[Paper Review] Attractors for two dimensional waves with homogeneous Hamiltonians of degree 0

Yves Colin de Verdìère, Laure Saint‐Raymond+1 more|arXiv (Cornell University)|Jan 16, 2018

Fluid Dynamics and Turbulent Flows3 citations

TL;DR

This paper analyzes the formation of wave attractors in two-dimensional domains with topography, using spectral theory and microlocal analysis to show that for generic forcing frequencies, inertial and internal waves concentrate on singular geometric structures due to the homogeneous Hamiltonian of degree zero. The key result is that such attractors emerge naturally from the wave dynamics when boundaries disrupt the standard Fourier decomposition, leading to energy localization on specific invariant manifolds.

ABSTRACT

The density stratification in an incompressible fluid is responsible for the propagation of internal waves. In domains with topography, these waves exhibit interesting features. In particular, numerical and lab experiments show that, in two dimensions, for generic forcing frequencies, these waves concentrate on attractors. The goal of this paper is to analyze mathematically this behavior, using tools from spectral theory and microlocal analysis. The same results apply also to inertial waves in rotating fluids.

Motivation & Objective

• To mathematically explain the formation of wave attractors observed in laboratory experiments involving inertial and internal waves in two-dimensional domains with topography.
• To analyze the behavior of waves governed by homogeneous Hamiltonians of degree zero, particularly when standard Fourier analysis fails due to boundary effects.
• To understand how energy concentrates on specific geometric structures (attractors) under generic forcing frequencies, despite the absence of resonant singularities in the linear system.
• To bridge the gap between physical observations of wave localization and rigorous mathematical analysis using spectral and microlocal tools.

Proposed method

• The authors model inertial and internal waves using linearized equations derived from the Coriolis and Boussinesq approximations, respectively, in bounded domains with non-periodic boundaries.
• They employ spectral theory to study the operator associated with the wave equation, focusing on the case where the Hamiltonian is homogeneous of degree zero.
• Microlocal analysis is applied to study the wavefront set and propagation of singularities, particularly in the presence of boundary reflections and topography.
• The analysis includes a detailed study of the forced wave equation in the integrable case on the torus, using Fourier series and eigenfunction expansions.
• A diophantine condition on the symbol of the Hamiltonian is introduced to characterize non-resonant behavior, linking spectral properties to dynamical conditions on the torus.
• The paper uses the Leray projection to handle the incompressibility constraint and derives effective wave equations in Fourier space, even when the domain is non-periodic.

Experimental results

Research questions

• RQ1 How do wave attractors emerge in two-dimensional domains with topography when the wave equation is governed by a homogeneous Hamiltonian of degree zero?
• RQ2 What is the role of the forcing frequency in determining whether wave energy concentrates on attractors rather than spreading uniformly?
• RQ3 Why does the standard Fourier-based analysis fail in non-periodic domains with boundaries, and how can spectral and microlocal methods overcome this?
• RQ4 Under what conditions does the solution to the forced wave equation remain bounded in L², and how does this relate to the Diophantine condition on the Hamiltonian symbol?
• RQ5 How does the geometry of the domain and the structure of the Hamiltonian influence the formation of singular wave patterns?

Key findings

• For generic forcing frequencies, wave energy concentrates on specific geometric structures known as attractors, even in the absence of resonant singularities, due to the homogeneous Hamiltonian of degree zero.
• In the resonant case where the Hamiltonian vanishes at some non-zero wave vector, the solution grows linearly in time in the L² norm if the forcing is in H¹, indicating energy accumulation.
• In the non-resonant case, a Diophantine condition on the Hamiltonian symbol ensures that the solution remains bounded in L², provided the forcing is smooth enough.
• The Diophantine condition on the symbol h(n) is equivalent to a Diophantine condition on the classical vector field Y associated with the dynamics on the level set h(p)=0.
• The wavefront set of the limiting solution u∞ is shown to be supported on the set where the Hamiltonian vanishes, indicating that singularities propagate along invariant tori and lead to energy concentration.
• The analysis reveals that the continuous spectrum and quasi-resonant mechanisms drive energy cascades to small scales, suggesting connections to wave turbulence theory despite the system being linear.