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Ferdinand Baer

Abstract

Three-dimensional structure functions characteristic of atmospheric energy are developed from data samples and presented. The horizontal dependence is based on known functions in spherical surfaces whereas the vertical dependence is derived from data. Three-dimensional scaling of these functions is determined from the properties of the functions as well as their ability to satisfy the potential vorticity equation. The distribution of observed planetary energy represented by these functions is presented in terms of their scale. Two-dimensional energy distributions in each of the vertical modes are also described. Power law relationships of energy versus scale with slope of −3 appear in the statistics, but not under all conditions. The results of the study may be useful in parameterizing non-resolved model scales for closure. The structure functions may be utilized for fully three-dimensional spectral modeling. The scaling of the structure functions indicates appropriate vertical resolution for given horizontal model truncation as well as the appropriate vertical levels to be selected.

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Ferdinand Baer
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Ferdinand Baer

Abstract

Analysis of hemispheric temperature variance data on five isobaric surfaces in terms of two-dimensional spectral decomposition shows that the available potential energy distributes with a slope in the neighborhood of −3 for the scale range 14≤n≤25. Although this slope varies with pressure, indications are that the observations substantiate the expectations of geostrophic turbulence theory. The noted deviations from −3 are discussed in terms of the distribution of energy in vertical modes which are not in the range in which −3 statistics should be expected. Vertical scales necessary for a three-dimensional spectral representation are considered with regard to the Brunt-Väisälä frequency distribution.

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Ferdinand Baer

Abstract

Although it is known that all spatial scales are nonlinearly interrelated in any prediction model of the atmosphere, truncation demands a limit to scale resolution. One is therefore compelled to parameterize sub-resolution scales, hopefully in such a manner that they describe observed statistics. Such statistics have been shown frequently as energy spectra of synoptic scales in terms of the planetary wavenumber. An alternate representation is the presentation of the energy in terms of the degree of a Legendre polynomial expansion; this representation may be more advantageous insofar as it presents a two-dimensional spectral index. Arguments are presented which indeed suggest the appropriateness of the index. Two months of atmospheric wind data at five pressure levels and on a hemispheric grid were analyzed to establish energy spectra. The spectra are described both as a function of time and as a function of wavenumber for time averages. Using a five-level linear baroclinic model, stability characteristics for each wave component for the observed zonal and vertical profiles were established. Based on these results, the energy data were fit logarithmically by least squares to the wavenumber (both planetary wavenumber and Legendre polynomial degree). Energy slopes show values close to −3 when utilizing the two-dimensional index in the non-baroclinically forced scale range. These results suggest the use of this index in studying scale parameterization.

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Ferdinand Baer
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Ferdinand Baer

Abstract

The barotropic vorticity equation in spectral form is integrated for a time period exceeding 80 days in two cases of hypothetical initial conditions with a time step of three hours without appreciable truncation error at the end of the period. The computational stability and truncation properties of the spectral system are discussed, the stability criterion for the two cases is computed, and the truncation errors are to some extent explained.

The results of the integrations show systematic periods of very pronounced energy exchange among the long waves, with little energy filtering down to the short waves. An analytic solution of a low-order spectral system suggests that the periodic exchange may be characteristic of the differential equation rather than dependent entirely on the initial conditions. The high-frequency components are examined for equilibrium of energy exchange after extended integration. Our results suggest that such an equilibrium does not exist in the model we have formulated.

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Ferdinand Baer

Abstract

A simple barotropic model is integrated for 39 days by one-hour time steps over 94 per cent of the earth's surface. By suitably smoothing the stream function at each time step, time truncation errors are controlled and the total energy of the system varies within only two per cent of the initial value over the entire integration period. By initiating the integration with energy in the mean flow and in planetary waves one and three, a pronounced periodic exchange of energy is observed over the integration period. The predominant periodicity is a six-day exchange between planetary wave number three and the mean flow. The energy distribution in the higher wave numbers appears to be capable of a statistical interpretation.

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Ming Ji and Ferdinand Baer

Abstract

A three-dimensional scale index based on spherical domain and quasigeostrophic scale analysis indicates a truncation limit of global atmospheric models that includes both horizontal and vertical dimensions. Applying such a scale index, a numerical experiment is designed using a simplified adiabatic version of the National Center for Atmospheric Research (NCAR) Community Climate Model (CCM0B) to examine, incorporating nonlinear dynamics alone, whether an optimal horizontal resolution for a nine-vertical-level (modes) global general circulation model can be achieved. In establishing appropriate vertical modes that can be uniquely scaled and are independent and physically relevant, an optimal distribution of levels, which has been developed, is utilized in the experiment.

The experimental results, which consist of a total of 110 individual integrations of the CCM0B with ten initial states for each of six horizontal truncations, appear to agree with the conclusions implied by the above referenced three-dimensional scale index; that is, a consistent horizontal resolution for a nine-vertical-level model should be in the range of triangular truncation T25 to T30 to yield optimal prediction results, considering, however, only the nonlinear dynamical aspect. It should be noted that due to the simplifications and idealizations made to carry out our experiment, additional studies under more realistic atmospheric conditions are necessary and are encouraged based on the results presented herein to further validate the existence of the consistency of three-dimensional model resolutions.

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Ferdinand Baer and Yuejian Zhu

Abstract

The National Center for Atmospheric Research Community Climate Model 1 was used as an experimental prediction model to assess the value of reassigning model levels in the vertical based on an optimizing hypothesis. The model was considered for T31 horizontal truncation and 12 vertical levels. The levels were relocated in a model called test, and the model with the conventional levels was denoted standard. Both models were integrated for 5 days with six independent initial states, and the results were composited. Analyses of the composites for both models were compared to actual observations. The results of the experiments indicate that the barotropic component of the flow was predicted with equal quality by both models but that the baroclinic component was predicted better by the test model. This observation may be explained by the increased fidelity of the vertical structure in the test model, since it has more resolution in the stratosphere.

Additional analyses were performed using a hypothesized three-dimensional scale index that relates the vertical to the horizontal truncation. The results of those analyses were sufficiently suggestive to encourage further studies to find optimum truncation in all three dimensions simultaneously.

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Ferdinand Baer and Fred N. Alyea

Abstract

A quasi-geostrophic, two-level baroclinic model with a fixed heating function and simple friction represented in spectral form was integrated with climatological initial conditions for 30 days. Three separate integrations were performed changing only the spectral truncation. Two of the experiments had 10 degrees of freedom in latitude with 16 and 20 planetary waves, while the third had 11 degrees of latitudinal freedom and 18 planetary waves. Comparison of the integration results indicates that the increase in latitudinal resolution caused pronounced changes in the predicted variables whereas the increase in the number of planetary waves had a negligible effect on the distribution of variables over the integration period.

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