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
 

Furthermore

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 = ψ₀e^{i(kx+ly - νt)} then

∇²ψ = -(k²+l²)ψ
and hence
-iν(-(k²+l^{2)ψ) + iku(-(k2+l2)ψ) + ikβψ = 0}
 

Dividing through by -iψ gives

(-ν + ku)(k²+l²) - kβ = 0
which 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

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

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