Geography Reference
In-Depth Information
resists meridional displacements and provides the restoring mechanism for Rossby
waves.
The speed of westward propagation, c, can be computed for this simple example
by letting δy
=
a sin [k (x
−
ct)], where a is the maximum northward displace-
D (δy)
Dt
ment. Then v
=
=−
kca cos [k (x
−
ct)], and
∂v
∂x
k
2
ca sin [k (x
ζ
=
=
−
ct)]
Substitution for δy and ζ in (7.87) then yields
k
2
ca sin [k (x
−
ct)]
=−
βa sin [k (x
−
ct)]
or
β
k
2
c
=−
(7.88)
Thus, the phase speed is westward relative to the mean flow and is inversely
proportional to the square of the zonal wave number.
7.7.1
Free Barotropic Rossby Waves
The dispersion relationship for barotropic Rossby waves may be derived formally
by finding wave-type solutions of the linearized barotropic vorticity equation. The
barotropic vorticity equation (4.27) states that the vertical component of absolute
vorticity is conserved following the horizontal motion. For a midlatitude β plane
this equation has the form
∂
∂t
+
ζ
∂
∂x
+
∂
∂y
u
v
+
βv
=
0
(7.89)
We now assume that the motion consists of a constant basic state zonal velocity
plus a small horizontal perturbation:
u
,v
v
,ζ
∂v
/∂x
∂u
/∂y
ζ
=
+
=
=
−
=
u
u
We define a perturbation streamfunction ψ
according to
u
=−
∂ψ
/∂y,
v
=
∂ψ
/∂x
from which ζ
=
∇
2
ψ
. The perturbation form of (7.89) is then
∂
∂t
+
β
∂ψ
∂
∂x
2
ψ
+
u
∇
∂x
=
0
(7.90)
