QG Analysis: Upper-Level Systems
Will this upper-level trough
intensify or weaken?
Where will the trough move?
Advanced Synoptic
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 Systems
• Evolution of Upper-level Systems
Advanced Synoptic
M. D. Eastin
QG Analysis: Upper-Level Systems
Goal:
We want to use QG analysis to diagnose and “predict” the formation,
evolution, and motion of upper-level troughs and ridges
Which QG Equation?
• We could use the QG omega equation
• Would require additional steps to convert vertical motions to structure change
• No prediction → diagnostic equation → we would still need more information!
• We can apply the QG height-tendency equation
• Ideal for evaluating structural change above the surface
• Prediction of future structure → exactly what we want!
 2 f 02  2 
  

2 
 p 

Vertical
Motion



f 0  Vg    g  f 
Vorticity
Advection
+
Advanced Synoptic
Diabatic
Forcing

  f o2 R

  
 Vg   T 
p   p

Differential Thermal
Advection
+
Topographic
Forcing
M. D. Eastin
QG Analysis: Upper-Level Systems
Evaluate Total Forcing:
 2 f 02  2 
  

2 
 p 

Vertical
Motion



f 0  Vg    g  f 
Vorticity
Advection
+
Diabatic
Forcing

  f o2 R

  
 Vg   T 
p   p

Differential Thermal
Advection
+
Topographic
Forcing
 You must consider the combined effects from each forcing type in order to infer the
expected total geopotential height change
• Sometimes one forcing will “precondition” the atmosphere for another forcing
and the combination will enhance amplification of the trough / ridge
• Other times, forcing types will oppose each other, inhibiting (or limiting) any
amplification of the trough / ridge
Note: Nature continuously provides us with a wide spectrum of favorable and unfavorable
combinations…see the case study and your homework
Advanced Synoptic
M. D. Eastin
QG Analysis: Upper-Level Systems
Evaluate Total Forcing:
 2 f 02  2 
  

2 
 p 


Vertical
Motion


f 0  Vg    g  f 
Vorticity
Advection
+
Diabatic
Forcing

  f o2 R

  
 Vg   T 
p   p

Differential Thermal
Advection
+
Topographic
Forcing
• Forcing for height falls:
(trough amplification)
→
→
→
→
PVA
Increase in WAA with height
Increase in diabatic heating with height
Increase in downslope flow with height
• Forcing for height rises:
(ridge amplification)
→
→
→
→
NVA
Increase in CAA with height
Increase in diabatic cooling with height
Increase in upslope flow with height
Advanced Synoptic
M. D. Eastin
QG Analysis: Upper-Level Systems
Important Aspects of Vorticity Advection:
 Vorticity maximum at trough axis:
• PVA and height falls downstream
• NVA and height rises upstream
• No height changes occur at the
trough axis
• Trough amplitude does not change
• Trough simply moves downstream
(to the east)
Advanced Synoptic
M. D. Eastin
QG Analysis: Upper-Level Systems
Important Aspects of Vorticity Advection:
 Vorticity maximum upstream of trough axis:
• PVA (or CVA) at the trough axis
• Height falls occur at the trough axis
• Trough amplitude increases
• Trough “digs” equatorward
Digs
 Vorticity maximum downstream of trough axis:
• NVA (or AVA) at the trough axis
• Height rises occur at the trough axis
• Trough amplitude decreases
• Trough ”lifts” poleward
Advanced Synoptic
Lifts
AVA
M. D. Eastin
QG Analysis: Upper-Level Systems
Important Aspects of Vorticity Advection: Digging Trough
500mb Wind Speeds
500mb Wind Speeds
t = 0 hr
t = 24 hr
500mb Absolute Vorticity
500mb Absolute Vorticity
t = 0 hr
t = 24 hr
Advanced Synoptic
M. D. Eastin
QG Analysis: Upper-Level Systems
Important Aspects of Vorticity Advection: Lifting Trough
500mb Wind Speeds
500mb Wind Speeds
t = 0 hr
t = 24 hr
500mb Absolute Vorticity
500mb Absolute Vorticity
t = 0 hr
t = 24 hr
Advanced Synoptic
M. D. Eastin
Example Case: Formation / Evolution
Will this upper-level trough
intensify or weaken?
Advanced Synoptic
M. D. Eastin
Example Case: Formation / Evolution
Vorticity Advection:
Trough Axis
Advanced Synoptic
M. D. Eastin
Example Case: Formation / Evolution
Vorticity Advection:
Trough Axis
NVA
Expect Height Rises
Χ>0
PVA
Expect Height Falls
Χ<0
Weak PVA at trough axis
Expect trough to “dig” slightly
Advanced Synoptic
M. D. Eastin
Example Case: Formation / Evolution
Differential Temperature Advection:
500mb
Trough Axis
Advanced Synoptic
System has westward
tilt with height
M. D. Eastin
Example Case: Formation / Evolution
Differential Temperature Advection:
500mb
Trough Axis
Low-level CAA
No temperature
advection aloft
Expect upper-level
Height Falls
Χ<0
System has westward
tilt with height
Low-level WAA
No temperature
advection aloft
Expect upper-level
Height Rises
Χ>0
Expect upper-level
trough to “dig”
Advanced Synoptic
M. D. Eastin
Example Case: Formation / Evolution
Diabatic Forcing:
500mb
Trough Axis
Advanced Synoptic
M. D. Eastin
Example Case: Formation / Evolution
Diabatic Forcing:
500mb
Trough Axis
Deep Convection
Diabatic Heating
Expect upper-level
Height Falls
Χ<0
Expect northern portion
of upper-level
trough to “dig”
Advanced Synoptic
M. D. Eastin
Example Case: Formation / Evolution
Topographic Forcing:
500mb
Trough Axis
Advanced Synoptic
M. D. Eastin
Example Case: Formation / Evolution
Topographic Forcing:
500mb
Trough Axis
Expect northern portion
of upper-level
trough to “dig”
Advanced Synoptic
Upslope flow (CAA) at low-levels
No “topo” flow at 500mb
Increase in WAA with height
Expect upper-level
Height Falls
Χ<0
M. D. Eastin
Example Case: Formation / Evolution
Summary of Forcing Expectations:
Initial Time
Will this upper-level trough
intensify or weaken?
Trough Axis
Expect upper-level
trough to “dig”
Advanced Synoptic
M. D. Eastin
Example Case: Formation / Evolution
“Results”
6-hr Later
Initial
Trough Axis
Trough “dug”
(intensified)
Advanced Synoptic
M. D. Eastin
QG Analysis: Upper-Level System Motion
Initial
Trough Axis
Trough moved east
Why?
Current
Trough Axis
Advanced Synoptic
M. D. Eastin
QG Analysis: Upper-Level System Motion
 2 f 02  2 
  

2 
 p 




f 0  Vg    g  f 
Vertical
Motion
Differential Thermal
Advection
Vorticity
Advection
+

f0 Vg    g

Relative Vorticity Advection

  f o2 R

  
 Vg   T 
p   p

Diabatic
Forcing
Topographic
Forcing
+

f 0  Vg   f

OR
 vg 
Planetary Vorticity Advection
 Whether the relative or planetary vorticity advection dominates the height changes
determines if the wave will “progress” or “retrograde”
Advanced Synoptic
M. D. Eastin
QG Analysis: Upper-Level System Motion
Scale Analysis for a Synoptic Wave:
Term B
Absolute Vorticity Advection

f0 Vg    g


f 0  Vg   f
Relative Vorticity Advection

OR
 vg 
Planetary Vorticity Advection
• Assume waves are sinusoidal in structure:
U = basic current (zonal flow)
L = wavelength of wave
β = north-south Coriolis gradient
• Ratio of relative to planetary vorticity is:

f  V

f
f 0  Vg    g
0
Advanced Synoptic
g

U  2 



 L 
2
M. D. Eastin
QG Analysis: Upper-Level System Motion
Scale Analysis for a Synoptic Wave:

f  V

f
f 0  Vg    g
0
For the Mid-Latitudes:
g

U  2 



 L 
2
U ~ 10 m s-1
β ~ 10-11 s-1 m-1
 Whether relative or planetary vorticity advection dominates the height changes
is a function of the wavelength
Short Waves:
L < 6000 km
Relative vorticity dominates
U  2 

  1
 L 
Long Waves:
L > 6000 km
Planetary vorticity dominates
U  2 

  1
 L 
Advanced Synoptic
2
2
M. D. Eastin
QG Analysis: Upper-Level System Motion
Short Waves:
• Most synoptic waves are short waves
with wavelengths less than 6000 km
• Relative vorticity maxima (minima) are
often located near trough (ridge) axes
• PVA and height falls east of troughs
• NVA and height rises east of ridges
L < 6000 km
L < 6000 km
L
L
Ridge
Trough
• Short waves move eastward
Trough
Ridge
Note: Several “short waves” can
stretch across the entire US
at one time
Vort
Max
Vort
Min
Adapted from Bluestein (1993)
Advanced Synoptic
M. D. Eastin
QG Analysis: Upper-Level System Motion
Long Waves:
• Long waves, with wavelengths greater
than 6000 km, occur during stationary
weather patterns
• Planetary vorticity maxima (minima)
are located at ridge (trough) axes
L > 6000 km
• NVA and height rises west of ridges
• PVA and height falls west of troughs
• Long waves move westward
Vort
Max
Ridge
Trough
Note: A single “long wave” would
stretch across the entire US
and beyond
Vort
Min
Advanced Synoptic
Adapted from Bluestein (1993)
M. D. Eastin
QG Analysis: Upper-Level System Motion
The “Kicker”:
• Long waves are often associated with stationary weather patterns
• When a short wave “kicker” approaches a stationary long wave trough, the
wavelength associated with the long wave is effectively decreased
• Hence, the long wave becomes a short wave and begins to move eastward
• The short wave “kicked out” the long wave, and the stationary weather pattern ends
From Bluestein (1993)
Advanced Synoptic
M. D. Eastin
Example Case: Motion
Where will this upper-level
trough move?
Advanced Synoptic
M. D. Eastin
Example Case: Motion
Is it a “short wave” or a “long wave”?
Initial Time
L < 6000 km
Short Wave
Trough
Examine
relative
vorticity
advection
Trough
Ridge
Trough
Advanced Synoptic
Ridge
M. D. Eastin
Example Case: Motion
Effects of Vorticity Advection:
Assume “local” absolute vort max
are relative vort maxima
Initial Time
Trough Axis
NVA
Expect Height Rises
Χ>0
PVA
Expect Height Falls
Χ<0
Expect trough to move east
Advanced Synoptic
M. D. Eastin
Example Case: Motion
Results:
6-hr Later
Initial
Trough Axis
Current
Trough Axis
Trough
moved east
Advanced Synoptic
M. D. Eastin
QG Analysis: Upper-Level Systems
Application Tips: Evolution and Motion
• ALL relevant forcing terms should be analyzed in each situation!!!
• Differential vorticity advection and thermal advection are the dominant terms
in the majority of situations → weight these terms more
• Diabatic forcing can be important for system evolution when deep convection
or dry/clear air are present.
• Diabatic forcing can be important for system motion when the forcing
is asymmetric about the system center
• Topographic forcing is only relevant near large mountain ranges or rapid
elevation changes over a short horizontal distance
Advanced Synoptic
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.
Advanced Synoptic
M. D. Eastin