The Galerkin-truncated Burgers equation: Crossover from inviscid-thermalised to Kardar-Parisi-Zhang scaling

TL;DR

This paper investigates the spectral properties and scaling behaviors of the 1D Galerkin-truncated Burgers equation with dissipation and noise, revealing crossovers between different universality classes including Edwards-Wilkinson, KPZ, and inviscid thermalized regimes.

Contribution

It demonstrates the crossover of correlation time scaling from Edwards-Wilkinson to KPZ and inviscid regimes in the Galerkin-truncated Burgers equation with spectral methods.

Findings

Correlation time scales as k^{-2} at short distances.

Transition to KPZ scaling with k^{-3/2} at larger scales.

Inviscid limit shows a thermalized regime with k^{-1} scaling.

Abstract

The one-dimensional ($1D$) Galerkin-truncated Burgers equation, with both dissipation and noise terms included, is studied using spectral methods. When the truncation-scale Reynolds number $R_{\rm min}$ is varied, from very small values to order $1$ values, the scale-dependent correlation time $\tau(k)$ is shown to follow the expected crossover from the short-distance $\tau(k) \sim k^{-2}$ Edwards-Wilkinson scaling to the universal long-distance Kardar-Parisi-Zhang scaling $\tau(k) \sim k^{-3/2}$. In the inviscid limit: $R_{\rm min}\to \infty$, we show that the system displays {\it another} crossover to the Galerkin-truncated inviscid-Burgers regime that admits thermalised solutions with $\tau(k) \sim k^{-1}$. The scaling form of the time-correlation functions are shown to follow the known analytical laws and the skewness and excess kurtosis of the interface increments distributions are characterised.