16
Sverdrup Theory
For the large scale flow of interest to us here, it is the collective, average
effect of the eddies that is of interest. In the vorticity balance on the large scale
this may disturb the Sverdrup relation if the large-scale divergence of the eddy
flux of relative vorticity is as large as the advection of planetary vorticity. The
former is:
V' . i1( = O ( u;ddyi Leddy )
(1.4.6)
where the overbar refers to an average over the large-scale flow with length
scale L. In (1.4.6) it is assumed that the relative vorticity flux has the local value
Ueddy( Ueddy/ Leddy), and that this flux varies smoothly over the large-scale L.
When compared with the large-scale advection of planetary vorticity which is
0( up), this gives a ratio of the divergence of eddy flux of relative vorticity to
the planetary advection of vorticity which is apparently of the order:
u;ddy/ LeddyL = ( Ue:dy ) UeddyLeddy .
up
PLeddy
UL
(1.4.7)
If Leddy is of the order of 50 km, with f3 as before, an eddy velocity of only 5
cmfs would make the first bracket on the right-hand side of (1.4. 7) of order 1.
The second factor on the right-hand side of (1.4. 7) is the ratio, assuming the
large-scale and eddy flows are both in geostrophic balance, of the contribution
to the dynamic pressure field of both the mean and the eddies. It is likely that
this number is of order 1 as well since the geostrophic disturbance to the large
scale pressure field by the eddy is of the order of the background field itself if
the eddies spring from a local instability of the large-scale flow. This would
seem to imply that the eddies could easily upset the Sverdrup relation of
vorticity. However, our estimate of EH which is supposed to measure the effect
of smaller scale motions on the vorticity dynamics of the large scale implied
that the eddies would be negligible.
If nothing else, this illustrates the care which must be taken in making
scaling estimates. Let us return to our estimate of the relative vorticity flux i1(.
We estimated its average over the large scale by multiplying its local value by
the large scale area, L 2 , and then dividing by the same area to obtain the
estimate in the numerator of (1.4.6). However, each component of the vector
vorticity flux for the eddies can itself be written as a divergence. This occurs
because on the eddy scale the motion is quasi-geostrophic, and the eddy
velocity field is horizontally nearly nondivergent (Pedlosky 1987; Chap. 7). On
the smaller scales of the eddies a locally Cartesian coordinate frame is adequate
so that we can write:
i1( = u(ov _ ou)
ax ay
(1.4.8)
where x andy are coordinates to the east and north, respectively. The x andy
components of (1.4.8) can be written:
Sverdrup Theory
For the large scale flow of interest to us here, it is the collective, average
effect of the eddies that is of interest. In the vorticity balance on the large scale
this may disturb the Sverdrup relation if the large-scale divergence of the eddy
flux of relative vorticity is as large as the advection of planetary vorticity. The
former is:
V' . i1( = O ( u;ddyi Leddy )
(1.4.6)
where the overbar refers to an average over the large-scale flow with length
scale L. In (1.4.6) it is assumed that the relative vorticity flux has the local value
Ueddy( Ueddy/ Leddy), and that this flux varies smoothly over the large-scale L.
When compared with the large-scale advection of planetary vorticity which is
0( up), this gives a ratio of the divergence of eddy flux of relative vorticity to
the planetary advection of vorticity which is apparently of the order:
u;ddy/ LeddyL = ( Ue:dy ) UeddyLeddy .
up
PLeddy
UL
(1.4.7)
If Leddy is of the order of 50 km, with f3 as before, an eddy velocity of only 5
cmfs would make the first bracket on the right-hand side of (1.4. 7) of order 1.
The second factor on the right-hand side of (1.4. 7) is the ratio, assuming the
large-scale and eddy flows are both in geostrophic balance, of the contribution
to the dynamic pressure field of both the mean and the eddies. It is likely that
this number is of order 1 as well since the geostrophic disturbance to the large
scale pressure field by the eddy is of the order of the background field itself if
the eddies spring from a local instability of the large-scale flow. This would
seem to imply that the eddies could easily upset the Sverdrup relation of
vorticity. However, our estimate of EH which is supposed to measure the effect
of smaller scale motions on the vorticity dynamics of the large scale implied
that the eddies would be negligible.
If nothing else, this illustrates the care which must be taken in making
scaling estimates. Let us return to our estimate of the relative vorticity flux i1(.
We estimated its average over the large scale by multiplying its local value by
the large scale area, L 2 , and then dividing by the same area to obtain the
estimate in the numerator of (1.4.6). However, each component of the vector
vorticity flux for the eddies can itself be written as a divergence. This occurs
because on the eddy scale the motion is quasi-geostrophic, and the eddy
velocity field is horizontally nearly nondivergent (Pedlosky 1987; Chap. 7). On
the smaller scales of the eddies a locally Cartesian coordinate frame is adequate
so that we can write:
i1( = u(ov _ ou)
ax ay
(1.4.8)
where x andy are coordinates to the east and north, respectively. The x andy
components of (1.4.8) can be written: