Vorticity is a divergence-free vector field because it is the curl of velocity:

$$\boldsymbol\omega=\nabla\times\mathbf v,$$

so

$$\boxed{\nabla\cdot\boldsymbol\omega=0}.$$

This simple identity imposes strong geometric constraints on how concentrated vorticity can exist in a fluid.

Vortex lines and vortex tubes

A vortex line is a curve whose tangent is everywhere parallel to the local vorticity vector.

A bundle of neighboring vortex lines forms a vortex tube. The side surface of the tube is itself made of vortex lines, so vorticity is tangent to that side surface and has no normal flux through it.

For a cross-section $A$ cutting the tube, define the vorticity flux

$$\boxed{\Gamma_A=\iint_A\boldsymbol\omega\cdot\mathbf n,dA}.$$

By Stokes' theorem, this flux equals the circulation around the cross-section boundary:

$$\boxed{\Gamma_A =\oint_{\partial A}\mathbf v\cdot d\mathbf r}.$$

Thus the circulation carried by a vortex tube is its vorticity flux.

Vortex-tube strength is constant along the tube

Take a finite segment of a vortex tube bounded by two cross-sections $A_1$ and $A_2$ and by the tube's side surface.

Apply the divergence theorem to vorticity:

$$\iint_{\partial V}\boldsymbol\omega\cdot\mathbf n,dA =\iiint_V\nabla\cdot\boldsymbol\omega,dV=0.$$

There is no flux through the side surface because $\boldsymbol\omega$ is tangent to it. With outward normals on the two end sections, the remaining terms give

$$-\Gamma_1+\Gamma_2=0.$$

Therefore

$$\boxed{\Gamma_1=\Gamma_2}.$$

The circulation strength of a vortex tube is constant along its length as long as the field remains regular.

A tube can change cross-sectional area. If it narrows while carrying the same vorticity flux, its typical vorticity magnitude must increase.

A vortex tube cannot terminate inside a regular fluid region

Suppose a vortex tube ended abruptly at an interior point. Enclose the end with a small closed surface. Vorticity would enter that surface through the tube but would have nowhere to leave, producing nonzero net vorticity flux.

That would contradict

$$\nabla\cdot\boldsymbol\omega=0.$$

Therefore a vortex tube cannot simply terminate in a smooth interior region. It must instead

• form a closed loop;
• extend to a boundary;
• extend out of the modeled domain, for example far downstream.

This is the fluid-mechanical analogue of the fact that a divergence-free field has no isolated sources or sinks.

Vortex filament idealization

If a vortex tube becomes very thin compared with the distances of interest, it can be idealized as a vortex filament carrying circulation strength $\Gamma$ along a curve.

The detailed vorticity distribution inside the core is then replaced by

• the filament centerline;
• its orientation;
• its circulation $\Gamma$.

The filament can bend through space, but its circulation cannot vary arbitrarily along one connected regular filament. A change of circulation requires additional vortex structure to branch or leave the region.

Worked example: a contracting vortex tube

Suppose one cross-section of a vortex tube has area

$$A_1=4.0\times10^{-4},\mathrm{m^2}$$

and approximately uniform vorticity magnitude

$$\omega_1=20,\mathrm{s^{-1}}$$

normal to the section.

Its circulation strength is

$$\Gamma=\omega_1A_1 =(20)(4.0\times10^{-4}) =8.0\times10^{-3},\mathrm{m^2/s}.$$

Farther along the same tube, suppose the cross-sectional area is

$$A_2=1.0\times10^{-4},\mathrm{m^2}.$$

Conservation of vorticity flux along the tube requires

$$\omega_2A_2=\Gamma,$$

so

$$\omega_2 =\frac{8.0\times10^{-3}}{1.0\times10^{-4}} =\boxed{80,\mathrm{s^{-1}}}.$$

The area has decreased by a factor of four, so the representative vorticity magnitude has increased by the same factor.

Kinematic constraint versus dynamical evolution

The statements above follow from

$$\nabla\cdot\boldsymbol\omega=0$$

at one instant. They should not be confused with Kelvin's theorem, which describes how circulation of a material loop evolves in time under additional dynamical assumptions.

Here the key result is geometric: vorticity lines do not have sources or sinks inside a regular fluid region, so vortex-tube circulation is continuous along the tube. This constraint is what forces the bound vorticity of a finite lifting wing to continue into its wake.