Past exam of the advanced fluid dynamics course of the University of Oxford / 2024 / 1 / d / Solution

Linearized magnetic flux freezing gives the Eulerian magnetic field perturbationThe background is uniform, and incompressibility gives $\nabla \cdot \mathbf{\xi }=0$. For a horizontal displacement proportional to $e^{ikz}$, $\delta \mathbf{B}=ik{B}_0\mathbf{\xi }$ and $\delta B_z=0$. Thus the first-order magnetic pressure perturbation vanishes, and the magnetic-tension response to an incompressible displacement isBending a field line produces a restoring force proportional to displacement, like a spring obeying Hooke's law. Per unit mass, its squared spring frequency is $k^2{v}_A^2$, where $v_A=B_0/\sqrt{{\mu }_0\rho }$ is the Alfvén speed. The equilibrium can be independent of $z$ while the perturbation varies with $z$; neglect of background vertical structure does not set this perturbation's wavenumber to zero.