TY - CHAP

T1 - Transfer operator-based extraction of coherent features on surfaces

AU - Padberg-Gehle, Kathrin

AU - Reuther, Sebastian

AU - Praetorius, Simon

AU - Voigt, Axel

PY - 2017

Y1 - 2017

N2 - Transfer operator-based approaches have been successfully applied to the extraction of coherent features in flows. Transfer operators describe the evolution of densities under the action of the flow. They can be efficiently approximated within a set-oriented numerical framework and spectral properties of the resulting stochastic matrices are used to extract finite-time coherent sets. Also finite-time entropy, a density-based stretching quantity similar to finite-time Lyapunov exponents, is conveniently approximated by means of the discretized transfer operator. Transfer operator-based computational methods are purely probabilistic and derivative-free. Therefore, they can also be applied in settings where derivatives of the flow map are hardly accessible. In this paper, we summarize the theory and numerics behind the transfer operator approach and then introduce a straightforward extension, which allows us to study coherent structures in complex flows on triangulated surfaces. We illustrate our general computational framework with the well-known periodically driven double-gyre flow. To demonstrate the applicability of the approach for complex flows, we consider an approximation of the surface ocean flow, obtained by a numerical solution of the incompressible surface Navier-Stokes equation in a complicated geometry on the sphere.

AB - Transfer operator-based approaches have been successfully applied to the extraction of coherent features in flows. Transfer operators describe the evolution of densities under the action of the flow. They can be efficiently approximated within a set-oriented numerical framework and spectral properties of the resulting stochastic matrices are used to extract finite-time coherent sets. Also finite-time entropy, a density-based stretching quantity similar to finite-time Lyapunov exponents, is conveniently approximated by means of the discretized transfer operator. Transfer operator-based computational methods are purely probabilistic and derivative-free. Therefore, they can also be applied in settings where derivatives of the flow map are hardly accessible. In this paper, we summarize the theory and numerics behind the transfer operator approach and then introduce a straightforward extension, which allows us to study coherent structures in complex flows on triangulated surfaces. We illustrate our general computational framework with the well-known periodically driven double-gyre flow. To demonstrate the applicability of the approach for complex flows, we consider an approximation of the surface ocean flow, obtained by a numerical solution of the incompressible surface Navier-Stokes equation in a complicated geometry on the sphere.

KW - Mathematics

KW - Coherent Structure

KW - Transfer Operator

KW - Transport Barrier

KW - Lagrangian Coherent Structure

KW - Incompressible Surface

UR - https://tu-dresden.de/mn/math/wir/ressourcen/dateien/forschung/publikationen/pdf2017/Padberg-Gehle2017.pdf?lang=de

U2 - 10.1007/978-3-319-44684-4_17

DO - 10.1007/978-3-319-44684-4_17

M3 - Contributions to collected editions/anthologies

SN - 978-3-319-44682-0

T3 - Topological methods in data analysis and visualization

SP - 283

EP - 297

BT - Topological Methods in Data Analysis and Visualization IV

A2 - Carr, Hamish

A2 - Garth, Christoph

A2 - Weinkauf, Tino

PB - Springer International Publishing AG

CY - Cham

ER -