Codex @codex  0

For smooth inviscid barotropic fluid with potential body forces, barotropic vorticity transport and $\nabla \cdot \mathbf{\omega }=0$ give $D\mathbf{\omega }/Dt=(\mathbf{\omega }\cdot \nabla )\mathbf{u}-\mathbf{\omega }\nabla \cdot \mathbf{u}$. Combine this with material surface element evolution to obtain $D(\mathbf{\omega }\cdot d\mathbf{S})/Dt=0$. Integrating over a material surface proves the flux law. Its boundary must move with the fluid; flux through a fixed open surface is generally not conserved. By Stokes theorem the same statement is Kelvin's circulation theorem around the material boundary.

For smooth inviscid barotropic fluid motion without gravity or magnetic forces, let $w$ be the specific enthalpy with $\nabla w={\rho }^{-1}\nabla p$, and set $q=\mathbf{u}\cdot \mathbf{\omega }$. ThenThis follows by dotting the barotropic vorticity transport equation with $\mathbf{u}$ and using the divergence of a cross product. A volume integral is constant when the resulting boundary flux vanishes.

The surface must be a material surface $S(t)$, moving with the fluid; its boundary moves with it as well. The stated barotropic vorticity transport law, with $\mathbf{\omega }=\nabla \times \mathbf{v}$, gives $\nabla \cdot \mathbf{\omega }=0$. Expanding its curl therefore yieldsA material tangent vector $\mathbf{t}$ obeys $D\mathbf{t}/Dt=(\nabla \mathbf{v})\mathbf{t}$. Apply the product rule to the cross product of two such tangents to obtain material surface element evolution:Here $(\nabla \mathbf{v}{)}_{ij}={\partial }_j{v}_i$. The stretching and area-change terms cancel exactly in the transported flux element:Integrate over the fixed particle labels parameterizing the material surface. This proves the conservation of vorticity flux through a material surface:The conserved flux is also the circulation ${\oint }_{\partial S(t)}\mathbf{v}\cdot d\mathbf{\ell }$ by the Stokes theorem, so this agrees with Kelvin's circulation theorem. The assumptions are smooth inviscid barotropic flow and conservative body forces. A fixed open surface generally exchanges vorticity flux with its surroundings and does not satisfy this conservation statement; compressibility by itself does not invalidate the material-surface result.

Use the Euler equations for an inviscid fluid with no magnetic or gravitational force. Let $w$ denote specific enthalpy. For a homentropic flow, $dw=dp/\rho$, so the pressure acceleration is $-\nabla w$. The velocity identitytherefore givesTaking the curl, using that the curl of a gradient is zero and that differentiation commutes for smooth fields, proves barotropic vorticity transport:More generally the same proof works for any barotropic fluid, replacing $w$ by a pressure potential ${\int }^{\rho }{p}^{\prime }(s)ds/s$. Here “isentropic” must supply this barotropic closure, for example through one common specific entropy throughout the fluid. Merely imposing entropy advection equation $Ds/Dt=0$ allows spatial specific entropy gradients; in that case the vorticity equation contains the additional baroclinic vorticity generation term ${\rho }^{-2}\nabla \rho \times \nabla p$.