# Factorization of the mean

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Three-dimensional model typically are unmanageable or demand huge computational efforts when applied to natural-scale geophysical mass flows, which can involve masses as large as $10^6$ to $10^{13}$ ${m}^3$. One way to make the problem more tractable is to assume that flows are long (or wide) relative to their depth, and to use depth-averaging in the direction normal to the sliding surface. This dramatically reduces the complexity associated with the flow from three-dimension to virtually shallow flow so that the computational cost is very low.

This is achieved by using the concept of factorization of the mean. This is a very fundamental concept widely accepted and used in shallow flow modelling, including the shallow water flow, granular avalanches, debris flows, flash flood, tsunami and other type geophysical mass flows.[1][2] It is assumed that the average of a product is approximated by the product of the averages. This simplification and approximation has great contributions in depth averaging the mass and momentum balance equations and associated expressions in mass flow modellings. This allows, for example, approximating the velocities by their means. This is mainly based on the observed phenomena that in smooth gravity- or pressure-driven geophysical and industrial mass flows, the momentum transfer in the direction normal to the main flow direction is negligible. Or in other words, the aspect ratio (typical flow depth divided by typical flow extent) is very small, or to say equivalently, the flow depth is much smaller than a typical wave length (long wave approximation).