1. The Boussinesq and the Quasi-geostrophic approximations
References:
1) Andrews “Atmospheric Physics” Chapter 5 (5.2;5.3)

2. General form of the fluid equations

3. Why not solve the full equations and be done with it?
Be my guest!
This is a system of non-linear second order differential equations.
Do we need to model the sound waves propagating in the air while we want to understand the Hadley Circulation?
Do we need to know that there is Hadley circulation when I model the sound wave propagation?

4. So, what do we do?
There is nothing better than an analytical solution to a problem!
The next best thing is to make some (educated) assumptions and to apply an approximation that leads to an analytical solution.
Solve the approximated equations numerically.

5. Approximations - a word of warning.
Useful but also dangerous!
We need to know:
What are the assumptions made;
What are they good for;
What are the limitations;
What is the price we pay;
And you need to be consistent throughout your solution.
Even the equations on the first slide have approximations already build in!
All this is especially true when you work on other planets atmospheres!!!
Some particular physics that might be negligible (and correctly so) for the Earth might be of vital importance for other planets!

6. Geostrophic Approximation
Large scale motions.
Coriolis force balances the pressure gradient force
Small Rossby number
It is good for a purely 2D flow.
It is a steady state approximation: no time evolution, it is not good for prognostics.
Not applicable where friction is important (boundary problems).
Breaks down near the equator.

7. The atmosphere as incompressible flow
What does “incompressible” mean?
The density does not change with pressure.
The density is conserved following the material motion of the parcel of air.
Why not
Note that the density can still vary with x,y,z.

8. Boussinesq approximation
In one sentence: We neglect the density changes in the fluid, except in the gravity term where r is multiplied by g.
This is not quite like:
This assumption is incorrect in the case of:
Unsteady flow.
Large Mach number.
When the vertical scale of the flow is more than a scale height. Then the hydrostatic pressure variations cause significant changes in the density.

9. Continuity equation The continuity equation
To a first approximation the atmosphere is hydrostatic, steady, incompressible, with no horizontal gradient and static.
The momentum equation becomes:

10. The full set of equations

11. Quasi-geostrophic approximation
Combination of Boussinesq approximation and the geostrophic approximation.
Substitute the velocity field in the Boussinesq approximation with a geostrophic wind ug plus some small correction, ua (ageostrophic wind).

12. …and we obtain the equations:

13. What is it good for?
Note that the system of equations is linear with respect to the ageostrophic wind.
For large scale motions.
For low frequency motions (a few hours).
It is a prognostic approximation, gives the time evolution of the flow.
Ro<<1
Not good for the tropics.

14. Vorticity equation General approach:
Use the momentum equations and take the corresponding partial derivatives to form the curl of the velocity field on the LHS and whatever is the result goes on the RHS.
And … you have a vorticity equation.

15. …and here how it is done:
Stretching term

16. Stretching a parcel of air
Stretching the column of air results in increased vorticity.
Regions of horizontal convergence are associated with increased vorticity.