1
Vorticity and the
Quasi-Geostrophic Approximation
Pressure Coordinates
It is sometime useful to use the fact the pressure, p, may be written as a function
of the height, z, so that we may use the pressure as a coordinate. Integrating the
hydrostatic equilibrium equation over z yields

p ( z )    gdz
(1)
z
We look at how an isobar varies with a
horizontal line.
p
Z ( p, z )  height of isobar

p = const.
We denote the magnitude of p with
p
,
s
p
cos    g
(2)
s
Z
And the horizontal components are
1 p
(3)

sin    g tan 
 s
tan however, is the slope of the isobaric
Fig 1
 Z Z 
surface  ,  . Therefore, in pressure
 x y 
coordinates, the horizontal component of the pressure gradient is given by
1 p
Z
1 p
Z

 g
; 
 g
(4) 
 x
x
 y
y
We define
Dp

pressure coordinate vertical velocity
(5)
Dt
So that we may write advection terms as
Q
Q
w

(6)
t
p
Applying the chain rule to the hydrostatic equilibrium equation indicates that the
geometric and the pressure coordinate vertical velocities are approximately related by,
p t
   g      gw
(7)
t z
Similarly, the continuity equation may be written as
2
w

 0  v 
0
(8)
z
p
(recall that v  (u, v,0) )
In geometrical coordinates, the lower surface boundary condition is simply
(9)
w  v  h
h being the geometrical height of the surface. In pressure coordinates, however, the
boundary conditions are much more complicated. The pressure at the boundaries
fluctuates and so the boundary moves. It is sometimes enough to use   0 at p=pR.
v 
Vorticity
The relative vorticity is defined as     u. The 'vorticity equation' is the curl of
the momentum (euler's) equation.
1

 u
(10)
  ((u )u)   ()  2 ( u)    p    F
t


Deriving the momentum is somewhat tedious but a worthy exercise. First, the terms
which involve the curl of a grad fall out.
(11)
  0
p
 ( p)  p   p
(12)
 


2
2



so that we're left with

 p
  ((u )u)  2 ( u) 
  F
t
2
next we expand the terms
2  (  u)  2  u         u      u   u(  ) 
 2  u  2 u
xˆ
    u   u  
easily verifiable using
(14)
zˆ


x
y
z
(u )u (u )v (u ) w
we look only at say, the z component
  

 
  

 
    u   u     u  v  w  v   u  v  w  u

 z x x
y
z 
y  x
y
z 

for reference, the z component of  is


  u z   z  v  u
x
y
so that () can be written as
   u   u    u   z   (  u)  z   ( )u  z


z

yˆ
(13)
(15)
(16)
(17)
3
 v u    u w    w v   
   
  w
 
 x y  z  z x  y  y z  x 
 ( )u z  
(18)
and
 v u  u v w 
(19)
   

 x y  x y z 
The vorticity equation is therefore

 p
(20)
 (u ) )  (2   ) u   2      u 
  F
t
2
It so happens, that only the vertical component of the vorticity equation is dominant in
large scale dynamics of the atmosphere. The last statement is equivalent to saying that
circulation in the horizontal plane is far greater than the circulation in the two remaining
vertical plane. However, when we think of VERY large scale circulation, for example a
stream surrounding the earth such that the wind is zonal all around the globe, locally
there is no vorticity because the wind is constant along the x axis. When we say large
scale, we therefore mean on the scale of a few ten's or hundreds of kilometers but not
encompassing the globe.
The vertical component of the vorticity equation is therefore


(21)
 u   ( f   )
(  F) z
t
p
here I use  z   . The second term on the right hand side of (20) is zero by virtue of the
continuity equation. The third term on the right of (20) is zero because  and p vary
mostly in z so that the product   p is essentially zero in the vertical direction.
The term ( f   ) is referred to as the 'absolute vorticity' and f (the coriolis
parameter) as the 'planetary vorticity'.
Physically, ( f   ) is simply twice the angular velocity of an air parcel around a
local vertical axis. Notice that conservation of angular momentum dictates that stretching
a column of air vertically would increase the vorticity and squashing a column of air will
decrease its vorticity. The friction term acts to reduce the relative vorticity. As before, if
we want an exponential decay of the vorticity we may use

(22)
  F z  
D
 (  u)  z  
The Quasi-Geostrophic Approximation
away from the equator, the large scale atmospheric flow is close to a 'geostrophic
balance'. i.e., the dominant terms in the momentum equation are the coriolis and the
pressure gradient terms. The 'Geostrophic velocity field' may be written in terms of the
geopotential height. We use (4) to write
g Z
g Z
(23)
ug  
; vg 
f y
f x
Using the hydrostatic equilibrium equation and the definition of the potential temperature
allows us to write the vertical variations of the geostrophic velocity fields as
4
ug
h( p) 
p
f y
where we define

;
vg
p

h( p) 
(dubious relationship)
f x
(24)
R
R  p  cp
(25)
h( p )  

p  pR 
From equation (24) we see that the gestrophic wind and temperature field are related in
what's called a 'thermal wind balance'. Another way of stating it is that a change in the
temperature field will create a change in the velocity field and vice versa.
In the same manner we may write the vertical change of the geostrophic vorticity as
 g h( p)   2  2 
(26)




p
f  x 2 y 2 
The set of equations (23)-(26) is known as the 'quasi-geostrophic' equation set. It should
be stressed however, that this set is not valid as we approach the equator. This set is most
useful in gaining insight into the dominant dynamical processes in the midlatitude and the
subtropical regions. The quasi-geostrophic set is also of no use in predicting the evolution
of the system because it involves no time dependency.
Momentum And Vorticity Equation Scale Analysis
In order to be able to determine the evolution of the geostrophic fields, we first
examine what are the conditions for geostrophic balance. Suppose that the typical
horizontal wind velocity, U, varies over a typical length scale, L. For the earth's
midlatitudes, U=10m/s and L=106m. The typical value of the advection terms, (u u) in
the momentum equation will therefore be U2/L. The typical value of the coriolis term is
fU. The ratio between the two is called the Rossby Number, R0:
U2
LU
(27)
R0 
fU
fL
Clearly, a necessary condition for geostrophic balance is that the Rossby number be very
small. Similarly, for geostrophic balance to be achieved the friction force must be very
small. We may also write
| | U
(28)

 R0
f
fL
meaning that another condition for geostrophic balance is that the trajectories of fluid
elements be only slightly curved (small relative vorticity). Static balance also implies that
horizontal advection roughly balances vortex stretching. From the vorticity equation we
find
U2
W
W  p 
(29)
 f


 R0
2
L
p
U  L 
where W is the typical vertical velocity. We know that p / L is not greater than unity and
we have found an important property of rotating fluid systems: rapid rotation (i.e. small
5
rosbby number) implies that vertical motion is suppressed. This observation is also
consistent with the assumptions of the Shallow water equations.