Volume 25, Issue 7, July 2015
Index of content:
- FOCUS ISSUE: PHYSICS OF SCALING AND SELF-SIMILARITY IN HYDROLOGIC DYNAMICS, HYDRODYNAMICS AND CLIMATE
A physical scaling model for aggregation and disaggregation of field-scale surface soil moisture dynamics25(2015); http://dx.doi.org/10.1063/1.4913235View Description Hide Description
Scaling relationships are needed as measurements and desired predictions are often not available at concurrent spatial support volumes or temporal discretizations. Surface soil moisture values of interest to hydrologic studies are estimated using ground based measurement techniques or utilizing remote sensing platforms. Remote sensing based techniques estimate field-scale surface soil moisture values, but are unable to provide the local-scale soil moisture information that is obtained from local measurements. Further, obtaining field-scale surface moisture values using ground-based measurements is exhaustive and time consuming. To bridge this scale mismatch, we develop analytical expressions for surface soil moisture based on sharp-front approximation of the Richards equation and assumed log-normal distribution of the spatial surface saturated hydraulic conductivity field. Analytical expressions for field-scale evolution of surface soil moisture to rainfall events are utilized to obtain aggregated and disaggregated response of surface soil moisture evolution with knowledge of the saturated hydraulic conductivity. The utility of the analytical model is demonstrated through numerical experiments involving 3-D simulations of soil moisture and Monte-Carlo simulations for 1-D renderings—with soil moisture dynamics being represented by the Richards equation in each instance. Results show that the analytical expressions developed here show promise for a principled way of scaling surface soil moisture.
25(2015); http://dx.doi.org/10.1063/1.4913236View Description Hide Description
The present article establishes connections between the structure of the deterministic Navier-Stokes equations and the structure of (similarity) equations that govern self-similar solutions as expected values of certain naturally associated stochastic cascades. A principle result is that explosion criteria for the stochastic cascades involved in the probabilistic representations of solutions to the respective equations coincide. While the uniqueness problem itself remains unresolved, these connections provide interesting problems and possible methods for investigating symmetry breaking and the uniqueness problem for Navier-Stokes equations. In particular, new branching Markov chains, including a dilogarithmic branching random walk on the multiplicative group (0, ∞), naturally arise as a result of this investigation.