theory Sverdrup's 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 1 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 ]0 zo zo zo x 0 p x - fM y = - dz + wind zo x - f vdz = -
0 0
f udz = -
zo
0
p dz + [ y ]0 zo zo y
0
fM x = -
p y dz + wind zo y
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 y
Notice that
M = v dz
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, , 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
y
p gradient, x
y If we ignore meridional winds, then x = 0 and
curl = -
x 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:
MX = 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 derived was 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 1 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
curl = -
x 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 ...