Atmospheric and Ocean Circulation
Graduate Student Mike Watson
Dates of Involvement 2006- 2008
Faculty Advisor Keith Julien
Background
The Atmospheric and Ocean Circulation Project focuses on computational geophysical fluid dynamics. Conceptual examples follow...
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Ocean Circulation |
Atmospheric Motions |
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Astrophysical Dynamics |
Planetary Dynamics / Proto-Planetary Disk Formation |
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Rayleigh Benard Convection: |
Theory
Navier-Stokes Equations, equations that describe all fluid motion, are used to explain the mathematical theory in this project. However, these equations can be very difficult to solve. To make these equations more tractable, we can look at specific geometries, and apply what we already know about motion in rotating systems. For example, take into consideration the following equations:
One can massage the equations a little to achieve the following:
At this point you may be thinking, "This doesn't look any better at all!" On the contrary, these equations are much better suited for analytic work and numerical simulation. Working on this project teaches how to understand what each component of these the equations do, and why they are important. Research studying these equations allows one to gain a better understanding of the types of phenomena shown above.
Simulation
Simulation provides a great way to gain intuition about the physics of the fluid equations and provide a visual confirmation that things are behaving as they should. The picture below left shows a numerical simulation of rotating Raleigh-Benard Convection. A thin fluid is heated from below and rotated about a central axis. Working in this group, students have the opportunity to do simulations in different geometries. Click here to watch the animation. (This file is large and loads slowly, please be patient.)
Experimentation
The group built a rotating convection experiment that was run in the Geophysical Fluid Dynamics Laboratory located in the Duane Physics building. The group used this experiment to verify that their analytic work and numerical simulations are in line with what happens in the real world. The experimental apparatus is shown below. The picture to the right displays some interesting dynamics with a clearly defined jet stream and noteworthy cyclonic motion.
What Our Students Do
Studying rotating fluids can be very exciting and rewarding. Additionally, the Applied Mathematics curriculum includes classes with direct relevance to these problems. Examples follow:
Differential Equations: Although the equations examined by this group are Partial Differential Equations, under certain assumptions, they reduce to standard Ordinary Differential Equations. A specific area where we would like to get students involved is linear stability analysis, which will rely heavily on working with ordinary differential equations.
Linear Algebra: In these numerical simulations, the group looks at discretizations of an equation set. This turns the continuous problem into a system of equations to be solved. If you know how to solve A x = b, one can go a long way.
Fourier Analysis: One can gain a lot of insight by studying an problem in spectral space, and this group does this all the time. In fact, the basis of our numerical manipulation is taking derivatives and inverting them in Fourier space for observation.
Mathematical Modeling: The work of this group begins with a physical problem, rotating convection in specific geometry, and builds a numerical simulation to model the dynamics. The group strives to get Applied Mathematics students involved in running these simulations on super computers along with performing data analysis.
Numerical Analysis: Numerical analysis is key to building the group's simulations. The group use a modified Runge-Kutta time-stepping algorithm to solve banded matrices so as to be able to invert the derivative operators. An understanding of discretization error is necessary to insure the accuracy of models.
Contact Information
If you are interested in fluid dynamics, numerical analysis, super computing, or data analysis and think this could be the right project for you, please contact:
Mike Watson
PhD Student, Applied Mathematics
University of Colorado
watson@colorado.edu
