Sverdrup balance
Encyclopedia
The Sverdrup balance, or Sverdrup relation, is a theoretical relationship between the wind
stress
exerted on the surface of the open ocean
and the vertically integrated meridional (north-south) transport of ocean water.
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
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 of the wind stress. This is known as the Sverdrup relation;
.
Here,
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.
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 Ekman layer.
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.
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
with latitude results in a narrow western boundary current in ocean basins. Walter Munk
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.
Wind
Wind is the flow of gases on a large scale. On Earth, wind consists of the bulk movement of air. In outer space, solar wind is the movement of gases or charged particles from the sun through space, while planetary wind is the outgassing of light chemical elements from a planet's atmosphere into space...
stress
Stress (physics)
In continuum mechanics, stress is a measure of the internal forces acting within a deformable body. Quantitatively, it is a measure of the average force per unit area of a surface within the body on which internal forces act. These internal forces are a reaction to external forces applied on the body...
exerted on the surface of the open ocean
Ocean
An ocean is a major body of saline water, and a principal component of the hydrosphere. Approximately 71% of the Earth's surface is covered by ocean, a continuous body of water that is customarily divided into several principal oceans and smaller seas.More than half of this area is over 3,000...
and the vertically integrated meridional (north-south) transport of ocean water.
History
Aside from the oscillatory motions associated with tidalTide
Tides are the rise and fall of sea levels caused by the combined effects of the gravitational forces exerted by the moon and the sun and the rotation of the Earth....
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
Temperature
Temperature is a physical property of matter that quantitatively expresses the common notions of hot and cold. Objects of low temperature are cold, while various degrees of higher temperatures are referred to as warm or hot...
and salinity
Salinity
Salinity is the saltiness or dissolved salt content of a body of water. It is a general term used to describe the levels of different salts such as sodium chloride, magnesium and calcium sulfates, and bicarbonates...
, and therefore in seawater
Seawater
Seawater is water from a sea or ocean. On average, seawater in the world's oceans has a salinity of about 3.5% . This means that every kilogram of seawater has approximately of dissolved salts . The average density of seawater at the ocean surface is 1.025 g/ml...
density
Density
The mass density or density of a material is defined as its mass per unit volume. The symbol most often used for density is ρ . In some cases , density is also defined as its weight per unit volume; although, this quantity is more properly called specific weight...
, and (2) wind forcing. In the 1940s, when Harald Sverdrup
Harald Sverdrup
Harald Ulrik Sverdrup was a Norwegian oceanographer and meteorologist who made a number of important theoretical discoveries in these fields. Having first worked in Bergen and Leipzig he was the scientific director of the North Polar expedition of Roald Amundsen aboard the Maud from 1918 to 1925...
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 of the wind stress. This is known as the Sverdrup relation;
.Here,
Beta planeIn geophysical fluid dynamics, an approximation whereby the Coriolis parameter, f, is set to vary linearly in space is called a beta plane approximation...
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;
is the wind stress vector.
Physical interpretation
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 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 equationBarotropic vorticity equation
This 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, no vertical wind shear of the geostrophic wind. It also implies that thickness contours are parallel to...
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 Ekman layer.The driving force behind the vertical velocity
is Ekman transportEkman transport
Ekman transport, part of Ekman motion theory first investigated in 1902 by Vagn Walfrid Ekman , is the term given for the 90 degree net transport of the surface layer due to wind forcings...
, 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.
Further development
In 1948 Henry StommelHenry Stommel
Henry Melson Stommel 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 that form the basis of physical oceanography today...
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
Coriolis effect
In physics, the Coriolis effect is a deflection of moving objects when they are viewed in a rotating reference frame. In a reference frame with clockwise rotation, the deflection is to the left of the motion of the object; in one with counter-clockwise rotation, the deflection is to the right...
with latitude results in a narrow western boundary current in ocean basins. Walter Munk
Walter Munk
Walter Heinrich Munk is an American physical oceanographer. He is professor of geophysics emeritus and holds the Secretary of the Navy/Chief of Naval Operations Oceanography Chair at Scripps Institution of Oceanography in La Jolla, California.-Early life:Born in 1917 in Vienna, Austria-Hungary,...
in 1950 combined the results of Rossby
Carl-Gustaf Rossby
Carl-Gustaf Arvid Rossby was a Swedish-U.S. meteorologist who first explained the large-scale motions of the atmosphere in terms of fluid mechanics....
(eddy viscosity), Sverdrup (upper ocean wind driven flow) and Stommel (western boundary current flow) and proposed a complete solution for the ocean circulation.