Ferdinand Baer

An Alternate Scale Representation of Atmospheric Energy Spectra

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 charac...

A PROCEDURE FOR NUMERICAL INTEGRATION OF THE SPECTRAL VORTICITY EQUATION

Abstract The vorticity equation may be integrated in spectral form by constructing a table of interaction coefficients. A method is given for computing and ordering this table and for performing the numerical integration. An advantage of the spectral form over the usual grid-point forms is that no mapping of the spherical surface is required. The results are discussed briefly for a 43-day trial integration in one-hour steps.

Integration with the Spectral Vorticity Equation

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.

Climate Modeling with Spectral Elements

As an effort toward improving climate model–component performance and accuracy, an atmosphericcomponent climate model has been developed, entitled the Spectral Element Atmospheric Climate Model and denoted as CAM_SEM. CAM_SEM includes a unique dynamical core coupled at this time to the physics component of the Community Atmosphere Model (CAM) as well as the Community Land Model. This model allows the inclusion of local mesh refinement to seamlessly study imbedded higher-resolution regional climate concurrently with the global climate. Additionally, the numerical structure of the model based on spectral elements allows for application of state-of-the-art computing hardware most effectively and economically to produce the best prediction/simulation results with minimal expenditure of computing resources. The model has been tested under various conditions beginning with the shallow water equations and ending with an Atmospheric Model Intercomparison Project (AMIP)-style run that uses initial conditions and physics comparable to the CAM2 (version 2 of the NCAR CAM climate model) experiments. For uniform resolution, the output of the model compares favorably with the published output from the CAM2 experiments. Further integrations with local mesh refinement included indicate that while greater detail in the prediction of mesh-refined regions—that is, regional climate—is observed, the remaining coarse-grid results are similar to results obtained from a uniform-grid integration of the model with identical conditions. It should be noted that in addition to spectral elements, other efficient schemes have lately been considered, in particular the finite-volume scheme. This scheme has not yet been incorporated into CAM_SEM. The two schemes—finite volume and spectral element—are quasi-independent and generally compatible, dealing with different aspects of the integration process. Their impact can be assessed separately and the omission of the finite-volume process herein will not detract from the evaluation of the results using the spectral-element method alone.

A Spectral Element Version of CAM2

The authors describe a recent development and some applications of a spectral element dynamical core. The improvements and development include the following: (i) the code was converted from FORTRAN 77 to FORTRAN 90; (ii) the dynamical core was extended to the generalized terrain-following, or hybrid , vertical coordinates; (iii) a fourth-order Runge–Kutta (RK4) method for time integration was implemented; (iv) moisture effects were added in the dynamical system and a semi-Lagrangian method for moisture transport was implemented; and (v) the improved dynamical core was coupled with the Community Atmosphere Model version 2 (CAM2) physical parameterizations and Community Land Model version 2 (CLM2) in such a way that it can be used as an alternative dynamical core in CAM2. This spectral element version of CAM2 is denoted as CAM-SEM. A mass fixer as used in the Eulerian version of CAM2 (CAM-EUL) is also implemented in CAM-SEM. Results from multiyear simulations with CAM-SEM (coupled with CLM2) with climatology SST are also presented and compared with simulations from CAMEUL. Close resemblances are shown in simulations from CAM-SEM and CAM-EUL. The authors found that contrary to what is suggested by some other studies, the high-order Lagrangian interpolation (with a limiter) using the spectral element basis functions may not be suitable for moisture and other strongly varying fields such as cloud and precipitation.

Hemispheric Spectral Statistics of Available Potential Energy

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-Vaisala frequency distribution.

Effects of Spectral Truncation on General Circulation and Long-Range Prediction

Abstract Although atmospheric prediction models appear to yield results similar to observation, both their detailed predictive capability and their time-averaged forecasts depend on space truncation. Such dependence may be readily studied with a spectral model because of the ease of modifying truncation. A simple, two-level, quasi-geostrophic, forced general circulation model was represented in spectral form and nine cases of different truncation were integrated for the same forcing, starting with initial conditions generated from a state of rest. The truncations ranged from six to sixteen meridional waves, from five to ten degrees of freedom with latitude, and the models were integrated for about 60 days with finite-amplitude nonlinearity. Considering the kinetic energy in the vertical mean flow, and separating this energy into zonal and eddy, the results show that the general circulation may be predicted with as few as twelve planetary waves and eight latitudinal degrees of freedom, whereas detailed pre...

THE EXTENDED NUMERICAL INTEGRATION OF A SIMPLE BAROTROPIC MODEL

Abstract A simple barotropic model is integrated for 39 days by one-hour time steps over 94 per cent of the earths 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.

EFFECTS OF ELECTRIC FIELDS ON WATER-DROPLET COALESCENCE

Abstract Growth of incipient precipitation particles by collision and coalescence with cloud droplets is one of the primary mechanisms of natural rain. Comparison of previous research shows wide divergence between various theoretical and laboratory values of collision efficiency and coalescence efficiency. In an effort to obtain additional laboratory measurements of droplet coalescence, high-speed photographs were taken of colliding droplets at the breakup point in a Rayleigh jet. With 700-micron diam droplets, less than 30 per cent of the collisions result in coalescence under no field condition. At fields of about 40 v per cm, the coalescence was about 100 per cent under all conditions of field.

Three-dimensional Scaling and Structure of Atmospheric Energetics

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 giv...