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
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(dv/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)/μo
Eqn. V, Faraday's Law
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