Fundamentals of Atmospheric, Ionospheric, and Magnetospheric Plasma Dynamics - The Earth's Ionosphere: Plasma Physics and Electrodynamics - page 63

Finally, we need an equation of state. Several possibilities exist depending on
the properties of the fluid. For a true conducting metallic fluid like mercury or
the earth's core, we could use the incompressibility condition

∇· V = 0

For a fluid with a high heat conductivity, we could use isothermal conditions

d p /ρ /

dT / dt = dt = 0

or for an adiabatic fluid
d p γ /ρ γ /

dt = 0

The particular equation of state to be used depends on the application.
To reduce the number of equations and gain physical insight, the current
density and electric field can be eliminated. Returning to the equation of motion
and ignoring viscosity and gravity, we have
ρ d V dt =−∇ / p + J × B =−∇ p + ( 1 /μ 0 )( ∇× B ) × B (2.57)

Using the vector identity

∇ ( X · Y ) = ( X ·∇ ) Y + ( Y ·∇ ) X + X × ( ∇× Y ) + Y × ( ∇× X )

with X = Y = B yields

B 2

∇ = ( ·∇ ) + × ( ∇× )

2 B B 2 B B

and thus the J × B term in (2.57) becomes

B 2

μ 0

f B = ( 1 /μ 0 ) ( ∇× B ) × B =−∇ / 2 + ( B ·∇ ) B /μ 0

and the equation of motion becomes

p

μ 0 ρ d V dt =−∇ B 2

/ + / 2 + ( B ·∇ ) B /μ 0 (2.58)

We study this equation in more detail following. In a similar fashion we may
eliminate E from Maxwell's magnetic field equation,

) =−∇× J B

∂ B /∂ t =− ( ∇× E /σ − V ×

/σ) ∇× J =∇× ( V × B ) − ( 1

=∇× ( V × B ) −∇× ( ∇× B )/σμ 0