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

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

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

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

CONVERGENCE AND DIVERGENCE Northern or Southern Hemisphere? Quasigeostrophic

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

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

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

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  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'  s -1 Density of Air:   1 kg m -3 Horizontal Eddy Viscosity: H  m 2 s -1 Vertical Eddy Viscosity: V  m 2 s ms ms ms ms -2

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  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'  s -1 Density of Air:   1 kg m -3 Horizontal Eddy Viscosity: H  m 2 s -1 Vertical Eddy Viscosity: V  m 2 s -1 Hydrostatic Balance

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

Vertical Momentum Balance: Hypsometric Equation rearranging where and ATMOSPHERIC PRESSURE

Vertical Momentum Balance: Hypsometric Equation rearranging where and

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  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'  s -1 Density of Air:   1 kg m -3 Horizontal Eddy Viscosity: H  m 2 s -1 Vertical Eddy Viscosity: V  m 2 s ms ms ms ms -2 ?

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  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'  s -1 Density of Air:   1 kg m -3 Horizontal Eddy Viscosity: H  m 2 s -1 Vertical Eddy Viscosity: V  m 2 s ms ms ms ms -2 ?

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

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

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

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)]

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

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

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

Vertical Momentum Balance (revisited): stable neutral unstable

Vertical Momentum Balance (revisited): stable neutral unstable