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## Magnetohydrodynamic Equilibria

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.

## Magnetohydrodynamic Reconnection

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.

## Magnetohydrodynamic Waves

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.

## Solar Cycle

The Sun’s magnetic field drives the solar wind and produces space weather. It also acts as the prototype for an understanding of other stars and their planetary environments. Plasma motions in the solar interior provide the dynamo action that generates the solar magnetic field. At the solar surface, this is evident as an approximately 11-year cycle in the number and position of visible sunspots. This solar cycle is manifest in virtually all observable solar parameters, from the occurrence of the smallest detected magnetic features on the Sun to the size of the bubble in interstellar space that is carved out by the solar wind. Moderate to severe space-weather effects show a strong solar cycle variation. However, it is a matter of debate whether extreme space-weather follows from the 11-year cycle. Each 11-year solar cycle is actually only half of a solar magnetic “Hale” cycle, with the configuration of the Sun’s large-scale magnetic field taking approximately 22 years to repeat. At the start of a new solar cycle, sunspots emerge at mid-latitude regions with an orientation that opposes the dominant large-scale field, leading to an erosion of the polar fields. As the cycle progresses, sunspots emerge at lower latitudes. Around solar maximum, the polar field polarity reverses, but the sunspot orientation remains the same, leading to a build-up of polar field strength that peaks at the start of the next cycle. Similar magnetic cyclicity has recently been inferred at other stars.

## Solar Dynamo

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.

## Solar Photosphere

The Sun is a G2V star with an effective temperature of 5780 K. As the nearest star to Earth and the biggest object in the solar system, it serves as a reference for fundamental astronomical parameters such as stellar mass, luminosity, and elemental abundances. It also serves as a plasma physics laboratory. A great deal of researchers’ understanding of the Sun comes from its electromagnetic radiation, which is close to that of a blackbody whose emission peaks at a wavelength of around 5,000 Å and extends into the near UV and infrared. The bulk of this radiation escapes from the solar surface, from a layer that is a mere 100 km thick. This surface from where the photons escape into the heliosphere and beyond, together with the roughly 400–500 km thick atmospheric layer immediately above it (where the temperature falls off monotonically with distance from the Sun), is termed the solar photosphere. Observations of the solar photosphere have led to some important discoveries in modern-day astronomy and astrophysics. At low spatial resolution, the photosphere is nearly featureless. However, naked-eye solar observations, the oldest of which can plausibly be dated back to 800 bc, have shown there to be occasional blemishes or spots. Systematic observations made with telescopes from the early 1600s onward have provided further information on the evolution of these sunspots whose typical spatial extent is 10,000 km at the solar surface. Continued observations of these sunspots later revealed that they increase and decrease in number with a period of about 11 years and that they actually are a manifestation of the Sun’s magnetic field (representing the first observation of an extraterrestrial magnetic field). This established the presence of magnetic cycles on the Sun responsible for the observed cyclic behavior of solar activity. Such magnetic activity is now known to exist in other stars as well. Superimposed on the solar blackbody spectrum are numerous spectral lines from different atomic species that arise due to the absorption of photons at certain wavelengths by those atoms, in the cooler photospheric plasma overlying the solar surface. These spectral lines provide diagnostics of the properties and dynamics of the underlying plasma (e.g., the granulation due to convection and the solar p-mode oscillations) and of the solar magnetic field. Since the early 20th century, researchers have used these spectral lines and the accompanying polarimetric signals to decode the physics of the solar photosphere and its magnetic structures, including sunspots. Modern observations with high spatial (0.15 arcsec, corresponding to 100 km on the solar surface) and spectral (10 mÅ) resolutions reveal a tapestry of the magnetized plasma with structures down to tens of kilometers at the photosphere (three orders of magnitude smaller than sunspots). Such observations, combined with advanced numerical models, provide further clues to the very important role of the magnetic field in solar and stellar structures and the variability in their brightness. Being the lowest directly observable layer of the Sun, the photosphere is also a window into the solar interior by means of helioseismology, which makes use of the p-mode oscillations. Furthermore, being the lowest layer of the solar atmosphere, the photosphere provides key insights into another long-standing mystery, that above the temperature-minimum (~500 km above the surface at ~4000 K), the plasma in the extended corona (invisible to the naked eye except during a total solar eclipse) is heated to temperatures up to 1,000 times higher than at the visible surface. The physics of the solar photosphere is thus central to the understanding of many solar and stellar phenomena.

## Solar Physics: Overview

Solar physics is one of the liveliest branches of astrophysics at the current time, with many major advances that have been stimulated by observations from a series of space satellites and ground-based telescopes as well as theoretical models and sophisticated computational experiments. Studying the Sun is of key importance in physics for two principal reasons. Firstly, the Sun has major effects on the Earth and on its climate and space weather, as well as other planets of the solar system. Secondly, it represents a Rosetta stone, where fundamental astrophysical processes can be investigated in great detail. Yet, there are still major unanswered questions in solar physics, such as how the magnetic field is generated in the interior by dynamo action, how magnetic flux emerges through the solar surface and interacts with the overlying atmosphere, how the chromosphere and corona are heated, how the solar wind is accelerated, how coronal mass ejections are initiated and how energy is released in solar flares and high-energy particles are accelerated. Huge progress has been made on each of these topics since the year 2000, but there is as yet no definitive answer to any of them. When the answers to such puzzles are found, they will have huge implications for similar processes elsewhere in the cosmos but under different parameter regimes.

## Solar-Wind Origin

The Sun continuously expels a fraction of its own mass in the form of a steadily accelerating outflow of ionized gas called the “solar wind.” The solar wind is the extension of the Sun’s hot (million-degree Kelvin) outer atmosphere that is visible during solar eclipses as the bright and wispy corona. In 1958, Eugene Parker theorized that a hot corona could not exist for very long without beginning to accelerate some of its gas into interplanetary space. After more than half a century, Parker’s idea of a gas-pressure-driven solar wind still is largely accepted, although many questions remain unanswered. Specifically, the physical processes that heat the corona have not yet been identified conclusively, and the importance of additional wind-acceleration mechanisms continue to be investigated. Variability in the solar wind also gives rise to a number of practical “space weather” effects on human life and technology, and there is still a need for more accurate forecasting. Fortunately, recent improvements in both observations (with telescopes and via direct sampling by space probes) and theory (with the help of ever more sophisticated computers) are leading to new generations of predictive and self-consistent simulations. Attempts to model the origin of the solar wind are also leading to new insights into long-standing mysteries about turbulent flows, magnetic reconnection, and kinetic wave-particle resonances.

## Sunspots

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.