  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.
