Magnetohydrodynamics


Magnetohydrodynamics is a model of electrically conducting fluids that treats all types of charged particles together as one continuous fluid. It is primarily concerned with the low-frequency, large-scale, magnetic behavior in plasmas and liquid metals and has applications in multiple fields including space physics, geophysics, astrophysics, and engineering.

Etymology

The word magnetohydrodynamics is derived from ' meaning magnetic field, ' meaning water, and meaning movement. The field of MHD was initiated by Hannes Alfvén, for which he received the Nobel Prize in Physics in 1970.

History

The MHD description of electrically conducting fluids was first developed by Hannes Alfvén in a 1942 paper published in Nature titled "Existence of Electromagnetic–Hydrodynamic Waves" which outlined his discovery of what are now referred to as Alfvén waves. Alfvén initially referred to these waves as "electromagnetic–hydrodynamic waves"; however, in a later paper he noted, "As the term 'electromagnetic–hydrodynamic waves' is somewhat complicated, it may be convenient to call this phenomenon 'magneto–hydrodynamic' waves."

Equations

In MHD, motion in the fluid is described using linear combinations of the mean motions of the individual species: the current density and the center of mass velocity. In a given fluid, each species has a number density, mass, electric charge, and a mean velocity. The fluid's total mass density is then, and the motion of the fluid can be described by the current density expressed as
and the center of mass velocity expressed as:
MHD can be described by a set of equations consisting of a continuity equation, an equation of motion, an equation of state, Ampère's law, Faraday's law, and Ohm's law. As with any fluid description to a kinetic system, a closure approximation must be applied to the highest moment of the particle distribution equation. This is often accomplished with approximations to the heat flux through a condition of adiabaticity or isothermality.
In the adiabatic limit, that is, the assumption of an isotropic pressure and isotropic temperature, a fluid with an adiabatic index electrical resistivity magnetic field and electric field can be described by the continuity equation
the equation of state
the equation of motion
the low-frequency Ampère's law
Faraday's law
and Ohm's law
Taking the curl of this equation and using Ampère's law and Faraday's law results in the induction equation,
where is the magnetic diffusivity.
In the equation of motion, the Lorentz force term can be expanded using Ampère's law and a vector calculus identity to give
where the first term on the right hand side is the magnetic tension force and the second term is the magnetic pressure force.

Ideal MHD

The simplest form of MHD, ideal MHD, assumes that the resistive term in Ohm's law is small relative to the other terms such that it can be taken to be equal to zero. This occurs in the limit of large magnetic Reynolds numbers during which magnetic induction dominates over magnetic diffusion at the velocity and length scales under consideration. Consequently, processes in ideal MHD that convert magnetic energy into kinetic energy, referred to as ideal processes, cannot generate heat and raise entropy.
A fundamental concept underlying ideal MHD is the frozen-in flux theorem which states that the bulk fluid and embedded magnetic field are constrained to move together such that one can be said to be "tied" or "frozen" to the other. Therefore, any two points that move with the bulk fluid velocity and lie on the same magnetic field line will continue to lie on the same field line even as the points are advected by fluid flows in the system. The connection between the fluid and magnetic field 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 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.

Ideal MHD equations

In ideal MHD, the resistive term vanishes in Ohm's law giving the ideal Ohm's law,
Similarly, the magnetic diffusion term in the induction equation vanishes giving the ideal induction equation,

Applicability of ideal MHD to plasmas

Ideal MHD is only strictly applicable when:
  1. The plasma is strongly collisional, so that the time scale of collisions is shorter than the other characteristic times in the system, and the particle distributions are therefore close to Maxwellian.
  2. The resistivity due to these collisions is small. In particular, the typical magnetic diffusion times over any scale length present in the system must be longer than any time scale of interest.
  3. Interest in length scales much longer than the ion skin depth and Larmor radius perpendicular to the field, long enough along the field to ignore Landau damping, and time scales much longer than the ion gyration time.

    Importance of resistivity

In an imperfectly conducting fluid the magnetic field can generally move through the fluid following a diffusion law with the resistivity of the plasma serving as a diffusion constant. This means that solutions to the ideal MHD equations are only applicable for a limited time for a region of a given size before diffusion becomes too important to ignore. One can estimate the diffusion time across a solar active region to be hundreds to thousands of years, much longer than the actual lifetime of a sunspot—so it would seem reasonable to ignore the resistivity. By contrast, a meter-sized volume of seawater has a magnetic diffusion time measured in milliseconds.
Even in physical systems—which are large and conductive enough that simple estimates of the Lundquist number suggest that the resistivity can be ignored—resistivity may still be important: many instabilities exist that can increase the effective resistivity of the plasma by factors of more than 109. The enhanced resistivity is usually the result of the formation of small scale structure like current sheets or fine scale magnetic turbulence, introducing small spatial scales into the system over which ideal MHD is broken and magnetic diffusion can occur quickly. When this happens, magnetic reconnection may occur in the plasma to release stored magnetic energy as waves, bulk mechanical acceleration of material, particle acceleration, and heat.
Magnetic reconnection in highly conductive systems is important because it concentrates energy in time and space, so that gentle forces applied to a plasma for long periods of time can cause violent explosions and bursts of radiation.
When the fluid cannot be considered as completely conductive, but the other conditions for ideal MHD are satisfied, it is possible to use an extended model called resistive MHD. This includes an extra term in Ohm's law which models the collisional resistivity. Generally MHD computer simulations are at least somewhat resistive because their computational grid introduces a numerical resistivity.

Structures in MHD systems

In many MHD systems most of the electric current is compressed into thin nearly-two-dimensional ribbons termed current sheets. These can divide the fluid into magnetic domains, inside of which the currents are relatively weak. Current sheets in
the solar corona are thought to be between a few meters and a few kilometers in thickness, which is quite thin compared to the magnetic domains. Another example is in the Earth's magnetosphere, where current sheets separate topologically distinct domains, isolating most of the Earth's ionosphere from the solar wind.

Waves

The wave modes derived using the MHD equations are called magnetohydrodynamic waves or MHD waves. There are three MHD wave modes that can be derived from the linearized ideal-MHD equations for a fluid with a uniform and constant magnetic field:
  • Alfvén waves
  • Slow magnetosonic waves
  • Fast magnetosonic waves
These modes have phase velocities that are independent of the magnitude of the wavevector, so they experience no dispersion. The phase velocity depends on the angle between the wave vector and the magnetic field. An MHD wave propagating at an arbitrary angle with respect to the time independent or bulk field will satisfy the dispersion relation
where
is the Alfvén speed. This branch corresponds to the shear Alfvén mode. Additionally the dispersion equation gives
where
is the ideal gas speed of sound. The plus branch corresponds to the fast-MHD wave mode and the minus branch corresponds to the slow-MHD wave mode. A summary of the properties of these waves is provided:
The MHD oscillations will be damped if the fluid is not perfectly conducting but has a finite conductivity, or if viscous effects are present.
MHD waves and oscillations are a popular tool for the remote diagnostics of laboratory and astrophysical plasmas, for example, the corona of the Sun.

Extensions

; Resistive
; Extended
; Two-fluid
; Hall
; Electron MHD
; Collisionless
; Reduced

Limitations

Importance of kinetic effects

Another limitation of MHD is that they depend on the assumption that the plasma is strongly collisional, so that the time scale of collisions is shorter than the other characteristic times in the system, and the particle distributions are Maxwellian. This is usually not the case in fusion, space and astrophysical plasmas. When this is not the case, or the interest is in smaller spatial scales, it may be necessary to use a kinetic model which properly accounts for the non-Maxwellian shape of the distribution function. However, because MHD is relatively simple and captures many of the important properties of plasma dynamics it is often qualitatively accurate and is therefore often the first model tried.
Effects which are essentially kinetic and not captured by fluid models include double layers, Landau damping, a wide range of instabilities, chemical separation in space plasmas and electron runaway. In the case of ultra-high intensity laser interactions, the incredibly short timescales of energy deposition mean that hydrodynamic codes fail to capture the essential physics.