1. QG Analysis: System Evolution
Advanced Synoptic
M. D. Eastin

2. QG Analysis QG Theory Basic Idea Approximations and Validity
QG Equations / Reference
QG Analysis
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

3. 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
and temperature advection
System Evolution: Predict changes in the local geopotential height patterns from
the observed distributions of geostrophic vorticity advection
and differential temperature advection
Advanced Synoptic
M. D. Eastin

4. 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.
Advanced Synoptic
M. D. Eastin

5. 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):
To do this, we need to eliminate the vertical motion (ω) from both equations
Step 1: Apply the operator to the thermodynamic equation
Step 2: Multiply the vorticity equation by
Step 3: Add the results of Steps 1 and 2
After a lot of math, we get the resulting prognostic equation……
Vorticity
Equation
Adiabatic
Thermodynamic
Equation
Advanced Synoptic
M. D. Eastin

6. QG Analysis: System Evolution
The QG Height Tendency Equation:
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…
Advanced Synoptic
M. D. Eastin

7. QG Analysis: System Evolution
The BASIC QG Height Tendency Equation:
Term A 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….
Advanced Synoptic
M. D. Eastin

8. QG Analysis: System Evolution
The BASIC QG Height Tendency Equation:
Term A Term B 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
Advanced Synoptic
M. D. Eastin

9. QG Analysis: System Evolution
The BASIC QG Height Tendency Equation:
Term A Term B Term C
Term B: Geostrophic Advection of Absolute Vorticity (Vorticity Advection)
Recall for a Single Pressure Level:
Positive vorticity advection (PVA) PVA →
causes local vorticity increases
From our relationship between ζg and χ, we know that PVA is equivalent to:
therefore: PVA → or, since: PVA →
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

10. Expect the trough to move east
QG Analysis: System Evolution
The BASIC QG Height Tendency Equation:
Term B: Geostrophic Advection of Absolute Vorticity (Vorticity Advection)
Initial Time
Trough Axis
Initial Time
Full-Physics
Model
Analysis
NVA
Expect
Height Rises
PVA
Expect
Height Falls
Expect the trough to move east
Advanced Synoptic
M. D. Eastin

11. QG Analysis: System Evolution
The BASIC QG Height Tendency Equation:
Term B: Geostrophic Advection of Absolute Vorticity (Vorticity Advection)
12 Hours Later
Trough Axis
Initial Time
Trough Axis
Generally
consistent
with
expectations!
Advanced Synoptic
M. D. Eastin

12. QG Analysis: System Evolution
The BASIC QG Height Tendency Equation:
Term B: Geostrophic Advection of Absolute Vorticity (Vorticity Advection)
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…]
Advanced Synoptic
M. D. Eastin

13. QG Analysis: System Evolution
The BASIC QG Height Tendency Equation:
Term A Term B Term C
Term C: Vertical Derivative of Geostrophic Temperature Advection
(Differential Thermal Advection)
Consider a three layer atmosphere where the warm air advection (WAA) is
strongest in the upper layer
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
WAA
Z-top
Z-400mb
Z-700mb
Z-bottom
ΔZ increases ΔZ
Height Falls
occur
below the
level of
maximum
Advanced Synoptic
M. D. Eastin

14. QG Analysis: System Evolution
The BASIC QG Height Tendency Equation:
Term A Term B Term C
Term C: Vertical Derivative of Geostrophic Temperature Advection
(Differential Thermal 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
Weak CAA in upper levels
Strong CAA in lower levels
Advanced Synoptic
M. D. Eastin

15. QG Analysis: System Evolution
The BASIC QG Height Tendency Equation:
Term A Term B Term C
Term C: Vertical Derivative of Geostrophic Temperature Advection
(Differential Thermal Advection)
Consider a three layer atmosphere where the warm air advection (CAA) is
strongest in the upper layer
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
Height rises
occur
below the
level of
maximum
CAA
Z-top
Z-400mb
Z-700mb
Z-bottom
ΔZ decreases ΔZ
Advanced Synoptic
M. D. Eastin

16. QG Analysis: System Evolution
The BASIC QG Height Tendency Equation:
Term A Term B Term C
Term C: Vertical Derivative of Geostrophic Temperature Advection
(Differential Thermal 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
Weak WAA in upper levels
Strong WAA in lower levels
Advanced Synoptic
M. D. Eastin

17. QG Analysis: System Evolution
The BASIC QG Height Tendency Equation:
Term C: Vertical Derivative of Geostrophic Temperature Advection
(Differential Thermal 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
Advanced Synoptic
M. D. Eastin

18. QG Analysis: System Evolution
The BASIC QG Height Tendency Equation:
Term C: Vertical Derivative of Geostrophic Temperature Advection
(Differential Thermal Advection)
12 Hours Later
850 mb
Initial
Trough Axis
Generally
consistent
with
expectations!
Ridge ”rose”
slightly
Trough “deepened”
Advanced Synoptic
M. D. Eastin

19. QG Analysis: System Evolution
The BASIC QG Height Tendency Equation:
Term C: Vertical Derivative of Geostrophic Temperature Advection
(Differential Thermal 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
Advanced Synoptic
M. D. Eastin

20. QG Analysis: System Evolution
The BASIC QG Height Tendency Equation:
Term C: Vertical Derivative of Geostrophic Temperature Advection
(Differential Thermal Advection)
Important: You should evaluate the vertical structure of temperature advection!!!
Advanced Synoptic
M. D. Eastin

21. N-S Cross-section of Temperature Advection
QG Analysis: System Evolution
The BASIC QG Height Tendency Equation:
Term C: Vertical Derivative of Geostrophic Temperature Advection
(Differential Thermal Advection)
Important: You should evaluate the vertical structure of temperature advection!!!
N-S Cross-section of Temperature Advection
WAA = Warm Colors
CAA = Cool Colors
Advanced Synoptic
M. D. Eastin

22. 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
Advanced Synoptic
M. D. Eastin

23. References 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,
Durran, D. R., and L. W. Snellman, 1987: The diagnosis of synoptic-scale vertical motionin an operational environment. Weather and Forecasting, 2,
Hoskins, B. J., I. Draghici, and H. C. Davis, 1978: A new look at the ω–equation. Quart. J. Roy. Meteor. Soc., 104,
Hoskins, B. J., and M. A. Pedder, 1980: The diagnosis of middle latitude synoptic development. Quart. J. Roy. Meteor.
Soc., 104,
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,
Advanced Synoptic
M. D. Eastin