Dynamical Meteorology in the Tropics: Asymptotic Nondivergence?
by Jun-Ichi Yanowith M. Bonazzola, S. Mulet, K. Delayen, S. Hagos, C. Zhang, D. Netherly
Large-Scale Tropical Tropsopheric Dynamics:•Vorticity is dominant more than Divergence•Deep Convection is secondary, and can be treated as a “catalytic” perturbation effect •Strongly Nonlinear
Large-Scale Tropical Tropsopheric Dynamics:•Vorticity is dominant more than Divergence•Deep Convection is secondary, and can be treated as a “catalytic” perturbation effect •Strongly Nonlinear
Scale Analysis (Charney 1963)
•L~1000km, U~10m/s: vorticity>>divergence i.e., nondiverget to the leading orderor asymtotically nondivergent
•cf., L~3000km, U~3m/s: Linear Equatorial Waves (cf., Yano and Bonazzola, 2009, JAS)
vorticity>>divergence ?
Scatter Plotsbetween Vorticity and Divergence:TOGA-COARELSA Data
vorticity
vorticity
vorticity
divergence
divergence
divergence
850hPa
500hPa
250hPa (cf., Yano, Multet, and Bonazzola, 2009, Tellus)
Measure of a Variability (RMS of a Moving Average):
where
RMS of Divergence/Vorticity (Transient)
Time scale (days)
ho
rizo
nta
l sc
ale
(km
)
Dry Equatorial Waves with hE=25 mOLR Spectrum:
(Wheeler & Kiladis 1999)Equatoriallyasymmetric
Equatoriallysymmetric
Zonal Wavenumber Zonal Wavenumber
Fre
qu
ency
Fre
qu
ency
Is this observational diagnosis consistent with (convectively-coupled) linear equatorial wave theories?
(cf., Delayen and Yano, 2009, Tellus)
cg=50m/s cg=12m/s
Linear Free Wave Solutions: RMS of divergence/vorticity
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
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)
Conclusions:Large-Scale Tropical Tropsopheric
Dynamics:• Asymptotically Nondivergent
•Strong Nonlinearty:
• Asymptotic Nondivergence is much weaker than those expected from linear wave theories(free and forced)
Is MJO a Modon?
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?
Balanced Dynamics (Asymptotic: Charney)
•vorticity equation (prognostic)
•thermodynamic balance: w~Q: 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)
Scale Analysis (Summary)•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)
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