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W. C. Thacker

Abstract

The purpose of this note is to point out that Hasselmann's optimal fingerprints for detecting climatic change follow from the geometrical interpretation of covariance as an inner product.

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W. C. Thacker

Abstract

The empirical relationship between salinity and temperature in the South Atlantic is quantified with the aid of local regression. To capture the spatial character of the TS relationship, models are fitted to data for each point on a three-dimensional grid with spacing of 1° in latitude, 2° in longitude, and 25 dbar in the vertical. To ensure sufficient data for statistical reliability each fit is to data from a region extending over several grid points weighted so that more remote data exert less influence than those closer to the target grid point. Both temperature and its square are used as regressors to capture the curvature seen in TS plots, and latitude and longitude are used to capture systematic spatial variations over the fitting regions. In addition to using statistics of residuals to characterize how well the models fit the data, errors for data not used in fitting are examined to verify the models’ abilities to simulate independent data. The best model overall for the entire region at all depths is quadratic in temperature and linear in longitude and latitude.

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W. C. Thacker

Abstract

Finite-difference techniques for irregular computational grids are presented. Successful simulations of transient normal mode oscillations in shallow circular basins, where analytic solutions are known, demonstrate that these techniques can yield accurate results, even in situations involving a curved boundary. These techniques should prove to be quite useful for numerically forecasting storm surges in bays and estuaries where calculations are complicated by the curving coastline.

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W. C. Thacker

Abstract

The finite-element scheme requires more computational expense because it is time-implicit and because it requires a smaller time step than the finite-difference scheme. Simulations of normal mode oscillations reveal that for cases where the basin depth and the numerical grid are uniform, the finite-difference scheme is more accurate, but for cases where the depth or the grid varies, the finite-element scheme is more accurate.

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W. C. Thacker

Abstract

No abstract available.

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W. C. Thacker

Abstract

Irregular grid finite-difference techniques lead to equations similar to those obtained using finite-element techniques. The simpler finite-difference equations offer the advantage of greater computational economy. The time-implicit finite-element equations must be inverted at each time step, and the maximum size of the time step is only half that which can be used with the finite-difference equations. Both techniques result in instabilities when used with highly irregular grids, and the finite-element equations are also unstable if the basin depth is variable. Although the finite-element results are better when compared with finite-difference results from the same grid, comparable finite-difference results can be obtained using a finer grid at less computational expense.

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W. C. Thacker

Abstract

Shear dispersion results from vertical shear of horizontal velocity and vertical mixing, features which cannot be included explicitly in one-layer, vertically integrated models. The parametric description of shear dispersion as effective horizontal diffusion in one-layer models is investigated by comparing analytic solutions of two-layer dispersion equations to the corresponding solutions of a one-layer diffusion equation. The diffusion description is found to be poor for times comparable with or shorter than the vertical mixing time but excellent for longer times.

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W. C. Thacker and R. Lewandowicz

Abstract

If indices are to be used as the variables predicted by linear statistical models, it is important to be able to recover as much local information as possible from the values forecast for the indices. Here it is shown that the indices that encapsulate the most information about the local climatic state are determined by a generalized (two-matrix) eigenvalue problem that is equivalent to the usual (one-matrix) eigenvalue problem involving the sample correlation matrix. Thus, the best indices in the sense of providing the most location-specific information are familiar principal-component indices.

Regarding the indices as predictors in linear statistical models similar to those routinely used for estimating meteorological fields from observations reveals the role of the Gauss-Markov theorem in EOF analyses. From this perspective each index can he characterized by two EOF-like maps: the first illustrating the linear combinations of the data used to define the index, and the second displaying the Gauss-Markov weights for the index to predict local variables, both of which are related to the eigenvectors of the sample correlation matrix. Other maps can be used to display information about sampling errors: one to characterize the uncertainty of the weights; another to display the skill with which the index accounts for the training data; and a third to show how well it explains independent data. Such maps are illustrated within the context of 43 years of North Atlantic seasonal sea surface temperature anomalies.

The analysis presented here underlines two additional points. First, any linear combination of the indices would result in an equivalent model yielding exactly the same forecast. Consequently, it may be desirable to use indices that are easier to interpret physically. Second, when indices are regarded as being variables of a linear statistical model, the analysis of sampling error can be formulated in terms of the uncertainty of the Gauss-Markov weights inferred from a limited training set rather than in terms of the sample-to-sample variability of eigenvalues and eigenvectors.

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Rafael C. Gonçalves, Mohamed Iskandarani, Tamay Özgökmen, and W. Carlisle Thacker

Abstract

The extensive drifter deployment during the Lagrangian Submesoscale Experiment (LASER) provided observations of the surface velocity field in the northern Gulf of Mexico with high resolution in space and time. Here, we estimate the submesoscale velocity field sampled by those drifters using a procedure that statistically interpolates these data both spatially and temporally. Because the spacing of the drifters evolves with the flow, causing the resolution that they provide to vary in space and time, it is important to be able to characterize where and when the estimated velocity field is more or less accurate, which we do by providing fields of interpolation errors. Our interpolation uses a squared-exponential covariance function characterizing correlations in latitude, longitude, and time. Two novelties in our approach are 1) the use of two scales of variation per dimension in the covariance function and 2) allowing the data to determine these scales along with the appropriate amplitude of observational noise at these scales. We present the evolution of the reconstructed velocity field along with maps of relative vorticity, horizontal divergence, and lateral strain rate. The reconstructed velocity field exhibits horizontal length scales of 0.4–3.5 km and time scales of 0.6–3 h, and features with convergence up to 8 times the planetary vorticity f, lateral strain rate up to 10f, and relative vorticity up to 13f. Our results point to the existence of a vigorous and substantial ageostrophic circulation in the submesoscale range.

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