We address the challenge of optimal incompressible stirring to mix an initially inhomogeneous distribution of passive tracers. As a measure for mixing we adopt the H-1 norm of the scalar fluctuation field. This 'mix-norm' is equivalent to (the square root of) the variance of a low-pass filtered image of the tracer concentration field, and is a useful gauge even in the absence of molecular diffusion. We show that the mix-norm's vanishing as time progresses is evidence of the stirring flow's mixing property in the sense of ergodic theory. For the case of a periodic spatial domain with a prescribed instantaneous energy or power budget for the stirring, we determine the flow field that instantaneously maximizes the decay of the mix-norm, i.e., the instantaneous optimal stirring --- when such a flow exists. When no such 'steepest descent' stirring exists, we determine the flow that maximizes that rate of increase of the rate of decrease of the norm. This local-in-time stirring strategy is implemented computationally on a benchmark problem and compared to an optimal control approach utilizing a restricted set of flows. This is joint work with Zhi Lin (University of Minnesota) and Jean-Luc Thiffeault (University of Wisconsin).
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