Axisymmetric incompressible epicyclic wave (${\omega }^2=4\Omega (\Omega -A)k_z^2/k^2$)

For $k_y=0$ and $k_z\ne 0$, pressure elimination gives ${\dot{v}}_x=2\Omega \chi v_y-\nu k^2{v}_x$ and ${\dot{v}}_y=-2(\Omega -A)v_x-\nu k^2{v}_y$, where $\chi =k_z^2/k^2$. The inviscid frequency is the radial epicyclic frequency times $\sqrt{\chi }$. With $w=v_x/\sqrt{\chi }$, $(\Omega -A)|w{|}^2+\Omega |v_y{|}^2$ is conserved when viscosity vanishes. In a Keplerian flow it bounds kinetic-energy amplification by four, whereas with zero rotation the same system has the algebraic lift-up effect.

// Target: turbulence.bigb