4.3 Low frequency magnetized plasma: Alfvén waves 135direction of propagation of
compressional Alfven wavecompressed field lines
rarified field lines
Figure 4.2: Compressional Alfvén waves4.3.5 Finite-pressure analysis of MHD waves
If the pressure is allowed to be finite, then the two modes become coupled and an acoustic
mode appears. Using the vector identity∇B^2 =2(B·∇B+B×∇×B)theJ×Bforce
in the MHD equation of motion can be written as
J×B=−∇
(
B^2
2 μ 0)
+
1
μ 0B·∇B. (4.65)
The MHD equation of motion thus becomes
ρDU
Dt=−∇
(
P+
B^2
2 μ 0)
+
1
μ 0B·∇B. (4.66)
Linearizing this equation about a stationary equilibrium where the pressure and the density
are uniform and constant, gives
ρ∂U 1
∂t=−∇
(
P 1 +
B·B 1
μ 0)
+
1
μ 0B·∇B 1. (4.67)
The curl of the linearized ideal MHD Ohm’s law,
E 1 +U 1 ×B=0, (4.68)gives the induction equation
∂B 1
∂t