Applied Math Seminar:
I am the primary organizer for the Applied Math Seminar in our department, typically held on Tuesdays from 2-3pm. In addition to outside visitors and speakers, we are very interested in having speakers from inside BYU, particularly from other departments. If you are interested in giving a seminar please contact me for more information.
I am interested in all areas of applied mathematics. This includes analysis as well as the development and use of numerical tools and scientific computing. I work on a wide spectrum of mathematically interesting problems that tie back in to the physics of interest. There are times when the beauty of the mathematics is of interest in itself, but I find this to be rare in my experience and prefer to work on mathematics that I can see and explain the application for it.
To date most of my research has been spent in classical physics, primarily in the field of classical fluid dynamics. My work has focused on several different sub-fields in this area, but for brevity I will only mention three here:
- Rigorous bounds and numerical verification of such for the transport of heat in Rayleigh-Benard convection. Convective processes surround us from stirring a cup of hot cocoa (I do work at BYU after all) to the circulation of the earth's oceans and atmosphere, convection is ubiquitous in nature and occurs frequently in the engineering sciences. My colleagues and I have utilized a variational technique to put rigorous bounds on the transport of heat in the Rayleigh-Benard model of this process in which the Boussinesq approximation is invoked. We have shown that these bounds depend highly on the type of boundary conditions, and on the dimension (2D or 3D) of the problem at hand, much to the chagrin of some traditionally held theories of asymptotically strongly driven convection. I am working with these and other colleagues to verify the veracity of these estimates via direct numerical simulation, a challenging problem that even in two dimensions requires some of the most advanced computational machinery available today.
- Quantification and verification of the dynamical core of a general circulation model of the Earth's atmosphere. I have been working with colleagues to come up with quantifiably accurate ways to measure and compare the dynamical core's (part of the model meant to integrate the fluid dynamics equations of motion) of different general circulation models. Each of these models is made up of several tens (if not hundreds) of thousands of lines of code, each conglomerated typically from several dozen different researchers from a variety of fields connected to atmospheric science. Such complicated models cannot easily be compared without careful thought and analysis of the underlying algorithms upon which the models are built. With several colleagues at the University of Michigan, I have explored a variety of different approaches to do just that. Recently I have also begun exploring the concept of carrying these comparisons a bit further by developing a way to compare the attracting state of each model in an idealized test case scenario.
- The role of asymptotics and different time scales in geophysical fluid dynamics. In most of the geophysical situations of interest to atmospheric scientists and oceanographers, there are two (typically strong) forces that play a vital role in assessing the dynamical behavior of the fluid. These two 'forces' are the rotation of the earth, and the stable stratification of most geophysical fluids (denser, warmer fluid lies below less dense, cooler fluid). If these two phenomenon are sufficiently strong they will induce a separation of time scales in the fluids evolution. Asymptotic analysis can then be used to find reduced systems of equations that are valid in these regimes in parameter space. With several colleagues I am using the tools of multiple time scale asymptotics and scientific computing to explore these different regimes for a variety of different models, ranging from partial differential equaitons to much simpler ordinary differential equation models.
My work spans a wide array of mathematical maturity, ranging from the study of simplified 'toy' models for the climate that include finite dimensional dynamical systems, to partial differential equations that are thought to govern the evolution of an incompressible fluid, to the analysis of the intimidating general circulation models that are in use today for weather and climate forecasts. In all of this, I try incorporate a balance of numerical and rigorous work, as I truly feel that one without the other may be misleading. Because there is such a variety of topics and levels of expertise in my work, I am interested in working with students at any level (undergraduates should probably have taken a first semester course in ODEs however). If you think any of these topics sound of interest, look through my research page, and send me an e-mail.
I am also a member of the Mathematics of Climate Research Netowrk, a network established to disseminate mathematical reseearch related to climate science, and provide connections between scientists studying this important subject.