Sverdrup relation

The Sverdrup balance , the Sverdrup relation or the Sverdrup regime is a principle of theoretical oceanography that explains the existence of steady wind-driven ocean currents in a zonally limited ocean. It states that in the eastern parts of the basin-wide oceanic eddies (gyren) the rotation of the wind shear stress on the sea surface is equal to the planetary eddy strength of the vertically integrated geostrophic current.

history

The stationary circulation in the oceans is essentially maintained by two causes: (1) thermohaline processes , which stimulate the movement by creating horizontal density differences due to different heat and freshwater flows through the sea surface, and (2) the wind shear stress, which is applied to those near the surface Sets layers of ocean in motion .

In the 1940s, Harald Ulrik Sverdrup , based on the fundamental work of Vagn Walfrid Ekman (1905) on the stimulation of near-surface currents on the rotating earth by wind shear stress, exclusively investigated wind-driven ocean currents. Starting from the assumption that outside the western boundary currents the dissipation due to turbulent friction is negligible, Sverdrup derived the relation named after him in 1947, which says that the meridional mass transport of the geostrophic flow is proportional to the vertical component of the rotation of the wind shear stress, namely

${\displaystyle V={\frac {\operatorname {rot} _{z}\mathbf {\tau } }{\beta \rho _{0}}}}$,

In a Cartesian coordinate system, which is referred to as a plane, is ${\displaystyle \beta }$

${\displaystyle V}$the vertically integrated meridional geostrophic flow ;

${\displaystyle \mathbf {\tau } }$ the vector of the wind shear stress on the sea surface.

${\displaystyle \beta }$the meridional derivative of the Coriolis parameter f;

${\displaystyle \rho _{0}}$ the density of sea water;

Physical interpretation

Scheme of the formation of a high pressure ridge in the ocean through the combined effect of Ekman transport from the west wind and trade wind belts .

Scheme of the elliptical deformation of a zonal high pressure ridge in the ocean due to the action of the Kelvin waves on the meridional coasts of the ocean.

Scheme of the asymmetrical deformation of a zonal high pressure ridge in the ocean by the action of the long Rossby waves radiating from the eastern coast .

The Sverdrup balance can be understood as a dynamic principle that enables a stationary ocean circulation on a rotating sphere.

In addition to the commonly used interpretation of the Sverdrup relation based on the conservation of potential vorticity , which is quite difficult to visualize, there is an alternative interpretation based on the conservation of mass (Tomscak and Godfrey, 2003).

Between the Westwind- and the Passat belt pushes Ekman transport along the surface water and produces a zonally extending back with increased water level. The constant effect of the ekmant transport causes the height of the ridge to increase linearly over time, while in the two coastal boundary layers the height of the water ridge is kept at the level outside the ridge by the action of Kelvin waves , so that it assumes an elliptical shape between the continents. The geostrophic adjustment creates an accelerating current parallel to the contour lines of the ridge in the form of an anticyclonic, basin-wide eddy, which is formed as an intensive marginal current in the bank zones due to the strong pressure gradient.

The convergence of the ekmant transport creates a permanent downward vertical speed, which is proportional to the local rotation of the wind shear stress and continuously transports the warm cover layer close to the surface into the depth. The resulting baroclinic pressure gradient is opposite to the barotropic pressure gradient and compensates for it at a certain depth below which the ocean remains at rest.

The dependence of the Coriolis parameter on the geographic latitude of the earth has the consequence that with the same pressure gradient, a lower geostrophic flow forms at a polar position than at a position closer to the equator. Thus, at the eastern edge of the elliptical vortex, there is a divergence of the geostrophic flow with an upward vertical component. At the points of the vortex where the oppositely directed vertical components of the convergence of the Ekman transport and the planetary divergence cancel each other, the increase in the water level and the decrease in the surface layer depth are stopped. This steady state is known as the Sverdrup regime.

The Sverdrup regime spreads from the east coast behind the westward propagating front of the long Rossby wave until it reaches the west coast of the ocean and the entire basin-wide vortex has become stationary (Gill, 1982). The oceanic vortex is asymmetrically deformed in the steady state in the east-west direction, since on its east side the zonal pressure gradient regulates the weak meridional geostrophic velocity, the planetary divergence of which balances the convergence of the Ekman transport. In addition, its depth increases continuously from the east to the west coast.

Derivation

When deriving the Sverdrup relation, we start from the physical interpretation described and will calculate the divergence of the sub-inertial flow on a rotating sphere in the approximation of the beta plane. For the mathematical description of the sub-inertial mass transport in the ocean, the local accelerations can be neglected, since the flow is geostrophically adapted to the pressure field with radiation from inertial waves ( Poincaré waves ) . We start from the linearized equations of motion. In a Cartesian coordinate system they are:

${\displaystyle -fv=-{\frac {1}{\rho _{0}}}{\frac {\partial p}{\partial x}}+{\frac {1}{\rho _{0}}}{\frac {\partial \tau ^{x}}{\partial z}}}$,

${\displaystyle +fu=-{\frac {1}{\rho _{0}}}{\frac {\partial p}{\partial y}}+{\frac {1}{\rho _{0}}}{\frac {\partial \tau ^{y}}{\partial z}}}$,

${\displaystyle 0=-{\frac {1}{\rho }}{\frac {\partial p}{\partial z}}-g}$.

For the continuity equation of the fluid considered incompressible, we get

${\displaystyle {\frac {\partial u}{\partial x}}+{\frac {\partial v}{\partial y}}+{\frac {\partial w}{\partial z}}=0}$.

In the equations are:

• ${\displaystyle x,y,z}$: the coordinates of a right-angled coordinate system with the zero point at sea level on the geographical reference latitude , e.g. B. positive to the east, positive to the north and positive directed against gravity. ${\displaystyle \varphi _{0}}$
• ${\displaystyle u,v,w}$: the horizontal and vertical components of the velocity vector in the direction of the x, y and z axes.
• ${\displaystyle p}$: the pressure in the ocean.
• ${\displaystyle \rho }$: the density of sea water.
• ${\displaystyle \tau ^{x},\tau ^{y}}$: the horizontal components of the turbulent shear stress vector.
• ${\displaystyle \eta }$: the deflection of the sea surface from its resting position.
• ${\displaystyle f=2\Omega \sin \varphi }$, the Coriolis parameter .
• ${\displaystyle t}$: the time

We can split the pressure into the sum of the barotropic and baroclinic pressure. The barotropic pressure results from the deflections of the sea surface from the resting position ( geoid ). It is and is constant over the entire water column, since the horizontal scales of sub-inertial processes are greater than the Rossby radius, which in turn is greater than the water depth . Thus, these processes are described by the dynamics of long waves, the pressure fields of which are constant from the sea surface to the sea floor. The baroclinic pressure results from the variability of the density in the water column and is determined by integrating the hydrostatic equation ${\displaystyle p(t,x,y,z)}$ ${\displaystyle \eta (t,x,y)}$ ${\displaystyle p_{bt}=\rho _{0}g\eta }$ ${\displaystyle H}$

${\displaystyle p_{bc}=\int _{z}^{0}g\rho (t,x,y,z')dz'}$.

The small portion that the water column in the area between and the deflection of the sea surface contributes to the baroclinic pressure can be neglected in large-scale processes. The barotropic component of the flow is defined by its vertical mean value above the water column ${\displaystyle z=0}$ ${\displaystyle \eta }$

${\displaystyle (u,v)_{bt}=\int _{-H}^{0}(u,v)dz={\frac {1}{H}}(U,V)}$.

where (U, V) are the components of mass transport. The baroclinic component of the flow is then . The baroclinic flow does not contribute anything to the mass transport. However, it turns out that the baroclinic pressure definitely influences the mass transport. To develop the equations of motion for mass transport, we integrate the equations of the momentum balance over the water column and split the pressure into the barotropic and baroclinic components. To calculate the vertical integral over the baroclinic pressure gradient, we use the relationship for the differentiation of an integral with variable limits according to a parameter, the rules of partial integration , and the equation for the hydrostatic pressure (Müller, 2006). This results in the equations of motion of the components of the horizontal mass transport: ${\displaystyle (u,v)_{bc}=(u,v)-(u,v)_{bt}}$

${\displaystyle -fV=-{\frac {1}{\rho _{0}}}\left(\rho _{0}gH{\frac {\partial \eta }{\partial x}}+H{\frac {\partial p_{bc}(z=-H)}{\partial x}}+{\frac {\partial E_{\mathrm {pot} }}{\partial x}}\right)+{\frac {1}{\rho _{0}}}(\tau _{0}^{x}-\tau _{H}^{x})}$ ,

${\displaystyle +fU=-{\frac {1}{\rho _{0}}}\left(\rho _{0}gH{\frac {\partial \eta }{\partial y}}+H{\frac {\partial p_{bc}(z=-H)}{\partial y}}+{\frac {\partial E_{\mathrm {pot} }}{\partial y}}\right)+{\frac {1}{\rho _{0}}}(\tau _{0}^{y}-\tau _{H}^{y})}$ ,

with . ${\displaystyle E_{\mathrm {pot} }=\int _{-H}^{0}zg\rho (t,x,y,z)dz}$

From the above equations it follows that the mass transport of the sub-inertial movements in the ocean is influenced by both the barotropic pressure gradient and the baroclinic pressure field. The wind shear stress on the sea surface drives the current, and the bottom shear stress slows it down. The latter is determined by the sum of the barotropic and baroclinic speed on the sea floor. This means that where the sum of baroclinic and barotropic speed on the ground is very small, the ground friction is negligible. This is generally true of the oceans outside the shelf areas . ${\displaystyle {\vec {\tau }}_{0}}$ ${\displaystyle {\vec {\tau }}_{H}}$

The divergence of the mass transport of the sub-inertial movements in the ocean can be calculated by deriving the equations for the components of the mass transport gradually and then subtracting the derived equations from each other. It then results, taking into account the width dependency of the Coriolis parameter and the vertically integrated continuity equation (Müller, 2006). ${\displaystyle y}$ ${\displaystyle x}$ ${\displaystyle f(y)}$

${\displaystyle {\frac {\partial \eta }{\partial t}}+{\frac {\partial U}{\partial x}}+{\frac {\partial V}{\partial y}}=0}$,

the following equation for the temporal change in sea level due to the various contributions to the divergence of the mass transport

${\displaystyle f{\frac {\partial \eta }{\partial t}}=\beta V-{\frac {1}{\rho _{0}}}\operatorname {rot} _{z}(\tau _{0}-\tau _{H})-{\frac {1}{\rho _{0}}}J[p_{H},H]}$.

Here is the determinant of the Jacobi matrix and physically describes the horizontal moment that the pressure exerts on the sea floor, referred to in English as bottom torque. Physically, the above relationship means that the sea level changes due to the planetary divergence of the meridional flow, the divergence of the ekman transport on the sea surface and the divergence of the ecumen transport in the ground friction layer, as well as a non-zero bottom torque. The bottom torque becomes zero when the sea floor is flat, the sum of the barotropic and baroclinic pressure gradients on the sea floor disappears, or the pressure gradient on the sea floor and the gradient of the sea floor are parallel. A steady state can only be established in the ocean by balancing the various terms on the right-hand side of the equation above. ${\displaystyle J[p_{H},H]={\frac {\partial p}{\partial x}}{\frac {\partial H}{\partial y}}-{\frac {\partial p}{\partial y}}{\frac {\partial H}{\partial x}}}$

For the central ocean, the barotropic pressure gradient is compensated for by the baroclinic one at greater depths. This means that the ground friction and the bottom torque are negligibly small there. The water level is then changed over time due to the divergence of the ekmant transport in the surface layer. Its growth stops when the front of the long Rossby wave emitted from the east bank arrives at a position in the interior of the ocean (Gill, 1982). With the arrival of the front, the divergence of the ekmant transport is compensated by the planetary divergence and the rise in sea level is stopped. Due to the later arrival of the Rossby wave front at positions further to the west, the rise in sea level there lasts longer. The sea level of the ocean increases linearly from east to west. This steady state of ocean circulation is the Sverdrup regime (Sverdrup, 1947), in which the Sverdrup relation applies:

${\displaystyle 0=\beta V-{\frac {1}{\rho _{0}}}\operatorname {rot} _{z}\tau _{0}}$.

Further development of the theory

In the area of ​​the western marginal current, the current is mainly meridional in the form of a strong jet stream down to the ground (poleward in the subtropical Gyre). The planetary divergence is then much greater than the divergence of the ekman transport excited by the wind. The strong ground current results in a divergence in the ground friction layer and a bottom torque other than zero. The planetary divergence is therefore essentially balanced in the area of ​​the western boundary flow by the rotation of the bottom shear stress and the bottom torque.

${\displaystyle 0=\beta V-{\frac {1}{\rho _{0}}}\operatorname {rot} _{z}(\tau _{0}-\tau _{H})-{\frac {1}{\rho _{0}}}J[p_{H},H]}$.

Stommel (1948) parameterized the terms for bottom friction and bottom torque in such a way that the above equation resulted in an equation for pressure that fulfilled the boundary conditions for geostrophic flow in a zonally delimited ocean basin with a flat bottom. His solution described for the first time a closed asymmetrical ocean vortex with a narrow, intense western marginal current and a broad, weak sverdrup current outside the western marginal current. However, the western marginal current described by Stommel (1948) was too narrow. Munk (1950) improved the Stommelian parameterization taking into account horizontal turbulent friction and obtained a solution with a western boundary current that corresponded to the observations to a greater extent.

In the following decades, the performance of supercomputers developed enormously. This enabled the numerical solution of the conservation equations for momentum, mass and energy in the ocean in the form of a system of non-linear partial differential equations with real ground topography and real meteorological drives. Such numerical circulation models result in temperature, salinity and current distributions in the ocean, which partly agree in great detail with the observations, on the other hand, due to various processes that are not resolved by the numerical solution methods used, deviate from the natural state (Gerdes et al., 2003) .

Since a close coupling between the ocean and the atmosphere determines the earth's climate, a climate simulation requires computers that are even more powerful than the operation of global circulation models of the ocean or the atmosphere. An example of a supercomputer that is dedicated to solving problems related to climate change is the Earth Simulator set up in Japan .

Observations of the ocean circulation

Dynamic topography of the Atlantic averaged over the years 1992–2002 from data by Nikolai Maximenko (IPRC) and Peter Niiler (SIO). Drawn with the APDRC LAS7 for public.

Vertical section of the climatically averaged water temperature between the sea surface and 1500 m depth in the North Atlantic at 30 ° N. The data come from the NODC World Ocean Atlas 2009 and were drawn with the APDRC LAS7 for public.

The theories of Sverdrup, Stommel and Munk describe a relatively simple current. However, the real flow in the ocean is far more complicated. In order to improve the existing theories, it is necessary to compare them with the observations, taking into account the errors of observation. Since the observation density in the Atlantic is relatively high and its circulation is similar to that of the other oceans, we select as an example observed properties of the circulation of middle latitudes in the Atlantic and in particular of the Gulf Stream.

The dynamic topography of the Atlantic Ocean was compiled from measurements of the topography of the sea surface from on board satellites and secured with the help of measured orbits of surface drifters. It provides information about the mean water level as well as, outside the immediate equatorial region of around 2 ° latitude, about the current, which is geostrophically adapted to the water level.

The figure shows in the subtropical latitudes a water level gradually rising by 0.75 m from the east bank to the west bank, as described by the Sverdruptheorie. The water level falls from its maximum, which is just off the American coast, to a value on the west coast that corresponds to that on the east coast. The respective intensive western marginal current is localized on this steep slope. The distribution of the water level shows broad, basin-wide eddies (gyren) in the middle latitudes in both the North Atlantic and the South Atlantic according to the Sverdruptheorie. On the western shores of the Atlantic each includes a western boundary current, the Gulf Stream in the North and the Brazil Current in the South Atlantic, the respective vertebrae. Towards the pole of these eddies are subpolar eddies, which contain the Labrador Current in the north and the Falkland Current in the south as the western boundary current . The subpolar eddies are caused by the divergence of the ekmant transport between the west wind belt and the area of ​​the polar east winds in the form of a lowered water level.

In the vicinity of the equator, at approximately ± 5 ° latitude, narrow ridges with a higher water level form due to the convergence of the ekman transport at the equatorial edge of the trade belt, on whose polar flank the north and south equatorial countercurrent flow. Between the ridges is the south equatorial current flowing westwards . It should be noted that a strong current on the NE coast of South America flows from the tropical area into the Caribbean .

The vertical temperature distribution along a circle of latitude through the core of the subtropical eddy provides information about its vertical expansion in the water column and about the baroclinic pressure gradients. Such a temperature section through the subtropical eddy in the North Atlantic, the Gulf Stream, shows a near-surface warm water layer thickening almost linearly from east to west, the thickness of which reaches its maximum of approximately 800 m just off the coast of Florida. This is the realm of the Sverdrup regime. From the point of maximum thickness, the warm water layer becomes thinner again to the Florida coast in the area of ​​the western boundary current.

literature

• Ekman, VW (1905): On the influence of the earth's rotation on ocean currents. Arch. Math. Astron. Phys. 2, No. 11.
• R.Gerdes, CWBöning, J.Willebrand: General circulation models. Ocean. Promet, 29, 2003, pp. 1-4, 15-28.
• Gill, AE (1982): Atmosphere-Ocean Dynamics. Academic Press. 662 pp. ISBN 0-12-283520-4
• Müller, P. (2006): The Equations of Oceanic Motions. Cambridge University Press, 291 pp. ISBN 0-521-85513-6
• Munk, WH (1950): On the Wind-Driven Ocean Circulation. J. Atmos. Sci., 7, 80-93.
• Stommel, H. (1948): The westward intensification of wind-driven ocean currents. Trans. Amer. Geophys. Union, 29 (2), 202-206.
• Sverdrup, HU (1947): Wind-Driven Currents in a Baroclinic Ocean; with Application to the Equatorial Currents of the Eastern Pacific. In: Proceedings of the National Academy of Sciences , 33 (11), 318-326.
• Tomczak, M. and JS Godfrey (2003): Regional Oceanography: an Introduction. 2nd edn 390pp. ISBN 81-7035-306-8 .