Editing Magnetohydrodynamics (section)

== Ideal MHD ==
{{qb
|quote=In view of the infinite conductivity, every motion (perpendicular to the field) of the liquid in relation to the lines of force is forbidden because it would give infinite [[eddy current]]s. Thus the matter of the liquid is "fastened" to the lines of force...
|author=[[Hannes Alfvén]]
|source=1943<ref>{{cite journal|last1=Alfvén|first1=Hannes|title=On the Existence of Electromagnetic-Hydrodynamic Waves|journal=Arkiv för matematik, astronomi och fysik|date=1943|volume=29B(2)|pages=1–7}}</ref>
|width=220px
}}
The simplest form of MHD, '''ideal MHD''', assumes that the resistive term <math>\eta\mathbf{J}</math> 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 number]]s during which [[Electromagnetic induction|magnetic induction]] dominates over [[magnetic diffusion]] at the velocity and [[length scale]]s under consideration.<ref name="bellan06" /> Consequently, processes in ideal MHD that convert magnetic energy into kinetic energy, referred to as ''ideal processes'', cannot generate [[heat]] and raise [[entropy]].<ref name="priest00" />{{rp|6}}

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.<ref>{{cite web | url=https://www.osti.gov/biblio/4329910 | osti=4329910 | title=Stability of the Pinch | date=April 1956 | last1=Rosenbluth | first1=M. }}</ref><ref name="priest00">{{cite book |first1=Eric |last1=Priest |first2=Terry |last2=Forbes |title=Magnetic Reconnection: MHD Theory and Applications |publisher=Cambridge University Press |edition=First |year=2000 |isbn=0-521-48179-1 }}</ref>{{rp|25}} The connection between the fluid and magnetic field fixes the [[Magnetic topology|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 <math>\eta\mathbf{J}</math> vanishes in Ohm's law giving the ideal Ohm's law,<ref name="bellan06" />
:<math>\mathbf{E} + \mathbf{v}\times\mathbf{B} = 0.</math>
Similarly, the magnetic diffusion term <math>\eta\nabla^2\mathbf{B}/\mu_0</math> in the induction equation vanishes giving the ideal induction equation,<ref name="priest00" />{{rp|23}}
:<math>\frac{\partial\mathbf{B}}{\partial t} = \nabla\times(\mathbf{v}\times\mathbf{B}).</math>

=== Applicability of ideal MHD to plasmas ===

Ideal MHD is only strictly applicable when:

# 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 [[Maxwell–Boltzmann distribution|Maxwellian]].
# 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.
# Interest in length scales much longer than the ion [[Plasma parameters#Lengths|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 (system is smooth and slowly evolving).

=== Importance of resistivity ===

In an imperfectly conducting fluid the magnetic field can generally move through the fluid following a [[Diffusion equation|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]] (from collisional resistivity) 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<ref>{{cite journal | url=https://iopscience.iop.org/article/10.1088/0029-5515/18/1/010 | doi=10.1088/0029-5515/18/1/010 | title=Hydromagnetic stability of tokamaks | year=1978 | last1=Wesson | first1=J.A. | journal=Nuclear Fusion | volume=18 | pages=87–132 | s2cid=122227433 }}</ref>—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 [[Instability|instabilities]] exist that can increase the effective resistivity of the plasma by factors of more than 10<sup>9</sup>. The enhanced resistivity is usually the result of the formation of small scale structure like current sheets or fine scale magnetic [[Magnetohydrodynamic turbulence|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]].