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For an incompressible shear flow $U(z){\mathbf{e}}_x$ with uniform magnetic field $B{\mathbf{e}}_x$, define $S=\sigma +k_{x}U$ for time dependence $e^{i\sigma t}$ and $\zeta =i{k}_{x}v-i{k}_{y}u$. Eliminating the magnetic perturbations from the horizontal momentum equations gives the displayed product for $S\ne 0$. Away from the Alfvén wave resonance, it yields $\zeta =k_y{U}^{\prime }w/S$. At resonance one must retain the original system: for constant $U$, $k_y=0$, $S^2=k_x^2{B}^2/\rho$, any smooth transverse amplitude $v(z)$ with $b_y=k_{x}Bv/S$ and all other amplitudes zero is an Alfvén wave. Its nonzero vorticity disproves the unrestricted zero-vorticity reduction. Exponentially growing modes with real background coefficients cannot encounter these real-frequency resonances.

For an inviscid barotropic fluid with only potential body forces, the vorticity $\mathbf{\omega }=\nabla \times \mathbf{u}$ obeysThe pressure force is a gradient because ${\rho }^{-1}\nabla p=\nabla {\int }^{\rho }{p}^{\prime }(s)ds/s$. A spatially uniform specific entropy makes a fixed-composition fluid barotropic. Merely having $Ds/Dt=0$ does not exclude baroclinic vorticity generation from spatial entropy gradients.

With the streamfunction convention $(u,v)=({\psi }_y,-{\psi }_x)$, constant scalar vorticity gives $\mathbf{u}\times \mathbf{\omega }=-\omega \nabla \psi$. For steady inviscid constant-density flow without body force, this integrates tothroughout a connected fluid region. Reversing the streamfunction convention changes the sign of its term; the declared convention matters.

For a spherical cell with its centre of mass displaced by $-{\ell }_g\mathbf{p}$ from its geometric centre, gravity produces torque $mg{\ell }_g\mathbf{p}\times \mathbf{k}$. Rotational resistance ${\zeta }_r$ gives angular velocity $\mathbf{\omega }/2+(mg{\ell }_g/{\zeta }_r)\mathbf{p}\times \mathbf{k}$. Taking its cross product with $\mathbf{p}$ gives the displayed orientation equation with $B={\zeta }_r/(mg{\ell }_g)$. Steady orientation therefore requires the two terms on the right to sum to zero; placing the vorticity term alone on the other side changes its sign.

The Brinkman equation augments Stokes flow with a local drag against a background matrix of velocity ${\mathbf{u}}_m$. The inverse screening length $\alpha$ has units of inverse length. With equal viscous coefficients the permeability is ${\alpha }^{-2}$. Taking the curl produces a screened vorticity equation.

For an upstream irrotational current on an f-plane in a rigid layer of height $h$, a circular region of height $h(1-ϵ)$ has the displayed relative vorticity if its fluid retains upstream potential vorticity $f/h$. To leading order in the height contrast, take $u=-{\Psi }_y$, $v={\Psi }_x$ and $\Psi =-Uy+\psi$. Solving the Poisson equation for $\psi$ gives $\psi =-C{r}^2/2$ inside and $\psi =C{a}^2[\log (a/r)-1/2]$ outside, where $C=fϵ/2$. The disturbance is clockwise for positive $fϵ$. Variable-depth transport and the transition across the rim introduce higher-order corrections to this leading columnar inversion.

Use circulation to summarize how much a velocity field runs along an oriented closed curve $C$, such as a circle around a fluid vortex. Its definition is the line integralwhere $d\mathbf{l}$ is tangent to $C$ in the chosen direction. Reversing that direction reverses the sign of circulation. Its units are ${\mathrm{m}}^2{\mathrm{s}}^{-1}$, because velocity is multiplied by a length. Counterclockwise circulation in the horizontal plane is positive when the chosen unit normal vector is $+{\mathbf{e}}_z$, according to the right-hand rule.

For solid-body rotation $\mathbf{u}=(-\Omega y,\Omega x,0)$, a counterclockwise circle of radius $R$ has constant tangential velocity $\Omega R$. Multiplying by its circumference givesFor the potential line vortex $\mathbf{u}=K{\mathbf{e}}_{\theta }/r$, the same circle instead gives ${\Gamma }_C=(K/R)(2\pi R)=2\pi K$, independent of $R$. Stipulate $\Omega =1{\mathrm{s}}^{-1}$, $R=1\mathrm{m}$ and $K=1{\mathrm{m}}^2{\mathrm{s}}^{-1}$: both velocity fields have ${\Gamma }_C=2\pi {\mathrm{m}}^2{\mathrm{s}}^{-1}\approx 6.283185{\mathrm{m}}^2{\mathrm{s}}^{-1}$, although their vorticities in the fluid are $2{\mathbf{e}}_z{\mathrm{s}}^{-1}$ and zero respectively. On a circle of radius $2\mathrm{m}$, their circulations are $8\pi {\mathrm{m}}^2{\mathrm{s}}^{-1}$ and $2\pi {\mathrm{m}}^2{\mathrm{s}}^{-1}$. The comparison plot in vorticity shows this difference in radius dependence.

The bridge between circulation and vorticity is Stokes theorem:Here the velocity field must be continuously differentiable on a neighborhood of the entire spanning surface $S$ inside the fluid, and its boundary orientation must match the unit normal vector $\mathbf{n}$. For solid-body rotation, the disk is admissible and its surface integral is $2\Omega \pi R^2$. For the potential line vortex, the origin is singular and excluded from the fluid: a disk crossing it is not admissible. On an annulus, Stokes theorem instead gives ${\Gamma }_{R_{\mathrm{out}}}-{\Gamma }_{R_{\mathrm{in}}}=0$, with both circulations here measured counterclockwise; the induced inner boundary direction is clockwise. Nonzero circulation therefore does not imply nonzero local curl in the available fluid. ^[ref]

This is useful because Stokes theorem lets a local vorticity measurement determine a loop's circulation when a suitable surface exists, while Kelvin's circulation theorem tracks circulation along a moving material curve. The beauty is the precise link between a local derivative and a global line integral, with the domain deciding which conclusions are valid.

Convective acceleration is the spatial part $(\mathbf{u}\cdot \nabla )\mathbf{u}$ of the material derivative of fluid velocity. It can remain nonzero in steady flow, because a moving particle passes through positions with different velocities. The identity $(\mathbf{u}\cdot \nabla )\mathbf{u}=\nabla (|\mathbf{u}{|}^2/2)-\mathbf{u}\times (\nabla \times \mathbf{u})$ relates it to vorticity.

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.

The curl of $F$ has components

For a continuously differentiable vector field in three-dimensional Euclidean space, the component of its curl along a unit normal vector is the limiting circulation per unit area around a small loop in the perpendicular plane. The loop direction and the normal obey the right-hand rule. Stokes theorem gives the corresponding relation for a finite surface.

For the rigidly rotating velocity field $G=(-\Omega y,\Omega x,0)$, direct differentiation gives $\nabla \times G=(0,0,2\Omega )$. A circle of radius $R$ has tangential speed $\Omega R$ and circumference $2\pi R$, so its circulation is $2\pi \Omega R^2$, twice the angular velocity times the enclosed area. The factor of two distinguishes vorticity from rigid-body angular velocity.

Curved integral curves alone do not demonstrate nonzero curl. The field $H=(x,-y,0)$ has zero curl, yet its trajectories satisfy $x=x_0{e}^t$, $y=y_0{e}^{-t}$ and hence $xy=x_0{y}_0$: away from the axes they are hyperbolas. The curl measures local circulation of the field, not the curvature of an individual trajectory.

In a planar force-driven encounter with a fixed equal sphere and positive impact parameter $b\gg a$, the mobility correction from a fixed distant sphere gives $Y-b=27a^{2}b/[32(X^2+b^2)]+O(a^4/b^3)$. The moving sphere reaches maximum deflection $27a^2/(32b)$ and rotates clockwise through $9\pi a^2/(32b^2)$ to leading order. The spin is obtained by integrating half the reflected Stokeslet vorticity, as prescribed by Faxén rotation law. These numerical formulas require a distant encounter.

A localized vorticity distribution with nonzero hydrodynamic impulse induces $\mathbf{u}\sim [3\mathbf{n}(\mathbf{I}\cdot \mathbf{n})-\mathbf{I}]/(4\pi r^3)$. This curl-free exterior velocity can remain long-ranged even when the vorticity is localized. In a random ensemble it produces dipolar longitudinal and transverse correlation tails.

Enstrophy measures squared vorticity: a local density is ${\omega }^2/2$, and its spatial integral or ensemble mean gives a global measure. In homogeneous incompressible flow, viscous dissipation satisfies $ϵ=\nu ⟨{\omega }^2⟩$. Vortex stretching can produce enstrophy in three dimensions; kinematic viscosity destroys it.

For a sphere of radius $a$ in a slowly varying ambient Stokes flow of vorticity ${\mathbf{\omega }}_{\infty }$, its angular velocity and applied torque obeyat the sphere centre.

In a nonrotating ideal Boussinesq layer with uniform current $U>0$, constant buoyancy frequency $N$ and rigid walls, any stationary superposition of channel wave modes with $k_n^2+(n\pi /H{)}^2=N^2/U^2$ obeys the displayed relations. With $(u,w)=({\psi }_z,-{\psi }_x)$, the total streamfunction is $\Psi =Uz+{\psi }^{\prime }$ and total buoyancy is $\sigma =(N^2/U)\Psi$. Its material derivative vanishes. The perturbation vorticity is proportional to ${\psi }^{\prime }$, so its nonlinear self-advection vanishes as well. The modes are therefore exact finite-amplitude solutions, although large amplitudes may violate physical approximation assumptions or become unstable.

In nearly columnar flow between rigid boundaries, a geostrophic contour is a contour of column height divided by the Coriolis parameter. If relative vorticity is negligible, conservation of shallow-water potential vorticity constrains leading geostrophic flow to these contours. Motion across them stretches or squashes columns, generating relative vorticity; for constant $f$, the contours are simply contours of $H$.

At an isolated interface with continuous base velocity, a stable density drop $\Delta \rho /2$ and a jump $[U^{\prime }]$ of base shear, decaying disturbances $w\propto e^{-k|z-z_i|}$ obey the displayed dispersion relation. It follows by inserting their derivative jump $[w^{\prime }]=-2kw$ into the jump conditions for stratified inviscid shear flow. Both the density jump and the vorticity jump contribute to the intrinsic phase velocity $c-U_i$.

Gyrotaxis is the orientation response of swimming cells to the competition between gravity-induced torque and fluid rotation. A bottom-heavy spherical cell tends to point upward in still fluid, while ambient vorticity rotates it. The response changes the mean swimming direction and can focus cells or contribute to bioconvection.

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 integral measures linkage and twisting of vortex lines through the velocity and vorticity fields.

In an unbounded incompressible fluid with localized vorticity and no added harmonic background, velocity is reconstructed by $\mathbf{u}(\mathbf{x})=(4\pi {)}^{-1}\int \mathbf{\omega }(\mathbf{y})\times (\mathbf{x}-\mathbf{y})/|\mathbf{x}-\mathbf{y}{|}^3{d}^{3}y$. Its multipole expansion relates hydrodynamic impulse to a dipolar far-field velocity.