Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 65 / 2

Write $Z=\tilde{z}$ and introduce an arbitrary constant streamfunction scale ${\Psi }_0$, with $\hat{\psi }={\Psi }_0\tilde{\psi }$. The displacement jump condition acquires only a constant prefactor $2{\Psi }_0/\Delta U$, which can be divided out:For the second of the jump conditions for stratified inviscid shear flow, the two derivative terms have common scale $\Delta U{\Psi }_0/h$. Dividing by it givesThe term proportional to the reference density has zero jump by the first condition. ThusNo physical reference-density force has been discarded inside either layer; its continuous interface contribution has canceled. The layer equation becomes ${\tilde{\psi }}_{ZZ}={\alpha }^2\tilde{\psi }$. This explains why only the density anomaly appears in the nondimensional jump condition.

Take $k>0$ without loss of generality, so $\alpha =kh/2>0$, and require decay as $Z\to \pm \infty$. The base velocity is $\tilde{U}=Z$ everywhere; there is no jump in its derivative. Let $C=\tilde{c}$ and $E=e^{-2\alpha }$. A decaying representation that already solves the layer equations isIt is continuous at both interfaces. At $Z=-1$ the value is $A+BE$ and the derivative jump is $-2\alpha A$; at $Z=1$ they are $B+AE$ and $-2\alpha B$. The density anomaly drops by one across each interface. The jump conditions for stratified inviscid shear flow therefore reduce toFor nonreal $C$ the displacement denominators do not vanish. Neutral limiting values are obtained by continuation, rather than by dividing by zero at an interface.

Set $q=J/(2\alpha )$. A nonzero mode requires the determinant to vanish:Expansion gives the dispersion relation for two density interfaces in uniform shear,Uniform shear throughout the exterior is important: replacing it with constant exterior velocities would introduce vorticity jumps and give a different dispersion relation.

For stable density jumps, $J>0$ and $q=J/(2\alpha )>0$. Let $y=C^2$. The two roots of the dispersion relation for two density interfaces in uniform shear areThey are real, and $y_{+}>0$. An exponentially growing normal mode exists precisely when $y_{-}<0$, giving a pair of imaginary phase speeds; the member with positive imaginary part grows in $e^{ik(x-ct)}$ for $k>0$.

Since $y_{+}{y}_{-}=(1-q{)}^2-q^2{E}^2$, this condition is $|1-q|<qE$. Solving both inequalities yieldsAt either endpoint $y_{-}=0$, so the exponential growth rate vanishes. Outside the open band both squared phase speeds are nonnegative and the interfacial modes have real frequencies. This is a modal instability test for the stably stratified configuration; unstable density inversions would require a separate analysis. Within the band the dimensional growth rate is $k\Delta U\sqrt{-y_{-}}/2$.

For large $\alpha$, the evanescent interaction between the two interfaces is proportional to $E=e^{-2\alpha }$. Ignoring it first gives isolated interfacial gravity waves with phase speedsThe lower wave propagating positively relative to its local current and the upper wave propagating negatively have equal laboratory speed whenTheir common speed is zero. This is counterpropagating wave instability: the interfaces communicate through their decaying velocity fields, and near resonance their wave phases can lock while extracting energy from the mean shear.

The instability interval has asymptotic endpoints $J_{\pm }=2\alpha [1\pm E+O(E^2)]$ and widthNear exact resonance the interaction is weak but destabilizing: at $q=1$, $C_{-}^2=2-\sqrt{4+E^2}=-E^2/4+O(E^4)$, so $\mathrm{Im}C\sim E/2$. Thus the resonant band narrows and growth weakens exponentially with separation measured in decay lengths. The resonance here is between two density interfaces in globally uniform shear, not a resonance modified by additional vorticity jumps.