Rossby Waves
109
vorticity in the above equations can be neglected with respect to the terms
involving Gn and Fn. The speeds are given below but for now we need to
observe only that the group velocity of the baroclinic waves in this limit is
always westward (Pedlosky 1987). After all the wave modes pass the
observation point (x, y) on their westward journey, the solution becomes
steady and under the influence of the steady Ekman pumping. Hence after the
passage of the waves the solutions must satisfy:
f3 81/JI = fo WE
8x
HI
f3 81/12 = 0
8x
f38t2 = 0.
Each of the l/1 n must satisfy:
l/Jn = 0, X =Xe n = 1,2,3.
(3.4.3a,b,c)
(3.4.4)
Hence in the steady state only l /1 1 differs from zero. The upper layer carries
all the Sverdrup transport while the lower two layers are at rest. The baroclinic
Ross by waves arriving from the eastern boundary "turn off'' the initial surge of
motion in the lower layers. The long baroclinic Rossby waves are the
messengers which, arriving from the east, signal the fluid of the presence of the
blocking action of the eastern boundary. The planetary potential vorticity
contours in this linear limit are just the lines of constant latitude along which in
linear, steady-state theory the streamfunction must be constant, and the lower,
unforced layers must therefore be at rest in the steady state since these contours
strike the eastern boundary where l /1 2 and l /1 3 are zero. Thus, the motion in the
lower two layers is "blocked" by the intersection of the potential vorticity
contours in those layers with the eastern boundary.
The baroclinic Rossby modes are relatively slow (see below), and their
speeds differ little from the speeds associated with the fluid motion in the gyre
circulation. To consider the qualitative effect on the propagation of the waves
by the gyre's mean flow, we can linearize the motion about a mean zonal flow
rather than a state of rest. For simplicity, and in preparation for the results of
our investigation below of the nonlinear steady flow problem, we choose a
mean flow consisting of an eastward velocity U, which is the same for the upper
two layers. The third layer has no mean velocity. That is, we now consider the
streamfunction written as:
109
vorticity in the above equations can be neglected with respect to the terms
involving Gn and Fn. The speeds are given below but for now we need to
observe only that the group velocity of the baroclinic waves in this limit is
always westward (Pedlosky 1987). After all the wave modes pass the
observation point (x, y) on their westward journey, the solution becomes
steady and under the influence of the steady Ekman pumping. Hence after the
passage of the waves the solutions must satisfy:
f3 81/JI = fo WE
8x
HI
f3 81/12 = 0
8x
f38t2 = 0.
Each of the l/1 n must satisfy:
l/Jn = 0, X =Xe n = 1,2,3.
(3.4.3a,b,c)
(3.4.4)
Hence in the steady state only l /1 1 differs from zero. The upper layer carries
all the Sverdrup transport while the lower two layers are at rest. The baroclinic
Ross by waves arriving from the eastern boundary "turn off'' the initial surge of
motion in the lower layers. The long baroclinic Rossby waves are the
messengers which, arriving from the east, signal the fluid of the presence of the
blocking action of the eastern boundary. The planetary potential vorticity
contours in this linear limit are just the lines of constant latitude along which in
linear, steady-state theory the streamfunction must be constant, and the lower,
unforced layers must therefore be at rest in the steady state since these contours
strike the eastern boundary where l /1 2 and l /1 3 are zero. Thus, the motion in the
lower two layers is "blocked" by the intersection of the potential vorticity
contours in those layers with the eastern boundary.
The baroclinic Rossby modes are relatively slow (see below), and their
speeds differ little from the speeds associated with the fluid motion in the gyre
circulation. To consider the qualitative effect on the propagation of the waves
by the gyre's mean flow, we can linearize the motion about a mean zonal flow
rather than a state of rest. For simplicity, and in preparation for the results of
our investigation below of the nonlinear steady flow problem, we choose a
mean flow consisting of an eastward velocity U, which is the same for the upper
two layers. The third layer has no mean velocity. That is, we now consider the
streamfunction written as: