Past exam of the advanced fluid dynamics course of the University of Oxford / 2016 / 1 / b / Solution

Set $\Delta p_0=p_{\perp 0}-p_{\parallel 0}$ and ${\mathcal{T}}_0=\Delta p_0+B_0^2/(4\pi )$. Distinguish the scalar magnetic field strength perturbation $\delta B$ from the vector perturbation $\delta \mathbf{B}$. To first order,where the last identity is the solenoidal magnetic-field constraint. For a Fourier mode with the stated time convention, putThe linear momentum conservation equation, after division by $i$, isUse the solenoidal magnetic-field constraint in its $z$ component. The coefficient of $\hat{\mathbf{z}}$ reduces to $-k_{\parallel }[\delta p_{\parallel }+\Delta p_0\delta B/B_0]$. ThusBoth scalar pressure perturbations act within the $xz$ plane. The $y$ component therefore decouples, independently of their closure. The ideal magnetohydrodynamic equations giveEliminating the amplitudes gives the Alfvén wave dispersion relationFor this Shear Alfvén wave, the velocity and fluid displacement are along $y$, perpendicular to both $\mathbf{k}$ and the equilibrium magnetic field. Let $\mathbf{\xi }={\xi }_y\hat{\mathbf{y}}$; thenA real snapshot ${\xi }_y={\xi }_0\cos (\mathbf{k}\cdot \mathbf{r}-\omega t)$ bends the magnetic field lines sideways, with $\delta B_y=B_0{\partial }_z{\xi }_y$. There is no first-order compression or change of magnetic field strength: $\nabla \cdot \mathbf{\xi }=0$, $\delta \rho =0$ and $\delta B=0$. Scalar pressure perturbations are unnecessary in this transverse subsystem. For a traveling stable branch, $\delta B_y/B_0=-k_{\parallel }{u}_y/\omega$, so the velocity and transverse magnetic field perturbations are in phase or opposite phase according to the branch. At $k_{\parallel }=0$ the bending force and this angular frequency vanish.