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

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