Intermittency exponents and energy spectrum of the Burgers and KPZ equations with correlated noise

Abstract

We numerically calculate the energy spectrum, intermittency exponents, and probability density $P(u^{\prime })$ of the one-dimensional Burgers and KPZ equations with correlated noise. We have used pseudo-spectral method for our analysis. When $\sigma$ of the noise variance of the Burgers equation (variance $\propto k^{-2\sigma }$) exceeds 3/2, large shocks appear in the velocity profile leading to $<|u(k){|}^2>\propto k^{-2}$, and structure function $<|u(x+r,t)-u(x,t){|}^q>\propto r$ suggesting that the Burgers equation is intermittent for this range of $\sigma$. For $-1\le \sigma \le 0$, the profile is dominated by noise, and the spectrum $<|h(k){|}^2>$ of the corresponding KPZ equation is in close agreement with Medina et al.'s renormalization group predictions. In the intermediate range $0<\sigma <3/2$, both noise and well-developed shocks are seen, consequently the exponents slowly vary from RG regime to a shock-dominated regime. The probability density $P(h)$ and $P(u)$ are gaussian for all $\sigma$, while $P(u^{\prime })$ is gaussian for $\sigma =-1$, but steadily becomes nongaussian for larger $\sigma$; for negative $u^{\prime }$, $P(u^{\prime })\propto \exp (-ax)$ for $\sigma =0$, and approximately $\propto u^{\prime -5/2}$ for $\sigma >1/2$. We have also calculated the energy cascade rates for all $\sigma$ and found a constant flux for all $\sigma \ge 1/2$.

Revised version combining chao-dyn/9904020v1 & chao-dyn/9904021, To appear in Physica A

References in corpus (2)

• Asymptotic Theory for the Probability Density Functions in Burgers Turbulence
• Burgers turbulence with pressure

Cited by in corpus (17)

• Kolmogorov-Burgers Model for Star Forming Turbulence
• Local shell-to-shell energy transfer via nonlocal Interactions in fluid turbulence
• Global fluctuations in physical systems: a subtle interplay between sum and extreme value statistics
• The Kardar-Parisi-Zhang equation with spatially correlated noise: a unified picture from nonperturbative renormalization group
• Variable energy flux in turbulence
• The Fractal Geometry of the Cosmic Web and its Formation
• KPZ equation with short-range correlated noise: emergent symmetries and non-universal observables
• The one-dimensional Kardar-Parisi-Zhang and Kuramoto-Sivashinsky universality class: limit distributions
• Kardar-Parisi-Zhang Equation with temporally correlated noise: a non-perturbative renormalization group approach
• Faceted patterns and anomalous surface roughening driven by long-term correlated noise
• Statistical Theory for the Kardar-Parisi-Zhang Equation in 1+1 Dimension
• Intermittency of Height Fluctuations and Velocity Increment of The Kardar-Parisi-Zhang and Burgers Equations with infinitesimal surface tension and Viscosity in 1+1 Dimensions
• A Hybrid Monte Carlo algorithm for sampling rare events in space-time histories of stochastic fields
• Equilibrium states of Burgers and KdV equations
• The inviscid fixed point of the multi-dimensional Burgers-KPZ equation
• Small-scale intermittency of premixed turbulent flames
• Incompressible Turbulence as Nonlocal Field Theory