Dissipation identity for a homogeneous Squire mode ($\mathrm{Im}\omega =-\frac{\Vert {\eta }^{\prime }{\Vert }_2^2+(k^2+l^2)\Vert \eta {\Vert }_2^2}{R\Vert \eta {\Vert }_2^2}$)

For real base-flow velocity and positive Reynolds number, multiply the homogeneous Squire equation by the conjugate mode and integrate with zero endpoint values. Taking real parts gives the displayed negative temporal growth rate. This proves decay of homogeneous Squire modes; the coupled shear-flow system can still be unstable or show transient growth because wall-normal velocity can force the Squire variable.