Past exam of the advanced fluid dynamics course of the University of Oxford / 2025 / 1

To first order in fluid velocity divided by the speed of light, the Lorentz transformation of electromagnetic fields gives ${\mathbf{E}}^{\prime }=\mathbf{E}+\mathbf{u}\times \mathbf{B}$. Applying Ohm's law in the local fluid frame gives $\mathbf{E}+\mathbf{u}\times \mathbf{B}=\eta \mathbf{J}$. Neglecting the displacement current in Ampère's law, $\mathbf{J}=\nabla \times \mathbf{B}/{\mu }_0$. Substitution in Faraday's law givesHere ${\eta }_m$ is the magnetic diffusivity; the paper's $\eta$ is electrical resistivity, rather than magnetic diffusivity. Expanding the curl with the solenoidal magnetic-field constraint gives the resistive induction equation in its compressible form:The left side is the material derivative of the magnetic field. The three right-hand terms respectively describe field-line stretching and reorientation, amplification or dilution by compression or expansion, and magnetic diffusion. In particular, compression has $\nabla \cdot \mathbf{u}<0$ and amplifies a field component perpendicular to the compression.

The assumptions are nonrelativistic bulk motion, the low-frequency approximation permitting omission of the displacement current, a conducting-fluid continuum, isotropic local Ohm's law with spatially constant scalar electrical resistivity, constant magnetic permeability ${\mu }_0$, and $\nabla \cdot \mathbf{B}=0$. Hall, electron-pressure and other non-Ohmic corrections to the generalized Ohm's law are excluded. Incompressible flow is not required: imposing it would remove the compression term.

Let $E_B={\int }_V{B}^2/(2{\mu }_0)dV$ be the total magnetic energy. Multiply the curl form of the resistive induction equation by $\mathbf{B}/{\mu }_0$ and integrate over the fixed volume. The identityfollows from the divergence theorem. The stated zero boundary fields remove its surface term. Applying it to the motional and resistive contributions yields the magnetic energy balance in resistive magnetohydrodynamics:The first term is the negative of the mechanical power delivered to the fluid by the Lorentz force density. It transfers energy between fluid motion and the magnetic field, with either sign. The second is the nonpositive magnetic contribution to Joule heating. A nonzero positive electrical resistivity therefore dissipates magnetic energy at rate $\eta {\int }_V{J}^{2}dV$; it does not force the total magnetic energy to decrease when fluid motion supplies a larger positive power.

Choose a magnetic vector potential with $\mathbf{B}=\nabla \times \mathbf{A}$. The resistive induction equation permits the evolution equationA gauge transformation changes $H={\int }_V\mathbf{A}\cdot \mathbf{B}dV$ by a surface integral of the normal magnetic field, which is zero here. Thus the integrated magnetic helicity has gauge invariance under the stated boundary conditions.

Differentiate both factors in $H$. Use the stated vanishing-boundary assumption to remove the helicity surface flux; this can be implemented in the usual closed-boundary gauge $\mathbf{n}\times \mathbf{A}=0$, or by extending the localized field and integrating over all space with sufficient decay. The curl integration by parts identity then givesThe motional term is zero pointwise because $(\mathbf{u}\times \mathbf{B})\cdot \mathbf{B}=0$. The scalar-potential term integrates to zero by $\nabla \cdot \mathbf{B}=0$ and the boundary condition. The resistive magnetic-helicity balance is thereforeUnlike the resistive magnetic energy contribution, this signed integral need not have a fixed sign.

A magnetic field varying over a length $\ell$ has $|\nabla \times \mathbf{B}|$ of order $B/\ell$. Consequently, the resistive magnetic energy loss contains two inverse powers of $\ell$, while the resistive magnetic-helicity balance contains only one. Small-scale gradients can therefore dissipate substantial magnetic energy while retaining an approximately conserved large-scale magnetic helicity.

A dimensionally consistent version is the selective decay bound for magnetic helicity. With $D_B={\eta }_m{\int }_V|\nabla \times \mathbf{B}{|}^{2}dV/{\mu }_0$, the Cauchy-Schwarz inequality givesFor bounded $E_B$ and bounded, possibly nonvanishing $D_B$ as ${\eta }_m\to 0$, the resistive change in magnetic helicity tends to zero, even though resistive magnetic energy loss may remain finite. Magnetic energy is preferentially dissipated, while magnetic helicity is approximately conserved in this bounded-energy, bounded-dissipation limit.

This is not a comparison of quantities with different physical units or a universal ordering of fractional decay rates. For a single Beltrami field satisfying $\nabla \times \mathbf{B}=k\mathbf{B}$, both integrated quantities have fractional decay rate $2{\eta }_m{k}^2$ without fluid motion. The boundedness assumptions and distribution of magnetic helicity across spatial scales matter.