The Galerkin-truncated Burgers equation: Crossover from inviscid-thermalised to Kardar-Parisi-Zhang 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 $R_{\min }$ 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: $R_{\min }\to \infty$, 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.

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• Burgers Turbulence
• Exact scaling functions for one-dimensional stationary KPZ growth
• An exact solution for the KPZ equation with flat initial conditions
• The one-dimensional KPZ equation and its universality class
• Transition from dissipative to conservative dynamics in equations of hydrodynamics