Computer modeling pervades today all fields of science. For the study of com­ plex systems, such as the environment, it has become an indispensable tool. But it is also a tool that is often misunderstood and misinterpreted. These dangers are particularly pronounced in the environmental sciences, an area of interest and concern not only to scientists, but also to the general public, the media, policy makers and powerful interest groups. We cannot experiment with our planet. The only quantitative tool available for the assessment of the impact of our actions today on the future environment and living conditions of later generations is numerical modeling. The better the general understanding of the potential and limitations of numerical models, the better the chances for a rational analysis and discussion of environmental problems and poli­ cies. But in addition to the more recent political issue of human impacts on the environment, numerical models play an important role for the forecasting of natural environmental variability, such as tides and storm surges or the weathcr, or for the interpretation of environmental changes in the past, such as the relation between the Late Maunder Minimum of the sunspot cycle from 1675 to 1710 and the winter half year cooling at the end of the 17th century. The reasons for misunderstandings and misinterpretations of numerical model results are manifold.

Table of Contents:
1 Introduction.- 2 Computer Models.- 3 Models and Data.- 4 The Dynamics of Tides and Climate.- 5 Modeling in Applied Environmental Sciences — Forecasting, Analysis and Scenarios.- 6 Modeling in Fundamental Environmental Sciences — Simulation and Hypothesis Testing.- 7 Issues and Conclusions.- Appendices.- A Fluid Dynamics.- A.1 The Balance Equations.- A.1.1 Mass Balances.- A.1.2 Momentum Balance.- A.1.3 Energy Balance.- A.2 Thermodynamic Specification.- A.3 The Phenomenological Flux Laws.- A.4 Boundary Conditions.- A.5 A Closer Look at the Balance Equations.- A.5.1 Cloud Formation.- A.5.2 Radiation.- A.5.3 Photochemical Reactions.- A.6 Reynolds Decomposition.- A.7 Parameterization of Interior Fluxes.- A.7.1 Eddy Diffusivities.- A.7.2 Eddy Viscosities.- A.8 Parameterization of Boundary Layer Fluxes.- A.8.1 The Constant Flux Layer.- A.8.2 The Planetary Boundary Layer.- A.9 Approximations.- A.9.1 Anelastic Approximation.- A.9.2 Shallow Water Approximation.- A.10 Representations.- A.10.1 Vertical Coordinates.- A.10.2 Decoupling.- B Numerics.- B.1 Discretization.- B.2 Partial Differential Equations.- B.2.1 Elliptic Problems.- B.2.2 Parabolic Problems.- B.2.3 Hyperbolic Problems.- B.3 Staggered Grids.- B.4 Spectral Models.- B.5 Finite Element Models.- C Statistical Analysis.- C.1 Random Variables and Processes.- C.1.1 Probability Function.- C.1.2 Bivariate Random Variables.- C.1.3 Random Processes.- C.2 Characteristic Parameters.- C.2.1 Expectation Values.- C.2.2 Empirical Orthogonal Functions.- C.2.3 Decomposition of Variance.- C.2.4 Skill Scores.- C.3 Inference.- C.3.1 Basic Aspects of Estimation.- C.3.2 Estimation of Auto-covariance Functions.- C.3.3 Estimation of Spectra.- C.3.4 Estimation of EOFs.- C.3.5 Hypothesis Testing.- D Data Assimilation.- D.1 Estimation.- D.2Filtering.- D.2.1 Kalman Filter.- D.2.2 Optimal or Statistical Interpolation.- D.2.3 Nudging.- D.2.4 Blending and Direct Insertion.- D.2.5 Minimization.- D.3 Smoothing.- D.3.1 Adjoint Method.- D.3.2 Inverse Method.- D.3.3 Parameter Estimation.- References.