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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

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 asymptotic nondivergence self organization

Large-Scale Tropical Atmospheric Dynamics:

Strongly Divergent ?

or

Asymptotically Nondivergent

?


Large scale tropical atmospheric dynamics asymptotic nondivergence self organization

Strongly Divergent?:Global Satellite Image (IR)


Large scale tropical atmospheric dynamics asymptotic nondivergence self organization

Madden-Julian Oscillation (MJO)

:Madden & Julian (1972)

30-60 days

Dominantly

Divergent-Flow

Circulations

?


Large scale tropical atmospheric dynamics asymptotic nondivergence self organization

MJO is Vorticity Dominant?

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


Large scale tropical atmospheric dynamics asymptotic nondivergence self organization

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


Large scale tropical atmospheric dynamics asymptotic nondivergence self organization

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


Large scale tropical atmospheric dynamics asymptotic nondivergence self organization

Observatinoal Evidences?

TOGA-COARE LSA data set

(Yano, Mulet, Bonazzola 2009, Tellus)



Large scale tropical atmospheric dynamics asymptotic nondivergence self organization

Temporal Evolution

of Longitude-Height Section:

Divergence

vorticity


Large scale tropical atmospheric dynamics asymptotic nondivergence self organization

850hPa

Scatter Plots

between

Vorticity and

Divergence

divergence

vorticity

500hPa

divergence

vorticity

250hPa

divergence

vorticity


Large scale tropical atmospheric dynamics asymptotic nondivergence self organization

Cumulative Probability

for

|divergence/vorticity| :

i.e.,

at majority of points:

Divergence < Vorticity


Large scale tropical atmospheric dynamics asymptotic nondivergence self organization

Quantification:

Measure of a Variability

(RMS of a Moving Average):

where


Large scale tropical atmospheric dynamics asymptotic nondivergence self organization

Asymptotic Tendency for Non-Divergence:

Divergence/Vorticity(Total)

horizontal scale (km)

Time scale (days)


Large scale tropical atmospheric dynamics asymptotic nondivergence self organization

Asymptotic Tendency for Non-Divergence:

Divergence/Vorticity(Transient)

horizontal scale (km)

Time scale (days)


Large scale tropical atmospheric dynamics asymptotic nondivergence self organization

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)!



Large scale tropical atmospheric dynamics asymptotic nondivergence self organization

OLR Spectrum:

Dry Equatorial Waves with hE=25 m

(Wheeler & Kiladis 1999)

Equatorially

asymmetric

Equatorially

symmetric

Frequency

Frequency

Zonal Wavenumber

Zonal Wavenumber


Large scale tropical atmospheric dynamics asymptotic nondivergence self organization


Large scale tropical atmospheric dynamics asymptotic nondivergence self organization

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


Large scale tropical atmospheric dynamics asymptotic nondivergence self organization

Question:

Are the Equatorial Wave Theories consistent with the Asymptotic Nondivergence?


Large scale tropical atmospheric dynamics asymptotic nondivergence self organization

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)


Large scale tropical atmospheric dynamics asymptotic nondivergence self organization

Linear Free Wave Solutions:

RMS of divergence/vorticity

cg=50m/s

cg=12m/s



Large scale tropical atmospheric dynamics asymptotic nondivergence self organization

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

RMS of divergence/vorticity

n=0

n=1


Large scale tropical atmospheric dynamics asymptotic nondivergence self organization

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)


Large scale tropical atmospheric dynamics asymptotic nondivergence self organization

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


Large scale tropical atmospheric dynamics asymptotic nondivergence self organization

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


Large scale tropical atmospheric dynamics asymptotic nondivergence self organization

Asymptotically Nondivergent Dynamics

(Formulation):

  • Leading-Order Dynamics:

    Conservation of Absolute Vorticity:

:Modon Solution?


Large scale tropical atmospheric dynamics asymptotic nondivergence self organization

Is MJO a Modon?:

Absolute

Vorticity

Streamfunction

A snap shot from TOGA-COARE (Indian Ocean):

40-140E, 20S-20N

?

(Yano, S. Hagos, C. Zhang)


Large scale tropical atmospheric dynamics asymptotic nondivergence self organization

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)


Large scale tropical atmospheric dynamics asymptotic nondivergence self organization

Convective Organizaton?:

(Yano, Liu, Moncrieff 2012 JAS)


Large scale tropical atmospheric dynamics asymptotic nondivergence self organization

Convective Organizaton?:

Point of view of Water Budget

Precipitation

Rate, P

?

Column-Integrated Water, I


Large scale tropical atmospheric dynamics asymptotic nondivergence self organization

Convective Organizaton?:

(Yano, Liu, Moncrieff 2012 JAS)

?

Self-Organized Criticality

Homeistasis

(Self-Regulation)


Large scale tropical atmospheric dynamics asymptotic nondivergence self organization

Convective Organizaton?:

(Yano, Liu, Moncrieff 2012 JAS)


Large scale tropical atmospheric dynamics asymptotic nondivergence self organization

Convective organization?:

(Yano, Liu, Moncrieff, 2012, JAS)

with spatial averaging for 4-128km:


Large scale tropical atmospheric dynamics asymptotic nondivergence self organization

Convective organization?:

(Yano, Liu, Moncrieff, 2012, JAS)


Large scale tropical atmospheric dynamics asymptotic nondivergence self organization

Convective organization?:

(Yano, Liu, Moncrieff, 2012, JAS):

dI/dt = F - P


Large scale tropical atmospheric dynamics asymptotic nondivergence self organization

Convective organization?:

(Yano, Liu, Moncrieff, 2012, JAS)


Large scale tropical atmospheric dynamics asymptotic nondivergence self organization

Self-Organized Criticality

and

Homeostasis:

Backgrounds


Large scale tropical atmospheric dynamics asymptotic nondivergence self organization

Self-Organized Criticality:

  • Criticality (Stanley 1972)

  • Bak et al (1987, 1996)

  • Dissipative Structure (Gladsdorff and Prigogine 1971)

  • Synergetics (Haken 1983)

  • Butterfly effect (Lorenz 1963)


Large scale tropical atmospheric dynamics asymptotic nondivergence self organization

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)