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

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

Strongly Divergent?: Global Satellite Image (IR)

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

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

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

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

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

Vorticity >> Divergence with MJO:

Temporal Evolution of Longitude-Height Section: Divergencevorticity

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

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

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

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

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

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

Waves ?

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

Equivalent depth: h E Vertical Scale of the wave: D Gravity-Wave Speed: c g =(gh E ) 1/2 ~ND

Scale Analysis (Summary): Yano and Bonazzola (2009, JAS) 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  L 2 /aU R.2. Vertical Advection: (Simple) (Asymptotic)

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

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

c g =50m/s c g =12m/s Linear Free Wave Solutions: RMS of divergence/vorticity

Forced Problem

Linear Forced Wave Solutions(c g =50m/s): RMS of divergence/vorticity n=0n=1

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)

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

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

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

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

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

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

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

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

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

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

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

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

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

Self-Organized Criticality and Homeostasis: Backgrounds

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

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)