A simplified form of the vorticity equation for an inviscid, divergence-free flow, the barotropic vorticity equation can simply be stated as

$\backslash frac\{D\; \backslash eta\}\{D\; t\}\; =\; 0,$

where $\backslash frac\{D\}\{D\; t\}$ is the material derivative and

$\backslash eta\; =\; \backslash zeta\; +\; f$

is absolute vorticity, with $\backslash zeta$ being relative vorticity, defined as the vertical component of the curl of the fluid velocity and f is the Coriolis parameter

$f\; =\; 2\; \backslash Omega\; \backslash sin\; \backslash phi,$

where $\backslash Omega$ is the angular frequency of the planet's rotation ($\backslash Omega$=0.7272*10^-4 s^-1 for the earth) and $\backslash phi$ is latitude.

In terms of relative vorticity, the equation can be rewritten as

$\backslash frac\{D\; \backslash zeta\}\{D\; t\}\; =\; -v\; \backslash beta,$

where $\backslash beta\; =\; \backslash partial\; f\; /\; \backslash partial\; y$ is the variation of the Coriolis parameter with distance $y$ in the north-south direction and $v$ is the component of velocity in this direction.

In 1950, Charney, Fjorloft, and von Neumann integrated this equation (with an added diffusion term on the RHS) on a computer for the first time, using an observed field of 50 kPa geopotential for the first timestep. This was the one of the first successful instances of numerical weather forecasting.

External links

• http://www.met.rdg.ac.uk/~ross/BarVor.html

Fluid dynamics | Equations