Steady-state Burgers turbulence with large-scale forcing

Steady-state Burgers turbulence supported by white-in-time random forcing at low wave numbers is studied analytically and by computer simulation. The peak of the probability distribution function (pdf) Q(xi) of velocity gradient xi is at xi=O(xi(f)), where xi(f) is a forcing parameter. It is concluded that Q(xi) displays four asymptotic regimes at Reynolds number R much greater than 1: (A) Q(xi)similar to xi(f)(-2)xi x exp(-xi(3)/3 xi(f)(3)) for xi much greater than xi(f) (reduction of large positive xi by stretching); (B) Q(xi)similar to xi(f)(2)\xi\(-3) for xi(f) much less than-xi much less than R(1/2)xi(f) (transient inviscid steepening of negative xi); (C) Q(xi)similar to\R xi\(-1) for R(1/2)xi(f)much less than-xi much less than R xi(f) (shoulders of mature shocks); (D) very rapid decay of Q for -xi greater than or equal to O(R xi(f)) (interior of mature shocks). The typical shock width is O(1/Rk(f)). If R(-1/2)much greater than rk(f)much greater than R-1, the pdf of velocity difference across an interval r is found to be P(Delta u,r)proportional to r(-1)Q(Delta u/r) throughout regimes A and B and into the middle of C. (C) 1998 American Institute of Physics. [S1070-6631(98)00111-1].