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

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)/μo

Another of Maxwell's equations.

δB/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

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