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The work of J. A. Krommes and R. A. Smith on rigorous upper bounds for the turbulent transport of a passively advected scalar is extended in two directions: (1) For their “reference model,” improved upper bounds are obtained by utilizing more sophisticated two-time constraints which include the effects of cross-correlations up to fourth order. Numerical solutions of the model stochastic differential equation are also obtained; they show that the new bounds compare quite favorably with the exact results, even at large Reynolds and Kubo numbers. (2) The theory is extended to take account of afinite spatial autocorrelation lengthLc. As a reasonably generic example, the problem of particle transport due to statistically specified stochastic magnetic fields in a collisionless turbulent plasma is revisited. A bound is obtained which reduces for smallLc to the quasilinear limit and for largeLc to the strong turbulence limit, and which provides a reasonable and rigorous interpolation for intermediate values ofLc.

Improved rigorous upper bounds for transport due to passive advection described by simple models of bounded systems