TY - JOUR
T1 - Geodesic theory of transport barriers in two-dimensional flows
AU - Haller, George
AU - Beron-Vera, Francisco J.
N1 - Funding Information:
We thank Ronny Peikert and Xavier Tricochet for their remarks and for the references they provided on tensor lines. We also thank Darryl Holm, Gábor Stépán, Alan Weinstein and the anonymous referees for their helpful comments and suggestions. We are grateful to Mohammad Farazmand for checking the details of several derivations in this paper. G.H. acknowledges support by the Canadian NSERC under grant 401839-11 . F.J.B.V. acknowledges support by the US NSF under grant CMG0825547 , by NASA under grant NNX10AE99G , and by British Petroleum under the Gulf of Mexico Research Initiative.
PY - 2012/10/15
Y1 - 2012/10/15
N2 - We introduce a new approach to locating key material transport barriers in two-dimensional, non-autonomous dynamical systems, such as unsteady planar fluid flows. Seeking transport barriers as minimally stretching material lines, we obtain that such barriers must be shadowed by minimal geodesics under the Riemannian metric induced by the Cauchy-Green strain tensor. As a result, snapshots of transport barriers can be explicitly computed as trajectories of ordinary differential equations. Using this approach, we locate hyperbolic barriers (generalized stable and unstable manifolds), elliptic barriers (generalized KAM curves) and parabolic barriers (generalized shear jets) in temporally aperiodic flows defined over a finite time interval. Our approach also yields a metric (geodesic deviation) that determines the minimal computational time scale needed for a robust numerical identification of generalized Lagrangian Coherent Structures (LCSs). As we show, an extension of our transport barrier theory to non-Euclidean flow domains, such as a sphere, follows directly. We illustrate our main results by computing key transport barriers in a chaotic advection map, and in a geophysical model flow with chaotic time dependence.
AB - We introduce a new approach to locating key material transport barriers in two-dimensional, non-autonomous dynamical systems, such as unsteady planar fluid flows. Seeking transport barriers as minimally stretching material lines, we obtain that such barriers must be shadowed by minimal geodesics under the Riemannian metric induced by the Cauchy-Green strain tensor. As a result, snapshots of transport barriers can be explicitly computed as trajectories of ordinary differential equations. Using this approach, we locate hyperbolic barriers (generalized stable and unstable manifolds), elliptic barriers (generalized KAM curves) and parabolic barriers (generalized shear jets) in temporally aperiodic flows defined over a finite time interval. Our approach also yields a metric (geodesic deviation) that determines the minimal computational time scale needed for a robust numerical identification of generalized Lagrangian Coherent Structures (LCSs). As we show, an extension of our transport barrier theory to non-Euclidean flow domains, such as a sphere, follows directly. We illustrate our main results by computing key transport barriers in a chaotic advection map, and in a geophysical model flow with chaotic time dependence.
KW - Coherent structures
KW - Invariant tori
KW - Manifolds
KW - Non-autonomous dynamical systems
KW - Transport
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U2 - 10.1016/j.physd.2012.06.012
DO - 10.1016/j.physd.2012.06.012
M3 - Article
AN - SCOPUS:84865998679
VL - 241
SP - 1680
EP - 1702
JO - Physica D: Nonlinear Phenomena
JF - Physica D: Nonlinear Phenomena
SN - 0167-2789
IS - 20
ER -