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# Barotropic vorticity equation

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

$\frac{d \eta}{d t} = 0,$

where $\frac{d}{d t}$ is the material derivative and

η = ζ + f

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

f = 2Ωsinφ,

where Ω is the angular frequency of the planet's rotation (Ω=0.7272*10-4 s-1 for the earth) and φ is latitude.

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

$\frac{d \zeta}{d t} = -v \beta,$

where $\beta = \partial f / \partial y$ 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 500 mb geopotential for the first timestep. This was the one of the first successful instances of numerical weather forecasting .