Shengqian (Chessy) Chen
Fundamental physics of incompressible stratified Euler fluids
By idealizing and decomposing complex problems from the real world, we obtain mathematical models
simpler but capable for applications. On the other hand, the extreme cases in such models deserve
attention (and further study) as they may capture relevant physical mechanism helpful in explaining
some observations as well as in elucidating conceptual points. The long wave models, in particular,
work under the shallow water limit assumption when the ratio of water depth and wave length goes
to zero, which leads to a horizontal infinite channel idealization. What crisis in physics we encounter
under this extreme case?
Geometrical methods in the theory of nonlinear waves and applications
The strongly nonlinear internal wave model has the merit of capturing the natural nonlinear behavior of internal ocean waves and reduces the computational effort as for the full Euler equation. Yet applied mathematicians' curiosity motivates us to touch the baseline analytically; we attempt to obtain a system that is analytically solvable for arbitrary compact-support initial conditions.
Together with our collaborators Dr. Gregorio Falqui, Dr. Giovanni Ortenzi and Dr. Marco Pedroni in Italy, we are working on to obtain a reduced Hamiltonian structure of the unconstrained completely integrable system which is solvable by inverse scattering methods.
Strongly nonlinear two-layer models for continuously stratified fluid
Strongly nonlinear long wave theories of internal waves are asymptotic models of shallow water for arbitrary wave amplitudes, in contrast with weakly nonlinear models such as KdV-type theories which restrict the amplitude too severely to be applicable to the large amplitudes that can be attained by internal waves. Strongly nonlinear internal wave theories have been developed by several authors. In particular I focus on those developed by Miyata (1985), and Choi and Camassa (1999).
These models are described by a coupled set of nonlinear equations evolving the layer-mean velocities, the interface displacement and the interface pressure. Unlike the KdV-type models as a single equation, where several mathematical results are known and been thoroughly studied, reports of strongly nonlinear internal wave models have been mostly limited to traveling wave solutions and not broadly explored in dynamical time-dependent regimes.
This project aims to broaden the application of strongly nonlinear internal wave models and assess their quantitative and predictive capabilities with respect to experiment and direct numerical simulations.