Ph.D. in Mathematics, August 2007.

Rensselaer Polytechnic Institute
Mathematical Sciences Dept.

Curriculum Vitae

Dissertation

Trapped Slender Vortex Filaments in Statistical Equilibrium

Successfully defended with no revisions on 14 May 2007

Dissertation Abstract:

Rotating fluids or vortices are ubiquitous. Animals and machines both rely on rotating or vortex fluids for propulsion. When birds flap their wings,  air rotates in vortices resulting forward motion. Jet engines turn superheated air with turbines. Ocean currents, like the Gulf Stream, are made up of numerous vortices. Vortices also form in magnetically confined plasmas. One of the main properties of vortices that we would like to understand is the length scale of a system of vortices under conservation of angular momentum on the unbounded plane (similar to the deep ocean, far from land, or in a solar atmosphere). Essentially, how big does a rotating fluid (meaning a system of vortices under an angular momentum constraint) get under given conditions? Much has been done with two dimensional point vortex studies, but these sweep away three dimensional effects. We study a more realistic model called nearly parallel vortex filaments (Lions and Majda, 2000), which presents vorticity as slender filaments that are nearly but not quite parallel to each other. This model is, in contrast to the perfectly parallel model of point vortices, slightly 3D, but not so much as to be intractable. We use a subset of the mean field statistical model of Lions and Majda to obtain a novel explicit formula for the size of a system of these filaments in terms of input parameters like temperature. We validate this formula with Path Integral Monte Carlo simulations of the non-mean field statistical system. Furthermore, we calculate that, for the case of an isolated system, the specific heat of the system is negative, similar to that of globular clusters. We have two applications for our model, the first is deep ocean convection and the second is for neutral plasmas, such as may be found in magnetic nuclear fusion experiments.

Papers

• Negative Specific Heat in a Quasi-2D Generalized Vorticity Model,
  Andersen, T. D. and Lim, C. C., Physical Review Letters, vol. 99, no. 165001, Oct. 2007.
• Explicit mean-field radius for nearly parallel vortex filaments in statistical equilibrium with applications to deep ocean convection
  Andersen, T. D. and Lim, C. C., Geophysical and Astrophysical Fluid Dynamics, vol. 102, no. 3, pp. 265-280, Jun. 2008.
• A length scale formula for confined Quasi-2D plasmas,
  Andersen, T. D. and Lim, C. C., J. Plasma Physics, vol. 75, no. 4, pp. 437-454, 2009.
  

Currently Working On

• A Yang-Mills theory of gravity
  
• A book on vortex statistics with (former) adviser Chjan Lim
• A demon algorithm of microcanonical quasi-2D vortex statistics

Research Interests

• Applied probability
• Statistics of log-potential and gravitational particle systems
• Stochastic algorithms such as Monte Carlo and Genetic
• Turbulence in confined systems
• Quantum gravity
  

Teaching Assistance

• MATH 1620, Spring 2007

Links

• Advisor, Chjan C. Lim, where there are details about his, my co-student Ding
  Xueru's, and my research.
• Tamer Elbatt's page with whom I had such rewarding collaborations at HRL Laboratories.
• RAND Corporation where I spent the Summer of 2005 working with a great group of
  people, notably my mentor Chris Pernin.
• Former HRL manager Bo Ryu's bio at his new company, SDRC, Inc. A wonderful
  mentor when I was working in industrial research.
• Senior Thesis Advisor's page at UT-Austin, Risto Miikkulainen
• The new page of Ken Stanley who was a great help in introducing me to genetic
  algorithms
• Dad, Per H. Andersen, at Computer Science Dept. at Texas Tech
• Mom, J. Susan Andersen at Texas Tech Health Sciences Center

Updated 12 August 2009.