This document discusses using concepts of chaotic mixing to understand the South American monsoon. It presents a methodology to identify Lagrangian Coherent Structures (LCSs) in atmospheric flows using finite-time Lyapunov exponents calculated from back-trajectories. The method is applied to the South American monsoon to analyze seasonal and intraseasonal mechanisms. Seasonally, LCSs vary in key regions like the ITCZ and SACZ depending on interactions between circulations. Intraseasonally, LCS events are linked to rainfall anomalies during events like the MJO or SALLJ. The placement and orientation of LCSs may influence moisture transport and monsoon rainfall on different timescales.

1. How ideas of chaotic mixing can advance the
understanding of the South American monsoon
Gabriel Perez
Pier Luigi Vidale
Helen Dacre
gabrielmpp@protonmail.com
Department of meteorology
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2. Presentation structure
1. Motivation
2. Methodology
• Definition of mixing
• Lagrangian Coherent Structures (LCSs)
• Implementation
3. Application in the South American Monsoon
• Seasonal mechanisms
• Intraseasonal mechanisms
4. Summary
5. Future steps
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3. Effective precipitation from ERA-Interim (Wills and Schneider 2015).
Time averaged atmospheric
moisture budget
Sources and sinks of atmospheric moisture
• The horizontal flow transports water from source to sink regions
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4. • Sink regions are usually associated with long moisture residence times
• Particularly tropical and subtropical convergence zones
• Points to the importance of the history of air parcels (i.e. Lagrangian framework)
Loderach and Sodeman (2016)
Residence time of water vapour in the atmosphere
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5. • How can we identify objectively
structures that shape moisture
distribution in the atmosphere?
• Do they help us understand the SAMS
and its intraseasonal variability?
Science questions
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6. Presentation structure
1. Motivation
2. Methodology
• Definition of mixing
• Lagrangian Coherent Structures (LCSs)
• Implementation
3. Application in the South American Monsoon
• Seasonal mechanisms
• Intraseasonal mechanisms
4. Summary
5. Future steps
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7. Mixing of continuous materials
Mixing is stretching and folding (Ottino,
1989).
• Stretching increases the distance
between neighboring material elements
• Folding decreases de distance between
material elements that were far apart
• Filaments of material arise after some
time
Example: candy gum preparation
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8. Mixing in fluid flows: filamentation is pervasive
• Advection in unsteady flows causes tracers to evolve
as complex fractal filaments, intensifying spatial
gradients (Welander, 1955)
• Even in incompressible 2D flows
• Turbulence at scale L creates tracer patterns at
a scale l << L. Why?
• Because of the chaotic nature of parcel trajectories
even in very simple flows
• Hence the term "chaotic advection" (Aref, 1983)
• In chaotic systems, the exponential separation of
trajectories is expressed by the Lyapunov exponent
Deformation of air by a barotropic cyclone (Welander, 1955)
Attracting axis
Repelling axis
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9. Finite-time Lyapunov exponent
• The Lagrangian deformation we just discussed is generally
expressed by the Finite-time Lyapunov exponent (FTLE):
Where C is the Cauchy-Green strain tensor
Mesoscale oceanic eddies (Beron-Vera et al., 2008)
Structures associated with pollution transport
(Nolan et al., 2020) 9

10. Lagrangian Coherent Structures (LCSs)
• Ridges of the FTLE characrerise Lagrangian Coherent
Structures
• Locally strongest attracting structures (Haller, 2001)
• Transport barriers (Shadden et al., 2005)
• Material structures advected with the flow
• No perpendicular flux
Finite-time Lyapunov exponent in a Von Kármán
vortex street (Kasten et al., 2010)
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11. Case study 06-Feb-2021
• How can we use the chaotic mixing
framework to identify flow structures
responsible for this particular moisture
distribution
• Note: ZCAS event in SA
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12. Algorithm step 1: Flow
1. Define a flow whereby the relevant tracer is
transported (dataset: ERA5 0.25 deg)
2. Define a time window
• Shorter than the mean residence
time of water vapour
• Longer than the life-cycle of local
circulations
• 2 days is a good compromise
06 Feb
04 Feb
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13. Algorithm step 2 - Backtrajectories
2. Compute the back-trajectories for sliding
windows of 2 days )
Implementation details
• 2-time-level iterative scheme (similar to what is used in
IFS/ERA5 advection scheme)
• Cubic spheric spline interpolation
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14. Algorithm step 3 – Deformation/FTLE
3. Calculate the deformation of trajectories departed
from the same neighborhood (I.e., Cauhy-Green strain
tensor)
Implementation details
• 4th order finite-differences in spherical
coordinates (Cython)
The largest eigenvalue of C is used to compute the
FTLE:
Finite-time Lyapunov exponent
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15. Algorithm step 4 - Identifying ridges
4. Find ridges based on the transport barrier criterion
of Shadden (2005)
• Strong negative curvature
• Local maxima
Finite-time Lyapunov exponent and Lagrangian Coherent
Strucutres (LCSs)
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16. 2. Calculate the back-trajectories
06 Feb
1. Define a horizontal flow 3. Compute the FTLE
5. Repeat the steps above for all times
4. Find ridges (LCSs)
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17. A closer look on the ZCAS
1. Initially homegeneous moisture distribution
2. Horizontal gradients are increased around the LCS
3. LCS is a transport barrier preventing tropical moisture
to reach South Brazil/South SA
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18. Presentation structure
1. Motivation
2. Methodology
• Definition of mixing
• Lagrangian Coherent Structures (LCSs)
• Implementation
3. Application in the South American Monsoon
• Seasonal mechanisms
• Intraseasonal mechanisms
4. Summary
5. Future steps
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19. FTLE climatology
• Large scale mixing varies seasonally in three
key regions
1 – ITCZ
• JJA: Stronger and more zonal
• DJF: Weaker and diagonally oriented
• Suggests that the mixing in the ITCZ depends
on the the interaction of the subtropical highs
1.
1.
1.
1.
2 – SACZ region
• DJF: Contours NW-SE oriented
• JJA: Contours are more zonal and stronger
• Possibly depends on the relative position of
ETCs /SASH
2.
2.
2.
2.
3 – N/NEBR coast
• Poorly mixed; parallel trajectories
• Varies according the southeasterlies
3.
3.
3.
3.
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20. LCSs climatology
• More significant changes in the tropics
• Relative position of barriers may
influence moisture input in the
continent
• It seems to be related with the seasonal
marching of the subtropical anticyclones
• Could there be a feedback
mechanism through the Hadley
overturning?
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21. LCSs along the NEBR coast
and SAM rainfall
• LCSs in NEBR seem to respond to the
marching of the Hadley cell (ITCZ and
SASH)
• Orientation (angle) highly in phase
with rainfall
• Links SAM rainfall and horizontal
features of the Hadley circulation
Precip (mm/month)
SAM rainfall index (red) and LCS orientation in NEBR (blue)
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22. Climatology key points
• Strong mixing and LCS occurrence seem to be associated with the
interaction of vortices
• ITCZ -> subtropical anticyclones (feedback Hadley overturning?)
• SACZ -> ETCs and the SASH
• Tropical kinematics seems more responsive to the seasonal cycle than
subtropical/extratropical
• Orientation of LCSs along the coast of NEBR is approx. in phase with
SAM rainfall
• What about intraseasonal variability?
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23. Intraseasonal rainfall and moisture
flux anomalies during LCS events
• LCSs in the Tiete basin
• Anomalies characterise the ZCAS
• Rainfall in the ITCZ is slightly
supressed
• Negative feedbacks the
SASH downwelling?
• LCSs in the Uruguay basin
• Rainfall anomalies resemble
SALLJ events
Summer/Tiete Winter/Tiete
Summer/Uruguay Winter/Uruguay
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24. Large-scale mechanisms associated with LCSs in SEBR
• From chaotic advection: Shifting the energy
spectrum to lower wavenumbers increases
the probability of coherent tracer filaments
(Babiano et al. 1985).
• Example: Mixing sugar in a
teacup (wavenumber 1) versus shaking
the teacup (very high wavenumbers)
• Thus, we expect that the formation
of moisture filaments in these regions can
be affected by teleconnection mechanisms
(e.g., MJO)
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25. Rainfall anomalies in phases 8, 1 and 4 of the MJO in January
Precip
anomaly
(mm/h)
• Phases 8 and 1:
• Stronger SAM rainfall, weaker
Atlantic ITCZ
• Phase 4:
• Weaker SAM rainfall, stronger
Atlantic ITCZ
• But why? Is this caused by how
circulation changes the moisture
distribution?
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26. FTLE anomalies:
• Phases 8 and 1
• SACZ: Stronger mixing increases
moisture accumulation
• ITCZ: Weaker mixing reduces
moisture content
• Potential negative feedback on
the Hadley overtnurning
• Phase 4 (opposite)
• Stronger ITCZ and SASH
• Potential positive feedback on
the Hadley overturning
Precip
anomaly
(mm/h)
FTLE
anomaly
(1/day)
Role of circulation in creating MJO rainfall anomalies
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27. Conclusions
• Our framework provides a diagnostics of features
shaping moisture, particularly tropical and
subtropical convergence zones
• By using a general a priori metric, as opposed to a
posteriori region-specific metrics (e.g. definitions of
ZCAS), it is possible to uncover new mechanisms
and links between structures
• It is a process-based approach that help us to
understand why the SA rainfall varies in different
scales
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28. Next steps
• Investigating connections with the Northern Hemisphere through the
Hadley overturning feedback (e.g. NAO)
• Employ the framework in GCMs to understand rainfall changes in past
and future climate
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29. Paper describing the method and preliminary results
Code for anyone interesting in using or
collaborating
https://github.com/gabrielmpp/LagrangianCoherence
Thanks!
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