The Ekman dynamics of the ocean surface circulation is known to contain attracting regions such as the great oceanic gyres and the associated garbage patches. Less well-known are the extents of the basins of attractions of these regions and how strongly attracting they are. Understanding the shape and extent of the basins of attraction sheds light on the question of the strength of connectivity of different regions of the ocean, which helps in understanding the flow of buoyant material like plastic litter. Using short flow time trajectory data from a global ocean model, we create a Markov chain model of the surface ocean dynamics. The surface ocean is not a conservative dynamical system as water in the ocean follows three-dimensional pathways, with upwelling and downwelling in certain regions. Using our Markov chain model, we easily compute net surface upwelling and downwelling, and verify that it matches observed patterns of upwelling and downwelling in the real ocean. We analyze the Markov chain to determine multiple attracting regions. Finally, using an eigenvector approach, we (i) identify the five major ocean garbage patches, (ii) partition the ocean into basins of attraction for each of the garbage patches, and (iii) partition the ocean into regions that demonstrate transient dynamics modulo the attracting garbage patches.
References
1.
M.
Kubota
, “A mechanism for the accumulation of floating marine debris north of Hawaii
,” J. Phys. Oceanogr.
24
, 1059
–1064
(1994
).2.
C.
Moore
, “Synthetic polymers in the marine environment: a rapidly increasing, long-term threat
,” Environ. Res.
108
, 131
–139
(2008
).3.
N.
Maximenko
, J.
Hafner
, and P.
Niiler
, “Pathways of marine debris derived from trajectories of Lagrangian drifters
,” Mar. Pollut. Bull.
65
, 51
–62
(2012
).4.
E.
van Sebille
, M.
England
, and G.
Froyland
, “Origin, dynamics and evolution of ocean garbage patches from observed surface drifters
,” Environ. Res. Lett.
7
, 044040
(2012
).5.
G.
Froyland
, K.
Padberg
, M.
England
, and A.-M.
Treguier
, “Detection of coherent oceanic structures via transfer operators
,” Phys. Rev. Lett.
98
, 224503
(2007
).6.
M.
Dellnitz
, G.
Froyland
, C.
Horenkamp
, K.
Padberg-Gehle
, and A. S.
Gupta
, “Seasonal variability of the subpolar gyres in the Southern Ocean: A numerical investigation based on transfer operators
,” Nonlinear Process. Geophys.
16
, 655
–663
(2009
).7.
V.
Rossi
, E.
Ser Giacomi
, C.
López
, and E.
Hernández García
, “Hydrodynamic provinces and oceanic connectivity from a transport network help designing marine reserves
,” Geophys. Res. Lett.
41
(8
), 2883
–2891, doi: (2014
).8.
E.
Kazantsev
, “Unstable periodic orbits and attractor of the barotropic ocean model
,” Nonlinear processes in Geophysics
5
, 193
–208
(1998
).9.
E.
Kazantsev
, “Sensitivity of the attractor of the barotropic ocean model to external influences: Approach by unstable periodic orbits
,” Nonlinear process. Geophys.
8
, 281
–300
(2001
).10.
S.
Khatiwala
, M.
Visbeck
, and M.
Cane
, “Accelerated simulation of passive tracers in ocean circulation models
,” Ocean Modell.
9
, 51
–69
(2005
).11.
S.
Khatiwala
, “A computational framework for simulation of biogeochemical tracers in the ocean
,” Global Biogeochem. Cycles
21
, GB3001
, doi: (2007
).12.
M.
Dellnitz
and O.
Junge
, “On the approximation of complicated dynamical behavior
,” SIAM J. Numer. Anal.
36
, 491
–515
(1999
).13.
M.
Demers
and L.-S.
Young
, “Escape rates and conditionally invariant measures
,” Nonlinearity
19
, 377
–397
(2006
).14.
Y.
Masumoto
, H.
Sasaki
, T.
Kagimoto
, N.
Komori
, A.
Ishida
, Y.
Sasai
, T.
Miyama
, T.
Motoi
, H.
Mitsudera
, K.
Takahashi
, et al., “A fifty-year eddy-resolving simulation of the world ocean: Preliminary outcomes of OFES (OGCM for the Earth simulator)
,” J. Earth Simul.
1
, 35
–56
(2004
), see http://www.jamstec.go.jp/esc/publication/journal/jes_vol.1/pdf/JES1-3.2-masumoto.pdf.15.
H.
Sasaki
, M.
Nonaka
, Y.
Masumoto
, Y.
Sasai
, H.
Uehara
, and H.
Sakuma
, “An eddy-resolving hindcast simulation of the quasiglobal ocean from 1950 to 2003 on the earth simulator
,” in High Resolution Numerical Modelling of the Atmosphere and Ocean
(Springer
, 2008
) pp. 157
–185
.16.
17.
C.
Hsu
, Cell-to-Cell Mapping: A Method of Global Analysis for Nonlinear Systems
(Springer-Verlag
, New York
, 1987
).18.
G.
Froyland
, “Extracting dynamical behaviour via Markov models
,” in Nonlinear Dynamics and Statistics: Proceedings of the Newton Institute, Cambridge, 1998
, edited by A.
Mees
(Birkhauser
, 2001
), pp. 283
–324
.19.
C. B.
Paris
, J.
Helgers
, E.
van Sebille
, and A.
Srinivasan
, “Connectivity modeling system: A probabilistic modeling tool for the multi-scale tracking of biotic and abiotic variability in the ocean
,” Environ. Modell. Softw.
42
, 47
–54
(2013
).20.
21.
F.
Roquet
, C.
Wunsch
, and G.
Madec
, “On the patterns of wind-power input to the ocean circulation
,” J. Phys. Oceanogr.
41
, 2328
–2342
(2011
).22.
J.
Milnor
, “On the concept of attractor, the theory of chaotic attractors
,” Commun. Math. Phys.
99
, 177
–195
(1985
).23.
24.
P.
Koltai
, “A stochastic approach for computing the domain of attraction without trajectory simulation
,” Discrete Contin. Dyn. Syst.
2011
, 854
–863
.25.
R.
Tarjan
, “Depth-first search and linear graph algorithms
,” SIAM J. Comput.
1
, 146
–160
(1972
).26.
T.
Kato
, Perturbation Theory for Linear Operators
, 2nd ed. (Springer
, Berlin
, 1995
).27.
P.
Deuflhard
, W.
Huisinga
, A.
Fischer
, and C.
Schütte
, “Identification of almost invariant aggregates in reversible nearly uncoupled Markov chains
,” Linear Algebra Appl.
315
, 39
–59
(2000
).28.
P.
Deuflhard
and M.
Weber
, “Robust Perron cluster analysis in conformation dynamics
,” Linear Algebra Appl.
398
, 161
–184
(2004
).29.
B.
Gaveau
and L.
Schulman
, “Multiple phases in stochastic dynamics: Geometry and probabilities
,” Phys. Rev. E
73
, 036124
(2006
).30.
G.
Froyland
, “On Ulam approximation of the isolated spectrum and eigenfunctions of hyperbolic maps
,” Discrete Contin. Dyn. Syst.
17
, 671
–689
(2007
).31.
E.
van Sebille
, L. M.
Beal
, and W. E.
Johns
, “Advective time scales of Agulhas leakage to the North Atlantic in surface drifter observations and the 3D OFES model
,” J. Phys. Oceanogr.
41
, 1026
–1034
(2011
).32.
P.
Brémaud
, Markov Chains: Gibbs Fields, Monte Carlo simulation, and Queues
(Springer-Verlag
, 1999
).33.
G.
Froyland
, “Statistically optimal almost-invariant sets
,” Physica D
200
, 205
–219
(2005
).34.
J. N.
Moum
, A.
Perlin
, J. D.
Nash
, and M. J.
McPhaden
, “Seasonal sea surface cooling in the equatorial Pacific cold tongue controlled by ocean mixing
,” Nature
500
, 64
–67
(2013
).35.
G.
Nelson
and L.
Hutchings
, “The Benguela upwelling area
,” Progress Oceanogr.
12
, 333
–356
(1983
).36.
J.
Sprintall
, S. E.
Wijffels
, R.
Molcard
, and I.
Jaya
, “Direct estimates of the Indonesian Throughflow entering the Indian Ocean: 2004–2006
,” J. Geophys. Res.
114
, C07001
, doi: (2009
).© 2014 AIP Publishing LLC.
2014
AIP Publishing LLC
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