Question 4.2.2: The magnetic field perturbation due to a shear Alfven wave, ......
The magnetic field perturbation due to a shear Alfven wave, propagating in a uniform magnetic field \vec{B}_{0} = B_{0}\vec{e}_{z}, is equal to b_{y} = \epsilon B_{0} \cos(kz−ωt).
Consider the magnetic field line which in the absence of the wave is defined by (x = 0, y = 0). Derive the coordinates of this field line when the wave is present, and verify that the magnetic field is frozen into the fluid, i.e., that the fluid particle displacements indeed follow the magnetic field line.
A magnetic field line is defined by the following equations:
dx/B_{x} = dy/B_{y} = dz/B_{z},
or
dx/dz = B_{x}/B_{z}, dy/dz = B_{y}/B_{z}.
In our case B_{x} = 0, B_{y} = \epsilon B_{0} \cos(kz − ωt), B_{z} = B_{0}. Hence,
dx/dz = 0, dy/dz = \epsilon \cos(kz − ωt)
Integrating gives the sought after coordinates,
x = 0, y = \frac{\epsilon}{k} \sin(kz − ωt) = λ \frac{\epsilon}{2π} \sin(kz − ωt),
where λ = 2π/k is the perturbation wavelength. Also, it follows from equation (4.22) that the fluid velocity is equal to
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)
v_{y} = − \frac{\epsilon kB^{2}_{0}}{4πρω} \cos(kz − ωt).
The latter results in the fluid particle displacement
ξ_{y} = \frac{\epsilon kB^{2}_{0}}{4πρω^{2}} \sin(kz − ωt),
which is equal to the magnetic field line y-coordinate since ω^{2}/k^{2} = B^{2}_{0}/4πρ.