent statement of a relation between two classic equations of physics: the relativistic Navier–Stokes equations and Einstein’s equations. The general structure of relativistic fluid dynamics is formulated in terms of conservation equations of energy–momentum and charges, supplemented with constitutive relations for the corresponding current densities. One can view this construction as an effective field theory for these conserved currents. This intuition applied to the gravitational equations of motion allows the solutions of relativistic hydrodynamics to be embedded as inhomogeneous, dynamical black holes in AdS spacetime.

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.

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.

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.

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.

The solar dynamo is the action of flows inside the Sun to maintain its magnetic field against Ohmic decay. On small scales the magnetic field is seen at the solar surface as a ubiquitous “salt-and-pepper” disorganized field that may be generated directly by the turbulent convection. On large scales, the magnetic field is remarkably organized, with an 11-year activity cycle. During each cycle the field emerging in each hemisphere has a specific East–West alignment (known as Hale’s law) that alternates from cycle to cycle, and a statistical tendency for a North-South alignment (Joy’s law). The polar fields reverse sign during the period of maximum activity of each cycle. The relevant flows for the large-scale dynamo are those of convection, the bulk rotation of the Sun, and motions driven by magnetic fields, as well as flows produced by the interaction of these. Particularly important are the Sun’s large-scale differential rotation (for example, the equator rotates faster than the poles), and small-scale helical motions resulting from the Coriolis force acting on convective motions or on the motions associated with buoyantly rising magnetic flux. These two types of motions result in a magnetic cycle. In one phase of the cycle, differential rotation winds up a poloidal magnetic field to produce a toroidal field. Subsequently, helical motions are thought to bend the toroidal field to create new poloidal magnetic flux that reverses and replaces the poloidal field that was present at the start of the cycle. It is now clear that both small- and large-scale dynamo action are in principle possible, and the challenge is to understand which combination of flows and driving mechanisms are responsible for the time-dependent magnetic fields seen on the Sun.

Sunspots are the most prominent manifestations of magnetic fields on the visible surface of the Sun (photosphere). While historic records mention sunspot observations by eye more than two thousand years ago, the physical nature of sunspots has been unraveled only in the past century starting with the pioneering work of Hale and Evershed. Sunspots are compact magnetic-field concentrations with a field strength exceeding 3,000 G in their center, a horizontal extent of about 30 Mm and typical lifetimes on the order of weeks. Research during the past few decades has focused on characterizing their stunning fine structure that became evident in high-resolution observations. The central part of sunspots (umbra) appears, at visible wavelengths, dark due to strongly suppressed convection (about 20% of the brightness of unperturbed solar granulation); the surrounding penumbra with a brightness of more than 75% of solar granulation shows efficient convective energy transport, while at the same time the constraining effects of magnetic field are visible in the filamentary fine structure of this region. The developments of the past 100 years have led to a deep understanding of the physical structure of sunspots. Key developments were the parallel advance of instrumentation; the advance in the interpretation of polarized light, leading to reliable inversions of physical parameters in the solar atmosphere; and the advance of modeling capabilities enabling radiation magnetohydrodynamic (MHD) simulations of the solar photosphere on the scale of entire sunspots. These developments turned sunspots into a unique plasma laboratory for studying the interaction of strong magnetic field with convection. The combination of refined observation and data analysis techniques provide detailed physical constraints, while numerical modeling has advanced to a level where a direct comparison with remote sensing observations through forward modeling of synthetic observations is now feasible. While substantial progress has been made in understanding the sunspot fine structure, fundamental questions regarding the formation of sunspots and sunspot penumbrae are still not answered.