TITLE: Exploitation Of Parallelism In Climate Models
PI/INSTITUTION: Ferdinand Baer, University
of Maryland; Joseph J. Tribbia and David L. Williamson, NCAR
PROJECT GRANT NO.: DEFG02-95ER62022
PROJECT START DATE: 9 SEPT., 1991
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:
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
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
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
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
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.
GRADUATE STUDENT THESES AND DISSERTATIONS:
Eric DeWeaver, MS, 1994, On the dynamics of extratropical low-frequency
atmospheric variability. Expected Ph. D. graduation date, Dec.
Bing Zhang, MS, 1993, Use of Taylor expnsion in computation of
the simple spectral barotropic vorticity equation. Expected Ph.
D. graduation date, Dec. 1997.