The simplest theoretical basis for understanding the location of
significant vertical motions in an Eulerian framework is
QUASI-GEOSTROPHIC THEORY
QG Theory: Developed in late 1940s (before the age of computers) to help meteorologists
diagnose the future state of the atmosphere.
Addressed the fundamental problem that divergence could not be calculated
with the standard rawinsonde network
Is the theoretical framework for synoptic-scale dynamics taught to generations
of meteorologists
We will look at standard development of the QG equations retaining diabatic
and friction terms
We will look at Trenberth (1978) simplifications to QG equations
We will study in detail the Q vector form of Q-G Theory
We will first derive the vorticity equation
du


 fv  Fx
dt
x
dv


 fu  Fy
dt
y
(1)
(2)
Expand total derivative
u
u
u
u

u
 v 

 fv  Fx
t
x
y
P
x
Take
  v u 
  v u
    u   
t  x y 
x  x y
v
v
v
v

u  v 

 fu  Fx
t
x
y
P
y
(2) (1)

x
y

  v u
f v   
y  x y


  v u 
f  
 

P  x y 

 v u
 u v   u  v    Fx Fy 
    f     

   x  y 

x

y

x

y

P

y

P

x


 
 

write relative vorticity
v u

x y
as 
 u v   u  v    Fx Fy 




 u   f   v   f   
   f     





t
x
y
P

x

y

P

y

P

x

x

y

 
 

The vorticity equation
 u v   u  v    Fx Fy 




 u   f   v   f   
   f     





t
x
y
P

x

y

P

y

P

x

x

y

 
 

Local rate of
change of relative
vorticity
Horizontal advection
of absolute vorticity
on a pressure surface
Divergence acting on
Absolute vorticity
(twirling skater effect)
Tilting of vertically
sheared flow
Gradients in force
Of friction
Vertical advection
of relative vorticity
In English: Horizontal relative vorticity is increased at a point if
1) positive vorticity is advected to the point along the pressure surface,
2) or advected vertically to the point,
3) if air rotating about the point undergoes convergence (like a skater twirling up),
4) if vertically sheared wind is tilted into the horizontal due a gradient in vertical motion
5) if the force of friction varies in the horizontal.
Next we will derive the “Quasi-Geostrophic” Vorticity Equation
Start with vorticity equation

 u v   u  v    Fx Fy 
 



  u   f   v   f   
   f     





t  x
y

P

x

y

P

y

P

x

x

y


 
 

1
2
3
4
5
6
1.
2.
3.
4.
5.
6.
Based on scale analysis, we will ignore 3, 5, 6 and  compared to f in 4
Assume relative vorticity in 1, 2 can be replaced by its geostrophic value g
Replace divergence in 4 using continuity equation
Assume that the velocity (u,v) in 2 can be replaced by its geostrophic value
Assume Coriolis force varies linearly across mid-latitudes (f = f0 + y)
Ignore y where f is not differentiated.
 g
t
1
 V g   g  f   f 0
2
4

P
Now derive the “Quasi-Geostrophic” Thermodynamic Equation
Start with conservation of energy equation
T  T
T 
T  1 dQ
  u
 v  

t 
x
y 
 p c p dt
1.
2.
1
2
Ignore 4, diabatic heating
p 
T


Use Hydrostatic equation
3
4
3.
Move outside derivatives (P is held constant in derivatives in P coordinate system)
4.
Multiple equation by p
to replace T in 1 and 2.
R p
p
R
R
  
    
TR 
    V g     
t  p  
p p
 p 
5. Write TR/P as the specific volume , and then write 
 
 , the static stability.
 p
  
    
    V g     
t  p  
 p 
1
2
3
 g
t
 V g   g  f   f 0

p
Q-G vorticity equation
  
    
    V g     
t  p  
 p 
Q-G thermodynamic equation
Now we will use the geostrophic wind relationships u g  
vg 
1 
to write g in terms of 
f 0 x
vg
u g
1 
f 0 y
and
  1     1   1   2  2  1 2
  
   2  2    
g 

 
x
y x  f 0 x  y  f 0 y  f 0  x
y  f 0
g 
1 2

f0
1  2
1

    V g    2  f  f 0
f 0 t
f0
p


    
  
    V g     
t  P  
 P 
Q-G vorticity equation
Q-G thermodynamic equation
We now have two equations in two unknowns,  and 
We will solve these to find an equation for , the vertical motion in pressure
coordinates, and for

, the change of geopotential height with time.
t
Derive the Q-G omega equation
(equation for vertical velocity in pressure coordinates)
1  2
1

2
    V g      f  f0
f 0 t
f0
p
1
    
  
    V g     
t  p  
 P 
2

1.
2.
3.
3.
4.
5.

Assume  is constant
1 2
Take f  Eqn. 2
0

Take
of Eq. 1.
p
Reverse order of differentiation on left side of equation
Subtract: (1) – (2)
Rearrange terms
1 2
 2 f 02  2 
f0  
  


V g     
2 
 p 
 p 

 f0
 1 2 
 

f      V g   
p 

 
The QG omega equation can be derived including the friction term and the diabatic
heating term. We will not do this here, but I will simply show the result.
QG OMEGA EQUATION
1 2
 2 f 02  2 
f0  
  


V g     
2 
 p 
 p 

 f0

 1 2 
 

f      V g   
p 

 
 1 dQ 
f0 
R

 K g   2 
 p
p  CP dt 


BEFORE WE PROCEED TO TRY TO UNDERSTAND THIS EQUATIONS
ANSWER THIS QUESTION:
WHAT PHENOMENA OF
IMPORTANCE ARE WE IMPROPERLY
REPRESENTING IN THE DERIVATION
OF THESE EQUATIONS?
By assuming that the static stability, 
 
  is constant
 p
WE HAVE ELIMINATED A TERM RELATED TO
FRONTS!
We took x, y, and P derivatives assuming  is constant
Real Atmosphere ()
QG Atmosphere ()
Where does all the precipitation occur?
Where does all the precipitation occur?
ALONG FRONTS!
By assuming that the static stability, 
 
  is constant
 p
WE HAVE ELIMINATED A TERM RELATED TO
JETSTREAKS!
We took x, y, and P derivatives assuming  is constant – this implies that the
vertical wind shear is constant, which can’t happen in a jetstreak environment
To illustrate quasi-geostrophic vertical motion in a real system
consider the cyclone below
700 mb height and rainwater field simulated by MM5 model
Vertical motion calculated at 700 mb by MM5 Model
Vertical motion attributed to terms in QG Omega equation
Residual vertical motion (not attributable to QG forcing)
Quasi-Geostrophic interpretation of the atmosphere:
What the quasi-geostrophic solutions do:
Quasi-Geostrophic Equations describe the broad scale ascent and descent
in troughs and ridges and the propagation of these troughs and ridges.
What the quasi-geostrophic solutions do not do:
Quasi-Geostrophic Equations do not describe the mesoscale ascent and descent
associated with ageostrophic circulation near fronts and within jetstreaks,
nor the propagation of low and high pressure systems due to jetstreaks.
(These are the motions are most responsible for significant weather)
So what does the QG system  equation tell us about the atmosphere
and how should we use it to diagnose vertical motions?
QG OMEGA EQUATION
1 2
 2 f 02  2 
f0  
  
 
V g     
2 
 p 
 p 

 f0
 1 2 
 
f      V g   
p 

 
f0 
R 2  1 dQ 


 K g   
 p
p  cP dt 


We are taking two derivatives of the vertical motion 

Let’s write  as a Fourier series in x:
   an cosnx  bn sin nx
n 1
When we take two x derivatives of this series we get

    an n 2 cosnx  bn n 2 sin nx
2
n 1
Therefore: First term is proportional to - 
Read the first term as “Rising motion”
QG OMEGA EQUATION
1 2
 2 f 02  2 
f0  
  
 
V g     
2 
 p 
 p 

 f0
 1 2 
 
f      V g  



P



f0 
R 2  1 dQ 


 K g   
 p
p  cP dt 


The second term is the vertical derivative of the absolute vorticity advection:
Rising motion is proportional to
“Differential Positive Vorticity Advection”
1 2
f0  
   
V


 g
 p 
 f0

f 

The geostrophic wind is strong at
jetstream level and the height
gradients are greatest.
Vertical gradient in vorticity
advection is strongest, on
average, in middle troposphere
(so absolute vorticity is typically
Plotted on the 500 mb surface)
The geostrophic wind is weak at
low levels and the height
gradients are weak.
Relationship between
Shear and curvature
in the Jetstream
Absolute vorticity
and Abs. Vorticity Advection
and
Vertical Motion
1 2
f0  
V g     
 p 
 f0

f 

QG OMEGA EQUATION
1 2
 2 f 02  2 
f0  
  
 
V g     
2 
 p 
 p 

 f0
 1 2 
 
f      V g  



P



f0 
R 2  1 dQ 


 K g   
 p
p  cP dt 



is the depth, or thickness, of the layer between two pressure surfaces. This depth
p
is proportional to the mean temperature of the layer.

We are taking two derivatives of the thickness advection,   V g  

Let’s write A as a Fourier series in x:

 
  A
p 
A   an cosnx  bn sin nx
n 1
When we take two derivatives of this series we get

 A   an n 2 cosnx  bn n 2 sin nx
2
n 1
Multiply term by -1 twice to get in form consistent with temperature advection

 
  
  
2
 Vg        Vg    
 p 
 p 


2
Read the term inside the parentheses as
temperature advection
However, it is the Laplacian of temperature advection we are interested in…
Let’s rewrite this term:
 
 
  
  
   Vg          Vg    
 p 
 p 


2
If the gradient of temperature advection
is constant, then the term is zero!
THEREFORE:  will be non-zero only in areas where
the thermal advection pattern in non uniform
Heterogeneity in warm advection on a pressure surface implies upward motion
When air is ascending on an isentropic surface…
…the projection of the wind on to a constant pressure surface
appears as warm air flowing toward cold air (warm advection)
But what if isentropic surface is moving northward at the same time?
Warm advection involves both flow on the isentropic surface and movement of the surface
Heterogeneity in cold advection on a pressure surface implies downward motion
When air is descending on an isentropic surface…
…the projection of the wind on to a constant pressure surface
appears as cold air flowing toward warm air (cold advection)
But what if isentropic surface is moving southward at the same time?
Cold advection involves both flow on the isentropic surface and movement of the surface
QG OMEGA EQUATION
1 2
 2 f 02  2 
f0  
  
 
V g     
2 
 p 
 p 

 f0
 1 2 
 
f      V g   
p 

 
f0 
R 2  1 dQ 


 K g   
 p
p  cP dt 


The vertical derivative of friction
acting on the geostrophic vorticity
Or more simply: The rate that friction
decreases with height in the
presence of cyclonic vorticity in the
boundary layer
QG OMEGA EQUATION
1 2
 2 f 02  2 
f0  
  
 
V g     
2 
 p 
 p 

 f0
 1 2 
 
f      V g   
p 

 
f0 
R 2  1 dQ 


 K g   
 p
p  cP dt 


 1 dQ 

 cP dt 
Again: Two derivatives in x,y ……….proportional to negative 
With additional minus sign:
Rising motion is proportional to the Laplacian of the
diabatic heating rate (modulated by static stability)
Sinking motion is proportional to the Laplacian of the
diabatic cooling rate (modulated by static stability)
Heterogeneity in the diabatic heating leads to rising motion
From our earlier discussion of isentropic coordinates:
135
   75K
  
 0.65 K / mb

p
115
mb
 
  

 p 
high static stability 
250
large
335
650
  

 p 
low static stability 
small
  
20K
  
 0.06 K / m b

p
315
m
b
 
From our earlier discussion of isentropic coordinates:
 d 


dt


  
 
 p 
For a given amount of diabatic heating, a
parcel in a layer with high static stability
Will have a smaller vertical displacement
than a parcel in a layer with low static
stability
QG OMEGA EQUATION
1 2
 2 f 02  2 
f0  
  
 
V g     
2 
 p 
 p 

 f0
 1 2 
 
f      V g   
p 

 
f0 
R 2  1 dQ 


 K g   
 p
p  cP dt 


SUMMARY
Broad scale (synoptic scale) rising motion in the atmosphere is proportional to:
Differential positive vorticity advection
Heterogeneity in the warm advection field
The rate of decrease with height of friction in the presence of vortex
Heterogeneity in the diabatic heating rate
Broad scale (synoptic scale) desending motion in the atmosphere is proportional to:
Differential negative vorticity advection
Heterogeneity in the cold advection field
The rate of decrease with height of friction in the presence of vortex
Heterogeneity in the diabatic cooling rate
Let’s compare the QG Omega Equation* with the
Omega equation we derived a long time ago
when we considered isentropic coordinates
Recall for isentropic coordinates:
dp

dt

p d
 p 
     V   p 
 dt
 t 
Vertical
Motion
1 2
 2 f 02  2 
f0  
   
  



V


 g
2 
 p 
 p 

 f0
Temperature
Advection
 1 
 
f    2   V g   
p 

 
Diabatic term

R 2  1 dQ 

 
P  C P dt 
*Note… I dropped friction term since we did not consider it in our discussion of isentropic coordinates
Let’s compare the QG Omega Equation* with the
Omega equation we derived a long time ago
when we considered isentropic coordinates
Recall for isentropic coordinates:
dp

dt

p d
 p 
     V   p 
 dt
 t 
Vertical
Motion
Must be
Related!
1 2
 2 f 02  2 
f0  
   
  



V


 g
2 
 p 
 p 

 f0
Temperature
Advection
 1 
 
f    2   V g   
p 

 
Diabatic term

R 2  1 dQ 

 
P  C P dt 
*Note… I dropped friction term since we did not consider it in our discussion of isentropic coordinates
850 mb heights, temperature, winds
TEMPERATURE ADVECTION TERM
Warm advection
Cold advection
Pressure, winds and RH on 290 K isentropic surface
TEMPERATURE ADVECTION TERM
Ascent on
isentropic surface
Descent on isentropic surface

p d
 p 
     V   p 
 dt
 t 
1 2
 2 f 02  2 
f0  
   
  



V


g

2 
 p 
 p 

 f0
 1 2 
 
f      V g   
p 

 
So how are orange terms related???

R 2  1 dQ 

 
P  C P dt 
Vertical displacement of isentrope =
 P 


 t 
(Air must rise for isentrope to be displaced upward)
Differential absolute vorticity advection measures the rate at which
potential temperature surfaces rise or fall as ridges and troughs propagate along!
500 mb w
700 mb w