Note: Intro -- Preface -- Acknowledgements -- Contents -- Contributors -- 1 Multi-scale Methods for Geophysical Flows -- 1.1 Introduction -- 1.2 The Governing Equations -- 1.2.1 Rotating Boussinesq Equations -- 1.2.2 Imbalance Variables -- 1.2.3 Mid-latitude Scalings -- 1.2.4 Hydrostatic Approximation -- 1.2.5 The Quasi-geostrophic Approximation on the β-plane -- 1.2.6 Rotating Shallow Water Equations -- 1.2.7 Geostrophic Scalings -- 1.2.8 Equatorial Scalings -- 1.3 Variational Principles and Hamiltonian Mechanics -- 1.3.1 Variational Principles -- 1.3.2 Variational Model Reduction -- 1.3.3 Poisson Formulation -- 1.3.4 Nambu Formulation -- 1.4 Dissipation, Turbulence, and Nonlinear Waves -- 1.4.1 Viscosity and Dissipation -- 1.4.2 Nonlinear Waves and Dynamical Systems Methods -- 1.5 Stochastic Model Reduction -- 1.5.1 Basic Setup -- 1.5.2 Slow Dynamics via the Kolmogorov Backward Equation -- 1.5.3 Direct Averaging -- 1.6 Outlook -- References -- 2 The Interior Energy Pathway: Inertia-Gravity Wave Emission by Oceanic Flows -- 2.1 Introduction -- 2.2 Rotating Shallow Water Equations and Spontaneous Emission -- 2.2.1 Shallow Water on the f-Plane -- 2.2.2 Spontaneous Emission -- 2.2.3 Beyond Shallow Water -- 2.3 Ray Equations and Wave Capture -- 2.4 Interactions Between IGWs and Density Fronts -- 2.4.1 Wave Capture in Frontal Strain -- 2.4.2 Role of IGWs in Frontal Geostrophic Adjustment -- 2.5 Diagnostics -- 2.5.1 Characterization of Flow Regimes via the Rossby Number -- 2.5.2 Linear Filters -- 2.5.3 Optimal Potential Vorticity Balance -- 2.5.4 A Simple Model for Optimal Balance -- 2.6 High-resolution Ocean General Circulation Models as a Novel Tool for Studying Spontaneous Emission -- 2.7 Discussion -- References -- 3 The IDEMIX Model: Parameterization of Internal Gravity Waves for Circulation Models of Ocean and Atmosphere. , 3.1 Internal Waves in Ocean and Atmosphere -- 3.2 The IDEMIX Model -- 3.2.1 Details of the Oceanic IDEMIX -- 3.2.2 The IDEMIX Concept Applied to Atmospheric Gravity Waves -- 3.3 Oceanic Processes in Present and Future IDEMIX -- 3.3.1 Including Energy Transfers from Mesoscale Eddies to Internal Waves -- 3.3.2 Including Wave-Mean Flow Interaction -- 3.3.3 Including Anisotropic Tidal Forcing -- 3.3.4 Including High-Frequency Compartments -- 3.3.5 Evaluation with Available Observations -- 3.4 Atmospheric Processes in IDEMIX -- 3.5 Summary -- References -- 4 Observations and Models of Low-Mode Internal Waves in the Ocean -- 4.1 Introduction -- 4.2 Numerical Modeling -- 4.2.1 Wind -- 4.2.2 Tides -- 4.3 Dissipation -- 4.4 Observations -- 4.4.1 Satellite Altimetry -- 4.4.2 Shipboard Observations -- 4.4.3 Moorings -- 4.5 Summary and Outlook -- References -- 5 Toward Consistent Subgrid Momentum Closures in Ocean Models -- 5.1 Introduction -- 5.2 Subgrid Momentum Closures -- 5.3 Quasigeostrophic Turbulence and Ocean Eddies -- 5.3.1 Two-Dimensional Turbulence -- 5.3.2 Two-Layer Geostrophic Flows -- 5.3.3 Continuously Stratified and Surface QG Dynamics -- 5.3.4 Ocean Models and Observational Evidence -- 5.4 Energy Backscatter -- 5.4.1 Models with Scalar Subgrid Energy Budget -- 5.4.2 Stochastic Superparameterizations -- 5.5 Other Closures -- 5.5.1 The Mana-Zanna Parameterization of Ocean Mesoscale Eddies -- 5.5.2 α-Models -- 5.6 Concluding Remarks -- References -- 6 Diagnosing and Parameterizing the Effects of Oceanic Eddies -- 6.1 Introduction -- 6.1.1 Isopycnal and Diapycnal Diffusion -- 6.1.2 Skew Diffusion -- 6.1.3 Diagnosing and Parameterizing the Diffusivities -- 6.2 Eddy Diffusivity Diagnostics -- 6.2.1 Lagrangian Particle Dispersion -- 6.2.2 Quasigeostrophic Linear Stability Analysis -- 6.2.3 Diffusivities from Eulerian Eddy Fluxes. , 6.3 Eddy Diffusivity Estimates in the Global Ocean -- 6.4 Limits of the Eddy Diffusion Model and Anomalous Diffusion -- 6.5 Eddy Diffusivity Parameterization -- 6.5.1 EKE Equation -- 6.6 Conclusions -- References -- 7 Entropy Production in Turbulence Parameterizations -- 7.1 The Numerically Modeled Atmosphere as a Forced-Dissipative System -- 7.2 The Entropy Budget Equation in Numerical Models of the Atmosphere -- 7.3 Moisture and Precipitation Fluxes -- 7.4 Thermal Fluxes -- 7.5 Momentum Fluxes -- 7.6 Fluctuation Theorem -- 7.7 Applicability of the Fluctuation Theorem in Geophysical Flows -- References -- 8 Reducing Spurious Diapycnal Mixing in Ocean Models -- 8.1 Introduction -- 8.2 Diagnosing Spurious Mixing -- 8.2.1 An Analytical Example -- 8.2.2 Variance Decay as a Measure for Mixing and Dissipation -- 8.2.3 Discrete Variance Decay -- 8.2.4 Applications -- 8.3 Arbitrary Lagrangian Eulerian Vertical Coordinate -- 8.3.1 tildez-Vertical Coordinate and its Effect on Spurious Mixing -- 8.3.2 Additional Techniques for Adaptive Vertical Model Layers -- 8.4 Advection Algorithms Stabilized with Isoneutral Mixing -- 8.5 ADER High Order Flux Evaluation and WENO Reconstruction -- 8.5.1 The Generalized Riemann Problem -- 8.5.2 Kernel-Based WENO Reconstruction -- 8.6 Discussion and Conclusions -- References -- 9 Diffuse Interface Approaches in Atmosphere and Ocean-Modeling and Numerical Implementation -- 9.1 Introduction -- 9.2 Diffuse Interface Approach -- 9.2.1 Notation -- 9.2.2 The Mathematical Model -- 9.3 Discretization -- 9.3.1 The Temporal Discretization -- 9.3.2 The Spatial Discretization and Energy Inequalities -- 9.3.3 A posteriori Error Estimation -- 9.4 Numerics -- 9.5 Outlook on the Direction of Research -- References -- Index.