Management and Optimal Use of Soil and Water Resources in Ecohydrological Systems by Norman F. Pelak III Department of Civil and Environmental Engineering Duke University Date: Approved: Amilcare M Porporato, Co-Supervisor Zbigniew J. Kabala, Co-Supervisor Gabriel G. Katul Roberto Revelli Dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Department of Civil and Environmental Engineering in the Graduate School of Duke University 2019 Abstract Management and Optimal Use of Soil and Water Resources in Ecohydrological Systems by Norman F. Pelak III Department of Civil and Environmental Engineering Duke University Date: Approved: Amilcare M Porporato, Co-Supervisor Zbigniew J. Kabala, Co-Supervisor Gabriel G. Katul Roberto Revelli An abstract of a dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Department of Civil and Environmental Engineering in the Graduate School of Duke University 2019 Copyright c 2019 by Norman F. Pelak III All rights reserved except the rights granted by the Creative Commons Attribution-Noncommercial License Abstract Human activities are shifting hydrological and biogeochemical cycles further from their natural states, often resulting in negative impacts on the environment. Because of increased pressures due to climate change and population growth, it is important to understand how human activities affect soil and water resources and how these resources can be managed sustainably. This dissertation presents a series of works which relate to the sustainable management of soil and water resources. In general, we make use of parsimonious ecohydrological models to describe key components of the soil and water system, and random hydroclimatic variability is accounted for with stochastic forcing. Methods from dynamical systems theory are also applied to further the analysis of these systems. Initially we focus on soil resources, the impacts of vegetation on soil production and erosion and the feedbacks between soil formation and vegetation growth are ex- plored with a minimal model of the soil-plant system, which includes key feedbacks, such as plant-driven soil production and erosion inhibition. Vegetation removal re- duces the stabilizing effect of vegetation and lowers the system resilience, thereby increasing the likelihood of transition to a degraded soil state. We then turn our at- tention to water resources. Rainwater harvesting (RWH) has the potential to reduce water-related costs by providing an alternate source of water, in addition to relieving pressure on other water sources and reducing runoff. An analytical formulation is developed for the optimal cistern volume as a function of the roof area, water use iv rate, climate parameters, and costs of the cistern and of the external water source, and an analysis of the rainfall partitioning characterizes the efficiency of a particular RWH system configuration. Then we consider nutrient management in addition to sustainable soil and water resources. Crop models, though typically constructed as a set of dynamical equations, are not often analyzed from a specifically dynamical systems point of view, and so we develop a minimal dynamical systems framework for crop models, which describes the evolution of canopy cover, soil moisture, and soil nitrogen. Important crop model responses, such as biomass and yield, are calcu- lated, and optimal yield and profitability under differing climate scenarios, irrigation strategies, and fertilization strategies are examined within the developed framework. Important in the use of crop and other ecohydrological models and studies on soil and water resources is the representation of soil properties. Soil properties are determined by a complex arrangement of pores, particles, and aggregates, which may change in time, as a result of both ecohydrological dynamics and land management processes. The soil pore size distribution (PSD) is a key determinant of soil properties, and its accurate representation has the potential to improve hydrological and crop models. A modeling framework is proposed for the time evolution of the PSD which takes into account processes such as tillage, consolidation, and changes in organic matter. This model is used to show how soil properties such as the water retention curve and the hydraulic conductivity curve evolve in time. Finally, in order to explore the coupled evolution of soil properties, ecohydrological processes, and crop growth, we couple a dynamic crop model with a soil biogeochemistry model and the previously developed model for the evolution of the soil PSD. v To my parents vi Contents Abstract iv List of Tables xi List of Figures xii Acknowledgements xiv 1 Introduction1 2 Bistable plant{soil dynamics and biogenic controls on the soil pro- duction function5 2.1 Introduction................................5 2.2 Coupled model of soil and vegetation dynamics............7 2.2.1 Soil production and erosion...................8 2.2.2 Vegetation growth, turnover, and harvest............ 10 2.3 Model analysis.............................. 11 2.3.1 The abiotic case.......................... 11 2.3.2 Vegetation influence on soil stability without harvest..... 12 2.3.3 Soil stability and resilience under land use change....... 16 2.4 Discussion and Conclusion........................ 19 3 Sizing a rainwater harvesting cistern by minimizing costs 22 3.1 Introduction................................ 22 3.2 Stochastic Water Balance in Cisterns.................. 26 vii 3.3 Non-dimensional parameters....................... 28 3.3.1 Steady-State Solutions...................... 29 3.4 Harvested Water Partitioning...................... 30 3.5 Optimal Cistern Size........................... 32 3.5.1 Fixed Costs............................ 33 3.5.2 Distributed Costs......................... 34 3.5.3 Solution for the Optimal Cistern Size.............. 35 3.5.4 Expected financial gain and maximum loss........... 38 3.5.5 Optimal size including runoff reduction............. 39 3.6 Conclusions................................ 40 4 A dynamical systems framework for crop models: toward optimal fertilization and irrigation strategies under climatic variability 42 4.1 Introduction................................ 42 4.2 Model components............................ 46 4.2.1 Canopy cover dynamics...................... 46 4.2.2 Soil moisture balance equation.................. 47 4.2.3 Soil nitrogen content....................... 51 4.2.4 Crop biomass and yield...................... 53 4.3 Reduced versions of the model...................... 54 4.3.1 Canopy growth equation and its parameterization....... 54 4.3.2 N and C system......................... 58 4.4 Soil moisture dynamics and hydrologic forcing............. 62 4.4.1 Soil moisture dry-down...................... 63 4.4.2 Stochastic forcing......................... 63 4.4.3 Impact of rainfall regimes on rain-fed agriculture....... 64 4.5 Optimal strategies............................ 67 viii 4.5.1 Optimization under stochastic rainfall conditions....... 70 4.6 Conclusion................................. 74 5 Dynamic evolution of the soil pore size distribution and its connec- tion to soil biogeochemical processes 76 5.1 Introduction................................ 76 5.2 Dynamic pore size distribution...................... 78 5.2.1 Evolution equation........................ 78 5.2.2 Power law pore size distribution................. 78 5.2.3 Terms of the evolution equation................. 80 5.3 Soil hydraulic properties......................... 80 5.3.1 Model parameterization..................... 82 5.4 Connection of parameters to temporal and biogeochemical processes. 83 5.4.1 Tillage and consolidation term.................. 84 5.4.2 SOM relationship......................... 85 5.5 Evolution of soil hydraulic properties and parameters......... 86 5.6 Conclusion................................. 92 6 Exploring the evolution of soil properties with a coupled agroe- cosystem model 94 6.1 Introduction................................ 94 6.2 Soil component.............................. 96 6.2.1 Soil moisture balance....................... 96 6.2.2 Soil carbon balance........................ 97 6.2.3 Soil nitrogen balance....................... 99 6.3 Plant carbon dynamics.......................... 100 6.4 Plant nitrogen dynamics......................... 104 6.4.1 Critical curve: pre- and post- vegetative stage......... 110 ix 6.5 Evolution of soil properties........................ 110 6.6 Conclusion................................. 116 7 Conclusion 118 A Analytical results for the stochastic water balance 120 A.1 Chapman-Kolmogorov equation and derivation of steady-state PDF. 120 A.2 Crossing Time Analysis.......................... 121 B Method of characteristics 122 C Water retention curve 124 Bibliography 126 Biography 141 x List of Tables 2.1 Typical Model Parameters........................ 13 3.1 A guide to the symbols and abbreviations used in this study...... 23 3.2 Parameters used in figures for the Duke Smart Home (SH) and Durham, NC, USA rainfall data.......................... 33 4.1 The model parameters used in this study................ 55 4.2 The climate and soil parameters used in this study.......... 56 xi List of Figures 2.1 Soil depth bifurcation diagram as a function of the control parameter ρ = p0=(e0 + e1).............................. 14 2.2 Soil production function and stability regimes for various values of the parameter σ ...............................