This book which is the outcome of a NATO-Advanced Study Institute on Mod­ elling the Ocean Circulation and Geochemical Tracer Transport is concerned with using models to infer the ocean circulation. Understanding our climate is one of the major problems of the late twentieth century. The possible climatic changes resulting from the rise in atmospheric carbon dioxide and other trace gases are of primary interest and the ocean pla. ys a ma. jor role in determining the magnitude, temporal evolution and regional distribution of those changes. Because of the poor observational basis the ocean general circulation is not well understood. The World Ocean Circulation Experiment (WOCE) which is now underway is an attempt to improve our knowledge of ocean dynamics and thermodynamics on global scales relevant to climate change. Despite those efforts, the oceanic data base is likely to remain scarce and it is crucial to use appropriate methods in order to extract the maximum amount of information from observations. The book contains a thorough analysis of methods to combine data of val'ious types with dynamical concepts, and to assimilate data directly into ocean models. The properties of geocl;temical tracers such as HC, He, Tritium and Freons and how they may be used to impose integral constraints on the ocean circulation are discussed.

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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.…