Question 4.2.1: Consider small-amplitude ideal MHD waves supported by a zero......

For a small amplitude wave, the linearized equation of motion and magnetic induction equation for the magnetic field perturbation \vec{b} and the plasma velocity \vec{v} take the following form:

ρ\frac{∂\vec{v}}{∂t} = \frac{1}{4π} [−\vec{∇}(\vec{B} · \vec{b}) + (\vec{B}  · \vec{∇})\vec{b}],        (4.19)

\frac{∂\vec{b}}{∂t} = \vec{∇} × (\vec{v} × \vec{B} )        (4.20)

The two terms on the right-hand side of (4.19) represent magnetic pressure and magnetic tension forces, respectively. Assuming that \vec{v} and \vec{b} vary in space and time as \exp[i(\vec{k}  · \vec{r}−ωt)], that the initial magnetic field \vec{B} is directed along the z-axis, and that the wave-vector, \vec{k}, is lying in the (x, z) plane, eqs. (4.19- 4.20) reduce to

4πωρ\vec{v} = \vec{k}Bb_{z} − k_{z}B\vec{b},        (4.21)

−ω\vec{b} = \vec{k} × (\vec{v} × \vec{B})

Writing these in component form gives

v_{z} = 0, v_{y} = − \frac{k_{z}}{4π}\frac{B}{ρω}b_{y}, v_{x} = \frac{B}{4πρω} (k_{x}b_{z} − k_{z}b_{x})        (4.22)

b_{x} = − \frac{k_{z}B}{ω} v_{x}, b_{y} = − \frac{k_{z}B}{ω} v_{y}, b_{z} = \frac{k_{x}B}{ω} v_{x}      (4.23)

As seen from equations (4.22-4.23), these perturbations make two separate groups, (v_{y}, b_{y}) and (v_{x}, b_{x}, b_{z}), indicating that this system supports two types of waves. The first type, called the shear Alfven wave, involves (v_{y}, b_{y}) and has dispersion relation

ω = \frac{k_{z}B}{4πρ} = \vec{k}  ·  \vec{V}_{A},  \vec{V}_{A} = \vec{B}/4πρ        (4.24)

Hence, its group velocity \vec{v}_{g} = ∂ω/∂\vec{k} = \vec{V}_{A}, i.e., such a wave propagates along the initial magnetic field with the Alfven speed. As seen from equation (4.21), the restoring force in this case is entirely due to magnetic tension.

For the second type of wave, which involves (v_{x}, b_{x}, b_{z}), the dispersion relation reads:

ω = kV_{A}        (4.25)

The group velocity, \vec{v}_{g} = V_{A}\vec{k}/k, is directed along the wave-vector \vec{k}. In such a wave, called the compressional Alfven wave, both magnetic pressure and magnetic tension contribute to the restoring force (see equation (4.21)). In the limit k_{x} = 0 this wave becomes identical to the shear Alfven wave, while for k_{z} = 0 its propagation is entirely due to magnetic pressure.