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The '''Sverdrup balance''', or '''Sverdrup relation''', is a theoretical relationship between the [[wind]] [[stress (physics)|stress]] exerted on the surface of the open [[ocean]] and the vertically integrated [[meridional]] (north-south) transport of ocean water.

== History ==

Aside from the oscillatory motions associated with [[tide|tidal]] flow, there are two primary causes of large scale flow in the ocean: ''(1)'' [[thermohaline]] processes, which induce motion by introducing changes at the surface in [[temperature]] and [[salinity]], and therefore in [[seawater]] [[density]], and ''(2)'' wind forcing. In the 1940s, when [[Harald Sverdrup (oceanographer)|Harald Sverdrup]] was 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 (mathematics)|curl]] of the wind stress. This is known as the Sverdrup relation;

:<math>V =\hat{\mathbf{k}} \cdot \frac{\nabla\times\mathbf{\tau}}{\beta}</math>.

Here,

:[[Beta plane|<math>\beta</math>]] 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 vector]] in the ''z'' (vertical) direction;
:<math>\tau</math> is the wind stress vector.

== Physical interpretation ==

Sverdrup balance may be thought of as a consistency relationship for flow which is dominated
by the Earth's rotation. Such flow will be characterized by weak rates of spin compared
to 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, while
the right represents the applied force of the wind.

== Derivation ==

The Sverdrup relation can be derived from the linearized [[barotropic vorticity equation]] for steady motion:

:<math>\beta v=f \, \partial{w}/\partial{z} \ </math>.

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:

:<math>\beta V=\rho f w_E \ </math>,

where <math>\rho</math> is seawater density, and <math>w_E</math> is the vertical velocity at the base of the [[Ekman layer]].

The driving force behind the vertical velocity <math>w_E</math> 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

:<math>\rho w_E = \hat{ \mathbf{k}} \cdot (\nabla\times\tau)/f \ </math>,

which, when combined with the previous equation, yields the Sverdrup relation.

== Further development ==

In 1948 [[Henry Stommel]] proposed 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 effect|Coriolis parameter]] with latitude results in a narrow [[western boundary current]] in [[ocean basin]]s. [[Walter Munk]] in 1950 combined the results of [[Carl-Gustaf Rossby|Rossby]] (eddy viscosity), Sverdrup (upper ocean wind driven flow) and Stommel (western boundary current flow) and proposed a complete solution for the ocean circulation.

== References ==
*{{cite journal |last=Sverdrup |first=H.U. |title=Wind-Driven Currents in a Baroclinic Ocean; with Application to the Equatorial Currents of the Eastern Pacific |journal=Proc. Natl. Acad. Sci. U.S.A. |volume=33 |issue=11 |pages=318–26 |date=November 1947 |pmid=16588757 |pmc=1079064 |doi=10.1073/pnas.33.11.318}}
*{{cite book |last=Gill |first=A.E. |title=Atmosphere-Ocean Dynamics |publisher=Academic Press |year=1982 }}

== External links ==
* [http://stommel.tamu.edu/~baum/paleo/ocean/node36.html#Sverdrupbalance Glossary of Physical Oceanography and Related Disciplines Sverdrup balance ]

{{physical oceanography}}

[[Category:Ocean currents]]
[[Category:Physical oceanography]]

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en>SmackBot: remove Erik9bot category,outdated, tag and general fixes