Set of equations that describe a plasma when it is assumed to be a fluid of a single ion species. This is a valid assumption to make for the study of many global plasma processes, particularly the study of large-scale plasma perturbations (MHD instabilities).

The equations of magnetohydrodynamics are derived from the fluid equations and from Maxwell's equations.

Note that the derivative d/dt will be defined as follows

d/dt= δ/δt + **v.div**

where

v is the velocity of the plasma fluid. This is to simplify the equations. The following

notation is used:

magnetic field **B**, plasma

pressure *p*, plasma

current **j**, plasma

density ρ, plasma

resistivity η and the

electric field **E**
**Eqn. I**, the continuity equation

This is the first fluid equation and implies the conservation of the total number of particles in the plasma

dρ/dt= -ρ div.**v**

**Eqn. II**, the equation of motion

This is the second fluid equation. Since the pressure is assumed to be isotropic the pressure tensor **P** is now a scalar *p*. The forces which drive the momentum change are the electromagnetic force **jxB** and that due to the pressure gradient

ρ(d**v**/dt)= **jxB** - grad p

**Eqn. III**, pressure equation

In order to deal with the pressure it is assumed that adiabatic behaviour applies.

dp/dt= -γ p div.**v**

**Eqn. IV**, Ampere's Law

This is one of Maxwell's equations where the displacement current is neglected.

**j**= (curl **B**)/μ_{o}

**Eqn. V**, Faraday's Law

Another of Maxwell's equations.

δ**B**/δt = -curl **E**

**Eqn. VI**, Ohm's Law

In *ideal* MHD, the plasma ions never collide so the right hand side of this equation will be zero. In *resistive* MHD the resistivity of the plasma is included

**E** + **v** X **B**= η **j**

Source: 'Tokamaks' by John Wesson, 'Tokamak Plasma: A complex physical system' by B.B Kadomtsev