Stratified flow over topography: the role of small-scale entrainment and mixing in flow establishment

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Abstract
Consideration of boundary–value problems in mechanics of materials with disordered microstructures leads to the introduction of an intermediate scale, a mesoscale, which specifies the resolution of a finite–element mesh relative to the microscale. The effective elastic mesoscale response is bounded by the Dirichlet and Neumann boundary–value problems. The two estimates, separately, provide inputs to two finite–element schemes (based on minimum potential and complementary energy principles, respectively) for bounding the global response. While in the classical case of a homogeneous material, these bounds are convergent with the finite elements becoming infinitesimal, the presence of a disordered non–periodic microstructure prevents such a convergence and leads to a possibility of an optimal mesoscale. The method is demonstrated through an example of torsion of a bar having a percolating two–phase microstructure of over 100 000 grains. By passing to an ensemble setting, we arrive at a hierarchy of two random continuum fields, which provide inputs to a stochastic finite–element method.