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 For