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The Boussinesq
and
the Quasi-geostrophic
approximations
References:
1) Andrews “Atmospheric Physics”
Chapter 5 (5.2;5.3)
General form of the fluid
equations

D
   u  0
Dt
Du
1 p
 fv 
 F ( x)
Dt
 x
Dv
1 p
 fu 
 F ( y)
Dt
 y
Dw 1 p

g 0
Dt  z
DT 1 Dp
cp

Q
Dt  Dt
p  RT
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?
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.
•
•
•
•
•
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!
Geostrophic Approximation
•
•
•
Large scale motions.
Coriolis force balances the pressure gradient force
Small Rossby number
Ro 
•
•
•
•
Non linear accelarati on U

 1
Coriolis force
fL
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.
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.

D
 0   u  0
Dt
• Why not


 0    u  0
t
• Note that the density can still vary with x,y,z.
Boussinesq approximation
• In one sentence: We neglect the density
changes in the fluid, except in the gravity
term where  is multiplied by g.
1 D
u v w
 ; ;
 Dt
x y z
1 D
0
 Dt
• This is not quite like:
• This assumption is incorrect in the case of:
• Unsteady flow.
u
• Large Mach number. M  c  0.3
• When the vertical scale of the flow is more than a
scale height. Then the hydrostatic pressure
variations cause significant changes in the density.
Continuity equation
• The continuity equation

D
 0   u  0
Dt
• To a first approximation the atmosphere is
hydrostatic, steady, incompressible, with no
horizontal gradient and static.
p( x, y, z, t )  p ( z )  p' ( x, y, z, t ),
 ( x , y , z , t )   ( z )   ' ( x, y , z , t )

 p   g  0
• The momentum equation
becomes:

 ' Du
1
' 
2
(1  )
  p '
g   u
 0 Dt
0
0
The full set of equations

.u  0
D  g ' 
 
  N B2 w  0
Dt  0 
g d
N 
 dz
2
B
u
Du
1 p'
 fv 
 F ( x)
Dt
 0 x
v
Dv
1 p '
 fu 
 F ( y)
Dt
 0 y
p'
  g '
z
2




 1  2 2
g ' 
    (up' )  0
0 u  v  
t 2 
0 N B  



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).
ua  u  u g , va  v  vg , wa  w
…and we obtain the equations:
Dg u g  f 0 va  yvg  0
Dg vg  f 0ua  yu g  0
ua va wa


0
x y
z
 g ' 
2


Dg  
 N B wa  0

 0 



Dg   u g
 vg
t
x
y
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.
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.
  w v  ˆ  u w  ˆ  v u  ˆ 
    u    i     j    k ,   (0,0,  )
 x z   z x   x y 

a  r  f
…and here how it is done:
y
Dg u g  f 0 va  yvg  0
x
Dg vg  f 0ua  yu g  0
ua va
wa


x y
z
wa
Dg   f 0
z
  f 0  y 
u g
y

vg
x
,
Stretching term
f  f 0  y,   absolute
Stretching a parcel of air
• Stretching the column of air results in
increased vorticity.
• Regions of horizontal convergence are
associated with increased vorticity.