1. EVAT 554 OCEAN-ATMOSPHERE DYNAMICS GYRE-SCALE OCEAN CIRCULATION LECTURE 16 (Reference: Peixoto & Oort, Chapter 8,10)

2. Sverdrup Transport What about the western boundary??? We are not conserving mass (note the behavior at the western boundary!)

3. Sverdrup Transport Problem is that we cannot satisfy two lateral boundary conditions with a solution to a first order equation We need to take into account missing physics Bottom Friction!

4. Sverdrup Transport Problem is that we cannot satisfy two lateral boundary conditions with a solution to a first order equation Bottom Friction!

5. Stommel ‘Bottom Friction’ model Bottom Friction! Assume a “Rayleigh” law for frictional stresses In areas of moderate flow, this will reduce to zero bottom stress, yielding the previous result

6. Assume a “Rayleigh” law for frictional stresses We might anticipate, however, that this solution could breakdown where we know the Sverdrup solution must break down… Stommel ‘Bottom Friction’ model

7. We thus assume the existence of a boundary layer of zonal width ‘  ’ that provides the return flow of the interior Sverdrup transport Stommel ‘Bottom Friction’ model Note that these expressions satisfy the requirement of no basin-integrated meridional transport at any latitude!

8. We thus assume the existence of a boundary layer of zonal width ‘  ’ that provides the return flow of the interior Sverdrup transport Stommel ‘Bottom Friction’ model Note that these expressions satisfy the requirement of no basin-integrated meridional transport at any latitude! =0

9. Useful to interpret the circulation in terms of ‘Vorticity’ (spin) Stommel ‘Bottom Friction’ model Absolute Vorticity=Planetary Vorticity ( f) +Relative Vorticity (curl of velocity field) Only friction can take away this vorticity (i.e., add negative vorticity) once it has been added Windstress adds positive vorticity

10. Stommel ‘Bottom Friction’ model Consider the fundamental equations Add these together, Differentiate with respect to x and y respectively Useful to interpret the circulation in terms of ‘Vorticity’ (spin)

11. Stommel ‘Bottom Friction’ model Useful to interpret the circulation in terms of ‘Vorticity’ (spin) Relative Vorticity Absolute Vorticity

12. Stommel ‘Bottom Friction’ model Consider the fundamental equations Now, differentiate with respect to y and x respectively Subtract second from first, Differentiate with respect to x and y respectively

13. Stommel ‘Bottom Friction’ model Divergence Equation If horizontal flow is non-divergent Assume Rayleigh friction

14. Stommel ‘Bottom Friction’ model This is only the boundary layer solution Recall the interior (Sverdrup) solution Assuming vertically uniform flow (an idealization), The full solution is thus,

15. Stommel ‘Bottom Friction’ model We require no net meridional transport!

16. Stommel ‘Bottom Friction’ model We can use continuity of the horizontal flow field to derive an expression for the zonal velocity We can thus define a streamfunction:

17. Stommel ‘Bottom Friction’ model We can use continuity of the horizontal flow field to derive an expression for the zonal velocity We can thus define a streamfunction:  =0 00

18. In reality, Western Boundary Current Separates Eddy-Resolving Ocean GCM Stommel Model Stommel ‘Bottom Friction’ model Stommel Model obviously an idealization, but it captures the essence of westward intensification of ocean currents