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

Q-G Theory:Q-G Theory:Using the Q-VectorUsing the Q-Vector

Patrick MarketDepartment of Atmospheric

ScienceUniversity of Missouri-Columbia

Introduction• Q-G forcing for

– 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

Q-vector Strengths• Eliminates competition between terms

in the Q-G 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.

Q-vector Weaknesses• Diabatic heating/cooling are neglected• Variations in f are neglected• Variations in static stability are

neglected• 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?

Q-vector: Choosing a layer…• RECALL: Q-G forcing for

– 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

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: 400-700 mb– Brackets LND (~550 mb)– Deep enough to

• Capture low level thermal advection• Significant differential vorticity advection

A Definition of QQ

• Q is the time rate of change of the potential temperature gradient vector of a parcel in geostrophic motion

(after Thaler)

pg

dt

dQ

Simple Example 1

• True gradient vector points L H• Equivalent barotropic environment

(after Thaler)

Z

Z+Z

Z+2Z

T

T+T

1 tT

2 tT

3 tT

121 tt TTQ

232 tt TTQ

Simple Example 2

(after Thaler)

Z

Z+Z

Z+2Z

T

T+T

1 tT

2 tT

3 tT

121 tt TTQ

232 tt TTQ

A Purpose for QQ• 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

Known Behaviors of Q

• 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 can reveal

regions of F

–FQ-G

• Q points toward warm air – frontogenesis• Q points toward cold air – frontolysis

Q

QQ Components• Qn – component normal to contours

• Qs – component parallel to contours

Qn

Qs

Q

Aspects of QQn

• Indicates whether geostrophic motion is frontolytic or frontogenetic– Qn points ColdWarm Frontogenesis

– Qn points WarmCold Frontolysis

• For f=f0, the geostrophic wind is purely non-divergent– Q-G frontogenesis is due entirely to

deformation

Aspects of QQs

• 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

A Case Study:A Case Study:27-28 April 200227-28 April 2002

27/23Z Synopsis

27/23Z PMSL & Thickness

27/23Z 850 mb Hght & Vag

27/23Z 300 mb Hght & |V |

27/23Z 300 mb Hght & Vag

J

27/23Z Cross section of , Normal |V|, Vag, &

27/23Z Layer Q and 550 mb

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

27/23Z Divergence of Qand 550 mb

27/23Z Layer Qn, and 550 mb

27/23Z Layer Qs, and 550 mb

27/23Z 550 mb Heights and

27/23Z 550 mb F

LDF (THTA)

27/23Z Stability (d/dp)

vs

ls

ADV(LDF (THTA), OBS)

27/23Z Advection of Stability by the Wind

OMEG b s-1

27/23Z 700 mb

27/23Z 900-700 Layer Mean RH

OutcomeOutcome

• 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

– Relative low stability– Modest low-level moisture

28/00-06Z IR Satellite

28/00-06Z RADAR Summary

Summary• Q aligns along low-level VVag in well-

developed systems• Div(Q)

– Portrays forcing well– Plotting stability may highlight regions where

Q under-represents total forcing– Plotting moisture helps refine regions of

inclement weather

• Q proportional to Q-G F

Quasi-geostrophic theory (Continued)

John R. Gyakum

The quasi-geostrophic omega equation:

(2 + f022/∂p2) =

f0/p{vg(1/f02 + f)}

+ 2{vg (- /p)}+ 2(heating)

+friction

The Q-vector form of the quasi-geostrophic omega equation

(p2 + (f0

2/)2/∂p2) =

(f0/)/p{vgp(1/f02 + f)}

+ (1/)p2{vg p(- /p)}

= -2p Q - (R/p)T/x)

Excepting the effect for adiabatic and frictionless

processes:

• Where Q vectors converge, there is forcing for ascent

• Where Q vectors diverge, there is forcing for descent

5340 m

5400 m

X-(R/p)T/x)>0

-(R/p)T/x)<0

WarmCold

Warm

The beta effect:

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

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

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:

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:

Xy

isothermscold

warm

Q = -(R/p)|T/y|k x (vg/x)

Therefore, the Q-vector is oriented 90 degrees clockwise to the geostrophic

change vector

(from Sanders and Hoskins 1990)

To see how this concept works, consider the case of only horizontalthermal advection forcing the quasi-geostrophic vertical motions:

Q = -(R/p)|T/y|k x (vg/x)

(from Sanders and Hoskins 1990)

Now, consider the case of an equivalent-barotropic atmosphere (heightsand isotherms are parallel to one another, in which the only forcing forquasi-geostrophic vertical motions comes from horizontal vorticityadvections: Q = -(R/p)|T/y|k x (vg/x)

(from Sanders and Hoskins 1990):

Q-vectors in a zone of geostrophicfrontogenesis:

Q-vectors in the entrance regionof an upper-level jet

Q = -(R/p)|T/y|k x (vg/x)

Static stability influence on QG omega

• Consider the QG omega equation: (2 + f0

22/∂p2) = f0/p{vg(1/f02 + f)} +2{vg (- /p)} + 2(heating)+friction

• The static stability parameter ln/p

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

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 adiabat2. 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 months3. The effect is strongest for relatively short wavelength disturbances and in warmer temperatures

Static stability

• Conditional instability occurs when the environmental lapse rate lies between the moist and dry adiabatic lapse rates: d > > m

• 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)

The shaded zone illustrates thetransition zone between the uppertroposphere’s weak stratificationand the relatively strong stratification of the lower stratosphere(Morgan and Nielsen-Gammon1998).

temperature (degrees C)

Cross-sectional analyses:

theta (dashed) and wind speed(solid; m per second)

What is the shaded zone? Staytuned!

References:• Bluestein, H. B., 1992: Synoptic-dynamic

meteorology in midlatitudes. Volume I: Principles of kinematics and dynamics. Oxford University Press. 431 pp.

• Morgan, M. C., and J. W. Nielsen-Gammon, 1998: Using tropopause maps to diagnose midlatitude weather systems. Mon. Wea. Rev., 126, 2555-2579.

• Sanders, F., and B. J. Hoskins, 1990: An easy method for estimation of Q-vectors from weather maps. Wea. and Forecasting, 5, 346-353.

Advanced Synoptic M. D. Eastin

Definition:

• Recall the quasi-geostrophic omega equation:

• An alternative form of the omega equation can be derived (see your dynamics book)

where:

Q-vectors

Tp

Rf

p

f

p

fppgpp

gg vv 202

2202

Q

pp p

f2

2

2202

2

1

Q

Q

Ty

Tx

p

R

pg

pg

v

v

Q

Q-vector Formof the QG Omega Equation

Advanced Synoptic M. D. Eastin

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

Q-vectors

Q

pp p

f2

2

2202

2

1

Q

Q

Ty

Tx

p

R

pg

pg

v

v

Q

02 Qp

02 Qp

Advanced Synoptic M. D. Eastin

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

Q-vectors

Q

pp p

f2

2

2202

2

1

Q

Q

Ty

Tx

p

R

pg

pg

v

v

Q

Advanced Synoptic M. D. Eastin

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

Q-vectorsSurface Systems

Upper-Level Systems

Advanced Synoptic M. D. Eastin

Examples:

Q-vectors

850 mb Analysis – 29 July 1997 at 00ZIsentropes (red), Q-vectors, Vertical motion (shading, upward only)

Advanced Synoptic M. D. Eastin

Examples:

Note: The broad region of Q-vector convergence (expected upward motion) and radar reflectivity correspond fairly well

Q-vectors

850mb Q-vector Analysis22 March 2007 at 1200 Z

Radar Reflectivity Summary22 March 2007 at 1215 Z

Advanced Synoptic M. D. Eastin

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 crosses the temperature gradient, the 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

Q-vectors and Frontogenesis

θw

θc

Q-vectors

Frontogenesis

Advanced Synoptic M. D. Eastin

Examples:

850 mb Analysis – 29 July 1997 at 00ZIsentropes (red), Q-vectors, Vertical motion (shading, upward only)

ExpectFrontogenesis

ExpectFrontolysis

ColdAir

WarmAir

Q-vectors and Frontogenesis

Advanced Synoptic M. D. Eastin

850mb Q-vectors and Potential Temperatures22 March 2007 at 1200 Z

Q-vectors and Frontogenesis

850mb Potential Temperatures23 March 2007 at 0000 Z

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

Advanced Synoptic M. D. Eastin

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

Dynamics of Frontogenesis

Advanced Synoptic M. D. Eastin

ReferencesBluestein, 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, 762-780.

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, 1767-1791.

http://www.crh.noaa.gov/lmk/soo/docu/forcing2.php

Why Not Look Only at Model Output?

Good Web Site to Explore VV

• http://www.twisterdata.com/

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.

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

QG Diagnostics Online

• Classic approach:http://www.mmm.ucar.edu/people/tomjr/files/realtime/qg_diag/Omeg700Tot-NorAmer/res.html

• Sutcliffe-Trenberth:http://www.mmm.ucar.edu/people/tomjr/files/realtime/qg_diag/OmegSutTren700Tot-NorAmer/res.html

• Q-vectorhttp://www.mmm.ucar.edu/people/tomjr/files/realtime/qg_diag/OmegQvec700Tot-NorAmer/res.html

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

Jet Streak Vertical Motions

For unusually straight jets, it might be reasonable

But usually there is much more going on, so be VERY careful in application of simple jet streak

model• Garp example…

• 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.