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Abstract

Scalar fields which are subject to turbulent mixing typically feature a broad range of scales. When focusing on the large-scale dynamics, it remains a question how to effectively parametrize the small scales. Here, we address this question within the framework of a stochastic, one-dimensional passive scalar model. We show that small-scale averaging, i.e., an ensemble average over small-scale velocity fluctuations, results in an effective diffusivity reminiscent of phenomenological eddy viscosity models, while reducing the effective Reynolds number of the advecting velocity field. Based on that, we establish a filtering procedure that exactly maps second-order statistics of the fully resolved passive scalar field to the one obtained by small-scale averaging. Using fully resolved simulations, we show that small-scale averaging also captures higher-order large-scale statistics of passive scalar fields.