Past exam of the astrophysical fluid dynamics course of the University of Cambridge / 2021 / Section A

The first argument mixes a material fluid element with a fixed spatial volume. For a material volume element following the fluid, its volume is generally time dependent:Therefore the material derivative cannot be taken outside the product with $\delta V$ as in the proposed cancellation. Instead, mass conservation statesThe momentum balance for a material fluid element, using Newton's second law, is consequentlyFor a fixed control volume, fluid crosses its boundary and carries momentum; a momentum flux term must be included. Its local conservative form is ${\partial }_t(\rho \mathbf{u})+\nabla \cdot (\rho \mathbf{u}\otimes \mathbf{u})=-\nabla p$, equivalent to the displayed Euler equations after using the continuity equation. The erroneous equation omits the term $\rho \mathbf{u}\nabla \cdot \mathbf{u}$ needed to convert a material derivative of momentum density into this conservative form.

For the second task, take a homogeneous fluid with constant mass density $\rho$, prescribed gravitational acceleration $\mathbf{g}$, incompressible flow and inviscid flow. Its equations areUnder the replacement $\rho \mapsto \tilde{\rho }=C\rho$, the same velocity solves these equations if $p\mapsto \tilde{p}=Cp$, up to a spatially constant pressure offset. This is density scaling invariance of incompressible Euler flow:The comparison assumes the same geometry and compatible velocity or gravitational free surface conditions; an independently specified nonuniform boundary pressure would supply additional information. The gravitational field is taken as prescribed. If the fluid's own self-gravity must be included, its field changes with mass density, so the invariance need not hold. A single freely moving particle's density-independent gravitational acceleration does not justify neglecting that distinction.

A highly conducting solar wind carries the Sun's magnetic field outward by magnetic flux freezing. This can open magnetic field lines that would form closed loops in a static dipole model. Beyond an opening radius $r_o$, approximate the field as radial. Constancy of the magnetic flux in a fixed solid-angle flux tube gives the radial solar-wind magnetic flux lawwhich decays more slowly than the vacuum magnetic dipole field, proportional to $r^{-3}$.

For example, if the field becomes radial already near the solar radius $R_{\odot }$, normalizing both estimates to the same field there givesThus a dipole estimate $10^{-11}\mathrm{T}$ becomes of order $2\times 10^{-9}\mathrm{T}$, comparable to the stated hundredfold enhancement. More generally, matching dipolar decay inside $r_o$ to radial decay outside givesThis is a plausible scaling explanation, not a determination of the actual opening radius. The magnetic field need not be outward everywhere: opposite-polarity sectors can preserve zero net magnetic flux through a closed surface, consistent with Gauss's law for magnetism. The argument concerns the changed geometry of the field in a flowing plasma, rather than changing the dipole exponent by assumption.

Take the outward wavenumber $k>0$; the absolute value in the source makes the two propagation directions equivalent. The phase velocity of the gravity-capillary wave obeysDifferentiate with respect to $k$:The second derivative is positive, and $c_p\to \infty$ at both ends of the positive-$k$ range, so this is the global minimum. The minimum phase speed of a gravity-capillary wave isFor $g=9.81\mathrm{m}{\mathrm{s}}^{-2}$ and the supplied surface tension and mass density,The corresponding wavenumber is $k_{\ast }\simeq 369{\mathrm{m}}^{-1}$, giving wavelength ${\lambda }_{\ast }=2\pi /k_{\ast }\simeq 1.70\mathrm{cm}$. At this minimum, $c_g=d\omega /dk=c_p+kd{c}_p/dk=c_p$. Thus the group velocity and phase velocity agree for this particular ripple, although they differ for general gravity-capillary waves.