Analytic structure of solutions of the one-dimensional Burgers equation with modified dissipation

TL;DR

This paper investigates how replacing standard dissipation with hyperviscosity or similar functions affects the analytic structure of solutions to the one-dimensional Burgers equation, combining theoretical analysis with numerical validation.

Contribution

It introduces a novel approach to analyze the asymptotic Fourier space structure of solutions with generalized dissipation in hydrodynamical equations, including hyperviscosity.

Findings

Detailed analytic structure of solutions with hyperviscous dissipation

Validation of theoretical results through high-precision numerical simulations

Insight into the bottleneck phenomenon in Burgers turbulence

Abstract

We use the one-dimensional Burgers equation to illustrate the effect of replacing the standard Laplacian dissipation term by a more general function of the Laplacian -- of which hyperviscosity is the best known example -- in equations of hydrodynamics. We analyze the asymptotic structure of solutions in the Fourier space at very high wave-numbers by introducing an approach applicable to a wide class of hydrodynamical equations whose solutions are calculated in the limit of vanishing Reynolds numbers from algebraic recursion relations involving iterated integrations. We give a detailed analysis of their analytic structure for two different types of dissipation: a hyperviscous and an exponentially growing dissipation term. Our results, obtained in the limit of vanishing Reynolds numbers, are validated by high-precision numerical simulations at non-zero Reynolds numbers. We then study the bottleneck problem, an intermediate asymptotics phenomenon, which in the case of the Burgers equation arises when ones uses dissipation terms (such as hyperviscosity) growing faster at high wave-numbers than the standard Laplacian dissipation term. A linearized solution of the well-known boundary layer limit of the Burgers equation involving two numerically determined parameters gives a good description of the bottleneck region.