Vorticity
Definition of Vorticity:
Vorticity describes the local spinning motion of a fluid. It is a measure of the local rotation (or angular velocity) of fluid elements.
This is the common definition for Vorticity, other definitions can be discussed in the article
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Mathematical expression
In three dimensions, the vorticity [math]\vec{\zeta}[/math] is a vector defined by the curl of the velocity field:
[math]\vec{\zeta} = \vec{\nabla} \times \vec{v}[/math],
where [math]\vec{v}[/math] is the velocity vector with components [math](u, v, w)[/math] along the [math](x, y, z)[/math] directions.
In two dimensions [math](x, y)[/math], the vorticity is a scalar, [math]\zeta = \dfrac{\partial v}{\partial x} - \dfrac{\partial u}{\partial y}[/math].
For circular motion, [math]x = r \cos \omega t[/math], [math]y = r \sin \omega t[/math], the vorticity equals twice the angular rotation speed, [math]\zeta = 2\omega[/math].
Conservation of angular momentum
The importance of the vorticity concept is related to the conservation of angular momentum in fluid motions not subjected to external torques, such as friction. The angular momentum of a cylindrical fluid volume of radius [math]r[/math], height [math]h[/math], mass [math]m[/math], and uniform density [math]\rho[/math], rotating with angular speed [math]\omega[/math] about its axis, scales with [math]\rho r^2 \omega[/math] and thus with [math]m \omega / h[/math].
Conservation of angular momentum implies that when the same cylindrical fluid mass moves frictionlessly into a region with different depth [math]h[/math], its rotation rate changes as the fluid column is stretched or compressed. This principle extends to fluid volumes of arbitrary shape, in which case the angular rotation rate [math]2\omega[/math] is replaced by the relative vorticity [math]\zeta[/math].
Potential vorticity
When fluid volumes travel large distances, their vorticity is influenced by Earth’s rotation. The associated planetary vorticity is
[math]f = 2 \Omega_{earth} \sin \phi[/math],
where [math]\Omega_{earth}[/math] is Earth’s angular rotation rate and [math]\phi[/math] is latitude (see Coriolis acceleration).
The potential vorticity of ocean currents is given by
[math]\Pi = \dfrac{\zeta + f}{h}[/math],
where [math]\zeta[/math] is the relative vorticity. Ocean currents with negligible friction and relative vorticity much smaller than [math]f[/math] approximately conserve [math]f/h[/math]. Over distances where [math]f[/math] varies little, such currents tend to follow isobaths, since crossing depth contours would require changes in potential vorticity. Currents in geostrophic balance (between pressure gradients and the Coriolis force) often exhibit this behavior and are called geostrophic currents.
More formal derivations of geostrophic currents and potential vorticity are given in the article Geostrophic flow.
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