The Sverdrup balance, or Sverdrup relation, is a theoretical relationship between the
windstress exerted on the surface of the open oceanand the vertically integrated meridional(north-south) transport of ocean water.
Aside from the oscillatory motions associated with tidal flow, there are two primary causes of large scale flow in the ocean: "(1)"
thermohalineprocesses, which induce motion by introducing changes at the surface in temperatureand salinity, and therefore in seawater density, and "(2)" wind forcing. In the 1940s, when Harald Sverdrupwas thinking about calculating the gross features of ocean circulation, he chose to consider exclusively the wind stress component of the forcing. As he says in his 1947 paper, in which he presented the Sverdrup relation, this is probably the more important of the two. After making the assumption that frictional dissipation is negligible, Sverdrup obtained the simple result that the meridional mass transport (the "Sverdrup transport") is proportional to the curl of the wind stress. This is known as the Sverdrup relation;
: is the rate of change of the
Coriolis parameter, "f", with meridional distance;:"V" is the vertically integrated meridional mass transport;:k is the unit vectorin the "z" (vertical) direction;: is the wind stress.
Sverdrup balance may be thought of as a consistency relationship for flow which is dominatedby the earth's rotation. Such flow will be characterized by weak rates of spin comparedto that of the earth.Any parcel at rest with respect to the surface of the earth must match the spin of the earth underneath it. Looking down on the earth at the north pole, this spin is in a counterclockwise direction, which is defined as "positive" rotation or vorticity. At the south pole it is in a clockwise direction, corresponding to "negative" rotation. Thus to move a parcel of fluid from the south to the north without causing it to spin, it is necessary to add sufficient (positive)rotation so as to keep it matched with the rotation of the earth underneath it. The left-hand side of the Sverdrup equation represents the motion required to maintain this match between the absolute vorticity of a water column and the planetary vorticity, whilethe right represents the applied force of the wind.
The Sverdrup relation can be derived from the linearized
potential vorticityequation for steady motion:
Here "v" and "w" are the y- and z-components of the water velocity (northward and upward), respectively. In words, this equation says that as a vertical column of water is squashed, it moves toward the equator; as it's stretched, it moves toward the pole. Assuming, as did Sverdrup, that there is a level below which motion ceases, the PV equation can be integrated from this level to the surface to obtain:
where is seawater density, and is the vertical velocity at the base of the
The driving force behind the vertical velocity is
Ekman transport, which in the Northern (Southern) hemisphere is to the right (left) of the wind stress; thus a stress field with a positive (negative) curl leads to Ekman divergence (convergence), and water must rise from beneath to replace the old Ekman layer water. The expression for this "Ekman pumping" velocity is
which, when combined with the previous equation, yields the Sverdrup relation.
Henry Stommelproposed a circulation for the entire ocean depth by starting with the same equations as Sverdrup but adding bottom friction, and showed that the variation in Coriolis parameter with latitude results in a narrow western boundary currentin ocean basins. Walter Munkin 1950 combined the results of Rossby (eddy viscosity), Sverdrup (upper ocean wind driven flow) and Stommel (western boundary current flow) and proposed a complete solution for the ocean circulation.
* Sverdrup, HU "Wind-driven currents in a baroclinic ocean; with application to the equatorial currents of the eastern Pacific", Proc. N.A.S., 1947.
* Gill, AE "Atmosphere-Ocean Dynamics", Academic Press, 1982.
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