Simulation of an ocean jet stream

Ocean Jet

The pictures on the left are generated using a finite difference approximation of the shallow-water equations. The parameters used are "Earth like" at mid latitudes. The plot represents a 3000km by 2000km area of water at an average depth of 500 m. The plots span over a three-year interval. Contours of the sea surface are shown. Red is above sea level, blue is below.

One problem in which I am interested is to generate and analyze algorithms capable of capturing complicated eddy and jet current fields in geophysical fluids over long-time intervals.

For more details see the list of my publications with the co-authors L. Margolin, A. Poje.



PUBLISHED WORK:

  1. On the number of determining nodes for the 2D Navier-Stokes equations, (with E.S. Titi), J. Math. Anal. Appl., 168, (1992), 72-88.

  2. Determination of the solutions of the Navier-Stokes equations by finite volume elements , (with E.S. Titi), Physica D60, (1992), 165-174.

  3. An approximate inertial manifold for computing Burgers' equation , (with L.G. Margolin), Physica D60, (1992), 175-184.

  4. Upper bounds on the number of determining modes, nodes, and volume elements for the Navier-Stokes equations, (with E.S. Titi), Indiana Univ Math. J., 42, (1993), 875-887.

  5. A remark on quasi-stationary approximate inertial manifolds for the 2D Navier-Stokes equations (with E.S. Titi) SIAM J. Math. Anal., 25 , (1994), 894-914.

  6. On the effectiveness of the approximate inertial manifold- a computational study, (with L. G. Margolin, E. S. Titi), Theor. Comp. Fluid Dyn., 7, (1995), 243-260.

  7. Determining degrees of freedom for nonlinear dissipative equations , (with E.S. Titi, B. Cockburn), C.R. Acad. Sci Paris Se'r I, 321, (1995), 563-568.

  8. Attractive invariant manifolds under approximation: Inertial manifolds, (with A.M. Stuart), J. Diff. Eq., 123, (1995), 588-637.

  9. On the behavior of attractors under finite difference approximations , Num. Funt. Anal. and Opt., 16, (1995), 1155-1180.

  10. Enslaved finite difference schemes for nonlinear dissipative PDEs, (with A.C. Poje, L.G. Margolin), Num. Meth. for PDEs, 12, (1996), 13-40.

  11. C^1 approximations of inertial manifolds for dissipative nonlinear equations, (with E. S. Titi), J. Diff. Eq., 127, (1996), 54-86.

  12. Enslaved finite difference schemes for quasigeostrophic shallow flows, (with A.C. Poje, L.G. Margolin), Physica D98, (1996), 559-573.

  13. Local existence results for the generalized inverse of the vorticity equation in the plane, (with C. R. Hagelberg, A.F. Bennett), Inverse Problems, 12, (1996), 437-454.

  14. Estimating the asymptotic degrees of freedom for nonlinear dissipative PDEs, (with E.S. Titi, B. Cockburn), Math. Comp., 66, (1997), 1073-1087.

  15. Resolution effects and Enslaved finite difference schemes for a double gyre, shallow water model, (with A. C. Poje, L.G. Margolin), J. Theor. Comp Fluid Dynamics., 9, (1997), 269-280.

  16. Persistence of invariant sets for dissipative evolution equations, (with A. M. Stuart, E.S. Titi), J. Math. Anal. Appl., 219, (1998), 479-502.

  17. Persistence of invariant manifolds for nonlinear PDEs, (with Steve Shkoller), Studies in Applied Math., 102, (1999), 27-67.

  18. An operator splitting for the shallow-water equations with large ageostrophic initial data, (with A. Mahalov, B. Nicolaenko). J. Theor. Comp Fluid Dynamics., 13, (1999), 263.

  19. Effects of random motility on micobial growth and competition in a flow reactor, (with M. Ballyk, H. Smith, L. Dung), SIAM Applied Math., (to appear).

  20. Don A. Jones, Hal L. Smith, Microbial competition for nutrient and wall sites in plug flow, SIAM Applied Math., (to appear).

  21. M. Ballyk, Don A. Jones, H.L. Smith, Microbial competition in reactors with wall attachment: a comparison of chemostat and plug flow models, Microbial Ecology, (submitted).

  22. Don A. Jones, Mathematical analysis of geophysical balance models, J. Diff. Eq, (submitted).

PROCEEDINGS:

  1. Approximations of dissipative partial differential equations over long-time intervals, (with A.R. Humphries, A.M. Stuart), Proceedings of the 14th Biannual Numerical Analysis Conference, Dundee, Edited by D.F. Griffiths, G.A. Watson. Pitman, London, 1994.

  2. Nonlinear difference approximations for evolutionary PDEs, (with A.C. Poje, L.G. Margolin), Nonlinear Evolution Equations and Dynamical Systems, NEEDS, Edited by V.G. Makhankov, A.R. Bishop, D.D. Holm, (1994), World Scientific, 65-75.