Stochastic Transport in Upper Ocean Dynamics II: STUOD 2022 Workshop, London, UK, September 26–29 (Mathematics of Planet Earth, 11)

Preface
Contents
Internal Tides Energy Transfers and Interactions with the Mesoscale Circulation in Two Contrasted Areas of the North Atlantic
	1 Introduction
	2 Governing Equations and Energy Budget
	3 Data and Method
		3.1 eNATL60 Simulation
		3.2 Filtering and Computing Methods
	4 Results
		4.1 Life Cycle of the Internal Tide
		4.2 Importance of the Different Contributions in the Energy Transfers
			4.2.1 Detailed View of Coupling Terms
			4.2.2 Modal Energy Budget
	5 Conclusion
	References
Sparse-Stochastic Model Reduction for 2D Euler Equations
	1 Introduction
	2 Sparse-Stochastic Model Reduction
	3 Numerical Simulations
	4 Conclusions and Outlook
	References
Effect of Transport Noise on Kelvin–Helmholtz Instability
	1 Introduction
	2 Model Formulation
		2.1 Point Vortex Method for Inviscid Flows
		2.2 Point Vortex Method for Viscous Flows
	3 Point Vortex Method with Environmental Noise
		3.1 Transport Noise and Deterministic Scaling Limit
		3.2 A Digression on the Theoretical Selection of the Noise
	4 Numerical Results
		4.1 Setting: Kelvin–Helmholtz Instability
			4.1.1 The Role of Intrinsic Instability
			4.1.2 The Role of Viscosity and Stability Restoration
		4.2 Numerical Results on Environmental Noise
			4.2.1 Selection of Divergence Free Field
			4.2.2 Positions and Intensities of Fixed Vortices
			4.2.3 Effect of Small Scale Common Noise
		4.3 Diagnostics
	5 Concluding Remarks
	References
On the 3D Navier-Stokes Equations with Stochastic Lie Transport
	Introduction
	1 Introduction
	2 Preliminaries
		2.1 Elementary Notation
		2.2 Functional Framework
		2.3 The SALT Operator
	3 The Velocity Equation on the Torus
		3.1 Definitions and Results
		3.2 Operator Bounds
		3.3 Proof of Proposition 3.2
		3.4 Proofs of Theorems 3.1 and 3.6.
	4 The Vorticity Equation on a Bounded Main
		4.1 Deriving the Equation
		4.2 Definitions and Results
		4.3 Operator Bounds
		4.4 Proof of Theorem 4.3
	5 Appendices
		5.1 Proofs from Sects.2.3, 3.2, and 4.3
		5.2 A Conversion from Stratonovich to Itô
		5.3 Abstract Solution Criterion I
		5.4 Abstract Solution Criterion II
	References
On the Interactions Between Mean Flows and Inertial Gravity Waves in the WKB Approximation
	1 Introduction
	2 Deterministic 3D Euler–Boussinesq (EB) Internal Gravity Waves
		2.1 Lagrangian Formulation of the WMFI Equations at Leading Order
		2.2 Hamiltonian Structure for the WMFI Equations at Leading Order
	3 Stochastic WMFI
	4 Conclusion
	Appendix: Asymptotic Expansion
	References
Toward a Stochastic Parameterization for Oceanic Deep Convection
	1 Introduction
	2 Stochastic Formulation of Direct Non-hydrostatic Pressure Correction
	3 Numerical Implementation and Simulations
		3.1 Stochastic, Non-hydrostatic Pressure Correction
		3.2 Numerical Experiments
	4 Results
	5 Conclusion and Perspectives
	References
Comparison of Stochastic Parametrization Schemes Using Data Assimilation on Triad Models
	1 Introduction
	2 Reduced Order Models for Incompressible Fluids
		2.1 Reduced Order Models for the 3D Euler Equation
		2.2 Stochastic Parametrizations for the 3D Euler Equation
			2.2.1 Modelling Under the Stochastic Advection by Lie Transport Principle
			2.2.2 Modeling Under the Location Uncertainty Principle
		2.3 Triad Model Comparison
	3 Data Assimilation Comparison
		3.1 Numerical Studies
			3.1.1 Numerical Implementation
			3.1.2 Data Assimilation for the Deterministic Model
			3.1.3 Reduced Order Model Realisations
			3.1.4 Model Statistics
			3.1.5 Data Assimilation
	4 Conclusions
	Appendix 1: Notation and Basic Identities
		Notation
		Vector Identities
	Appendix 2: Derivation of Triad Models
		Deterministic Euler
		SALT Euler
		LU Euler
	Appendix 3: Supplementary Numerics
		Calibration of the Noise Amplitude
		Data Assimilation Verification
	References
An Explicit Method to Determine Casimirs in 2D Geophysical Flows
	1 Introduction
	2 Geophysical Flows
	3 Explicitly Determining the Casimirs
	4 Conclusion
	References
Correlated Structures in a Balanced Motion Interacting with an Internal Wave
	1 Introduction
	2 Model
	3 Methods
		3.1 Spectral Proper Orthogonal Decomposition
		3.2 Broadband Proper Orthogonal Decomposition
			3.2.1 Complex Demodulation of the Wave Field
			3.2.2 Link with SPOD
			3.2.3 Extended Broadband Proper Orthogonal Decomposition
	4 Results
	5 Summary and Perspectives
	References
Linear Wave Solutions of a Stochastic Shallow Water Model
	1 Introduction
	2 Review of RSW-LU
	3 Stationary Solution
	4 Stochastic Rotating Shallow Water Waves
		4.1 Ensemble-Mean Waves Under Homogeneous Noise
			4.1.1 Mean Poincaré Waves
			4.1.2 Mean Geostrophic Mode
		4.2 Path-Wise Waves Under Constant Noise
			4.2.1 Stochastic Poincaré Waves
			4.2.2 Stochastic Geostrophic Mode
		4.3 Approximation of Path-Wise Waves Under Homogeneous Noise
			4.3.1 Stochastic Poincaré Waves
			4.3.2 Stochastic Geostrophic Mode
		4.4 Numerical Illustrations
	5 Shallow Water PV Dynamics and Geostrophic Adjustment
	6 Conclusions
	References
Analysis of Sea Surface Temperature Variability Using Machine Learning
	1 Introduction
	2 Method
		2.1 Deterministic Model Hypothesis
		2.2 Stochastic Model Hypothesis: The Stochastic NbedDyn
	3 Numerical Experiments
		3.1 Data
		3.2 Analysis of the Deterministic Model
		3.3 Analysis of the Stochastic Model
	4 Conclusion
	Appendix 1: Training
	Appendix 2: Parameterization of the Diffusion Function
	References
Data Assimilation: A Dynamic Homotopy-Based Coupling Approach
	1 Introduction
	2 Problem Formulation and Background
	3 Schrödinger Bridge Approach
	4 Homotopy Induced Dynamic Coupling
	5 Numerical Implementation
		5.1 Ensemble Kalman Mean Field Approximation
		5.2 Particle Approximation and Time-Stepping
	6 Examples
		6.1 Pure Diffusion Processes
		6.2 Purely Deterministic Processes
		6.3 Linear Gaussian Case
		6.4 Nonlinear Diffusion Example
		6.5 Lorenz-63 Example
	7 Conclusions
	Appendix 1: Derivation of Control Term Equation
	Appendix 2: Ensemble Kalman Filter Approximations
	References
Constrained Random Diffeomorphisms for Data Assimilation
	1 Introduction
	2 Induced Stochastic PDE
	3 Comparison with Other Perturbation Schemes
		3.1 Comparison with the LU Equations
			3.1.1 0-Forms in the LU Framework
			3.1.2 n-Forms in the LU Framework
		3.2 The SALT Perturbation Scheme
	4 Conclusion
	Appendix: Expression of Tt*θ
	References
Stochastic Compressible Navier–Stokes Equations Under Location Uncertainty
	1 Introduction
	2 Stochastic Reynolds Transport Theorem
	3 Stochastic Compressible Navier–Stokes Equations
		3.1 Non-dimensioning
		3.2 Continuity
		3.3 Momentum
		3.4 Energy
		3.5 Equation of State
	4 Low Mach Approximation
	5 Boussinesq-Hydrostatic Approximation
	6 Extension to Non-Boussinesq
	7 Conclusion
	Appendix A: Stochastic Reynolds Transport Theorem from Stratonovich to Itō
	Appendix B: Calculation Rules
		Distributivity of the Stochastic Transport Operator
		Work of Random Forces
	Appendix C: Displacement of a Transported Control Surface
	References
Data Driven Stochastic Primitive Equations with Dynamic Modes Decomposition
	1 Introduction
	2 Location Uncertainty (LU)
	3 Stochastic Boussinesq Equations
	4 Methods
		4.1 High Resolution Data Filtering
		4.2 Off-Line Noise Modelling Through DMD
		4.3 On-Line Noise Reconstruction
	5 Results
	6 Conclusions
	References
Index