Vorticity

Vorticity is the curl of the velocity field, $\omega =\nabla \times u$.

The velocity induced by planar vorticity is a convolution with this kernel, whose sign depends on the stream function and vorticity conventions. Its magnitude is $(2\pi |z|{)}^{-1}$ and its derivative is bounded by $C|z{|}^{-2}$. Splitting the convolution at radius $R$ gives $\Vert K\ast \omega {\Vert }_{\infty }\le R\Vert \omega {\Vert }_{\infty }+\Vert \omega {\Vert }_1/(2\pi R)$. The same singularity yields a log-Lipschitz modulus, enough for a Yudovich characteristic flow.

Circular streamlines and constant vorticity imply $u_{\varphi }=\omega r/2+C/r$. Zero velocity at $2a$ and positive azimuthal velocity $V$ at $a$ giveThe radial Euler equations for an inviscid fluid give $p_r=\rho u_{\varphi }^2/r$, yielding outer-minus-inner pressure $(15-16\log 2)\rho V^2/18$. The pressure difference is independent of circulation orientation.

For differentiable vector fields, the displayed identity relates vorticity to the antisymmetric part of the velocity gradient. Together with the curl of a cross product, it converts the adjoint linearized Navier-Stokes evolution between component and rotational forms.

The local vector-calculus identityseparates the convective acceleration into a kinetic-energy gradient and a vorticity cross-product contribution. Substitution into the steady Euler equations for an inviscid fluid yields the vorticity form of the Bernoulli equation.

Kinetic helicity measures the alignment and linking of a velocity field and its vorticity. It is the volume integral of $\mathbf{u}\cdot \mathbf{\omega }$, with boundary conditions needed for its conservation.

The local density of kinetic helicity is $q=\mathbf{u}\cdot \mathbf{\omega }$. Conservation of its volume integral does not imply that $q$ is a materially advected scalar.

For a transverse complex velocity amplitude $\hat{\mathbf{u}}$ and real nonzero wavevector $\mathbf{k}$, write $\hat{\mathbf{u}}=\mathbf{a}+i\mathbf{c}$. Solenoidality makes both real vectors perpendicular to $\mathbf{k}$, so $\frac{1}{2}\mathrm{Re}(i{\hat{\mathbf{u}}}^{\ast }\times \hat{\mathbf{u}})=-\mathbf{a}\times \mathbf{c}=H\mathbf{k}$. The corresponding spatial average of kinetic helicity density is $-H{k}^2$. Cross-product order determines this sign.

For smooth inviscid barotropic fluid motion without gravity or magnetic forces, let $w$ be the specific enthalpy with $\nabla w={\rho }^{-1}\nabla p$, and set $q=\mathbf{u}\cdot \mathbf{\omega }$. ThenThis follows by dotting the barotropic vorticity transport equation with $\mathbf{u}$ and using the divergence of a cross product. A volume integral is constant when the resulting boundary flux vanishes.

The kinetic helicity conservation law impliesTherefore $q$ is materially constant exactly when the right side vanishes. Incompressible flow together with constancy of $w-u^2/2$ along vortex lines is sufficient. In compressible flow, the density-normalized quantity satisfies $D(q/\rho )/Dt=-(\mathbf{\omega }/\rho )\cdot \nabla (w-u^2/2)$.

Absolute vorticity is the vorticity measured in an inertial frame. In a frame rotating with angular velocity $\mathbf{\Omega }$, it is $\mathbf{\omega }+2\mathbf{\Omega }$.

The planetary contribution to absolute vorticity is supplied by the rotating frame. Its local vertical component is the Coriolis parameter $f=2\Omega \sin \varphi$. On a beta plane, motion with northward velocity $v$ changes this contribution at the rate $\beta v$.

For a flow satisfying incompressibility, this antisymmetric tensor writes advection and vortex stretching in conservative form. With its first index as transport direction, $(\nabla \cdot T{)}_j={\partial }_i{T}_{ij}$.

When the only nonpotential force points radially, its curl has no radial component. The radial absolute vorticity therefore obeys a local flux conservation law, although vortex stretching can change it along a fluid trajectory.

Relative vorticity is the curl of velocity measured in the chosen rotating frame. Adding the planetary vorticity gives absolute vorticity.

The circulation around a closed curve is $\Gamma =\oint u\cdot dl$.

An irrotational flow has zero vorticity and locally admits a velocity potential.