Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 73 / 1

Put $P=p/{\rho }_0$ and use the downward buoyancy anomaly $b=g{\rho }^{\prime }/{\rho }_0$. This sign convention is required by the printed potential vorticity. The linear rotating Boussinesq approximation givesLet $D=u_x+v_y$ and $\zeta =v_x-u_y$. Horizontal divergence and curl give $D_t-f\zeta =-{\nabla }_H^{2}P$ and ${\zeta }_t=-fD=f{w}_z$. HenceThus the linear potential vorticity is a time-independent field fixed by the initial data. Differentiate the pressure identity twice. Using ${\zeta }_{tt}=-f(P_{zz}+b_z)$ and $b_{ztt}=-N^2(P_{zz}+b_z)$ gives ${\nabla }^2{P}_{tt}=(N^2-f^2)(P_{zz}+b_z)$. Combining terms,The linear pressure equation for rotating stratified flow has source $Q={\rho }_{0}f{N}^{2}q$. “Arbitrary” means that different initial potential-vorticity fields give different time-independent sources; it is not independent forcing that may vary in time. If buoyancy were instead defined upward as $-g{\rho }^{\prime }/{\rho }_0$, both appearances of its sign would change consistently. The pressure source represents the balanced component alongside propagating inertia-gravity waves.

For a nonzero-frequency Fourier mode proportional to $e^{i(kx+mz-\omega t)}$, the time-independent source has no oscillatory part. The pressure equation givesTake $k,m\ne 0$ and ${\omega }^2\ne f^2,N^2$ for the following pressure-amplitude formulas. With $P_0$ the amplitude of $p/{\rho }_0$, the momentum and buoyancy equations implyConsequentlybecause the numerator is $(k^2+m^2){\omega }^2-N^2{k}^2-f^2{m}^2$. Propagating linear inertia-gravity waves carry zero perturbation potential vorticity. In degenerate directions the division formulas must be replaced by the original linear equations, or interpreted by a regular limit; the zero-PV property of the nonzero-frequency wave component still follows from $q_t=0$. A steady balanced mode need not have zero potential vorticity.

For the positive-frequency branch and $K^2=k^2+m^2$, the wavefront-normal phase velocity and the group velocity areTheir scalar product vanishes since $k(k{m}^2)+m(-m{k}^2)=0$. Equivalently, the frequency is homogeneous of degree zero in the wave vector, and hence $\mathbf{k}\cdot {\nabla }_{\mathbf{k}}\omega =0$.

The vertical sign relation isIt is negative for the usual geophysical ordering $N>|f|$ and nonzero $k,m$. Thus vertical phase and energy propagation are opposite under that ordering; negative-frequency waves obey the same product relation. The PDF does not state this ordering, so its unconditional direction assertion needs qualification. For example, $N=1$, $f=2$, $k=m=1$ gives positive vertical components on the positive-frequency branch. If $N=|f|$ the group velocity vanishes, and in axial limiting directions one vertical component can be zero. Orthogonality is valid throughout, while strictly opposite nonzero vertical components need the stated nondegeneracy and $N>|f|$.

Let the vertical-velocity amplitude $W$ be real. A convenient inertia-gravity wave polarization, avoiding pressure denominators, isThe physical velocities and buoyancy are their real parts. The period-mean kinetic energy density and available potential energy density areThe latter follows from $A={\rho }_0{b}^2/(2N^2)$, or from $A={\rho }_0{N}^2{\xi }_z^2/2$ for vertical fluid displacement. The factor $1/4$ includes both the energy definition and the mean of a squared harmonic.

Using the dispersion relation,This energy partition of rotating internal waves shows that the printed request for ordinary kinetic/potential equipartition in the rotating case is false in general. An explicit counterexample is $f=1$, $N=2$, $k=m=1$, $W=1$, giving ${\omega }^2=5/2$, $⟨K⟩=3{\rho }_0/5$ and $⟨A⟩=2{\rho }_0/5$.

The valid modified oscillator balance isThe transverse rotational velocity is in quadrature with the in-plane motion, so it supplies an additional positive energy term on the displacement side of this oscillator balance. It remains physically kinetic energy, not gravitational potential energy. Ordinary kinetic/potential equipartition is recovered when $f=0$ or when the transverse amplitude vanishes. For a real harmonic wave the instantaneous total $K+A$ is constant at a fixed point: the coefficient of ${\cos }^2(kx+mz-\omega t)$ from in-plane motion equals the coefficient of ${\sin }^2$ from transverse motion and buoyancy. This consistency does not imply equality of their separate period means.