Deposits of clastic carbonate-dominated (calciclastic) sedimentary slope systems in the rock record have been identified mostly as linearly-consistent carbonate apron deposits, even though most ancient clastic carbonate slope deposits fit the submarine fan systems better. Calciclastic submarine fans are consequently rarely described and are poorly understood. Subsequently, very little is known especially in mud-dominated calciclastic submarine fan systems. Presented in this study are a sedimentological core and petrographic characterisation of samples from eleven boreholes from the Lower Carboniferous of Bowland Basin (Northwest England) that reveals a >250 m thick calciturbidite complex deposited in a calciclastic submarine fan setting. Seven facies are recognised from core and thin section characterisation and are grouped into three carbonate turbidite sequences. They include: 1) Calciturbidites, comprising mostly of highto low-density, wavy-laminated bioclast-rich facies; 2) low-density densite mudstones which are characterised by planar laminated and unlaminated muddominated facies; and 3) Calcidebrites which are muddy or hyper-concentrated debrisflow deposits occurring as poorly-sorted, chaotic, mud-supported floatstones. These

Phd by thesis

The Koopman operator is a linear but infinite-dimensional operator that governs the evolution of scalar observables defined on the state space of an autonomous dynamical system and is a powerful tool for the analysis and decomposition of nonlinear dynamical systems In this manuscript, we present a data-driven method for approximating the leading eigenvalues, eigenfunctions, and modes of the Koopman operator The method requires a data set of snapshot pairs and a dictionary of scalar observables, but does not require explicit governing equations or interaction with a “black box” integrator We will show that this approach is, in effect, an extension of dynamic mode decomposition (DMD), which has been used to approximate the Koopman eigenvalues and modes Furthermore, if the data provided to the method are generated by a Markov process instead of a deterministic dynamical system, the algorithm approximates the eigenfunctions of the Kolmogorov backward equation, which could be considered as the “stochastic Koopman operator” (Mezic in Nonlinear Dynamics 41(1–3): 309–325, 2005) Finally, four illustrative examples are presented: two that highlight the quantitative performance of the method when presented with either deterministic or stochastic data and two that show potential applications of the Koopman eigenfunctions

A Data-Driven Approximation of the Koopman Operator: Extending Dynamic Mode Decomposition

The Koopman operator is a linear but infinite dimensional operator that governs the evolution of scalar observables defined on the state space of an autonomous dynamical system, and is a powerful tool for the analysis and decomposition of nonlinear dynamical systems. In this manuscript, we present a data driven method for approximating the leading eigenvalues, eigenfunctions, and modes of the Koopman operator. The method requires a data set of snapshot pairs and a dictionary of scalar observables, but does not require explicit governing equations or interaction with a "black box" integrator. We will show that this approach is, in effect, an extension of Dynamic Mode Decomposition (DMD), which has been used to approximate the Koopman eigenvalues and modes. Furthermore, if the data provided to the method are generated by a Markov process instead of a deterministic dynamical system, the algorithm approximates the eigenfunctions of the Kolmogorov backward equation, which could be considered as the "stochastic Koopman operator" [1]. Finally, four illustrative examples are presented: two that highlight the quantitative performance of the method when presented with either deterministic or stochastic data, and two that show potential applications of the Koopman eigenfunctions.

Much of the debris in the near-surface ocean collects in so-called garbage patches where, due to convergence of the surface flow, the debris is trapped for decades to millennia. Until now, studies modelling the pathways of surface marine debris have not included release from coasts or factored in the possibilities that release concentrations vary with region or that pathways may include seasonal cycles. Here, we use observational data from the Global Drifter Program in a particle-trajectory tracer approach that includes the seasonal cycle to study the fate of marine debris in the open ocean from coastal regions around the world on interannual to centennial timescales. We find that six major garbage patches emerge, one in each of the five subtropical basins and one previously unreported patch in the Barents Sea. The evolution of each of the six patches is markedly different. With the exception of the North Pacific, all patches are much more dispersive than expected from linear ocean circulation theory, suggesting that on centennial timescales the different basins are much better connected than previously thought and that inter-ocean exchanges play a large role in the spreading of marine debris. This study suggests that, over multi-millennial timescales, a significant amount of the debris released outside of the North Atlantic will eventually end up in the North Pacific patch, the main attractor of global marine debris.

https://iopscience.iop.org/article/10.1088/1748-9326/7/4/044040/pdf

Origin, dynamics and evolution of ocean garbage patches from observed surface drifters

Lagrangian ocean analysis: Fundamentals and practices

Coherent nondispersive structures are known to play a crucial role in explaining transport in nonautonomous dynamical systems such as ocean flows. These structures are difficult to extract from model output as they are Lagrangian by nature and not revealed by the underlying Eulerian velocity fields. In the last few years heuristic concepts such as finite-time Lyapunov exponents have been used in an attempt to detect barriers to oceanic transport and thus identify regions that trap material such as nutrients and phytoplankton. In this Letter we pursue a novel, more direct approach to uncover coherent regions in the surface ocean using high-resolution model velocity data. Our method is based upon numerically constructing a transfer operator that controls the surface transport of particles over a short period. We apply our technique to the polar latitudes of the Southern Ocean.

/pdf/detection-of-coherent-oceanic-structures-via-transfer-2dvxkk7vqs.pdf

Detection of coherent oceanic structures via transfer operators.

We combine the techniques of almost invariant sets (using tree structured box elimination and graph partitioning algorithms) with invariant manifold and lobe dynamics techniques. The result is a new computational technique for computing key dynamical features, including almost invariant sets, resonance regions as well as transport rates and bottlenecks between regions in dynamical systems. This methodology can be applied to a variety of multibody problems, including those in molecular modeling, chemical reaction rates and dynamical astronomy. In this paper we focus on problems in dynamical astronomy to illustrate the power of the combination of these different numerical tools and their applicability. In particular, we compute transport rates between two resonance regions for the three-body system consisting of the Sun, Jupiter and a third body (such as an asteroid). These resonance regions are appropriate for certain comets and asteroids.

/pdf/transport-in-dynamical-astronomy-and-multibody-problems-1jbkeg0yjj.pdf

Transport in Dynamical Astronomy and Multibody Problems

We employ set oriented methods in combination with graph partitioning algorithms to identify key dynamical regions in the Sun-Jupiter-particle three-body system. Transport rates from a region near the 3:2 Hilda resonance into the realm of orbits crossing Mars' orbit are computed. In contrast to common numerical approaches, our technique does not depend on single long term simulations of the underlying model. Thus, our statistical results are particularly reliable since they are not affected by a dynamical behavior which is almost nonergodic (i.e., dominated by strongly almost invariant sets).

/pdf/transport-of-mars-crossing-asteroids-from-the-quasi-hilda-3amkfxz0zx.pdf

Transport of Mars-crossing asteroids from the quasi-Hilda region.

We present a new method based on set oriented computations for the calculation of reaction rates in chemical systems. The method is demonstrated with the Rydberg atom, an example for which traditional Transition State Theory fails. Coupled with dynamical systems theory, the set oriented approach provides a global description of the dynamics. The main idea of the method is as follows. We construct a box covering of a Poincare section under consideration, use the Poincare first return time for the identification of those regions relevant for transport and then we apply an adaptation of recently developed techniques for the computation of transport rates ([12], [27]). The reaction rates in chemical systems are of great interest in chemistry, especially for realistic three and higher dimensional systems. Our approach is applied to the Rydberg atom in crossed electric and magnetic fields. Our methods are complementary to, but in common problems considered, agree with, the results of [14]. For the Rydberg atom, we consider the half and full scattering problems in both the 2- and the 3-degree of freedom systems. The ionization of such atoms is a system on which many experiments have been done and it serves to illustrate the elegance of our method.

Set oriented computation of transport rates in 3-degree of freedom systems: the rydberg atom in crossed fields

During the last decade set oriented methods have been developed for the approximation and analysis of complicated dynamical behavior These techniques do not only allow the computation of invariant sets such as attractors or invariant manifolds Also statistical quantities of the dynamics such as invariant measures, transition probabilities, or (finite‐time) Lyapunov exponents, can be efficiently approximated All these techniques have natural applications in the numerical treatment of problems in astrodynamics In this contribution we will give an overview of the set oriented numerical methods and how they are successfully used for the solution of astrodynamical tasks For the demonstration of our results we consider the (planar) circular restricted three body problem In particular, we approximate invariant manifolds of periodic orbits about the L1 and L2 equilibrium points and show an extension to the application of a continuous control force Moreover, we demonstrate that expansion rates (finite‐time Lyapunov exponents), which so far have mainly been applied in fluid dynamics, can provide useful information on the qualitative behavior of trajectories in the context of astrodynamics The set oriented numerical methods and their application to astrodynamical problems discussed in this contribution serve as further important steps towards understanding the pathways of comets or asteroids and the design of energy‐efficient trajectories for spacecraft

Kathrin Padberg

Papers

Detection of coherent oceanic structures via transfer operators.

Transport in Dynamical Astronomy and Multibody Problems

Transport of Mars-crossing asteroids from the quasi-Hilda region.

Set oriented computation of transport rates in 3-degree of freedom systems: the rydberg atom in crossed fields

Set Oriented Approximation of Invariant Manifolds: Review of Concepts for Astrodynamical Problems