Advanced Synoptic M. D. Eastin

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

Review: The BASIC QG Omega Equation


 2  f

0
2 
2
 p
2


  f

0

 p

V g
 

 g
 f
 

R
 p
 2

V g
 
T

Term A
Term B : Differential Vorticity Advection
Term B Term C
PVA
PVA
PVA
ΔZ
ΔZ
Z-top
ΔZ decreases
ΔZ decreases
Z-400mb
Z-700mb
Z-bottom
Hydrostatic
Balance
Thickness decreases must occur with cooling
Rising
Motions
Adiabatic
Cooling
Sinking
Motions
Adiabatic
Warming
• Therefore, in the absence of geostrophic vorticity advection and diabatic processes:

An increase in PVA with height will induce rising motion
 An increase in NVA with height will induce sinking motion
Advanced Synoptic M. D. Eastin

Review: The BASIC QG Omega Equation


 2  f

0
2 
2
 p
2


  f

0

 p

V g
 

 g
 f
 

R
 p
 2

V g
 
T

Term A Term B Term C
Term C : Thermal Advection
•
WAA ( CAA ) leads to local temperature / thickness increases (decreases)
• In order to maintain geostrophic flow , ageostrophic flows and mass continuity produce a vertical motion through the layer
Z-top
Z-400mb
Surface
Rose
Z-top
Z-400mb
WAA
ΔZ ΔZ increase
Z-700mb
Z-bottom
Surface
Fell
Z-700mb
Z-bottom
• Therefore, in the absence of geostrophic vorticity advection and diabatic processes:
 WAA will induce rising motion
 CAA will induce sinking motion
Advanced Synoptic M. D. Eastin

What effect does diabatic heating or cooling have?
Diabatic Heating : Latent heat release due to condensation (Ex: Cumulus convection)
Strong surfaces fluxes (Ex: CAA over the warm Gulf Stream)
(Ex: Intense solar heating in the desert)
• Heating always leads to temperature increases → thickness increases
• Consider the three-layer model with a deep cumulus cloud
Surface
Rose
Z-top
Z-400mb
ΔZ ΔZ increases
Surface
Fell
Z-700mb
Z-bottom
• Again, the maintenance of geostrophic flow requires rising motion through the layer
•
Identical to the physical response induced by WAA
• Therefore: Diabatic heating induces rising motion
Advanced Synoptic M. D. Eastin

What effect does diabatic heating or cooling have?
Diabatic Cooling : Evaporation (Ex: Precipitation falling through sub-saturated air)
Radiation (Ex: Large temperature decreases on clear nights)
Strong surface fluxes (Ex: WAA over snow/ice)
• Cooling always leads to temperature decreases → thickness decreases
• Consider the three-layer model with evaporational / radiational cooling
Z-top
ΔZ ΔZ decreases
Surface
Fell
Surface
Rose
Z-400mb
Z-700mb
Z-bottom
• Again, maintenance of geostrophic flow requires sinking motion through the layer
•
Identical to the physical response induced by CAA
• Therefore: Diabatic cooling aloft induces sinking motion
Advanced Synoptic M. D. Eastin

What effect does flow over topography have?
Downslope Motions : Flow away from the Rockies Mountains
Flow away from the Appalachian Mountains
• Subsiding air always adiabatically warms
• Subsidence leads to temperature increases → thickness increases
• Consider the three-layer model with downslope motion at mid-levels
Surface
Rose
ΔZ ΔZ increases
Surface
Fell
Z-top
Z-400mb
Z-700mb
Z-bottom
• Again, maintenance of geostrophic flow requires rising motion through the layer
•
Identical to the physical response induced by WAA and diabatic heating
• Therefore:
Downslope flow induces rising motion
Advanced Synoptic M. D. Eastin

What effect does flow over topography have?
Upslope Motions : Flow toward the Rockies Mountains
Flow toward the Appalachian Mountains
• Rising air always adiabatically cools
• Ascent leads to temperature decreases → thickness decreases
• Consider the three-layer model with upslope motion at mid-levels
ΔZ ΔZ decreases
Surface
Fell
Surface
Rose
Z-top
Z-400mb
Z-700mb
Z-bottom
• Again, maintenance of geostrophic flow requires sinking motion through the layer
•
Identical to the physical processes induced by CAA and diabatic cooling
• Therefore:
Upslope flow induces sinking motion
Advanced Synoptic M. D. Eastin

Update: The Modified QG Omega Equation


 2  f

0
2 
2
 p
2


  f

0

 p

V g
 

 g
 f
 

R
 p
 2

V g
 
T

Vertical
Motion
Differential Vorticity
Advection
+ Diabatic + Topographic
Forcing Forcing
Thermal
Advection
Note: The text includes a modified equation with only diabatic effects [Section 2.5]
Application Tips:
• Differential vorticity advection and thermal advection are the dominant terms in the majority of situations → weight these terms more
• Diabatic forcing can be important when deep convection or dry/clear air are present
• Topographic forcing is only relevant near large mountain ranges
Advanced Synoptic M. D. Eastin

Application Tips:
Diabatic Forcing
• Use radar → more intense convection → more vertical motion
• Use IR satellite → cold cloud tops → deep convection or high clouds?
→ warm cloud tops → shallow convection or low clouds?
• Use VIS satellite → clouds or clear air?
• Use WV satellite → clear air → dry or moist?
Topographic Forcing
• Topographic maps → Are the mountains high or low?
• Use surface winds → Is flow downslope, upslope, or along-slope?
Advanced Synoptic M. D. Eastin

Review: The BASIC QG Height Tendency Equation


 2  f

0
2 
2
 p
2


  f o


V g
 

 g
 f
 
 

 p


 f
 o
2
R p

V g
 
T
 

Term A
Term B : Vorticity Advection
Term B Term C
• Positive vorticity advection ( PVA ) causes local vorticity increases
PVA →
  g
 t

0
• From our relationship between ζ g and
χ
, we know that PVA is equivalent to:
 
 t g 
1 f
0
 2 p
 therefore: PVA →  2 p
 
0
 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
Advanced Synoptic M. D. Eastin

Review: The BASIC QG Height Tendency Equation


 2  f

0
2 
2
 p
2


  f o


V g
 

 g
 f
 
 

 p


 f
 o
2
R p

V g
 
T
 

Term A Term B Term C
Term C : Differential Thermal Advection
• Consider an atmosphere with an arbitrary vertical profile of temperature advection
• Thickness changes throughout the profile will result from the type (
WAA / CAA ) and magnitude of temperature advection though the profile
•Therefore:
An increase in WAA advection with height leads to height falls
An increase in CAA advection with height leads to height rises
Advanced Synoptic M. D. Eastin

Recall:
• Local diabatic heating produces the same response as local WAA
• Likewise local diabatic cooling is equivalent to local CAA
Evaluation:
• Examine / Estimate the vertical profile of diabatic heating / cooling from all available radar / satellite data
Clear Regions
Z
Diabatic Cooling max located in upper-levels due to radiational cooling
Diabatic heating max located near surface due to surface fluxes
Net Result: Increase in cooling with height
Height Rises
Regions of Deep Convection
Z
Diabatic Heating max located in upper-levels due to condensation
Diabatic cooling max located below cloud base due to evaporation
Net Result: Increase in heating with height
Height Falls
Regions of Shallow Convection
Z
Diabatic Cooling max located in upper-levels due to radiational cooling
Diabatic heating max located in lower-levels due to condensation
Net Result: Increase in cooling with height
Height Rises
Advanced Synoptic M. D. Eastin

Recall:
• Local downslop flow produces the same response as local WAA
• Likewise local upslope flow is equivalent to local CAA
Evaluation:
• Examine / Estimate the vertical profile of heating due to topographic effects
Z
Downslope Flow
No adiabatic heating
No topographic effects above the mountains
Adiabatic Heating due to downslope flow
Z
Upslope Flow
No adiabatic heating
No topographic effects above the mountains
Adiabatic Cooling due to upslope flow
Net Result: Decrease in heating with height above heating max → height rises
Decrease in heating with height below heating max → height falls
Advanced Synoptic
Net Result: Decrease in cooling with height above cooling max → height falls
Decrease in cooling with height below cooling max → height rises
M. D. Eastin

The Modified QG Height Tendency Equation


 2  f

0
2 
2
 p
2


  f
0


V g
 

 g
 f
 
 

 p

 f
 o
2
R p


V g
 
T
 

Height
Tendency
Vorticity
Advection
Differential Thermal
Advection
+ Diabatic
Forcing
+ Topographic
Forcing
Application Tips:
• Differential vorticity advection and thermal advection are the dominant terms in the majority of situations → weight these terms more
• Diabatic forcing can be important when deep convection or dry/clear air are present
• Topographic forcing is only relevant near large mountain ranges
Advanced Synoptic M. D. Eastin

Application Tips:
Diabatic Forcing
• Use radar → more intense convection → more vertical motion
• Use IR satellite → cold cloud tops → deep convection or high clouds?
→ warm cloud tops → shallow convection or low clouds?
• Use VIS satellite → clouds or clear air?
• Use WV satellite → clear air → dry or moist?
Topographic Forcing
• Topographic maps → Are the mountains high or low?
• Use surface winds → Is flow downslope, upslope, or along-slope?
Advanced Synoptic M. D. Eastin

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