Squire equation ($[(-i\omega +i\alpha U)-R{e}^{-1}(D^2-{\kappa }^2)]\hat{\eta }=-i\beta U^{\prime }\hat{v}$)

The Squire equation governs wall-normal vorticity $\eta ={\partial }_{z}u-{\partial }_{x}w$ in the same normal mode convention as the Orr-Sommerfeld equation. It follows by taking ${\partial }_z$ of the streamwise momentum equation minus ${\partial }_x$ of the spanwise momentum equation. At a rigid no-slip wall, $\hat{\eta }=0$.

A Squire mode has zero wall-normal velocity and nonzero wall-normal vorticity, satisfying the homogeneous Squire equation. For real base velocity, positive finite Reynolds number, and zero boundary terms,for a nontrivial mode in a finite channel with no-slip walls. This follows by multiplying the homogeneous equation by $\overline{\eta }$, integrating by parts and taking the real part. Such modes are damped even when the coupled velocity-vorticity system can exhibit transient growth from non-normal modes.