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Format: 1 online resource (347 pages)

Edition: 1st ed.

ISBN: 9783031400940

Series Statement: Mathematics of Planet Earth Series ; v.11

Note: 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.

Additional Edition: Print version: Chapron, Bertrand Stochastic Transport in Upper Ocean Dynamics II Cham : Springer,c2023 ISBN 9783031400933

Language: English

Keywords: Electronic books.

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