The Galerkin-truncated Burgers equation: crossover from inviscid-thermalized to Kardar–Parisi–Zhang scaling
Carlos Cartes, E. Tirapegui, Rahul Pandit, Marc Brächet
Abstract
The one-dimensional Galerkin-truncated Burgers equation, with both dissipation and noise terms included, is studied using spectral methods. When the truncation-scale Reynolds number <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>R</mml:mi> <mml:mrow> <mml:mrow> <mml:mtext>min</mml:mtext> </mml:mrow> </mml:mrow> </mml:msub> </mml:math> is varied, from very small values to order 1 values, the scale-dependent correlation time <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>τ</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>k</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:math> is shown to follow the expected crossover from the short-distance <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>τ</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>k</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>∼</mml:mo> <mml:msup> <mml:mi>k</mml:mi> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> </mml:math> Edwards–Wilkinson scaling to the universal long-distance Kardar–Parisi–Zhang scaling <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>τ</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>k</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>∼</mml:mo> <mml:msup> <mml:mi>k</mml:mi> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>3</mml:mn> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> </mml:math> . In the inviscid limit, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>R</mml:mi> <mml:mrow> <mml:mrow> <mml:mtext>min</mml:mtext> </mml:mrow> </mml:mrow> </mml:msub> <mml:mo stretchy="false">→</mml:mo> <mml:mi mathvariant="normal">∞</mml:mi> </mml:math> , we show that the system displays another crossover to the Galerkin-truncated inviscid-Burgers regime that admits thermalized solutions with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>τ</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>k</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>∼</mml:mo> <mml:msup> <mml:mi>k</mml:mi> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:math> . The scaling forms 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 characterized. This article is part of the theme issue ‘Scaling the turbulence edifice (part 2)’.