MINIPROGRAM on Mathematics and Fluids

Titles and Abstracts
June 2

Geometry and Analysis of the Euler and Average Euler Equations,
Jerrold Marsden CDS, Caltech

This talk will review some of the background on geometric fluid dynamics appropriate for the Euler and averaged Euler equations. In particular, we shall recall how one regards the equations as geodesics on the diffeomorphism group. Specifically, as Arnold showed, these are geodesics with respect to the $L^2$ metric for the Euler equations. We discuss how this approach gives insight into a variety of things, ranging from existence and uniqueness theorems, results on limits of zero viscosity and the variational and Hamiltonian structure of the equations and of vortex methods. Using Euler-Poincar\'e theory, we show how the equations may be regarded as geodesic equations for the $H^1$ metric on the volume preserving diffeomorphism group. Then we will present some of the analytical theorems including the convergence as viscosity tends to zero, even in the presense of boundaries. We will also briefly indicate some of the interesting computational aspects of the equations. The geometric mechanics approach to hydrodynamics also gives insight into the derivation of new classes of equations, namely the averaged Euler equations, both the isotropic and the nonisotropic forms; this derivation and related issues is the subject of the talk of Steve Shkoller at this meeting.

The Nonisotropic Averaged Euler Equations,
Steve Shkoller UC Davis.

I will present a derivation of a new model of incompressible hydrodynamics, called the nonisotropic averaged Euler equations, based on fuzzying-up the Lagrangian flow map, and averaging a new hybrid Eulerian-Lagrangian decomposition of the macroscopic velocity field. The new model is a coupled system of equations for the macroscopic velocity field u which is accurate down to some given length scale alpha and a symmetric fluctuation tensor F. Upon solving for (u,F), one can then solve for a "corrector" which improves the accuracy of the macroscopic velocity field to O(alpha^2). Some well-posedness results will be given.

Boundary layer theory and zero-viscosity limit of the Navier-Stokes equations,
Weinan E, Princeton University.

We will review the recent progress on mathematical analysis of the boundary layer equations and the related problem of the zero-viscosity limit of solutions of the Navier-Stokes equation in the presence of boundaries. We will then present detailed results of careful numerical experiments on flow past cylinder for Reynolds number in the range 10^3 - 10^5.

Vortex structure and local helicity in turbulence,
K. Ohkitani, RIMS Kyoto.

We investigate turbulence numerically by using the Kuzmin-Osedelets formalism. An attempt is made to characterize vortex structure via local helicity. Alignment statistics of 'velocity' w.r.t. rate-of-strain is studied. Two-dimensional flows will also be considered in this formulation.

Alpha models of classical and superfluid turbulence,
Darryl D. Holm, Los Alamos National Laboratory.

Recent developments in geometrical mechanics have led us to a new family of models of classical fluid turbulence called alpha models. We shall summarize the initial successes of the alpha models in mathematical analysis, comparison with experiments and direct numerical simulations. We shall then discuss potential improvements of these models based on combining the geometrical approach leading to the alpha-models with the physical principles underlying our new models of superfluid Helium turbulence, derived recently in the same geometrical framework as for the alpha models. Thus, the complementary applications of this approach to both classical and superfluid turbulence will each be used to illuminate the properties of the other.

Is there a Markov length in fully developed turbulence?
Tom Kambe, Tokyo University.

Fully developed turbulence (FDT) is structured with a number of elongated intense vortices. In addition to the various DNS analyses of coherent structures of FDT, there are observational evidences [1] of existence of a Markov length of the order of Taylor microscale, over which the turbulence cascade is regarded as a Markov process. Considering that this length is closely related with the coherent structures, a model of structured turbulence is proposed which is an ensemble of strained vortices (i.e. Burgers vortices) distributing randomly in space, in order to get leading order representation of statistics at small scales (less than the Taylor microscale) and higher order moments in FDT. It is found [2] that probability density functions of longitudinal velocity differences and higher-order structure functions thus obtained are in good agreement with known results. References: [1] Friedrich, Zeller and Peinke (1998) Europhys. Lett. vol.41, 143; Friedrich, Lueck, Renner and Peinke (2000) in 'Proc. of IUTAM Symposium on Geometry and Statistics of Turbulence'. [2] Kambe and Hatakeyama (2000) to appear in Fluid Dynamics Research.

On the Connection Between the Viscous Camassa-Holm Equations (Navier-Stokes -alpha model) and Turbulence Theory,
Edriss S. Titi, U.C. Irvine.

In this talk we will show the global well-posedness of the three dimensional viscous Camassa-Holm equations, also known as the Navier-Stokes-alpha model. The dimension of their global attractor will be esitmated and shown to be comparable with the number of degrees of freedom suggested by classical theory of turbulence. We will present semi-rigorous arguments showing that up to a certain wave number, in the inertial range, the translational energy power specturm obeys the Kolmogorov power law for the energy decay of the three dimensional turbulent flow. However for the rest the inertial range the energy spectrum of this model obeys the Kraichnan power law for the energy decay of the two dimensional turbulent follows. This observation makes the Navier-Stokes-alpha model more computable than the Navier-Stokes equations. Furthermore, we will show that by using the Camassa-Holm equations ( Navier-Stokes-alpha model) as a closure model to the Reynolds averaged equations of the Navier-Stokes one gets very good agreement with empirical and numerical data of turbulent flows in infinite pipes and channels.

June 3

Exact solution of a three constraints equilibrium statistical mechanics model for 2-D turbulence,
Chjan Lim, RPI.

This three constraints model is the third one in a family of few constraints equilibrium statistical mechanics models proposed by Turkington and Majda for the study of 2-D and quasi-geostrophic flows. This model is equivalent to the Batchelor-Lee-Kraichnan model for inverse cascades after a Fourier transform. The exact solution of this model is based on a simple but fundamental observation that the enstrophy corresponds exactly to the higher dimensional spherical constraint introduced by Kac. The exact solution is then obtained by extending Kac's and Berlin's solution of the spherical model to a long range logarithmic interaction. A negative critical temperature for phase transition to coherent structures is obtained.

Increasing subsequences in random permutations, random matrices, and integrable systems,
Andrei Okounkov, University of California, Berkeley.

I will survey recent progress in connecting combinatorial problems, such as distribution of increasing subsequences in random permutations, to random matrices and to related integrable systems.

On the geometry of solutions of the Quasi-geostrophic equation,
Diego Cordoba, The University of Chicago.

We study solutions of the 2D Quasi-geostrophic equation involving a simple hyperbolic saddle. There is a naturally associated notion of simple hyperbolic saddle breakdown. It is proved that such a breakdown cannot occur in finite time. At large time, these solutions may grow at most at a quadruple-exponential rate.

Alexandre Chorin, UC Berkeley