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__

_{}

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)

_{}

Notice
that if there were no pressure gradient we would just have Ekman transports –
because the depth *z _{o}* is
(much) deeper than the Ekman layer.

Now
take _{} of *x* equation and add to_{}of *y* equation

_{}

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:

_{}

where
_{}

Notice
that _{} has dimensions of:

density.velocity*.*depth = kg s^{-1} m^{-1 }

^{ }

*M ^{y}*

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 __ _{}__ and therefore

_{} = 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 *M ^{y} *westward from the eastern
boundary and map the streamlines of the depth-integrated flow.

[Apel fig. 6.36 – Schematic of zonal
winds and gyres]

As
presented here, the Sverdrup balance only describes the north-south component
of flow, and doesn’t immediately say anything about the east-west 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.

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, _{}, and its variation with longitude, _{}, 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 gradient, _{}

If we ignore
meridional winds, then _{} = 0 and

_{}

Then the continuity
equation

_{}

can be used to
calculate the zonal (west-east direction) transport *M ^{x}* from
the Sverdrup relation:

_{}

These terms depend
only on latitude, so integrating with respect to x gives:

_{}

Variations in the
wind stress dominate over variations in _{} in this analysis.

The point here is
that the zonal transport *M ^{x}* is roughly linearly proportional
to longitude

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.

If we can ignore
meridional (north-south direction) winds, then _{} = 0 and the wind
stress curl is simply

_{}

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

_{}= -0.1 Pa/1000 x 10^{3} m = -10^{-7} N/m^{3} (or kg m^{-1} s^{-1})

The
meridional transport per unit distance in the x direction is

_{} = -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

*M _{Ekman}*

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 *M ^{y}* is the total mass
transport in the y-direction per unit distance in x, and is the

*M ^{y}
= M^{y}_{Geostrophic} + M^{y}_{Ekman}*

In
the example above, we get

*M ^{y}_{Geostrophic}*
= 5500 kg m

These
transports are per unit width in the east-west direction. We can sum (integrate)
across all longitudes using the local values of *M ^{y}* 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

*M _{TOT}* = -5000 kg m

or,
dividing by a density of _{} = 1000 kg m^{-3}

*M _{TOT}* = -60 x 10

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,

M_{Ekman} =
6 x 10^{6} m^{3} 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

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, and I going to propose
average temperatures:

in the thermocline of T_{Thermocline}
= 15^{o}C

in the Kuroshio T_{Kuroshio} =
18^{o}C (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 T_{Ekman} =
22^{o}C

_{}

with units of

*c _{p}* is Joules C

^{ }

then multiply by transport m^{3} s^{-1} and temperature ^{o}C

and we get Joules per second, or

This estimate of
0.91 PetaWatts is of about the right magnitude for the annual mean oceanic heat
transport across 24^{o}N in the Pacific.