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 $P(u')$ of the one-dimensional Burgers and KPZ equations with correlated noise. We have used pseudo-spectral method for our analysis. When $σ$ of the noise variance of the Burgers equation (variance $\propto k^{-2 σ}$) 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 $σ$. For $-1 \le σ\le 0$, 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 $0 < σ<3/2$, 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 $σ$, while $P(u')$ is gaussian for $σ=-1$, but steadily becomes nongaussian for larger $σ$; for negative $u'$, $P(u') \propto \exp(-a x)$ for $σ=0$, and approximately $\propto u'^{-5/2}$ for $σ> 1/2$. We have also calculated the energy cascade rates for all $σ$ and found a constant flux for all $σ\ge 1/2$.