### QG Analysis

```QG Analysis: System Evolution
M. D. Eastin
QG Analysis
QG Theory
• Basic Idea
• Approximations and Validity
• QG Equations / Reference
QG Analysis
• Basic Idea
• Estimating Vertical Motion
• QG Omega Equation: Basic Form
• QG Omega Equation: Relation to Jet Streaks
• QG Omega Equation: Q-vector Form
• Estimating System Evolution
• QG Height Tendency Equation
• Diabatic and Orographic Processes
• Evolution of Low-level Cyclones
• Evolution of Upper-level Troughs
M. D. Eastin
QG Analysis: Basic Idea
Forecast Needs:
• The public desires information regarding temperature, humidity, precipitation,
and wind speed and direction up to 7 days in advance across the entire country
• Such information is largely a function of the evolving synoptic weather patterns
(i.e., surface pressure systems, fronts, and jet streams)
Forecast Method:
Kinematic Approach: Analyze current observations of wind, temperature, and moisture fields
Assume clouds and precipitation occur when there is upward motion
and an adequate supply of moisture
QG theory
QG Analysis:
• Vertical Motion:
Diagnose synoptic-scale vertical motion from the observed
distributions of differential geostrophic vorticity advection
• System Evolution: Predict changes in the local geopotential height patterns from
the observed distributions of geostrophic vorticity advection
M. D. Eastin
QG Analysis: A Closed System of Equations
Recall: Two Prognostic Equations –Two Unknowns:
• We defined geopotential height tendency (X) and then expressed geostrophic vorticity (ζg)
and temperature (T) in terms of the height tendency.


t
 g
t
T
t
 g
t

 1 2 
    
t  f 0


  p  
p 
 
  
t  R p 
R p
 Vg  ( g  f )  f 0

p
T
p
 Vg  T  
t
R
1 2
 
f0
g 
1 2

f0
T 
p 
R p
1
1 2
   Vg     2 
fo
 fo



f   f0
p

 p  
p 
p
 Vg  



R p
R
 R p 
M. D. Eastin
QG Analysis: System Evolution
The QG Height Tendency Equation:
 We can also derive a single prognostic equation for X by combining our modified
vorticity and thermodynamic equations (the height-tendency versions):
1
1 2
   Vg     2 
fo
 fo


f   f0
p

 p  
p 
p

 Vg  



R p
R

p
R


Vorticity
Equation
Thermodynamic
Equation
 To do this, we need to eliminate the vertical motion (ω) from both equations
Step 1:
2
 
f
Apply the operator  0   R  to the thermodynamic equation
 p  p 
Step 2:
Multiply the vorticity equation by f 0
Step 3:
Add the results of Steps 1 and 2
After a lot of math, we get the resulting prognostic equation……
M. D. Eastin
QG Analysis: System Evolution
The QG Height Tendency Equation:
 2 f 02  2 
  

2 
 p 

 1
  f o Vg     2 
 fo

  
  f o2
 

f   

V



g
p  
 p 

• This is (2.32) in the Lackmann text
• This form of the equation is not very intuitive since we transformed geostrophic
vorticity and temperature into terms of geopotential height.
• To make this equation more intuitive, let’s transform them back…
g 
 2 f 02  2 
  

2 
 p 



1 2

f0
T 

f o  Vg    g  f 
p 
R p

  f o2 R


 

V


T
g


p   p

M. D. Eastin
QG Analysis: System Evolution
The BASIC QG Height Tendency Equation:
 2 f 02  2 
  

2 
 p 

Term A



f o  Vg    g  f 

  f o2 R


 

V


T
g


p   p

Term B
Term C
• To obtain an actual value for X (the ideal goal), we would need to compute the
forcing terms (Terms B and C) from the three-dimensional wind and temperature fields,
and then invert the operator in Term A using a numerical procedure, called “successive
over-relaxation”, with appropriate boundary conditions
• This is NOT a simple task (forecasters never do this)…..
 Rather, we can infer the sign and relative magnitude of X simple inspection
of the three-dimensional absolute geostrophic vorticity and temperature fields
(forecasters do this all the time…)
 Thus, let’s examine the physical interpretation of each term….
M. D. Eastin
QG Analysis: System Evolution
The BASIC QG Height Tendency Equation:
 2 f 02  2 
  

2 
 p 

Term A



f o  Vg    g  f 
Term B

  f o2 R


 

V


T
g


p   p

Term C
Term A: Local Geopotential Height Tendency
 This term is our goal – a qualitative estimate of the synoptic-scale
geopotential height change at a particular location
• For synoptic-scale atmospheric waves, this term is proportional to –X
• Thus, if we incorporate the negative sign into our physical interpretation,
we can just think of this term as local geopotential height change
M. D. Eastin
QG Analysis: System Evolution
The BASIC QG Height Tendency Equation:
 2 f 02  2 
  

2 
 p 




f o  Vg    g  f 
Term A

  f o2 R


 

V


T
g


p   p

Term B
Term C
Recall for a Single Pressure Level:
causes local vorticity increases
PVA →
 g
t
0
• From our relationship between ζg and χ, we know that PVA is equivalent to:
 g
t

1 2
 p  therefore: PVA → 2p   0
f0
or, since: 2    
PVA →   0
 Thus, we know that PVA at a single level leads to height falls
 Using similar logic, NVA at a single level leads to height rises
M. D. Eastin
QG Analysis: System Evolution
The BASIC QG Height Tendency Equation:
Initial Time
Trough Axis
Initial Time
Full-Physics
Model
Analysis
NVA
PVA
Expect
Height Rises
Expect
Height Falls
Expect the trough to move east
M. D. Eastin
QG Analysis: System Evolution
The BASIC QG Height Tendency Equation:
12 Hours Later
Trough Axis
Initial Time
Generally
consistent
with
expectations!
M. D. Eastin
QG Analysis: System Evolution
The BASIC QG Height Tendency Equation:
Generally Consistent…BUT…Remember!
• Only evaluated one level (500mb) → should evaluate multiple levels
• Used full wind and vorticity fields → should use geostrophic wind and vorticity
• Mesoscale-convective processes → QG focuses on only synoptic-scale (small Ro)
• Condensation / Evaporation → neglected diabatic processes
• Did not consider differential temperature (thermal) advection (Term C)!!!
Application Tips:
 Often the primary forcing in the upper troposphere (500 mb and above)
• Term is equal to zero at local vorticity maxima / minima
• If the vorticity maxima / minima are collocated with trough / ridge axes,
(which is often the case) this term cannot change system strength by
increasing or decreasing the amplitude of the trough / ridge system
 Thus, this term is often responsible for system motion [more on this later…]
M. D. Eastin
QG Analysis: System Evolution
The BASIC QG Height Tendency Equation:
 2 f 02  2 
  

2 
 p 


Term A


f o  Vg    g  f 

  f o2 R


 

V


T
g


p   p

Term B
Term C
Term C: Vertical Derivative of Geostrophic Temperature Advection
• Consider a three layer atmosphere where the warm air advection (WAA) is
strongest in the upper layer
Z-top
WAA
WAA
WAA
ΔZ
ΔZ increases
Z-400mb
Z-700mb
Height Falls
occur
below the
level of
maximum
WAA
Z-bottom
• The greater temperature increase aloft will produce the greatest thickness increase
in the upper layer and lower the pressure surfaces (or heights) in the lower levels
 Therefore an increase in WAA advection with height leads to height falls
M. D. Eastin
QG Analysis: System Evolution
The BASIC QG Height Tendency Equation:
 2 f 02  2 
  

2 
 p 




f o  Vg    g  f 
Term A

  f o2 R


 

V


T
g


p   p

Term B
Term C
Term C: Vertical Derivative of Geostrophic Temperature Advection
• Possible height fall scenarios:
Strong WAA in upper levels
Weak WAA in lower levels
WAA in upper level
CAA in lower levels
No temperature advection in upper levels
CAA in lower levels
Weak CAA in upper levels
Strong CAA in lower levels
M. D. Eastin
QG Analysis: System Evolution
The BASIC QG Height Tendency Equation:
 2 f 02  2 
  

2 
 p 


Term A


f o  Vg    g  f 

  f o2 R


 

V


T
g


p   p

Term B
Term C
Term C: Vertical Derivative of Geostrophic Temperature Advection
• Consider a three layer atmosphere where the warm air advection (CAA) is
strongest in the upper layer
CAA
CAA
CAA
ΔZ
Z-top
ΔZ decreases Z-400mb
Z-700mb
Height rises
occur
below the
level of
maximum
CAA
Z-bottom
• The greater temperature increase aloft will produce the greatest thickness increase
in the upper layer and lower the pressure surfaces (or heights) in the lower levels
 Therefore an increase in CAA advection with height leads to height rises
M. D. Eastin
QG Analysis: System Evolution
The BASIC QG Height Tendency Equation:
 2 f 02  2 
  

2 
 p 




f o  Vg    g  f 
Term A

  f o2 R


 

V


T
g


p   p

Term B
Term C
Term C: Vertical Derivative of Geostrophic Temperature Advection
• Possible height rise scenarios:
Strong CAA in upper levels
Weak CAA in lower levels
CAA in upper level
WAA in lower levels
No temperature advection in upper levels
WAA in lower levels
Weak WAA in upper levels
Strong WAA in lower levels
M. D. Eastin
QG Analysis: System Evolution
The BASIC QG Height Tendency Equation:
Term C: Vertical Derivative of Geostrophic Temperature Advection
Initial Time
850 mb
Initial
Trough Axis
Full-Physics
Model
Analysis
Strong WWA
Weaker WAA aloft
(not shown)
Expect Height Rises
Strong CAA
Weaker CAA aloft
(not shown)
Expect Height Falls
M. D. Eastin
QG Analysis: System Evolution
The BASIC QG Height Tendency Equation:
Term C: Vertical Derivative of Geostrophic Temperature Advection
12 Hours Later
850 mb
Initial
Trough Axis
Ridge ”rose”
slightly
Generally
consistent
with
expectations!
Trough “deepened”
M. D. Eastin
QG Analysis: System Evolution
The BASIC QG Height Tendency Equation:
Term C: Vertical Derivative of Geostrophic Temperature Advection
Generally Consistent…BUT…Remember!
• Used full wind field → should use geostrophic wind
• Only evaluated one level (850mb) → should evaluate multiple levels/layers **
• Mesoscale-convective processes → QG focuses on only synoptic-scale (small Ro)
• Condensation / Evaporation → neglected diabatic processes
• Did not consider vorticity advection (Term B)!!!
Application Tips:
 Often the primary forcing in the lower troposphere (below 500 mb)
• Term is equal to zero at local temperature maxima / minima
 Since the temperature maxima / minima are often located between the
trough / ridge axes, significant temperature advection (or height changes)
can occur at the axes and thus amplify the system intensity
M. D. Eastin
QG Analysis: System Evolution
The BASIC QG Height Tendency Equation:
Term C: Vertical Derivative of Geostrophic Temperature Advection
Important: You should evaluate the vertical structure of temperature advection!!!
M. D. Eastin
QG Analysis: System Evolution
The BASIC QG Height Tendency Equation:
Term C: Vertical Derivative of Geostrophic Temperature Advection
Important: You should evaluate the vertical structure of temperature advection!!!
WAA = Warm Colors
CAA = Cool Colors
M. D. Eastin
QG Analysis: System Evolution
The BASIC QG Omega Equation:
Application Tips:
 Remember the underlying assumptions!!!
 You must consider the effects of both Term B and Term C at multiple levels!!!
 If the vorticity maxima/minima are not collocated with trough/ridge axes,
then Term B will contribute to system intensity change and motion
 If the vorticity advection patterns change with height, expect the system
“tilt” to change with time (become more “tilted” or more “stacked”)
 If differential temperature advection is large (small), then expect
Term C to produce large (small) changes in system intensity
 Opposing expectations from the two terms at a given location will weaken
the total vertical motion (and complicate the interpretation)!!!
 The QG height-tendency equation is a prognostic equation:
• Can be used to predict the future pattern of geopotential heights
• Diagnose the synoptic–scale contribution to the height field evolution
• Predict the formation, movement, and evolution of synoptic waves
M. D. Eastin
References
Bluestein, H. B, 1993: Synoptic-Dynamic Meteorology in Midlatitudes. Volume I: Principles of Kinematics and Dynamics.
Oxford University Press, New York, 431 pp.
Bluestein, H. B, 1993: Synoptic-Dynamic Meteorology in Midlatitudes. Volume II: Observations and Theory of Weather
Systems. Oxford University Press, New York, 594 pp.
Charney, J. G., B. Gilchrist, and F. G. Shuman, 1956: The prediction of general quasi-geostrophic motions. J. Meteor.,
13, 489-499.
Durran, D. R., and L. W. Snellman, 1987: The diagnosis of synoptic-scale vertical motionin an operational environment.
Weather and Forecasting, 2, 17-31.
Hoskins, B. J., I. Draghici, and H. C. Davis, 1978: A new look at the ω–equation. Quart. J. Roy. Meteor. Soc., 104, 31-38.
Hoskins, B. J., and M. A. Pedder, 1980: The diagnosis of middle latitude synoptic development. Quart. J. Roy. Meteor.
Soc., 104, 31-38.
Lackmann, G., 2011: Mid-latitude Synoptic Meteorology – Dynamics, Analysis and Forecasting, AMS, 343 pp.
Trenberth, K. E., 1978: On the interpretation of the diagnostic quasi-geostrophic omega equation. Mon. Wea. Rev., 106,
131-137.