San José State University |
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Thayer Watkins Silicon Valley & Tornado Alley USA |
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Consider the flow of fluid of constant depth on a latitudinal circle. On the latitudinal circle the relative vorticity ζ is zero but if the flow deviates from the latitudinal circle by an amount δy then a nonzero relative vorticity will develop to conserve absolute vorticity η. Along the path of the fluid flow η at time t1 is the same as η at t0. Since η=ζ+f it follows that if ζ at t0 is zero then
More generally the rate of change of the relative vorticity following a fluid parcel along its trajectory is given by
The path-following rate of change is the instaneous rate of change plus the advection term; i.e.,
The above equation may be linearized under the assumptions that
When these are substituted into the equation for vorticity and terms which are products of deviation terms are dropped the result is:
If a stream function ψ exists for the deviation flows then
Furthermore
The equation that ψ must satisfy is:
If the stream function ψ is assumed to be of the form ψ = ψ0ei(kx+ly - νt) then
Dividing through by -iψ gives
The phase speed c = ν/k is then
The group velocity is given by
A notable thing about Rossby waves is that they propagate to the west relative to the prevailing wind.
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