Stochastic representation of the Reynolds transport theorem: revisiting large-scale modeling

We explore the potential of a formulation of the Navier-Stokes equations incorporating a random description of the small-scale velocity component. This model, established from a version of the Reynolds transport theorem adapted to a stochastic representation of the flow, gives rise to a large-scale description of the flow dynamics in which emerges an anisotropic subgrid tensor, reminiscent to the Reynolds stress tensor, together with a drift correction due to an inhomogeneous turbulence. The corresponding subgrid model, which depends on the small scales velocity variance, generalizes the Boussinesq eddy viscosity assumption. However, it is not anymore obtained from an analogy with molecular dissipation but ensues rigorously from the random modeling of the flow. This principle allows us to propose several subgrid models defined directly on the resolved flow component. We assess and compare numerically those models on a standard Green-Taylor vortex flow at Reynolds 1600. The numerical simulations, carried out with an accurate divergence-free scheme, outperform classical large-eddies formulations and provides a simple demonstration of the pertinence of the proposed large-scale modeling

A future for intelligent autonomous ocean observing systems

Ocean scientists have dreamed of and recently started to realize an ocean observing revolution with autonomous observing platforms and sensors. Critical questions to be answered by such autonomous systems are where, when, and what to sample for optimal information, and how to optimally reach the sampling locations. Definitions, concepts, and progress towards answering these questions using quantitative predictions and fundamental principles are presented. Results in reachability and path planning, adaptive sampling, machine learning, and teaming machines with scientists are overviewed. The integrated use of differential equations and theory from varied disciplines is emphasized. The results provide an inference engine and knowledge base for expert autonomous observing systems. They are showcased using a set of recent at-sea campaigns and realistic simulations. Real-time experiments with identical autonomous underwater vehicles (AUVs) in the Buzzards Bay and Vineyard Sound region first show that our predicted time-optimal paths were faster than shortest distance paths. Deterministic and probabilistic reachability and path forecasts issued and validated for gliders and floats in the northern Arabian Sea are then presented. Novel Bayesian adaptive sampling for hypothesis testing and optimal learning are finally shown to forecast the observations most informative to estimate the accuracy of model formulations, the values of ecosystem parameters and dynamic fields, and the presence of Lagrangian Coherent Structures

Bayesian Learning of Coupled Biogeochemical-Physical Models

Predictive dynamical models for marine ecosystems are used for a variety of needs. Due to sparse measurements and limited understanding of the myriad of ocean processes, there is however significant uncertainty. There is model uncertainty in the parameter values, functional forms with diverse parameterizations, level of complexity needed, and thus in the state fields. We develop a Bayesian model learning methodology that allows interpolation in the space of candidate models and discovery of new models from noisy, sparse, and indirect observations, all while estimating state fields and parameter values, as well as the joint PDFs of all learned quantities. We address the challenges of high-dimensional and multidisciplinary dynamics governed by PDEs by using state augmentation and the computationally efficient GMM-DO filter. Our innovations include stochastic formulation and complexity parameters to unify candidate models into a single general model as well as stochastic expansion parameters within piecewise function approximations to generate dense candidate model spaces. These innovations allow handling many compatible and embedded candidate models, possibly none of which are accurate, and learning elusive unknown functional forms. Our new methodology is generalizable, interpretable, and extrapolates out of the space of models to discover new ones. We perform a series of twin experiments based on flows past a ridge coupled with three-to-five component ecosystem models, including flows with chaotic advection. The probabilities of known, uncertain, and unknown model formulations, and of state fields and parameters, are updated jointly using Bayes' law. Non-Gaussian statistics, ambiguity, and biases are captured. The parameter values and model formulations that best explain the data are identified. When observations are sufficiently informative, model complexity and functions are discovered.Comment: 45 pages; 18 figures; 2 table

Mini-Workshop: Innovative Trends in the Numerical Analysis and Simulation of Kinetic Equations

In multiscale modeling hierarchy, kinetic theory plays a vital role in connecting microscopic Newtonian mechanics and macroscopic continuum mechanics. As computing power grows, numerical simulation of kinetic equations has become possible and undergone rapid development over the past decade. Yet the unique challenges arising in these equations, such as highdimensionality, multiple scales, random inputs, positivity, entropy dissipation, etc., call for new advances of numerical methods. This mini-workshop brought together both senior and junior researchers working on various fastpaced growing numerical aspects of kinetic equations. The topics include, but were not limited to, uncertainty quantification, structure-preserving methods, phase transitions, asymptotic-preserving schemes, and fast methods for kinetic equations

Stochastic acoustic ray tracing with dynamically orthogonal equations

Submitted in partial fulfillment of the requirements for the degree of Master of Science in Mechanical Engineering at the Massachusetts Institute of Technology and the Woods Hole Oceanographic Institution May 2020.Developing accurate and computationally efficient models for ocean acoustics is inherently challenging due to several factors including the complex physical processes and the need to provide results on a large range of scales. Furthermore, the ocean itself is an inherently dynamic environment within the multiple scales. Even if we could measure the exact properties at a specific instant, the ocean will continue to change in the smallest temporal scales, ever increasing the uncertainty in the ocean prediction. In this work, we explore ocean acoustic prediction from the basics of the wave equation and its derivation. We then explain the deterministic implementations of the Parabolic Equation, Ray Theory, and Level Sets methods for ocean acoustic computation. We investigate methods for evolving stochastic fields using direct Monte Carlo, Empirical Orthogonal Functions, and adaptive Dynamically Orthogonal (DO) differential equations. As we evaluate the potential of Reduced-Order Models for stochastic ocean acoustics prediction, for the first time, we derive and implement the stochastic DO differential equations for Ray Tracing (DO-Ray), starting from the differential equations of Ray theory. With a stochastic DO-Ray implementation, we can start from non-Gaussian environmental uncertainties and compute the stochastic acoustic ray fields in a reduced order fashion, all while preserving the complex statistics of the ocean environment and the nonlinear relations with stochastic ray tracing. We outline a deterministic Ray-Tracing model, validate our implementation, and perform Monte Carlo stochastic computation as a basis for comparison. We then present the stochastic DO-Ray methodology with detailed derivations. We develop varied algorithms and discuss implementation challenges and solutions, using again direct Monte Carlo for comparison. We apply the stochastic DO-Ray methodology to three idealized cases of stochastic sound-speed profiles (SSPs): constant-gradients, uncertain deep-sound channel, and a varied sonic layer depth. Through this implementation with non-Gaussian examples, we observe the ability to represent the stochastic ray trace field in a reduced order fashion.Office of Naval Research Grants N00014-19-1-2664 (Task Force Ocean: DEEP-AI) and N00014-19-1-2693 (INBDA

DisPar Methods and Their Implementation on a Heterogeneous PC Cluster

Esta dissertação avalia duas áreas cruciais da simulação de advecção- difusão. A primeira parte é dedicada a estudos numéricos. Foi comprovado que existe uma relação directa entre os momentos de deslocamento de uma partícula de poluente e os erros de truncatura. Esta relação criou os fundamentos teóricos para criar uma nova família de métodos numéricos, DisPar. Foram introduzidos e avaliados três métodos. O primeiro é um método semi-Lagrangeano 2D baseado nos momentos de deslocamento de uma partícula para malhas regulares, DisPar-k. Com este método é possível controlar explicitamente o erro de truncatura desejado. O segundo método também se baseia nos momentos de deslocamento de uma partícula, sendo, contudo, desenvolvido para malhas uniformes não regulares, DisParV. Este método também apresentou uma forte robustez numérica. Ao contrário dos métodos DisPar-K e DisParV, o terceiro segue uma aproximação Eulereana com três regiões de destino da partícula. O método foi desenvolvido de forma a manter um perfil de concentração inicial homogéneo independentemente dos parâmetros usados. A comparação com o método DisPar-k em situações não lineares realçou as fortes limitações associadas aos métodos de advecção-difusão em cenários reais. A segunda parte da tese é dedicada à implementação destes métodos num Cluster de PCs heterogéneo. Para o fazer, foi desenvolvido um novo esquema de partição, AORDA. A aplicação, Scalable DisPar, foi implementada com a plataforma da Microsoft .Net, tendo sido totalmente escrita em C#. A aplicação foi testada no estuário do Tejo que se localiza perto de Lisboa, Portugal. Para superar os problemas de balanceamento de cargas provocados pelas marés, foram implementados diversos esquemas de partição: “Scatter Partitioning”, balanceamento dinâmico de cargas e uma mistura de ambos. Pelos testes elaborados, foi possível verificar que o número de máquinas vizinhas se apresentou como o mais limitativo em termos de escalabilidade, mesmo utilizando comunicações assíncronas. As ferramentas utilizadas para as comunicações foram a principal causa deste fenómeno. Aparentemente, o Microsoft .Net remoting 1.0 não funciona de forma apropriada nos ambientes de concorrência criados pelas comunicações assíncronas. Este facto não permitiu a obtenção de conclusões acerca dos níveis relativos de escalabilidade das diferentes estratégias de partição utilizadas. No entanto, é fortemente sugerido que a melhor estratégia irá ser “Scatter Partitioning” associada a balanceamento dinâmico de cargas e a comunicações assíncronas. A técnica de “Scatter Partitioning” mitiga os problemas de desbalanceamentos de cargas provocados pelas marés. Por outro lado, o balanceamento dinâmico será essencialmente activado no inicio da simulação para corrigir possíveis problemas nas previsões dos poderes de cada processador.This thesis assesses two main areas of the advection-diffusion simulation. The first part is dedicated to the numerical studies. It has been proved that there is a direct relation between pollutant particle displacement moments and truncation errors. This relation raised the theoretical foundations to create a new family of numerical methods, DisPar. Three methods have been introduced and appraised. The first is a 2D semi- Lagrangian method based on particle displacement moments for regular grids, DisPar-k. With this method one can explicitly control the desired truncation error. The second method is also based on particle displacement moments but it is targeted to regular/non-uniform grids, DisParV. The method has also shown a strong numerical capacity. Unlike DisPar-k and DisParV, the third method is a Eulerian approximation for three particle destination units. The method was developed so that an initial concentration profile will be kept homogeneous independently of the used parameters. The comparison with DisPar-k in non-linear situations has emphasized the strong shortcomings associated with numerical methods for advection-diffusion in real scenarios. The second part of the dissertation is dedicated to the implementation of these methods in a heterogeneous PC Cluster. To do so, a new partitioning method has been developed, AORDA. The application, Scalable DisPar, was implemented with the Microsoft .Net framework and was totally written in C#. The application was tested on the Tagus Estuary, near Lisbon (Portugal). To overcome the load imbalances caused by tides scatter partitioning was implemented, dynamic load balancing and a mix of both. By the tests made, it was possible to verify that the number of neighboring machines was the main factor affecting the application scalability, even with asynchronous communications. The tools used for communications mainly caused this. Microsoft .Net remoting 1.0 does not seem to properly work in environments with concurrency associated with the asynchronous communications. This did not allow taking conclusions about the relative efficiency between the partitioning strategies used. However, it is strongly suggested that the best approach will be to scatter partitioning with dynamic load balancing and with asynchronous communications. Scatter partitioning mitigates load imbalances caused by tides and dynamic load balancing is basically trigged at the begging of the simulation to correct possible problems in processor power predictions

Mathematical Theory and Modelling in Atmosphere-Ocean Science

Mathematical theory and modelling in atmosphere-ocean science combines a broad range of advanced mathematical and numerical techniques and research directions. This includes the asymptotic analysis of multiscale systems, the deterministic and stochastic modelling of sub-grid-scale processes, and the numerical analysis of nonlinear PDEs over a broad range of spatial and temporal scales. This workshop brought together applied mathematicians and experts in the disciplinary fields of meteorology and oceanography for a wide-ranging exchange of ideas and results in this area with the aim of fostering fundamental interdisciplinary work in this important science area

Particles and fields in fluid turbulence

The understanding of fluid turbulence has considerably progressed in recent years. The application of the methods of statistical mechanics to the description of the motion of fluid particles, i.e. to the Lagrangian dynamics, has led to a new quantitative theory of intermittency in turbulent transport. The first analytical description of anomalous scaling laws in turbulence has been obtained. The underlying physical mechanism reveals the role of statistical integrals of motion in non-equilibrium systems. For turbulent transport, the statistical conservation laws are hidden in the evolution of groups of fluid particles and arise from the competition between the expansion of a group and the change of its geometry. By breaking the scale-invariance symmetry, the statistically conserved quantities lead to the observed anomalous scaling of transported fields. Lagrangian methods also shed new light on some practical issues, such as mixing and turbulent magnetic dynamo.Comment: 165 pages, review article for Rev. Mod. Phy