From observations to models:

1. From observations to models: More on the western boundary current problem

2. Stommel-Munk model • Stommel (1948) and Munk (1950) solved the western boundary current problem by including friction in the equations of motion.

3. In the simplest case, friction is taken directly proportional to the current velocity (“Rayleigh friction”) .

4. This, according to the Sverdrup relation, a typical subtropical zonal wind stress t causes a depth integrated flow that is directed equatorward. • One cannot resolve the question whether the flow generates an eastern or western boundary current, or whether it will be east-west symmetric [Figure 5 from Stocker 2000].

5. Resulting streamlines of the wind-driven circulation with bottom (“Rayleigh”) friction. Only in the case of a latitudinal dependence of the Coriolis parameter is there a western intensification of the flow [Figure 6 fom Stocker 2000]. Western intensification is the result of wind stress acting on a rotating sphere.

6. Stommel considered three different situations: • The ocean is assumed to be on a non-rotating Earth. • The ocean is rotating but the Coriolis parameter f is constant. • The ocean is rotating and the Coriolis parameter varies with latitude (for the sake of simplicity, this variation is assumed to be linear).

7. Sea-surface height (cm) Streamfunction

8. Sea-surface topography: What topographic feature may be clearly seen in situation (2), but is not present in situation (1)? • Flow patterns: What do they tell us about the intensification of western boundary currents? Is it simply the result of existence of the Coriolis force?

9. Flow patterns indicated by streamlines Sea-surface height (cm) • Sea-surface topography: What topographic feature may be clearly seen in situation (b), but is not present in situation (a)? • Flow patterns: What do they tell us about the intensification of western boundary currents? Is it simply the result of existence of the Coriolis force? • no rotation • Coriolis force constant • Coriolis force increases linearly with latitude

10. Why does the intensification of western boundary currents result from the fact that the Coriolis force increases with latitude? • Intensification of western boundary currents may also be explained in terms of vorticity balance. • Water that has a rotary motion in relation to the surface of the Earth, caused by wind stress and/or frictional forces, is said to possess relative vorticity z. • The vorticity possessed by a parcel of fluid by reason of its being on the rotating Earth is known as planetary vorticity f. • The conservation of the absolute vorticity of a water parcel is expressed as:

11. Why does the intensification of western boundary currents result from the fact that the Coriolis force increases with latitude? Factors that affect relative vorticity are: • The wind field acts to supply negative (clockwise) vorticity. • The change in latitude: • Water moving northwards on the western side into regions of larger positive planetary vorticity acquires negative relative vorticity. • Water moving southwards on the eastern side into regions of smaller positive planetary vorticity loses negative relative vorticity (or gains positive relative vorticity). • Because as much water moves northwards as moves southwards, the net change in relative vorticity is zero.

12. Why does the intensification of western boundary currents result from the fact that the Coriolis force increases with latitude? • Friction: There will be significant friction with the coastal boundaries as a result of horizontal eddies. • Using appropriate values for the coefficient of eddy viscosity for horizontal motion Ah, we can make vorticity-balance calculations for a symmetrical and an asymmetrical ocean circulation. • The frictional force between moving water and a boundary is approximately proportional to the square of the current speed. • However, Stommel used a simpler relationship, with friction directly proportional to current speed (written as Ru for the x-direction and Rv for the y-direction, where R is a constant). • For the frictional forces to be large enough to balance the negative relative vorticity input by the wind field, the circulation of a symmetrical gyre would have to be many times faster than observed.

13. Contributions to the vorticity balance • Asymmetrical subtropical gyre with an intensified western boundary current: Flow on the western side must be much faster, and the effects of friction and change in latitude increase significantly. • Symmetrical subtropical gyre: Negative relative vorticity supplied by wind and that resulting from movement of water into higher latitudes far outweigh the positive relative vorticity provided by friction.

14. The ocean circulation model derived by Stommel (1948) already bears a strong resemblance to the circulatory systems of the subtropical gyre. Streamlines (in units of Sverdrup, 1 Sv = 106 m s-1) in a rectangular basin with (a) constant Coriolis parameter and (b) latitude-dependent Coriolis parameter [Figure 7.20 from Dietrich et al. (1975)]

15. We define a streamfunction such that contours of constant Y (streamlines) are everywhere parallel to the volume flow DY is equal to the volume flow rate confined between streamlines labeled Y and Y + DY . Streamfunction Stewart (2006, Figure 11.4)

16. Resulting streamlines of the wind-driven circulation with bottom (“Rayleigh”) friction [Figure 6 fom Stocker 2000].

17. Addendum: model of wind-driven ocean circulation (Shallow) • How to plot the streamfunction as a function of longitude: plot(xG(:), psi(:, 16), '-r')

18. Further Reading • Historical: Stommel, H. (1948), The westward intensification of wind-driven ocean currents, Transactions of American Geophysical Union 29, 202-206 • More mathematical: Stocker, T. (2000), The Ocean in the climate system: Observing and modeling its variability, 55 pp. (see website - the original version appeared as Chapter 2 in: Topics in Atmospheric and Interstellar Physics and Chemistry, European Research Course on Atmospheres, Volume 2, edited by C. F. Boutron, Les Editions de Physique, Les Ulis, France, 1996, pages 39-90) • Rather complete, but also more mathematical: Mellor, George L. (1996), Introduction to Physical Oceanography, American Institute of Physics, Woodbury, New York, 260 pp. • Nice web resource: http://minerva.simons-rock.edu/~gshel/geosci245/western/western.html