EVAT 554 OCEAN-ATMOSPHERE DYNAMICS LECTURE 5 EQUATION OF STATE; FILTERING OF EQUATIONS FOR ATMOSPHERE (Reference: Peixoto & Oort, Chapter 3)

We still do not have a closed system of equations! equation of state... Let us first consider the atmosphere…

Recall Ideal Gas Law: For Dry Air: Define Thus, we can write:

What about water vapor? Dalton’s Law of Partial Pressures: For water vapor we can write: Where Take Thus, we can write:

What about water vapor? Dalton’s Law of Partial Pressures: For water vapor we can write: Where: Take Now define:

What about water vapor? Take Now define:

What about water vapor?

What about water vapor? This equation of state has general validity for the atmosphere, for the current atmospheric composition

Governing Equations for Atmosphere Zonal Momentum Balance Meridional Momentum Balance Vertical Momentum Balance Continuity Equation of State Temperature Equation How many equations? How many variables?

The Earth's Atmosphere from Space The thin blue arc near the horizon shows the shallowness of the earth's atmosphere.

Zonal Momentum Balance: Length scale: L106m, l102m Depth scale: H104m, h 102m 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 105s or H/w 106s Radius of Earth: a=6.37x 106m Coriolis parameter: f,f' 10-4 s-1 Density of Air: r 1 kg m-3 Horizontal Eddy Viscosity: nH 10-1 m2s-1 Vertical Eddy Viscosity: nV 10-1 m2s-1 10-4 ms-2 10-3 ms-2 10-6 ms-2 10-3 ms-2 10-4 ms-2 10-4 ms-2

Meridional Momentum Balance: Length scale: L106m, l102m Depth scale: H104m, h 102m 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 105s or H/w 106s Radius of Earth: a=6.37x 106m Coriolis parameter: f,f' 10-4 s-1 Density of Air: r 1 kg m-3 Horizontal Eddy Viscosity: nH 10-1 m2s-1 Vertical Eddy Viscosity: nV 10-1 m2s-1 10-4 ms-2 10-3 ms-2 10-3 ms-2 10-4 ms-2 10-4 ms-2