Burgers turbulence with random forcing: Similarity functional solution of the Hopf equation.

Abstract

For the problem of Burgers turbulence with random forcing, a similarity functional solution of the Hopf equation is presented and compared with scaling arguments and replica Bethe-ansatz treatments. The corresponding field theory is almost nonanomalous. In one dimension the local fluctuations develop self-similar time-dependent behavior, while relative fluctuations within the correlation length form a steady state with Gaussian distribution. This is the precise meaning of the so-called fluctuation-dissipation theorem. The one-dimensional properties are also studied numerically. It is shown that the fluctuation-dissipation theorem is invalid above one dimension and higher-order cumulants are nonzero. In two dimensions the cumulants exhibit a logarithmic spatial dependence, which is close to but different from that in the Edwards-Wilkinson case. No other similarity functional solution is found, which may indicate that the ``strong-coupling'' results are not described by the forced Burgers equation.