Geography Reference
In-Depth Information
surfaces approximate material contours, and wavebreaking can best be illustrated
by considering the field of potential vorticity. Wavebreaking may occur when the
disturbance fields reach amplitudes for which nonlinear effects can no longer be
neglected in the dynamical equations. For example, if the flow is divided into
mean and disturbance parts, and nonlinear terms are included, the equation for
conservation of quasi-geostrophic potential vorticity (12.10) becomes
∂t +
q +
∂q
∂x
∂q
∂y
∂x
∂q
∂y =−
v
u
v
u
(12.17)
For steady waves propagating relative to the ground at zonal phase speed c x , the
variation of phase in time and space is given by φ
=
k (x
c x t), where k is the
zonal wave number, and it is readily verified that
∂t =−
∂x
c x
so that in the linearized version of (12.17) there is a balance between advection
of the disturbance potential vorticity q by the Doppler-shifted mean wind and the
advection of mean potential vorticity by the disturbance meridional wind:
c x ) ∂q
∂q
∂y
v
(u
∂x =−
(12.18)
The validity of the linear approximation can thus be assessed by comparing the
sizes of the two terms on the right side of (12.17) with either term in (12.18).
Linearity holds provided that
u
c x u
(12.19a)
and
∂q /∂y
∂q/∂y
(12.19b)
Basically these criteria require that the slope of the material contours in the x, y
plane must be small.
As indicated in (12.13), in an atmosphere with constant mean-zonal wind, ver-
tically propagating linear Rossby waves have amplitudes increasing exponentially
in height. Thus, at some altitude the disturbance amplitude will become sufficiently
large so that wavebreaking must occur. In the real atmosphere, however, the mean
zonal flow varies in both latitude and in height, and this variation is crucial for
understanding the distribution and mean-flow forcing provided by Rossby wave-
breaking. The simplest example of Rossby wavebreaking occurs in the presence
of a critical surface along which the Doppler-shifted phase speed of the wave
 
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