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EVAT 554 OCEAN-ATMOSPHERE DYNAMICS

EVAT 554 OCEAN-ATMOSPHERE DYNAMICS. FILTERING OF EQUATIONS FOR OCEAN. LECTURE 10. (Reference: Peixoto & Oort, Chapter 3,8). Note that the major horizontal ocean circulation systems mirror closely the semi-permanent high and low pressure systems. Scale Analysis. The Ocean.

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EVAT 554 OCEAN-ATMOSPHERE DYNAMICS

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  1. EVAT 554OCEAN-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: How many equations? Vertical Momentum Balance: How many variables? Continuity: (incompressible!) Equation of State: Heat Equation:

  5. Zonal Momentum Balance: Length scale: L106m, l105m Depth scale: H103m, h 102m Horizontal velocity scale: u,v 10-1 ms-1 Vertical velocity scale: w 10-4 ms-1 Horizontal pressure scale: p 100 mb = 104Pa Time Scale: L/u 107s H/w 107s Radius of Earth: a=6.37x 106m Coriolis parameter: f,f' 10-4 s-1 Density of Water: r 1000 kg m-3 Horizontal Eddy Viscosity: nH 105 m2s-1 Vertical Eddy Viscosity: nV 10-1 m2s-1 10-8 ms-2 10-5 ms-2 10-8 ms-2 10-5 ms-2 10-6 ms-2 10-6 ms-2

  6. Meridional Momentum Balance: Length scale: L106m, l105m Depth scale: H103m, h 102m Horizontal velocity scale: u,v 10-1 ms-1 Vertical velocity scale: w 10-4 ms-1 Horizontal pressure scale: p 100 mb = 104Pa Time Scale: L/u 107s H/w 107s Radius of Earth: a=6.37x 106m Coriolis parameter: f,f' 10-4 s-1 Density of Water: r 1000 kg m-3 Horizontal Eddy Viscosity: nH 105 m2s-1 Vertical Eddy Viscosity: nV 10-1 m2s-1 10-8 ms-2 10-5 ms-2 10-5 ms-2 10-6 ms-2 10-6 ms-2

  7. Horizontal Momentum Balance “Rossby Number” Geostrophic Balance Holds when Ro << 1 (zonal) (meridional) Geostrophic Balance Length scale: L106m, l105m Depth scale: H103m, h 102m Horizontal velocity scale: u,v 10-1 ms-1 Vertical velocity scale: w 10-4 ms-1 Horizontal pressure scale: p 100 mb = 104Pa Time Scale: L/u 107s H/w 107s Radius of Earth: a=6.37x 106m Coriolis parameter: f,f' 10-4 s-1 Density of Water: r 1000 kg m-3 Horizontal Eddy Viscosity: nH 105 m2s-1 Vertical Eddy Viscosity: nV 10-1 m2s-1 10-8 ms-2 10-5 ms-2 10-5 ms-2 10-6 ms-2 10-6 ms-2

  8. Horizontal Momentum Balance (zonal) (meridional) Geostrophic Balance Length scale: L106m, l105m Depth scale: H103m, h 102m Horizontal velocity scale: u,v 10-1 ms-1 Vertical velocity scale: w 10-4 ms-1 Horizontal pressure scale: p 100 mb = 104Pa Time Scale: L/u 107s H/w 107s Radius of Earth: a=6.37x 106m Coriolis parameter: f,f' 10-4 s-1 Density of Water: r 1000 kg m-3 Horizontal Eddy Viscosity: nH 105 m2s-1 Vertical Eddy Viscosity: nV 10-1 m2s-1 10-8 ms-2 10-5 ms-2 10-5 ms-2 10-6 ms-2 10-6 ms-2 “Ekman Number” Geostrophic Balance Holds when Ek << 1

  9. Horizontal Momentum Balance (zonal) (meridional) Geostrophic Balance Length scale: L106m, l105m Depth scale: H103m, h 102m Horizontal velocity scale: u,v 10-1 ms-1 Vertical velocity scale: w 10-4 ms-1 Horizontal pressure scale: p 100 mb = 104Pa Time Scale: L/u 107s H/w 107s Radius of Earth: a=6.37x 106m Coriolis parameter: f,f' 10-4 s-1 Density of Water: r 1000 kg m-3 Horizontal Eddy Viscosity: nH 105 m2s-1 Vertical Eddy Viscosity: nV 10-1 m2s-1 10-8 ms-2 10-5 ms-2 10-5 ms-2 10-6 ms-2 10-6 ms-2 “Ekman Number” Geostrophic Balance Holds when Ek << 1

  10. Horizontal Momentum Balance (zonal) (meridional) Geostrophic Balance Length scale: L106m, l105m Depth scale: H103m, h 102m Horizontal velocity scale: u,v 10-1 ms-1 Vertical velocity scale: w 10-4 ms-1 Horizontal pressure scale: p 100 mb = 104Pa Time Scale: L/u 107s H/w 107s Radius of Earth: a=6.37x 106m Coriolis parameter: f,f' 10-4 s-1 Density of Water: r 1000 kg m-3 Horizontal Eddy Viscosity: nH 105 m2s-1 Vertical Eddy Viscosity: nV 10-1 m2s-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 (zonal) (meridional) Geostrophic Balance

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

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

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

  15. Vertical Momentum Balance: Length scale: L106m, l105m Depth scale: H103m, h 102m Horizontal velocity scale: u,v 10-1 ms-1 Vertical velocity scale: w 10-4 ms-1 Horizontal pressure scale: p 100 mb = 104Pa Time Scale: L/u 107s H/w 107s Radius of Earth: a=6.37x 106m Coriolis parameter: f,f' 10-4 s-1 Density of Water: r 1000 kg m-3 Horizontal Eddy Viscosity: nH 105 m2s-1 Vertical Eddy Viscosity: nV 10-1 m2s-1 10-11 ms-2 10-11 ms-2 10-5 ms-2 10 ms-2 10 ms-2 10-11 ms-2

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

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