Stochastic Transport in Upper Ocean Dynamics II : : STUOD 2022 Workshop, London, UK, September 26-29.

• Superior document: Mathematics of Planet Earth Series ; v.11
• Author: Chapron, Bertrand.
• TeilnehmendeR: Crisan, Dan.; Holm, Darryl.; Mémin, Etienne.; Radomska, Anna.
• Place / Publishing House: Cham : : Springer,, 2023.; ©2024.
• Year of Publication: 2023
• Edition: 1st ed.
• Language: English
• Series: Mathematics of Planet Earth Series
• Physical Description: 1 online resource (347 pages)

Table of Contents

• Intro
• 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.