= Axisymmetric incompressible epicyclic wave
{title2=$\omega^2=4\Omega(\Omega-A)k_z^2/k^2$}
For $k_y=0$ and $k_z\ne0$, <pressure> elimination gives $\dot v_x=2\Omega\chi v_y-\nu k^2v_x$ and $\dot v_y=-2(\Omega-A)v_x-\nu k^2v_y$, 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
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