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 Sciences Purdue University West Lafayette, Indiana

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A COMPARISON OF VERTICAL MOTIONS OBTAINED FROM
DIFFERENT FORMS OF THE OMEGA EQUATIONChristopher
J. MelickandPhillip J. SmithDepartment of
Earth and Atmospheric SciencesPurdue
UniversityWest Lafayette, Indiana
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OBJECTIVE

• Comparison of vertical motions obtained from
  Four quasi-geostrophic (QG) forms and one
  ageostrophic form (extended) of the omega
  equation.
• Case study Strahl and Smith (MWR2001)
• Explosive cyclone development accompanied by
  upper-air wave merger over North America (Nov.
  2-3 1999)

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INTRODUCTION

• Scale Analysis
• V 10 m/sec
• W 1 cm/sec
• ?Too small to be measured directly.
• ?Computational methods required.
• NOTE Assumes synoptic-scale (large spatial and
  temporal) conditions.

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COMPUTATIONAL TECHNIQUES

• Kinematic Integrated Divergence/Convergence in
  Continuity Equation
• Vorticity Integrated Vorticity
  Advection/Tendency
• Adiabatic Temperature Advection/Tendency
• Isentropic Pressure Advection along Potential
  Temperature Surface
• Omega Equation Combine Vorticity Equation and
  1st Law of Thermodynamics

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QG THEORY

• Ageostrophic motions and hence vertical motions
  are produced as the atmosphere progresses from
  one geostrophically balanced state to another
• Geostrophic Balance PGF CF

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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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FORMS OF OMEGA EQUATION

• BASIC QUASIGEOSTROPHIC
• Q-VECTOR
• TRENBERTH
• (2) AND (3) ACCOUNTS FOR CANCELLATION FEATURE
  FOUND IN (1).
• APPROXIMATE TRENBERTH
• NEGLECTS DEFORMATION TERM
• EXTENDED
• INCLUDES AGEOSTROPHIC WIND

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• Surface/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

• LAYER EXAMINED 700-300mb AVERAGED VERTICAL
  MOTIONS
• CORRELATIONS/ MEAN ABSOLUTE VALUES
• SYNOPTIC PATTERNS
• OMEGA vs. SYNOPTIC FEATURES
• PRECIPITATION PATTERNS
• PRECIP IN UPWARD MOTION

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12 UTC 2 Nov. 1999 (TOP)SL Pressure
6hr. Precip. (MIDDLE)500-mb Height Abs.
Vort. (BOTTOM) 200-mb Height Winds
984 mb 0000 3rd
1000 mb 1200 2nd
1009 mb 0000 2nd
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• CC (omega patterns)
• ?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 (omega magnitudes)
• ?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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PRECIP IN UPWARD MOTION

• ?E ?QG ?Q ?T ?AT
• 93 87 97 91 82
• All results are similar in magnitude
• No clear evidence of one form being superior

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CONCLUSIONS

• ALL VERTICAL MOTION FIELDS COMPARE FAVORABLY
  WITH SYNOPTIC FEATURES AND PRECIPITATION FIELDS.
• CANNOT DETERMINE SUPERIOR METHOD USING SIMPLE
  SYNOPTIC AND PRECIPITATION COMPARISONS.
• QG APPROXIMATION WORKS QUITE WELL EVEN FOR A CASE
  OF EXPLOSIVE CYCLONE DEVELOPMENT.