Past exam of the mathematics course of the University of Cambridge / 2009 / ii / Paper 3 / 38A / Solution

Start from $\rho ({\partial }_{t}u+u\cdot \nabla u)=-\nabla p-\rho g{e}_z$, ${\partial }_t\rho +u\cdot \nabla \rho =0$, and $\nabla \cdot u=0$. The stationary background obeys $p_0^{\prime }=-g{\rho }_0$. Linearize in small perturbations and use the Boussinesq approximation: replace the density by a reference value ${\rho }_{\ast }$ in inertia but retain the density perturbation in the buoyancy force. With $\pi =p^{\prime }/{\rho }_{\ast }$ and $b=-g{\rho }^{\prime }/{\rho }_{\ast }$,This is the Brunt–Väisälä frequency. Its assumed constancy requires a locally uniform stable density gradient in this approximation; rotation and viscosity are neglected.

For a plane wave with wavevector $K=(k,\ell ,m)$, project the momentum equation onto the space perpendicular to $K$ to eliminate pressure. Its vertical component is $-i\omega w=b[1-m^2/|K{|}^2]$, while $-i\omega b+N^{2}w=0$. Eliminating $b$ yieldsFor $0<\omega <N$, a small oscillating body radiates upward and downward conical internal gravity wave beams. In a vertical section the beams make a St Andrew's cross, with angle $\alpha$ to the horizontal satisfying $\sin \alpha =\omega /N$. Indeed the group velocity is perpendicular to the wavevector and has ratio of vertical to horizontal magnitudes $\sqrt{k^2+{\ell }^2}/|m|$. Thus wave energy follows the cones, rather than the direction normal to a fixed-radius spherical wavefront. There are no such propagating beams for $\omega >N$.

For the shear-flow calculation choose the positive intrinsic-frequency branch and $k_0>0$, conditions needed for the printed critical-level formula. The slowly varying local dispersion relation isStationarity and horizontal homogeneity conserve $\Omega$ and $k=k_0$. The Hamiltonian ray equations for a local dispersion relation giveThus $m_0<0$ stays negative and the ray propagates upward. Conservation of frequency givesIf the shear reaches the required velocity difference, its first zero is the critical level of an internal gravity wave,As $z\uparrow z_c$, the intrinsic frequency tends to zero, $m\to -\infty$, the vertical wavelength tends to zero and the vertical group velocity tends to zero. The ideal ray cannot cross this level. For finite $U_c^{\prime }>0$, $\dot{z}\sim k_0(U_c^{\prime }{)}^2(z_c-z{)}^2/N$, so it approaches the level only asymptotically in time. The disturbance develops short scales; even weak viscosity can then absorb it, and nonlinear effects can invalidate the small-amplitude approximation.

The printed sign and existence hypotheses are incomplete. On the positive-frequency branch, $k_0<0$ instead gives $\dot{m}>0$ and eventually a turning point when the intrinsic frequency reaches $N$. As a concrete counterexample to the stated universal height bound, take $N=1$, $k_0=-1$, $m_0=-3$, and $U(z)=\epsilon z$ with any small $\epsilon >0$. At the printed height $z_c=1/(\epsilon \sqrt{10})$, the intrinsic frequency is $2/\sqrt{10}<1$ and $m=-\sqrt{3/2}<0$, so the vertical group velocity is still positive and the ray crosses that height. Also, $U^{\prime }>0$ alone does not ensure that the displayed velocity difference is ever reached; an increasing bounded $U$ may have no finite critical level. The qualified derivation above gives the intended positive-$k_0$ result whenever that level exists.