Vortensity ($\zeta$)

Vortensity is absolute vorticity divided by surface density. In a two-dimensional inviscid barotropic flow, it is conserved along fluid trajectories.

Linearized vortensity is the first-order perturbation of vortensity about a background flow. For a shearing wave in a uniform Keplerian sheet, its amplitude is constant.

Linearized vortensity conservation is the perturbative form of material vortensity conservation. In an inviscid barotropic shearing sheet, the linearized vortensity obeys $D{f}^{\prime }/Dt=0$.

For a radial mode with constant nonzero wavenumber in an inviscid razor-thin disc, eliminate velocity using continuity and the conserved linearized vortensity. The resulting oscillator has ${\omega }_k^2={\kappa }^2+c_s^2{k}^2-2\pi G{\Sigma }_0|k|$. The vortensity gives a constant forcing and hence the stationary balanced component when ${\omega }_k^2>0$. It is lost by an ansatz that divides every amplitude equation by a nonzero frequency. Negative squared frequency gives exponential instability; zero frequency allows secular growth.