Renormalized dissipation in the nonconservatively forced Burgers equation

In a famous calculation of the “large-distance and longtime properties of a randomly stirred fluid,” Forster, Nelson, and Stephen (FNS) analyzed the consequences of various forcing scenarios for the Navier–Stokes equation and, to some extent, Burgers equation. They considered both a conservative forcing (Model A) and a nonconservative one (Model B), and predicted nontrivial properties for the lowfrequency, small-wave-number limits of the two-point correlation and response functions. They did not explicitly consider Model B for Burgers equation; however, that was later studied in considerable detail by Hwa and Kardar (HK). Recently Diamond and Hahm (DH) attempted to use the Burgers Model B in support of a paradigm of self-organized criticality (SOC) for plasma transport. In the course of their discussion, they performed a calculation of the renormalized dissipation coefficient ηk that describes the mean propagation of small-amplitude pulses with Fourier components k. In the absence of macroscopic velocity shear V ′, they invoked a wave-number scaling for the turbulent dissipation (ηk ∼ k) that disagreed with that predicted by HK (ηk ∼ |k|). As a consequence, they encountered a catastrophic long-wavelength divergence ∼ ∫ kmin dq/q, where kmin is a minimum wave-number cutoff. Their result for V ′ 6= 0 also exhibited pathologies. In the present work, I reconsider the calculations. For V ′ = 0, I find only a benign logarithmic divergence and a wave-number scaling in agreement with HK. I remark that such scaling, although anomalous, does not by itself point to an SOC paradigm. Finally, I deduce a more satisfactory scaling formula for V ′ 6= 0. The 1-D forced Burgers equation is