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Temporal dynamics of soil moisture variability: 1. Theoretical basis

Coarse soil moisture fields and nonlinear relationships between fluxes and soil moisture combine to yield errors in both diagnostic and predictive estimates of large‐scale mass and energy fluxes. Efforts to empirically define the dynamics of subgrid spatial variance of soil moisture have led to contradictory results. Moreover, most reports of soil moisture variability range from qualitative to descriptively quantitative, owing to the lack of a robust theoretical framework for moisture variance dynamics. In this paper we derive a conservation equation for the spatial variance of subgrid root zone soil moisture, based on first principles of statistical fluid mechanics. We arrive at a variance budget in which explicit covariances between moisture fields and land surface flux fields act to produce or destroy variance through time (according to the sign of the correlation between the flux and state fields). A series of examples are used to explore how simple forms of soil, vegetation, precipitation, topography, and initial moisture variability lead to evolving covariances between spatial fields of soil moisture and particular land surface fluxes and how these covariances relate to the temporal trajectory of the spatial variance of soil moisture. We isolate a set of processes and conditions that demonstrate variance production through time and a set that demonstrate variance destruction. Of particular interest is the tendency for transpiration and infiltration‐runoff processes to either produce or destroy variance, depending on the background wetness regime. Field data are also employed and shown to demonstrate a temporal behavior of the spatial variance that is readily described by the proposed approach. Ultimately, this work should aid field data interpretation and, when supplemented with a closure model for the variance budget, lead to improved land surface flux predictability over coarse grids.