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 Incoming links: Conservation of vorticity flux through a material surface

Past exam of the astrophysical fluid dynamics course of the University of Cambridge / 2017 / 4 / b / Solution 2026-10-18

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

ω = \nabla \times v

, gives

\nabla \cdot ω = 0

. Expanding its curl therefore yields

\frac{D ω}{D t} = (ω \cdot \nabla) v - ω (\nabla \cdot v) .

A material tangent vector

t

obeys

D t / D t = (\nabla v) t

. Apply the product rule to the cross product of two such tangents to obtain material surface element evolution:

\frac{D ( d S )}{D t} = [(\nabla \cdot v) I - (\nabla v)^{T}] d S .

Here

(\nabla v)_{ij} = \partial_{j} v_{i}

. The stretching and area-change terms cancel exactly in the transported flux element:

\frac{D}{D t} (ω \cdot d S) = [(ω \cdot \nabla) v - ω \nabla \cdot v] \cdot d S + ω \cdot [(\nabla \cdot v) I - (\nabla v)^{T}] d S = 0.

Integrate over the fixed particle labels parameterizing the material surface. This proves the conservation of vorticity flux through a material surface:

\frac{d}{d t} \int_{S (t)} ω \cdot d S = 0.

The conserved flux is also the circulation

\oint_{\partial S (t)} v \cdot d ℓ

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.

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