Effect of Finite Mixed Layer Depth
235
The thermocline circulation has until now been identified with the
adiabatic, nearly frictionless region below the mixed layer. This assumes
implicitly that the mixed layer is so thin that the Sverdrup transport captured
within the mixed layer is negligible. Yet observations, as presented in
Fig. 4.1 0.1, show that the mixed layer constitutes a nonnegligible fraction of
the total water column involved in the circulation of the thermocline. We see,
for example, that in the North Atlantic the mixed layer depth in late winter is
several hundred meters. Moreover, the depth of the mixed layer varies strongly,
from about 50 m in lower, subtropical latitudes to nearly 400 m at the northern
boundary of the gyre. The variation of the mixed layer depth as well as its
substantial value is dynamically important.
Suppose we wish to determine the amount of fluid that ventilates the
thermocline from the mixed layer compared to the flux of fluid involved in the
horizontal recirculation in the gyre. Were the mixed layer of constant
thickness, and relatively thin, the total vertical mass flux from the Ekman
layer would be of the order of WELxLy, where Lx and Ly are the characteristic
scales in the x and y direction of the gyre. At the same time, the horizontal
mass flux in the gyre is of the order VsLxH where H is the depth of the
thermocline and Vs is the characteristic meridional Sverdrup velocity. Using the
Sverdrup balance, we have v.H = 0(/we//3), and therefore the ratio of the
recirculation transport to the flux pumped out of the Ekman layer is given by
the Rhines ratio:
f
Rh = {3Ly.
( 4.10.1)
This ratio is normally larger than 1 (the f3 plane approximation is, after all,
based on RJ; 1 being small) and one would conclude that most of the mass
transport in the gyre is associated with recirculation rather than ventilation.
However, the effect of the spatial variation of the mixed layer thickness
alters the argument radically. Consider the situation shown in Fig. 4.10.2.
When the mixed layer has a nonnegligible thickness, hm, which varies with
latitude, say, the outcropping of isopycnal layers produce a configuration of
the density interfaces as shown schematically in the figure. Within the mixed
layer the density surfaces are vertical since there can be, by definition, no
vertical variation of density in the layer. An isopycnal interface rises towards
the surface in the thermocline, makes contact with the base of the mixed layer,
and then rises vertically to the sea surface within the mixed layer. The Ekman
layer n~rmally occupies a small fraction of the mixed layer and pumps into the
top of the mixed layer a vertical velocity WE. Within the mixed layer and below
the Ekman layer the velocity is geostrophic. Therefore a horizontal, meridional
velocity in the mixed layer, Vm, is forced by the stretching of planetary vorticity
filaments as in the Sverdrup balance, and we can estimate this velocity as
(! / f3H)wE. That is, it is of the same order as the velocity in the thermocline.
Since the base of the mixed layer becomes shallower to the south, some of this
235
The thermocline circulation has until now been identified with the
adiabatic, nearly frictionless region below the mixed layer. This assumes
implicitly that the mixed layer is so thin that the Sverdrup transport captured
within the mixed layer is negligible. Yet observations, as presented in
Fig. 4.1 0.1, show that the mixed layer constitutes a nonnegligible fraction of
the total water column involved in the circulation of the thermocline. We see,
for example, that in the North Atlantic the mixed layer depth in late winter is
several hundred meters. Moreover, the depth of the mixed layer varies strongly,
from about 50 m in lower, subtropical latitudes to nearly 400 m at the northern
boundary of the gyre. The variation of the mixed layer depth as well as its
substantial value is dynamically important.
Suppose we wish to determine the amount of fluid that ventilates the
thermocline from the mixed layer compared to the flux of fluid involved in the
horizontal recirculation in the gyre. Were the mixed layer of constant
thickness, and relatively thin, the total vertical mass flux from the Ekman
layer would be of the order of WELxLy, where Lx and Ly are the characteristic
scales in the x and y direction of the gyre. At the same time, the horizontal
mass flux in the gyre is of the order VsLxH where H is the depth of the
thermocline and Vs is the characteristic meridional Sverdrup velocity. Using the
Sverdrup balance, we have v.H = 0(/we//3), and therefore the ratio of the
recirculation transport to the flux pumped out of the Ekman layer is given by
the Rhines ratio:
f
Rh = {3Ly.
( 4.10.1)
This ratio is normally larger than 1 (the f3 plane approximation is, after all,
based on RJ; 1 being small) and one would conclude that most of the mass
transport in the gyre is associated with recirculation rather than ventilation.
However, the effect of the spatial variation of the mixed layer thickness
alters the argument radically. Consider the situation shown in Fig. 4.10.2.
When the mixed layer has a nonnegligible thickness, hm, which varies with
latitude, say, the outcropping of isopycnal layers produce a configuration of
the density interfaces as shown schematically in the figure. Within the mixed
layer the density surfaces are vertical since there can be, by definition, no
vertical variation of density in the layer. An isopycnal interface rises towards
the surface in the thermocline, makes contact with the base of the mixed layer,
and then rises vertically to the sea surface within the mixed layer. The Ekman
layer n~rmally occupies a small fraction of the mixed layer and pumps into the
top of the mixed layer a vertical velocity WE. Within the mixed layer and below
the Ekman layer the velocity is geostrophic. Therefore a horizontal, meridional
velocity in the mixed layer, Vm, is forced by the stretching of planetary vorticity
filaments as in the Sverdrup balance, and we can estimate this velocity as
(! / f3H)wE. That is, it is of the same order as the velocity in the thermocline.
Since the base of the mixed layer becomes shallower to the south, some of this