20 research outputs found
Analysis of tidal flows through the Strait of Gibraltar using Dynamic Mode Decomposition
The Strait of Gibraltar is a region characterized by intricate oceanic
sub-mesoscale features, influenced by topography, tidal forces, instabilities,
and nonlinear hydraulic processes, all governed by the nonlinear equations of
fluid motion. In this study, we aim to uncover the underlying physics of these
phenomena within 3D MIT general circulation model simulations, including waves,
eddies, and gyres. To achieve this, we employ Dynamic Mode Decomposition (DMD)
to break down simulation snapshots into Koopman modes, with distinct
exponential growth/decay rates and oscillation frequencies. Our objectives
encompass evaluating DMD's efficacy in capturing known features, unveiling new
elements, ranking modes, and exploring order reduction. We also introduce
modifications to enhance DMD's robustness, numerical accuracy, and robustness
of eigenvalues. DMD analysis yields a comprehensive understanding of flow
patterns, internal wave formation, and the dynamics of the Strait of Gibraltar,
its meandering behaviors, and the formation of a secondary gyre, notably the
Western Alboran Gyre, as well as the propagation of Kelvin and coastal-trapped
waves along the African coast. In doing so, it significantly advances our
comprehension of intricate oceanographic phenomena and underscores the immense
utility of DMD as an analytical tool for such complex datasets, suggesting that
DMD could serve as a valuable addition to the toolkit of oceanographers
Geometry of the ergodic quotient reveals coherent structures in flows
Dynamical systems that exhibit diverse behaviors can rarely be completely
understood using a single approach. However, by identifying coherent structures
in their state spaces, i.e., regions of uniform and simpler behavior, we could
hope to study each of the structures separately and then form the understanding
of the system as a whole. The method we present in this paper uses trajectory
averages of scalar functions on the state space to: (a) identify invariant sets
in the state space, (b) form coherent structures by aggregating invariant sets
that are similar across multiple spatial scales. First, we construct the
ergodic quotient, the object obtained by mapping trajectories to the space of
trajectory averages of a function basis on the state space. Second, we endow
the ergodic quotient with a metric structure that successfully captures how
similar the invariant sets are in the state space. Finally, we parametrize the
ergodic quotient using intrinsic diffusion modes on it. By segmenting the
ergodic quotient based on the diffusion modes, we extract coherent features in
the state space of the dynamical system. The algorithm is validated by
analyzing the Arnold-Beltrami-Childress flow, which was the test-bed for
alternative approaches: the Ulam's approximation of the transfer operator and
the computation of Lagrangian Coherent Structures. Furthermore, we explain how
the method extends the Poincar\'e map analysis for periodic flows. As a
demonstration, we apply the method to a periodically-driven three-dimensional
Hill's vortex flow, discovering unknown coherent structures in its state space.
In the end, we discuss differences between the ergodic quotient and
alternatives, propose a generalization to analysis of (quasi-)periodic
structures, and lay out future research directions.Comment: Submitted to Elsevier Physica D: Nonlinear Phenomen