Magnetohydrodynamic Waves - Hydromagnetic Waves in the Magnetosphere and the Ionosphere - page 76

Here P is the gas kinetic pressure, j is the current density.
The hydrodynamic equations should be supplemented by Faraday's induc-
tion law

1
c

∂ B
∂t ,

∇ × E = − (4.3)

and by Ampere's law

B = 4 π ∇ ×

c j .

(4.4)

Applying operator ∇· to (4.3), we obtain

∂
∂t ∇ ·

B =0 , (4.5)

so that it is sucient to accept the condition of magnetic field solenoidality

∇ · B = 0 (4.6)

as an initial condition. The electric and magnetic fields are fully determined
by (4.3), (4.4) and by Ohm's law
j = σ E + 1 B ] ,

c [ u

× (4.7)

where σ is plasma conductivity. For an ideal plasma, when σ →∞ , it follows

from (4.7) that

1
c [ u

E = − × B ] . (4.8)

Equations (4.1)-(4.8) should be supplemented by the state equation

P = P ( ρ, T ) , (4.9)

as well as by the heat transfer equation, which for an ideal medium, when
energy dissipation can be neglected, is given by

ds
dt

= ∂s

∂t +( u

∇ ) s =0 , (4.10)

where s is the entropy density. Since P = P ( ρ, s ) , then s can be found as a
function of P and ρ and (4.10) for the isentropic conditions becomes

c s = ∂P
∂ρ

∂P
∂t

∂ρ
∂t ,

= c s

, (4.11)

s = const

where c s is the adiabatic sound velocity. The set of equations (4.1)-(4.11)
form a complete system allowing the description and comprehension of most
of the effects arising in the propagation of ULF-waves in the magnetosphere
and in the upper ionosphere.