Rossby Wave Equation

 

A.  Assumptions

 

1.    Non-divergence

2.    Zonal flow, with no variation constant west wind speed with longitude, and negligible vertical motions (or u>>v>>w)=0

3. Wind is geostrophic.

B.  Rossby's Equation

 

Using barotropic vorticity equation, by assumption 1 above:

 

                                                         (1)

 

and

 

                                                           (2)

 

Expansion of the individual derivative on the right hand side of (2), making the beta-plane approximation and making appropriate deletions (from last semester's discussion of the the expansion of the total derivative of "f")

 

                                                      (3)

 

Substitute (3) into (2) and expand left hand side.

 

                                  (4)

 

By assumption 2 above

 

                                                        (5)

 

From last semester, in a situation of non-divergence, all of the local changes vorticity will be due to the translation of existing vorticity patterns because in a truly barotropic atmosphere air parcels do not experience a change in vorticity.  This is expressed in equation (6).

 

                                                          (6)

 

By assumption 2 above

 

                                                            (7)

 

Put (5), (6), and (7) into (4)

 

                                                   (8)

 

The definition of relative vorticity is

 

                                                (9)

 

By assumption 2 above

 

                                                        (10)

 

Put  (10) into (8)

 

                                           (11)

 

Equation (11) is a second order differential equation with solution of form

 

                                               (12)

 

where L is the wavelength, A is the amplitude, and c is the phase speed of a sinusoidal wave. Substitution of (12) into (11) and remembering definition of beta yields (see box below)

 

                                                         (13a)

 

                                               (i)

 

                                                            (ii)

                             where R is radius of earth

                                          (iii)

 

                             Put (ii) and (iii) into (i)

from last semester

 

                                                  (iv)

 

put (iv) into (13a)

                                                          (13b)

 

The equation states that the phase speed of a wave is directly related to wind speed modified by effects due to the wavelength (and latitude).  For a given wavelength, the faster the zonal wind speed, the faster the motion of the waves.

 

For a given zonal wind speed, short waves will progress more faster than long waves. 

We can define a "critical speed" as that value of zonal wind speed in which waves of a given wavelength will become stationary. In other words,

 

            (14)

 

 

Solving (14) for "L" gives the so-called "critical wavelength" at which the phase speeds are zero.  Wavelengths larger than this will be associated with waves that retrogress.