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Abstract
Upper bounds are considered for steady-state transport in turbulent bounded systems, for both passive advection and self-consistent cylindrical plasmas. Specifically, the work of Krommes and Smith on passive advection (Ann. Phys. 177, 246 (1987)) is extended in two directions: (1) For their "reference model," improved upper bounds are obtained by including 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 problem of particle transport due to statistically specified stochastic magnetic fields in a collisionless turbulent plasma is revisited to take account of a finite spatial autocorrelation length $L\sb{c}$. A bound is obtained which reduces for small $L\sb{c}$ to the quasilinear limit and for large $L\sb{c}$ to the strong turbulence limit, and which provides a reasonable and rigorous interpolation for intermediate values of $L\sb{c}$.
Then, a self-consistent problem is treated by considering upper bounds for the turbulently generated electromotive force (emf) $\bar {\alpha}$ along the axial direction in a magnetized cylindrical plasma bounded by a perfectly conducting wall. One goal in particular is to obtain rigorous results about relaxation in reversed-field pinches. It is assumed that the plasma is driven by an external electric field and has reached a possibly turbulent steady state in which the total current ${\cal T}$ is fixed and in which the conservation laws of energy and magnetic helicity are enforced. For simplicity, incompressible, resistive MHD equations are used. A bounding curve ${\cal T}$($\bar {\alpha};H$) is obtained without the helicity constraint, where H is the Hartmann number. Features of the maximal-emf state corresponding to the reversed-field state are discussed; for instance, the $F-\Theta$ curve is presented and comparisons are made with the "Taylor state".
