Wind-Driven Circulation (Sverdrup, Stommel and Munk Theories)

Sverdrup Theory

definitions…

note…

1 Eq 1 Eq 1

Differentiate Eq 1 and 2 with respect to y and x respectively and then subtract Eq 1 from Eq 2…

gathering terms…

rearranging term and noting that df/dt = β…

using continuity requirement (see box to right)…

or…

This expression is known as Sverdrup Balance and it equates the curl of the surface to the north south transport over the integrated to the depth of no motion.

2 Note: wind stress is primarily zonal so the derivative of y-directed wind stress with respect to x is small and can be ignored to a first approximation.

Mx can be computed from My in two steps: recall… the first step is to use the continuity expression given on the previous slide to find the derivative of Mx with respect to x…

The second step is to integrate the above expression with respect to x starting from a western a moving east and assuming no x- directed mass flow at the western boundary, i.e., Mx=0 at x = 0

Bracketed terms refer to zonal averages of the wind stress

3 Stream Function

The stream function (ψ) is defined by…

The stream function is a scalar from which the vector field can be calculated. Values of constant ψ depict stream lines and for steady flow stream line will equal path lines, where path lines are the path taken by a fluid parcel moving within the fluid flow field.

The mass transport stream function is similarly defined…

4 Example of a Steam Function….

η d e p t h

Lines of constant surface height (η) are stream lines and the flow is along these lines…

5 Stommel’s Theory

Sverdrup Theory of wind driven circulation that was derived in the previous slides began with the following equations which were integrated to a depth of no motion (e.g., 1000 m) where is was assumed the stress (friction) went to zero). Note: Sverdrup’s Theory does not consider Western Boundary currents and does not address the issue of Western Intensification

Stommel’s Theory of wind driven circulation uses the same basic equations but integrates to the bottom of the and allows for bottom friction that is a simple linear function of velocity Fx=-Ru and Fy=-Rv. The friction at the top of the ocean is just the applied zonal wind stress τx

6 Stommel’s Theory

Munk’s Theory

Munk’s Theory of wind-driven circulation takes the same basic equations that Sverdrup and Stommel started with and adds a lateral (north/south) diffusion.

Sverdrup’s Original Equations…

Munk’s Starting Equations…

7 Munk replaced u and v velocity components in the starting equations and then followed Sversrup’s approach of taking x and y derivatives of respective equation and then subtracting one equation from another to get…

Munk’s Solution… Sverdrup’s Solution…

Sverdrup Balance which expresses the conservation of potential vorticity (in the absence of frictional loss of vorticity)

Loss of vorticity due to lateral friction which is large close to lateral boundaries such as continents.

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8 Vorticity Balance Vorticity Balance is NOT Vorticity Balance is IS Maintained with Symmetric Maintained with Asymmetric Currents East and West Currents East and West

Wind Stress Provides Negative Relative Vorticity. Moving North Increases Planetary Vorticity Requiring Relative Vorticity to Decrease (this the same as adding negative relative vorticity) -the opposite happens when moving south. Finally, Friction adds Positive Vorticity and this increased with Velocity near the boundary

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