Helicity invariant of an inviscid shearing wave ($H(t)|\mathbf{k}(t){|}^2=H(0)(k_x^2+k_z^2)$)

For a solenoidal velocity amplitude $\mathbf{v}$ with $\mathbf{k}=(k_x,-\Omega t{k}_x,k_z)$ and $\dot{\mathbf{v}}+\Omega v_y\hat{\mathbf{x}}=-i\mathbf{k}q$, differentiating $\mathbf{k}\cdot \mathbf{v}=0$ gives $q=2i\Omega k_x{v}_y/K^2$, where $K^2=|\mathbf{k}{|}^2$. Thus ${\dot{v}}_y=2\Omega k_x{k}_y{v}_y/K^2$ and ${\dot{v}}_z=2\Omega k_x{k}_z{v}_y/K^2$. For the real helicity quantity $H=i{k}_x(v_y{v}_z^{\ast }-v_z{v}_y^{\ast })$, the mixed forcing terms cancel, leaving $\dot{H}=2\Omega k_x{k}_{y}H/K^2=-{\dot{K}}^{2}H/K^2$. ConsequentlyThe invariant is a consequence of inviscid linear shear dynamics, not a claim that $H$ itself is constant under the shear.