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

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

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.

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?

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

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

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)

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

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

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

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

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

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

A Case Study: 27-28 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 Q, Normal |V|, Vag, & w

27/23Z Layer Q and 550 mb Q

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

27/23Z Divergence of Q and 550 mb Q

27/23Z Layer Qn, and 550 mb Q

27/23Z Layer Qs, and 550 mb Q

27/23Z 550 mb Heights and Q

27/23Z 550 mb F

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

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

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

27/23Z 900-700 Layer Mean RH

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

28/00-06Z IR Satellite

28/00-06Z RADAR Summary

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

Quasi-geostrophic theory (Continued) John R. Gyakum

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

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)

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

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

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: isotherms cold y X warm

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

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)

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)

(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

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

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 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

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)

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!

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.

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

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

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

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

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

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

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

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

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

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

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, 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. Advanced Synoptic M. D. Eastin

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-vector http://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.