TITLE: Exploitation Of Parallelism In Climate Models

PI/INSTITUTION: Ferdinand Baer, University of Maryland; Joseph J. Tribbia and David L. Williamson, NCAR



PROJECT END DATE: 15 April, 1998

CONTACT INFORMATION: Ferdinand Baer , Department of Meteorology, University of Maryland, College Park, MD 20742: Phone: 301 405-5397, Fax: 301 314-9482, email: baer@atmos.umd.edu.

WWW URL: http://metosrv2.umd.edu/~baer/


Global climate predictions currently made with coupled atmosphere and ocean components including the biosphere are so complex and require so much computing power on available computing hardware that using traditional numerical algorithms allow very few integrations with given resources. Moreover most current models do not include a sufficient breadth of spatial scales to realistically predict the long time scales which are essential for climate prediction. Additionally models do not include the details of some significant physical forcing (albeit not yet clearly understood) nor are they systematically coupled to those environments which have a substantial impact on the climate's evolution. Numerous scenarios must be computationally played out with the most comprehensive models to identify meaningful bounds on the range of influence of parametric variables. The need for computing resources many orders of magnitude larger than presently available, at least when applying current numerical methodology, is thus strikingly evident.

Experiments with systems in high rotation such as the atmosphere and including significant external energy input over a broad scale range to include phenomena such as boundary layer friction and convection, indicate that incorporating time independent statistics for subgrid scale forcing is not adequate to provide successful predictions.

For reliable climate predictions, numerical results must not be dependent on model spatial resolution. Models must be run at resolutions where dynamic scalability, or numerical convergence under mesh refinement, has been achieved. Note that this is distinct from computational scalability commonly discussed with regard to parallel computer performance. A lack of dynamic scalability calls into question model predictions since the effects of parameterized scales can be sufficient to render the climates of models with differing resolutions distinct. Such scalability is well established for purely dynamical models, but very little is known on the dynamic scalability of subgrid physical parameterizations used in GCMs. Subgrid resolution studies will help in the coupling of local, high resolution models to GCMs in addition to answering questions on scalability. Much of the forcing from subgrid parametrizations will ultimately be replaced by results from concurrently run local, high resolution models.

As stated in our proposal to the CHAMMP program, our research efforts are designed to meet the CHAMMP initiative which hopes to develop the capability to make meaningful regional climate forecasts on time scales exceeding a decade, based on numerical prediction type models. Our plan was outlined by three primary initiatives stated as follows:

a) To reconfigure the prediction equations such that the time iteration process can be compressed by use of MPP architecture, and to develop appropriate algorithms.

b) To develop local subgrid scale models which can provide time and space dependent parameterization for a state-of-the-art climate model to minimize the scale resolution necessary for a climate model, and to utilize MPP capability to simultaneously integrate those subgrid models and their statistics.

c) To capitalize on the MPP architecture to study the inherent ensemble nature of the climate problem. By careful choice of initial states, many realizations of the climate system can be determined concurrently and more realistic assessments of the climate prediction can be made in a realistic time frame.

To explore these initiatives, we have exploited all available computing technology, and in particular MPP machines. Our efforts have proceeded along the lines indicated above but as our research has progressed our focus has gradually taken a more cohesive perspective. Our approach has gradually coalesced on dealing with regional and global climate modeling concurrently from two different model perspectives, always considering the use of MPPs as the principal tool. These approaches to be elaborated in the sequel involve multitasking with an imbedded MM5 model and parallel algorithms with spectral accuracy. Concurrently we are searching for optimum realization statistics associated with concurrent climate simulations. Finally, we continue our search for a more efficient time truncation scheme based on various expansion techniques. As a corollary to our methodological approach, we strive for computational architectural independence.


A) Parallel algorithms with spectral accuracy;

Our interest in developing time and space dependent sub-grid parametrizations, as well as our interest in studying the resolution sensitivity of sub-grid parametrizations has lead us to develop a spectral element atmospheric model. This is a type of finite element method in which a high degree spectral method is used within each element. The method provides spectral accuracy while retaining both the parallel efficiency and geometric flexibility of finite elements. This geometric flexibility makes the model an ideal tool for studying sub-grid parametrizations. It provides a very natural way to incorporate local mesh refinement within a global model. Thus regional or sub-grid scale resolution can be achieved (over a limited region) within a global model in a truly interactive way and without resorting to any kind of nesting or interpolation between different models. Thus it will be feasible to model the interaction between large scale and subgrid scale phenomena. This will allow us to calculate detailed statistics of this interaction, leading to improved subgrid parameterizations. This work may also lead to improved climate simulations by allowing increased resolution in a few dynamically significant regions. Similarly, regional climate simulations may be improved by allowing regional resolution to be incorporated within a global model.

We first developed an MPP spectral element shallow water model and made use of the now standard shallow water test cases [Williamson et. al., J. Comput. Phys. 1992] to verify several points:

1) The spectral accuracy of the method, even for the difficult non-linear test cases in spherical geometry.

The shallow water test cases allowed us to compare the spectral element method with several other methods without having to obtain and run the other codes. The test cases prescribe several forms of error and how they are to be computed so that any method can be compared objectively to several other methods by simply consulting the literature. The results were as expected: On a per-grid point basis, the method achieves error levels similar to that of spherical harmonic spectral methods, and significantly more accurate than finite difference methods. The results also show the spectral element method has no problems with spherical geometry. This is due to the fact that the sphere can be tiled almost uniformly with quadrilateral elements. This work was described in Taylor et al, J. Comput. Phys. 1997, and Haidvogel et al, Atmospher-Ocean 1997.

2) The almost perfect parallel efficiency and scalability of the method.

The spectral element method is ideal for MPPs. The spectral elements are distributed equally among the processors. Communication costs are minimal since each element needs only to communicate its boundary data with neighboring elements. The computationally intensive spectral transforms which must be performed within each element are localized to each processor and require no communication. On the Cray T3D, for a wide variety of problem and machine sizes, the parallel efficiency of the algorithm never drops below 95%. By this we mean that the time spent on inter-process communication is never more than 5% of the total, resulting in almost perfect parallel scalability.

3) The effectiveness of the localized mesh refinement capability of the method.

Finally we used the test cases to show that the local mesh refinement capability of the method is quite effective. It improves the solution within the region of high resolution without degrading (and sometimes improving) the solution over the rest of the sphere. We then created a more extreme test by increasing the steepness of the topography in the flow past a mountain test case. With uniform spectral element grids, severe Gibbs type oscillations appeared upstream of the mountain which were only removed at T360 and higher resolution. However, results as good as the T360 results could be obtained using a spectral element grid with T42 resolution over most of the sphere and a 10-fold increase in resolution near the mountain. This run was 10 times faster than the T360 uniform resolution run.

The success of the spectral element method with the shallow water equations has prompted us to develop a primitive equation version of the model. We have recently completed this work and are now using the Held-Suarez test cases [Held and Suarez, Bull. Amer. Met. Soc. 1994] to compare the spectral element method with other GCM dynamical cores. Preliminary results indicate the model competes favorably with spherical harmonic spectral models. Our next step is to add an established physics package (such as that in the NCAR CCM3) to the model so that we may begin to address sub-grid scale resolution and parametrization questions mentioned above.

B. Concurrent climate simulations:

We are continuing our studies of the utility of ensemble methods in understanding the climate dynamics of models of the earth system which exhibit almost intransitive behavior. As an addition to our studies involving thermohaline reversals as examples of such bimodal systems, we are also pursuing a simpler atmospheric example of multiple jet structure and equatorial super-rotation as described in Saravanan (1992). The atmospheric dynamics of this example involve the balance between eddy forcing and zonal jet structure. Because only the atmosphere is involved, climate statistics equilibrate much more rapidly than in the coupled system and, thus, this is an example of a more rapidly evolving climate distribution. This example,which is believed to be relevant for the equatorial dynamics of the equable climate of 6000 y BP, serves to demonstrate the efficacy of this technique even for atmosphere alone climate studies and is a precurser to the examination of an interesting paleo-climatic period of reduced ENSO variability.

Planetary wave bimodality was also studied in the NCAR CCM and reported on by Tribbia at the Wiin-Nielsen Symposium (Tribbia 1995, abstract only). To further investigate these and the abovementioned atmosphere alone questions in a more realistic modeling context, an R15 version of CCM2 is currently being tested in ensemble mode on the Cray T3D at NCAR.

C. Model reconstruction efforts to speed up computations;

Currently climate prediction models are solved as a marching problem in time, and are limited in time by the number of steps needed to march into the future. Climate change requirements indicate that prediction times may reach beyond decades, including increased resolution to determine smaller scales events. Given the many experiments needed to develop a successful model (GCM), a more efficient integration scheme seems essential.

Our effort at reconstructing the prediction system to optimally use parallel processors involves time compression. If we define the "computing cycle" to be that time required to do once all the calculations which must systematically be repeated to complete an entire calculation we would in principle attempt to do everything necessary in such a cycle. The longer the time step, the more computations could be completed in one cycle. On a massively parallel processor with unlimited processors, the computing cycle would include all the calculations which would not need repetition by their dependence on previous calculations. For conventional marching problems, the minimum computing cycle could be one complete time step. It is this computing cycle that we attempt to approach.

We demonstrated in our original proposal a procedure to extend the time step in a climate model, i.e., to calculate more than one time step in a computing cycle by noting that the equations we must solve represent an initial value problem and that the ultimate solution depends only on the initial conditions. Hypothetically it should be possible, by suitable restructuring of the numerical equations, to come to the final solution in only one computing cycle, achievable only with an unlimited number of processors. The example used was a first order linear differential equation with constant coefficients written in finite difference form. Repeated substitution shows that the solution at NÆ can be represented as a function of the initial values which can be computed in one machine cycle on a parallel processor. This process clearly becomes complex when applied to a nonlinear system.

We concentrated on the simplest example to study the ramifications of time compression, the low order BVE. This model was first considered by Lorenz (1960) on the f-plane, and then by Baer (1970) on the sphere. We developed a formula by resubstitution to relate the variables at any time to the initial values which would allow a computation to be made in one machine cycle. However the number of calculations required grew astronomically as the time step grew large. The formula could also be represented by a Taylor Series and with a small error the number of terms was drastically reduced. The formula for the low order system in terms of the Taylor Series can be developed by repeated differentiation. This was accomplished by use of Mathematica (Wolfram, 1991). In addition this system has an analytic solution thereby allowing an accurate test of any numeric approximation.

The most interesting feature of this system involves the interaction of some planetary wave with a zonal current. A number of experiments were performed with different wavelengths, various shapes of the zonal current, different relative energy distributions between the wave and the zonal field and various amounts of total energy in the system. In each experiment, the number of terms of the Taylor Series required to give a good approximation to the exact solution was recorded.

Since we presume that all calculations for a given time step could be performed in one machine cycle for this simple system, we compared the solution by the Taylor Series method to the leapfrog scheme, requiring both methods to yield results within ten percent of the exact solution. We also tested the Taylor Series scheme using one sided and centered differences. Although there was considerable variability based on the initial conditions selected, the time step achievable with the Taylor Series was seven to ten times longer than the time step used for the leapfrog scheme. Although the Taylor Series method is bounded in time by the true oscillation period, that constraint was never met by any of our experiments.

Based on these positive results, the experiments were redone using the model on the sphere. Although the calculations were considerably more complex and involved many more computations, the results were comparable to those with the Lorenz model. The spherical model was expanded to include two independent waves each with its own latitudinal structure and both interacting with an arbitrary zonal flow and comparable experiments were performed. The results were similar to those achieved with the Lorenz model although we did not use as many terms in the Taylor Series expansion.

These encouraging results with experiments using the Taylor Series (TS) approach applied to the BVE both on the grid and on the sphere, and in particular with the low order systems studies to date, led us to proceed with the shallow water equations. This system is considerably more realistic insofar as it allows for gravity waves. From the knowledge gained so far, shorter frequencies implied by gravity motions could restrict the benefits of the TS approach. However since some smoothing is always applied to counteract the high frequency low amplitude gravity waves, we first applied the bounded derivative method which seems ideally suited to the TS process. We used the balancing conditions from the initialization as the forecast system. Since we were unable to push the time step beyond the stability criterion, we did not make much headway with this procedure. Our next attempt involved applying the shallow water equations in the spectral mode using an expansion in Hough modes, first suggested by Kasahara. Following a recommendation by Daley, we split the spectrum of waves into the high and low frequency domains, predicting the low frequencies and balancing the high frequencies. For time prediction we again tested the TS approach.

In this experiment we included 20 planetary waves and 20 each for eastward and westward gravity waves as well as 20 Rossby waves. A pretested initial state was used with an equivalent depth of 10km. A number of techniques were tested which balance the gravity modes. A reference calculation was made using the leapfrog scheme with a time step of three minutes which is much smaller than any of the linear periods of the model. Because of limited computer resources, only cases with one and three terms of the Taylor Series expansion were used. Integrations for short periods (six days) and for longer times (up to 360 days) were performed. The results showed that there was sensitivity to the balancing technique selected. More sensitivity was noted to the mode at which the balancing was started. The more gravity modes included in the prediction, the more accurate the result. This of course limited the maximum tine step that could be used, but when the additional term in the Taylor Series was added, the accuracy improved. Alternately, for a given order of accuracy, the additional term in the expansion allowed for a longer time step.

The complexity of using the Taylor Series expansion method by calculating higher derivatives led us to explore an alternate technique which requires only the initial values of the model dependent variables and their first time derivatives. We call this a multi-level time integration scheme and it requires the calculation of a number of initial values which is straightforward by using the prediction system itself. In this case we can use these values instead of the higher time derivatives to determine the coefficients for the Taylor Series. We tested this method with the Lorenz system. In this case we set a much more rigorous accuracy condition. We discovered that as we increase the desired accuracy of the solution, the computational advantage increased. Thus for accuracy in excess of one percent we were able to gain a threefold advantage in the basic time step. Given this success, we are now testing the method with the shallow water equations. Given the nature of the process, there appears to be no disadvantage to solving more complex problems.

D. Optimum realization statistics;

As indicated by our interest in the ensemble nature of the climate problem and that meaningful climate statistics require many realizations, we undertook to examine the possibility that by suitable representation of model output, some reduction in the number of realizations might be feasible. This was stimulated by the repeated observation that if model variables are expressed spectrally, amplitude variations seem to be more robust than phase and that phase variations for smaller scales tend to be exceptionally variable. We set out to check this hypothesis from archived model simulations represented in spectral space and we chose the expansion in Solid Harmonics since often these are the actual functions used in the numerical integration. The presumption behind this endeavor is that if amplitude does not vary substantially amongst the realizations, then if amplitude is an adequate measure of climate, many fewer realizations might be needed Clearly this reduction in computer time would be welcome in resolving the climate prediction problem.

From a many year integration archive of a two level baroclinic model, we selected 80 realizations of the mean and shear stream field as our ensemble data set. To test sensitivity to the climate period itself, we took a number of climate averages from one day to two months, but focused most attention on the monthly averages. Each spatial field of the ensemble was converted to spectral form and stored as amplitude and phase. The amplitudes were averaged over the ensemble and a standard deviation field was constructed. The ratio of the standard deviation of each spectral coefficient amplitude to its mean value was recorded. The spatial patterns of the same realizations were then averaged and a standard deviation pattern was created. Both the mean spatial pattern and its standard deviation field were spectrally decomposed and the ratio of the amplitudes of the standard deviations to the means were calculated. The results of the experiment highlighted the fact that the ratios developed from the amplitudes of the spatial fields were almost an order of magnitude larger than the ratios determined from the averaged amplitudes. This was true for each of the climate averaging periods we considered .Since averaging the spatial fields incorporates changes in phase as well as amplitude, the implication of the results corroborates our hypothesis insofar as many ensemble calculations are needed primarily to overcome variability in phase predictability. If amplitude patterns are sufficient to determine climate statistics, only very few realizations may suffice to yield meaningful results.

We confirmed that these results were did not depend significantly on model truncation, repeating the analysis with archives from the same model running at truncations of T31, T42 and T63. However it was still necessary to establish if our findings would be corroborated with more climatically realistic models. For this purpose, we have available 9 realizations of 45 year integrations (as well as 12 realizations of 11 years) of the NMC forecast model. We processed this data similarly to the way we analyzed the archived model data described above, concentrating on the surface temperature and the 200 and 500 hPa geopotential heights. Our conclusions on climate variability in amplitude and phase are confirmed from the analysis of output from this more realistic model. Moreover the large variability in phase was also confirmed from independent analyses of the spectral phase data.

Noting the dependence of the results on representation, and the difficulty in predicting phase, we have begun to develop EOFs of sample data sets with available model data and we plan to project the model realizations onto these functions. Since the EOFs are undoubtedly robust, we anticipate that the statistics with regard to EOFs may well show up with the advantageous characteristics we have found for amplitude. Additionally, we will continue to study the phase results and how we might develop a statistical phase pattern which will mimic the norm of many realizations without actually requiring the development of statistics from a large sample of realizations.

E. Parameterization studies;

As an adjunct to our efforts to impfove models from a representational and computational viewpoint, we have also studied a number of issues relating to forcing in the atmosphere which may assist us in parameterizing those processes which cannot be developed by a model in complete detail. Since the tropics are primary source for such forcing, we have focussed most attention on them, including the EL NINO phenomenon.

1) El Nino-Generated Streamfunction Anticyclones over The Equatorial Pacific:

Perhaps the most prominent of the upper-tropospheric flow patterns associated with El NINO is the anticyclone pair straddling the equator in the central Pacific. Gill's (1980) simple linear model of the tropical circulation produces similar anticyclones in response to an idealized heat source centered on the equator. However, when the source is placed at an appropriate longitude for El NINO, the modeled anticyclones are well to the west of their observed position. To explain this longitudinal discrepancy, several authors (e.g., Hendon 1986, Sardeshmukh and Hoskins 1985) have noted that an eastward shift of the anticyclones could be caused by dynamical nonlinearity. Although these studies regard the Gill model as an oversimplification, they accept Gill's hypothesis that the anticyclones are generated by local atmospheric heating in the equatorial belt. In our research we test Gill's hypothesis during the summer months, when the nonlinearities should be less severe. Specifically, we simulate the El NINO anticyclones for May and August taken from a 10-year composite using a diagnostic model forced by heatin diagnosed residually from observations. In August, the Gill hypothesis is found to be essentially correct, although the longitude of the anticyclones is determined equally by local anomalous heating and cooling. However, in May the principle forcing of the anticyclones is neither local nor equatorial. Rather, the anticyclones are forced remotely from the northern tropics and subtropics, by heating over the subtropical Atlantic and cooling over the western Pacific and Southeast Asia.

2) Effects of upstream energy propagation on the response of barotropic diagnostic models to tropical heating:

Many studies have claimed that El NINO-related tropical heating anomalies influence the midlatitude atmospheric circulation by generating stationary Rossby waves which propagate over great distances. Such studies typically use ray-tracing and wave refraction arguments to show the paths along which stationary wave energy radiates into the extratropics. In this literature, it is almost universally assumed that such energy propagation has an eastward, or downstream, component. In our barotropic simulations, we find that upstream energy propagation also plays a significant role in determining the extratropical response to tropical heating (i.e. divergence) anomalies. This upstream propagation manifests itself primarily in the form of large plumes in the streamfunction field which extend poleward and westward in both hemispheres from the divergence source. However, comparison shows that the upstream plumes produced by the barotropic model are larger both in amplitude and meridional scale than those produced in a comparable baroclinic simulation. In the simplest case of a resting atmosphere on an infinite equatorial beta-plane, we derive an analytical solution which demonstrates the general form of the upstream response. Using this analytical solution, it can be shown that barotropic simulations are susceptible to a form of resonance which is not present in the baroclinic calculations. Our finding, which will be submitted for publication in the near future, is thus that barotropic models overestimate the extent of upstream energy propagation. We believe that this finding has important consequences for barotropic simulations of both climatological and anomalous flows.

3) Effect of mountain ranges on the midlatitude atmospheric response to el nino events:

To gain better insight into how parameterization might be effectively used in Global Climate prediction and how efficiencies in computation in this arena might be applied, we have considered the barotropic modeling of the interaction of tropical-heating induced by stationary waves with mid-latitude orography. Tropical heating associated with El NINO events can excite large amplitude stationary waves that extend well into the extratropics. However, these directly forced stationary waves do not explain the upper level streamfunction anomalies that occur during El NINO winters. To generate the observed anomalies, the directly forced patterns must undergo secondary interactions with other components of the climate system. We have thus investigated the modification of the extra-tropical response to tropical heating by a secondary interaction with mid-latitude orography. Using a steady barotropic anomaly model, we found that this interaction, occurring primarily in the Himalayan-Tibetan region, generates circulation anomalies which resemble the observed circulation anomalies, including a four-celled pattern over the Pacific/North American section with height anomalies in excess of 40 geopotential meters. The associated zonal wind anomalies reach 6m/sec in the central Pacific, where they act to extend the jet southeastward, as is typical in El NINO winters. Results were published in Nature (see DeWeaver and Nigam, 1995).

As a follow-up to this study (see Nigam and DeWeaver, 1997) we test the sensitivity of the above orographic interaction to a variety of factors. The principal finding is that the orographic interaction is sensitive to the specification of the extratropical convergence in the model. This convergence is required to offset the tropical divergence so that the net global mean divergence anomaly is zero. The sensitivity is partly a consequence of the interaction between the eddy and zonal-mean components of the flow anomaly simulated by the model.

4) Dynamics of zonal-mean flow assimilation:

Further insights on potential parameterization were gained from a study of the causes and dynamical implications of differences between NASA/GEOS assimilated and ECMWF analyzed 200mb divergent circulations during recent El NINO /La NINA winters. The Data Assimilation Office at the Goddard Laboratory for Atmospheres has recently produced an atmospheric data set for the years 1985 to 1993 using a fixed assimilation system known as the Goddard Earth Observing System Data Assimilation System (GEOS-DAS). The data set is produced using a novel method, the Incremental Analysis Update (IAU) procedure, in which observed data are fed gradually into a model integration as external forcing functions. We diagnosed the seasonally averaged 200mb circulations for recent El NINO (1987/88) and La NINA (1988/89) winters (DJF) using both GEOS-DAS and ECMWF data to determine whether there are significant differences in the dynamics of the zonal-mean circulation revealed by the two data sets, and whether these differences are related to the IAU method. We find that The IAU method is not adequate to assimilate the Hadley circulation. This result is demonstrated using an idealized model of a zonally symmetric assimilation in log-pressure coordinates. The finding is borne out by a comparison of GEOS-DAS assimilation, analysis, and first-guess fields, in addition to the comparison with the ECMWF wintertime Hadley cell. An alternative assimilation method is proposed for the zonally symmetric circulation, in which thermal and mechanical forcing functions are constructed from the difference between first guess and analysis estimates of potential vorticity and secondary circulation. For results see DeWeaver and Nigam, 1997.



Zhang, Bing, 1993: Use of Taylor expnsion in computation of the simple spectral barotropic vorticity equation. MS scholarly paper, Dept. of Meteorology, U. Of Maryland, 14pp + tables and figures.


DeWeaver, E., 1994: On the dynamics of extratropical low-frequency atmospheric variability. MS scholarly paper, Dept. of Meteorology, U. Of Maryland, 25pp + figures.

Baer, F., and Wm. Campbell, 1994: Experiments In Optimum Three-Dimensional Model Truncation. PROCEEDINGS OF THE TENTH CONFERENCE ON NWP, 17-22 July 1994, Portland, OR, Boston, MA, p. 41-43.

Baer, F., and E. DeWeaver, 1994: The impact of subgrid scale motions and their parameterization in atmospheric models. PROCEEDINGS OF THE TENTH CONFERENCE ON NWP, 17-22 July 1994, Portland, OR, Boston, MA, p. 389-391.


DeWeaver, E., and S. Nigam, 1995: Influence of mountain ranges on the mid-latitude atmospheric response to El NiÒo events. NATURE, 378, 706-708.

Taylor, M., 1995: Cubature for the sphere and the discrete spherical harmonic transform. SIAM J. NUMER. ANAL., 32, 667-670.

Mizzi, A., J. Tribbia and J. Curry, 1995: Vertical Spectral Representation in Primitive Equation Models of the Atmosphere. MON. WEA. REV., 123, 2426--2446.


Baer, F., 1996: Experiments in optimum three-dimensional truncation. MODERN DYNAMICAL METEOROLOGY, Proceedings from a Symposium in honour of Professor Aksel Wiin-Nielsen, Copenhagen, Denmark, 1995, Ed. by Peter D. Ditlevsen, pp 9-14.


Taylor, M., J. J. Tribbia and M. Iskandarani, 1995: The spectral element method for the shallow water equations on the sphere. J. COMPUT. PHYS., 130, 92-108.

Haidvogel, D., E. Curchitser, M. Iskandarani, R. Hughes and M. Taylor, 1997: Global modeling of the ocean and atmosphere using the spectral element method. To appear in ATMOSPHERE-OCEAN.

Ehrendorfer, M. and J. Tribbia, 1997: Optimal Prediction of Covariances through Singular Vectors. To Appear, J. ATMOS. SCI.

DeWeaver, E., and S. Nigam, 1997: Dynamics of zonal-mean flow assimilation and implications for winter circulation anomalies. J. ATMOS. SCI., in press.

Nigam, S., and E. DeWeaver, 1997: Influence of orography on the extratropical response to El NINO events. Submitted, J. CLIM.


Eric DeWeaver, MS, 1994, On the dynamics of extratropical low-frequency atmospheric variability. Expected Ph. D. graduation date, Dec. 1997.

Bing Zhang, MS, 1993, Use of Taylor expnsion in computation of the simple spectral barotropic vorticity equation. Expected Ph. D. graduation date, Dec. 1997.


Baer (27)

Tribbia (10)

Taylor (8)

Thibaud (3)

DeWeaver (10)

Zhang (1)