Sverdrup balance

The Sverdrup balance, the Sverdrup relation, or the Sverdrup regime is a principle of theoretical oceanography, explaining the existence of stationary wind-driven ocean circulation in a zonally limited Ocean. It says that in the eastern parts of the basin-wide oceanic eddy ( Gyren ) the rotation of the wind stress at the sea surface is equal to the planetary vorticity of the vertically integrated geostrophic flow.

History

The stationary circulation in the oceans is maintained by two factors mainly: (1) thermohaline processes, stimulate movement by generating horizontal density differences due to different heat and freshwater fluxes through the sea surface, and (2) is the wind stress, the near-surface layers of the ocean in motion.

Harald Ulrik Sverdrup studied in the 1940s, building on the fundamental work of Vagn Walfrid Ekman (1905 ) on the excitation of the near-surface flow on the rotating earth by the wind stress, only the wind-driven ocean circulation. Starting from the assumption that outside the western boundary currents the dissipation is negligible due to turbulent friction, headed Sverdrup (1947 ) from the eponymous relation, which states that the meridional mass transport of the geostrophic flow is proportional to the vertical component of the rotation of the wind stress is namely

Here, in a Cartesian coordinate system as a - is referred to plane,

Physical interpretation

The Sverdrup balance can be understood as a dynamic principle which allows on a rotating ball a stationary ocean circulation.

In addition to the generally accepted interpretation of the Sverdrup relation on the basis of the conservation of potential vorticity, which is quite non-descriptive, there is an alternative interpretation based on the conservation of mass, see Tomscak and Godfrey ( 2003).

Between the west wind and the Passat Belt of Ekmantransport pushes along surface water and produces a zonally running back with increased water level. The constant action of the Ekmantransports allows the height of the back linearly increase with time, while in the two coast boundary layers the amount of the water back is held by the action of Kelvin wave at the level outside of the back, so that it assumes an elliptical shape between the continents. Due to the geostrophic adjustment an accelerating flow arises parallel to the contour lines of the back in the form of a basin-wide anticyclonic vortex formed in the riparian zones by the strong pressure gradient as intense boundary current. The convergence of the Ekmantransports produces a permanent downward vertical velocity, which is proportional to the local rotation of the wind shear stress and transports the warm outer layer near the surface continuously in depth. The resulting baroclinic pressure gradient is opposite to the barotropic pressure gradient and compensated him at a certain depth, below which the ocean remains at rest.

The dependence of the Coriolis parameter on latitude of the earth has the consequence that for the same pressure gradient at a position lower poleward geostrophic flow forms than at a position closer to the equator. Thus arises on the eastern edge of the elliptical vortex divergence of the geostrophic flow with an upward vertical component. At the points of the vertebra to which cancel out the oppositely directed vertical component of the convergence of the Ekman transport and planetary divergence, the increase of the water level and the decrease of the surface layer depth is stopped. This steady state is called Sverdrup regime. The Sverdrup regime spreads from the east coast, starting behind the westward propagating front of the long Rossbywelle off until it has reached the West Coast of the ocean and the whole basin -wide vortex has become stationary, see Gill ( 1982). The oceanic eddy is deformed asymmetrically in the steady state in east-west direction, since the weak meridional geostrophic speed controls on the eastern side of the zonal pressure gradient, whose planetary divergence balances the convergence of the Ekman transport. Moreover, its depth from the East to the West Coast continues to grow.

Derivation

In the derivation of the Sverdrup relation, we assume that the described physical interpretation and calculate the divergence of the subinertialen flow on a rotating sphere in the approximation of the beta- plane. For the mathematical description of the subinertialen mass transport in the ocean, the local accelerations can be neglected, since the flow under radiation of inertial waves ( Poincaré waves) is adapted to the geostrophic pressure field. We proceed from the linearized equations of motion. They read in a Cartesian coordinate system

For the continuity equation of the respected as incompressible fluid, we obtain

In the equations are:

• T: time
• X, y, z: coordinates of a rectangular coordinate system with the origin at the sea level at the geographical reference width, for example, positively directed to the east, north positively and positively against the force of gravity.
• U, v, w: the horizontal and vertical components of the velocity vector in the direction of the x -, y -and z- axis.
• P: the pressure in the ocean.
• : The density of the seawater.
• : The horizontal components of the turbulent shear stress vector.
• : The displacement of the sea surface from the rest position.
• , The Coriolis parameter.

The pressure p ( t, x, y, z) we can split up into the sum of and the barotropic baroclinic pressure. The barotropic pressure resulting from the displacements

The sea surface from the equilibrium position ( geoid ). It is, and is constant over the entire column of water, as the horizontal scale subinertialen processes are greater than the Rossbyradius which in turn is greater than the water depth H. thus

These processes are described by the dynamics of long waves, the pressure fields being constant from the sea surface to the sea floor. Baroclinic the pressure arising from the variability of the density of the water column, and is determined by the integration of the hydrostatic equation

The small share that the water column in the area between z = 0 and the displacement of the sea surface η contributes to the baroclinic pressure can be neglected in large-scale processes. The barotropic component of the flow is defined by its vertical average over the water column

Wherein ( U, V) are the components of the mass transport. The baroclinic component of the flow is then. The baroclinic flow thus contributes nothing to the mass transport. It appears, however, that the baroclinic pressure certainly affects the mass transport. To develop the equations of motion for the mass transport, we integrate the equations of momentum balance over the water column and split the pressure in the barotropic and baroclinic share on. To calculate the vertical integral of the baroclinic pressure gradient, we use the relation to the differentiation of an integral with variable limits for a parameter, the rules of integration by parts and the equation for the hydrostatic pressure, see Müller (2006 ) for details. This results in the equations of motion for the components of the horizontal mass transport

With. From the above equations it follows that the mass transport of the subinertialen movements is influenced both by the barotropic pressure gradient and by the baroclinic pressure field in the ocean. Wind shear stress at the sea surface drives the flow and the bottom stress slows them down. The latter is determined by the sum of the barotropic and baroclinic velocity at the seabed. This means, where the sum of baroclinic and barotropic velocity at the bottom is very small, the bottom friction is negligible. This generally applies to the oceans outside the shelf areas. The divergence of the mass transport of the subinertialen movement in the ocean can be calculated from each other by differentiating the equations for the components of the mass transport to y and for x and the subsequent subtraction of derived equations. The result is then taking into account the dependence of the width Coriolisparameters f ( y) and the vertically-integrated continuity equation, see, for example Müller ( 2006)

The following equation for the temporal change in sea level by the various contributions to the divergence of the fair transport

This is the determinant of the Jacobian matrix, and describes the physical horizontal factor which exerts the pressure on the sea bottom, known in English as a bottom torque. Physically, the above relationship that the sea level changed by the planetary divergence of the meridional flow, the divergence of the Ekmantransport at the sea surface and the divergence of the Ekmantransports in the bottom boundary layer, and by a non-zero bottom torque. The bottom torque is zero when the seabed is flat on the sea floor, the sum of barotropic and baroclinic pressure gradient disappears, or the pressure gradient at the seabed and the gradient of the sea floor are parallel. A steady state can be set in the ocean only by a balancing of the various terms on the right side of the above equation.

For the central ocean that is compensated at greater depths the barotropic pressure gradient through the baroclinic. Thus, the bottom friction and the Bottom Torque there are negligibly small. The water level will then be changed in time by the divergence of the Ekmantransports in the top layer. Its growth is terminated when the front of the radiated from the eastern shore long Rossby wave arrives at a position in the interior of the ocean, see Gill ( 1982). With the arrival of the front, the divergence of the Ekmantransports is compensated by the planetary divergence and stopped the rise in sea level. By the later arrival of the Rossby wave front further west in the positions the increase in sea level stops there longer. The sea level of the ocean increases from east to west linearly. This stationary state of the ocean circulation is the Sverdrupregime, Sverdrup (1947 ), in which the Sverdruprelation

Applies.

Further development of the theory

In the area of the western boundary current, the flow is in the form of a strong jet stream to the ground mainly meridionally aligned ( poleward in the subtropical gyre ). The planetary divergence is then substantially larger than the divergence of the excited by the wind Ekmantransports. The heavy bottoms stream has a divergence in the surface friction layer and a non-zero torque to the bottom row. The planetary divergence is thus in the region of the western edge of the flow balance through the rotation of the bottom stress and bottom torque substantially.

Stommel (1948 ) parameterized the terms for bottom friction and bottom torque so that the above equation an equation for the pressure was that filled the boundary conditions for a geostrophic flow in a zonally limited ocean basin with a flat bottom. His solution first described a closed asymmetric eddy with a narrow, intense western boundary current and a broad, weak Sverdrupstrom outside the western boundary current. However, the western boundary current of Stommel (1948 ) described was too narrow. Munk (1950 ) improved the Stommelsche parameterization taking into account the horizontal turbulent friction and obtain a solution with a western boundary current of the observations corresponded 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 nonlinear partial differential equations with real bottom topography and real meteorological drives. Such numerical circulation models yield temperature, salinity and current distributions in the ocean that match some detail with the observations in part, on the other hand, because of various unresolved by the applied numerical methods processes that exhibit divergent from the natural state, see Gerdes et al. (2003). Since a tight coupling between ocean and atmosphere determines the Earth's climate, a climate simulation requires more powerful computer than running a global circulation models of the ocean or the atmosphere. An example of a supercomputer, which is devoted to the solution of problems in the context of climate change, established in Japan 's Earth Simulator.

Observations of ocean circulation

The theories of Sverdrup, Stommel and Munk describe a relatively simple flow. The real flow in the ocean, however, is far more complicated. To improve the existing theories, their comparison with the observations taking into account the errors of observation is required. Since the observation density in the Atlantic is relatively high and similar to its circulation of the other oceans, we choose as an example of observed characteristics of the midlatitude circulation in the Atlantic and in particular the Gulf Stream from.

The dynamic topography of the Atlantic Ocean was compiled from measurements of the topography of the sea surface of board satellites and validated with the aid of measured trajectories of Oberflächendriftern. It provides information about both the mean water level as well, outside the immediate equatorial region of about 2 ° width over which flow is geostrophic adapted to the water level. The figure shows in the subtropical latitudes gradually rising to 0.75 m from the east bank to the west bank water level, as it describes the Sverdruptheorie. The water level drops from its maximum, which is located just below the American coast to the west coast to a value that corresponds to the on the east coast. On this steep slope of the respective intensive western boundary current is localized. The distribution of the water level shows broad, basin- wide vortex ( Gyren ) in the mid-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 ( Gyren ). Poleward of these vortices align themselves with sub-polar vortex, which contain as western boundary current of the Labrador Current in the north and the Falkland Current in the south. The subpolar gyre caused by the divergence of the Ekmantransports between the westerly wind belt and the area of the polar easterlies in the form of a lowered water level. Near the equator, at about ± 5 ° of latitude, to narrow back with higher water level are emerging through the convergence of Ekmantransports at the equatorward edge of the Passat belt, flowing at the polwärtiger edge of the Northern or Südäquatoriale counterflow. Westward flowing South Equatorial Current is the Between the back. It should be noted that a strong current on the NE coast of South America from the tropical area flows into the Caribbean.

The vertical temperature distribution along a circle of latitude through because the core of the subtropical vortex provides information about its vertical distribution in the water column and on the baroclinic pressure gradient. Such a temperature section through the subtropical eddy in the North Atlantic, the Gulf Stream, shows a nearly linear thickening from east to west near surface layer of hot water, the thickness of which reaches just off the coast of Florida a maximum of approximately 800 m. This is the area of ​​the Sverdrupregimes. From the point of maximum thickness of the layer of hot water is up to the coast of Florida in the area of the western boundary current thin again.