= Helicity invariant of an inviscid shearing wave
{title2=$H(t)|\mathbf k(t)|^2=H(0)(k_x^2+k_z^2)$}
For a solenoidal velocity amplitude $\mathbf v$ with $\mathbf k=(k_x,-\Omega tk_x,k_z)$ and $\dot{\mathbf v}+\Omega v_y\widehat{\mathbf x}=-i\mathbf kq$, differentiating $\mathbf k\cdot\mathbf v=0$ gives $q=2i\Omega k_xv_y/K^2$, where $K^2=|\mathbf k|^2$. Thus $\dot v_y=2\Omega k_xk_yv_y/K^2$ and $\dot v_z=2\Omega k_xk_zv_y/K^2$. For the real helicity quantity $H=ik_x(v_yv_z^*-v_zv_y^*)$, the mixed forcing terms cancel, leaving $\dot H=2\Omega k_xk_yH/K^2=-\dot K^2H/K^2$. Consequently
$$
H(t)K(t)^2=H(0)(k_x^2+k_z^2).
$$
The invariant is a consequence of inviscid linear shear dynamics, not a claim that $H$ itself is constant under the shear.
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