Wave-cyclones from the Q-G perspective
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 height tendency equation
(equation for height changes of a pressure surface)
1  2
1

    V g   2  f   f 0
f 0 t
f0
p
1
  
    
    V g     
t  p  
 p 
2
1.
2.
Assume  is constant
Change order of differentiation on left side of (2)
3.
4.
4.
Multiply (2) by 0 and (1) by f0

Differentiate (2) with respect to p
Add to resultant equation to (1)
f

1 2
 2 f 02  2  
  

  f 0 V g     
2 
 p  t

 f0

 f 02  
 
  V g   
f  
p 
  p 
Both the QG omega equation and the QG height tendency 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 


QG HEIGHT TENDENCY EQUATION

1 2
 2 f 02  2  
  

  f 0 V g     
2 
 p  t

 f0

 f 02  
 
f  

V


g




p

P



f 02   R 1 dQ 


 f 0  K g 
 p  p cP dt 


QG HEIGHT TENDENCY EQUATION

1 2
 2 f 02  2  
  

  f 0 V g     
2 
 P  t

 f0

 f 02  
 
  V g   
f  
p 
  p 
f 02   R 1 dQ 


 f 0  K g 
 p  P cP dt 


First term is proportional to


t
First term represents the rate at which geopotential height decreases with time
First term represents the rate at which geopotential height decreases with time
500 mb maps -- 24 hours between panels
trough propagating
trough deepening
First term represents the rate at which geopotential height decreases with time
500 mb maps -- 24 hours between panels
trough propagating

0
t

0
t

0
t
trough deepening

0
t
QG HEIGHT TENDENCY EQUATION

1 2
 2 f 02  2  
  

  f 0 V g     
2 
 p  t

 f0

 f 02  
 
  V g   
f  
p 
  p 
f 02   R 1 dQ 


 f 0  K g 
 p  p cP dt 


The first term on the right side of the equation describes the
propagation of the height field.
Propagation of the height field depends on :
the advection of the relative vorticity
(spin imparted by shear and curvature)
1 2

f0
and the advection of the planetary vorticity
(spin imparted by the earth rotation)
f
A “battle” between advection of planetary vorticity and
relative vorticity occurs in ridge-trough systems.
High Latitudes
f
Height rises will occur
due to the advection
of relative vorticity
Height falls will occur
due to the advection
of planetary vorticity
Height falls will occur
due to the advection
of relative vorticity
1 2

f0
Low Latitudes
Height rises will occur
due to the advection
of planetary vorticity
Which process will win the battle?
Short waves in flow
f doesn’t change much –
little deviation in latitude
Due to strong shear and sharp curvature change
Relative vorticity changes substantially from ridge to trough
Advection of absolute vorticity is dominated by advection of relative vorticity:
The troughs and ridges will propagate rapidly eastward
Long waves in flow
f changes significantly –
large deviation in latitude
weak shear and wide curvature:
relative vorticity changes small
from ridge to trough
As a result, advection of absolute vorticity is nearly equal to zero
Long waves are stationary, or propagate
very slowly eastward (g > f) or westward (g < f)
00 Hours
+12 Hours
Note the speed of propagation of the waves on these maps: the shorter the
wavelength becomes, the faster the wave propagates
+24 Hours
+36 Hours
QG HEIGHT TENDENCY EQUATION

1 2
 2 f 02  2  
  

  f 0 V g     
2 
 p  t

 f0

 f 02  
 
f  
 V g  

P 
  p 
f 02   R 1 dQ 


 f 0  K g 
 p  p cP dt 


We will examine these two terms together:
The vertical derivative of thickness advection
(Differential thickness advection)
The vertical derivative of diabatic heating
(Differential diabatic heating)
Differential thickness (mean layer temperature) advection
or Diabatic Heating
Za
Za
Za
A
A
A
Z
Z
Z
B
Zb
B
B
Zb
Suppose we have a
layer of air bounded
by height Za and Zb
with level Z in the
middle.
Zb
Warm advection or
diabatic heating in A
causes layer to expand
Cold advection or
diabatic cooling in A
causes layer to contract
Cold advection or
diabatic heating in B
causes layer to contract
Warm advection or
diabatic heating in B
causes layer to expand
Height Z falls to lower
altitude
Height Z rises to higher
altitude
In the real atmosphere
warm and cold advection and diabatic heating and cooling
decrease with height
and are always strongest in the lower atmosphere
850 mb 2 Mar 03 00 UTC
700 mb 2 Mar 03 00 UTC
500 mb 2 Mar 03 00 UTC
Red circles: Strong warm advection pattern at 850, weaker at 700, very weak at 500 mb
Blue circles: Strong cold advection pattern at 850, weaker at 700, very weak at 500 mb
QG HEIGHT TENDENCY EQUATION

1 2
 2 f 02  2  
  

  f 0 V g     
2 
 p  t

 f0

 f 02  
 
  V g   
f  
p 
  p 
f 02   R 1 dQ 


 f 0  K g 
 p  p cP dt 


Summary
Cold advection in the lower atmosphere will produce height falls
amplifying trough aloft
Warm advection in the lower atmosphere will produce height rises
amplifying ridge aloft
QG HEIGHT TENDENCY EQUATION

1 2
 2 f 02  2  
  

  f 0 V g     
2 
 p  t

 f0

 f 02  
 
  V g   
f  
p 
  p 
f 02   R 1 dQ 


 f 0  K g 
 P  P CP dt 


Surface cyclone (term outlined < 0)
Convergence, ascending air and height rises
Surface anticyclone (term outlined > 0)
Divergence, descending air and height falls