Burgers Turbulence with Large-scale Forcing
  Burgers turbulence supported by white-in-time random forcing at low wavenumbers is studied analytically and by computer simulation. It is concluded that the probability density Q of velocity gradient displays four asymptotic regimes at very large Reynolds number: (A) a region of large positive gradient where Q decays rapidly (reduction of gradient by stretching); (B) an intermediate region of negative gradient where Q falls off as the inverse third power of gradient (transient inviscid steepening of negative gradient); (C) an outer power-law region of negative gradient where Q falls off as the reciprocal of gradient (shoulders of mature shocks); (D) a final region of large gradient where Q decays very rapidly (interior of mature shocks). The probability density of velocity difference across an interval r, divided by r, lies on Q throughout regions A and B and into the middle of C, for small enough inertial-range r. 
Toshiyuki Gotoh, Robert H. Kraichnan