Stochastic Transport in Upper Ocean Dynamics : STUOD 2021 Workshop, London, UK, September 20–23

This open access proceedings volume brings selected, peer-reviewed contributions presented at the Stochastic Transport in Upper Ocean Dynamics (STUOD) 2021 Workshop, held virtually and in person at the Imperial College London, UK, September 20–23, 2021. The STUOD project is supported by an ERC Synergy Grant, and led by Imperial College London, the National Institute for Research in Computer Science and Automatic Control (INRIA) and the French Research Institute for Exploitation of the Sea (IFREMER). The project aims to deliver new capabilities for assessing variability and uncertainty in upper ocean dynamics. It will provide decision makers a means of quantifying the effects of local patterns of sea level rise, heat uptake, carbon storage and change of oxygen content and pH in the ocean. Its multimodal monitoring will enhance the scientific understanding of marine debris transport, tracking of oil spills and accumulation of plastic in the sea. All topics of these proceedings are essential to the scientific foundations of oceanography which has a vital role in climate science. Studies convened in this volume focus on a range of fundamental areas, including: Observations at a high resolution of upper ocean properties such as temperature, salinity, topography, wind, waves and velocity; Large scale numerical simulations; Data-based stochastic equations for upper ocean dynamics that quantify simulation error; Stochastic data assimilation to reduce uncertainty. These fundamental subjects in modern science and technology are urgently required in order to meet the challenges of climate change faced today by human society. This proceedings volume represents a lasting legacy of crucial scientific expertise to help meet this ongoing challenge, for the benefit of academics and professionals in pure and applied mathematics, computational science, data analysis, data assimilation and oceanography. Preface Organization Contents Blow-Up of Strong Solutions of the Thermal Quasi-GeostrophicEquation 1 Introduction 1.1 Notations 1.2 Main Result 2 Blow-Up 2.1 Estimate for the 2D Modified Helmholtz Equation or the Screened Poisson Equation 2.2 Log-Sobolev Estimate for Velocity Gradient 2.3 A Priori Estimate References Modeling Under Location Uncertainty: A Convergent Large-Scale Representation of the Navier-Stokes Equations 1 Introduction 2 Modelling Under Location Uncertainty 3 Notations and Main Result 4 Proofs of the Main Result References A Stochastic Benjamin-Bona-Mahony Type Equation 1 Introduction 2 Truncation 3 Proof of the Main Result References Observation-Based Noise Calibration: An Efficient Dynamics for the Ensemble Kalman Filter 1 Introduction 2 The Stochastic SQG Model Under Location Uncertainty (LU) 3 Girsanov Theorem and Noise Calibration 3.1 Change of Measure 3.2 Computation of the Girsanov Drift 4 Experiments 5 Conclusion References A Two-Step Numerical Scheme in Time for Surface Quasi Geostrophic Equations Under Location Uncertainty 1 Introduction 2 Numerical Schemes 2.1 Derivation of a Milstein Scheme 2.1.1 Lévy Area Simulation 2.2 Multi-Step Schemes 3 Numerical Results 4 Conclusion and Perspectives Appendix: Convergence of Euler-Maruyama Scheme Under Moderate Noise References The Dissipation Properties of Transport Noise 1 Introduction 2 Well-Posedness and Motivations 2.1 Notations and Definitions 2.2 Motivations 3 Main Results 4 Explicit Computations 4.1 Explicit Construction 4.2 Numerical Simulation References Existence and Uniqueness of Maximal Solutions to a 3D Navier-Stokes Equation with Stochastic Lie Transport 1 Introduction 2 SALT Navier-Stokes and Results 2.1 Preliminaries from Stochastic Analysis 2.2 SALT Navier-Stokes Equation 2.3 Notions of Solution and Results 3 Abstract Framework and Results 3.1 Assumption Set 1 3.2 Assumption Set 2 3.3 Notions of Solution and Results 4 Abstract Solution Method and Application 4.1 Abstract Solution Method 4.2 SALT Navier-Stokes in the Abstract Framework Appendix References Coupling of Waves to Sea Surface Currents Via Horizontal Density Gradients 1 Introduction 1.1 Submesoscale Sea Surface Dynamics 2 Submesoscale Thermal Wave-Current Dynamics on a Free Surface 2.1 Surface Waves as Symmetry-Breaking Features of Local Force Imbalances 2.2 A Tale of Two Maps: Currents and Waves 2.3 Thermal Potential Vorticity (TPV) Dynamics on a Free Surface 2.4 CM Equations in the Slowly Varying Envelope (SVE) Approximation 2.5 Thermal Potential Vorticity Dynamics with SVE on a Free Surface 3 Numerical Implementation 4 Conclusion and Outlook References Variational Stochastic Parameterisations and Their Applications to Primitive Equation Models 1 Introduction 2 Stochastic Primitive Equations 2.1 Variational Principles for Stochastic Primitive Equations 2.2 Conservation Laws 3 Calibration of the Stochastic Parameters 3.1 Lagrangian Paths 3.2 Eulerian Differences 4 Results 5 Summary and Discussion Appendix: Numerical Implementation Bibliography A Pathwise Parameterisation for Stochastic Transport 1 Introduction 2 Problem Formulation 3 Methodology 4 Robustness 5 Numerical Results Appendix References Stochastic Parameterization with Dynamic Mode Decomposition 1 Introduction 2 Modelling Under Location Uncertainty 2.1 Stochastic Flow 2.2 Stochastic QG Model 3 Numerical Parameterization of Unresolved Flow 3.1 EOF-Based Method 3.2 DMD-Based Method 4 Numerical Experiments 4.1 Configurations 4.2 Diagnostics 4.3 Discussion 5 Conclusions References Deep Learning for the Benes Filter 1 Introduction 1.1 Nonlinear Stochastic Filtering Problem 1.2 Filtering Equation and General Splitting Method 2 Derivation and Outline of the Deep Learning Algorithm 2.1 Feynman–Kac Representation 2.2 The Benes Filtering Model 2.3 Neural Network Model for the Prediction Step 2.4 Monte-Carlo Normalisation Step 3 Numerical Results for the Benes Filter 3.1 No Domain Adaptation 3.2 With Domain Adaptation 4 Conclusion and Outlook References End-to-End Kalman Filter in a High Dimensional Linear Embedding of the Observations 1 Introduction 2 Method 3 Numerical Experiments 3.1 Preliminary Analysis on SST Anomaly Data 3.2 Shallow Water Equation (SWE) Case-Study 4 Conclusion References Dynamical Properties of Weather Regime Transitions 1 Introduction 2 European-Atlantic Weather Regime Transitions 3 Dimensionality Around Transitions 4 Persistence Around Transitions 5 Conclusion and Perspectives Appendix 1: Data Description: Twentieth Century Reanalysis Appendix 2: Statistical Descriptors Empirical Orthogonal Functions Gaussian Mixture Model Appendix 3: Dynamical Indicators Local Dimensions Inverse Persistence θ References Frequentist Perspective on Robust Parameter Estimation Using the Ensemble Kalman Filter 1 Introduction 2 Ensemble Kalman Parameter Estimation 3 Frequentist Analysis 4 Multi-Scale Data 5 Numerical Example 6 Conclusions References Random Ocean Swell-Rays: A Stochastic Framework 1 Introduction 2 Wave-Current Interaction in the Literature 3 The Time-Decorrelation Assumption 3.1 The Ray Lagrangian Correlation Time 3.2 Ray Absolute Diffusivity 3.3 A Practical Estimation 4 Numerical Simulations 5 Conclusion References Modified (Hyper-)Viscosity for Coarse-Resolution Ocean Models 1 Introduction 2 Double Gyre Quasi-Geostrophic Model 2.1 Governing Equations 2.2 Pytorch Implementation 2.3 Eddy-Resolving and Eddy-Permitting Regimes 3 Proposed Modified Viscosity 3.1 Motivation 3.2 Modified Viscosity 3.3 Modified Viscosity Regularization 3.4 Iterative Procedure 4 Results and Discussion 4.1 Statistics 4.2 Iterative Procedure 5 Conclusion Appendix Downsampling Procedure Parameter Tables References Primitive Equations Under Location Uncertainty: Analytical Description and Model Development 1 Introduction 2 Location Uncertainty (LU) 3 Stochastic Transport Theorem 4 Boussinesq Equations 5 Methods 6 Results 7 Conclusions References Bridging Koopman Operator and Time-Series Auto-Correlation Based Hilbert–Schmidt Operator 1 Introduction 2 Preliminaries and the Main Result 3 Proof of the Main Result References Index