John B. Drake

A standard test set for numerical approximations to the shallow water equations in spherical geometry

A suite of seven test cases is proposed for the evaluation of numerical methods intended for the solution of the shallow water equations in spherical geometry. The shallow water equations exhibit the major difficulties associated with the horizontal dynamical aspects of atmospheric modeling on the spherical earth. These cases are designed for use in the evaluation of numerical methods proposed for climate modeling and to identify the potential trade-offs which must always be made in numerical modeling. Before a proposed scheme is applied to a full baroclinic atmospheric model it must perform well on these problems in comparison with other currently accepted numerical methods. The cases are presented in order of complexity. They consist of advection across the poles, steady state geostrophically balanced flow of both global and local scales, forced nonlinear advection of an isolated low, zonal flow impinging on an isolated mountain, Rossby-Haurwitz waves, and observed atmospheric states. One of the cases is also identified as a computer performance/algorithm efficiency benchmark for assessing the performance of algorithms adapted to massively parallel computers.

Design and performance of a scalable parallel community climate model

Abstract We describe the design of a parallel global atmospheric circulation model, PCCM2. This parallel model is functionally equivalent to the National Center for Atmospheric Researchs Community Climate Model, CCM2, but is structured to exploit distributed memory multi-computers. PCCM2 incorporates parallel spectral transform, semi-Lagrangian transport, and load balancing algorithms. We present detailed performance results on the IBM SP2 and Intel Paragon. These results provide insights into the scalability of the individual parallel algorithms and of the parallel model as a whole.

Higher trends but larger uncertainty and geographic variability in 21st century temperature and heat waves

Generating credible climate change and extremes projections remains a high-priority challenge, especially since recent observed emissions are above the worst-case scenario. Bias and uncertainty analyses of ensemble simulations from a global earth systems model show increased warming and more intense heat waves combined with greater uncertainty and large regional variability in the 21st century. Global warming trends are statistically validated across ensembles and investigated at regional scales. Observed heat wave intensities in the current decade are larger than worst-case projections. Model projections are relatively insensitive to initial conditions, while uncertainty bounds obtained by comparison with recent observations are wider than ensemble ranges. Increased trends in temperature and heat waves, concurrent with larger uncertainty and variability, suggest greater urgency and complexity of adaptation or mitigation decisions.

Algorithm 888: Spherical Harmonic Transform Algorithms

A collection of MATLAB classes for computing and using spherical harmonic transforms is presented. Methods of these classes compute differential operators on the sphere and are used to solve simple partial differential equations in a spherical geometry. The spectral synthesis and analysis algorithms using fast Fourier transforms and Legendre transforms with the associated Legendre functions are presented in detail. A set of methods associated with a spectral_field class provides spectral approximation to the differential operators ∇ ⋯, ∇ ×, ∇, and ∇2 in spherical geometry. Laplace inversion and Helmholtz equation solvers are also methods for this class. The use of the class and methods in MATLAB is demonstrated by the solution of the barotropic vorticity equation on the sphere. A survey of alternative algorithms is given and implementations for parallel high performance computers are discussed in the context of global climate and weather models.

A scalable parallel Strassen's matrix multiplication algorithm for distributed-memory computers

The authors present a scalable parallel Strassen`s matrix multiply algorithm for distributed memory, message passing computers. Strassen`s algorithm to multiply two N x N matrices reduces the asymptotic operation count from O(N{sup 3}) of the traditional algorithm to O(N{sup 2.81}). In a sequential implementation the Strassen`s algorithm offers better performance even for relatively low order matrices. However, due to its complexity, the parallel Strassen`s algorithm is less than straight forward. Here a scalable parallel Strassen`s algorithm is presented and compared with several other parallel algorithms. Performances of these algorithms are tested on a 128-processor Intel iPSC/860.

The Cartesian method for solving partial differential equations in spherical geometry

Abstract Cartesian coordinates are used to solve the nonlinear shallow-water equations on the sphere. The two-dimensional equations, in spherical coordinates, are first embedded in a three-dimensional system in a manner that preserves solutions of the two-dimensional system. That is, solutions of the three-dimensional system, with appropriate initial conditions, also solve the two-dimensional system on the surface of the sphere. The higher dimensional system is then transformed to Cartesian coordinates. Computations are limited to the surface of the sphere by projecting the equations, gradients, and solution onto the surface. The projected gradients are approximated by a weighted sum of function values on a neighboring stencil. The weights are determined by collocation using the spherical harmonics in trivariate polynomial form. That is, the weights are computed from the requirement that the projected gradients are near exact for a small set of spherical harmonics. The method is applicable to any distribution of points and two test cases are implemented on an icosahedral geodesic grid. The method is both vectorizable and parallelizable.

Overview of the Software Design of the Community Climate System Model

The Community Climate System Model (CCSM) is a computer model for simulating the Earth’s climate. The CCSM is built from four individual component models for the atmosphere, ocean, land surface, and sea ice. The notion of a physical/dynamical component of the climate system translates directly to the software component structure. Software design of the CCSM is focused on the goals of modularity, extensibility, and performance portability. These goals are met at both the component level and within the individual component models. Performance portability is the ability of a code to achieve good performance across a variety of computer architectures while maintaining a single source code. As a community model, the CCSM must run on a variety of machine architectures and must perform well on all these architectures for computationally intensive climate simulations.

Performance Portability in the Physical Parameterizations of the Community Atmospheric Model

Community models for global climate research, such as the Community Atmospheric Model, must perform well on a variety of computing systems. Supporting diverse research interests, these computationally demanding models must be efficient for a range of problem sizes and processor counts. In this paper we describe the data structures and associated infrastructure developed for the physical parameterizations that allow the Community Atmospheric Model to be tuned for vector or non-vector systems, to provide load balancing while minimizing communication overhead, and to exploit the optimal mix of distributed Message Passing Interface (MPI) processes and shared OpenMP threads.

Accuracy Analysis of a Spectral Element Atmospheric Model Using a Fully Implicit Solution Framework

Abstract A fully implicit (FI) time integration method has been implemented into a spectral finite-element shallow-water equation model on a sphere, and it is compared to existing fully explicit leapfrog and semi-implicit methods for a suite of test cases. This experiment is designed to determine the time step sizes that minimize simulation time while maintaining sufficient accuracy for these problems. For test cases without an analytical solution from which to compare, it is demonstrated that time step sizes 30–60 times larger than the gravity wave stability limits and 6–20 times larger than the advective-scale stability limits are possible using the FI method without a loss in accuracy, depending on the problem being solved. For a steady-state test case, the FI method produces error within machine accuracy limits as with existing methods, but using an arbitrarily large time step size.

Introduction to the special issue on parallel computing in climate and weather modeling

Abstract In this introduction to the special issue on ‘Parallel computing in climate and weather modeling’, we review the historical development of computer models of the weather and climate system, and the application of novel, high performance parallel computing technologies to the execution of these models. We also provide some context for the articles that follow by summarizing the structure of typical models and the numerical methods used to implement them. Finally, we describe the eight articles in the special issue, and outline challenges that must be addressed in future research.