Lagrangian transport in the dynamical systems approach has so far been investigated disregarding the connection between the whole state space and the concept of observability. Key issues such as the definitions of Lagrangian and chaotic mixing are revisited under this light, establishing the importance of rewriting nonautonomous flow systems derived from a stream function in autonomous form, and of not restricting the characterization of their dynamics in subspaces. The observability of Lagrangian chaos from a reduced set of measurements is illustrated with two canonical examples: the Lorenz system derived as a low-dimensional truncation of the Rayleigh-Bénard convection equations and the driven double-gyre system introduced as a kinematic model of configurations observed in the ocean. A symmetrized version of the driven double-gyre model is proposed.
REFERENCES
1.
G. K.
Batchelor
, An Introduction to Fluid Dynamics
(Cambridge University Press
, 2000
).2.
S.
Wiggins
and J. M.
Ottino
, “Foundations of chaotic mixing
,” Philos. Trans. R. Soc. London
362
, 937
–970
(2004
). 3.
J. M.
Ottino
, “Mixing, chaotic advection and turbulence
,” Annu. Rev. Fluid Mech.
22
, 207
–253
(1990
). 4.
T.
Bohr
, M. H.
Jensen
, G.
Paladin
, and A.
Vulpiani
, Dynamical Systems Approach to Turbulence
(Cambridge University Press
, 2005
).5.
O.
Ménard
, C.
Letellier
, J.
Maquet
, L. L.
Sceller
, and G.
Gouesbet
, “Analysis of a non-synchronized sinusoidally driven dynamical system
,” Int. J. Bifurcat. Chaos
10
, 1759
–1772
(2000
). 6.
A.
Carrassi
, M.
Ghil
, A.
Trevisan
, and F.
Uboldi
, “Data assimilation as a nonlinear dynamical systems problem: Stability and convergence of the prediction-assimilation system
,” Chaos
18
, 023112
(2008
). 7.
M.
Ghil
, “The compatible balancing approach to initialization, and four-dimensional data assimilation
,” Tellus
32
, 198
–206
(1980
). 8.
M.
Ghil
and P.
Malanotte-Rizzoli
, Data Assimilation in Meteorology and Oceanography
(Elsevier
, 1991
), pp. 141
–266
.9.
M.
Ghil
, “Advances in sequential estimation for atmospheric and oceanic flows
,” J. Meteorolog. Soc. Japan Ser. II
75
, 289
–304
(1997
). 10.
R.
Hermann
and A.
Krener
, “Nonlinear controllability and observability
,” IEEE Trans. Automat. Contr.
22
, 728
–740
(1977
). 11.
C.
Letellier
, L. A.
Aguirre
, and J.
Maquet
, “Relation between observability and differential embeddings for nonlinear dynamics
,” Phys. Rev. E
71
, 066213
(2005
). 12.
P.
Bergé
, Y.
Pomeau
, and C.
Vidal
, L’ordre dans le Chaos—Vers une Approche Déterministe de la Turbulence
(Hermann
, 1997
).13.
E. N.
Lorenz
, “Deterministic nonperiodic flow
,” J. Atmos. Sci.
20
, 130
–141
(1963
). 14.
H.
Yang
and Z.
Liu
, “Chaotic transport in a double gyre ocean
,” Geophys. Res. Lett.
21
, 545
–548
, (1994
). 15.
A. C.
Poje
and G.
Haller
, “Geometry of cross-stream mixing in a double-gyre ocean model
,” J. Phys. Oceanogr.
29
, 1649
–1665
(1999
). 16.
E.
Simonnet
, M.
Ghil
, and H.
Dijkstra
, “Homoclinic bifurcations in the quasi-geostrophic double-gyre circulation
,” J. Marine Res.
63
, 931
–956
(2005
). 17.
T.
Matsuura
and M.
Fujita
, “Two different aperiodic phases of wind-driven ocean circulation in a double-gyre, two-layer shallow-water model
,” J. Phys. Oceanogr.
36
, 1265
–1286
(2006
). 18.
S. C.
Shadden
, F.
Lekien
, and J. E.
Marsden
, “Definition and properties of Lagrangian coherent structures from finite-time Lyapunov exponents in two-dimensional aperiodic flows
,” Physica D
212
, 271
–304
(2005
). 19.
K. G. D. S.
Priyankara
, S.
Balasuriya
, and E.
Bollt
, “Quantifying the role of folding in nonautonomous flows: The unsteady double-gyre
,” Int. J. Bifurcat. Chaos
27
, 1750156
(2017
). 20.
D.
Lipinski
and K.
Mohseni
, “A ridge tracking algorithm and error estimate for efficient computation of Lagrangian coherent structures
,” Chaos
20
, 017504
(2010
). 21.
M. R.
Allshouse
and T.
Peacock
, “Lagrangian based methods for coherent structure detection
,” Chaos
25
, 097617
(2015
). 22.
K. R.
Pratt
, J. D.
Meiss
, and J. P.
Crimaldi
, “Reaction enhancement of initially distant scalars by Lagrangian coherent structures
,” Phys. Fluids
27
, 035106
(2015
). 23.
M. O.
Williams
, I. I.
Rypina
, and C. W.
Rowley
, “Identifying finite-time coherent sets from limited quantities of Lagrangian data
,” Chaos
25
, 087408
(2015
). 24.
H.
Aref
, J. R.
Blake
, M.
Budišić
, S. S. S.
Cardoso
, J. H. E.
Cartwright
, H. J. H.
Clercx
, K.
El Omari
, U.
Feudel
, R.
Golestanian
, E.
Gouillart
, G. F.
van Heijst
, T. S.
Krasnopolskaya
, Y.
Le Guer
, R. S.
MacKay
, V. V.
Meleshko
, G.
Metcalfe
, I.
Mezić
, A. P. S.
de Moura
, O.
Piro
, M. F. M.
Speetjens
, R.
Sturman
, J.-L.
Thiffeault
, and I.
Tuval
, “Frontiers of chaotic advection
,” Rev. Mod. Phys.
89
, 025007
(2017
). 25.
26.
M.
Hénon
, “Sur la topologie des lignes de courant dans un cas particulier
,” C. R. Acad. Sci.
262
, 312
–314
(1966
).27.
H.
Aref
, “Stirring by chaotic advection
,” J. Fluid Mech.
143
, 1
–21
(1984
). 28.
H.
Kantz
and T.
Schreiber
, Nonlinear Time Series Analysis
(Cambridge University Press
, 2010
).29.
30.
H.
Jeffreys
, “The stability of a layer of fluid heated below
,” Philos. Mag. VII
2
, 833
–844
(1926
). 31.
32.
C.
Letellier
, P.
Dutertre
, and G.
Gouesbet
, “Characterization of the Lorenz system, taking into account the equivariance of the vector field
,” Phys. Rev. E
49
, 3492
–3495
(1994
). 33.
G. P.
Massimo Falcioni
and A.
Vulpiani
, “Regular and chaotic motion of fluid particles in a two-dimensional fluid
,” J. Phys. A
21
, 3451
–3462
(1988
). 34.
N. H.
Packard
, J. P.
Crutchfield
, J. D.
Farmer
, and R. S.
Shaw
, “Geometry from a time series
,” Phys. Rev. Lett.
45
, 712
–716
(1980
). 35.
F.
Takens
, “Detecting strange attractors in turbulence
,” Lect. Notes Math.
898
, 366
–381
(1981
). 36.
I.
Sendiña Nadal
, S.
Boccaletti
, and C.
Letellier
, “Observability coefficients for predicting the class of synchronizability from the algebraic structure of the local oscillators
,” Phys. Rev. E
94
, 042205
(2016
). 37.
C.
Letellier
, I.
Sendiña-Nadal
, E.
Bianco-Martinez
, and M. S.
Baptista
, “A symbolic network-based nonlinear theory for dynamical systems observability
,” Sci. Rep.
8
, 3785
(2018
). 38.
T. H.
Solomon
and J. P.
Gollub
, “Chaotic particle transport in time-dependent Rayleigh-Bénard convection
,” Phys. Rev. A
38
, 6280
–6286
(1988
). 39.
C.
Couliette
and S.
Wiggins
, “Intergyre transport in a wind-driven, quasigeostrophic double gyre: An application of lobe dynamics
,” Nonlinear Process. Geophys.
7
, 59
–85
(2000
). 40.
H. A.
Dijkstra
and M.
Ghil
, “Low-frequency variability of the large-scale ocean circulation: A dynamical systems approach
,” Rev. Geophys.
43
, RG3002
, (2005
). 41.
S.
Wiggins
, Introduction to Applied Nonlinear Dynamical Systems and Chaos
(Springer-Verlag
, New York
, 2003
).42.
C.
Letellier
and R.
Gilmore
, “Covering dynamical systems: Twofold covers
,” Phys. Rev. E
63
, 016206
(2000
). 43.
C.
Letellier
and L. A.
Aguirre
, “Investigating nonlinear dynamics from time series: The influence of symmetries and the choice of observables
,” Chaos
12
, 549
–558
(2002
). 44.
T.
Kailath
, Linear Systems, Information and System Sciences Series (Prentice-Hall, 1980).45.
L. A.
Aguirre
, “Controllability and observability of linear systems: Some noninvariant aspects
,” IEEE Trans. Educ.
38
, 33
–39
(1995
). 46.
C.
Letellier
, J.
Maquet
, L. L.
Sceller
, G.
Gouesbet
, and L. A.
Aguirre
, “On the non-equivalence of observables in phase-space reconstructions from recorded time series
,” J. Phys. A
31
, 7913
–7927
(1998
). 47.
C.
Letellier
and L. A.
Aguirre
, “Symbolic observability coefficients for univariate and multivariate analysis
,” Phys. Rev. E
79
, 066210
(2009
). 48.
L. A.
Aguirre
, L. L.
Portes
, and C.
Letellier
, “Structural, dynamical and symbolic observability: From dynamical systems to networks
,” PLoS One
13
, e0206180
(2018
). 49.
E.
Bianco-Martinez
, M. S.
Baptista
, and C.
Letellier
, “Symbolic computations of nonlinear observability
,” Phys. Rev. E
91
, 062912
(2015
). © 2019 Author(s).
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