1. EVAT 554 OCEAN-ATMOSPHERE DYNAMICS FILTERING OF EQUATIONS FOR ATMOSPHERE (CONT) LECTURE 6 (Reference: Peixoto & Oort, Chapter 3)

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

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

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

5. Horizontal Momentum Balance Geostrophic Balance (zonal)(meridional) “Geostrophic Wind” d PGF CF V

6. Horizontal Momentum Balance “Geostrophic Wind” d PGF CF V f=2  sin     7.27x10 -5 s -1 d=600 km =5.6 m/s Example:

7. Let us Revisit e.g. the Meridional Momentum Balance: Length scale: L  10 6 m, l  10 2 m Depth scale: H  10 4 m, h  10 2 m Horizontal velocity scale: u,v  10 ms -1 Vertical velocity scale: w  10 -2 ms -1 Horizontal pressure scale:  p  10 mb = 1000 Pa Time Scale: L/u  10 5 s or H/w  10 6 s Radius of Earth: a=6.37x 10 6 m Coriolis parameter: f,f'  10 -4 s -1 Density of Air:   1 kg m -3 Horizontal Eddy Viscosity: H  10 -1 m 2 s -1 Vertical Eddy Viscosity: V  10 -1 m 2 s -1 10 -4 ms -2 10 -3 ms -2 10 -4 ms -2 What if the acceleration/non-linear term cannot be neglected? i.e., Ro  1

8. Let us Revisit e.g. the Meridional Momentum Balance: Length scale: L  10 6 m, l  10 2 m Depth scale: H  10 4 m, h  10 2 m Horizontal velocity scale: u,v  10 ms -1 Vertical velocity scale: w  10 -2 ms -1 Horizontal pressure scale:  p  10 mb = 1000 Pa Time Scale: L/u  10 5 s or H/w  10 6 s Radius of Earth: a=6.37x 10 6 m Coriolis parameter: f,f'  10 -4 s -1 Density of Air:   1 kg m -3 Horizontal Eddy Viscosity: H  10 -1 m 2 s -1 Vertical Eddy Viscosity: V  10 -1 m 2 s -1 10 -4 ms -2 10 -3 ms -2 10 -4 ms -2 This applies to flows with strong curvature What if the acceleration/non-linear term cannot be neglected? i.e., Ro  1

9. Horizontal Momentum Balance: This applies to flows with strong curvature “Gradient Wind Balance” Centripetal acceleration (zonal) (meridional) a c =V 2 /R R V=R  V=(u 2 + v 2 ) 1/2

10. Horizontal Momentum Balance: This applies to flows with strong curvature “Gradient Wind Balance” Centripetal acceleration (zonal) (meridional) a c =V 2 /R R V=R  V=(u 2 + v 2 ) 1/2

11. Horizontal Momentum Balance: What about flow near the equator? “Gradient Wind Balance” Centripetal acceleration (zonal) (meridional) a c =V 2 /R R V=R  V=(u 2 + v 2 ) 1/2

12. Horizontal Momentum Balance: “Cyclostrophic Balance” What about flow near the equator? a c =V 2 /R V=R  V=(u 2 + v 2 ) 1/2 Centripetal acceleration (zonal) (meridional) R Near equator (e.g. Hurricane), Coriolis Force is negligible, and balance is between PGF and Centripetal acceleration

13. Horizontal Momentum Balance Geostrophic Balance (zonal)(meridional) Generally an excellent approximation for ‘upper level winds’ Any evidence of breakdown of Geostrophy?

14. Horizontal Momentum Balance Geostrophic Balance (zonal)(meridional) Relationship Between Temperature and Winds? Advection

15. Horizontal Momentum Balance Geostrophic Balance (zonal)(meridional) Relationship Between Temperature and Winds? Advection

16. Horizontal Momentum Balance Geostrophic Balance (zonal)(meridional) What can we say about this term?