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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)
Madden-Julian Oscillation (MJO) :Madden & Julian (1972) 30-60 days
Dominantly Divergent-FlowCirculations?
(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
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
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)!
Dry Equatorial Waves with hE=25 mOLR Spectrum:
(Wheeler & Kiladis 1999)Equatoriallyasymmetric
Equatoriallysymmetric
Zonal Wavenumber Zonal Wavenumber
Freq
uen
cy
Freq
uen
cy
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)
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)
?
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?: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)
?
Self-Organized Criticality:
•Bak et al (1987, 1996)•Criticality (Stanley 1972)
•Dissipative Structure (Gladsdorff and Prigogine 1971)
•Butterfly effect (Lorenz 1963)
•Synergetics (Haken 1983)