Steady-state Burgers turbulence with large-scale forcing

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Abstract
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(ξ) of velocity gradient ξ is at ξ=O(ξf), where ξf is a forcing parameter. It is concluded that Q(ξ) displays four asymptotic regimes at Reynolds number R≫1: (A) Q(ξ)∼ξf−2ξexp(−ξ3/3ξf3) for ξ≫ξf (reduction of large positive ξ by stretching); (B) Q(ξ)∼ξf2|ξ|−3 for ξf≪−ξ≪R1/2ξf (transient inviscid steepening of negative ξ); (C) Q(ξ)∼|Rξ|−1 for R1/2ξf≪−ξ≪Rξf (shoulders of mature shocks); (D) very rapid decay of Q for −ξ⩾O(Rξf) (interior of mature shocks). The typical shock width is O(1/Rkf). If R−1/2≫rkf≫R−1, the pdf of velocity difference across an interval r is found to be P(Δu,r)∝r−1Q(Δu/r) throughout regimes A and B and into the middle of C.
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