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
- <math>\frac{d \eta}{d t} = 0,<math>
where <math>\frac{d}{d t}<math> is the material derivative and
- <math>\eta = \zeta + f<math>
is absolute vorticity, with <math>\zeta<math> being relative vorticity, defined as the vertical component of the curl of the fluid velocity and f is the Coriolis parameter
- <math>f = 2 \Omega \sin \phi,<math>
where <math>\Omega<math> is the angular frequency of the planet's rotation (<math>\Omega<math>=0.7272*10-4 s-1 for the earth) and <math>\phi<math> is latitude.
In terms of relative vorticity, the equation can be rewritten as
- <math>\frac{d \zeta}{d t} = -v \beta,<math>
where <math>\beta = \partial f / \partial y<math> is the variation of the Coriolis parameter with distance <math>y<math> in the north-south direction and <math>v<math> 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.