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.
