San José State University
Thayer Watkins
Silicon Valley
& Tornado Alley

The Analytical Derivation of Rossby Waves

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

ζ(t1) + f(y(t1)) = f(y(t0))
and thus
ζ(t1) = -[f(y(t1))-f(y(t0))]
= -(df/dy)δy = -βδy(t1)

More generally the rate of change of the relative vorticity following a fluid parcel along its trajectory is given by

dζ/dt = -βd(δy)/dt = -βv

The path-following rate of change is the instaneous rate of change plus the advection term; i.e.,

dζ/dt = ∂ζ/∂t + u∂ζ/∂x + v∂ζ/∂y + βv = 0

The above equation may be linearized under the assumptions that

ζ = ζ + ζ'
u = u + u'
v = v'

When these are substituted into the equation for vorticity and terms which are products of deviation terms are dropped the result is:

∂ζ'/∂t + u∂ζ'/∂x + βv' = 0

If a stream function ψ exists for the deviation flows then

u' = -∂ψ/∂y and v' = ∂ψ/∂x


ζ' = ∇2ψ

The equation that ψ must satisfy is:

∂(∇2ψ)/∂t + u∂(∇2ψ)/∂x + β∂ψ/∂x = 0

If the stream function ψ is assumed to be of the form ψ = ψ0ei(kx+ly - νt) then

2ψ = -(k2+l2
and hence
-iν(-(k2+l2)ψ) + iku(-(k2+l2)ψ) + ikβψ = 0

Dividing through by -iψ gives

(-ν + ku)(k2+l2) - kβ = 0
which when solved gives
ν = ku - βk/(k2+l2)

The phase speed c = ν/k is then

c = u - β/(k2+l2))

The group velocity is given by

vg = ∂ν/∂k = u - β/(k2+l2)) + β(2k)/(k2+l2)2
= u - β[(k2-2k+l2)/(k2+l2)2)

A notable thing about Rossby waves is that they propagate to the west relative to the prevailing wind.

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