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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. Large-Scale Tropical Atmospheric Dynamics:.

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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


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-Flow

Circulations

?


MJO is Vorticity Dominant?

(e.g., Yanai et al., 2000)


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


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)



Temporal Evolution

of Longitude-Height Section:

Divergence

vorticity


850hPa

Scatter Plots

between

Vorticity and

Divergence

divergence

vorticity

500hPa

divergence

vorticity

250hPa

divergence

vorticity


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)

horizontal scale (km)

Time scale (days)


Asymptotic Tendency for Non-Divergence:

Divergence/Vorticity(Transient)

horizontal scale (km)

Time scale (days)


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)!



OLR Spectrum:

Dry Equatorial Waves with hE=25 m

(Wheeler & Kiladis 1999)

Equatorially

asymmetric

Equatorially

symmetric

Frequency

Frequency

Zonal Wavenumber

Zonal Wavenumber



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


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)


Linear Free Wave Solutions:

RMS of divergence/vorticity

cg=50m/s

cg=12m/s



Linear Forced Wave Solutions(cg=50m/s):

RMS of divergence/vorticity

n=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 Convection

    Slow Modulation of the Amplitude of the Vorticity


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


Asymptotically Nondivergent Dynamics

(Formulation):

  • Leading-Order Dynamics:

    Conservation of Absolute Vorticity:

:Modon Solution?


Is MJO a Modon?:

Absolute

Vorticity

Streamfunction

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.

(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)


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:

  • Criticality (Stanley 1972)

  • Bak et al (1987, 1996)

  • Dissipative Structure (Gladsdorff and Prigogine 1971)

  • Synergetics (Haken 1983)

  • Butterfly effect (Lorenz 1963)


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)


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