Magnetohydrodynamics

Magnetohydrodynamics (MHD; also magneto-fluid dynamics or hydro­magnetics) is the study of the magnetic properties and behaviour of electrically conducting fluids. Examples of such magneto­fluids include plasmas, liquid metals, salt water, and electrolytes. The word 'magneto­hydro­dynamics' is derived from magneto- meaning magnetic field, hydro- meaning water, and dynamics meaning movement. The field of MHD was initiated by Hannes Alfvén, for which he received the Nobel Prize in Physics in 1970. Magnetohydrodynamics (MHD; also magneto-fluid dynamics or hydro­magnetics) is the study of the magnetic properties and behaviour of electrically conducting fluids. Examples of such magneto­fluids include plasmas, liquid metals, salt water, and electrolytes. The word 'magneto­hydro­dynamics' is derived from magneto- meaning magnetic field, hydro- meaning water, and dynamics meaning movement. The field of MHD was initiated by Hannes Alfvén, for which he received the Nobel Prize in Physics in 1970. The fundamental concept behind MHD is that magnetic fields can induce currents in a moving conductive fluid, which in turn polarizes the fluid and reciprocally changes the magnetic field itself. The set of equations that describe MHD are a combination of the Navier–Stokes equations of fluid dynamics and Maxwell’s equations of electro­magnetism. These differential equations must be solved simultaneously, either analytically or numerically. The first recorded use of the word magnetohydrodynamics is by Hannes Alfvén in 1942: The ebbing salty water flowing past London's Waterloo Bridge interacts with the Earth's magnetic field to produce a potential difference between the two river-banks. Michael Faraday called this effect 'magneto-electric induction' and tried this experiment in 1832 but the current was too small to measure with the equipment at the time, and the river bed contributed to short-circuit the signal. However, by a similar process the voltage induced by the tide in the English Channel was measured in 1851. The simplest form of MHD, Ideal MHD, assumes that the fluid has so little resistivity that it can be treated as a perfect conductor. This is the limit of infinite magnetic Reynolds number. In ideal MHD, Lenz's law dictates that the fluid is in a sense tied to the magnetic field lines. To explain, in ideal MHD a small rope-like volume of fluid surrounding a field line will continue to lie along a magnetic field line,even as it is twisted and distorted by fluid flows in the system. This is sometimes referred to as the magnetic field lines being 'frozen' in the fluid.The connection between magnetic field lines and fluid in ideal MHD fixes the topology of the magnetic field in the fluid—for example, if a set of magnetic field lines are tied into a knot, then they will remain so as long as the fluid/plasma has negligible resistivity. This difficulty in reconnecting magnetic field lines makes it possible to store energy by moving the fluid or the source of the magnetic field. The energy can then become available if the conditions for ideal MHD break down, allowing magnetic reconnection that releases the stored energy from the magnetic field. The ideal MHD equations consist of the continuity equation, the Cauchy momentum equation, Ampere's Law neglecting displacement current, and a temperature evolution equation. As with any fluid description to a kinetic system, a closure approximation must be applied to highest moment of the particle distribution equation. This is often accomplished with approximations to the heat flux through a condition of adiabaticity or isothermality. The main quantities which characterize the electrically conducting fluid are the bulk plasma velocity field v, the current density J, the mass density ρ, and the plasma pressure p. The flowing electric charge in the plasma is the source of a magnetic field B and electric field E. All quantities generally vary with time t. Vector operator notation will be used, in particular ∇ is gradient, ∇⋅ is divergence, and ∇× is curl. The mass continuity equation is The Cauchy momentum equation is