1. 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

2. 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 (MWR:2001) –Explosive cyclone development accompanied by upper-air wave merger over North America (Nov. 2-3 1999)

3. 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.

4. COMPUTATIONAL TECHNIQUES 1.Kinematic – Integrated Divergence/Convergence in Continuity Equation 2.Vorticity – Integrated Vorticity Advection/Tendency 3.Adiabatic – Temperature Advection/Tendency 4.Isentropic – Pressure Advection along Potential Temperature Surface 5.Omega Equation – Combine Vorticity Equation and 1 st Law of Thermodynamics

5. 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

6. 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).

7. FORMS OF OMEGA EQUATION 1.BASIC QUASIGEOSTROPHIC 2.Q-VECTOR 3.TRENBERTH (2) AND (3) ACCOUNTS FOR CANCELLATION FEATURE FOUND IN (1). 4.APPROXIMATE TRENBERTH NEGLECTS DEFORMATION TERM 5.EXTENDED INCLUDES AGEOSTROPHIC WIND

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

9. 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

10. 12 UTC 2 Nov. 1999 (TOP): SL Pressure & 6hr. Precip. (MIDDLE): 500-mb Height & Abs. Vort. (BOTTOM): 200-mb Height & Winds 984 mb 0000 3 rd 1000 mb 1200 2 nd 1009 mb 0000 2 nd

11. 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

12. 500MB Z: SOLID 700-300MB OMEGA: DASHED SHADED: UPWARD MOTION EXT QG Q TREN AT

13. % 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

14. 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.