1. Q-G Theory: Using the Q-Vector
Patrick Market
Department of Atmospheric Science
University of Missouri-Columbia

2. Introduction Q-G forcing for w Evaluation Q-vector
Vertical motions (particularly in an ETC) complete a secondary ageostrophic circulation forced by geostrophic and hydrostatic adjustments on the synoptic-scale.
Evaluation
Traditional
Laplacian of thickness advection
Differential vorticity advection
PIVA/NIVA (Trenberth Approximation)
Q-vector

3. Q-vector Strengths
Eliminates competition between terms in the Q-G w equation
Unlike PIVA/NIVA, deformation is retained as a forcing mechanism
Q-vectors are proportional in strength and lie along the low level Vag.
Analysis of Q-vectors with isentropes can reveal areas of frontogenesis/frontolysis.
Only one isobaric level is needed to compute forcing.

4. Q-vector Weaknesses Diabatic heating/cooling are neglected
Variations in f are neglected
Variations in static stability are neglected
w is still not calculated; its forcing is
Although one may employ a single level for the process, layers are thought to be better for Q evaluation
So, which layer to use?

5. Q-vector: Choosing a layer…
RECALL: Q-G forcing for w
Vertical motions complete a secondary ageostrophic circulation…
Inertial-advective adjustments with the ULJ
Isallobaric adjustments with the LLJ
Deep layers can be useful
Max vertical motion should be near LND (~550 mb)
Ideally that layer will be included

6. Q-vector: Choosing a layer…
Avoid very low levels (PBL)
Friction
Radiative/sensible heating/cooling
Look
low enough to account for CAA/WAA
deep enough to account for vertical change in vorticity advection
Typical layer: mb
Brackets LND (~550 mb)
Deep enough to
Capture low level thermal advection
Significant differential vorticity advection

7. A Definition of Q
Q is the time rate of change of the potential temperature gradient vector of a parcel in geostrophic motion
(after Thaler)

8. Simple Example 1 Z Z+DZ Z+2DZ True gradient vector points L  H
T+DT
Z+2DZ
True gradient vector points L  H
Equivalent barotropic environment
(after Thaler)

9. Simple Example 2 Z T
Z+DZ
T+DT
Z+2DZ
(after Thaler)

10. A Purpose for Q
If Q exists, then the thermal gradient is changing following the motion…
So… thermal wind balance is compromised
So… the thermal wind is no longer proportional to the thickness gradient
So… geostrophic and hydrostatic balance are compromised
So… forcing for vertical motion ensues as the atmosphere seeks balance

11. Known Behaviors of Q FQ-G
Q often points along Vag in the lower branch of a transverse, secondary circulation
Q often proportional to low-level |Vag|
Q points toward rising motion
Q plotted with a field of q can reveal regions of F
FQ-G
Q points toward warm air – frontogenesis
Q points toward cold air – frontolysis

12. Q Components Q Qn – component normal to q contours
Qs – component parallel to q contours q Qn Q
q+Dq Qs
q+2Dq

13. Aspects of Qn
Indicates whether geostrophic motion is frontolytic or frontogenetic
Qn points ColdWarm Frontogenesis
Qn points WarmCold Frontolysis
For f=f0, the geostrophic wind is purely non-divergent
Q-G frontogenesis is due entirely to deformation

14. Aspects of Qs
Determines if the geostrophic deformation is rotating the isentropes cyclonically or anticyclonically
Qs points with cold air on left Q rotates cyclonically
Qs points with cold air on right Q rotates anticyclonically
Rotation is manifested by vorticity and deformation fields

15. A Case Study: April 2002

16. 27/23Z Synopsis

17. 27/23Z PMSL & Thickness

18. 27/23Z mb Hght & Vag

19. 27/23Z mb Hght & |V |

20. 27/23Z mb Hght & Vag

21. J
27/23Z Cross section of Q, Normal |V|, Vag, & w

22. 27/23Z Layer Q and 550 mb Q

23. 27/23Z Layer Q, Qn, Qs, and 550 mb Q

24. 27/23Z Divergence of Q and 550 mb Q

25. 27/23Z Layer Qn, and 550 mb Q

26. 27/23Z Layer Qs, and 550 mb Q

27. 27/23Z 550 mb Heights and Q

28. 27/23Z 550 mb F

29. 27/23Z Stability (dQ/dp) vs ls
LDF (THTA)

30. 27/23Z Advection of Stability by the Wind
ADV(LDF (THTA), OBS)

31. 27/23Z 700 mb w
OMEG m b s-1

32. 27/23Z Layer Mean RH

33. Outcome Convection initiates in western MO
Left exit region of ~linear jet streak
Qn points cold  warm
Frontogenesis present but weak
Qs points with cold to left  cyclonic rotation of q
Relative low stability
Modest low-level moisture

34. 28/00-06Z IR Satellite

35. 28/00-06Z RADAR Summary

36. Summary Q aligns along low-level Vag in well-developed systems Div(Q)
Portrays w forcing well
Plotting stability may highlight regions where Q under-represents total w forcing
Plotting moisture helps refine regions of inclement weather
Q proportional to Q-G F

37. Quasi-geostrophic theory (Continued)
John R. Gyakum

38. The quasi-geostrophic omega equation:
(s2 + f022/∂p2) =
f0/p{vg(1/f02 + f)}
+ 2{vg  (- /p)}+ 2(heating)
+friction

39. The Q-vector form of the quasi-geostrophic omega equation
(p2 + (f02/)2/∂p2) =
(f0/)/p{vgp(1/f02 + f)}
+ (1/)p2{vg  p(- /p)}
= -2p  Q - (R/p)b(T/x)

40. Excepting the b effect for adiabatic and frictionless processes:
Where Q vectors converge, there is forcing for ascent
Where Q vectors diverge, there is forcing for descent

42. 5340 m 5400 m Warm Cold Warm X The beta effect: -(R/p)b(T/x)<0

43. Advantages of the Q-vector approach:
Forcing functions can be evaluated on a constant pressure surface
Forcing functions are “Galilean Invariant” (the functions do not depend on the reference frame in which they are being measured)…although the temperature advection and vorticity advection terms are each not Galilean Invariant, the sum of these two terms is Galilean Invariant
There is not partial cancellation between terms as there typically is with the traditional formulation

44. Advantages of the Q-vector approach (continued):
The Q-vector forcing function is exact, under the adiabatic, frictionless, and quasi-geostrophic approximation; no terms have been neglected
Q-vectors may be plotted on analyses of height and temperature to obtain a representation of vertical motions and ageostrophic wind

45. However:
One key disadvantage of the Q-vector approach is that Q-vector divergence is not as physically meaningful as is seen in either horizontal temperature advection or vorticity advection
To remedy this conceptual difficulty, Hoskins and Sanders (1990) have proposed the following analysis:

46. Q = -(R/p)|T/y|k x (vg/x) where the x, y axes follow respectively, the isotherms, and the opposite of the temperature gradient:
isotherms
cold y X
warm

47. Q = -(R/p)|T/y|k x (vg/x)
Therefore, the Q-vector is oriented 90 degrees clockwise to the geostrophic change vector

48. To see how this concept works, consider the case of only horizontal
thermal advection forcing the quasi-geostrophic vertical motions:
Q = -(R/p)|T/y|k x (vg/x)
(from Sanders and Hoskins 1990)

49. Now, consider the case of an equivalent-barotropic atmosphere (heights
and isotherms are parallel to one another, in which the only forcing for
quasi-geostrophic vertical motions comes from horizontal vorticity
advections:
Q = -(R/p)|T/y|k x (vg/x)
(from Sanders and Hoskins 1990)

50. (from Sanders and Hoskins 1990):
Q = -(R/p)|T/y|k x (vg/x)
Q-vectors in a zone of geostrophic
frontogenesis:
Q-vectors in the entrance region
of an upper-level jet

51. Static stability influence on QG omega
Consider the QG omega equation: (s2 + f022/∂p2) = f0/p{vg(1/f02 + f)} +2{vg  (- /p)} 2(heating)+friction
The static stability parameter s=-T ln/p

52. Static stability (continued)
1. Weaker static stability produces more vertical motion for a given forcing
2. Especially important examples of this effect occur when cold air flows over relatively warm waters (e.g.; Great Lakes and Gulf Stream) during late fall and winter months
3. The effect is strongest for relatively short wavelength disturbances

53. Static stability (Continued) 1
Static stability (Continued) 1. The ‘effective’ static stability is reduced for saturated conditions, when the lapse rate is referenced to the moist adiabatic, rather than the dry adiabat 2. Especially important examples of this effect occur in saturated when cold air flows over relatively warm waters (e.g.; Great Lakes and Gulf Stream) during late fall and winter months 3. The effect is strongest for relatively short wavelength disturbances and in warmer temperatures

54. Static stability
Conditional instability occurs when the environmental lapse rate lies between the moist and dry adiabatic lapse rates: Gd > g > Gm
Potential (or convective) instability occurs when the equivalent potential temperature decreases with elevation (quite possible for such an instability to occur in an inversion or absolutely stable conditions)

55. Cross-sectional analyses:
temperature (degrees C)
The shaded zone illustrates the
transition zone between the upper
troposphere’s weak stratification
and the relatively strong stratification of the lower stratosphere
(Morgan and Nielsen-Gammon
1998).
theta (dashed) and wind speed
(solid; m per second)
What is the shaded zone? Stay
tuned!

56. References:
Bluestein, H. B., 1992: Synoptic-dynamic meteorology in midlatitudes. Volume I: Principles of kinematics and dynamics. Oxford University Press pp.
Morgan, M. C., and J. W. Nielsen-Gammon, 1998: Using tropopause maps to diagnose midlatitude weather systems. Mon. Wea. Rev., 126,
Sanders, F., and B. J. Hoskins, 1990: An easy method for estimation of Q-vectors from weather maps. Wea. and Forecasting, 5,

57. Q-vectors Definition: Recall the quasi-geostrophic omega equation:
An alternative form of the omega equation can be derived (see your dynamics book)
where:
Q-vector Form
of the
QG Omega Equation
Advanced Synoptic
M. D. Eastin

58. Q-vectors Physical Interpretation:
The components Q1 and Q2 provide a measure of the horizontal wind shear
across a temperature gradient in the zonal and meridional directions
The two components can be combined to produce a horizontal “Q-vector”
Q-vectors are oriented parallel to the ageostrophic wind vector
Q-vectors are proportional to the magnitude of the ageostrophic wind
Q-vectors point toward rising motion
In regions where:
Q-vectors converge Upward vertical motion
Q-vectors diverge Downward vertical motion
Advanced Synoptic
M. D. Eastin

59. Q-vectors Advantages:
All forcing on the right hand side can be evaluated on a single isobaric surface
(before vorticity advection was inferred from differences between two levels)
Can be easily computed from 3-D data fields (quantitative)
Only one forcing term, so no partial cancellation of forcing terms
(before vorticity and temperature advection often offset one another)
The forcing is exact under the QG constraints
(before a few terms were neglected)
The Q-vectors computed from numerical model output can be plotted on maps
to obtain a clear representation of synoptic-scale vertical motion
Disadvantages:
Can be very difficult to estimate from standard upper-air observations
Neglects diabatic heating, orographic, and frictional effects
Advanced Synoptic
M. D. Eastin

60. Q-vectors Typical Synoptic Systems:
In synoptic-scale systems the Q-vectors
often point toward regions of WAA
located to the east of surface cyclones
and upper troughs
The converging Q-vectors suggest
rising (sinking) motion should occur
to the east of troughs (ridges) and
surface cyclones (anticyclones)
Thus, Q-vector analysis is consistent
with analyses of the “traditional” QG
omega equation
Surface Systems
Upper-Level Systems
Advanced Synoptic
M. D. Eastin

61. Q-vectors Examples: 850 mb Analysis – 29 July 1997 at 00Z
Isentropes (red), Q-vectors, Vertical motion (shading, upward only)
Advanced Synoptic
M. D. Eastin

62. Q-vectors
Examples:
Note: The broad region of Q-vector convergence (expected upward motion) and
radar reflectivity correspond fairly well
850mb Q-vector Analysis
22 March 2007 at 1200 Z
Radar Reflectivity Summary
22 March 2007 at 1215 Z
Advanced Synoptic
M. D. Eastin

63. Q-vectors and Frontogenesis
Application of Q-Vectors:
The orientation of the Q-vector to the
potential temperature gradient provides
any easy method to infer frontogenesis
or frontolyisis
If the Q-vector points toward warm air and
crosses the temperature gradient, the
ageostrophic flow will produce frontogenesis
If the Q-vector points toward cold air and
ageostrophic flow will produce frontolysis
If the Q-vector points along the temperature
gradient, the ageostrophic flow will have
no impact on the temperature gradient θc
Q-vectors θw
Frontogenesis
Advanced Synoptic
M. D. Eastin

64. Q-vectors and Frontogenesis
Examples:
850 mb Analysis – 29 July 1997 at 00Z
Isentropes (red), Q-vectors, Vertical motion (shading, upward only)
Expect
Frontolysis
Expect
Frontogenesis
Cold
Air
Warm
Air
Advanced Synoptic
M. D. Eastin

65. Q-vectors and Frontogenesis
Examples:
Note: The regions of expected and observed frontogenesis / frontolysis generally agree
Part of the observed evolution is due to system motion and diabatic effects
850mb Q-vectors and Potential Temperatures
22 March 2007 at 1200 Z
850mb Potential Temperatures
23 March 2007 at 0000 Z
Advanced Synoptic
M. D. Eastin

66. Dynamics of Frontogenesis
Summary:
Review of Kinematic Frontogenesis
Basic Dynamic Response (physical processes)
Conceptual Model (physical processes)
Impact of Ageostrophic Advection
Q-vectors (physical interpretation, advantages / disadvantages)
Application of Q-vectors to Frontgenesis
Advanced Synoptic
M. D. Eastin

67. 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.
Keyser, D., M. J. Reeder, and R. J. Reed, 1988: A generalization of Pettersen’s frontogenesis function and its relation to
the forcing of vertical motion. Mon. Wea. Rev., 116,
Schultz, D. M., D. Keyser, and L. F. Bosart, 1998: The effect of large-scale flow on low-level frontal structure and evolution
in midlatitude cyclones. Mon. Wea. Rev., 126,
Advanced Synoptic
M. D. Eastin

69. Why Not Look Only at Model Output?

73. Good Web Site to Explore VV

74. Comparison of Various Forms of the Q-G Omega Equation
Classic (Two terms: differential vorticity advection, Laplacian of the thermal wind)
The result is the difference between two large terms resulting in large truncation error.
Cannot estimate reliably from vorticity advection at a single level or from warm advection alone.
Using at a single level, best done at 500 hPa for strong events. Really need a 3D solution for an accurate answer.
Trenberth/Sutcliffe formulation (advection of absolute vorticity by the thermal wind) is more accurate since no cancellation problem.

75. Q-Vector approach is the best in many ways
No cancellation problems
Includes deformation term
Provides insights into the lower branch of the ageostrophic circulation forced by the geostrophic forcing
Provides insights into frontogenesis
Allows rapid and intuitive graphical interpretation

76. QG Diagnostics Online
Classic approach:
Sutcliffe-Trenberth:
Q-vector

77. Vertical Motion Can be as the complex sum of:
QG motions (relatively large scale and smooth)
Orographic forcing
Convective forcing
Gravity waves and other small-scale stuff
QG diagnostics helpful for seeing the big picture

78. Jet Streak Vertical Motions

80. For unusually straight jets, it might be reasonable

81. But usually there is much more going on, so be VERY careful in application of simple jet streak model
Garp example…

84. You may be familiar with Q-vectors. Q-vectors calculate the effect
that the geostrophic wind is having on the flow. Specifically, the orientation of Q points in
the same direction as the low-level branch of the secondary circulation, and the magnitude
of Q is proportional to the magnitude of this branch. Through the QG omega equation,
the divergence of Q can be used to diagnose the forcing for vertical motion. One
partitioning of the Q-vector yields Qn and Qs. Qn is the component of the Q-vector normal
to the local orientation of the isentropes. Qs is the component of the Q-vector parallel
to the local orientation of the isentropes. Thus, Qn represents the frontogenesis due to
the geostrophic wind alone. As we previously argued, this is generally inappropriate for
ascertaining frontal circulations. In AWIPS, you may find some functions called F-vectors.
F-vectors have two components: Fn and As. F-vectors are the total-wind generalization
of Q-vectors and the magnitude of Fn is the same as Petterssen frontogenesis.
While no similar expression relating F-vectors to forcing for vertical motion (as in the Qvectors
in QG theory), the divergence of F-vectors can be used to diagnose the forcing
for vertical motion due to the total wind. Thus, Fn and its divergence are the preferred
methods for diagnosing frontal circulations, not Qn and its divergence. Because F uses
the total wind, the convergence field is much noisier than seen with Q-vector convergence.
Therefore forecasters should look for temporal continuity in the divergence of Fn
and overlay frontogenesis in order to help identify areas where there is persistent forcing
for ascent.