Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 345 / 1

Let $u=ps{i}_z$ and $w=-ps{i}_x$, so mass conservation for the two-dimensional incompressible flow is automatic. Write $b$ for buoyancy and introduce the diffusive operatorsThe Linearized Boussinesq equations areTaking the curl of the momentum conservation equations eliminates the pressure and givesApply $D_{\kappa }$ and use $D_{\kappa }b=N^2{\psi }_x$. Since the constant-coefficient linear partial differential operators commute,This is the viscous-diffusive internal gravity wave equation.

Let the group velocity make an angle $\theta$ above the horizontal. A convenient right-handed choice of rotated unit vectors isThe first vector is parallel to the wavevector, the second is parallel to the group velocity, and their perpendicularity is the defining geometry of an internal-wave phase and group velocity. ThusWriting ${\Delta }_{\xi \zeta }={\partial }_{\xi }^2+{\partial }_{\zeta }^2$, the equation becomesFor an inviscid plane wave proportional to $e^{i(k\xi -\omega t)}$, its dispersion relation is $\omega =N\sin \theta$.

Put $Z=ϵ\zeta$ and seek the slowly attenuating wave envelope $A(Z)$ inAt order ${ϵ}^0$, the equation gives the internal gravity wave dispersion relationAt order ${ϵ}^1$, retaining one derivative of the slowly varying amplitude givesUsing $\omega =N\sin \theta$ therefore yieldsHence the leading viscous attenuation of an internal-wave beam isThe wave energy density, being quadratic in the amplitude, decays twice as rapidly in the exponent.

Setwhere $r$ is the magnitude of the internal-wave ray slope and $s$ is the bottom slope. A rightward ray remains in the triangular basin precisely when its bottom reflection is subcritical internal-wave reflection, namely $r>s$. Therefore

Conservation of frequency and of the component of the wavevector tangent to the slope gives the focusing power of internal-wave reflectionwhere $\tan \alpha =s$. In this range $\gamma >1$, so the reflected wavelength is shorter by the factor $\gamma$. The reflected normal group velocity is smaller by ${\gamma }^2$; conservation of normal energy flux therefore increases the wavelength-averaged energy density by

The incident internal-wave ray has equation $z=-rx$. Intersecting it with the bottom $z=-H_0+sx$ givesIts distance to the slope is consequentlyThe subcritically reflected ray rises through the same vertical distance at angle $\theta$, so its next free-surface reflection is at

On the first leg, viscous attenuation of an internal-wave beam multiplies the energy density by $e^{-2k\ell /\mathrm{Re}}$. The bottom reflection multiplies it by ${\gamma }^2$ and changes the wavenumber to $\gamma k$. Since the corresponding Reynolds number is $\mathrm{Re}/{\gamma }^2$, attenuation on the second leg contributes $e^{-2{\gamma }^{3}k\ell /\mathrm{Re}}$. Ignoring boundary-layer enhancement as requested,