Editing Magnetohydrodynamics (section)

== Equations ==
In MHD, motion in the fluid is described using linear combinations of the mean motions of the individual [[Chemical species|species]]: the [[current density]] <math>\mathbf{J}</math> and the [[center of mass]] velocity <math>\mathbf{v}</math>. In a given fluid, each species <math>\sigma</math> has a [[number density]] <math>n_\sigma</math>, mass <math>m_\sigma</math>, electric charge <math>q_\sigma</math>, and a mean velocity <math>\mathbf{u}_\sigma</math>. The fluid's total mass density is then <math display=inline>\rho = \sum_\sigma m_\sigma n_\sigma</math>, and the motion of the fluid can be described by the current density expressed as
:<math>\mathbf{J} = \sum_\sigma n_\sigma q_\sigma \mathbf{u}_\sigma</math>
and the center of mass velocity expressed as:
:<math>\mathbf{v} = \frac{1}{\rho} \sum_\sigma m_\sigma n_\sigma \mathbf{u}_\sigma .</math>

MHD can be described by a set of equations consisting of a [[continuity equation]], an equation of motion (the [[Cauchy momentum equation]]), an [[equation of state]], [[Ampère's Law]], [[Faraday's law of induction#Maxwell–Faraday equation|Faraday's law]], and [[Ohm's law]]. As with any fluid description to a kinetic system, a [[Moment closure|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 [[Adiabatic process|adiabaticity]] or [[Isothermal process|isothermality]].

In the adiabatic limit, that is, the assumption of an [[isotropic]] pressure <math>p</math> and isotropic temperature, a fluid with an [[adiabatic index]] <math>\gamma</math>, [[electrical resistivity]] <math>\eta</math>, magnetic field <math>\mathbf{B}</math>, and electric field <math>\mathbf{E}</math> can be described by the continuity equation
:<math> \frac{\partial \rho}{\partial t} + \nabla \cdot \left(\rho\mathbf{v}\right)=0,</math>
the equation of state
:<math>\frac{\mathrm{d}}{\mathrm{d}t} \left(\frac{p}{\rho^\gamma}\right) = 0,</math>
the equation of motion
:<math> \rho\left(\frac{\partial }{\partial t} + \mathbf{v}\cdot\nabla \right)\mathbf{v} = \mathbf{J}\times\mathbf{B} - \nabla p,</math>
the low-frequency Ampère's law
:<math>\mu_0 \mathbf{J} = \nabla\times\mathbf{B},</math>
Faraday's law
:<math>\frac{\partial \mathbf{B}}{\partial t} = - \nabla \times \mathbf{E},</math>
and Ohm's law
:<math>\mathbf{E} + \mathbf{v}\times\mathbf{B} = \eta\mathbf{J}.</math>
Taking the curl of this equation and using Ampère's law and Faraday's law results in the [[induction equation]],
:<math>\frac{\partial\mathbf{B}}{\partial t} = \nabla\times(\mathbf{v}\times\mathbf{B}) + \frac{\eta}{\mu_0}\nabla^2\mathbf{B},</math>
where <math>\eta/\mu_0</math> is the [[magnetic diffusivity]].

In the equation of motion, the [[Lorentz force]] term <math>\mathbf{J}\times\mathbf{B}</math> can be expanded using Ampère's law and a [[vector calculus identity]] to give
:<math>\mathbf{J}\times\mathbf{B} =  \frac{\left(\mathbf{B}\cdot\nabla\right)\mathbf{B}}{\mu_0} - \nabla\left(\frac{B^2}{2\mu_0}\right),</math>
where the first term on the right hand side is the [[magnetic tension force]] and the second term is the [[magnetic pressure]] force.<ref name="bellan06">{{cite book |last1=Bellan |first1=Paul Murray |title=Fundamentals of plasma physics |date=2006 |publisher=Cambridge University Press |location=Cambridge |isbn=0521528003}}</ref>