The vorticity equation and vortex stretching

The pressure disappears if we take the curl of the momentum equation. Start from the Lamb form, \(\partial_t\mathbf{u} - \mathbf{u}\times\boldsymbol{\omega} = -\nabla(p + \tfrac12|\mathbf{u}|^2) + \nu\Delta\mathbf{u}\). The curl of a gradient is zero, and the identity \(\nabla\times(\mathbf{u}\times\boldsymbol{\omega}) = (\boldsymbol{\omega}\cdot\nabla)\mathbf{u} - (\mathbf{u}\cdot\nabla)\boldsymbol{\omega}\), valid because both fields are divergence-free, gives the vorticity equation \[ \frac{\partial\boldsymbol{\omega}}{\partial t} + (\mathbf{u}\cdot\nabla)\boldsymbol{\omega} = (\boldsymbol{\omega}\cdot\nabla)\mathbf{u} + \nu\Delta\boldsymbol{\omega}. \] Vorticity is carried by the flow (left side), diffused by viscosity (last term), and changed by one new term, \((\boldsymbol{\omega}\cdot\nabla)\mathbf{u}\), the vortex stretching term. The velocity is recovered from the vorticity by the Biot-Savart law, \(\mathbf{u}(\mathbf{x}) = \frac{1}{4\pi}\int\frac{\boldsymbol{\omega}(\mathbf{y})\times(\mathbf{x}-\mathbf{y})}{|\mathbf{x}-\mathbf{y}|^3}\,d\mathbf{y}\), so the equation is closed in \(\boldsymbol{\omega}\) alone.

Write \(\nabla\mathbf{u} = S + W\). Since \(W\boldsymbol{\omega} = \tfrac12\boldsymbol{\omega}\times\boldsymbol{\omega} = 0\), the stretching term is \((\boldsymbol{\omega}\cdot\nabla)\mathbf{u} = S\boldsymbol{\omega}\): only the strain acts on vorticity. If \(\boldsymbol{\omega}\) points along an eigenvector of \(S\) with positive eigenvalue, the vortex line is stretched and \(|\boldsymbol{\omega}|\) grows, like a figure skater spinning faster as she pulls in her arms. Kelvin's theorem says the same thing: a thinning vortex tube keeps its circulation, so the vorticity inside it must rise. In the strain \(\mathbf{u} = (-x,-y,2z)\), a vertical vortex \(\boldsymbol{\omega} = \omega\,\mathbf{e}_z\) feels \((\boldsymbol{\omega}\cdot\nabla)\mathbf{u} = 2\omega\,\mathbf{e}_z\), exponential growth. Burgers (1948) found the steady vortex in which this stretching is exactly balanced by viscous diffusion, with vorticity \(\omega(r) = \frac{\Gamma\alpha}{4\pi\nu}e^{-\alpha r^2/(4\nu)}\).

Now the crucial dichotomy. In two dimensions, \(\mathbf{u} = (u_1(x,y),u_2(x,y),0)\) and \(\boldsymbol{\omega} = (0,0,\omega)\), so \((\boldsymbol{\omega}\cdot\nabla)\mathbf{u} = \omega\,\partial_z\mathbf{u} = 0\). The vorticity equation becomes a scalar advection-diffusion equation, \[ \frac{D\omega}{Dt} = \nu\Delta\omega, \] which obeys the maximum principle: \(\|\omega(t)\|_{L^\infty} \le \|\omega_0\|_{L^\infty}\) for all time, with equality for the Euler equations (\(\nu = 0\)), where vorticity is simply transported by particles. This uniform bound on vorticity is the heart of the proof that smooth 2D solutions exist for all time, for Euler (Wolibner and Hölder, 1933) and for Navier-Stokes (Leray, 1933; Ladyzhenskaya, 1959).

In three dimensions nothing prevents stretching from amplifying vorticity. Heuristically the stretching term is \(S\boldsymbol{\omega}\), and \(S\) is as large as \(\boldsymbol{\omega}\) through Biot-Savart, so \(|\boldsymbol{\omega}|\) might obey something like \(\dot y = y^2\), whose solutions become infinite in finite time. This is only a heuristic (the Biot-Savart relation is non-local and the geometry of vortex lines matters), and viscosity fights back. Whether smooth solutions of the 3D Navier-Stokes equations remain smooth for all time is one of the major open questions of mathematics, one of the Clay Millennium Prize Problems (2000). Vortex stretching is the reason it is hard.

Picture it: a tornado's funnel narrowing as warm air is pulled up through it, spinning ever faster. Think it: in 2D vorticity is a scalar density carried by an area-preserving flow, so all its \(L^p\) norms are controlled; in 3D it is a vector field carried and deformed by the flow map, and its size is controlled by nothing we know how to bound.