Three-layer stratified piecewise-linear shear flow ($U_i=\pm \Delta U/2,J=g\Delta \rho h/({\rho }_0\Delta U^2)$)

This three-layer shear flow has $U=\Delta Uz/h$ and density ${\rho }_0$ between $z=\pm h/2$. Above and below, the velocity is the corresponding constant $\pm \Delta U/2$, and the mass density is ${\rho }_0\mp \Delta \rho /2$. The density drops and vorticity jumps coincide at both interfaces. Its hydrodynamic stability involves coupled gravity-vorticity interface waves, unlike a profile with uniform shear extending through all three layers.

For the bounded-shear three-layer profile with density contrasts minus one, zero and plus one, the localized-mode matching determinant is $[a(1-c{)}^2-(1-c)-J][a(1+c{)}^2-(1+c)-J]-b^2(1-c^2{)}^2=0$, where $a=\alpha [1+\coth (2\alpha )]$ and $b=\alpha \mathrm{csch}(2\alpha )$. Dividing by $a^2-b^2=2\alpha a$ gives a quartic in $c$ and a quadratic in $c^2$. Its constant term is $\{[2\alpha -(1+J){]}^2-e^{-4\alpha }(1+J{)}^2\}/(4{\alpha }^2)$, whose negative sign proves the unstable band of a three-layer stratified shear flow.

For the three-layer stratified piecewise-linear shear flow, set $\alpha =kh/2$. The dispersion relation is quadratic in squared nondimensional phase velocity. Its constant term is $[2\alpha -(1+J)(1+e^{-2\alpha })][2\alpha -(1+J)(1-e^{-2\alpha })]/(4{\alpha }^2)$. Throughout the displayed band this is negative, so one squared speed is negative and gives an exponentially growing normal mode.