These exact solutions correspond to time-dependent motions in parabolic basins. A characteristic feature is that the shoreline is not fixed. It is free to move and must be determined as part of the solution. In general, the motion is oscillatory and has the appropriate small-amplitude limit. For the case in which the parabolic basin reduces to a flat plane, there is a solution for a flood wave. These solutions provide a valuable test for numerical models of inundating storm tides.

A numerical model can be fit to data by minimizing a positive quadratic function of the differences between the data and their model counterparts. The rate at which algorithms for computing the best fit to data converge depends on the size of the condition number and the distribution of eigenvalues of the Hessian matrix, which contains the second derivatives of this quadratic function. The inverse of the Hessian can be identified as the covariance matrix that establishes the accuracy to which the model state is determined by the data; the reciprocals of the Hessian's eigenvalues represent the variances of linear combinations of variables determined by its eigenvectors. The aspect of the model state that are most difficult to compute are those about which the data provide the least information. A unified formalism is presented in which the model may be treated as providing either strong or weak constraints, and methods for computing and inverting the Hessian matrix are discussed. Examples are given of the uncertainties resulting from fitting an oceanographic model to several different sets of hypothetical data.

A formalism is presented for fitting dynamic forecast models to asynoptic data. Because of the importance of wind stress forcing in oceanic models and of the inadequacies of wind stress observations, the formalism allows an oceanic model to be fit to both Oceanographic and meteorological data. Within the context of this formalism the important question of whether an asynoptic data set contains sufficient information to determine the model state completely and unambiguously is discussed. Because the information travels along wave characteristics, it is clear that for the data to be sufficient to determine the model state, they must be distributed so that every feature of the flow is seen at some time or another. Such widespread coverage of the oceans requires a data collection system that relies heavily on satellites. The formalism is illustrated using a highly truncated model of the wind‐driven equatorial ocean and computational examples demonstrate how surface elevation and wind stress observations might be used to recover the model state.

Polynomial chaos (PC) expansions are used to propagate parametric uncertainties in ocean global circulation model. The computations focus on short-time, high-resolution simulations of the Gulf of Mexico, using the hybrid coordinate ocean model, with wind stresses corresponding to hurricane Ivan. A sparse quadrature approach is used to determine the PC coefficients which provides a detailed representation of the stochastic model response. The quality of the PC representation is first examined through a systematic refinement of the number of resolution levels. The PC representation of the stochastic model response is then utilized to compute distributions of quantities of interest (QoIs) and to analyze the local and global sensitivity of these QoIs to uncertain parameters. Conclusions are finally drawn regarding limitations of local perturbations and variancebased assessment and concerning potential application of the present methodology to inverse problems and to uncertainty management.