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EVAT 554 OCEAN-ATMOSPHERE DYNAMICS FILTERING OF EQUATIONS OF MOTION FOR ATMOSPHERE (CONT) LECTURE 7 (Reference: Peixoto & Oort, Chapter 3,7)

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Presentation on theme: "EVAT 554 OCEAN-ATMOSPHERE DYNAMICS FILTERING OF EQUATIONS OF MOTION FOR ATMOSPHERE (CONT) LECTURE 7 (Reference: Peixoto & Oort, Chapter 3,7)"— Presentation transcript:

1 EVAT 554 OCEAN-ATMOSPHERE DYNAMICS FILTERING OF EQUATIONS OF MOTION FOR ATMOSPHERE (CONT) LECTURE 7 (Reference: Peixoto & Oort, Chapter 3,7)

2 Geostrophic Balance “Geostrophic Wind” d PGF CF V Recall from previous lecture… Exercise: under Boussinesq approximation and assumption f  [constant] show Defines a ‘streamfunction’

3 Winds don’t parallel the streamfunction! under Boussinesq approximation and assumption f  [constant]

4 CONVERGENCE AND DIVERGENCE Northern or Southern Hemisphere? Winds don’t parallel the streamfunction!

5 CONVERGENCE AND DIVERGENCE Northern or Southern Hemisphere? Quasigeostrophic

6 CONVERGENCE AND DIVERGENCE Northern or Southern Hemisphere? Near the surface, friction leads to horizontal convergence

7 CONVERGENCE AND DIVERGENCE Near the surface, friction leads to horizontal convergence Quasigeostrophic

8 CONVERGENCE AND DIVERGENCE Near the surface, friction leads to horizontal convergence Relationship between horizontal convergence and vertical motion Quasigeostrophic

9 Vertical 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 -7 ms -2 10 -3 ms -2 10 ms -2 10 -7 ms -2

10 Vertical 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 Hydrostatic Balance

11 Vertical Momentum Balance: ATMOSPHERIC PRESSURE Hypsometric Equation “Scale height” What’s the solution? where Hydrostatic Balance and Combining these, rearranging,

12 Vertical Momentum Balance: Hypsometric Equation rearranging where and ATMOSPHERIC PRESSURE

13 Vertical Momentum Balance: Hypsometric Equation rearranging where and

14 Vertical Momentum Balance (revisited): 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 -7 ms -2 10 -3 ms -2 10 ms -2 10 -7 ms -2 ?

15 Vertical Momentum Balance (revisited): 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 -7 ms -2 10 -3 ms -2 10 ms -2 10 -7 ms -2 ?

16 Vertical Momentum Balance (revisited): Consider a parcel displaced displaced from hydrostatic equilibrium: (1) (2) (2)-(1) Buoyancy Force

17 Vertical Momentum Balance (revisited): Consider a parcel displaced displaced from hydrostatic equilibrium: Buoyancy Force

18 Vertical Momentum Balance (revisited): Consider a parcel displaced displaced from hydrostatic equilibrium: Buoyancy Force

19 Vertical Momentum Balance (revisited): Now, consider the Thermodynamics: We consider parcel motion with no diffusion of heat and no fluxes of heat across the parcel boundary (Q=0): “Adiabatic” For an ideal gas we can rewrite this: [or “isentropic” (since ds/dt=Q/T)]

20 Vertical Momentum Balance (revisited): Now, consider the Thermodynamics: For an ideal gas we can rewrite this: Potential Temperature Define What is useful about this quantity?  is conserved for adiabatic motion

21 Vertical Momentum Balance (revisited): Now, consider the Thermodynamics: Dry Adiabatic lapse rate for adiabatic motion But Recall Stability Properties? Assume

22 Vertical Momentum Balance (revisited): Now, consider the Thermodynamics: Recall Stability Properties? Assume Exercise: Show Thus: stable neutral unstable

23 Vertical Momentum Balance (revisited): stable neutral unstable

24 Vertical Momentum Balance (revisited): stable neutral unstable


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