1. Dynamical Meteorology in the Tropics: Asymptotic Nondivergence?by
Jun-Ichi Yano
with
M. Bonazzola, S. Mulet, K. Delayen,
S. Hagos, C. Zhang, D. Netherly

2. 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

3. 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

4. Scale Analysis (Charney 1963)
L~1000km, U~10m/s: vorticity>>divergence
i.e., nondiverget to the leading order
or asymtotically nondivergent
cf., L~3000km, U~3m/s:
Linear Equatorial Waves
(cf., Yano and Bonazzola, 2009, JAS)

5. vorticity>>divergence?

6. Scatter Plots between Vorticity and Divergence: TOGA-COARE LSA Data
850hPa
Scatter Plots
between
Vorticity and
Divergence:
TOGA-COARE
LSA Data
divergence
vorticity
500hPa
divergence
vorticity
(cf., Yano,
Multet,
and
Bonazzola,
2009, Tellus)
250hPa
divergence
vorticity

7. Measure of a Variability
(RMS of a Moving Average):
where

8. RMS of Divergence/Vorticity (Transient)
Time scale (days)
horizontal scale (km)

9. Dry Equatorial Waves with hE=25 m (Wheeler & Kiladis 1999)
OLR Spectrum:
Dry Equatorial Waves with hE=25 m
(Wheeler & Kiladis 1999)
Equatorially
asymmetric
symmetric
Zonal Wavenumber
Frequency

10. Is this observational diagnosis consistent with (convectively-coupled)
linear equatorial wave theories?
(cf., Delayen and Yano, 2009, Tellus)

11. Linear Free Wave Solutions:
RMS of divergence/vorticity
cg=50m/s
cg=12m/s

12. Linear Forced Wave Solutions(cg=50m/s):
RMS of divergence/vorticity
n=0
n=1

13. 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)

14. 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

15. Asymptotically Nondivergent Dynamics (Formulation):
Leading-Order Dynamics:
Conservation of Absolute Vorticity:
:Modon Solution?

16. ? Is MJO a Modon?: A snap shot from TOGA-COARE (Indian Ocean):
40-140E, 20S-20N
Streamfunction
Absolute
Vorticity ?
(Yano, S. Hagos, C. Zhang)

17. Large-Scale Tropical Tropsopheric Dynamics:
Conclusions:
Large-Scale Tropical Tropsopheric Dynamics:
Asymptotically Nondivergent
Asymptotic Nondivergence is much weaker than those expected from linear wave theories
(free and forced)
Is MJO a Modon?
Strong Nonlinearty:

18. Last Theorem Last Remark
“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)

19. } Balanced Dynamics (Asymptotic: Charney) thermodynamic balance: w~Q:
Q w
Q=Q(q, q,… ) }
dynamic balance: non-divergent
divergence equation (diagnostic) f
vorticity equation (prognostic)
barotropics b-plane vorticity equation
Rossby waves (without geostrophy): vH(0)
hydrostatic balance: f q
moisture equation (prognostic): q
continuity: w weak divergence
weak forcing on vorticity (slow time-scale)

20. Scale Analysis (Summary)
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: b =2WL2/aU
R.2. Vertical
Advection:

21. 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