Rate-Independent Hysteresis in Terrestrial Hydrology

The goal of this article is twofold. The first objective is to demonstrate the role of hysteresis in hydrology. The second objective is to describe a class of simple and convenient mathematical models of hydrological systems with hysteresis and to briefly discuss methods for the quantitative and qualitative analysis of such models. This article is organized as follows. In the first section we discuss the role of the Preisach model. We then present an explicit expression for analogues of wetting and drying curves in the Preisach model context. The central role in this section is occupied by special one-parameter classes of Preisach operators which are useful as models of soil-water hysteresis for particular soils. Fitting is discussed in the context of the GRIZZLY database of hysteretic soil-water characteristics. The second section introduces a simple but instructive example of a differential equation coupled with a Preisach nonlinearity. This equation serves as a basis for modeling vegetated soil in terrestrial hydrology. The third section is dedicated to the FEST model, where FEST stands for "fully vegetated slab of soil with transpiring plants". This model describes drainage, transpiration, and infiltration flows in a fully vegetated slab of soil. We demonstrate that hysteresis effects are critical, both on qualitative and quantitative levels, for modeling hydrological systems and flows in soil, while theoretical analysis and numerical implementation of the models raise interesting mathematical questions. To demonstrate the application of this model we use rainfall data gathered in the Feale region in County Kerry, Ireland, covering a period of 19 months between July 2002 and February 2004. The FEST model includes a quantitative description of the dynamics of the plant biomass, which depends on the amount and regularity of precipitation and soil type, including details of soil-water hysteresis. Hence, the FEST model can be used as a component of ecological system models. Finally, the qualitative properties of the FEST model are discussed.