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