A dynamical system approach to soil salinity and sodicity
Introduction
Soil salinity and sodicity impose stress on agricultural crops, reducing yields when critical thresholds are surpassed [1], [2]. While soil salinity refers to high concentrations of salt in the soil, soil sodicity is the condition in which sodium constitutes a large portion of overall cations, not necessarily accompanied by high salinity levels. Increased osmotic pressure of the soil water, which impedes its uptake by the roots, and nutrient imbalances, which in turn lead to toxicities and deficiencies, are the major causes for the adverse effects of salinity and sodicity on plant growth [3]. Other detrimental factors include poor physical soil conditions, like low permeability to air and water, caused by clay swelling and dispersion, associated with a complex interplay between sodicity and electrolyte concentration [4].
Irrigation with poor-quality water and insufficient leaching of soils are major factors contributing to secondary salinization, or the accumulation of salt in soils by means of human intervention. Ghassemi et al. [5] estimated that about 20% of irrigated land are salt-affected, and nearly four million acres of farmland are lost to excessive salt every year [6].
Detailed numerical models have been proposed to simulate the dynamics of water and salt in the root zone [7], [8]. However, they demand solving Richards’ equation for the water flow and partial differential equations for the transport of chemical species. Thus, while being very detailed, they require cumbersome calculations if long-term predictions are required, and the lack of analytical solutions somewhat masks the relationships among governing variables. As a result, other simple approaches have been proposed: some analytical models are based on a balance equation for the soil salt with a stochastic term for rain induced leaching events [9], [10], but do not model explicitly soil water nor sodium; another account solves balance equations of water and salt cations (sodium and calcium) [11], but its solutions are strictly numerical. Partly with the exception of the latter, these simple models do not analyze in detail the interplay between cations in the soil complex and the solution, as well as the nonlinear dynamics resulting from the thermodynamic equilibrium of differently charged cations (e.g., and ).
To investigate such dynamics, in this paper we present a simple analytical model of salinity and sodicity based on the balance equations for soil water and salt (sodium and calcium cations), coupled to an equation for their chemical thermodynamic equilibrium. In deterministic conditions (i.e., no stochastic forcing), these equations are amenable to analysis. We derive here the trajectories in phase space, the time scales for the dynamics, and the dependence of steady-state salinity and sodicity levels on irrigation water quality. We also discuss alternative choices of cation exchange equations, and their effects on phase–space dynamics.
The paper is structured as follows. Section 2 develops a dynamical equation for salt concentration in soil water, Section 3 then develops a dynamical equation for the fraction of sodium adsorbed in the soil, and Section 4 investigates the coupled dynamics of the sodium fraction and the salt concentration in phase space. Finally, we synthesize our results and present conclusions in Section 5. Table 1 shows the definition of all symbols defined throughout this paper.
Section snippets
Soil water balance
We begin by considering the water and salt balances in a unit area of soil subject to irrigation. The balance equation for the relative soil moisture s is [12]where Zr is the rooting soil depth and n is the dimensionless porosity. The water inputs are the precipitation P and irrigation I, while water leaves the system through evapotranspiration ET, deep percolation L and surface runoff Q. For simplicity, the water table is considered to be deep with respect to
Sodicity dynamics
Having described how the total salinity evolves via Eq. (8), the next step is to describe the partitioning between the cations in the soil solution and in the adsorbed phase, as a function of the properties of the soil solution and thus of the quality of irrigation water.
The main cations found in saline soils are and, to a lesser degree, . All of them are involved in an intricate dynamical process of adsorption and desorption, whose mathematical modeling would be too cumbersome
Study of the dynamical system
From (1), (7) and (25) we have a dynamical system of three variables: soil moisture s, salt concentration in the soil water C, and equivalent fraction of sodium in the exchange complex E. However, assuming constant soil moisture, we are left with two dynamical equations only, for C and E. This is justified by the fact that the soil moisture reaches its steady-state value s⋆ in a time scale of days, while salinization and sodification processes occur in the order of a few weeks to months.
This
Conclusions
We presented here a simple system of differential equations to describe the dynamics of soil salinity and sodicity.
One main result is the determination of time scales for the convergence of C and E to their steady-state values. The time scale τC associated with the dynamics of soil salinity is a function only of the ratio between the soil water content w⋆ and the percolation rate L⋆: higher irrigation rates imply higher percolation rates, which in turn mean shorter convergence times τC. We
Acknowledgments
The authors would like to thank SEATM van der Zee for useful discussion. YM acknowledges support from BARD, the United States–Israel Binational Agricultural Research and Development Fund, Vaadia–BARD Postdoctoral Fellowship award no. FI-517-14. AP acknowledges NSF grants: CBET 1033467, EAR 1331846, EAR 1316258, FESD 1338694, as well as the US DOE through the Office of Biological and Environmental Research, Terrestrial Carbon Processes program (de-sc0006967), the Agriculture and Food Research
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