Applied Mathematics

Mathematical calculationsApplied mathematics is a branch of mathematics that deals with the practical application of mathematical knowledge to other fields. Applied mathematics is especially important in scientific computing, where CISL brings mathematical tools to bear on fundamental problems in the geosciences.

CISL research in this area includes developing numerical methods and parallel algorithms, applying statistical analyses to the Earth system, creating new techniques for data assimilation, investigating geophysical turbulence, and producing mathematical software for the research community.



Numerical methods



Numerical methods and parallel algorithms

In geoscientific research, physical systems such as the atmosphere and the ocean are simulated in terms of mathematical equations — linear, nonlinear, ordinary differential, partial differential, or some combination. These equations are solved by using various numerical methods, depending on the problem. Once a suitable numerical method is selected, the final step is to chose or design an algorithm — that is, a step-by-step implementation of the numerical method. Designing algorithms for the geosciences requires a specialized and detailed understanding of mathematics, the scientific domain, and the computer hardware that will run the simulation.

CISL computer scientists develop algorithms that will improve the fidelity, performance, and capability of scientific simulations on today’s high-performance, parallel computers. Their work includes developing scalable mass-conserving advection schemes, adaptive-mesh technologies for climate simulation, and techniques for simulating multiscale phenomena. One new area of investigation is radial basis functions, a methodology that offers high-order accuracy when dealing with scattered points on irregular modeling grids.





Geophysical visualization

Geophysical visualization


Geophysical statistics

CISL pursues the innovative application and development of statistics and probability theory to problems in the Earth sciences. CISL research in this area includes applying mathematical regression and model selection to the analysis of geophysical data, deriving statistical bases for forecasting, simulating complex physical processes through Bayesian hierarchical models, and understanding physical processes through the use of dynamical systems and nonlinear time series. In facilitating this research, CISL serves as a bridge between the statistical-probabilistic and the atmospheric-oceanographic research communities.

CISL has recently developed a method for extrapolating precipitation extremes to areas where observational station data are not available. Several CISL projects are supporting the analysis of data from the fourth report of the Intergovernmental Panel for Climate Change. This analysis includes developing a statistical framework to combine different model results and producing an integrated estimate of climate change at regional scales. Most of the statistical methods developed in CISL are distributed to the community in the form of open-source packages in R, a software environment for statistical computing and graphics.

CISL also sponsors a unique postdoctoral appointment program in statistics, encouraging new Ph.D.s to develop solid research programs of their own, branch into new areas, and develop statistical applications in geophysical and environmental sciences.




Observational data

DART data



Data assimilation

Data assimilation is the set of statistical and computational methods for combining observational data with numerical models — a process that is critical to understanding and predicting geophysical systems like the Earth’s atmosphere. Although its most visible application is in operational weather forecasting, data assimilation can also be used to validate and understand the features of different geophysical and chemical models, as well as to interpolate among observations that are irregularly observed over space and time.

CISL has developed a data assimilation facility that allows scientists to prototype the impact of data assimilation on different geophysical models and observation sets. The facility, called the Data Assimilation Research Testbed (DART), is an open-source, publicly available software environment that combines assimilation algorithms, models, and observation sets to provide a flexible, extensible framework for data assimilation research.

DART can now be used to validate and improve many climate models, including the Weather Research and Forecasting Model; the Community Atmosphere Model 2 and 3; ROSE; and models from the Massachusetts Institute of Technology, the Geophysical Fluid Dynamics Laboratory, and the National Centers for Environmental Prediction.





Turbulence visualization



Turbulence refers to the complex behavior of fluid flow in liquids or gases. From sea breezes and plumes of smoke to drifting snow and billowing fog banks, turbulence is found everywhere in the Earth systems. Because turbulence is responsible for mixing and transporting elements such as moisture, heat, salinity, and energy across the globe, it plays a key role in many environmental processes.

CISL facilitates fundamental research on geophysical turbulence using both theory and numerical simulation with experimental validation. Research topics include the interaction of shear forces on stratified environments, nonlocal aspects of turbulence in magnetohydrodynamic flows, and condensation under diffusive mixing. A key activity is the development of an object oriented code for Geophysical and Astrophysical Spectral Element Adaptive Refinement (GASpAR), which provides a framework for the accurate simulation of turbulent systems.

CISL hosts regular seminars and workshops on turbulence that bring together atmospheric scientists, computational scientists, mathematicians, physicists, and engineers to share experiments and observations.





Spherepack capability


Mathematical software

One area of applied mathematics in which CISL is involved is the development of efficient, portable, high-performance, open-source numerical libraries for the mathematical and geosciences community. One such library is the Spectral Toolkit, spectral-transform software that runs fast and efficiently on platforms ranging from laptops and PCs to the IBM Power4. Other packages include SPHEREPACK 3.0, MUDPACK, FISHPACK90, and FFTPACK5.