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?
Condensation(K/day)
Vertical Advection
=Diabatic Heating
Heat Budget
Convective
Heating(K/day)
(Free-Ride,
Fraedrich &
McBride 1989):
(TOGA-COARE IFA Observation)
Vertical Advection+Radiation
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:
Divergence
vorticity
850hPa
divergence
vorticity
500hPa
divergence
vorticity
250hPa
divergence
vorticity
Scatter Plots
between
Vorticity and
Divergence
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?
Effectively Neutral
Stratification:hE=0 :
:No Waves (Gravity)!
Condensation(K/day)
1. Vertical Advection
=Diabatic Heating
Heat Budget
Convective
Heating(K/day)
(Free-Ride,
Fraedrich &
McBride 1989):
(TOGA-COARE IFA Observation)
Vertical Advection+Radiation
Waves ?
OLR Spectrum:
Dry Equatorial Waves with hE=25 m
(Wheeler & Kiladis 1999)
Equatorially
symmetric
Equatorially
asymmetric
Frequency
Frequency
Zonal Wavenumber
Zonal Wavenumber
•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)
(Simple)
(Asymptotic)
R.1. Nondimensional: =2L2/aU
R.2. Vertical
Advection:
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
Forced Problem
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)
•thermodynamic balance: w~Q: Q w
Q=Q(q,… )
(free ride)
•dynamic balance: non-divergent
•divergence equation (diagnostic)

•vorticity equation (prognostic)
barotropics -plane vorticity equation
Rossby waves (without geostrophy): vH(0)
•hydrostatic balance: 
•moisture equation (prognostic): q
•continuity: w weak divergence
weak forcing on vorticity (slow time-scale)
}
Asymptotically Nondivergent Dynamics
(Formulation):
•Leading-Order Dynamics:
Conservation of Absolute Vorticity:
:Modon Solution?
Is MJO a Modon?:
A snap shot from TOGA-COARE (Indian Ocean):
40-140E, 20S-20N
?
Absolute
Vorticity
Streamfunction
(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)