The inviscid Burgers equation with random and spatially smooth forcing is considered in the limit when the size of the system tends to infinity. For the one-dimensional problem, it is shown both theoretically and numerically that many of the features of the space-periodic case carry over to infinite domains as intermediate time asymptotics. In particular, for large time T we introduce the concept of T-global shocks replacing the notion of main shock which was considered earlier in the periodic case (1997, E et al., Phys. Rev. Lett. 78, 1904). In the case of spatially extended systems these objects are no anymore global. They can be defined only for a given time scale and their spatial density behaves as ?(T) ~ T-2/3 for large T. The probability density function p(A) of the age A of shocks behaves asymptotically as A -5/3. We also suggest a simple statistical model for the dynamics and interaction of shocks and discuss an analogy with the problem of distribution of instability islands for a simple first-order stochastic differential equation.
|Number of pages||19|
|Journal||Journal of Statistical Physics|
|Publication status||Published - Dec 2003|
- Burgers turbulence
- Shock discontinuities
- Stochastic forcing