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 ∝k−2σ) exceeds 3/2, large shocks appear in the velocity profile leading to 〈|u(k)|2〉∝k−2, and structure function 〈|u(x+r,t)−u(x,t)|q〉∝r suggesting that the Burgers equation is intermittent for this range of σ. For −1⩽σ⩽0, the profile is dominated by noise, and the spectrum 〈|h(k)|2〉 of the corresponding KPZ equation is in close agreement with the Medina et al. 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 non-Gaussian for larger σ; for negative u′, P(u′)∝exp(−ax) for σ=0, and approximately ∝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⩾1/2.