Nonlinear water waves over strongly varying bottom topography

Speaker: 

Professor John Grue

Institution: 

University of Oslo, Norway

Time: 

Tuesday, November 27, 2007 - 2:00pm

Location: 

MSTB 254

A fully nonlinear time-stepping model for water wave motion over strongly varying topography
in three dimensions is presented. The modl is fully dispersive, fully nonlinear and, and also very rapid. The kinematic and dynamic boundary
condition at the free surface are used to derive the prognostic equations. Conservation of mass yields two integral equations for the normal velocity at the free surface and the wave potential at the sea floor. These are inverted analytically be means of Fourier transform. Various levels of nonlinearity of the equations are derived. A highly efficient computational scheme is obtained by the FFT-part of the formulation. Computations exemplify how a very long tsunami with leading depression running into very shallow water develop very short waves, that in the beginning are linear, developing then into a train of solitary waves of
large amplitude. Numerical examples on the formation of very strong ocean surface waves - rogue waves - are given.

3D Euler in a 2D Symmetry Plane: Preliminary Computations

Speaker: 

Dr. Miguel Bustamante

Institution: 

University of Warwick, UK

Time: 

Friday, November 9, 2007 - 2:00pm

Location: 

MSTB 254

Initial results from new calculations of interacting anti-parallel Euler vortices are presented with the objective of understanding the origins of singular scaling presented by Kerr (1993) and the lack thereof by Hou and Li (2006). Core profiles designed to reproduce the two results are presented, new more robust
analysis is proposed, and new criteria for when calculations should be terminated are given. Most of the analysis is on a $512\times 128 \times 2048$ mesh, with new analysis on a just completed $1024\times 256\times 2048$ used to confirm trends. The qualitative conclusions of Kerr (1993) are supported, but most of the proposed scaling laws will have to be modified. Assume enstrophy growth like $\Omega\sim (T_c-t)^{-\gamma_\Omega}$ and vorticity growth like $||\omega||_\infty \sim (T_c-t)^{-\gamma}$. Present results would support $\gamma_\Omega\rightarrow 1/4-1/2$ and $\gamma>$. The results are not conclusive since they require higher resolution calculations (work in progress) to further confirm the trends.

Global Well-posedness of Viscoelastic Fluids with Partial Dissipation and Small Initial Data

Speaker: 

Dr. Zhen Lei

Institution: 

CALTECH

Time: 

Friday, November 16, 2007 - 4:00pm

Location: 

MSTB 254

Classical ideal fluid motion is described by Euler and Navier-Stokes equations. For real fluids, their motions are more complicated and governed by Euler and Navier-Stokes equations coupled with various constitutive equations. We study viscoelastic models whose motions are
carried out by the competition between the kinetic energies and internal elastic energies. The deformation tensor plays an essential role in our studies. We will present how to use the heuristics coming from the special
structure of the deformation tensor to establish the global well-posedness results for several viscoelastic models, but will focus on a 2D Strain-Rotation model.

The Gross-Pitaevskii equation in the presence of random potential.

Speaker: 

Professor Ziad Muslimani

Institution: 

Florida State University

Time: 

Friday, October 26, 2007 - 4:00pm

Location: 

MSTB 254

In this talk, I will present recent results on wave localization in nonlinear random media in the frame work of the stochastic Gross-Pitaevskii equation (describing Bose-Einstein condensation). In particular, it is shown numerically that the disorder average spatial extension of the stationary density profile decreases with
an increasing strength of the disordered potential both for repulsive and attractive interactions.

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