On the Validity of Sverdrup Theory
15
The first of these is certainly met for the large-scale flow, and the second,
we have argued, is most likely to be true. These conditions are required so that,
below the mixed layer, the ocean is in geostrophic balance. That is, that the
advection of momentum is then negligible compared to the Coriolis
acceleration or equivalently, that the relative vorticity:
( = k · "V x if= O(U/L)
(1.4.2)
be small compared to the planetary vorticity, f
The Sverdrup relation is an approximation to the vorticity equation and
comes from taking the curl of the momentum equation. This involves a
comparison of the advection of the relative vorticity and the advection of the
planetary vorticity rather than a comparison of the relative and planetary
vorticities themselves. The relative vorticity, (, varies on the length scale L.
However, the planetary vorticity, by (1.2.13), varies on the planetary scaleR. As
long as R exceeds L, this implies thatfvaries more slowly than(. Thus, in order
to neglect the advection of relative vorticity compared with the advection of
planetary vorticity we must have:
if. "V( =a(~~)« if. "Vf= O(UfJ) or
u
c = {JL2 « 1.
(1.4.3)
(1.4.4)
Since usually fJL the smallness of the Rossby number. For L = 1000 km, fJ = 2.1 x w- 13 cm- 1
s- 1 , and U = 1 cm/s, c: is 5x to-4, which is larger than R0 but is still small
enough to imply that we may neglect relative vorticity advection.
A more subtle requirement for the validity of the Sverdrup relation has
been somewhat hidden by our scaling argument. When we make a scaling
argument and attribute to the motion field a scale L, say, for the length scale of
the motion, we are tacitly assuming that there is only one length scale that
characterizes the flow. The same is true for the velocity scale U. If, as in the
case of the ocean, there are several scales of motion coexisting in space, the
argument becomes more subtle. Since the 1960s it has been evident that the
ocean is populated by small- and meso-scale eddies. The eddies are variable in
intensity, and their geographical distribution is nonuniform. The character of
the eddy field has been the subject of intense observational and theoretical
study, and its description is beyond the scope of this book. However, we need
to recognize that the eddies are distributed over the whole oceanic basin, and
that they possess scales for length and velocity Leddy and Ueddy, respectively,
such that:
(1.4.5)
so that on the eddy scale the advection of relative vorticity cannot be ignored.
15
The first of these is certainly met for the large-scale flow, and the second,
we have argued, is most likely to be true. These conditions are required so that,
below the mixed layer, the ocean is in geostrophic balance. That is, that the
advection of momentum is then negligible compared to the Coriolis
acceleration or equivalently, that the relative vorticity:
( = k · "V x if= O(U/L)
(1.4.2)
be small compared to the planetary vorticity, f
The Sverdrup relation is an approximation to the vorticity equation and
comes from taking the curl of the momentum equation. This involves a
comparison of the advection of the relative vorticity and the advection of the
planetary vorticity rather than a comparison of the relative and planetary
vorticities themselves. The relative vorticity, (, varies on the length scale L.
However, the planetary vorticity, by (1.2.13), varies on the planetary scaleR. As
long as R exceeds L, this implies thatfvaries more slowly than(. Thus, in order
to neglect the advection of relative vorticity compared with the advection of
planetary vorticity we must have:
if. "V( =a(~~)« if. "Vf= O(UfJ) or
u
c = {JL2 « 1.
(1.4.3)
(1.4.4)
Since usually fJL the smallness of the Rossby number. For L = 1000 km, fJ = 2.1 x w- 13 cm- 1
s- 1 , and U = 1 cm/s, c: is 5x to-4, which is larger than R0 but is still small
enough to imply that we may neglect relative vorticity advection.
A more subtle requirement for the validity of the Sverdrup relation has
been somewhat hidden by our scaling argument. When we make a scaling
argument and attribute to the motion field a scale L, say, for the length scale of
the motion, we are tacitly assuming that there is only one length scale that
characterizes the flow. The same is true for the velocity scale U. If, as in the
case of the ocean, there are several scales of motion coexisting in space, the
argument becomes more subtle. Since the 1960s it has been evident that the
ocean is populated by small- and meso-scale eddies. The eddies are variable in
intensity, and their geographical distribution is nonuniform. The character of
the eddy field has been the subject of intense observational and theoretical
study, and its description is beyond the scope of this book. However, we need
to recognize that the eddies are distributed over the whole oceanic basin, and
that they possess scales for length and velocity Leddy and Ueddy, respectively,
such that:
(1.4.5)
so that on the eddy scale the advection of relative vorticity cannot be ignored.