A Comparison of Vertical Motions Obtained from Different Forms of the Omega Equation Christopher J'

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A Comparison of Vertical Motions Obtained from
Different Forms of the Omega Equation
Christopher J. Melick and Phillip J. Smith
Department of Earth and Atmospheric
SciencesPurdue UniversityWest Lafayette, IN
February 9th, 2004Current Affiliation
Department of Soils, Environmental, and
Atmospheric SciencesUniversity of
Missouri-ColumbiaColumbia, MO
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OBJECTIVE

• Comparison of vertical motions Four
  quasigeostrophic (QG) forms of the omega equation
  and one with ageostrophic forcing implicitly
  included (extended)
• Case study Explosive cyclone development
  accompanied by upper-air wave merger over North
  America (Nov. 2-3 1999)

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PRESENTATION FORMAT

• 1.) Introduction and Theory
• Basic Description of Vertical Motions
• Review of QG Theory (Holton 1992)
• Computational Methods
• 2.) Data and Methodology
• Comparison Techniques
• 3.) Synoptic Discussion
• Case Study 2-3 November 1999
• Strahl and Smith (2001) Mon. Wea. Rev.
• 4.) Results
• 5.) Summary and Conclusions

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INTRODUCTION

• Importance of Vertical Air Motions (Smith 1971)
• 1) Vertical transport of atmospheric properties
  
• 2) Temperature changes ? clouds/precipitation
• Scale Analysis Equation of Motion (Holton 1992,
  p.39)
• Synoptic-Scale (spatial 106 m temporal 105
  sec)
• Horizontal 10 m/s
• Vertical w 1 cm/s (height) ? -1 µb/s
  (pressure)
• Vertical motions Too small to be measured
  directly.
• Computational methods required.

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QUASI-GEOSTROPHIC (QG) THEORY

• Geostrophic Balance
• Coriolis Force Pressure Gradient Force
• For midlatitude synoptic-scale systems, the flow
  is approximately (quasi) geostrophic.
• Ageostrophic motions, Divergence fields, and
  vertical motions permitted as the atmosphere
  progresses from one balanced state to another.

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COMPUTATIONAL TECHNIQUES

• Kinematic Integrate Continuity Equation
• Vorticity Integrate Vorticity Equation
• Adiabatic Isolate ? in 1st Law of
  Thermodynamics
• Isentropic Pressure Advection along Potential
  Temperature Surface
• Omega Equation Combine Vorticity Equation and
• 1st Law of Thermodynamics

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QG OMEGA EQUATION (BASIC FORM)

• (A) (B)
• Regions of UPWARD MOTION
• Vorticity Advection increasing w/height (A) WAA
  (B).
• Regions of DOWNWARD MOTION
• Vorticity Advection decreasing w/height (A) CAA
  (B).

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DIFFERENT REPRESENTATIONS

• MOTIVATION Differential vorticity and
  temperature advection terms are out of phase in
  the middle to upper atmosphere.
• This cancellation produces uncertainty in the
  total forcing, especially when both terms are of
  similar magnitude.
• In order to account for this feature, different
  representations of the omega equation were
  developed (Hoskins et al. 1978, Trenberth 1978)

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• ALTERNATE FORMS OF THE OMEGA EQUATION
• Q-VECTOR (Hoskins et al. 1978)
• Q-vector Convergence/Divergence
• TRENBERTH (1978)
• APPROXIMATE TRENBERTH
• Neglects Deformation term
• Assumed to be significant only near frontal
  zones/strong jets
• However, Martin (1998) found a case where
  deformation term was significant in
  mid-troposphere
• EXTENDED (Tsou et al. 1987)
• Replaces the QG wind and relative vorticity with
  observed or analyzed values.

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• Sfc/Upper-Air Data 0000 UTC Nov. 2nd to 0000 UTC
  Nov. 3rd
• 2-pass Barnes Analysis Scheme
• 25x17 grid
• Dashed Box Computational Domain
• Solid Box
• Display Domain

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COMPARISON METHODS

• Correlation Coefficients vertical motion fields
• Mean Absolute Values vertical motion magnitudes
• Map comparisons Omega vs. Synoptic wave patterns
  
• Precipitation Relationships Test superiority of
  method
• Precipitation occurs in regions of large-scale
  ascent.
• RESULTS ? Expressed as Averages
• 24-h (NOV 2nd 3rd 1999) and 700-300mb

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Precipitation Scoring ProcedureSimilar to Smith
and Lin (1978)

• 1 Point given to method(s)
• 1 Largest of precipitation with upward motion
• 2 Weak (W) precipitation class
• Largest (smallest) in upward (downward) motion
  categories
• 3 Intense (I) precipitation class
• Largest (smallest) in the two strongest (weakest)
  upward motion categories
• 4 Tie between multiple forms of the omega
  equation

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COMPARISON METHODS

• AVAILABLE POTENTIAL ENERGY(APE)
• Measure of baroclinicity in atmosphere (Lorenz
  1955)
• Conversion between APE Kinetic Energy (KE)
• Horizontal pressure gradients are altered as
  warm air rises and cold air sinks. Thus, KE
  increases at the expense of APE.
• APE Release Mathematical Representation
• Mass integral of omega and specific volume
  (1000-100mb)
• Question Why is released APE important?
• Answer Represents a useful measure of the
  intensity of cyclone development

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12 UTC 2 Nov. 1999 (TOP)Pressure 6hr.
Precip. (MIDDLE)500mb Hght Abs. Vort.
(BOTTOM) 200mb Height Winds
984 mb 0000 3rd
1000 mb 1200 2nd
1009 mb 0000 2nd
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Satellite Infrared Images

• 0000 UTC 2 Nov 1999 1200 UTC 2 Nov 1999
  0000 UTC 3 1999

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• Correlation Coefficients
• ?QG ?Q ?T ?AT
• ?E 0.66 0.59 0.59
  0.62
• ?QG 1.00 0.91 0.96
  0.90
• ?Q xxxxx 1.00 0.95
  0.85
• ?T xxxxx xxxxx 1.00 0.90
• MAV (microbars/sec)
• ?QG ?Q ?T ?AT
  ?E
• 0.69 0.90 0.75 0.94
  0.89

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TREN
EXT
AT
QG
500MB Z SOLID 700-300MB OMEGA DASHED SHADED
UPWARD MOTION
Q
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Precipitation Scoring Results

• ?E ?QG ?Q ?T ?AT
• 5 4 11 4 2
• Majority of precipitation reports occurring in
  areas of synoptic-scale ascent
• However, quantitative tabulation indicates
• Q-vector form to be superior method
• Poorest performance Approximate Trenberth

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APE Release (W m-2)

• EXTENDED HAS MAXIMUM AT MIDPOINT TIME
• BEGINNING OF EXPLOSIVE CYCLONE DEVELOPMENT
• KENNEY AND SMITH (1983) RESULTS ARE SIMILAR
• COMPARABLE TEMPORAL DISTRIBUTION SUGGESTS
  EXTENDED PROVIDES BETTER REPRESENTATION OF APE
  RELEASE

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CONCLUSIONS

• INCLUSION OF AGEOSTROPHIC FORCING ALTERS THE
  VERTICAL MOTION PATTERN MORE THEN OVERALL
  MAGNITUDES OF VERTICAL MOTION.
• HOWEVER, QG APPROXIMATION GIVES A SUFFICIENT
  ESTIMATE OF SYNOPTIC-SCALE VERTICAL MOTION.
• PRECIPITATION FIELD COMPARISONS MOST FAVOR THE
  Q-VECTOR FORM OF THE OMEGA EQUATION
• HOWEVER, CYCLONE SYSTEM DEVELOPMENT REPRESENTED
  BETTER BY EXTENDED METHOD.

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FUTURE WORK

• INCLUDE MORE CASE STUDIES
• RELIABILITY IN RESULTS INCREASES
• EXAMINE AFFECT OF INCREASING SPATIAL RESOLUTION
  (i.e. FINER DETAIL RESOLVED)
• QG APPROACHES SHOULD BECOME LESS RELIABLE

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• LIST OF REFERENCES
• Barnes, S.L., 1964 A technique for maximizing
  details in numerical weather map analysis. J.
  App. Meteor., 3, 396-409.
• _____, 1973 Mesoscale objective map analysis
  using weighted time series observations. NOAA
  Technical Memorandum, ERL NSSL-62. 60pp.
• Bluestein, H.B., 1992 Synoptic-Dynamic
  Meteorology in Midlatitudes Principles of
  Kinematics and Dynamics, Vol. 1. Oxford
  University Press, New York. 431pp.
• Carlson, T.N., 1998 Mid-Latitude Weather
  Systems. American Meteorological Society, Boston.
  507pp.
• Holton, J.R., 1992 An Introduction to Dynamic
  Meteorology. 3rd ed. Academic Press, San Diego.
  511pp.
• Hoskins, B.J., I. Draghici, and H.C. Davies,
  1978 A new look at the ?-equation. Quart. J. R.
  Met. Soc., 104, 31-38.
• Kenney, S.E., and P.J. Smith, 1983 On the
  release of eddy available potential energy in an
  extratropical cyclone system. Mon. Wea. Rev.,
  111, 745-755.
• Krishnamurti, T.N., 1968 A diagnostic balance
  model for studies of weather systems of low and
  high latitudes, Rossby Number less than 1. Mon.
  Wea Rev., 96, 197-207.

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• Lorenz, E.N., 1955 Available potential energy
  and the maintenance of the general circulation.
  Tellus., 7, 157-167.
• Martin, J.E., 1998 On the deformation term in
  the quasigeostrophic omega equation. Mon. Wea.
  Rev., 126, 2000-2007.
• Rolfson, D.M., and P.J. Smith, 1996 A composite
  diagnosis of synoptic-scale extratropical cyclone
  development over the United States. Mon. Wea.
  Rev., 124, 1084-1099.
• Smith, P.J., 1970 A note on energy conversions
  in open atmospheric systems. J. Atmos. Sci., 27,
  518-521.
• ____, and C.P. Lin, 1978 A comparison of
  synoptic-scale vertical motions computed by the
  kinemtic method and two forms of the omega
  equation. Mon. Wea. Rev., 106, 1687-1694.
• Strahl, J.L.S., and P.J. Smith, 2001 A
  diagnostic study of an explosively developing
  extratropical cyclone and an associated 500-hPa
  trough merger. Mon. Wea. Rev., 129, 2310-2328.
• Trenberth, K.E., 1978 On the interpretation of
  the diagnostic quasi-geostrophic omega equation.
  Mon. Wea. Rev., 106, 131-137.
• Tsou, C.-H., P.J. Smith, and P.M. Pauley, 1987
  A comparison of adiabatic and diabatic forcing in
  an intense extratropical cyclone system. Mon.
  Wea. Rev., 115, 763-786.

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THANK YOU!!QUESTIONS/COMMENTS??Special Thanks
Drs. Phillip J. Smith, Anthony R. Lupo, and
Patrick S. Market