Burgers vortex

A Burgers vortex balances axial vortex stretching against radial viscous diffusion, producing a Gaussian vorticity profile.

A planar strained shear layer is a Burgers vortex sheet. For strain $u_x=-\alpha x$, $u_z=\alpha z$ and transverse velocity $u_y=U\mathrm{erf}(x/{\delta }_s)$, viscous balance gives ${\delta }_s^2=2\nu /\alpha$. At fixed velocity jump its excess viscous dissipation per sheet area scales as $U^2\sqrt{\nu \alpha }$, which vanishes as kinematic viscosity tends to zero.

The swirl viscous dissipation of a steady Burgers vortex, integrated per unit length and per unit density, is $\nu \int {\omega }_z^{2}dA=\alpha {\Gamma }^2/(8\pi )$. It remains finite as kinematic viscosity tends to zero at fixed strain and circulation. This is excess over the background strain: the uniform strain itself has infinite total viscous dissipation over an infinite cross-section.

An axisymmetric Gaussian Burgers vortex with constant circulation in axial strain $\alpha$ has core area parameter $s={\delta }^2$ satisfying $\dot{s}+\alpha s=4\nu$. Thus $s=4\nu /\alpha +(s_0-4\nu /\alpha )e^{-\alpha t}$, an attracting balance of transverse compression and viscous diffusion for every positive finite initial radius.