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For a Burgers vortex, integrating the axial vorticity gives the azimuthal velocity:The swirl contribution to the viscous dissipation per unit length and per unit density isFor the equality, expand the two squares using ${\omega }_z=u_{\theta }^{\prime }+u_{\theta }/r$; their integrated difference is proportional to $[u_{\theta }^2{]}_0^{\infty }$, which vanishes. Substitution of the Gaussian vorticity profile givesThis excess dissipation of a Burgers vortex is independent of kinematic viscosity at fixed strain $\alpha$ and circulation $\Gamma$. Multiply by the mass density if a dimensional power per length is wanted.

The qualification "excess" matters. The imposed uniform strain itself has viscous dissipation density $3\nu {\alpha }^2$ per unit mass, so its integral over an infinite cross-section diverges. One must subtract that background or restrict to a finite cross-section. In a core-sized area, its contribution is of order ${\nu }^2\alpha$, smaller than the swirl contribution by order $(\nu /\Gamma {)}^2$ as $\Gamma /\nu \to \infty$.

Meanwhile $\delta \propto {\nu }^{1/2}$ shrinks and the central vorticity $\Gamma /(\pi {\delta }^2)$ grows. The finite dissipation becomes concentrated in a narrow tube: a useful local model of internal intermittency and the turbulent dissipation anomaly, though not a proof that an entire turbulent flow consists of such vortices.