Oceanic Circulation Models: Combining Data and Dynamics

Tracer Inverse Problems.- 1. Introduction.- 1.1 The General Problem.- 1.2 On Determinateness.- 2. Interpolation and Map Making.- 2.1 Interpolation.- 2.2 The Gauss-Markov Theorem.- 2.3 Determining a Mean Value.- 2.4 A Priori Information.- 3. Simple Estimation.- 3.1 Elementary Least-Squares.- 3.2 Underdetermined Systems.- 3.3 Errors in the Observations.- 3.4 Resolution.- 3.5 Row and Column Scaling.- 3.6 Generalized Inverses.- 3.7 Other Estimation Procedures.- 3.8 Inverse Methods and Inverse Problems.- 4. Using Steady Tracers.- 4.1 The Background.- 4.2 Where Steady Tracer Inverse Methods are Going.- 4.3 Eclectic Modelling.- 4.4 Remaining Steady Tracer Issues.- 5. Time Dependent Problems.- 5.1 Observational Realities and Boundary Controls.- 5.2 Modifying the Model.- Appendix. Some Notes on the History of Inverse Methods in Ocean Circulation Problems.- A Geometrical Interpretation of Inverse Problems.- 1. The Overdetermined Case.- 2. The Underdetermined Case.- 3. The Singular Value Decomposition (SVD).- 3.1 The Overdetermined-Underconstrained Case.- 3.2 Again the Underdetermined Case.- 3.3 The Tapered Cut-off Solution.- Determining Diffusivities from Hydrographic Data by Inverse Methods with Applications to the Circumpolar Current.- 1. Introduction to Ill-posed and Noisy Problems.- 1.1 A Singular Example.- 1.2 Some Noisy Examples.- 2. The Physical Model.- 2.1 Equations of Motion and the Level-of-no-motion Problem.- 2.2 Parametrization of Mixing.- 3. The Inverse Model.- 3.1 The ?-Spiral Method.- 3.2 Mass Conservation.- 4. Examples: Circulation and mixing in the Southern Ocean.- 4.1 A Singular and Some Well-posed Problems.- 4.2 The Diabatic Model.- Ocean Acoustic Tomography: a Primer.- 1. Introduction.- 2. Elementary Hydrodynamics and Acoustics.- 3. Ocean Sound Speed Distribution.- 4. Rays and Modes.- 5. Capsule Description of the Tomographic Method.- 6. A Simple Ray Example.- 7. Notes on Hardware Limitations and Observational Errors.- 8. Inversions.- 9. Some Results and Future Plans.- 10. Conclusions.- 11. Literature.- The Circulation in the Western North Atlantic Determined by a Nonlinear Inverse Method..- 1. Introduction.- 2. A Nonlinear Inverse Formalism.- 3. The Circulation in the Western North Atlantic.- 3.1 Motivation.- 3.2 The Data Base.- 3.3 The Dynamical Model.- 3.4 A Priori Assumptions.- 3.5 The Results.- Altimeter Data Assimilation into Ocean Circulation Models — Some Preliminary Results.- 1. Introduction.- 2. A Dynamic Initialization Scheme.- 3. Time and Space Dependence of the Observations.- 4. Geosat Assimilation in the Agulhas Retroflection Region.- 5. Discussion.- Assimilation of Data into Ocean Models.- 1. Introduction.- 2. Theory.- 2.1 The Kalman Filter.- 2.2 Non-Linear Systems.- 2.3 Alternative Approaches.- 2.4 Adjoint System.- 3. Applications.- 3.1 Kalman Filters.- 3.2 Objective Analysis.- 3.3 Oceanographic Applications.- 3.4 Projection Schemes.- 3.5 Adjoint Schemes.- 4. Conclusions.- Driving of Non-linear Time-dependent Ocean Models by Observation of Transient Tracers — a Problem of Constrained Optimisation.- 1. Introduction.- 2. A Control Problem.- 3. An Elegant and Efficient Way to Calculate the Gradient of the Cost Function.- 4. The Ocean Model and its Adjoint.- 5. Results.- 6. Model Parameters as Control Variables.- 7. Sensitivity.- 7.1 Error Covariance and Resolution.- 7.2 Observational Analysis.- 8. Conclusions.- Assimilation of XBT Data Using a Variational Technique.- 1. Introduction.- 2. Implementation of Variational Assimilation Using Lagrange Multipliers.- 3. Results.- The Role of Real-Time Four-Dimensional Data Assimilation in the Quality Control, Interpretation, and Synthesis of Climate Data.- 1. Introduction.- 2. The Importance of Accurate Data Assimilation for NWP.- 3. The Analysis Module.- 3.1 The O/I Algorithm.- 3.2 The O/I Filter.- 3.3 The O/I Interpolator.- 3.4 The Relationship Between the Filter and Interpolator.- 3.5 General Comments.- 4. Non-linear Normal Mode Initialization.- 5. The Forecast Model.- 6. Quality Control and Data Monitoring.- 6.1 Aireps.- 6.2 Radiosonde Monitoring.- 6.3 Remotely Sensed Wind Data.- 6.4 Scatterometer Winds.- 6.5 Temperature Soundings from Satellites.- 7. The Value and Limitations of Global NWP Datasets for Climate Studies.- 7.1 Advantages.- 7.2 Limitations.- 8. Inversion and Quality Control of Remotely Sensed Data.- 8.1 Unified Variational Retrieval/Analysis Procedures.- 8.2 Coupled Assimilation Systems.- 8.3 Quality Control of Remotely-Sensed Data.- 9. Real-Time Integration and Synthesis of WCRP Observations.- to Chemical Tracers of the Ocean Circulation.- 1. Dimensional Analysis of a Tracer Conservation Equation.- 1.1 Oxygen in the Deep Northwest Atlantic Ocean.- 1.2 Oxygen Near the Surface Northeast Atlantic Ocean.- 1.3 Conclusion.- 2. General Information on Tracers of Ocean Circulation.- 2.1 Dissolved Oxygen and Nutrients.- 2.2 Carbon Species.- 2.3 14Carbon.- 2.3.1 Natural 14Carbon.- 2.3.2 Anthropogenic 14Carbon.- 2.4 Tritium and 3Helium.- 2.5 Chlorofluorocarbons.- 2.6 Conclusion.- 3. Surface Boundary Conditions.- 3.1 Seasonal Evolution of the CO2 Air-sea Gas Exchange Coefficient.- 3.2 Seasonal Evolution of the Surface Partial Pressure.- 3.3 Discussion.- 4. Mixing in the Deep Ocean.- 4.1 The Two-degree Discontinuity as Explained by Boundary Mixing.- 4.2 Purposeful Tracer Experiment.- 4.3 Discussion.- 5. Conclusion.- On Oceanic Boundary Conditions for Tritium, on Tritiugenic 3He, and on the Tritium-3He Age Concept.- 1. Introduction.- 2. Tritium Ocean Surface Boundary Condition.- 3. Separation of Tritiugenic 3He.- 4. The Tritium-3He Age Concept.- 5. Tritium-3He Age Distributions on Isopycnal Surfaces in the Lower Northeast Atlantic Main Thermocline.- 6. Conclusions.- Ocean Carbon Models and Inverse Methods.- 1. Introduction.- 2. Diagnostic Equations.- 2.1 The General Setting.- 2.2 The Condition of Geostrophy.- 2.3 Water Continuity; Definition of ‘Loops’ as Advective Variables.- 2.4 Continuity Equations for Tracers.- 2.5 The Set of Diagnostic Equations and the Inequality Constraints.- 2.6 Basic Characteristics of the System.- 3. Parameterization of the Model: Methodological Issues for Inverse Methods.- 3.1 Linear Inversion.- 3.2 Nonlinear Inversion.- 4. Some Preliminary Experiments.- Model of the Nutrient and Carbon Cycles in the North Atlantic. An Application of Linear Programming Methods.- 1. Introduction.- 2. The Model.- 3. Results for the ‘Inorganic’ Case.- 4. Results for the ‘Inorganic and Organic’ Case.- 5. Summary.- The Design of Numerical Models of the Ocean Circulation.- 1. Fundamentals of Model Design.- 1.1 Linearized Equations.- 1.2 Decomposition into Vertical Modes.- 1.3 Dispersion Diagrams.- 1.4 Staggered Grids.- 2. Stability.- 2.1 Inertia-Gravity Waves in ‘B’ and ‘C’ Grids.- 2.2 Filtering.- 2.3 Implicit Treatment.- 2.3.1 Inertial Motion — An Example.- 2.3.2 Gravity Waves — A Second Example.- 3. Stability of Advective Schemes and Vertical Coordinates.- 3.1 Time Differencing Continued: A Prototype Equation.- 3.2 Nonlinear Instability.- 3.3 A Variance Conserving Form.- 3.4 First and Second Order Advection Schemes.- 3.5 Choice of Vertical Coordinate.- 3.5.1 z-coordinate System.- 3.5.2 Depth-Normalized Coordinate.- 3.5.3 Isopycnal Coordinates.- 4. The Application of Ocean General Circulation Models.- 4.1 Observations.- 4.2 A Quasi-Geostrophic Model.- 4.3 An Isopycnal Model.- 4.4 A z-coordinate, Eddy-Resolving Model.- 4.5 Discussion.- Instabilities and Multiple Steady States of the Thermohaline Circulation.- 1. Introduction.- 2. Box Models.- 3. 2-D Model.- 4. GFDL Model.- 5. Conclusions.- Subgridscale Representation.- 1. Introduction.- 2. Organization.- 3. Quasi-horizontal Stirring.- 3.1 Basis for Fickian Diffusion?.- 3.2 Geophysical Influences (Rossby Waves).- 3.2.1 A Numerical Empirical Approach.- 3.2.2 A Closure Assumption.- 3.2.3 What a Lot of Bother!.- 3.3 Negative Diffusion?.- 3.3.1 Positive Diffusion in Disguise.- 3.3.2 Momentum Transport in 2D Turbulence.- 3.3.3 Upper Ocean Organic Carbon.- 3.3.4 On Further Consideration.- 3.4 Role of Coherent Vortices.- 3.5 Statistical Inhomogeneity.- 3.5.1 Inhomogenous Fickian Assumption.- 3.5.2 The Inhomogenous Random Walk.- 3.6 Horizontal or Isopycnal?.- 3.7 SGS within Partly Resolved Eddy Fields.- 3.7.1 Biharmonic (?4).- 3.7.2 Anticipated Potential Vorticity.- 3.7.3 Eddy Noise.- 4. Quasi-vertical Mixing.- 4.1 Shear Dispersion.- 4.2 Internal Wave Breaking.- 4.3 Buoyant Turbulence.- 4.3.1 More Numerical Empiricism.- 4.3.2 More Closure Theory.- 5. Benthic Boundary Processes.- Appendix A.- Appendix B.