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Joseph J. Tribbia

Abstract

Using a low-order, spectral, shallow-water model on an f-plane, the conditions under which height-constrained nonlinear normal mode initialization fails and the existence of realizable balancing wind fields are examined. The relationship of this nonrealizability condition and the ellipticity condition for the standard nonlinear balance equation is also examined. A conclusion from this analysis is that non-elliptic geopotential regions must be accompanied by transient gravity wave motion if there is no forcing mechanism.

The low-order results are extended through the use of a global shallow-water model. The relationship between the local f-plane results and the global results is analyzed and a strong correlation between the appearance of non-elliptic geopotential regions and the breakdown of the iteration scheme used in non-linear normal mode balancing is noted.

It is concluded that moderately weak anticyclonic disturbances in equatorial areas may act as regions of energy exchange between rotational and gravitational modes. Also, the climatological existence of these regions implies the necessity forcing to maintain them in the atmosphere and numerical forecast models.

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Joseph J. Tribbia

Abstract

The variational problem of initial data specification from observations with the strong constraint of the elimination of transient gravity waves through nonlinear normal mode balancing is reconsidered. The exact formulation of this problem is contrasted to the approximate formulation previously given by Daley.

Through the judicious use of model normal modes, an alternative algorithm is developed which allows the use of confidence weights which reflect the fidelity of observations in a more realistic manner than previously possible. In particular, longitudinally varying confidence weights can be utilized. Examples of the use of this technique using the Machenhauer and the second-order Baer-Tribbia initialization are given. An attempt to ascertain the validity of Daley's method demonstrates the accuracy of this approximation and justifies its continued use.

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Joseph J. Tribbia

Abstract

The efficacy of the nonlinear initialization technique for use on global-scale numerical models is tested using a normal-mode, spectral model of the shallow-water equations on an equatorial beta-plane. Despite the nonexistence of strong, frequency separation for the ultralong, equatorially trapped modes, test integrations show that the nonlinear initialization scheme acts to smooth the most rapid oscillations in the system. Further integrations involving only spectral components associated with low-frequency, rotational modes show that the rotational mode trajectories are nearly unaffected by the presence of the balanced gravitational modes. The likely distortion of the divergence field obtained from a rotational-mode-only calculation makes this filtering-through-truncation technique appear unattractive, so an alternative scheme which uses both the truncation and nonlinear initialization schemes is proposed.

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Joseph J. Tribbia

Abstract

An algorithm for obtaining high-order mode initialization of the type first proposed by Baer and Tribbia is developed which is free from the major difficulty of previous methods—the necessity of calculating Frechet derivatives of the nonlinear terms. This new method is shown to be a logical extension of the technique proposed by Machenhauer; thus the asymptotic equivalence of the Machenhauer and Baer-Tribbia initialization methods is accomplished. A comparison between the new algorithm and the older method of calculating second-order initialization demonstrates the accuracy and ease of implementation of the new technique.

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Roger Temam and Joseph Tribbia

Abstract

It was shown by Oliger and Sundström, in 1978, that the initial boundary value problems for the hydrostatic primitive equations of meteorology and oceanography are ill posed if the boundaries are open and fixed in space. In this article it is shown, with theory and computation, that the same problems are well posed for a suitable set of local (pointwise applied) boundary conditions, if a mild vertical viscosity is added to the hydrostatic equation. Some indications on the behavior of the solutions, as the vertical viscosity parameter goes to zero, are also given. The Boussinesq equations are shown to be well posed in the same context of boundaries open and fixed in space. Finally, numerical simulations supporting the analysis are included.

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Martin Ehrendorfer and Joseph J. Tribbia

Abstract

Optimal perturbations, also referred to as singular vectors (SVs), currently constitute an important guideline for the generation of initial ensembles to be used for ensemble prediction. The optimality of these perturbations refers to their property of maximizing prespecified quadratic measures of error growth, given that tangent-linear error evolution is assumed. The goal of ensemble prediction is the accurate prediction of the uncertainty of forecasts made with dynamical numerical weather prediction models.

In the present paper the theoretical justification for the use of SVs in ensemble prediction systems is investigated. It is shown that, in a tangent-linear framework, SVs—constructed using covariance information valid at the initial time—evolve into the eigenvectors of the forecast error covariance matrix valid for the end of the optimization interval. As such, SVs represent the most efficient means for predicting the forecast error covariance matrix, given a prespecified number of allowable (tangent-linear) model integrations. Such optimal prediction is of particular importance in light of the fact that the forecast error covariance matrix is summarizing important information about the probability density function of the model state at a given future time.

Based on the above result, optimal covariance prediction through appropriately determined SVs is demonstrated here for a three-dimensional Lorenz model, as well as for a barotropic model of intermediate dimensionality, both within a perfect-model framework. In the case of the barotropic model it is found that less than 15% of the SVs suffice to account for more than 95% of the total final error variance. Viewed differently, at least 80% of the final error variance is accounted for by retaining those SVs that are amplifying in terms of an enstrophy norm. In addition, variances and covariances predicted through SVs agree closely with independently obtained Monte Carlo estimates, as long as the tangent-linear approximation is sufficiently accurate.

Further, the problem of approximating the forecast error covariance matrix in the presence of a state-independent model-error representation is briefly considered. The paper is concluded with a summary of the results and a discussion of their possible implications on data assimilation procedures and on the further development of ensemble prediction systems.

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Tsing-Chang Chen and Joseph J. Tribbia

Abstract

The multiyear climate simulations performed by the Community Climate Model of the National Center for Atmospheric Research with three horizontal resolutions [a 1 5-wave rhomboidal truncation (R15) and 31- and 42-wave triangular truncations, (T31 and T42)] were used to examine the effect of the model's horizontal resolution on the simulated stationary long waves. It was found that through increasing the model's horizontal resolution, the major parts of changes in streamfunction and velocity potential of stationary long waves became spatially in quadrature with these two fields of the R15 resolution; the stationary long waves of the R15 resolution were closer to the observed than those of the other two higher resolutions, and amplitudes of stationary long waves generated by the T31 and T42 models were amplified. A systematic enhancement of the tropical diabatic heating is noted over the three tropical continents when the model's horizontal resolution is increased. Based upon a possible chain relation between diabatic beating, divergent circulation, and rotational motion, it is suggested that the aforementioned systematic changes in streamfunction and velocity potential may be a part of the response of the model atmosphere to those changes in tropical diabatic heating that are induced by the increase of the model's horizontal resolution.

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Tsing-Chang Chen and Joseph J. Tribbia

Abstract

Diagnostic computations of nonlinear cascades of enstrophy have been performed in the wavenumber domain for two northern summers. Attention is focused on the interactions among the waves, the interaction between the zonal flow and a given wave and the exchanges due to the beta effect. It is found that two wave ranges (low and intermediate wavenumbers) cascade enstrophy to two ranges of wavenumbers. Calculations are also performed to evaluate the contributions from the standing (92-day mean) and transient modes to the nonlinear enstrophy cascade.

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Joseph J. Tribbia and David P. Baumhefner

Abstract

This paper presents the results of an ensemble of 20 predictability experiments derived from the NCAR Community Climate Model (CCM). Particular emphasis is placed on the question of the predictability of dynamically driven low-frequency components of the model atmosphere. The conclusion drawn, using time averaging alone as a means of isolating low-frequency variability, is that in the ensemble mean there is little skill in a 30-day mean forecast. Examination of the variability of skill among the ensemble members indicates that approximately 40 percent of the perturbed monthly mean forecasts would be useful. Examples of skillful and poor monthly mean predictions am shown and conclusions are drawn as to the implications of the results with regard to the likelihood of success of extended range deterministic forecasts.

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Peter R. Gent and Joseph J. Tribbia

Abstract

A model of the tropical ocean and global atmosphere is described. It consists of an aqua-planet form of version one of the NCAR Community Climate Model coupled to a primitive equation model for the upper tropical ocean in a rectangular basin. A 24-year simulation is described that has almost no climate drift, a good simulation of the mean temperature gradient across the ocean, but smaller than observed annual and interannual variability. The coupled model is analyzed to see where it occurs on the schematic bifurcation diagram of Neelin. In years 9–16 of the simulation there is a dominant oscillation with a period of two years. The spatial pattern of this oscillation shows up clearly in the first empirical orthogonal function calculated from monthly averages of sea surface temperature anomalies. A series of 19 model-twin predictability experiments were carried out with the initial perturbation being a very small change in the ocean temperature field. The correlation coefficient of monthly sea surface temperature anomalies from these model-twin experiments decreases rapidly over the first 6 months and after that, more slowly, showing that there is some predictability out to a year. The predictability times are marginally increased if only the coefficient of the first empirical orthogonal function of monthly averaged sea surface temperature anomalies or NIN03 sea surface temperature is predicted. There is some evidence to indicate that it is easier to predict the onset of a model warm event than to predict the onset of a model cold event. More detailed analysis of the first model-twin experiment shows that the initial divergence in the integrations is a change at day 6 in the incoming solar radiation due to a change in the atmospheric model clouds. The dominant early change in sea surface temperature occurs by this change in radiative heat flux. If the cloud feedback is set to zero, then the first changes are delayed to day 12 and occur in the evaporative and sensible heat fluxes and in the atmospheric wind stress. In this case the dominant early change to sea surface temperature is by advection due to the changed wind stress.

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