Sverdrup’s theory of the oceanic circulation
We can arrive at a very robust version of the key features of the Sverdrup
balance without needing to make the 1½ layer assumption, or consider the
details of the vertical stratification.
The approach is similar to the way we derived the Ekman transport relation
by integrating the momentum equations over a large enough depth to cover
the entire Ekman layer (the near surface region where vertical mixing of the
momentum imparted by the wind is significant).
It turned out we didn’t need to explicitly know the Ekman layer depth, or
indeed any details about the vertical profile of the turbulent mixing
coefficient (the eddy viscosity). All that mattered was that over some several
tens of meters (a depth range estimated from a simple scale analysis) it had
to be that all the wind momentum was transferred to the ocean.
Start with the steady (no time derivative) momentum equations at small
Rossby number (advection terms are negligible) with both friction and
Coriolis
p  x
 fv   
x z
p  y
fu   
y
z
Sverdrup integrated these equations from the surface to a depth at which the
horizontal pressure gradient becomes zero (i.e. our level of no motion)
p
dz  [ x ]0zo
zo
zo x
0 p
x
 fM y   
dz   wind
zo x
0
 f  vdz   
0
p
dz  [ y ]0zo
zo y
0
f  udz   
zo
0
p
y
dz   wind
zo y
fM x   
0
Notice that if there were no pressure gradient we would just have Ekman
transports – because the depth zo is (much) deeper than the Ekman layer.
Now take   y of x equation and add to  x of y equation


 x  y
y
x
( fM )  ( fM )  

y
x
y
x
f
M x M y
 y  x
y
M  f(

)

y
x
y
x
y
The second term on the left-hand-side is the mass conservation equation
integrated over depth from the surface to the level of no motion. It is
therefore zero. This leaves:
 x  y
M 

 curl ( )
y
x
y

where
f
y
o
M   v dz
y
Notice that
z0
has dimensions of:
density.velocity.depth = kg s-1 m-1
My is the mass transport in the y direction per unit distance in the x direction.
Integrated across the whole width of the ocean basin this will be the total
north-south direction mass transport of the gyre (i.e. not including the
western boundary current), and it is driven by the wind stress curl.
At some latitudes curl   0 and therefore My = 0, i.e. there is no north-south
transport.
curl 
= 0 lines are the natural boundaries that divide the ocean up into the
subtropical and subpolar gyres.
With wind stress curl computed from observations, we can integrate My
westward from the eastern boundary and map the streamlines of the depthintegrated flow.
As presented here, the Sverdrup balance only describes the north-south
component of flow, and doesn’t immediately say anything about the eastwest flow.
Consider the outcome of having Sverdrup transport that changes with
latitude.
This is typically the case, because the westerlies change smoothly to the
Trades. (Often the maximum in wind stress curl is close to the minimum
wind speed, but this is not necessarily so).
Between the maximum of the westerlies and the maximum of the wind stress
curl, there is increasing equatorward Sverdrup transport as one goes toward
the equator. This has to come from somewhere, and is fed from the west
 consider a box up against the eastern boundary
o there is more flow out the south face than in through the north
o flow must enter from the west to balance mass
 consider the next box to the west
o mass is lost out the eastern face to the eastern box
o so even more flow must enter though the west face
o so the inflow from the west builds are we move westward,
implying the streamlines becomes closer together going west.
This gives the distinctive westward distorted ellipse pattern to the
circulation.
Southward of the latitude of the maximum wind stress curl, the equatorward
flow is weakening. More must flow out the west face of each box than in.
The Sverdup balance flow pattern that corresponds to the observed mean
zonal (west-east) winds in the Pacific was computed in 1950 by Walter
Munk.
Stewart figure 11.6: Munk’s calculation of the Sverdrup circulation of the
North Pacific calculated from wind stress curl
The streamlines of the flow that show this distorted ellipse pattern are
computed using mass conservation to evaluate the east-west part of the
transport that balances the north-south transport given by the Sverdrup
relation.
Typically, the north-south component of the wind,  y , and its variation with
y
longitude,  x , are negligible compared to the zonal winds. In fact, the
very large x-scale compared to y-scale means that x-variations are generally
negligible in the equatorial region in almost all terms except the pressure
p
gradient, x
y
If we ignore meridional winds, then  x = 0 and
 x
curl  
y
Then the continuity equation
M X M y

0
x
y
can be used to calculate the zonal (west-east direction) transport Mx from the
Sverdrup relation:
M X
M y
 1

  ( curl )
x
y
y 

 1 
(
)
y  y
These terms depend only on latitude, so integrating with respect to x gives:
M
X

M X
 1 
 1 
dx  
(
)dx  x( (
))
x
y  y
y  y
Variations in the wind stress dominate over variations in  in this analysis.
The point here is that the zonal transport Mx is roughly linearly proportional
to longitude x, recognizing that Mx=0 at the eastern boundary.
Key concepts of the Sverdrup solution
The Sverdrup solution was derived without needing to consider any details
about how the oceanic density field arranges itself.
We integrated momentum equations vertically over the whole water column
from the surface to the level of no motion.
We kept the Coriolis, pressure gradient, and wind stress terms in the
momentum equations.
The assumed dynamics is that there is a steady state geostrophic balance to
the net influence of the Ekman pumping.
The general solution for the pattern of streamlines of the Sverdrup flow can
be obtained by integrating the wind stress curl westward starting from the
eastern boundary.
The Sverdrup transport is the combination of geostrophic and Ekman
transports together. The individual contributions of geostrophic transport and
Ekman can be in different directions.
 The direction of the Ekman flow depends on the sign of the zonal
wind stress
 The direction of the total Sverdrup=Ekman+Geostrophic depends on
the sign of the wind stress curl
The Sverdrup transport result still holds for a continuously stratified ocean.
What we have lost (by integrating over a large depth range) is any
information about the shape of the thermocline, but we know from the 1½
layer model that net equatorward flow would be balanced by a thermocline
deepening toward the west (to give higher dynamic height or geopotential in
the west). This is consistent with thermal wind, which says the southward
flow in the subtropical gyre requires “light water on the right” so density
surfaces slope downward toward the west across the entire basin where the
Sverdrup balance holds.
Only in the western boundary current does this slope of the isopycnals and
isotherms reverse. In the boundary current the Sverdrup balance doesn’t
hold, but we do know from the principle of mass conservation that the gyre
scale Sverdrup transports tells us the total mass transport of the (equal and
opposite) western boundary current.
Orders of magnitude: a simple Sverdrup calculation for typical values
y
If we can ignore meridional (north-south direction) winds, then  x = 0
and the wind stress curl is simply
 x
curl  
y
Say  is a maximum of 0.05 Pa in the maximum of the westerlies, and
similarly -0.05 Pa in the center of the Trades, and the meridional (north
south) length scale between these latitudes is 1000 km. Then
curl   
 x
y = -0.1 Pa/1000 x 103 m = -10-7 N/m3 (or kg m-1 s-1)
The meridional transport per unit distance in the x direction is
My 
1

curl 
= -10-7 / 2 x 10-11 = - 5000 kg m-1 s-1 (southward)
in kg s-1 per meter zonal (west-east) width.
We can compare this to the directly wind-driven Ekman transport:
For  of -0.05 Pa in the center of the Trades, the Ekman mass transport is
simply

MEkman =  f
= 0.05 / 10-4 = 500 kg m-1 s-1
(northward)
in kg s-1 per meter zonal distance. (This is the volume transport multiplied
by density).
We see that the magnitude of the Sverdrup transport is 10 times greater than
the Ekman transport itself. This is typical of the mid-latitude gyres.
Note than My is the total mass transport in the y-direction per unit distance in
x, and is the sum of Ekman and geostrophic (thermocline) components. We
can break My into the separate contributions:
My = MyGeostrophic + MyEkman
In the example above, we get
MyGeostrophic = 5500 kg m-1 s-1
These transports are per unit width in the east-west direction. We can sum
(integrate) across all longitudes using the local values of My to determine the
total southward transport.
In the example above, if the wind stress is uniform across an ocean basin
12,000 km wide, we would get a total southward Sverdrup transport of
MTOT = -5000 kg m-1 s-1 x 12000 x 103 m = 60 x 109 kg s-1
or, dividing by a density of  = 1000 kg m-3
MTOT = -60 x 106 m3 s-1 = -60 Sv
In a closed basin such as the North Pacific, all this southward transport has
to be balanced by northward flow somewhere else; namely, the western
boundary current (Kuroshio).
Similarly,
MEkman = 6 x 106 m3 s-1 = 6 Sv
and geostrophic interior flow, not including the Ekman layer, is 66 Sv
southward.
Now, the Pacific is a closed basin with virtually no flow out through the
Bering Strait (it’s actually about 1-2 Sv southward).
To conserve mass, the Sverdrup flow must be balanced by …?
the western boundary current (Kuroshio) with a transport of?
60 Sv northward
Now we can make an approximate heat transport estimate by looking at the
temperatures in the hydrographic data.
The interior of the ocean doesn’t fluctuate all that much seasonally, so we
propose average temperatures:
in the thermocline of TThermocline = 15oC
in the Kuroshio TKuroshio = 18oC (warmer because it is moving
equatorial water northward subtropical)
In the Ekman layer, it’s important to remember that there is a strong
seasonal cycle, so use a value typical of annual mean conditions, say
in the Ekman layer TEkman = 22oC
H f   c p vT dx dz  c p  vT dx dz
 c p (VEkmanTEkman  VKuroshioTKuroshio  VThermoclineTThermocline )
 4.1x10 6 [(6)( 22)  (60)(18)  (66)(15)]
 0.91x1015
with units of Watts (= power = energy per second = Joules per second)
cp is Joules C-1 kg-1
so  cp is J C-1 m-3
then multiply by transport m3 s-1 and temperature oC
and we get Joules per second, or Watts, which is heat transport.
This estimate of 0.91 PetaWatts is of about the right magnitude for the
annual mean oceanic heat transport across 24oN in the Pacific.