For more information about this meeting, contact Jinchao Xu, Stephanie Zerby, Xiantao Li, Yuxi Zheng, Kris Jenssen, Hope Shaffer.

Title: | Waves at an air-water interface when the air and the water are Navier-Stokes fluids |

Seminar: | Computational and Applied Mathematics Colloquium |

Speaker: | Diane Henderson, Penn State Mathematics |

Abstract: |

It is know that the classical, inviscid, Stokes wavetrain is unstable to long-wave perturbations. The instability is called the "Benjamin-Feir instability" in water waves and "modulational instability" in other systems. The (inviscid) nonlinear Schroedinger (NLS) equation, derived from Euler's equations, describes both the instability and its subsequent evolution. However, the addition of linear dissipation into the NLS equation, no matter how small, stabilizes this modulational (Benjamin-Feir) instability in finite time. And predictions of nine independent parameters measured in the lab are predicted well by the dissipative NLS equation. Two questions arise, which are considered here. (i) Is this stability result relevant to the ocean, where dissipation is considered negligible with respect to the global energy balance? (ii) What is the correct model for dissipation? If one considers a free air-water interface, allows the water to be governed by the Navier-Stokes equations, and looks in the limit of small viscosity, then the resulting dissipation rate is much smaller than what is measured in either the ocean or the lab. Dore (1978)
allowed for both the viscosity of the water and the air and found that the dynamics of the air make a significant difference in the dissipation of water waves. Here we consider a 2-layer system of Navier-Stokes fluids with arbitrary viscosities and densities and consider linear waves at the interface. We derive the dispersion relation for waves at the interface and look at three special cases to find predicted damping rates that are closer to values measured in waves in the ocean and the lab. |

### Room Reservation Information

Room Number: | MB106 |

Date: | 03 / 23 / 2012 |

Time: | 03:35pm - 04:25pm |