Q-G Theory: Using the Q-Vector

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Q-G TheoryUsing the Q-Vector

• Patrick Market
• Department of Atmospheric Science
• University of Missouri-Columbia

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Introduction

• Q-G forcing for w
• 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

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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.

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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?

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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

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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

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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)

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Simple Example 1
Z
T
ZDZ
TDT
Z2DZ

• True gradient vector points L ? H
• Equivalent barotropic environment
• (after Thaler)

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Simple Example 2
Z
T
ZDZ
TDT
Z2DZ

• (after Thaler)

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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

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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 q can reveal regions of
  F
• FQ-G
• Q points toward warm air frontogenesis
• Q points toward cold air frontolysis

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Q Components

• Qn component normal to q contours
• Qs component parallel to q contours

q
Qn
Q
qDq
Qs
q2Dq
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Aspects of Qn

• Indicates whether geostrophic motion is
  frontolytic or frontogenetic
• Qn points Cold?Warm Frontogenesis
• Qn points Warm?Cold Frontolysis
• For ff0, the geostrophic wind is purely
  non-divergent
• Q-G frontogenesis is due entirely to deformation

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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

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A Case Study27-28 April 2002
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27/23Z Synopsis
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27/23Z PMSL Thickness
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27/23Z 850 mb Hght Vag
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27/23Z 300 mb Hght V
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27/23Z 300 mb Hght Vag
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J
27/23Z Cross section of Q, Normal V, Vag, w
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27/23Z Layer Q and 550 mb Q
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27/23Z Layer Q, Qn, Qs, and 550 mb
Q
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27/23Z Divergence of Qand 550 mb Q
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27/23Z Layer Qn, and 550 mb Q
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27/23Z Layer Qs, and 550 mb Q
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27/23Z 550 mb Heights and Q
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27/23Z 550 mb F
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27/23Z Stability (dQ/dp)
vs
ls
LDF (THTA)
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27/23Z Advection of Stability by the Wind
ADV(LDF (THTA), OBS)
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27/23Z 700 mb w
OMEG m b s-1
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27/23Z 900-700 Layer Mean RH
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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

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28/00-06Z IR Satellite
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28/00-06Z RADAR Summary
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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

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Quasi-geostrophic theory (Continued)

• John R. Gyakum

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The quasi-geostrophic omega equation

• (s?2 f02?2/?p2)?
• f0?/?pvg??(1/f0?2? f)
• ?2vg ? ?(- ??/?p) ?2(heating)
• friction

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

• (?p2 (f02/?)?2/?p2)?
• (f0/?)?/?pvg??p(1/f0?2? f)
• (1/?)?p2vg ? ?p(- ??/?p)
• -2?p ? Q - (R/?p)b(?T/?x)

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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

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5340 m
The beta effect
5400 m
Warm
Cold
Warm
-(R/?p)b(?T/?x)lt0
X
-(R/?p)b(?T/?x)gt0
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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

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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

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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

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Q -(R/?p)?T/?yk x (?vg/?x)where the x, y
axes follow respectively, the isotherms, and the
opposite of the temperature gradient
isotherms
cold
y
X
warm
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Q -(R/?p)?T/?yk x (?vg/?x)

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

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To see how this concept works, consider the case
of only horizontal thermal advection forcing the
quasi-geostrophic vertical motions
Q -(R/?p)?T/?yk x (?vg/?x)
(from Sanders and Hoskins 1990)
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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/?yk x (?vg/?x)
(from Sanders and Hoskins 1990)
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(from Sanders and Hoskins 1990)
Q -(R/?p)?T/?yk x (?vg/?x)
Q-vectors in a zone of geostrophic frontogenesis
Q-vectors in the entrance region of an
upper-level jet
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Static stability influence on QG omega

• Consider the QG omega equation (s?2
  f02?2/?p2)? f0?/?pvg??(1/f0?2? f) ?2vg
  ? ?(- ??/?p)
  ?2(heating)friction
• The static stability parameter s-T ?ln?/?p

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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

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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
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Static stability

• Conditional instability occurs when the
  environmental lapse rate lies between the moist
  and dry adiabatic lapse rates Gd
  gt g gt 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)

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Cross-sectional analyses
temperature (degrees C)
The shaded zone illustrates the transition zone
between the upper tropospheres 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!
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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.

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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
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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

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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)

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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
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Q-vectors
Examples
850 mb Analysis 29 July 1997 at 00Z Isentropes
(red), Q-vectors, Vertical motion (shading,
upward only)
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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
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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
• crosses the temperature gradient, the
• ageostrophic flow will produce frontolysis

?c
Q-vectors
?w
Frontogenesis
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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
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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
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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

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References
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 Pettersens
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.
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http//www.crh.noaa.gov/lmk/soo/docu/forcing2.php
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Why Not Look Only at Model Output?
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Good Web Site to Explore VV

• http//www.twisterdata.com/

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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.

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• 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

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QG Diagnostics Online

• Classic approach http//www.mmm.ucar.edu/people/t
  omjr/files/realtime/qg_diag/Omeg700Tot-NorAmer/res
  .html
• Sutcliffe-Trenberthhttp//www.mmm.ucar.edu/people
  /tomjr/files/realtime/qg_diag/OmegSutTren700Tot-No
  rAmer/res.html
• Q-vector http//www.mmm.ucar.edu/people/tomjr/file
  s/realtime/qg_diag/OmegQvec700Tot-NorAmer/res.html

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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

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Jet Streak Vertical Motions
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For unusually straight jets, it might be
reasonable
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But usually there is much more going on, so be
VERY careful in application of simple jet streak
model

• Garp example

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• 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

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