Long-time evolution of forced waves in the low viscosity regime

TL;DR

This paper analyzes the long-term behavior of forced internal waves in a low viscosity regime, revealing that dissipation effects become significant only after a specific time scale related to viscosity.

Contribution

It provides a qualitative description of the wave evolution under certain dynamical conditions, highlighting the timing of dissipation effects in low viscosity regimes.

Findings

Dissipation effects appear around time scale ^{-1/3} in small viscosity regimes.

The study applies Morse-Smale or similar conditions to classical dynamics.

Long-time behavior of solutions with smooth forcing is characterized.

Abstract

We consider a model for internal waves described by a zero order pseudo-differential Hamiltonian $P$ damped by a second order viscosity term $i \nu Q$. Under Morse-Smale or similar weaker global conditions on the classical dynamics, we describe qualitatively the long-time behavior of solutions of the corresponding evolution equation with smooth forcing in a small $\nu$ regime. We show that dissipation effects arise no earlier than at the $t\sim \nu^{-1/3-}$ time scale.