The Sverdrup balance, or Sverdrup relation, is a theoretical relationship between the
wind stress
In physical oceanography and fluid dynamics, the wind stress is the shear stress exerted by the wind on the surface of large bodies of water – such as oceans, seas, estuaries and lakes. Stress is the quantity that describes the magnitude of a for ...
exerted on the surface of the open
ocean
The ocean (also the sea or the world ocean) is the body of salt water that covers approximately 70.8% of the surface of Earth and contains 97% of Earth's water. An ocean can also refer to any of the large bodies of water into which the wo ...
and the vertically integrated
meridional (north-south) transport of ocean water.
History
Aside from the oscillatory motions associated with
tidal
Tidal is the adjectival form of tide.
Tidal may also refer to:
* ''Tidal'' (album), a 1996 album by Fiona Apple
* Tidal (king), a king involved in the Battle of the Vale of Siddim
* TidalCycles, a live coding environment for music
* Tidal (servic ...
flow, there are two primary causes of large scale flow in the ocean: ''(1)''
thermohaline
Thermohaline circulation (THC) is a part of the large-scale ocean circulation that is driven by global density gradients created by surface heat and freshwater fluxes. The adjective ''thermohaline'' derives from '' thermo-'' referring to temper ...
processes, which induce motion by introducing changes at the surface in
temperature
Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measured with a thermometer.
Thermometers are calibrated in various temperature scales that historically have relied o ...
and
salinity
Salinity () is the saltiness or amount of salt dissolved in a body of water, called saline water (see also soil salinity). It is usually measured in g/L or g/kg (grams of salt per liter/kilogram of water; the latter is dimensionless and equal ...
, and therefore in
seawater
Seawater, or salt water, is water from a sea or ocean. On average, seawater in the world's oceans has a salinity of about 3.5% (35 g/L, 35 ppt, 600 mM). This means that every kilogram (roughly one liter by volume) of seawater has appro ...
density
Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematical ...
, and ''(2)'' wind forcing. In the 1940s, when
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
cURL (pronounced like "curl", UK: , US: ) is a computer software project providing a library (libcurl) and command-line tool (curl) for transferring data using various network protocols. The name stands for "Client URL".
History
cURL was fi ...
of the wind stress. This is known as the Sverdrup relation;
:
.
Here,
:
is the rate of change of the
Coriolis parameter
The Coriolis frequency ''ƒ'', also called the Coriolis parameter or Coriolis coefficient, is equal to twice the rotation rate ''Ω'' of the Earth multiplied by the sine of the latitude \varphi.
:f = 2 \Omega \sin \varphi.\,
The rotation rate ...
, ''f'', with meridional distance;
:''V'' is the vertically integrated meridional mass transport including the geostrophic interior mass transport and the Ekman mass transport;
:k is the
unit vector
In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in \hat (pronounced "v-hat").
The term ''direction vecto ...
in the ''z'' (vertical) direction;
:
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 The barotropic vorticity equation assumes the atmosphere is nearly barotropic, which means that the direction and speed of the geostrophic wind are independent of height. In other words, there is no vertical wind shear of the geostrophic wind. I ...
for steady motion:
:
.
Here
is the geostrophic interior y-component (northward) and
is the z-component (upward) of the water velocity. In words, this equation says that as a vertical column of water is squashed, it moves toward the Equator; as it is stretched, it moves toward the pole. Assuming, as did Sverdrup, that there is a level below which motion ceases, the vorticity equation can be integrated from this level to the base of the Ekman surface layer to obtain:
:
,
where
is seawater density,
is the geostrophic meridional mass transport and
is the vertical velocity at the base of the
Ekman layer
The Ekman layer is the layer in a fluid where there is a force balance between pressure gradient force, Coriolis force and turbulent drag. It was first described by Vagn Walfrid Ekman. Ekman layers occur both in the atmosphere and in the ocean. ...
.
The driving force behind the vertical velocity
is the
Ekman transport
Ekman transport is part of Ekman motion theory, first investigated in 1902 by Vagn Walfrid Ekman. Winds are the main source of energy for ocean circulation, and Ekman Transport is a component of wind-driven ocean current. Ekman transport occurs w ...
, 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 and adding the Ekman transport, yields the Sverdrup relation.
Further development
In 1948
Henry Stommel
Henry Melson Stommel (September 27, 1920 – January 17, 1992) was a major contributor to the field of physical oceanography. Beginning in the 1940s, he advanced theories about global ocean circulation patterns and the behavior of the Gulf Stream ...
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 parameter
The Coriolis frequency ''ƒ'', also called the Coriolis parameter or Coriolis coefficient, is equal to twice the rotation rate ''Ω'' of the Earth multiplied by the sine of the latitude \varphi.
:f = 2 \Omega \sin \varphi.\,
The rotation rate ...
with latitude results in a narrow
western boundary current
Boundary currents are ocean currents with dynamics determined by the presence of a coastline, and fall into two distinct categories: western boundary currents and eastern boundary currents.
Eastern boundary currents
Eastern boundary currents are ...
in
ocean basin
In hydrology, an oceanic basin (or ocean basin) is anywhere on Earth that is covered by seawater. Geologically, ocean basins are large geologic basins that are below sea level.
Most commonly the ocean is divided into basins foll ...
s.
Walter Munk
Walter Heinrich Munk (October 19, 1917 – February 8, 2019) was an American physical oceanographer. He was one of the first scientists to bring statistical methods to the analysis of oceanographic data. His work won awards including the Nation ...
in 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.
References
*
*
External links
Glossary of Physical Oceanography and Related Disciplines Sverdrup balance
{{physical oceanography
Ocean currents
Physical oceanography