Wind-Driven Circulation Sverdrup Theory

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 wind stress to the north south transport over the water
column 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:
the first step is to use the continuity expression given
on the previous slide to find the derivative of Mx with
respect to x…
recall…
The second step is to integrate the above expression with respect to x
starting from a western boundary current a moving east and assuming no xdirected mass flow at the western boundary, i.e., Mx=0 at x = 0
Bracketed terms refer to zonal
averages of the wind stress
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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…
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Example of a Steam Function….
η
depth
Lines of constant sea surface height (η) are stream lines and the flow is
along these lines…
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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 ocean 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
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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) eddy diffusion.
Sverdrup’s Original Equations…
Munk’s Starting Equations…
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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)
Munk’s Theory
Loss of vorticity due to lateral friction
which is large close to lateral boundaries
such as continents.
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Vorticity Balance
Vorticity Balance is NOT
Maintained with Symmetric
Currents East and West
Vorticity Balance is IS
Maintained with Asymmetric
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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