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date: 18 October 2019

Magnetohydrodynamic Equilibria

This is an advance summary of a forthcoming article in the Oxford Research Encyclopedia of Physics. Please check back later for the full article.

Magnetohydrodynamic Equilibria are solutions of the full MHD equations, which are time independent. An important class is static equilibria without plasma flow, leading to the magnetohydrostatic equations.

B is the magnetic field, j the electric current density, P the plasma pressure, ρ the mass density, Ψ the gravitational potential, and µ0 the permeability of free space. Under equilibrium conditions the Lorentz force j×B is compensated by plasma forces, namely the plasma pressure gradient force and the gravity force.

Despite the relatively simple-looking 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 translation or axial symmetry. The magnetohydrostatic 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, 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, for example, the solar corona, the magnetic pressure is orders of magnitudes higher than the plasma pressures, which allows the neglect of the plasma in lowest order and to compute force-free configurations. Here the plasma pressure and gravity term in (1) are neglected so that the Lorentz force has to vanish.

Generalizations of magnetohydrostatic equilibria are stationary equilibria, including a stationary plasma flow, for example, 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. 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 modelling the corona) or external sources (e.g., mass loading in planetary magnetospheres). Finally, MHD equilibria can be used as an initial condition for time-dependent MHD simulations.

j×B=P+ρΨ,
(1)

×B=µ0j,
(2)

B=0.
(3)

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