Past exam of the advanced fluid dynamics course of the University of Oxford / 2023 / 1 / d / Solution

In this question, the coefficient multiplying ${\nabla }^2\mathbf{B}$ is the magnetic diffusivity. Denote it ${\eta }_m$; it is the paper's $\eta$, with dimensions of length squared per time. If ${\eta }_{\mathrm{Ohm}}$ denotes electrical resistivity, then ${\eta }_m={\eta }_{\mathrm{Ohm}}/{\mu }_0$. The two conventions must not be interchanged.

For an imposed real positive $\omega$, retain the momentum equation and useElimination gives the exact Resistive damping of an Alfvén wave relationHere $k$ is complex because the driving frequency is real. Introduce the dimensionless diffusion parameter $ϵ={\eta }_m\omega /v_A^2\ll 1$. For propagation and attenuation along $+z$, choose the square-root branch with $\mathrm{Re}k,\mathrm{Im}k>0$. The spatial damping of a driven Alfvén wave isWith the convention $e^{i(kz-\omega t)}$, the amplitude decays as $e^{-\mathrm{Im}kz}$. Define its e-folding length $L_A=1/\mathrm{Im}k$ and the wavelength $\lambda =2\pi /\mathrm{Re}k$. ThenIf damping length instead refers to wave energy, the squared amplitude decays twice as fast: $L_E=L_A/2$ and $L_E/\lambda \sim 1/(2\pi ϵ)$. Stating the convention avoids a factor-of-two ambiguity. The temporal damping rate ${\eta }_m{k}^2/2$ at fixed real $k$ is a different specification from this imposed-frequency problem.