Note: Intro -- Preface -- Contents -- Part I Core Climate Issues -- A Mathematical Perspective on Microbial Processes in Earth's Biogeochemical Cycles -- 1 Introduction -- 2 Microbially Mediated Biogeochemical Cycling: The Challenge of Scale -- 3 Microbially Mediated Biogeochemical Cycling: The Challenge of Complexity -- 4 Merging Data and Models: The Challenge of Heterogeneous and Sparse Observations -- 5 Summary and Mathematical Challenges -- References -- Turbulence, Horizontal Convection, and the Ocean's Meridional Overturning Circulation -- 1 On the Defining Properties of Turbulence -- 2 Horizontal Convection -- 3 A Brief Survey of Other Results on Horizontal Convection -- 4 Implications for Oceanography and Climate -- References -- Part II Mathematical Techniques -- Source Reconstruction by Partial Measurements for a Class of Hyperbolic Systems in Cascade -- 1 Introduction -- 2 Preliminaries and Set-Up -- 2.1 Notation -- 2.2 Hidden Regularity and Observability Assumptions for the Scalar Wave Equation -- 2.3 Geometric Assumptions -- 2.3.1 Multiplier Geometric Conditions -- 2.3.2 Geometric Control Conditions -- 2.4 Observability Results for Homogeneous Systems -- 3 Well-Posedness -- 4 Stability Estimates for the Inverse Source Problem -- 5 A Negative Result -- References -- Remarks on Stochastic Navier-Stokes Equations -- 1 Introduction -- 2 Preliminary: The Deterministic Navier-Stokes Equations -- 3 Stochastic Perturbations -- 3.1 Motivations for Additive Noise -- 3.2 Motivations for Transport Noise -- 4 Remarks on Well-Posedness -- 5 Practical Issues -- 5.1 Perturbative Approach by Girsanov (Cameron-Martin) -- 5.2 Perturbative Approach by Kolmogorov Equation -- References -- Remarks on Long Time Versus Steady State Optimal Control -- 1 Introduction -- 2 Preliminaries on the Linear Heat Equation -- 3 Local Results for the Semilinear Heat Equation. , 3.1 The System of Optimality -- 3.2 Small Solutions -- 4 Time Independent Controls by Γ-Convergence -- 5 Further Comments and Open Problems -- References -- Part III Paleoclimate -- Effects of Additive Noise on the Stability of Glacial Cycles -- 1 Introduction -- 2 Methods -- 3 Results -- 3.1 Dispersions in Each Model -- 3.2 Order in Dispersions -- 3.3 Implication for Ice Ages -- 4 Concluding Discussion -- Appendix -- References -- On the Calibration of Lévy Driven Time Series with Coupling Distances and an Application in Paleoclimate -- 1 Introduction -- 2 Lévy Processes and Coupling Distances -- 2.1 Lévy Processes -- 2.2 Coupling Distances Between Lévy Measures -- 3 Statistical Considerations -- 3.1 Basic Notions -- 3.2 Quantiles -- 3.3 The Empirical Wasserstein Distance -- 3.4 Asymptotic Distribution and Rate of Convergence -- 4 The Procedure in Detail -- 5 Case Study -- 5.1 Concrete Formulas -- 5.2 Simulations -- 5.3 Paleoclimatic Time Series -- References -- Part IV Data Assimilation -- Data Assimilation for Geophysical Fluids: The Diffusive Back and Forth Nudging -- 1 Introduction -- 2 Variational Method: 4D-VAR -- 2.1 Model and Observations -- 2.2 Cost Function -- 2.3 Gradient Step -- 2.4 Optimality System -- 2.5 4D-Var Algorithm Computation -- 2.6 Computational Issues -- 3 Sequential Methods: Kalman Filter -- 3.1 The Extended Kalman Filter -- 3.2 Ensemble Kalman Filter -- 4 Back and Forth Nudging Schemes: BFN and Diffusive BFN (DBFN) -- 4.1 The Nudging Algorithm -- 4.2 Backward Nudging -- 4.3 The BFN Algorithm -- 4.4 DBFN: Diffusive Back and Forth Nudging Algorithm -- 4.5 Theoretical Considerations -- 5 Numerical Results on a 2D Shallow-Water Model -- 5.1 Description of the Model -- 5.2 Experimental Setup -- 5.3 Convergence of DBFN -- 5.4 Comparison Between BFN, DBFN and 4DVAR. , 5.5 Sensitivity of the BFN and DBFN with Respect to the Diffusion Coefficient -- 5.6 Sensitivity of the BFN and DBFN with Respect to Observation Noise -- 6 Numerical Results on a 3D Primitive Equation Ocean Model -- 6.1 Description of the Model -- 6.2 Model Configuration -- 6.3 Experimental Setup -- 6.3.1 Model Aspects -- 6.3.2 Observation Network -- 6.4 Convergence of DBFN -- 6.5 Comparison Between DBFN and BFN -- 6.6 Concluding Remarks -- 7 Conclusion -- References -- Deterministic Treatment of Model Error in Geophysical Data Assimilation -- 1 Introduction -- 2 Formulation -- 2.1 General Description of Model Error Dynamics -- 2.2 Model Error Due to Parameter Uncertainties -- 3 Deterministic Model Error Treatment in the Extended Kalman Filter -- 3.1 Short Time Extended Kalman Filter: ST-EKF -- 3.1.1 Numerical Results with ST-EKF: Error Due to Unresolved Scale -- 3.2 Short Time Augmented Extended Kalman Filter: ST-AEKF -- 3.2.1 Numerical Results with ST-EKF and ST-AEKF -- 4 Deterministic Model Error Treatment in Variational Data Assimilation -- 4.1 Numerical Results with ST-w4DVar -- 5 Discussion -- References -- Decreasing Flow Uncertainty in Bayesian Inverse Problems Through Lagrangian Drifter Control -- 1 Introduction -- 2 Setup -- 3 Variance Reduction Using a Proof-of-Concept Control -- 4 Posterior Informed Control -- 5 Conclusion -- References.