1. EVAT 554 OCEAN-ATMOSPHERE DYNAMICS FILTERING OF EQUATIONS FOR OCEAN LECTURE 10 (Reference: Peixoto & Oort, Chapter 3,8)

2. Note that the major horizontal ocean circulation systems mirror closely the semi- permanent high and low pressure systems

3. Scale Analysis The Ocean

4. Zonal Momentum Balance: Meridional Momentum Balance: Vertical Momentum Balance: Continuity: Equation of State: Heat Equation: (incompressible!) How many equations? How many variables?

5. Zonal Momentum Balance: Length scale: L  10 6 m, l  10 5 m Depth scale: H  10 3 m, h  10 2 m Horizontal velocity scale: u,v  10 -1 ms -1 Vertical velocity scale: w  10 -4 ms -1 Horizontal pressure scale:  p  100 mb = 10 4 Pa Time Scale: L/u  10 7 s  H/w  10 7 s Radius of Earth: a=6.37x 10 6 m Coriolis parameter: f,f'  10 -4 s -1 Density of Water:   1000 kg m -3 Horizontal Eddy Viscosity: H  10 5 m 2 s -1 Vertical Eddy Viscosity: V  10 -1 m 2 s -1 10 -8 ms -2 10 -5 ms -2 10 -8 ms -2 10 -5 ms -2 10 -6 ms -2

6. Meridional Momentum Balance: 10 -8 ms -2 10 -5 ms -2 10 -6 ms -2 Length scale: L  10 6 m, l  10 5 m Depth scale: H  10 3 m, h  10 2 m Horizontal velocity scale: u,v  10 -1 ms -1 Vertical velocity scale: w  10 -4 ms -1 Horizontal pressure scale:  p  100 mb = 10 4 Pa Time Scale: L/u  10 7 s  H/w  10 7 s Radius of Earth: a=6.37x 10 6 m Coriolis parameter: f,f'  10 -4 s -1 Density of Water:   1000 kg m -3 Horizontal Eddy Viscosity: H  10 5 m 2 s -1 Vertical Eddy Viscosity: V  10 -1 m 2 s -1

7. Horizontal Momentum Balance 10 -8 ms -2 10 -5 ms -2 10 -6 ms -2 Geostrophic Balance “Rossby Number” Geostrophic Balance Holds when Ro << 1 (zonal)(meridional) Length scale: L  10 6 m, l  10 5 m Depth scale: H  10 3 m, h  10 2 m Horizontal velocity scale: u,v  10 -1 ms -1 Vertical velocity scale: w  10 -4 ms -1 Horizontal pressure scale:  p  100 mb = 10 4 Pa Time Scale: L/u  10 7 s  H/w  10 7 s Radius of Earth: a=6.37x 10 6 m Coriolis parameter: f,f'  10 -4 s -1 Density of Water:   1000 kg m -3 Horizontal Eddy Viscosity: H  10 5 m 2 s -1 Vertical Eddy Viscosity: V  10 -1 m 2 s -1

8. Horizontal Momentum Balance 10 -8 ms -2 10 -5 ms -2 10 -6 ms -2 Geostrophic Balance Geostrophic Balance Holds when Ek << 1 (zonal)(meridional) “Ekman Number” Length scale: L  10 6 m, l  10 5 m Depth scale: H  10 3 m, h  10 2 m Horizontal velocity scale: u,v  10 -1 ms -1 Vertical velocity scale: w  10 -4 ms -1 Horizontal pressure scale:  p  100 mb = 10 4 Pa Time Scale: L/u  10 7 s  H/w  10 7 s Radius of Earth: a=6.37x 10 6 m Coriolis parameter: f,f'  10 -4 s -1 Density of Water:   1000 kg m -3 Horizontal Eddy Viscosity: H  10 5 m 2 s -1 Vertical Eddy Viscosity: V  10 -1 m 2 s -1

9. Horizontal Momentum Balance 10 -8 ms -2 10 -5 ms -2 10 -6 ms -2 Geostrophic Balance Geostrophic Balance Holds when Ek << 1 (zonal)(meridional) “Ekman Number” Length scale: L  10 6 m, l  10 5 m Depth scale: H  10 3 m, h  10 2 m Horizontal velocity scale: u,v  10 -1 ms -1 Vertical velocity scale: w  10 -4 ms -1 Horizontal pressure scale:  p  100 mb = 10 4 Pa Time Scale: L/u  10 7 s  H/w  10 7 s Radius of Earth: a=6.37x 10 6 m Coriolis parameter: f,f'  10 -4 s -1 Density of Water:   1000 kg m -3 Horizontal Eddy Viscosity: H  10 5 m 2 s -1 Vertical Eddy Viscosity: V  10 -1 m 2 s -1

10. Horizontal Momentum Balance Geostrophic Balance (zonal)(meridional) Length scale: L  10 6 m, l  10 5 m Depth scale: H  10 3 m, h  10 2 m Horizontal velocity scale: u,v  10 -1 ms -1 Vertical velocity scale: w  10 -4 ms -1 Horizontal pressure scale:  p  100 mb = 10 4 Pa Time Scale: L/u  10 7 s  H/w  10 7 s Radius of Earth: a=6.37x 10 6 m Coriolis parameter: f,f'  10 -4 s -1 Density of Water:   1000 kg m -3 Horizontal Eddy Viscosity: H  10 5 m 2 s -1 Vertical Eddy Viscosity: V  10 -1 m 2 s -1 Note that these approximations are only appropriate for “interior solutions” and will break down in boundary layers, where horizontal or vertical shear are large! Or near the equator!!

11. Horizontal Momentum Balance Geostrophic Balance (zonal)(meridional)

12. Horizontal Momentum Balance Geostrophic Balance (zonal)(meridional) Dynamic Topography

13. Horizontal Momentum Balance Geostrophic Balance (zonal)(meridional) Dynamic Topography

14. Horizontal Momentum Balance Geostrophic Balance (zonal)(meridional) Dynamic Topography the dynamic typography is not a simple consequence of the overlying sea level pressure requires an understanding of ocean dynamics and its relation with atmospheric windstress

15. Vertical Momentum Balance: Length scale: L  10 6 m, l  10 5 m Depth scale: H  10 3 m, h  10 2 m Horizontal velocity scale: u,v  10 -1 ms -1 Vertical velocity scale: w  10 -4 ms -1 Horizontal pressure scale:  p  100 mb = 10 4 Pa Time Scale: L/u  10 7 s  H/w  10 7 s Radius of Earth: a=6.37x 10 6 m Coriolis parameter: f,f'  10 -4 s -1 Density of Water:   1000 kg m -3 Horizontal Eddy Viscosity: H  10 5 m 2 s -1 Vertical Eddy Viscosity: V  10 -1 m 2 s -1 10 -11 ms -2 10 -5 ms -2 10 ms -2 10 -11 ms -2

16. But now,  depends on T,S,p As with the atmosphere, we can combine geostrophic and hydrostatic balance to get We can’t go proceed until we develop the equation of state for ocean water… Thermal Wind Balance