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

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 a of the noise variance of the Burgers equation (variance proportion al to k(-2 sigma)) exceeds 3/2, large shocks appear in the velocity profile leading to [\u(k)\(2)] proportional to k(-2), and structure function [\u(x + r, t)- u(x, t)\(q)] proportional to r suggesting that the Burgers equati on is intermittent for this range of sigma. For -1 less than or equal to si gma less than or equal to 0, the profile is dominated by noise, and the spe ctrum [\h(k)\(2)] of the corresponding KPZ equation is in close agreement w ith the Medina et al. renormalization group predictions. In the intermediat e range 0 < a < 3/2, both noise and well-developed shocks are seen, consequ ently 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') is Gaussian for sigma = -1, but steadily becomes non-Gaussian for large r a, for negative u', P(u') proportional to exp(-ax) for sigma = 0, and app roximately proportional to u'-5/2 for sigma > 1/2. We have also calculated the energy cascade rates for all a and found a constant flux for all sigma greater than or equal to 1/2. (C) 2000 Published by Elsevier Science B.V. A ll rights reserved.