Barotropic vorticity equation

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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 hPa geopotential height for the first timestep. This was the one of the first successful instances of numerical weather forecasting.