Large-Scale Tropical Atmospheric Dynamics: Asymptotic Nondivergence

1. Large-Scale Tropical Atmospheric Dynamics: Asymptotic Nondivergence & Self-Organization (& Self-Organization) by Jun-Ichi Yano With Sandrine Mulet, Marine Bonazzola, Kevin Delayen, S. Hagos, C. Zhang, Changhai Liu, M. Moncrieff

2. Large-Scale Tropical Atmospheric Dynamics: Strongly Divergent ? or Asymptotically Nondivergent ?

3. Strongly Divergent?:Global Satellite Image (IR)

4. Madden-Julian Oscillation (MJO) :Madden & Julian (1972) 30-60 days Dominantly Divergent-Flow Circulations ?

5. MJO is Vorticity Dominant? (e.g., Yanai et al., 2000)

6. Balanced? (Free-Ride, Fraedrich & McBride 1989): (TOGA-COARE IFA Observation) moisture Heat Budget Condensation(K/day) Convective Heating(K/day) Vertical Advection+Radiation Vertical Advection Vertical Advection =Diabatic Heating

7. Scale Analysis (Charney 1963) Thermodynamic equaton: i.e., the vertical velocity vanishes to leading order i.e., the horizontal divergence vanishes to leading order of asymptotic expansion i.e., Asymptotic Nondivergence

9. Observatinoal Evidences? TOGA-COARE LSA data set (Yano, Mulet, Bonazzola 2009, Tellus)

10. Vorticity >> Divergence with MJO:

11. Temporal Evolution of Longitude-Height Section: Divergence vorticity

12. 850hPa Scatter Plots between Vorticity and Divergence divergence vorticity 500hPa divergence vorticity 250hPa divergence vorticity

13. Cumulative Probability for |divergence/vorticity| : i.e., at majority of points: Divergence < Vorticity

14. Quantification: Measure of a Variability (RMS of a Moving Average): where

15. Asymptotic Tendency for Non-Divergence: Divergence/Vorticity(Total) horizontal scale (km) Time scale (days)

16. Asymptotic Tendency for Non-Divergence: Divergence/Vorticity(Transient) horizontal scale (km) Time scale (days)

17. Balanced? (Free-Ride, Fraedrich & McBride 1989): (TOGA-COARE IFA Observation) moisture Heat Budget Condensation(K/day) Convective Heating(K/day) Effectively Neutral Stratification:hE=0 : Vertical Advection+Radiation Vertical Advection 1. Vertical Advection =Diabatic Heating :No Waves (Gravity)!

18. Waves ?

19. OLR Spectrum: Dry Equatorial Waves with hE=25 m (Wheeler & Kiladis 1999) Equatorially asymmetric Equatorially symmetric Frequency Frequency Zonal Wavenumber Zonal Wavenumber

20. Equivalent depth: hE • Vertical Scale of the wave: D • Gravity-Wave Speed: cg=(ghE)1/2~ND

21. Scale Analysis (Summary):Yano and Bonazzola (2009, JAS) (Simple) (Asymptotic) R.2. Vertical Advection: • L~3000km, U~3m/s (cf., Gill 1980): Wave Dynamics (Linear) • L~1000km, U~10m/s (Charney 1963): Balanced Dynamics (Nonlinear) R.1. Nondimensional: =2L2/aU

22. Question: Are the Equatorial Wave Theories consistent with the Asymptotic Nondivergence?

23. A simple theoretical analysis: RMS Ratio between the Vorticity and the Divergence for Linear Equaotorial Wave Modes: <(divergence)2>1/2/<(vorticity)2>1/2 ? (Delayen and Yano, 2009, Tellus)

25. Linear Free Wave Solutions: RMS of divergence/vorticity cg=50m/s cg=12m/s

26. Forced Problem

27. Linear Forced Wave Solutions(cg=50m/s): RMS of divergence/vorticity n=0 n=1

28. Asymptotically Nondivergent but Asymptotic Nondivergence is much weaker than those expected from linear wave theories (free and forced) Nonlinearity defines the divergence/vorticity ratio (Strongly Nonlinear)

29. Asymptotically Nondivergent Dynamics (Formulation): • Leading-Order Dynamics: Conservation of Absolute Vorticity • Higher-Order: Perturbation“Catalytic” Effect of Deep Convection Slow Modulation of the Amplitude of the Vorticity

30. Balanced Dynamics (Asymptotic: Charney) Qw Q=Q(q,… ) • divergence equation (diagnostic)  barotropics -plane vorticity equation Rossby waves (without geostrophy): vH(0) } • hydrostatic balance:  • continuity: w weak divergence weak forcing on vorticity (slow time-scale) • thermodynamic balance: w~Q: (free ride) • dynamic balance: non-divergent • vorticity equation (prognostic) • moisture equation (prognostic): q

31. Asymptotically Nondivergent Dynamics (Formulation): • Leading-Order Dynamics: Conservation of Absolute Vorticity: :Modon Solution?

32. Is MJO a Modon?: Absolute Vorticity Streamfunction A snap shot from TOGA-COARE (Indian Ocean): 40-140E, 20S-20N ? (Yano, S. Hagos, C. Zhang)

33. Last Theorem “Asymptotic nondivergence” is equivalent to “Longwave approximation” to the linear limit. (man. rejected by Tellus 2010, JAS 2011) Last Question: What is wrong with this theorem? Last Remark However, “Asymptotic nondivergence” provides a qualitatively different dynamical regime under Strong Nonlinearity. Reference: Wedi and Smarkowiscz (2010, JAS)

34. Convective Organizaton?: (Yano, Liu, Moncrieff 2012 JAS)

35. Convective Organizaton?: Point of view of Water Budget Precipitation Rate, P ? Column-Integrated Water, I

36. Convective Organizaton?: (Yano, Liu, Moncrieff 2012 JAS) ? Self-Organized Criticality Homeistasis (Self-Regulation)

37. Convective Organizaton?: (Yano, Liu, Moncrieff 2012 JAS)

38. Convective organization?: (Yano, Liu, Moncrieff, 2012, JAS) with spatial averaging for 4-128km:

39. Convective organization?: (Yano, Liu, Moncrieff, 2012, JAS)

40. Convective organization?: (Yano, Liu, Moncrieff, 2012, JAS): dI/dt = F - P

41. Convective organization?: (Yano, Liu, Moncrieff, 2012, JAS)

42. Self-Organized Criticality and Homeostasis: Backgrounds

43. Self-Organized Criticality: • Criticality (Stanley 1972) • Bak et al (1987, 1996) • Dissipative Structure (Gladsdorff and Prigogine 1971) • Synergetics (Haken 1983) • Butterfly effect (Lorenz 1963)

44. Homeostasis: • etimology: homeo (like)+stasis(standstill) • Psyology: Cannon (1929, 1932) • Quasi-Equilibrium (Arakawa and Schubert 1974) • Gaia (Lovelock and Margulis 1974) • Self-Regulation (Raymond 2000) • cybernetics (Wiener 1948) • Buffering (Stevens and Feingold 2009) • Lesiliance (Morrison et al., 2011)