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The first-order perturbation of the vortensity isTaking the curl of the linearized momentum equation and using the linearized continuity equation yieldsThis is linearized vortensity conservation. It is the perturbative form of material conservation of potential vorticity in an inviscid barotropic fluid; the forcing $-\nabla \Psi$ is a gradient and therefore creates no vorticity.

For number density $n$, rest-frame energy density $\rho$ and pressure $p$, the enthalpy per particle is $ϵ=(\rho +p)/n$. In an isentropic barotropic fluid, $d\rho =ϵdn$ and $dp=ndϵ$. It differs from enthalpy per unit mass by the particle mass when that mass is fixed. Using $ϵ$ as thermodynamic variable gives $n=p^{\prime }(ϵ)$ and $\rho =ϵp^{\prime }(ϵ)-p(ϵ)$.

Taking the curl of axisymmetric hydrostatic equilibrium with centrifugal acceleration gives $\varpi {\partial }_z{\Omega }^2=(\nabla p\times \nabla \rho {)}_{\varphi }/{\rho }^2$. For an approximately spherical ideal gas with constant composition, use $\nabla \rho =\nabla p/c_s^2-\rho \nabla S/c_p$ to obtain the displayed relation. Latitudinal entropy differences can support noncylindrical differential rotation. A barotropic fluid gives cylindrical rotation in this leading balance; the relation does not determine the transport stresses or circulation rate.

In a connected inviscid uniformly rotating barotropic fluid, momentum balance makes the displayed sum constant. The pressure term is the barotropic enthalpy function. Perturbations experience Coriolis acceleration, and their rotating-frame frequency depends on the chosen exponential sign convention.