The vorticity equation is an important prognostic equation in the atmospheric sciences. Vorticity is a vector, therefore, there are three components. The equation of vorticity (three components in the canonical form) describes the total derivative (that is, the local change due to local change with time and advection) of vorticity, and thus can be stated in either relative or absolute form.

The more compact version is that for absolute vorticity, component $\backslash eta$, using the pressure system:

$\backslash frac\{d\; \backslash eta\}\{d\; t\}\; =\; -\backslash eta\; \backslash cdot\; \backslash nabla\_h\; \backslash mathbf\{v\}\_h\; -\; \backslash left(\; \backslash frac\{\backslash partial\; \backslash omega\}\{\backslash partial\; x\}\; \backslash frac\{\backslash partial\; v\}\{\backslash partial\; z\}\; -\; \backslash frac\{\backslash partial\; \backslash omega\}\{\backslash partial\; y\}\; \backslash frac\{\backslash partial\; u\}\{\backslash partial\; z\}\; \backslash right)\; -\; \backslash frac\{1\}\{\backslash rho^2\}\; \backslash mathbf\{k\}\; \backslash cdot\; (\; \backslash nabla\_h\; p\; \backslash times\; \backslash nabla\_h\; \backslash rho\; )$

Here, $\backslash rho$ is density, u, v, and $\backslash omega$ are the components of wind velocity, and $\backslash nabla\_h$ is the 2-dimensional (i.e. horizontal-component-only) del.

The terms on the RHS denote the positive or negative generation of absolute vorticity by divergence of air, twisting of the axis of rotation, and baroclinity, respectively.

See also

• Barotropic vorticity equation

Fluid dynamics | Equations