Dynamics of partially thermalized solutions of the Burgers equation

TL;DR

This paper investigates how thermalization occurs in spectrally truncated Burgers equation, revealing that localized structures called tygers initiate the process in specific regions before spreading throughout the domain, with their propagation characterized by energy conservation.

Contribution

It demonstrates that thermalization in truncated Burgers flows begins locally with tygers and propagates at a predictable speed, advancing understanding of energy transfer in inviscid flow models.

Findings

Thermalization starts in localized regions with tygers.

Tygers propagate at a mean speed derived from energy conservation.

Thermalization eventually engulfs the entire domain.

Abstract

The spectrally truncated, or finite dimensional, versions of several equations of inviscid flows display transient solutions which match their viscous counterparts, but which eventually lead to thermalized states in which energy is in equipartition between all modes. Recent advances in the study of the Burgers equation show that the thermalization process is triggered after the formation of sharp localized structures within the flow called "tygers". We show that the process of thermalization first takes place in well defined subdomains, before engulfing the whole space. Using spatio-temporal analysis on data from numerical simulations, we study propagation of tygers and find that they move at a well defined mean speed that can be obtained from energy conservation arguments.