# 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:$eta\; =\; zeta\; +\; f$

is "absolute

vorticity ", with $zeta$ being "relative vorticity", defined as the vertical component of the curl of the fluidvelocity and "f" is the "Coriolis parameter ":$f\; =\; 2\; Omega\; sin\; phi,$

where $Omega$ is the

angular frequency of the planet'srotation ($Omega$=0.7272*10^{-4}s^{-1}for the earth) and $phi$ islatitude .In terms of "relative vorticity", the equation can be rewritten as

:$frac\{D\; zeta\}\{D\; t\}\; =\; -v\; eta,$

where $eta\; =\; partial\; f\; /\; partial\; y$ is the variation of the "Coriolis parameter" with distance $y$ in the north-south directionand $v$ 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 acomputer for the first time, using an observed field of 500 hPageopotential height for the first timestep. This was the one of the first successful instances ofnumerical weather forecasting .**ee also****References****External links*** http://www.met.reading.ac.uk/~ross/Science/BarVor.html

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