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) 30-60 days

Dominantly Divergent-FlowCirculations?

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

(TOGA-COARE IFA Observation)Heat Budget

Con

vective H

eating(K

/day)

Vertical Advection+Radiation

Con

den

sation(K

/d

ay)Balanced?(Free-Ride,Fraedrich & McBride 1989):Vertical Advection=Diabatic Heating

Scale Analysis (Charney 1963)

Thermodynamic equaton:

i.e., the vertical velocity vanishes to leading orderi.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:

Divergence vorticity

Scatter Plotsbetween Vorticity and Divergencevorticity

vorticity

vorticity

divergence

divergence

divergence

850hPa

500hPa

250hPa

Cumulative Probabilityfor |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)

ho

rizon

tal scale (km

)

Asymptotic Tendency for Non-Divergence:

Divergence/Vorticity(Transient)

Time scale (days)

ho

rizon

tal scale (km

)

(TOGA-COARE IFA Observation)Heat Budget

Con

vective H

eating(K

/day)

Vertical Advection+Radiation

Con

den

sation(K

/d

ay)Balanced?(Free-Ride,Fraedrich & McBride 1989):1. Vertical Advection=Diabatic Heating

Effectively Neutral Stratification:hE=0 :

:No Waves (Gravity)!

Waves ?

Dry Equatorial Waves with hE=25 mOLR Spectrum:

(Wheeler & Kiladis 1999)Equatoriallyasymmetric

Equatoriallysymmetric

Zonal Wavenumber Zonal Wavenumber

Freq

uen

cy

Freq

uen

cy

•Equivalent depth: hE

•Vertical Scale of the wave: D

•Gravity-Wave Speed: cg=(ghE)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: =2L2/aUR.2. VerticalAdvection:

(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:

<(divergence)2>1/2/<(vorticity)2>1/2

(Delayen and Yano, 2009, Tellus)

?

cg=50m/s cg=12m/s

Linear Free Wave Solutions: RMS of divergence/vorticity

Forced Problem

Linear Forced Wave Solutions(cg=50m/s): RMS of divergence/vorticityn=0 n=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 ConvectionSlow 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): vH

(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):40-140E, 20S-20N

(Yano, S. Hagos, C. Zhang)

Last Theorem

“Asymptotic nondivergence” is equivalent to “Longwave approximation” to the linear limit.

Last RemarkHowever, “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)(Yano, Liu, Moncrieff 2012 JAS)

Convective Organizaton?:Point of view of Water BudgetPoint of view of Water Budget

PrecipitationRate, P

Column-Integrated Water, I

?

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

Self-Organized CriticalityHomeistasis(Self-Regulation)

?

Convective Organizaton?:(Yano, Liu, Moncrieff 2012 JAS)(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 CriticalityandHomeostasis: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 andSchubert 1974)•Gaia (Lovelock and Margulis 1974)•Self-Regulation (Raymond 2000)•cybernetics (Wiener 1948)•Buffering (Stevens and Feingold 2009)•Lesiliance (Morrison et al., 2011)