Management and Optimal Use of Soil and Water Resources in Ecohydrological Systems

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

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 explored with a minimal model of the soil-plant system, which includes key feedbacks, such as plant-driven soil production and erosion inhibition. Vegetation removal reduces 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 attention 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 calculated, 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 production 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 connection 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 agroecosystem 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 ...... 15

2.3 Soil potential function and soil depth timeseries...... 17

2.4 A description of resilience for the coupled plant-soil system...... 18

3.1 PDFs and soil moisture timeseries for contrasting roof areas and cistern volumes...... 31

3.2 The ratio of water demand to rainfall rate as a function of the cistern demand index, Dc, for diﬀerent values of γ ...... 32 3.3 Cost ($) vs. cistern volume (m3) for commercially available cisterns. 34

3.4 The unit monthly cost per m3 of municipally-supplied water in Durham, NC, USA...... 35

3.5 Fixed, distributed, and combined cistern costs...... 37

3.6 The optimal cistern size as a function of the climate parameters α and λ 38

4.1 Control of soil moisture and canopy cover on evapotranspiration... 51

4.2 Growth of canopy cover in the C-only model with experimental data 58

4.3 Timeseries and phase portrait for the C-N model...... 60

4.4 Timeseries of model state variables...... 64

4.5 Timeseries of model ﬂuxes...... 65

4.6 Three dimensional phase space of C, S, and N ...... 66

4.7 Averaged model outputs for varying values of λ ...... 67

xii 4.8 Model responses with constant values of S and N ...... 69

4.9 Yield and proﬁt as a function of the constant fertilization rate.... 72

4.10 Crop yield as a function of ξ and τ for three soil depths...... 73

5.1 Example of using the Ψ − θ data to obtain the model parameters.. 82

5.2 The PSD for the tilled and untilled soils of Teiwes(1988)...... 85

5.3 The three model parameters as a function of soil organic matter (C). 87

5.4 Timeseries for the varying SOM levels representing low, middle, and high inputs...... 88

5.5 Timeseries for sw and Ksat, as well as the WRC and the HCC, for varying tillage and SOM levels...... 89

5.6 Timeseries for the model parameters and PSD at various time points 91

6.1 Model results and data for above ground carbon content of fully irrigated maize...... 101

6.2 Kcb(t) during the growing season compared to normalized CS and the transpiration ﬂux T r ...... 102

6.3 Canopy cover (CC) and above ground carbon CS ...... 103 6.4 Data and ﬁtted relationships from Anderson(1988)...... 105

6.5 Figure from Lemaire and Gastal(1997)...... 107

6.6 A comparison of model predictions under unstressed growing conditions and data from Anderson(1988)...... 108

∗ 6.7 The growth limitation term, C , as a function of NS/NC ...... 109

6.8 Eﬀects on the labile soil carbon CL of tillage and crop residue management strategies...... 112

6.9 Eﬀects on the porosity φ of tillage and crop residue management strategies...... 114

6.10 Eﬀects on the wilting point sw of tillage and crop residue management strategies...... 115

6.11 Eﬀects on the saturated hydraulic conductivity Ksat of tillage and crop residue management strategies...... 117

xiii Acknowledgements

First, I would like to thank my family for their inextinguishable support during my time in graduate school: my mom, Joan Pelak, for her perseverance in sending care packages even unto my tenth year of college, my dad, Norm Pelak Jr., for going out of his way to pass through Princeton to buy me beer, Ryan Pelak, for the steady stream of early 2000s pop culture references, Katelyn Pelak, for reclaiming the ‘favorite child’ title which burdened me for many years, and Anthony Nielsen, for giving me the confidence to say ‘chief’ and ‘mate’ in casual conversation. I also want to thank my advisor, Amilcare Porporato, for his guidance, especially the mountain climbing metaphors, and also for assembling a research group which has been composed, somewhat implausibly, entirely of high-quality people. Of those, I’d like to thank in particular the group with whom I moved to Princeton: Salvo Calabrese, for teaching me how to make the perfect coffee, Sara Bonetti, for appreci- ating a good burger as much as I do, Jun Yin, for the fishing lessons, Sam Hartzell, for tending the CAM garden, and Mark Bartlett, for the great stories. Speaking of Princeton, I’d like to acknowledge Khaled Ghannam, for being the first friend I had in both Durham and Princeton, Ben Schaffer, for letting me draft on biking trips, and all the others who made it feel like home. I am also thankful to Weston Ross, for helping me get past tough mountain faces, Dane Emmerling, for many unbilled hours of therapy and for letting me listen to his records, Jon and Iris Holt, for their outstanding hospitality, and all the others that

xiv made Durham great. I’d also like to thank Coston and Janie Rowe for a decade of friendship and for seeing my PhD and raising me a baby (I fold). Finally, moving on to people I’ve never actually met, I want to thank J.R.R. Tolkien, Randall Munroe, and John Darnielle, whose works have been a source of inspiration and a welcome distraction from research for many years. This work was partially funded by the US National Science Foundation through grants CBET-1033467, EAR-1331846, FESD-1338694, EAR-1316258, and EAR-0838301, by the US Department of Agriculture through grant 2011-67003-30222, by the U.S. Department of Energy through grant DE-SC0006967, by the U.S. Department of Defense through the NDSEG Fellowship program, and by the Duke WISENET program through grant DGE-1068871. I would also like to thank the Duke Graduate School and the SAVI International Scholars Program for travel support, and Prince- ton University for hosting me during the ﬁnal two years of my PhD.

xv 1

Introduction

The Earth’s hydrological and biogeochemical processes are increasingly being altered by human activities, and projected changes in climate, including increased hydroclimatic variability and population growth (Easterling et al., 2000; Vörösmarty et al., 2000; Feng et al., 2013) make it even more important to understand how soil and water resources are being affected and how these resources can be managed sustainably (Porporato et al., 2015). This dissertation presents a series of works on these topics, with the overall aim of understanding the dynamics of ecohydrological systems under different management practices and hydroclimatic variability, and how these systems can be optimally managed. The dissertation chapters are presented chronologically in the order in which they were published. Chapters2,3, and4 are based on Pelak et al.(2016), Pelak and Porporato(2016), Pelak et al.(2017), respectively, while the work of Chapters5 and6 will be submitted for future publication. Minimal ecohydrological models are a useful tool for understanding the interactions between human activities, ecohyrological dynamics, and variability, as they are capable of representing the main components of the soil-plant system and its relation to water and nutrient cycles in a parsimonious manner. In combination with stochas-

1 tic forcing by random hydroclimatic variables (especially the amount and timing of rainfall), relatively complex ecohydrological systems, the states of which are determined by the interactions between soil moisture, biogeochemical processes, and plant growth, can be represented with only a small number of key variables. Such models are also often amenable to analysis with dynamical systems theory, which provides a useful set of tools for understanding the behavior of these systems. Understanding the impacts of plant-soil interactions on soil stability and resilience in managed ecosystems is key to the sustainable use of soil resources (Montgomery, 2007; Porporato and Rodriguez-Iturbe, 2013; Porporato et al., 2015). These impacts are explored in Chapter2, in which we develop a minimal model for soil-vegetation interaction which allows us to analyze the stability regimes of the soil vegetation system as controlled by the soil erosion, production, and harvesting rates. We found bistability in the soil-vegetation system, the presence of which leads to a critical soil depth. When harvesting pressure brings the soil below this threshold, a collapse of the soil-vegetation system to a degraded state occurs. Similar tipping points resulting from excessive anthropogenic pressure were found in other ecosystem contexts by Walker et al.(2004) and Folke et al.(2004). The problem of optimization under stochastic hydroclimatic forcing is studied using both analytical and numerical methods in Chapters3 and4. Rainwater harvesting has potential as an alternative water supply in water scarce regions and as a means to reduce runoﬀ (Pandey et al., 2003; Steﬀen et al., 2013; Sample and Liu, 2014). The cistern volume is the most critical design criteria in such systems, and so in Chapter3, we obtain an analytical solution for the optimal size of a rainwater harvesting cistern as a function of the roof area, demand, mean rainfall frequency and event depth, and the costs of water and the cistern. To do this, we extend methods which were originally developed to describe the stochastic soil water balance (Milly, 1993; Rodr´ıguez-Iturbe and Porporato, 2004a), and we also characterize

2 the efficiency of a RWH by examining the rainfall partitioning. In Chapter4, we consider not only the sustainable management of water resources but also of nutrient cycles. Fertilizer use is a primary source of nutrient contamination in US water bodies (Puckett, 1995), which can lead to major environmental problems such as eutrophication (Rabalais et al., 2002; Conley et al., 2009). Additionally, excessive fertilizer use leads to high levels of greenhouse gas emissions (Shcherbak et al., 2014). Therefore, to better understand the optimal rates and timing of fertilization and irrigation for agroecosystems, we develop a dynamic crop model which is forced by stochastic rainfall. Methods from dynamical systems theory are used to analyze a simplified version of the model, without stochastic forcing, and then the full system is solved numerically to account for the impact of random hydroclimatic variability. Soil properties represent an important and often overlooked component of ecohydrological models, and their accurate representation, especially the water retention curve (WRC) and hydraulic conductivity curve (HCC), is needed, as they play an important role in the hydrologic cycle (Rodr´ıguez-Iturbe and Porporato, 2004a; Por- porato et al., 2015). Soil properties typically have a static description within ecohydrological models, but they do change in time, and may do so rapidly in response to human activities (Vereecken et al., 2010). The soil pore size distribution (PSD) is an important determinant of these properties, and in order to understand how soil processes change under management and natural processes, in Chapter5 we develop a model for the dynamic evolution of the soil PSD. The power law description of the PSD is shown to be the solution to an evolution equation with particular drift and source/sink terms. The parameters are linked to key temporal and ecohydrological processes such as tillage, consolidation, and soil biogeochemical cycling, and we show how the WRC, HCC, and other soil properties evolve in time. To demonstrate how this model may be applied, in Chapter6 we implement the dynamic PSD model with an agroecosystem model. We first develop a crop model which is an extended version

3 of the crop model of Chapter4, with a more complex nutrient dynamics and crop growth component. By coupling this crop model to a soil biogeochemical model and the dynamic PSD model, we perform an initial exploration of the coupled evolution of soil properties, ecohydrological processes, and crop growth.

4 2

Bistable plant–soil dynamics and biogenic controls on the soil production function

This chapter is adapted from Pelak, N. F., Parolari, A. J., and Porporato, A. (2016), Bistable plant–soil dynamics and biogenic controls on the soil production function, Earth Surface Processes and Landforms, 41, 10111017.

2.1 Introduction

Soil degradation associated with deforestation, cultivation, and grazing is a major challenge to land management for ecological and agricultural objectives, and the consequences of such human-induced degradation include increased water and wind erosion, changes in soil biogeochemistry, and soil compaction. With respect to soil erosion and biogeochemistry, interactions between soils and vegetation have been implicated as mechanisms causing potentially irreversible loss of soil function, e.g., Schlesinger et al.(1990). The 1930s Dust Bowl in the United States central plains is an especially vivid example of an abrupt transition to a degraded state brought about by the conﬂuence of intensive vegetation removal, soil erosion, and drought

5 (Worster, 1982; Cook et al., 2009), and demonstrates that a full understanding of the role of plant-soil interactions on soil stability and resilience in managed ecosystems is key to the sustainable use of soil resources (Montgomery, 2007; Porporato and Rodriguez-Iturbe, 2013; Porporato et al., 2015). Interaction between soil and vegetation enhances soil formation and stability. It has long been known that soil is more stable in the presence of vegetation (Lyell, 1834) and, more recently, the active role of plants in “engineering” the soil environment to improve water and nutrient access has been emphasized (Jones et al., 1997; Gilad et al., 2004; Cuddington et al., 2009). In addition to stability provided by root cohesion, plants enhance soil production by the mechanical action of roots (Gabet et al., 2003; Wilkinson et al., 2009), soil acidification, which stimulates chemical weathering (Schwartzman and Volk, 1989; Berthrong et al., 2009; Brady and Weil, 2010), and organic matter inputs (Jobbágy and Jackson, 2000). In turn, soil provides vegetation with water and nutrient storage, as well as a suitable living environment. These interactions give rise to a coupled plant-soil system with a positive feedback loop. Positive feedback and related non-linear dynamics often lead to non-trivial stability scenarios in which alternate stable states may be reached depending on the external forcing or initial conditions. In geomorphological systems, soil depth can exhibit two stable states, or bistability, when the soil production function has a maximum at some nonzero soil depth (Dietrich et al., 1995; D’Odorico, 2000; Furbish and Fagher- azzi, 2001). This is the so-called ‘humped’ soil production function, demonstrated in at least one field experiment where soil production rates from exposed bedrock were found to be approximately half of that from soil-mantled bedrock (Heimsath et al., 2009). Mechanisms through which vegetation impacts on soil evolution lead to bistability have been explored in the context of episodic mass wasting events such as landslides and tree throw in humid forests on steep terrain. Gabet and Mudd (2010) considered processes of root fracture and tree throw along a one-dimensional

6 model hillslope, suggesting that contrasting increased tree density and decreased soil production with soil depth results in a humped soil production function. On the other hand, Runyan and D’Odorico(2014) showed that interaction between vegetation density, soil depth, and landslide frequency in a lumped model introduces a minimum into the erosion rate as a function of soil depth. A minimum erosion rate may also lead to soil depth bistability, even when the soil production function is constant or monotonic. However, neither study considered simultaneously the role of vegetation in increasing soil production and enhancing protection from erosion, and the important role of vegetation in these processes makes it logical to examine this case. This study follows this line of work by analyzing the role of plant-soil feedbacks on soil depth bistability in upland agricultural landscapes with moderate slopes, where overland ﬂow and ﬂuvial processes dominate soil dynamics. It is not intended as a predictor of soil depths or rates of production and erosion but rather as a means to explore relationships between and generate hypotheses about the feedbacks between soil formation and vegetation in such environments. A minimal model of plant-soil dynamics is developed and analyzed, demonstrating that positive plant-soil feedback underlies the humped soil production function and, therefore, one condition leading to soil depth bistability. The soil and vegetation properties that control transitions between soil-mantled, exposed bedrock, and bistable regions are derived. Finally, we characterize the impacts of vegetation loss associated with agriculture on transitions between fertile and degraded plant-soil states and relate agricultural pressure to soil resilience and catastrophic soil loss.

2.2 Coupled model of soil and vegetation dynamics

The use of dynamical systems theory to study soil formation was ﬁrst suggested by Phillips(1993, 1998), who put forth a framework for a set of coupled nonlinear

7 differential equations, each corresponding to a different soil formation factor. Here we take a similar approach, but with a focus on only the soil and vegetation components. The soil is modeled as an aggregated soil column of depth, h (m), whose time rate of change is the difference between production (P ) and erosion (E),

dh = P (h, b) − E(h, b), (2.1) dt

and vegetation is modeled as the plant biomass density, b (kg m−2), which is balanced by growth (G) and harvest (H),

db = G(h, b) − H(b, t). (2.2) dt

Equations (1) and (2) are coupled through the dependencies of P , E, and G on the states h and b. A precise deﬁnition of the soil depth relevant to the coupling of soil and vegetation production is diﬃcult to obtain, given the discrepancy between root depth and transitions from active soil to saprolite and bedrock. However, here we take the geomorphic perspective and interpret h as the depth of the near surface soil that is mobile and contributes to vegetation stability and productivity (Amundson et al., 2015). Therefore, P represents production of this soil, encompassing all of the preceding bedrock weathering processes and organic matter inputs.

2.2.1 Soil production and erosion

Soil production depends on a wide variety of environmental factors, including the weathering of bedrock, deposition of organic matter, temperature, water availability and chemistry, the presence of vegetation and animals, and abrasive water or wind flows. Climatic and hydrologic factors are typically accounted for by assuming their effects on bedrock weathering decrease exponentially with the distance between the soil-bedrock interface and soil surface (Dietrich et al., 1995; Heimsath et al., 1997, 2000). For example, the amplitude of temperature fluctuations and the frequency of

8 infiltration pulses both decrease with depth from the surface, thereby resulting in reduced weathering (Ahnert, 1988). On the other hand, vegetation typically stimulates physical and chemical weathering. Roots break apart rocks, tree throw displaces rock and soil (Gabet et al., 2003), and stiff-stemmed plants can transmit above-ground stresses below-ground, e.g., by shaking during storms (Hole, 1988). Further, vegetation improves conditions for chemical weathering by enhancing infiltration and moisture storage and acidifying the soil through the addition of organic matter and respired CO2 and nutrient depletion (Brady and Weil, 2010). These effects justify a soil production function that increases linearly with vegetation biomass and that decreases exponentially with soil depth,

P (h, b) = (p0 + pvb) exp (−ksh). (2.3)

In (3), p0 is the abiotic soil production rate in the absence of vegetation and soil, pv describes the sensitivity of soil production to vegetation, and ks represents the rate at which increasing soil depth decouples soil production from surface weathering processes. Note that the vegetation eﬀect also decreases exponentially with h, which may account for decreased root contact with the weathering zone. In addition to its eﬀects on soil production, vegetation decreases erosion by root cohesion and soil armoring, as well as by reducing the intensity of precipitation kinetic energy and wind speed (Wolfe and Nickling, 1993; Gabet et al., 2003; Gyssels et al., 2005; Wilkinson et al., 2009; Brady and Weil, 2010). Castillo et al.(1997) reported that one-time removal of vegetation increased soil loss over a 5-year period by a factor of 2.27 and Montgomery(2007) found that when native vegetation is harvested and landscapes are converted to conventional agricultural uses, erosion rates can increase by 1-2 orders of magnitude. This sensitivity of erosion rates to vegetation cover is frequently modelled using an exponential relation (Lal et al., 1994; Gyssels et al., 2005; Zuazo and Pleguezuelo, 2009), an approach we adopt here. While this approach

9 does not take into account the episodic nature of erosion events, it does capture the macroscopic eﬀect of vegetation on erosion, considered as a long-term average rate integrated over individual erosion events at geomorphic time scales. Further, we assume that the only direct eﬀect of h on erosion is that there must be soil available to supply erosion, a condition modeled with the Heaviside step function, θ(h)—i.e., θ(0) = 0 and θ = 1 for all h > 0 (Bracewell and Bracewell, 2000). The expression for soil- and vegetation-modulated erosion is thus,

E(h, b) = [e0 + e1 exp (−kvb)] · θ(h). (2.4)

where e0 is the rate of erosion that occurs in the presence of full vegetative cover, e1 is the range of erosion rates between the fully vegetated and the unvegetated states, and kv controls the sensitivity of erosion to biomass. The erosion rate in the absence of vegetation (i.e., for b = 0) is e0 + e1 and is controlled by climate and geologic factors. This approach does not account for the impact of bioturbation on erosion rates and limits the applicability of the model to arid or semiarid ecosystems.

2.2.2 Vegetation growth, turnover, and harvest

Vegetation dynamics are assumed to follow a logistic-type model, with linear growth and harvest terms and a quadratic mortality term (Murray, 2002). The logistic vegetation model is expanded by linking the carrying capacity to soil depth (Runyan and D’Odorico, 2014). Soil depth plays an important role in vegetation growth due to the provision of water and nutrient storage and structural stability. Further, it is reasonable that growth saturates with soil depth as rainfall or above-ground competition for light become important limitations to productivity. Vegetation losses occur by natural turnover processes or harvest and are assumed to be independent of soil depth h. The growth of biomass over time is therefore modeled as

2 G(h, b) = r r1 − exp (−kgh)s b − mb , (2.5) 10 where r is the maximum growth rate, kg controls the eﬀect of soil depth on growth, and m is the plant mortality rate. In addition, a simple model of biomass harvest is

H(b, t) = fh(t)b (2.6)

where fh(t), which here represents agricultural pressure, is the fraction of biomass

harvested per unit time. Generally, agricultural pressure fh(t) could be speciﬁed as a function in time or linked dynamically to the ecological states, h or b. The latter case deﬁnes a coupled social-ecological system (e.g., Brander and Taylor(1998)), which may further interesting dynamics. These will be explored in subsequent contributions. With these assumptions, the full coupled dynamical system of h and b is

dh = (p + p b) exp (−k h) − re + e exp (−k b)s · θ(h) (2.7a) dt 0 v s 0 1 v

db = r r1 − exp (−k h)s b − mb2 − f (t)b. (2.7b) dt g h

In Section 3.1, the system is analyzed in the absence of vegetation. In Section 3.2, the full system is examined without harvest to elucidate the role of vegetation on soil stability, and in Section 3.3, the eﬀects of agricultural pressure on plant-soil feedback and soil stability are quantiﬁed with respect to measures of resilience and sustainability.

2.3 Model analysis

2.3.1 The abiotic case

We begin by analyzing soil formation in the absence of vegetation, where soil formation is driven entirely by abiotic processes. Apart from its inherent interest related to extreme landscapes on Earth and planets with no known life (e.g., Mars, Dietrich

11 and Perron(2006)), this case is also useful as a point of comparison to delineate the role of vegetation. The abiotic case is entirely described by Equations (2.7) with b = 0, dh = p exp (−k h) − pe + e q · θ(h), (2.8) dt 0 h 0 1

which has a single stable ﬁxed point,

 −1 −1  − kh ln ρ ρ > 1 h = (2.9)  0 ` ˘ ρ ≤ 1,

where ρ = p0/(e0 + e1) is the ratio of the abiotic production and erosion rates. The ratio ρ integrates the climatic and geologic factors that control soil production and erosion in abiotic landscapes and acts as an external environmental control parameter with inﬂuence on system stability. The ﬁxed point in (2.9) exhibits weathering-limited and transport-limited soil production regimes across a gradient of relative production and erosion in the environment (Figure 1a). In the case where the abiotic production rate is greater than the abiotic erosion rate (i.e., ρ > 1), there is one stable soil depth at a nonzero value.

This is the transport-limited regime where erosion is dictated by e0 + e1. In the case where erosion exceeds production (i.e., ρ ≤ 1), there is one stable soil depth at h = 0, and soil production is weathering-limited. Due to the nature of the Heaviside step function, the soil production rate in this weathering-limited regime is equal to p0, but any soil produced is instantaneously eroded, also at a rate e0 + e1 ≥ p0.

2.3.2 Vegetation inﬂuence on soil stability without harvest

Returning to the full system of equations (2.7), we now consider the coupled plant-soil dynamics in the absence of harvest. Because the typical timescale for plants to reach a stable biomass is on the order of decades (Likens et al., 1978; Brown and Lugo,

12 Table 2.1: Typical Model Parameters Parameter Symbol Units Value −1 Abiotic production rate p0 mm yr 0.05 −1 −1 2 Production response to vegetation pv mm yr kg m 0.4 Vegetation growth rate r yr−1 0.2 Vegetation turnover rate m yr−1 kg−1 m2 0.05 −1 Vegetation growth response to soil depth kg mm 0.2 −1 Production decay with soil depth ks mm 0.8 −1 Erosion rate under full vegetative cover e0 mm yr 0.02 −1 Erosion rate range e1 mm yr 1.0 2 −1 Erosion response to vegetation kv m kg 0.4

1982; Hughes et al., 1999; Silver et al., 2000), while that for soil depth may be on the order of centuries or millenia (Raich et al., 1997; Torn et al., 1997; Brady and Weil, 2010; Richter and Yaalon, 2012), we assume plant biomass to be in instantaneous equilibrium with the corresponding soil conditions. Under this assumption, the plant biomass is found as a function of soil depth by imposing db/dt = 0,

r b(h) = r1 − exp (−k h)s . (2.10) m g

Inserting Equation (10) into Equation (7a) then gives a single evolution equation for the soil depth,

dh n p r o = p + v r1 − exp (−k h)s exp (−k h) dt 0 m g s

(2.11)

k r − e + e exp − v r1 − exp (−k h)s · θ(h). 0 1 m g ˆ ˙ Analysis of Equation (11) shows that biotic systems with plant-soil feedback may contain diﬀerent dynamical regimes as a function of ρ (Figure 2). Similar to the abiotic case, the coupled system has a single nonzero stable state (h > 0, b > 0) when ρ > 1. Soil stability when ρ ≤ 1, however, depends on the shape of the soil production and erosion functions, P [h, b(h)] and E[h, b(h)].

13 The shape of P is related to a parameter that controls the soil depth at maximum soil production (Figure 2a),

k p m σ = s 0 + 1 . (2.12) k + k p r s g ˆ v ˙ When σ ≥ 1, P is maximum at h = 0 and P decreases monotonically with h. In this case, plant-soil feedback is weak and there is a single stable soil depth h = 0 for ρ ≤ 1. On the other hand, when σ < 1, P is maximum at a nonzero soil depth h > 0, indicating strong plant-soil feedback in soil production. A transition from a

(a)

1 (m) ¯ h 0.5

(b)

4 (m) ¯ h 2

0 0 0.5 1 1.5 2 ρ Figure 2.1: Soil depth bifurcation diagram as a function of the control parameter ρ = p0/(e0 + e1): (a) abiotic and (b) biotic cases, corresponding to Equations (8) and (11), respectively. ρ is varied by varying e0, σ = 0.825, and other parameters are listed in Table 1.

14 mono-stable bare state to a bistable regime occurs in erosion-dominated systems at

a critical value ρc(σ), where Equation (11) is tangent to the line dh/dt = 0. ρc(σ) can be evaluated numerically and marks the location of a fold bifurcation (Strogatz, 2001) (Figures 1b and 2b). In the bistable regime, there are two stable soil depths, one zero and one non-zero. Comparison of the abiotic and biotic systems demonstrates that positive plant- soil feedback stabilizes a vegetated, soil-mantled state in what would otherwise be

(a) 0.06 σ > 1 σ = 1 σ 0.05 < 1 )

1 0.04 −

0.03

(mm yr 0.02 P

0.01

0 0 2h(m) 4 6

1.5 (b)

single h = 0 single h > 0

ρc(σ) σ

0.5 bistable

0 0 0.5 1 1.5 2 ρ Figure 2.2: (a) The soil production function P (h, b) for various values of σ, which controls the effect of vegetation on soil production (σ was varied by changing the value of pv). This figure assumes steady state biomass (see section 3.2). (b) Stability regimes for the biotic system (Equation 11) as a function of the control parameters ρ and σ. σ is varied by varying the parameter kg. The solid line corresponds to the parameters in Table 1 and the dashed line corresponds to a case with stronger −1 vegetation influence, pv = 0.8 mm yr .

15 a weathering-limited, exposed bedrock state in the absence of vegetation. In other words, multiple stable soil depths are predicted to occur in erosion-dominated environments with a strong positive plant-soil feedback. This regime is characterized by a soil production function that peaks at an intermediate soil depth. In the bistable regime, external perturbations may cause the system to shift between zero and nonzero soil states.

2.3.3 Soil stability and resilience under land use change

Transitions between soil-mantled and degraded states may be driven by external factors that lead to large and rapid changes in soil depth, such as landslides, or by changes to the vegetation dynamics, such as harvest or drought, which are subse- quently propagated to the soil dynamics. Here the model is used to quantify soil stability in relation to internal dynamical factors (i.e., plant-soil feedback) and dis-

turbances (e.g., harvest), and we examine the critical time τc that the soil-biomass

system can withstand a given harvest rate fh. Harvest is conceptualized as the removal of native vegetation in favor of agriculture, an occurrence which greatly increases erosion rates and thus decreases the stability of the plant-soil system and increases its susceptibility to a catastrophic shift from a vegetated to a bare state,

R h dh which can be quantiﬁed from the potential function V (h, t) = − 0 dt (h, t)dh. Stable and unstable states appear as valleys and peaks in V (h, t), respectively. An external perturbation must increase the potential above a peak (called the potential barrier) to shift the system between two attractors (the valleys). To show the impact of harvest on the soil dynamics, V (h, t) is drawn in Figure 3a for three harvest levels: a zero harvest level, a critical harvest level in which the stable and unstable states have merged to create a half-stable point, and an intense harvest level in which the only stable state is the degraded one. In the absence of harvest, the system is bistable, as indicated by two local minima

16 (the zero and nonzero stable states) and a local maxima (the potential barrier) in the potential function (thick solid line, Figure 3a). Under the critical harvest level (the dashed line in Figure 3a), the stable and unstable states have merged. This results in a half-stable state, which occurs at the point in which the dashed line in Figure 3a is horizontal. At this point, any perturbation to a lower soil depth will result in a collapse to the degraded state. This demonstrates that harvest brings the stable state of the system and the potential barrier closer together, reducing the

2500 (a) No Harvest Critical Harvest Intense Harvest )

1 2000 − yr

2 1500 ) (cm

h 1000 ( V 500

0 0 2 4 6 8 h (m)

0 (b)

-0.5

(m) -1 h ∆ -1.5 No Harvest Critical Harvest Intense Harvest -2 100 101 102 103 104 105 106 t (yr) Figure 2.3: (a) Soil potential function V (h, t) with no harvest (thick line), a critical harvest level in which the stable and unstable states have joined together (dashed line), and intense harvest that pushes the system toward a bare stable state (solid line). Valleys correspond to stable steady states and peaks correspond to unstable steady states. (b) A time series illustrating the change in soil depth (∆h) for zero, critical, and intense harvest regimes. Erosion rates were chosen within the ranges reported by Montgomery(2007).

17 perturbation magnitude required to shift the system to the degraded state and, thus, reducing the system resilience (Walker et al., 2004). Further increases in the harvest rate shift the system to a monostable regime, driving the system toward a degraded state (the thin solid line in Figure 3a). Soil-depth time-series for no harvest, critical harvest, and intense harvest are shown in Figure 3b.

Harvest-induced transitions from soil-mantled to degraded states depend on fh

as well as the time the system is under harvest. We calculate the critical time τc as the time required for the system to cross the potential barrier and escape the basin of attraction surrounding the soil-mantled state for a given fh. After a period

τc, an irreversible collapse to h = 0 is initiated; however, if harvest is discontinued prior to τc (i.e., before the system has crossed the original potential barrier of the zero-harvest system), the soil-mantled stable soil state will eventually return. The sustainability of a given harvest strategy can then be deﬁned by an intensity fh over a time τc (Figure 4).

1000 6 m 800 3 m 1 m

600 Unsustainable (yr) c τ 400

Sustainable 200

0 0 0.1 0.2 0.3 0.4 fh Figure 2.4: A description of resilience for the coupled plant-soil system. Each line separates sustainable (left side) and unsustainable (right side) harvest pressures, for initial stable soil depths of 6, 3, and 1 meters.

18 2.4 Discussion and Conclusion

This model and analysis further generalize previous quantitative work on soil depth dynamics, demonstrating that soil bistability emerges from positive plant-soil feedback that is destabilized by anthropogenic vegetation removal. Many studies hy- pothesized a humped soil production function (Carson and Kirkby, 1972; Dietrich et al., 1995; D’Odorico, 2000; Furbish and Fagherazzi, 2001; Humphreys and Wilkin- son, 2007; Heimsath et al., 2009) and D’Odorico(2000) showed that the humped soil production function is necessary to produce bistability when the erosion ﬂux is assumed constant. We found that the humped soil production function, and therefore bistability, emerges from positive plant-soil feedback when the vegetation dynamics are explicitly coupled to the soil production and erosion functions. Further, the coupled plant-soil model deﬁnes the conditions amenable to bistability: erosion- dominated environments where vegetation is a strong control on soil production (i.e.,

ρc(σ) < ρ < 1). The occurrence of bistability emphasizes the role of plants as “obligate” ecosystem engineers in these environments (Cuddington et al., 2009). That is, plant biomass and associated positive plant-soil feedbacks through increased production and decreased erosion are necessary to provide a viable soil environment. Because the bare steady- state is also stable in the bistable regime, an external perturbation to the soil depth that raises it beyond the potential barrier is required to initiate stabilizing plant- soil interactions over bare, unproductive sites. Transitions from exposed bedrock to a soil-mantled state may be generated by a rapid input of soil (e.g., landslide or tree throw) or through weathering and soil production processes that dynamically link the two states over time-scales much slower than, and therefore external to, the system modeled here (Kuehn, 2015) (e.g., bedrock weathering (Aghamiri and Schwartzman, 2002; Heimsath et al., 2009)). In contrast, plants are non-obligate

19 engineers in monostable, production-dominated regimes where soil stability is independent of plant activity, although soil depth (and thus stability) increases when vegetation is present in these environments. The proposed interpretation of the soil production function and its relationship to biologic activity suggest that increasing the intensity of agricultural pressure or altering other factors which control this relationship could strongly impact geomorphologic processes in agroecosystems ex- hibiting bistable plant-soil behavior. As large portions of the earth’s surface are affected in some way by agricultural activity, such impacts on the interplay between soil production, erosion, and vegetation have the potential to affect geomorphic processes on a global scale. Soil bistability generated by positive plant-soil feedback suggests that changes to the vegetation dynamics can directly cause transitions to degraded states or reduce system resilience to other external perturbations. As shown in Figure 3a, vegetation removal weakens the plant-soil feedback, decreasing the distance between the potential barrier and the stable state (a measure of resilience (Walker et al., 2004)) and, therefore, the magnitude of external perturbation required for irreversible soil degradation. As the harvest rate increases further, the plant-soil feedback effect on soil stability disappears, the system is no longer resilient, and the system contains a single, degraded soil state. Therefore, soil degradation may result from rapid soil loss by external processes (e.g., landslide or drought) or from destabilization of the internal dynamics through decreased vegetation biomass. The interplay between soil production, erosion, and vegetation has a vital role in landscape development and stability. Results from the model studied here improve our understanding of the feedbacks in eco-geomorphic processes that control the evolution of soil depth in coupled plant-soil systems. Soil bistability was explained by simultaneous soil and vegetation control of the soil production function shape, leading to the conclusion that the risk of catastrophic soil loss is greatest in highly

20 erosive environments with a strong feedback between vegetation and soil production. Further, in addition to external perturbations, state transitions in the bistable system were found to be induced by changes to the internal vegetation dynamics as characterized by harvest intensity and duration. Future work may explore the explicit roles of climate, soil moisture, and biogeochemical cycles on the plant-soil processes which have been studied here. Presently, their impact is embedded implicitly in the parameters σ and ρ, but these parameters could be more directly tied to the physical processes which underly them. Additionally, spatial and stochastic components could be incorporated to account for the eﬀect of spatial interactions and the episodic nature of erosive events.

21 3

Sizing a rainwater harvesting cistern by minimizing costs

This chapter is adapted from Pelak, N. and Porporato, A. (2016), Sizing a rainwater harvesting cistern by minimizing costs, Journal of Hydrology, 541, 13401347.

3.1 Introduction

In a period of rapidly rising populations and climate uncertainty, rainwater harvesting (RWH) is seen as an increasingly attractive option to reduce the pressure on diminishing water supplies in many regions of the world (Pandey et al., 2003). Its potential for runoff reduction also has benefits even in areas where water is relatively abundant (Steffen et al., 2013; Sample and Liu, 2014; Walsh et al., 2014; Wang and Zimmerman, 2015). When used as an alternative water supply, captured rainwater may be used in nonpotable applications such as agriculture, car washing, and toilet flushing. Less commonly due to the cost of treatment, it may be used as a potable water source. In this study we focus on designing RWH systems for domestic, nonpotable uses with a relatively constant demand, thereby avoiding complications such as the dependence of agricultural demand on soil moisture levels and the additional

22 Table 3.1: A guide to the symbols and abbreviations used in this study. Symbol Units Parameter Name A m2 Roof Area c - Normalized Storage Volume Dc - Cistern Demand Index G $ Total Cost Gd $ Distributed Costs Gf $ Fixed Costs h m3d−1 Constant Water Demand Rate H m3d−1 Water Demand Rate (from Cistern) 3 −1 Hm m d Water Demand Rate (from Municipal Source) p0 - Probability that c = 0 m3 Q d Overflow Rate $ q m3 Storage Capacity Unit Cost m R d Rainfall Rate RWH - Rainwater Harvesting $ r m3 Water Unit Cost SH - Smart Home T d Lifetime of Cistern V m3 Cistern Volume α mm Mean Rainfall Event Depth φ - Runoff Coefficient γ - Dimensionless Cistern Volume λ d−1 Mean Rainfall Event Frequency Θ - Heaviside Step Function cost and complexity of incorporating a water treatment system. The volume of the storage tank or cistern is perhaps the most important aspect in the design of a RWH system, but the methods to determine this volume have various drawbacks. Basic methods which could be applied by anyone exist (Ball, 2001; Krishna et al., 2005; Jones and Hunt, 2008) and may be as simple as sizing the cistern based on a fraction of the total average annual rainfall, but this approach ignores much of the information that could be obtained to better inform the design. Numerous methods of a more technical nature have been employed to study this question, focusing to a large degree on design criteria related to reliability, which is generally defined as the percentage of time that the RWH system was able to meet

23 the desired demand (e.g., Basinger et al.(2010)). Reliability is naturally of concern in the design of any RWH system, and is likely to be the dominant one in situations where it is the primary or the only water source. However, other water sources (such as municipal supplies or water trucks) may be available, each with an associated cost. Considering these sources in the design process allows for a more complete picture of the utility and benefits of the system. Economic considerations must also be examined, as the appeal of RWH will be limited if it is not cost-effective. Government regulations encouraging or mandating such systems also provide a political dimension which could increase their attractiveness. Many previous studies (e.g. Lee et al.(2000); Ghisi et al.(2007); Cowden et al. (2008); Basinger et al.(2010); Jones and Hunt(2010); Khastagir and Jayasuriya (2010); Steffen et al.(2013); Fernandes et al.(2015)) utilized numerical approaches to study optimal cistern design. Some, including Lee et al.(2000), Ghisi et al. (2007), Khastagir and Jayasuriya(2010), and Steffen et al.(2013), developed regres- sion equations which are simple to use, but dependent on numerical data and thus highly specific to certain areas. These investigations also often had applications to particular regions or cities (Lee et al., 2000; Ghisi et al., 2007; Guo and Baetz, 2007; Cowden et al., 2008; Abdulla and Al-Shareef, 2009; Basinger et al., 2010; Jones and Hunt, 2010; Khastagir and Jayasuriya, 2010; Palla et al., 2012; Assayed et al., 2013; Mehrabadi et al., 2013; Steffen et al., 2013; Hanson and Vogel, 2014; Wang and Zim- merman, 2015; Rostad et al., 2016) and typically used available local rainfall records to optimize their designs. The drawbacks of relying too heavily on specific rainfall observations are that rainfall records are insufficient to capture the true variability, and moreover the results may not hold in other regions. Climate change is likely to increase the variability in rainfall patterns and amounts in the future (Easterling et al., 2000; Feng et al., 2013), adding further uncertainty and necessitating methods which can account for such changes, such as synthetic analytical representations of

24 rainfall. Here we develop an analytical expression for the optimal cistern size of a RWH system based on economic considerations. We use a parametric description of rainfall, in which rainfall occurs as a marked Poisson process (e.g., Rodr´ıguez-Iturbe and Porporato, 2004b). This simple representation of rainfall is parsimonious, mathemat- ically tractable, and may be easily adapted to future climate scenarios with different rainfall patterns by adjusting the climate parameters. In areas where rainfall cannot be modeled exponentially, seasonal variation in rainfall could be incorporated by employing a time-dependent Poisson process, a route which will be explored in future contributions. A similar approach, which also took into account the length of individual storm events, was used by Guo and Baetz(2007) to derive an analytic expression for optimal cistern size. However, as with most of the studies discussed previously, it focused on optimizing the cistern volume to obtain a desired reliability. Here we provide an expression that optimizes the cistern volume by minimizing the total cost, and make an application of the results to the Duke Smart Home in Durham, NC. Minimizing total cost to find the optimal cistern storage volume has been used by several other authors. Okoye et al.(2015) used linear programming methods to minimize an objective cost function. Other studies used nonlinear programming methods to explore the tradeoffs between maximizing rainwater capture and reducing or delaying runoff (Sample and Liu, 2014), to design water networks for RWH in residential developments (Bocanegra-Mart´ınezet al., 2014) and for water capture-reuse systems in a housing complex (Garc´ıa-Montoya et al., 2015). These latter two studies had the dual goals of minimizing freshwater consumption and total cost. Walsh et al.(2014) examined the interesting case of a RWH system which was operated with the primary goal of runoff reduction. Liaw and Tsai(2004) developed curves by which an optimal cistern size could be selected which met a given reliability level

25 and also minimized a cost function. Campisano and Modica(2012) used historical rainfall data to simulate the water balance in the cistern, developed empirical equations to describe the water savings efficiency in terms of dimensionless parameters, and combined these with a cost function which was then minimized. In contrast, our analysis is not dependent on historical rainfall data or simulations and is backed by mathematical models which have been widely applied to hydrological problems (see Section 3.2). We also note that DeBusk et al.(2013) examined four existing RWH systems in North Carolina and found that they were not cost-effective over their expected lifetimes, but it is not clear that these systems were operated or sized in such a way as to minimize the owner’s water-related costs. Our goal here is to assess the conditions of optimal cistern size as a function of rainfall regime as well as fixed and distributed costs under ideal operating conditions to provide a benchmark for further study and analysis of the impact of pricing policies and sustainability incentives on water consumption. In the following sections, we will first describe the water balance in the cistern and formulate the problem in mathematical terms. Then, general solutions to related problems obtained by other researchers will be shown, along with the partitioning of the water balance. Finally, we will develop a cost function for the cistern over its lifetime and find an exact expression for the cistern volume which minimizes the total cost.

3.2 Stochastic Water Balance in Cisterns

The mass balance of water in a cistern which is ﬁlled by means of rooftop rainwater harvesting is as follows

dc V = φAR(t; α, λ) − H(c) − Q(c), (3.1) dt 26 where V [m3] is the cistern volume, c is the normalized cistern storage volume ranging from zero to one, φ is a runoﬀ coeﬃcient, A [m2] is the area of the roof capable of

m collecting water, R [ d ] is the stochastic rainfall rate with mean event depth α [m]

1 m3 and mean inter-arrival time λ [d], H [ d ] is the rate of demand of water from the

m3 cistern, and Q [ d ] is the loss rate due to overflow. Here we assume that the cistern is covered, so that losses to evapotranspiration can be neglected, and that there are no losses to leakage. In using a runoff coefficient, we are essentially adapting the rational method (Farreny et al., 2011). The coefficient φ depends on factors such as the roof material and slope as well as the efficiency of the water collection system. A review by Farreny et al.(2011) found that they can range from 0.7 to 0.95, but the value may be much lower for green roofs (Fassman-Beck et al., 2015). The latter two references contain tables which can assist in the selection of the proper runoff coefficient for different roof properties. As it does not qualitatively affect the results, we do not include the diversion of the first flush of rainfall events from the cistern, which is found in many RWH systems. However, first flush diversion can be incorporated using the censoring approach found in Rodr´ıguez-Iturbe and Porporato (2004b). We use the following form for the loss rate

0 if c = 0, H(c) = (3.2) h if c > 0,

where h is the constant rate of water demand. When c = 0, water is assumed to be provided by another source, such as a municipal supply. The rate of losses due to overﬂow take the form

Q(c) = (φRA − V (1 − c)) · Θ(φRA − V (1 − c)), (3.3)

where Θ is the Heaviside step function (Abramowitz and Stegun, 1965), deﬁned here such that Θ (and thus Q) is equal to 0 unless a rainfall event causes c to be greater

27 than 1. We assume that the RWH system is equipped with an adequately sized overflow pipe which allows the overflow to enter into the property’s external storm sewer system. Sizing guidelines for such a pipe can be found in local plumbing and building codes, and typically require the pipe to be sized to handle a design storm rate of a particular duration and return period. In North Carolina, for example, if the overflow pipe must be sized for the 100 year, 1-hour storm (Jones and Hunt, 2008). Equation (3.1) is similar to the minimalistic model for soil moisture balance developed by Milly(1993) and Porporato et al.(2004) (see also Porporato et al. (2002) and Rodr´ıguez-Iturbe and Porporato(2004b)). The underlying mathematical process is the Takacs process with an upper bound (Cox and Miller, 1977). The main results are presented below, but more details can be found in the Appendix.

3.3 Non-dimensional parameters

V h The system can be described by four variables (α, λ, φA , and V ) which contain two base units (L and T). Applying the Buckingham-Pi Theorem (Logan, 2013) results in two dimensionless groups, for which we choose

V γ = , (3.4) φAα

and h D = , (3.5) c φAhRi

where hRi = λα is the mean rainfall rate. The brackets h·i here and in the following analysis denote the ensemble average of a random variable.

We deﬁne γ as the dimensionless cistern volume and Dc as the cistern demand index. γ characterizes the cistern volume, V , in relation to the volume which enters

28 it in an average event, φAα. Dc is the ratio of the maximum water demand rate, h, to the rate at which rainfall can be captured, φAhRi. Non-dimensional parameters were also used by Palla et al.(2011) and Campisano and Modica(2012) to characterize RWH systems. The two parameters chosen in the former study were the ratio of the cistern storage volume to the annual water supply (storage fraction) and the ratio of the annual water demand to the annual water supply (demand fraction), while the latter study had a modiﬁed storage fraction. The parameter γ can be transformed to the storage fraction by dividing it by the number of rainfall events per year, λT , where T =365 days. The demand fraction is similar to the cistern demand index, except that the former considers total losses to total supply over a year while the latter considers the corresponding rates. While both formulations provide similar information, we use (3.4) and (3.5) as they are not tied to a particular timeframe.

3.3.1 Steady-State Solutions

Because the losses are constant as long as c 6= 0, there is an atom of probability, p0, that the cistern will be empty (c = 0). The steady-state solutions for the probability density function (PDF) and the atom of probability were obtained (Milly, 1993; Rodr´ıguez-Iturbe and Porporato, 2004b) (see Appendix) as

γ γp 1 −1qc p(c) = p0e Dc , (3.6) Dc

1 − Dc p0 = (3.7) γ( 1 −1) e Dc − Dc

Figure 3.1 shows the analytical PDF (right axes) and atom solutions (left axes), along with sample time series for each case. For the special case of Dc = 1, the PDF and atom are given by γ p(c) = , (3.8) 1 + γ 29 1 p = . (3.9) 0 1 + γ

We note that the atom of probability in Equations (3.7) and (3.9) are dependent on V through γ.

3.4 Harvested Water Partitioning

The amount of water captured, φAhRi, is either consumed by the demand or lost to overﬂow. The long-term water balance in the cistern can therefore be partitioned as follows: φAhRi = hHi + hQi, (3.10) or hHi hQi 1 = + , (3.11) φAhRi φAhRi where hHi is the expected loss rate due to water demand that is supplied by the cistern and hQi is the expected loss rate due to overﬂow. The expected loss rate hHi is equal to the constant loss rate h times the fraction of time that the cistern is full

hHi = h(1 − p0), (3.12)

where 1 − p0 is the fraction of time that the cistern is not empty. We can express the non-dimensionalized partitioning of harvested water by dividing Equation (3.12) by the expected rate at which rainfall can be captured

hHi = D (1 − p ). (3.13) φAhRi c 0

In this expression, Dc is analogous to the Budyko dryness index (Budyko, 1974), which gives the ratio of the mean potential evapotranspiration to the mean rainfall rate. The comparison to the Budyko dryness index allows us to assess the eﬃciency of a particular RWH system conﬁguration. For any location where climatic data and

30 0.012

1 9 0.01 8

c 0.5 7 0.008 6 0 0 5 0.006 p

p 0 20 40 60 80 100 t [d] 4 0.004 3

2 0.002 1

0 0 0 0.2 0.4 0.6 0.8 1 c

0.16

0.6 9 0.14 8 0.4 0.12 c 7 0.2 0.1 6 0 0 5 0.08 p

p 0 20 40 60 80 100 t [d] 4 0.06 3 0.04 2 0.02 1

0 0 0 0.2 0.4 0.6 0.8 1 c Figure 3.1: Two PDFs with diﬀerent cistern storage volumes and a roof with A=267 m2, the approximate roof area of the Duke Smart Home in Durham, NC, USA (top plot), and a roof with half this area, A=133.5 m2 (bottom plot), in order to show diﬀerent possible shapes of the PDF. Gray lines correspond to a cistern volume of 7.95 m3 (the volume of the indoor cisterns at the Smart Home) and black lines correspond to a cistern volume of 19.2 m3 (the combined volume of the indoor and outdoor cisterns at the Smart Home). Other parameters are as in Table 3.2. The inset plots contain sample time series for each combination of roof area and cistern volume.

31 1.2 γ = 4.54 γ = 1.51 1 γ = 0.454

0.8 i i R D h 0.6 H A h φ

0.4

0.2

0 0 0.5 1 1.5 2 Dc Figure 3.2: The ratio of water demand to rainfall rate as a function of the cistern demand index, Dc, for diﬀerent values of γ. γ ≈ 4.54 for the Duke Smart Home, and 1 1 other values which are 3 and 10 of the SH value were chosen to illustrate diﬀerent possible shapes of the curve.

information about water use rates are available, we can quantify the potential for RWH to reduce the demand on local water sources. In the cistern partitioning curve,

when Dc < 1 it is not the available energy which prevents all of the water from being consumed by the demand but rather the available storage capacity. When Dc > 1, however, the available water is the limit as in the original Budyko diagram. The expected use rate of water in the cistern comes closer to the maximum possible rate for larger values of the dimensionless cistern volume γ, as can be seen in Figure 3.2, which plots the ratio of the water demand over the rainfall rate vs. the cistern demand index.

3.5 Optimal Cistern Size

Here we derive an expression for determining the optimal size of a cistern based on cost minimization. We consider a typical building, for which the total cost of a RWH system, G[$], is comprised of two parts: the initial or ﬁxed cost of building and maintaining the cistern, Gf [$], and the distributed cost, Gd[$], of supplying the water

32 Table 3.2: Parameters used in figures for the Duke Smart Home (SH) and Durham, NC, USA rainfall data. The cistern volume was obtained from the SH website (http://smarthome.duke.edu/) and the roof area from SH engineering plans. Water demand is for toilet flushing only, and is based on 6 L/flush for 10 residents flushing 5 times per day (http://www.home-water-works.org/indoor-use/toilets). Rainfall data for Durham, NC was obtained from Daly and Porporato(2006). Parameter Symbol Units Value Roof Area A m2 267 Water Demand h m3d−1 0.3 Estimated Lifetime of Cistern T d 10950 Cistern Volume (Indoor Use) V m3 7.95 Mean Rainfall Event Depth α mm 8.2 Runoff Coefficient φ - 0.8 Mean Rainfall Event Frequency λ d−1 0.31 from an external source when there is an insufficient volume of water in the cistern to meet the demand. Below we will examine particular forms for each, and apply these forms to the Duke Smart Home in Durham, NC, for which the parameters can be found in Table 3.2.

3.5.1 Fixed Costs

We can reasonably assume that the cost of building and maintaining a cistern will increase with its volume V . We do not explicitly consider maintenance costs, but by assuming either that they are independent of the cistern size or that they scale with cistern volume in the same way as do the building costs, they can easily be incorporated. Figure 3.3 plots commercial cistern prices against storage capacity for above-ground plastic cisterns and suggests that a linear relationship is reasonable, although it is possible that an economy-of-scale eﬀect produces a nonlinear relationship for larger cistern volumes. A similar relationship was used by Campisano and Modica(2012) and can also be found in Table 3 of Fernandes et al.(2015). The ﬁxed cost function takes the following form

Gf = qV · Θ(V ), (3.14)

33 9000 Price Data 8000 q= $157.4 / m 3 7000

6000

5000

4000 Price, $

3000

2000

1000

0 0 10 20 30 40 50 Cistern Volume, m3 Figure 3.3: Cost [$] vs. cistern volume [m3] for commercially available cisterns. Data are for a variety of manufacturers and were taken from www.rainharvest.com. We note that while there is a roughly linear price trend across all types of cisterns, it is most pronounced within individual cistern types, e.g. above ground plastic, below-ground plastic, steel, etc. The data in the ﬁgure are for above-ground plastic cisterns.

$ where q[ m3 ] is the unit cost per storage capacity and as before Θ is the Heaviside step function (Abramowitz and Stegun, 1965), deﬁned here such that Θ(0) = 0, in contrast to the step function found in Equation (3.3).

3.5.2 Distributed Costs

$ The distributed cost depends on r[ m3 ], the unit price of water, which varies between locations and is a function of the expected rate at which water must be supplied by

the municipal system, hHmi = hp0. As an example, the monthly price per unit of water in Durham, NC increases in steps (Figure 3.4), as is typical of municipalities in the US. Under such a stepped monthly pricing scheme, the unit cost of water can be written as n=k 1 X r = r (v − v ), (3.15) 30hp n n n−1 0 n=1

34 2.2

1.8

1.6 ] 3 1.4 r [$/m 1.2

0.8

0.6 0 10 20 30 40 50 60 Monthly Water Use [m 3] Figure 3.4: The unit monthly cost per m3 of municipally-supplied water in Durham, NC, USA (www.durhamnc.gov). The price increases occur up to 15 ccf (43 m3) and are thereafter constant, although the plot is only shown up to 21 ccf (60 m3). Once a new increment is reached, the unit price applies only to that increment and not to the entire volume.

where the rn are the monthly unit prices for the nth pricing tier, the vn are the monthly volumes at which the nth pricing tier ends, and the monthly demand falls

into the kth pricing tier, such that vk = 30hp0. We also deﬁne T as the lifetime of the RWH system in days, and assume that the cistern begins at steady state conditions, so that the transient period to reach steady state can be neglected. The average

distributed cost is then the water unit cost multiplied by hHmiT , the total expected municipal demand over the lifetime of the cistern

Gd = rhT p0, (3.16)

where p0 is given by Equations (3.7) and (3.9).

3.5.3 Solution for the Optimal Cistern Size

The total cost can be found as the sum of the ﬁxed and distributed costs, G =

Gf +Gd. For the cost functions in Equations (3.14) and (3.16), an analytical solution for the cistern volume V is obtained by minimizing the total cost. The cost function

35 G is

G = qV · Θ(V ) + rhT p0. (3.17)

Figure 3.5 plots the cost functions in Equations (3.14), (3.16), and (3.17) as a function of the cistern volume V . The ﬁxed costs (3.14) always increase with V while the distributed costs (3.16) always decrease (due to the dependence of p0 on V ), yielding a minimum of the cost function G. Such a minimum can clearly be seen in Figure 3.5. We can solve for the optimal cistern volume V ∗ by taking the derivative with respect to V and solving for V . The part of this solution containing positive values is expressed below as

∗ αφAh n h hrT h αφAλ−h V = αφAλ−h ln αφAλ − 2q p1 − αφAλ qp αφAλ q o (3.18) αφAλ−h 2 + 2α2φ2A2λq 4αφAhqrT + [rT (αφAλ − h)] , a when V ∗ ≥ 0 and 0 otherwise. We note that the only r-value from Equation (3.15) which in appears in the solution for V ∗ is the one corresponding to the tier under which the monthly demand falls, so we have termed this value as simply ‘r’ with no subscript in Equation (3.18) and below in Equation (3.19). We also note that the volume given by Equation (3.18) is negative for rainfall event inter-arrival frequencies

q below a critical value λc = rT . For such frequencies the incremental cost per unit dG volume of the cistern (i.e. the derivative dV ) does not yield a minimum of G for any positive values of V , and therefore a RWH system is not ﬁnancially viable over the time period T and the optimal volume is then equal to 0. For the special case of

Dc = 1 (see Equation (3.9)),

αφArhT V ∗ = − αφA. (3.19) d q

It is interesting to observe how the optimal cistern size changes according to the two climate parameters α and λ. Figure 3.6 shows the variation in optimal cistern size as

36 2500 ﬁxed costs Gf distributed costs Gd total costs G 2000

1500

Cost [$] 1000

500

0 0 1 2 3 4 5 6 7 8 Cistern Volume [m3] Figure 3.5: The ﬁxed cistern costs (dashed line) increase linearly with increasing capacity, while the distributed costs (dotted line) of tapping into the municipal supply decrease. The minimum of the total cost (solid line) is taken as the optimal cistern volume. a function of α and λ (keeping the roof area and household demand from Table 3.2). For frequent but relatively small events, a larger volume is optimal (the upper left), as even though the events are small they are numerous enough to keep a larger cistern full and thereby oﬀset its cost. A larger cistern volume is also optimal for infrequent but large events (the lower right), as here the additional volume allows for the system to take advantage of the large size of the events and store the resulting rainfall during the longer dry periods. The inset of Figure 3.6 (with λ=0.31 d−1 and α=8.2 mm) shows cross-sections of this plot at the α and λ values corresponding to Durham, NC. A ridge of relatively large optimal cistern volumes peaks approximately where

Dc=1 (the solid white line in Figure 3.6). The optimal cistern values are a maximum near this line because there the system is neither on the water-limited RHS of the partitioning diagram (there is exactly enough supply to meet the demand) nor is it in the storage-limited region on the LHS.

37 λ [d-1] 4 1 B 0 0.5 1 4 3.5 A-A

0.8 ] 3 3 2 2.5 ] V* [m

-1 0.6 B-B

[d 2

λ 0 0 1 2 3 0.4 α [cm] 1.5 A A 1 0.2 D = 1 c 0.5 B 0 0 0 0.5 1 1.5 2 2.5 3 α [cm] Figure 3.6: The optimal cistern size as a function of the climate parameters α and λ. The cross sections of the optimal cistern volume solution, A-A and B-B, are taken at constant λ and α values corresponding to Durham, NC. The peak cistern values for a particular α or λ value occur near the solid white line, which denotes the section of the parameter space where the cistern demand index Dc=1.

3.5.4 Expected ﬁnancial gain and maximum loss

As a consequence of the steady state condition, if an extreme climate event such as a drought were to occur during the time period T the total cost would not be as predicted in Equation (3.17). To quantify the financial impact of such an event, both the expected financial benefit from building the cistern and the maximum amount of money that can be lost on the investment can be determined. The expected financial benefit may be quantified as the cost of the cistern subtracted from the cost of obtaining the expected water demand from municipal sources

∗ ∗ hBi = rhHiT − G(V ) = rhT (1 − p0) − qV . (3.20)

The maximum loss, Lmax, which corresponds to the extreme case of no rainfall col- lected at all during the lifetime of the cistern, is calculated as the cost of building the cistern added to the cost of obtaining the total water demand from municipal sources

∗ Lmax = qV + rhT. (3.21) 38 The values hBi and Lmax could help decision-makers by providing information about the ﬁnancial risk and potential beneﬁt of investing in a RWH system.

3.5.5 Optimal size including runoﬀ reduction

It should also be noted that in addition to its ability to reduce reliance on municipal supplies, rainwater harvesting may provide another important benefit, namely the reduction of runoff. This is true from an environmental as well as a cost standpoint, although in this analysis, we consider water supply to be the primary benefit of RWH, with reduced runoff as a potential supplemental benefit. Many municipalities (including Durham, NC) assess a storm water fee which may be (partially or wholly) based on the amount of impervious surface on a property (Wang and Zimmerman, 2015). A RWH system effectively removes a portion of the roof area from the total impervious area of a property, and as the roof area may comprise a sizable fraction of the total land area for some properties (especially in urban settings), it could provide a financial benefit to homeowners via a reduction in storm water fees. This assumes, of course, that municipalities would be willing to consider the installation of a RWH system when assessing such fees. From the partitioning in Equation (3.10), an expression for the ratio of the rate of expected losses from overflow to the rate of the amount of rainfall captured can also be obtained

hQi = 1 − D (1 − p ). (3.22) φAhRi c 0

In the absence of a RWH system, all of the rainwater falling on the impervious roof surface, occurring at an average rate φAhRi, becomes runoﬀ. As such, Equation (3.22) can also be interpreted as the ratio of the runoﬀ rates after and before the installation of a RWH system. If this fraction is multiplied by the roof area A, an adjusted roof area is obtained, which could then be used in place of the true roof area when measuring the impervious surface area of a property.

39 We note that while we have provided an expression for V ∗ which is useful for determining an optimal cistern volume, local governing bodies may have additional requirements and guidelines which must be complied with in the design of a RWH system. For example, the North Carolina Department of Environmental Quality re- quires property owners who wish to earn runoff reduction credits to design according to specific requirements, including that the system can handle a 10 year, 24-hour storm event, among others (NCDENR, 2014). Other types of regulations, such as local plumbing codes, must also be followed. Properties are normally divided into different billing tiers based on the amount of impervious area on the property, and then charged a corresponding stormwater fee. This component of the cost function has not been explicitly included in the preceding cost analysis for two reasons. First, it is not typical for municipalities to offer a reduction in the stormwater fee in exchange for installing a RWH system, though it is certainly possible. Second, the tiered pricing structure does not qualitatively alter the previous results. As noted by Steffen et al.(2013), Sample and Liu(2014), and Walsh et al.(2014), an additional challenge is that RWH systems must be operated differently depending on whether the goal of the system is providing a water supply or storm water management (or some combination of these), requiring an operator to decide how to balance the two. Nevertheless, the ability of a RWH system to reduce runoff is worth considering for both its environmental and cost-saving benefits.

3.6 Conclusions

An analytically tractable means of representing the water balance in the storage tank of a RWH system has been developed and used to ﬁnd the expected volume of additional water which would need to be supplied from a municipal source. When combined with the unit cost of water and an estimate of the cost of building a cistern, an expression for the total cost of the RWH system as a function of the

40 cistern tank volume was obtained. A solution was found for this volume, allowing for a direct calculation of the desired storage volume in terms of the roof area, water demand rate, rainfall parameters, and cistern and unit water costs, making clear the role each parameter plays in the optimization. The mathematical models upon which this solution is based have been widely applied to hydrological problems (e.g. in Rodr´ıguez-Iturbe and Porporato(2004b)), and we have included a cost function so that optimization is based on more than reliability alone. This is a necessary consideration, as most property owners interested in installing a RWH system will be be concerned with its cost-saving potential as well as its reliability. By partitioning the water into that which is consumed by the demand and that which is lost to overflow and comparing it to the prior runoff, the potential runoff reduction provided by the system can be found. As many municipalities in the US assess a stormwater fee, this represents another potential component of the cost function that could be incorporated, if municipalities are willing to incentivize the installation of a RWH system. The link to climate parameters also allows for the expansion of this work to compare the performance of RWH systems across different regions. Finally, including the various costs associated with a RWH system as part of this optimization could assist in the reduction of municipal water use and/or stormwater runoff. Government planners and water managers could use this information to provide incentives for owners to install RWH systems via subsidies or changes in the price structure of municipal water.

41 4

A dynamical systems framework for crop models: toward optimal fertilization and irrigation strategies under climatic variability

This chapter is adapted from Pelak, N., Revelli, R., and Porporato, A. (2017), A dynamical systems framework for crop models: Toward optimal fertilization and irrigation strategies under climatic variability, Ecological Modelling, 365, 8092.

4.1 Introduction

As tools to forecast or backcast crop yields, improve management strategies, and better understand the physical processes underlying crop production, crop models are important tools from both a research and an engineering viewpoint (Wallach et al., 2006; Steduto et al., 2009). The model outputs, structure, parameterization, and data assimilation are all active areas of crop modeling research. Because different users have different goals, several types of crop models have been proposed, which can be categorized in a number of ways. One of the most basic distinctions is between dynamic crop models, which are comprised of a set of differential equations,

42 which are then integrated in time to simulate the crop responses of interest at each time point (often daily), and crop response models, which, though they may be built on dynamic models, relate crop responses directly to inputs (Thornley and Johnson, 1990; Wallach et al., 2006). Most crop models have as their main state variables above-ground biomass, leaf area index (LAI), harvestable yield, and water and nitrogen balances, though the choice and precise number of state variables varies (Wallach et al., 2006). Virtually all crop models are process-based, but necessarily involve empirical components, and are of varying levels of complexity, depending on the particular goals of the model and on the availability of input data. Some are specific to certain crops or groups of crops, such as CERES (Ritchie et al., 1998) and AZODYN (Jeuffroy and Recous, 1999), while others are more generic, such as CROPGRO (Boote et al., 1998), CROPSYST (Stöckle et al., 2003), STICS (Brisson et al., 2003), and some focus on particular regions (e.g., INFOCROP (Aggarwal et al., 2006) for tropical regions). Also in the category of generic models, but with a more parsimonious framework, is AquaCrop (Steduto et al., 2009). Despite the abundance of crop models which have dynamical systems at their core, they are not often analyzed as dynamical systems per se–that is, using the wide array of tools and methods provided by dynamical systems theory to understand the mathematical behavior and properties of the models (Strogatz, 2014). There are a number of potential reasons for this, such as the difficulty of applying these methods to complex models and the aims of modelers, which may be focused toward other goals. Although they tend to be considerably more complex and serve different purposes, crop models share many features and describe many of the same processes as do minimal ecohydrological models. The use of such models, which are typically formulated as dynamical systems, has provided many insights into soil moisture dynamics, plant-water interactions, and nutrient cycling (Rodriguez-Iturbe et al., 1999; Porporato et al., 2002, 2003; Rodr´ıguez-Iturbe and Porporato, 2004a). Some features

43 of this type of ecohydrological model, such as the parsimonious representation of processes and stochastic and dynamic coupling between state variables, are well-suited to study the feedbacks, nonlinearities, and effect of random hydroclimatic forcing on agroecosystems (Porporato et al., 2015). Indeed, the underlying assumptions of many dynamic ecohydrological models are better met in agroecosystems than in the natural ecosystems where they are normally applied. Such assumptions include homogenous soil depth and plant spacing, as well as good drainage, which describe well an agricultural field with tillage, uniform crop spacing, and tile drains. Various studies have used a dynamical systems framework to examine grass ecosystems (Thornley and Verberne, 1989; Tilman and Wedin, 1991), grass growth modulated by competition with legumes (Thornley et al., 1995) and grazing (Johnson and Parsons, 1985), forest ecosystems (Thornley and Cannell, 1992), forest ecosystems under harvest (Parolari and Porporato, 2016), soil salinity and sodicity (Mau and Porporato, 2015), and the cycles themselves, including feedbacks and nonlinearities (Porporato et al., 2003; Manzoni et al., 2004; Manzoni and Porporato, 2007). Studying crop models with dynamical systems theory allows for the more ready exploration of many interesting aspects of crop systems, including their stability with respect to parameter change, the feedbacks between water, carbon, and nutrient cycling, the optimal conditions for growth, and the impact of external inputs such as changes in climate patterns and management choices (i.e. fertilization and irrigation). With the goal of taking advantage of the tools of dynamical systems theory, in this work we develop a dynamic crop model which captures the main crop fluxes and responses of interest without being overly complex. The model has three main variables which interact dynamically: the canopy cover, the relative soil moisture, and the soil nitrogen. The differential equations which account for these components are coupled via the crop growth, nitrogen uptake and leaching, and evapotranspiration

44 terms. Biomass and yield, which are not considered to interact dynamically with the other state variables but rather are determined by them, are also included as derived variables of agroecologic interest. The model is used to examine the crop response to water and nutrient availability and varying climatic conditions in order to examine questions of optimal fertilization and irrigation and reduction of nutrient leaching. Several aspects of the model are derived from AquaCrop (Steduto et al., 2009; Raes et al., 2009; Hsiao et al., 2009), which is the existing generic crop model that, in addition to its parsimony, can perhaps most easily be viewed as a dynamical system. It is also physically based, validated for a variety of crops, and widely known. AquaCrop itself is largely based on earlier FAO publications, in particular through its use of crop coefficients (Allen et al., 1998) and in the relation between crop water uptake and yield (Doorenbos and Kassam, 1998). The most notable similarities between the model developed here and AquaCrop are that canopy cover is used rather than the more typical LAI, that evapotranspiration is represented by crop coefficients, and in the dependence of the partitioning of transpiration and evaporation on the canopy cover. Some key differences involve the soil moisture balance (the model developed here makes use of a single vertically averaged soil moisture value rather than a soil column consisting of multiple layers, and it uses the same soil moisture stress thresholds throughout) and the nitrogen balance (a balance of total mineral nitrogen in the soil is used here rather than the empirical fertility coefficient employed in AquaCrop). Here a different viewpoint and set of tools is emphasized for studying dynamic crop models, and we also aim to place crop models in a dynamical systems context and to discuss the application of the associated methods to crop models. We hope that this contribution will be of interest to both the crop modeling community and to researchers in the area of theoretical ecohydrology as a means to explore the response of agroecosystems to uncertain climatic conditions and optimal management

45 strategies.

4.2 Model components

In this section a dynamical system is constructed which describes the interaction of three main components: canopy cover C(t), relative soil moisture S(t), and total nitrogen content in the soil N(t). We also consider two related variables, namely the crop biomass B(t) and the crop yield Y (t) (hereafter we drop the t-dependence of the state variables). The model is interpreted at the daily timescale (no diurnal dynamics are considered) and applied over the course of a single growing season. It can be forced by random rainfall inputs (Rodr´ıguez-Iturbe and Porporato, 2004a), and is assumed to apply to an agricultural ﬁeld which is homogenous in terms of soil composition, climatic forcing, and management.

4.2.1 Canopy cover dynamics

We deﬁne the canopy cover to be the fraction of ground covered by a crop. The beneﬁt of using this alternative to the LAI, which was also employed by AquaCrop (Steduto et al., 2009), is that it combines multiple attributes of the crop canopy into a single, easily measured or estimated variable. The rate of change in canopy cover is modeled as a balance between the increase due to canopy growth and the decrease due to metabolic limitations and senescence, so that

dC = G(C, S, N, t) − M(C, t), (4.1) dt

where G is the canopy growth rate, and M is a term which combines the eﬀects of metabolic limitation and senescence. The growth rate is assumed to be proportional to the rate of nitrogen uptake, U (discussed further in Section 4.2.3), giving

G(C, S, N, t) = rG · U(C, S, N, t), (4.2)

46 where rG is the canopy cover increase per amount of nitrogen taken up (the value for this and other crop growth parameters can be found in Table 4.1). The combined metabolic limitation and mortality/senescence term is

2 M(C, t) = rM + γ(t − tsen) · Θ(t − tsen) · C , (4.3) ´ ¯ where the ﬁrst term, rM , is a constant metabolic limitation term, and the next term is a time-dependent mortality and senescence term. For the latter, a linear function is used which increases with a slope of γ after the senescence onset time, tsen, at which point the Heaviside step function, Θ, causes the senescence term to begin to aﬀect the equation. This form recalls somewhat the Gompertz-Makeham law (Makeham, 1860), which includes an age-independent mortality term and an age-dependent mortality term, although here the constant term is conceptualized as a metabolic limitation term and the time-dependent term as a senescence term.

For unstressed conditions (suﬃciently high S and N) prior to tsen, Equation (6.19) is the logistic growth equation (Murray, 2002), and it includes the approximately exponential growth of C in the initial growth stage, the slowing of growth as a limit is reached, and the negligible growth rate near the carrying capacity. This compares well with the data for canopy cover presented by Hsiao et al.(2009) (see Section 4.3.1).

4.2.2 Soil moisture balance equation

Soil moisture is modeled as a balance between gains from rainfall and irrigation and losses to evapotranspiration and leakage (Rodr´ıguez-Iturbe and Porporato, 2004a; Vico and Porporato, 2010)

dS φZ = R(t) + I(S, t) − T (S, C, t) − E(S, C, t) − Q(S), (4.4) dt

47 where S is the vertically averaged relative soil moisture, φ is porosity, and Z is a soil depth with homogenous characteristics (Table 4.2 contains values for the soil parameters). φZ is defined as the active soil depth (Laio et al., 2001a), the volume per surface area available for water storage. In agricultural soils, tilling tends to rearrange soil profiles so that the top layer of soil is relatively uniform in composition and depth. We assume that the root growth (which we do not explicitly model) is constricted to Z, and that hydraulic redistribution over this depth allows water to easily move to areas of lower soil moisture, making the vertically-averaged soil moisture a good description of the amount of water available for evapotranspiration (Guswa et al., 2002). R is the rainfall rate. For the purposes of a probabilistic analysis, here it is modeled as a marked Poisson process with mean event frequency λ and exponentially distributed rainfall events depths α (Rodr´ıguez-Iturbe and Porporato, 2004a). This stochastic components allows for the model to include the effect of unpredictable external forcing via rainfall, which is especially important in arid and semi-arid ecosystems, and for rain-fed agriculture. In the case of irrigated agriculture, a term I gives the irrigation rate, which may be a function of S and/or t depending on the irrigation strategy employed (e.g., stress avoidance or microirrigation) (Vico and Porporato, 2010, 2011a,b). The transpiration rate T is assumed to be proportional to C and is given by

T (S, C, t) = Ks(S) · C · Kcb · ET0(t), (4.5)

where Ks(S) is a water stress coeﬃcient, Kcb is a basal crop coeﬃcient (essentially

the midseason basal crop coeﬃcient of Allen et al.(1998)), and ET0(t) is the reference evapotranspiration, which is calculated using the Penman-Monteith equation for a reference crop (normally grass, but occasionally alfalfa) (Allen et al., 1998). As no

diurnal variation is considered, ET0(t) is a mean daily rate and thus the model should 48 be interpreted at the daily timescale. The water stress coeﬃcient is given by

 0 S ≤ S ,  w S−Sw ∗ Ks(S) = ∗ S < S ≤ S , (4.6) S −Sw w  1 S∗ < S,

∗ where Sw is the wilting point and S is the point of incipient stomatal closure.

Ks(S) therefore captures the plant stomatal response to soil moisture conditions. As mentioned previously, the plant is assumed to be able to easily compensate for areas of low soil moisture in the soil column by drawing more water from areas of high soil moisture, making S a good indicator of the amount of water available to the plant. However, this assumption is weakened if the plant cannot do so because of high root resistance or spatial heterogeneities in the soil properties (Guswa et al., 2002). The evaporation rate E is assumed to be proportional to (1 − C) and is given by

E(S, C, t) = Kr(S) · (1 − C) · Kec · ET0(t), (4.7)

where Kr(S) reduces evaporation according to soil moisture and Kec is a baseline evaporation coeﬃcient. A similar dependence of evaporation on 1 − C was used by Steduto et al.(2009). The evaporation reduction coeﬃcient is given by

( 0 S ≤ Sh, Kr(S) = (4.8) S−Sh S ≥ Sh, 1−Sh

where Sh is the hygroscopic point, below which no soil moisture losses occur. A

diagram of Ks and Kr as a function of S is shown in the upper panel of Figure 4.1, and the increase of evapotranspiration as a whole with increasing S can be seen from top to bottom in the lower panel of Figure 4.1. Evaporation draws primarily from a thin top layer of soil, drawing from lower soil layers only when potential gradients drive water from lower soil depths upward. This is often modeled using the two stage method for soil evaporation (Ritchie, 1972; Brutsaert and Chen, 1995). The

49 dependence of E on the average soil moisture value over a depth Z simpliﬁes the actual relationship, but it does capture the high rates of evaporation at saturation

(S = 1) and the trend toward a rate of zero evaporation as S approaches Sh. The form that is used for Ks is essentially equivalent to the expression for transpiration used in Laio et al.(2001a), while the form for Kr is quite diﬀerent from that used for evaporation in the same paper. Laio et al.(2001a) considered evaporation and transpiration separately, with the former being very small due to the presence of the plant canopy. However, as we are interested in the crop canopy as it develops throughout the growing season (from left to right in the lower panel of Figure 4.1), the maximum values for T and E must be of somewhat similar magnitude to capture the dominance of E shortly after planting and that of T later in the growing season

(this is reflected in the fact that Kcb and Kce are indeed nearly the same) (Kelliher et al., 1995). The combined percolation and runoff rate is denoted as Q, and as we are considering well-drained agricultural fields, subsurface percolation is assumed to dominate compared to overland runoff and to be equal to the hydraulic conductivity, i.e.,

d Q(S) = k(S) = ksat · S , (4.9)

where k is the hydraulic conductivity, ksat is the saturated hydraulic conductivity, and d is an empirically based parameter (Brooks and Corey, 1964; Rodr´ıguez-Iturbe and Porporato, 2004a).

∗ Calculation of Sw and S

Using data for silty loam (a common agricultural soil) and methods from Clapp and

Hornberger(1978) and Laio et al.(2001b), the wilting point Sw of was calculated as the soil moisture level corresponding to a matric potential of -1.5 MPa. Corn begins to suﬀer water stress when approximately 50% of the total available water

50 1

0.8

[-] 0.6 r ,K s K 0.4

0.2 K s K r 0 0 S S S* 1 h w S [-] 10-3 1 5

4.5

3.5

2.5 S [-] S*

2 S w 1.5

1 S h 0.5

0 0 0 0.2 0.4 0.6 0.8 1 C [-] Figure 4.1: Top: the water stress coeﬃcient (dashed line) and the evaporation reduction coeﬃcient (solid line) as a function of soil moisture S. Bottom: evapotranspiration [m/d] as a function of S and C, with values of ET0 and the soil moisture thresholds as in Table 4.1.

(which is the water content at ﬁeld capacity minus that at the wilting point) is depleted (Rhoads and Yonts, 2000). Therefore, we calculate the point of incipient

∗ ∗ ∗ stomatal closure S as S = (Sw + Sfc)/2. For silty loam, Sw = 0.35, S = 0.47, and

Sfc = 0.59.

4.2.3 Soil nitrogen content

While AquaCrop (Steduto et al., 2009) makes use of an empirical measure of soil fertility that allows the model to be used even if detailed soil nitrogen data are not available, most crop models consider a nitrogen balance (Ritchie et al., 1998;

51 Jeuﬀroy and Recous, 1999; Boote et al., 1998; St¨ockle et al., 2003; Brisson et al., 2003; Aggarwal et al., 2006) due to its key role in the growth and development of crops. In order to better examine crop growth and yield under optimal fertilization and irrigation strategies, a soil nitrogen balance is also included here. The evolution of total mineral nitrogen in the soil is given by the balance between deposition and fertilization as inputs and leaching and plant uptake as outputs (Porporato et al., 2003) dN = D(t) + F (N, t) − L(S,N) − U(S, N, C, t), (4.10) dt

where N is nitrogen content per unit area of soil, D is the rate of natural nitrogen addition to the soil, and F is the fertilization rate. For all ﬁgures in this study, the average annual rate of nitrogen deposition for a heavily agricultural region has been used as a constant deposition rate D (, NRSP-3). Unless otherwise noted, the

fertilization rate F is considered to be the maximum potential uptake of nitrogen Ft

divided by the length of the growing season, tGS. The total mineral nitrogen content in the soil, rather than the individual nitrate and ammonium components, is used because plants are able to take up both forms, making the separation of the two unnecessary in the case of this model, which aims for a general picture of nitrogen ﬂuxes. The leaching term L is proportional to the percolation from the hydrologic balance, Q, and the nitrogen concentration as

aN L(S,N) = Q(S), (4.11) SφZ

where a is the fraction of N which is dissolved in the soil moisture (a ≈ 1 for nitrate, while a ≤ 1 for ammonium). The nitrogen concentration in the soil moisture is given

aN by the quantity SφZ , which is denoted by η.

52 Plant uptake of nitrogen, U, is given by

U(S, N, C, t) = f(η) · T (S, C, t), (4.12)

in which f(η) is a function which limits the nitrogen uptake above a certain critical

concentration ηc, with the form

( aN aN SφZ SφZ < ηc, f(η) = aN (4.13) ηc SφZ ≥ ηc.

The physical reasoning for this limitation is that beyond a certain point, taking up more nitrogen is not useful for the plant to increase its growth rate, and extremely high nitrogen concentrations in plant tissue are toxic to the plant. The above limitation is meant to account in a parsimonious way for the plant’s ability to exclude nitrogen from transpired water (i.e. active uptake (Porporato et al., 2003)). It is worth noting that a reduction in S can either increase or decrease the N uptake. As long as S > S∗, a reduction in S increases the concentration η, thereby increasing N

∗ uptake if initially η < ηc. However, if S drops below S , transpiration decreases and therefore so does N uptake.

4.2.4 Crop biomass and yield

While the dynamics of the model are contained in the equations for C, S, and N, other variables which depend on one of the three main variables are also of interest. Speciﬁcally, we consider the crop biomass B and crop yield Y . The accumulation of plant biomass is modeled using the normalized daily water productivity W ∗ (e.g., Steduto et al.(2009)), which is typically multiplied by the ratio of transpiration to reference evaporation to model biomass accumulation. However, in place of transpiration we us the nitrogen uptake divided by the nitrogen uptake threshold ηc, giving

∗ dB ∗ U(S, N, C, t) W = W = Ks(S)Kcbf(η)C. (4.14) dt ηcET0(t) ηc 53 The use of U rather than T allows us to extend the concept of water productivity ηc to also consider the eﬀects of nitrogen limitation. When η ≥ ηc, one recovers the biomass growth equation used by Steduto et al.(2009) and others, which considered transpiration rather than nitrogen uptake for biomass accumulation. The biomass and yield are related through a harvest index, h, which is the fraction of the biomass which makes up the yield. The harvest index is often modeled as an increasing function in time which is modulated by various stresses (Steduto et al., 2009; Raes et al., 2009). Here we instead utilize a reference value for h and assume that stress limitations are suﬃciently accounted for elsewhere in the crop growth equations, recognizing that this limits the validity of the crop yield calculations to the end of the growing season. The yield is then

Y = h · B. (4.15)

4.3 Reduced versions of the model

The complete model is defined by the balance equations for C, S, and N (Equations (6.19), (6.1), and (4.10)) and their component fluxes, with Equations (4.14) and (4.15) defining additional variables. Two reduced versions of the model will now be examined. The first, in which S and N are held constant, is compared to canopy cover data to estimate parameters for Equation (6.19). The second, in which only S is held constant (and there is therefore no stochastic forcing), is analyzed as a typical deterministic dynamical system in order to demonstrate some of the insights which can be gained from this approach.

4.3.1 Canopy growth equation and its parameterization

We begin by examining the simplest version of the model, in which S and N are ﬁxed

∗ (at S ≥ S and with N such that η ≥ ηc) but C is allowed to vary. In these conditions

54 Table 4.1: The model parameters used in this study. Parameter Value Units Name/Description Source 2 rG 560 m /kg N Canopy growth per unit N uptake Calculated using data from Hsiao et al.(2009) rM 0.2 1/d Canopy decline due to metabolic limitation Calculated using data from Hsiao et al.(2009) 2 γ 0.005 1/d Slope of increase of senescence after tsen Calculated using data from Hsiao et al.(2009) Kcb 1.03 - Max. T/ET0 Allen et al.(1998); Hsiao et al.(2009) Kce 1.1 - Max. E/ET0 Hsiao et al.(2009) tsen 110 d Days until onset of senescence Mean of values in Table 2 of Hsiao et al.(2009)

55 tGS 140 d Length of growing season Mean of values in Table 2 of Hsiao et al.(2009) W ∗ 3.37·10−2 kg B/m2/d Normalized daily water productivity Hsiao et al.(2009) h 0.5 kg Y /kg B Maximum harvest index Hsiao et al.(2009) 3 ηc 0.054 kg N/m water Maximum N concentration taken up Derived from model parameters D 5.5·10−6 kg/m2/d N deposition rate (NRSP-3) 2 Ft 0.0286 kg N/m Maximum N uptake Bender et al.(2013) pY 0.12 $/kg Corn price per kg of yield Lamm et al.(2007) pF 0.639 $/kg Fertilizer unit price N Lamm et al.(2007) 3 pI 0.0148 $/m Irrigation water unit price Vico and Porporato(2011b) pL 0 $/kg Cost of leached N Set to 0 in current simulations 2 pfix 0.109 $/m Fixed costs Lamm et al.(2007); Vico and Porporato(2011b) Table 4.2: The climate and soil parameters used in this study. Parameter Value Units Name/Description Source α 1.5 cm Mean rainfall depth Sample values λ 0.3 1/d Mean rainfall frequency Sample values −3 ET0 5×10 m/d Reference evapotranspiration Approximated from Hsiao et al.(2009) Sh 0.14 - Hygroscopic point Rodr´ıguez-Iturbe and Porporato(2004a) 56 Sw 0.17 - Wilting point Rodr´ıguez-Iturbe and Porporato(2004a) S∗ 0.35 - Point of incipient stomatal closure Rodr´ıguez-Iturbe and Porporato(2004a) Sfc 0.59 - Field capacity Rodr´ıguez-Iturbe and Porporato(2004a) ksat 0.33 m/d Saturated hydraulic conductivity Rodr´ıguez-Iturbe and Porporato(2004a) d 13 - Leakage parameter Rodr´ıguez-Iturbe and Porporato(2004a) a 1 - Fraction of N dissolved Porporato et al.(2003) φ 0.43 - Soil porosity Rodr´ıguez-Iturbe and Porporato(2004a) Z 1.0 m d Soil depth Irmak and Rudnick(2014) (and also with ET0 constant, to maintain analytical tractability), Equation (6.19) reduces to

dC = r K ET η · C − r + γ(t − t ) · Θ(t − t ) · C2, (4.16) dt G cb 0 c M sen sen ´ ¯ which is simply the logistic model if t < tsen, in which case this equation can be solved analytically as

rGKcbET0ηct rGKcbET0ηcC0e C(t) = r K ET η t , (4.17) rGKcbET0ηc + C0rM pe G cb 0 c − 1q

which is the logistic equation (Murray, 2002). In order to parameterize Equation (4.16), it was necessary to use data from a growing season in which the crop did not experience water or nitrogen stress. Values

for rG, rM , and γ have therefore been obtained by minimizing the RMSE of the model compared to the data from Hsiao et al.(2009) for fully irrigated and fertilized conditions. The data come from 6 seasons of experiments spread over 22 years in Davis, CA. The ﬁrst three years used slightly diﬀerent maize cultivars, while the last three used the same cultivars, but in order to include more data they have been assumed to be similar enough to consider together. An approximate mean reference

evapotranspiration rate ET0 and the value for Kcb were also taken from Hsiao et al.

(2009). These experiments did not report soil N or uptake rates, and so ηc has been estimated by averaging the cumulative N uptake for maize found in Bender et al.(2013) across the growing season. While it would be preferable to use a more complete single dataset for the model parameterization, the emphasis here is not on predicting crop growth but on reproducing the general crop behavior. Figure 4.2 shows the model vs. the data against which it was parameterized, demonstrating

a good ﬁt particularly prior to the time of senescence, tsen. The value for tsen is

cultivar-speciﬁc and for this ﬁgure was taken as the average tsen over the 6 years of

57 1

0.9

0.8

0.7

0.6

0.5 C [-] 0.4

0.3

0.2

0.1

0 0 20 40 60 80 100 t 120 140 sen t [d] Figure 4.2: Growth of canopy cover in the C-only model, using the parameterization as described in the text (black line). Data are from 6 years of maize experiments in Davis, CA (Hsiao et al., 2009) (open circles).

experiments. The value of this parameter and all others discussed in this section can be found in Table 4.1.

4.3.2 N and C system

An interesting 2-D dynamical system is obtained when C and N are free to vary in time, but S is kept constant, which approximates the conditions in an agricultural ﬁeld with a microirrigation system and constant fertilization and deposition rate,

F + D = F0. The top panel of Figure 4.3 shows the evolution of the two state variables N and C in time, while the bottom panel is a phase space diagram which shows sample trajectories in the C-N phase space. It is easy to see the development of the state variables for different initial conditions, and the effects of parameter changes on the vector field, which determines the direction the system moves for a given condition, can also be examined using this type of diagram. In the bottom of Figure 4.3, the ηc threshold of Equation (4.13) can be seen as the solid gray line

58 which separates the two parts of the solution–on the left side, η < ηc, and on the

right, η > ηc. Optimization will be further discussed in a later section, but for now we point out that in order to maximize crop growth, the system should be kept on the right side of this threshold, as the trajectories of the vector ﬁeld here point to higher values of C and thereby greater rates of crop growth.

Diﬀerent solutions exist above and below ηc because when η ≥ ηc, suﬃcient nitrogen is available for crop growth, and Equation (6.19) is decoupled from N. An

analytic expression for C(t) can be obtained (when t < tsen) due to this decoupling, which is shown in Equation (4.17). An exact expression can also be found for N(t),

but as it is rather involved it is not included here. When η < ηc, the crop experiences nitrogen stress and Equation (6.19) is again coupled to N. Analytical expressions for C(t) and N(t) are unavailable in this case.

Fixed points and stability, η ≥ ηc

For the simpler case of η ≥ ηc and t < tsen, the ﬁrst ﬁxed point is given by

∗ C1 = 0, (4.18)

∗ F0 · SφZ N1 = d , (4.19) a · ksatS

while the expressions for the second are

∗ rG C2 = KcbET0ηc, (4.20) rM

rM ∗2 rG 2 2 2 SφZ F0 − C2 SφZ F0 − KcbET0 ηc ∗ rG rM N2 = d = d . (4.21) ´a · ksatS ¯ ´ a · ksatS ¯

dC Recalling Equation (6.19) with dt = 0, we note that the steady-state uptake of nitrogen is given by rM (C∗2) = rG K2 ET 2η2, a quantity which can also been seen rG 2 rM cb 0 c inside the parentheses in Equation (4.20).The ﬁrst ﬁxed point is an unstable node

59 1.0 0.02

0.8 0.015 ] 2

0.6 m / [-]

kg C [ Ncrit 0.4 N

0.005 0.2

0.0 0 0 20 40 60 80 100

t [d] 1.0

0.8 ●

0.6 [-] C

0.4

0.2

0.0 0.000 0.005 0.010 0.015 0.020 N [kg/m2] Figure 4.3: Top: timeseries of canopy cover and soil nitrogen for the C-N model, for two differing initial conditions of N (solid lines represent C; dashed lines represent N). Bottom: C-N vector plot and phase portrait, for two initial conditions. The black dot represents the stable fixed point (N2,C2), and the solid gray line is the ηc threshold. Note that in this figure, the combined fertilization and deposition rate has been reduced slightly in order to better show the impact of varying the initial conditions, and the simulation has only been performed until tsen because after this point in time the trajectories no longer converge towards the same fixed point.

60 and the second is a stable node (a third exists, but it is always negative and thus not physical). The eigenvalues of the ﬁrst ﬁxed point are

a · k Sd λ = − sat , (4.22) 1a SφZ

λ1b = ET0ηcKcbrG, (4.23) while those of the second ﬁxed point are

a · k Sd λ = − sat , (4.24) 2a SφZ

λ2b = −ET0ηcKcbrG. (4.25)

The ﬁrst ﬁxed point is always an unstable node, and in the second is always a stable node. This is unsurprising, as the standard logistic equation, which is contained

within the system dynamics when η ≥ ηc, likewise has one stable and one unstable node as its ﬁxed points.

Fixed points and stability, η < ηc

If the system were allowed to develop to steady state (t → ∞), the explicitly time- dependent part of the mortality M(C, t) term would ultimately drive the canopy cover to a value of zero, and the soil nitrogen content would approach a value determined by the balance between the fertilization/deposition and leaching terms. The fixed points are obtained assuming no senescence term (e.g., if a perennial crop rather than an annual one were considered). In this condition, there are two fixed points. The first has the same expressions as Equations (4.18) and (4.19), while the second is

2 2 d d 2 4F0·ET0 KcbrG −ksatS + (ksatS ) + ∗ rM C2 = , (4.26) b 2ET0Kcb

61 −r Zφk Sd+1 + SφZ (r k Sd)2 + 4F · ET 2K2 r N ∗ = M sat M sat 0 0 cb G . (4.27) 2 2aET 2K2 r a 0 cb G The stability can more easily be seen when it is put in terms of the fertilization and deposition term, the critical value of which is derived from the expression for the eigenvalues and is given by

2d 2 Fc = 2aS ksatrM paET0Kcb − SZφrM q

2 2 2 2 2 2 2 · a ET0 Kcb + 4aET0SZφKcbrM − S Z φ rM

` 2 2 2 2˘ 2 2 2 2 ÷ ET0KcbrG a ET0 Kcb − 6aET0SZφKcbrM + S Z φ rM , (4.28) ` ˘ so that the fixed point is a spiral when F0 < Fc and a node when F0 > Fc, pointing to the possibility of damped oscillations. Oscillations related to nitrogen cycling were also observed by Thornley et al.(1995) (oscillations of LAI and soil nitrogen in a model of grass-legume dynamics), Tilman and Wedin(1991) (oscillations and possible chaos in interannual dynamics of a perennial grass), Manzoni and Porporato (2007) (shifts in stability in a model of substrate carbon and nitrogen dynamics), and Parolari and Porporato(2016) (stability shifts in a model of forest carbon and nitrogen cycles under harvesting). The interpretation of such oscillations is not entirely clear, as it is possible that they are merely artifacts of simplified models or the results of overfitting the available data, but their presence in such models is intriguing and deserves further attention.

4.4 Soil moisture dynamics and hydrologic forcing

The addition of the soil moisture dynamics greatly increases the model complexity, especially when the rainfall stochastic forcing is considered. This forcing adds considerable interest to the dynamics of the model, as it allows us to consider the eﬀect of varying rainfall parameters as well as to examine the model in a probabilistic sense.

62 While it is possible to obtain some analytic results regarding soil moisture probability distributions for statistically steady states under stochastic rainfall forcing (see for example Rodr´ıguez-Iturbe and Porporato(2004a)), the complexity of the crop growth function and nitrogen balance employed here make it necessary to proceed numerically (though see Schaﬀer et al.(2015) for a special case in which analytical results were obtained for stochastically driven soil moisture and plant biomass).

4.4.1 Soil moisture dry-down

Many important agroecosytems have some form of the Mediterranean climate, in which the rainfall occurs out of phase with the growing season. In this case, the soil moisture dynamics occur as a deterministic dry-down, with the exception of whatever small amounts of precipitation may occur during the growing season. Therefore, all other factors being equal, the crop yield of rain-fed (i.e., non-irrigated) agriculture in this type of climate depends greatly on the initial condition of soil moisture that is available at the beginning of the growing season. Of course, this initial supply may also be supplemented by irrigation, which is similar to the case considered in Section 4.3.2.

4.4.2 Stochastic forcing

Figures 4.4 and 4.5 show the development of the three main state variables and their associated fluxes over the course of a growing season, with t = 0 taken as the start of the growing season. Note that in these and the proceeding figures, a constant rate of nitrogen fertilization/deposition was imposed. As compared to the deterministic scenarios discussed in Section 4.3, the variables in the full model show much greater variability, due to the direct dependence of the fluxes on the stochastically driven soil moisture balance. Figure 4.6 shows the main dynamic variables in the three- dimensional phase space. Observing this sample time series, excursions below the

63 0.8 1 0.08 0.7 s Rainfall 0.07 0.8 0.6 0.06 0.5 0.6 0.05 0.4 0.04 S [ ] C [-] 0.3 0.4 0.03 Rainfall [mm] 0.2 0.02 0.2 0.1 0.01 0 0 0 0 20 40 60 80 100 120 140 0 20 40 60 80 100 120 140 t [d] t [d] (b) (a)

0.017 3 B Y 0.016 2.5

0.015 ] ] 2 2 2 0.014 1.5 0.013 N [kg N/m B, Y [kg/m 1 0.012

0.011 0.5

0.01 0 0 20 40 60 80 100 120 140 0 20 40 60 80 100 120 140 t [d] t [d] (c) (d)

Figure 4.4: Time series of (a) canopy cover C, (b) soil moisture S and rainfall R, (c) soil nitrogen N, (d) crop biomass and yield over a growing season. soil moisture threshold S∗ (the dotted lines that are perpendicular to the S axis) and

below the ηc threshold (the diagonal line on the S-N plane) can be seen to coincide with reductions in C. The case of water stress depends on S only, while the latter case of nitrogen stress involves the interaction of S and N because of their joint eﬀect on the f(η) limitation function.

4.4.3 Impact of rainfall regimes on rain-fed agriculture

The timing and amount of rainfall exerts a strong control on crop growth in rain-fed agriculture. We first examine the effect of different rainfall regimes when associated

64 10-3 6 10-4 T 2.5 5 E U 0.058 ET f( ) 2 0.056

0.054 ] 3 /d]

2 1.5 0.052 3 0.05

1 0.048 ) [kg N/m

2 U [kg N/m f ( T, E, ET [m/d] 0.046 0.5 0.044 1 0.042 0 0 20 40 60 80 100 120 140 0 0 20 40 60 80 100 120 140 t [d] t [d] (b) (a)

10-3 0.07 3

0.06 2.5 0.05 2 d] 2

d] 0.04 3 1.5 0.03 Q [m

0.02 L [kg N/m 1

0.01 0.5

0 0 20 40 60 80 100 120 140 0 0 20 40 60 80 100 120 140 t [d] t [d] (c) (d)

Figure 4.5: Time series of (a) transpiration T , evaporation E, and evapotranspiration ET , (b) nitrogen uptake U and limitation function f(η), (c) leakage Q, and (d) nitrogen leaching L. parameters (mean event depth α and mean frequency λ) are constant throughout a growing season, which is a reasonable approximation for growing season conditions in many regions of the world. Despite the fact that the parameters are constant in time, there remains a strong intra-seasonal time dependence, primarily due to the growth of canopy cover, C. This is due to both its growth in time and more explicitly through the time-dependence of the M(C, t) term. This pattern in time can be seen not only in Figures 4.4 and 4.5 but also in

65 1

0.5 C [-]

0 0 0 0.5 0.01 0.02 S [-] 1 0.03 N [kg N/m 2] Figure 4.6: A sample trajectory shown in the 3-dimensional phase space of C, S, and N (black line), and projections onto the three planes (gray lines). Dashed lines ∗ denote the dynamics which occur after tsen, and the dotted lines the S threshold (S-C and S-N planes) and the ηc threshold (S-N plane).

Figure 4.7, which shows the ensemble average over many simulations of canopy cover (top left), soil nitrogen (top right), soil moisture (bottom left), and nitrogen leaching (bottom right). In each simulation, the mean rainfall rate was kept constant but α and λ were changed, to demonstrate the interaction of rainfall event frequency and event size. Simulations with larger, less frequent events are characterized by reduced canopy cover, and higher rates of nitrogen leaching. However, there are also slightly higher levels of soil nitrogen which remain, as the reduced soil moisture and canopy cover led to a low nitrogen uptake and thus higher soil moisture levels. The system shown in Figure 4.7 undergoes a typical fertilization schedule for corn, in which some fraction ξ of the total fertilization Ft is applied at the beginning of the growing season, and the remainder is applied after a period τ, resulting in the jump which can be observed in N in Figure 4.7b. Here, ξ = 0.3 and τ = 40 d (see Section 4.5.1 and Figure 4.10 for a discussion of the optimization of ξ and τ).

66 0.9 0.04

0.8 =0.1 0.035 0.7 =0.3 =0.5 0.6 ] 0.03 2

0.5 0.025

C [ ] 0.4

0.3 N [kg N/m 0.02 =0.1 0.2 =0.3 0.015 0.1 =0.5 0 0.01 0 20 40 60 80 100 120 140 0 20 40 60 80 100 120 140 t [d] t [d] (a) (b)

10-4 0.55 7 =0.1 =0.1 6 0.5 =0.3 =0.3 =0.5 5 =0.5 /d]

0.45 2 4 S [ ] 0.4 3

L [kg N/m 2 0.35

0.3 0 20 40 60 80 100 120 140 0 0 20 40 60 80 100 120 140 t [d] t [d] (c) (d)

Figure 4.7: A comparison of the mean (a) canopy cover, (b) soil nitrogen, (c), soil moisture, and (d) leaching across diﬀerent precipitation regimes for λ = 0.1 d−1 (light gray), λ = 0.3 d−1 (gray), λ = 0.5 d−1 (black), with α altered to keep a constant mean rainfall rate of 4.5 mm/d for all ﬁgures. A typical fertilization treatment for corn has been applied, resulting in the observed jump in N.

4.5 Optimal strategies

Crop models represent an important tool to study the impact of different management strategies aimed at maximizing yield, minimizing water and fertilizer use, reducing the leaching of fertilizers, and optimizing the timing of irrigation and fertilization treatments under hydroclimatic variability (Wallach et al., 2006). Toward this goal we develop a first-order objective function, which considers the profit from

67 the sale of produce, costs of fertilizer and irrigation, ‘environmental cost’ of nitrogen leaching, and ﬁxed costs,

Pnet = pY · Y (tGS) − pF · Ftot − pI · Itot − pL · Ltot − pfix, (4.29)

where pY [$/kg Y] is the unit sale price of the crop yield at the end of the growing

3 season, Y (tGS). pF [$/kg N] and pI [$/m ] are the unit prices of fertilizer and irrigation water, respectively, while the cumulative fertilization and irrigation are given

R tGS R tGS by Ftot = 0 F (N, t)dt and Itot = 0 I(S, t)dt. The unit ‘environmental cost’ of

leached nitrogen is given by pL [$/kg N], here conceptualized as the cost necessary to mitigate these losses or to pay associated ﬁnes, while the cumulative nitrogen leach-

R tGS 2 ing is given by Ltot = 0 L(S,N)dt. Finally, pfix [$/m ] is a fixed cost representing distribution and energy costs, here estimated following Vico and Porporato(2011b). Estimated values for these parameters can be found in Table 4.1. This objective function should be thought of as a means to quantify the relative financial impact and importance of various components of the crop system, rather than as a way to obtain firm predictions about the profitability of various management strategies. Figure 4.8 shows several key crop responses under idealized, non-stochastic conditions. The responses of crop yield Y and the objective function Pnet to different mean soil moisture and soil nitrogen conditions are illustrated in Figures 4.8a and 4.8b. Figures 4.8c and 4.8d show the cumulative amounts of irrigation and fertilization, respectively, that would be needed to keep the S and N at the designated mean values. The horizontal lines mark the S∗ threshold, below which point the transpiration begins to decrease, and the diagonal line marks the ηc concentration threshold. Nitrogen Use Efficiency (NUE) is the ratio of the amount of nitrogen which is taken up by the crop to the amount which is applied, and is an important metric by which to judge fertilization strategies. Similarly, we can define the Irrigation Efficiency (IE) as the ratio of the irrigation water applied to the amount which is transpired

68 Yield [kg/m 2] Profit [$/m 2] 1.2 0.00 0.7 0.7 -0.07 0.02 0 1 0.65 0.93 0.65 -0.02

-0.05 0.8 0.6 0.78 0.6 -0.04

1.09 -0.04 0.55 0.55 0.62 0.6 -0.06 0.5 0.5 -0.02 0.47 1.24 0.4 -0.08 0.45 0.45 -0.1 0.31 0.2 0.4 0.16 0.4 -0.09 -0.11 -0.12 0.005 0.01 0.015 0.02 0.025 0.03 0.005 0.01 0.015 0.02 0.025 0.03

(a) (b)

Cumulative Fertilization [kg/m 2] Cumulative Irrigation [m] 1.32 0.06 0.07 0.08 0.09 1.18 1.04 0.7 0.7 1.2 0.05 0.08 0.90 0.04 0.65 0.65 0.76 0.07 1

0.6 0.06 0.6 0.62 0.05 0.8 0.55 0.02 0.55 0.04 0.5 0.5 0.6 0.03 0.48 0.45 0.02 0.45 0.01 0.34 0.4 0.4 0.01 0.4 0.2 0.005 0.01 0.015 0.02 0.025 0.03 0.005 0.01 0.015 0.02 0.025 0.03

NUE [-] IE [-] 0.34 0.23 0.34 0.7 0.45 0.8 0.7 0.26 0.42 0.6 0.56 0.17 0.12 0.67 0.7 0.51 0.65 0.65 0.5 0.78 0.09 0.59 0.6 0.6 0.89 0.6 0.4 0.5 0.68 0.55 0.55 0.4 0.3 0.5 0.5 0.3 0.2 0.45 0.2 0.45 0.1 0.4 0.1 0.4

0.005 0.01 0.015 0.02 0.025 0.03 0.005 0.01 0.015 0.02 0.025 0.03

(e) (f)

Figure 4.8: Response to the necessary rates of irrigation and fertilization to keep S and N at the designated constant values of (a) crop yield, (b) profit, (c) cumulative fertilization [kg/m2], (d) cumulative irrigation [m], (e) Nitrogen use efficiency (NUE), the cumulative nitrogen uptake as a fraction of total fertilization, and (f) irrigation efficiency (IE), the cumulative transpiration as a fraction of the total irrigation. The horizontal dashed line represents the S∗ threshold, while the diagonal dashed line is the ηc limit. All figures are for deterministic conditions (i.e., no stochastic forcing in the rainfall). 69 by the crop. These two values are shown in Figures 4.8e and 4.8f, respectively, as a function of the mean soil moisture and mean soil nitrogen. Each of the panels in Figure 4.8 sheds light on a different consideration for the optimal use of water and nitrogen resources. However, the common thread between them is that in each case, the ‘best’ scenario from the perspective of using water and nitrogen resources in the most efficient manner (i.e. maximizing IE and NUE to produce the highest possible

∗ yield and proﬁt), occurs at the intersection of the S and ηc lines. At this point neither water nor nitrogen is limiting, and no extra irrigation or fertilization beyond what is needed to keep the system at these S and N values is used. However, with the addition of the random rainfall in the next section, additional concerns such as the robustness of the optimal strategies under stochastic forcing must also be considered.

4.5.1 Optimization under stochastic rainfall conditions

Random hydroclimatic forcing adds uncertainty to the expected value of the objective function. This is illustrated in Figure 4.9, which shows the numerical probability distribution functions of yield and profit for varying fertilization rates. Note that as the fertilization rate increases, both the mean of the yield and its variance increase. The mean increases because the higher fertilization rates lead to less likelihood that the crop will experience a shortage of nitrogen, while the variance increases because the field of possible yields expands–the extra nitrogen raises the maximum possible yield, while a low-rainfall growing season could still occur, so lower yields are still possible. The probability distributions in Figure 4.9 highlight the fact that under stochastic rainfall conditions, the question of optimization must be examined from a probabilistic point of view. Unlike in the previous section, optimal strategies for a system undergoing stochastic forcing must attempt to maximize profit while also being robust to adverse conditions, such as drought or flood years. This necessarily involves tradeoffs between maximizing yield and profit on the one hand and miti-

70 gating risk on the other. For example, we point out that in the bottom of Figure 4.9, the theoretical maximum (which refers to the value which would be obtained if the crop experienced no water stress and took up as much N as possible) for proﬁt (dashed line) occurs at a much lower fertilization rate than the maximum of the actual proﬁt under stochastic rainfall conditions (black line), demonstrating in a simple way the necessity of accounting for the possibility of adverse conditions. A detailed analysis of such concerns is beyond the scope of this work, though we point out that many studies have examined the related concept of resilience in ecological and social systems (see for example Walker et al.(2004)). In order to examine the impact of stochastic forcing on optimal fertilization, we

first suppose that the total fertilization over the course of a growing season Ft (the value of which can be found in Table 4.1) is to be divided into two treatments, which corresponds to a typical fertilization schedule for corn (e.g., Brady et al.(1996)). The placement of the fertilizer applications in time is varied by changing the fraction of Ft which is applied in each application and the amount of time between the two applications. In order to focus on optimal fertilization timing and amounts under stochastic conditions, we do not consider the other potential degrees of freedom in the fertilization scheduling, such as varying the total amount of fertilizer used or using more than two applications. The effect of varying ξ and τ (the fraction of Ft in the first application and the time between the first and second applications, respectively) on crop yield can be seen in Figure 4.10, which shows the yield response to varying ξ and τ for three different soil depths. Larger soil depths lead to less variation in S and thus less percolation and leaching, thereby increasing the fraction of ξ − τ space in which higher yields can occur. The exact location of the peak yield is a result of the balance which maximizes the uptake of nitrogen and minimizes the loss due to leaching.

71 Figure 4.9: Yield (top) and proﬁt (bottom) as a function of the constant fertilization rate. The dashed line represents a theoretical maximum while the solid line is the mean of many simulations. The inset plots show the numerical probability density functions at each point.

72 1

0.9 0.9 0.8 0.90

0.7 0.8 1.00 0.6 0.7 0.5 0.6 0.4 0.80

0.3 0.5 0.2 0.60 0.4 0.1 0.70 0.50 0 20 40 60 80 100

1.1

0.9 1 0.8 0.9 0.7 0.8 0.6 0.7 0.5 0.90 0.4 1.10 0.6

0.3 0.5 0.70 0.2 0.4 1.00 0.80 0.1 0.600.50 0 20 40 60 80 100

1.1

0.9 1 0.8 0.9 0.7 0.8 0.6 0.7 0.5 0.90 0.4 1.10 0.6

0.3 0.5 0.70 0.2 0.4 1.00 0.50 0.1 0.80 0.60 0 20 40 60 80 100

Figure 4.10: Crop yield as a function of ξ, the fraction of the total fertilization amount which is applied at the beginning of the growing season, and τ, the time between the ﬁrst and second fertilizer applications, for three soil depths: Z=33 cm (top), Z=67 cm (middle), and Z=100 cm (bottom). In this ﬁgure, microirrigation was used to prevent the soil moisture from73 dropping below S∗, and therefore the crop can experience only nitrogen stress, not water stress. 4.6 Conclusion

We have presented a dynamical system for crop evolution, based on the AquaCrop model (Steduto et al., 2009) and minimal models for soil moisture and nitrogen cycling used in ecohydrology (Rodr´ıguez-Iturbe and Porporato, 2004a). It includes canopy cover, soil moisture, and soil nitrogen as its main state variables and tracks fluxes of water and nitrogen from evapotranspiration, nitrogen uptake, and leaching. This parsimonious model, with its reduced number of parameters, may be useful for evaluating the impact of different fertilization and irrigation strategies as well as different precipitation and climate regimes on crop yield, expected profit, and other outputs of interest. A simple objective function was used to compare optimal strategies of fertilization and irrigation. The results highlight the importance of considering, from a quantitative and theoretical point of view, the optimization of these agricultural inputs, and also provide a direct connection with climate parameters. Hydroclimatic forcing is a major driver of variability in agricultural systems, which has implications not only for crop yield and profitability but also for environmental impact. The model developed here is capable of characterizing the variability in the model outputs and provides a link to the random processes which drive this variability. Agroecosystems cover a large portion of the Earth’s surface and provide essentially all of the global food supply. It is thus crucial to have a more complete understanding of the fluxes of water and nutrients in such systems, and their dependence on potentially changing hydroclimatic inputs and human activities. To this regard, the model presented here may be useful to explore scenarios and generate hypotheses. The framework can be extended in a number of directions. In order to emphasize the dynamical systems point of view, the model presented here necessarily included certain simplifications. However, including more detailed plant and

74 soil models and performing a comparison with more complete crop models would provide ﬁrmer ground from which to make predictions. Moreover, the model could easily account for periodic seasonal variations in temperature, radiation, or rainfall, which alter the water and nutrient cycles and therefore the optimal fertilization and irrigation strategies. Finally, the nature of agroecosystems is that they are heavily intertwined with human activities (e.g., Sivapalan et al.(2012); Porporato et al. (2015); Assouline et al.(2015), suggesting the need to couple models for ecological systems and landscape evolution with social and behavioral models (e.g., harvesting in Parolari and Porporato(2016) and Pelak et al.(2016)). We hope that these considerations will be accounted for in future contributions, providing a quantitative framework for the sustainable use of soil and water resources while ensuring food security.

75 5

Dynamic evolution of the soil pore size distribution and its connection to soil biogeochemical processes

5.1 Introduction

The properties of a soil are determined by a complex arrangement of pores, particles, and aggregates, but the temporal evolution of these properties and their links to biogeochemical cycles are typically not taken into account in current biogeochemical and ecohydrological models. An accurate representation of soil properties such as the water retention curve (WRC) and hydraulic conductivity curve (HCC) is needed for such models in both natural and agricultural environments, as they largely control the soil moisture balance, which plays a key role in the hydrologic cycle (e.g. Manzoni and Porporato, 2009; Porporato et al., 2015). However, direct measurement of these properties for a particular soil is often infeasible, leading to the need to estimate them based on more easily obtained information, such as the silt/sand/clay fractions, bulk density, and organic matter. Pedotransfer functions (PTFs), which use large soil databases to statistically relate such basic data to the soil properties of interest, have been widely used to estimate useful soil properties from more readily available

76 data (e.g. Wöstenet al., 2001; Vereecken et al., 2010), but they are largely empirical and limited in their physical information content. Among their main limitations are that their ability to predict soil properties outside of the regions or soil types which were used to create the functions is unknown and uncertain. Also, they provide little information about soil structure, such as the shape of the pore size distribution (PSD), and although more recently Vereecken et al.(2010) suggested the development of temporal PTFs to better represent the time-evolution of soil properties, they still implicitly assume static conditions and do not account for time-dependent processes. Although they are often assumed to be constant for modeling applications, soil properties do change over short timescales, especially under the influence of human activities (Vereecken et al., 2010). Combined with the well-recognized importance of the PSD in determining soil properties, this has led to efforts to model the time- evolution of the PSD and thereby the dynamics of the WRC and HCC (Or et al., 2000; Leij et al., 2002a,b). Maggi and Porporato(2007) also coupled the PSD with biological processes to understand the effect of bioclogging on soil hydraulic properties. We follow this work by developing a PSD model which evolves in time as a result of time-dependent soil processes and biogeochemical cycling. Our model differs from previous efforts in that we make use of a simple power law description of the PSD which connects to other widely used models (e.g. Brooks and Corey, 1964; Clapp and Hornberger, 1978), which is in some ways a more physically consistent mathematical representation of the PSD, and in that we also consider biogeochemical processes by connecting the parameters to the soil organic matter (SOM) content. We first develop an evolution equation for the dynamics of the pore size distribution and derive the power law distribution as one of its solutions. We then derive expressions for key soil properties from the power law PSD, and with a careful consideration of experimental data from the literature, link its parameters to key soil processes and management activities such as tillage, soil consolidation, and changes in SOM.

77 Finally, we show how these processes alter the soil properties.

5.2 Dynamic pore size distribution

5.2.1 Evolution equation

We model the time-evolution of the pore size distribution f = f(r, t) as a function of the pore radius r and the time t via a generic transport equation of the form

∂f ∂ = ρf − mf, (5.1) ∂t ∂r ” ı where ρ = ρ(r, t) is a drift term, and m = m(r, t) is a source/sink term. The term ρ is called the drift and represents the shrinking of a pore radii as a function of time, while m represents the instantaneous gain or loss of pores of radius r. This type of equation is typically used in nonequilibrium statistical mechanics and kinetic theories to describe the evolution of statistical distributions of multiparticle systems (Balescu, 1997). Or et al.(2000) applied a related evolution equation (the Fokker- Planck equation) to the time evolution of the PSD, and obtained a lognormal PSD. The lognormal distribution, which was first used for this purpose by Brutsaert(1966) and later by Kosugi(1994, 1996), has the advantage of being flexible. However, unlike in that study, we do not include a diffusion term. The presence of such a term implies a random walk in r, which is not physically justified, despite the fact that reasonable forms of the PSD may be obtained in this way.

5.2.2 Power law pore size distribution

Another commonly used model is the power law PSD, which has the advantage of a clear connection to the well known power law WRC of Brooks and Corey(1964), and has a basis in the fractal fragmentation of soils (Tyler and Wheatcraft, 1992; Perrier et al., 1996). Not all soils exhibit fractal scaling, so the power law model is not applicable in all cases, and the power law form imposes certain limitations

78 to the shape of the distribution, in that it is not able to represent a PSD with a nonzero mode. However, due to its widespread use, physical basis, and the ease with which it can be linked to bulk soil properties, we adopt the power law PSD in this study. We assume that in its evolution the PSD retains its mathematical form with time-varying parameters

−b(t) f(r, t) = a(t)r 0 ≤ r ≤ Rm(t), (5.2)

in which a(t) is a scaling parameter, b(t) is the power law exponent, and Rm(t) is the maximum eﬀective pore radius. Rm is related to the air-entry or bubbling pressure, which can be deﬁned as the matric pressure at which air enters the soil pores (Brooks and Corey, 1964). The integral of the PSD over r is equal to the porosity, φ(t), which is equivalent to the soil water content at saturation. We can therefore solve for φ(t) as

Z Rm(t) a(t)R (t)1−b(t) φ(t) = a(t) r−b(t)dr = m , (5.3) 0 1 − b(t) from which the scaling parameter a(t) is

φ(t)(1 − b(t)) a(t) = 1−b(t) . (5.4) Rm(t)

The full expression for the PSD is then

φ(t)(1 − b(t)) −b(t) f(r, t) = 1−b(t) r . (5.5) ˜ Rm(t) ¸

In order for the integral to converge, it is necessary that b(t) < 1, which we assume is a physically imposed condition. In the next section we develop expressions for ρ and m which allow us to obtain a power law form of the PSD.

79 5.2.3 Terms of the evolution equation

We now return to the as yet undeﬁned terms of the evolution equation, Equation (5.1), the drift ρ(t) and the source/sink m(t). To follow our program of obtaining a power law PSD which evolves in time, these terms must be

r ρ(r, t) = a0(t) − a(t)b0(t) ln(r) , (5.6) a(t)b(t) ´ ¯ and a0(t) b0(t) m(r, t) = − 1 + ln(r) , (5.7) a(t)b(t) b(t) ´ ¯ where the time-derivative of the parameters is indicated with a prime (0) symbol, we obtain a form of Equation (5.1) which can be solved analytically using the method of characteristics (Logan, 2013) (see the Appendix for details). The initial condition for this problem is

−b(0) f(r, 0) = f0(r) = a(0)r , 0 < r < Rm(0), t = 0. (5.8)

In the following section, we will discuss the model parameterization and the dependence of the parameters on both time and soil biogeochemical cycling.

5.3 Soil hydraulic properties

In order to utilize the results of this study in an ecohydrological model, we obtain expressions for the evolution of key soil properties, especially the WRC and the HCC. Note that the PSD is intended as an average over the soil depth, as we do not consider diﬀerent soil horizons, and so these expressions should likewise be taken as average values. Using the methods outlined by Mualem and Dagan(1978) and Brutsaert(2005), we model the ﬂow of water through the soil by assuming that we assume that the soil pores act as capillary tubes, with a distribution of pore sizes

80 determined by the normalized PSD (i.e. the probability density function), and derive the following expression for the matric potential Ψ

Cs −1/(1−b(t)) Ψs(s, t) = − s , (5.9) ˜Rm(t)¸ where s is the relative soil moisture and Cs is a constant related to the surface tension of water (a more complete derivation can be found in the Appendix). We can invert this expression for the WRC to solve for the relative soil moisture at a particular matric potential. For example, we can obtain the soil moisture at the wilting point

sw by solving for s with the wilting matric potential, Ψsw , as

b(t)−1

−Ψsw Rm(t) sw(t) = . (5.10) ˜ Cs ¸

The same procedure can be followed to obtain other values of interest for soil moisture, such as the hygroscopic point, the point of incipient water stress, and the ﬁeld capacity. Values for the wilting point vary with the plant type, but in this study we

use Ψsw = −1.5 MPa (this is a typical value for a crop such as maize, though a value of −3 MPa is often used for stress tolerant plants) (Rodr´ıguez-Iturbe and Porporato, 2004a). As the this tends to lead to an unrealistically high value for the saturated hydraulic conductivity, a so-called series-parallel model can be used, in which the parallel tubes are cut normal to the ﬂow direction and randomly rearranged (Brutsaert, 2005). This leads to the following expression for the hydraulic conductivity

2 2 2 γwGeφ(t) Rm(t) (1 − b(t)) 4−2b(t) K(s, t) = s 1−b(t) , (5.11) µ(3 − b(t))(2 − b(t))

where γw is the speciﬁc weight and µ the dynamic viscosity of water. If the ﬂow in

the pores follows the Hagen-Poiseuille equation, Ge = 1/8. This expression can be 81 6 Mid 5 NoTill

3 [pF]

0 0 0.1 0.2 0.3 0.4 0.5 [-] Figure 5.1: Example of using the Ψ−θ data to obtain the model parameters. Here, the data is from Teiwes(1988). The ‘NoTill’ data are from the long term plot without tillage, and the ‘Mid’ data are from samples which were taken at approximately the midpoint between annual tillage, roughly 180 days after tillage occurred. easily adjusted to account for tortuosity in the soil pores and more complex models of soil structure.

5.3.1 Model parameterization

As the WRC curve is generally easier to measure than pore size data, it is useful to estimate the latter from the former. We can relate the relative soil moisture s to the water content θ and the model parameters

θ s(θ, t) = , (5.12) φ(t) which allows us to rewrite Equation (5.9) as

−1/(1−b(t)) Cs θ Ψs(θ, t) = − . (5.13) ˜Rm(t)¸˜φ(t)¸

We estimated the parameter values by using the Matlab constrained minimization function ‘fmincon’ to minimize the root mean square error (RMSE) between Equation

82 (5.13) and the data. Figure 5.1 shows an example of this ﬁtting to data of Teiwes (1988), which was obtained from Leij et al.(2002b), which compares data from two experimental sites, one which had been tilled annually and one which was managed using no-till methods.

5.4 Connection of parameters to temporal and biogeochemical processes

We separate the important factors which affect the PSD parameters into those which are explicitly time-dependent (including external forcing such as soil tillage and internal processes such as consolidation) and those which depend on the soil biogeochemistry. The effect of soil biogeochemistry on soil structure and properties are complex. To account for these effects in our model, we adopt the soil organic matter content, C(t), as a means to encapsulate the effect of soil biogeochemical changes on the soil. While this is certainly a simplification, it is a necessary one to maintain the parsimonious description of the soil PSD. In order to incorporate the effects of time dependent processes and soil organic matter, we posit that these changes are sufficiently independent that they can be accounted for with two multiplicative terms. In reality, of course, they are not likely to be completely independent, but this is a useful first step to separate these two important influences. In this way, it represents an improvement on a typical pedotransfer function, for which, all else being equal, a given value of SOM will always result in the same soil properties, because time- dependent information such as the time when the soil was last tilled are not taken into account. Below, we present only the power law exponent b, but the same procedure was followed for all parameters. We separate the factors which control the value of b into two parts: first, a function γb(t) which is used to account for explicitly time- dependent changes in the PSD, and second, a component bC (C(t)), which relates the

83 value of b to the SOM content. These terms are then multiplied together to obtain an expression for b(t, C(t))

b(t, C(t)) = γb(t) · bC (C(t)). (5.14)

The next subsections discuss γb(t) and bC (C(t)), respectively.

5.4.1 Tillage and consolidation term

Differences in the soil properties of tilled and untilled soils are evident in many studies (Teiwes, 1988; Arshad et al., 1999; Or et al., 2000; Leij et al., 2002b) and can be seen for example in Figure 5.1. While tillage induces an immediate change in the PSD, there is also a consolidation process which occurs over a larger timescale as larger pores close and the distribution of pore sizes changes (Jiang et al., 2018). In order to incorporate the effect of tillage and consolidation into our model, we utilized data of Teiwes(1988), which presents the contrasting WRC curve from plots which were tilled annually and those which were left untilled for an extended period (in this case, 18 years). Figure 5.2 shows the fitted PSD model before and after tillage. The samples were taken approximately 6 months (≈ 180 d) after the soil was tilled. For each parameter, we obtained the ratio between the no-till and the tilled value. Based on the empirical differences between the PSD of tilled and no-till soils (Figure (5.1) we assume that there is a consistent ratio of the value of each parameter at two given points in time (e.g., the ratio of a parameter immediately after tillage and after maximum consolidation is constant), and that this ratio is independent of the current SOM content. Given the time scales of interest in this work, ranging from days to multiple growing seasons, tillage is modeled as an instantaneous jump of the PSD to the tilled state. This is followed by consolidation, which is modeled by an exponential decline towards the untilled state. Therefore, we write the settling

84 100 NoTill Mid

10-1 m]

10-2 f [1/

10-3

10-2 10-1 100 101 102 r [ m] Figure 5.2: The PSD for the tilled and untilled soils of Teiwes(1988).

term γb as

∗ γb(t) = rb + (1 − rb) exp{−kb(t − 180)}, (5.15) where rb is the ratio of the no-till parameter value to the base value, kb is the time- rate of settling, and t∗ is the time since the soil was tilled. In this way, when t∗ = 180

d (the soil has been tilled 180 days prior), the consolidation term γb = 1 and the modeled PSD matches the data. As t∗ → ∞ (after a long period without tillage) the

consolidation term γb = rb and the PSD approaches the untilled state. By assuming

a constant value for kb, the parameter values at tillage can be obtained by setting t∗ = 0 d. The eﬀect of this term on the parameter values can be seen in the black lines in Figure 5.6, which show the parameter evolution with constant C.

5.4.2 SOM relationship

Organic matter is an important indicator of soil structure and function (e.g. Dexter, 2004; Dexter et al., 2008)), which is demonstrated by the inclusion of SOM as a predictor variable in many PTFs (e.g. Wöstenet al., 2001; Vereecken et al., 2010). Soils which have different levels of organic matter clearly have different properties.

85 In reality, there is likely some overlap between the eﬀect on the PSD of management practices which tend to increase or decrease SOM content and the physical properties of SOM itself, but in this study we take the amount of SOM as a proxy for these

joint eﬀects. The expression for bC is

bC (C(t)) = b0 + σbC(t), (5.16)

where b0 is the parameter value at C = 0 and σb is the slope of the b-C relationship. Here we have used a linear fit, though other relationships may be supported by additional data. We also note that the data to which this relationship was fit extends over a limited range, and it therefore may not hold outside of it (for example, it may not be valid as low as C = 0, despite the fact that b0 appears in the equation). The magnitude of parameter changes over time due to changing C can be seen in Figure 5.6. The three lines in each panel correspond to increasing, constant, and decreasing C. All the data used in this study is from silt loam soils, and it is possible (or even likely) that other types of soils would produce substantially different results. Additionally, the tillage method used in Teiwes(1988) was moldboard plowing, while multiple tillage types were used in Naveed et al.(2014), including an offset disc plow and rotary harrow. Here, we do not consider different effects for different types of tillage, though there are differences (Heard et al., 1988), which could possibly be taken into account by adjusting the ratios between till and no till parameters.

5.5 Evolution of soil hydraulic properties and parameters

In the preceding sections we have developed a model which accounts for the evolution of the PSD as a result of both temporal (tillage and consolidation) and biogeochemical (SOM content) changes in the soil, and which connects the resulting dynamic PSD to important soil hydraulic properties. We will now see the joint eﬀects of these

86 0.83 2 0.828 R =0.87

0.826

0.824

0.822 b [-] 0.82

0.818

0.816

0.814

40 45 50 55 60 65 C [kg/m 3] 140 R2=0.37

130

m] 120 [ m R 110

100

40 45 50 55 60 65 C [kg/m 3]

0.46 R2=0.94

0.44

0.42 [-]

0.4

0.38

0.36 40 45 50 55 60 65 C [kg/m 3] Figure 5.3: The three model parameters as a function of soil organic matter (C), from Naveed et al.(2014).

87 60 B

] 55 3 High input A Constant C Low input

C [kg/m 50

45 C

0 5 10 15 20 25 t [yr] Figure 5.4: Timeseries for the varying SOM levels representing low, middle, and high inputs (of fertilizer, manure, or other amendments which tend to increase SOM) and which were used to drive the results of Figures 5.5 and 5.6. changes on the model. The effects of biogeochemical processes are are shown by varying the SOM levels (see Figure 5.4). We have imposed three theoretical regimes of SOM dynamics, in which SOM declines, remains constant, and increases, respectively. These regimes are intended to represent low, medium and high input levels of fertilizer, manure, or other amendments which tend to promote SOM accumulation. The upper and lower limits of SOM are chosen so that the results are shown over the range of SOM values contained in the data from Naveed et al.(2014) which were used to parameterize the model. The effects of temporal processes are demonstrated by imposing a ‘Till’ period of several years of annual tillage followed by a longer ‘No Till’ period. These periods are separated by the gray lines in Figures 5.5 and 5.6. Additionally, we have denoted 3 points on Figure 5.4 as A, B, and C, which we will use to indicate the differences between the soil PSD and hydraulic properties at each of these points. As previously discussed, two of the most important hydraulic properties which are largely controlled by the PSD are the WRC and the HCC. In Figure 5.5, the effects

88 102 0.4 Till No Till C A B 1 0.38 10 C 0.36 B 100 0.34 [-] w s

0.32 (s) [MPa] 10-1 0.3 Low Input -2 0.28 10 A Constant C High Input 0.26 0 5 10 15 20 25 0 0.2 0.4 0.6 0.8 1 t [yr] s [-]

5 A 5 Low Input B Constant C 4 C 4 A High Input

3 3 [m/d]

sat 2 K(s) [m/d] K 2

1 1 B Till No Till C 0 0 5 10 15 20 25 0.6 0.7 0.8 0.9 1 t [yr] s [-]

Figure 5.5: Timeseries for the wilting point sw (top left) and the saturated hydraulic conductivity Ksat (bottom left) for seven years of tillage followed by a no till period and varying SOM levels. Also shown are the WRC (top right) and the HCC (bottom right) at time points A, B, and C. of tillage on these two properties are shown with a timeseries of the soil wilting point sw (top left) and the saturated hydraulic conductivity Ksat (bottom left) which were estimated from Equations (5.9) and (5.11). Additionally, the right side of Figure 5.5 shows the WRC and the HCC at points A, B, and C, to demonstrate how these changing soil properties can be integrated into an ecohydrological model. In the top panel of Figure 5.5, the wilting point decreases with tillage and for higher levels of SOM. This reﬂects the general purpose of tillage, which is to improve soil structure for crop growth (Hillel, 1980), and lowered wilting points would allow a crop to continue transpiring at lower soil moisture levels. The negative relationship between sw and SOM is also consistent with other studies which considered

89 the impact of SOM on soil properties (for example, Dexter(2004) found a positive correlation between SOM and a measure of soil physical quality).

In the bottom panel, we observe that the hydraulic conductivity Ksat is increased with tillage and for higher values of SOM. Some studies show an increase of soil hydraulic conductivity with tillage (Mielke et al., 1986; Heard et al., 1988; Pikul Jr et al., 1990), which may represent an increase in the number of macropores (although macropores are more continuous in no-till soils, Heard et al.(1988) found them to be more numerous in tilled soils). Macroporosity is not separated from total porosity in our model, but is most closely related to the Rm parameter, which also increases with tillage (see the middle panel of Figure 5.6). However, it is also possible that undisturbed soils could have a higher hydraulic conductivity because their macropores formed by earthworms, insects, or roots are more continuous and are not disrupted during tillage (Ehlers and Van Der Ploeg, 1976; Klute, 1982; Wu et al., 1992). The positive relationship between hydraulic conductivity and SOM can be attributed to an associated increase in macro- and mesoporosity (Mahmood- ul Hassan et al., 2013). The study from which we obtained SOM data, Naveed et al. (2014), did not present hydraulic conductivity data, though it did show a general trend of increasing air permeability with increasing SOM, and it is to be expected that hydraulic conductivity would trend in the same direction as air permeability. Figure 5.6 shows timeseries of the model parameters for a tilled followed by a no-till period and for contrasting SOM levels, as in Figure 5.5. Although the power law PSD has certain limitations in the variety of soil types that it can adequately represent, a decrease in the power law exponent b generally means that for a particular matric pressure, the soil is capable of maintaining a higher water content, which is favorable for crops. The top left panel of Figure 5.6 shows that b decreases with tillage and with higher levels of SOM, which is consistent with the positive correlation of SOM with a soil quality index found elsewhere (Dexter, 2004). The positive

90 0.86 Till No Till C 130 Low Input A Constant C 120 0.85 B High Input 110 0.84

[-] 100 m b [-] R 90 0.83

Low Input 80 0.82 Constant C B A 70 High Input Till No Till C

0 5 10 15 20 25 0 5 10 15 20 25 t [yr] t [yr]

10-1 0.44 Low Input A Constant C B 0.42 A High Input C -2 0.4 10 m]

[-] 0.38 B

f (r) [1/ -3 0.36 10

0.34 Till No Till C 10-4 0 5 10 15 20 25 0 20 40 60 80 100 120 140 t [yr] r [ m]

Figure 5.6: In the top row and bottom left are shown timeseries for the model parameters b, Rm, and φ with exponential decay from the tilled to the untilled state and with tillage causing instantaneous jumps back to the tilled state, for seven years of tillage followed by a no till period. In the bottom right is the PSD at time points A, B, and C.

relationship of Rm with both tillage and SOM (top right panel) can be connected to the associations of both with increased macroporosity, as discussed previously, and is shown in the middle of Figure 5.6. The positive relationships between porosity and SOM and the decrease in φ in the absence of tillage (bottom left panel) can be seen in the bottom panel of Figure 5.6 and have support in the literature (Teiwes, 1988; Dexter et al., 2008; Naveed et al., 2014). The bottom right panel shows the the combined eﬀects of diﬀerent parameter combinations on the PSD at timepoints A, B, and C. We emphasize that the results shown in Figures 5.4-5.6 do not represent a full coupling of the dynamic PSD model to an ecohydrological model. This would include feedbacks between, for example, the soil hydraulic properties and the SOM

91 dynamics, and will also require estimations of other soil properties such as the soil depth from the changes in the PSD. Such a coupling will be explored in a follow-up publication.

5.6 Conclusion

The model described in this study represents realistic trends of key soil properties as they evolve in time and change with soil biogeochemistry in a physically consistent manner. Changes in the key soil properties (indicated here by sw and Ksat) and the parameters which control them (b, Rm, φ) are shown in Figures 5.5 and 5.6. Each panel shows a timeseries of the parameter or variable, with the soil undergoing several years of annual tillage followed by a longer period of consolidation (corresponding to a no till period), for increasing, decreasing, and constant SOM. The general trends shown by the soil properties in Figures 5.5 and 5.6 have support in the literature. Via their dependence on SOM, these dynamic variables can be readily coupled to an ecohydrological model, such as the minimalist models utilized by Rodr´ıguez-Iturbe and Porporato(2004a). We note that the model presented here does not account for very short term changes in soil properties, such as those due to wetting an drying cycles. However, when coupled to an ecohydrological model, the long-term effects of such hydrologic forcing on soil properties could be indirectly accounted for via the changes in SOM which would result from different hydrologic regimes or irrigation strategies. To create Figures 5.5 and 5.6, we imposed three contrasting SOM regimes, which were intended to represent low, middle, and high input of amendments which tend to promote the accumulation of SOM. This model does not consider possible negative effects of tillage such as erosion or increased greenhouse gas emissions. Therefore, the effects of tillage in this model would be considered mostly positive from an agroecosystem point of view, in that (for example) increased porosity and drainage are more likely to improve agricul-

92 tural conditions than degrade them. However, such factors must also be taken into account to properly understand the effects of different management strategies. The results in Figures 5.5 and 5.6 point toward a means to account for the considerable changes which can be brought about in soil properties as a result of contrasting management strategies–specifically, those which increase or decrease SOM, and the effect of tillage. Future contributions will explore the impact of contrasting management strategies (e.g. irrigation, fertilization, organic matter inputs) on the soil structure and properties.

93 6

Exploring the evolution of soil properties with a coupled agroecosystem model

6.1 Introduction

Human activities shape soil properties, which change dynamically, although they are often described statically within ecohydrological models (Vereecken et al., 2010). Soil properties are coupled to the ecohydrological and biogeochemical cycles which are mediated by the soil itself. Agroecosystems contain complex, often nonlinear interactions and feedbacks between the crop and soil components of the system and are also subject to random hydro-climatic forcing (Porporato et al., 2015). Because of the complexity of the coupling between soil properties and agroecosystem water and nutrient cycles, there is a need for ecohydrological models which capture the key feedbacks and dynamics within these systems, including those of soil properties. Agroecosystems cover approximately 30% of global land area (Lambin and Meyfroidt, 2011), so changes in management strategies can have a major impact on global water and nutrient cycles. This chapter aims to better understand the impacts of contrasting management strategies on soil properties and feedback between changes

94 in soil properties and important outputs of agroecosystems. To examine these impacts, we develop a coupled dynamical system describing crop-soil dynamics in agroecosystems, building on related work by other authors in the areas of ecohydrological and biogeochemical modeling (Porporato et al., 2003; Rodr´ıguez-Iturbe and Porporato, 2004a), the dynamic evolution of soil properties (Or et al., 2000; Leij et al., 2002b), and crop modeling (Pelak et al., 2017; Steduto et al., 2009). The soil component of the model includes soil moisture as well as labile and non-labile soil carbon and nitrogen, and is based on a simpliﬁed version of the soil carbon and nitrogen cycling model of Porporato et al.(2003). In order to model the dynamic nature of soil properties, we apply the method described in Chapter5 and which follows the work of Or et al.(2000) and Leij et al.(2002b). This model consists of a power law pore size distribution (PSD) with time-varying parameters, which are connected to natural processes and management interventions such as tillage, consolidation, and changes in organic matter. The crop component of the model includes the balance of carbon and nitrogen aboveground and belowground (roots and shoots), while the concept of critical dilution curves (e.g. Lemaire and Gastal(1997)) is used to track the impact of nitrogen on crop growth, which provides a simple yet realistic representation of the dynamics of crop growth with respect to nitrogen. The model is an expansion of Pelak et al.(2017) and is also based in part on the AquaCrop model of Steduto et al.(2009). We ﬁrst give an overview of the soil component of the model, including the soil biogeochemistry, and then then discuss the crop component and the coupling between the two. Then, we present use the combined model to explore initial results for the the evolution of soil properties under contrasting management strategies. All the results in this chapter are parameterized for maize, though the model can be generally applied to other crops.

95 6.2 Soil component

The soil component of the model is made up of a soil moisture and biogeochemistry

model, which is simpliﬁed into non-labile and labile carbon (CNL and CL) and non- labile and labile nitrogen (NNL and NL). The soil properties evolve in time according the the dynamic PSD model of Chapter5.

6.2.1 Soil moisture balance

The general soil balance equation for soil moisture are

ds φ(t)Z (t) = R(t) + I(s, t) − T r(s, C , t) − Ev(s, C , t) − Q(s, t), (6.1) r dt S S

where s is soil moisture in a soil with porosity φ(t) over a rooting zone with depth

Zr(t). R is the rainfall, modeled as a marked Poisson process with rate λ(t) and exponentially distributed depth α(t) and I is the irrigation rate, which may be a function of s and/or t depending on the strategy employed. T r is the crop transpiration, modeled as

CS T r(s, CS, t) = Ks(s, t)ET0(t)Kcb(t) 1 − exp − β , (6.2) ˜ ˜ CS,max ¸¸

where Ks(s, t) is a factor which deﬁnes the limitations of soil moisture on transpiration capacity, ET0 is the reference evapotranspiration, Kcb(t) is the crop coeﬃcient,

CS is the above ground plant carbon, β is a coeﬃcient which controls the eﬀect of

CS on transpiration, and CS,max is the maximum above ground carbon, which can be speciﬁed for a particular crop (see Section 6.3 for a complete description of the

transpiration and crop component of this model). Ks(s, t) is deﬁned as

 0 s ≤ s (t),  w s−sw(t) ∗ K (s, t) = ∗ s (t) < s ≤ s (t), (6.3) s s (t)−sw(t) w  1 s∗(t) < s,

96 ∗ where sw(t) and s (t) are the wilting point and the point of incipient water stress, respectively. A similar dependence of the transpiration rate on soil moisture can be found in Allen et al.(1998) and Rodr´ıguez-Iturbe and Porporato(2004a). The evaporation is modeled as

CS Ev(s, CS, t) = Kr(s, t)ET0(t) exp − β , (6.4) ˜ CS,max ¸

where Kr is the evaporation reduction coeﬃcient, deﬁned as

( 0 s ≤ s (t), K (s, t) = h (6.5) r s−sh(t) s > sh(t), 1−sh(t)

where sh(t) is the hygroscopic point (Rodr´ıguez-Iturbe and Porporato, 2004a). The

same form for Kr was used in Pelak et al.(2017). This form gives Ev a maximum

at CS = 0 (i.e. bare soil) and then decreases linearly as CS increases. Finally, the leakage of soil moisture out of the rooting zone, Q is given by

c(t) Q(s, t) = Ksat(t)s , (6.6)

in which Ksat(t) is the saturated hydraulic conductivity and c(t) is the power law coeﬃcient (Rodr´ıguez-Iturbe and Porporato, 2004a). We note that all of the soil

∗ properties in this and later sections (e.g. φ, Zr, sh, sw, s , Ksat, and c) have a time-dependence because they vary in time according to the dynamic PSD model of Chapter5.

6.2.2 Soil carbon balance

The non-labile soil carbon (CNL) balance is

dC NL = H (C ,C , t) − DEC (s, C , t), (6.7) dt C S R C NL

97 where HC is the carbon addition to the soil due to harvest residue and DECC is decomposition of non-labile soil carbon, making the non-labile component essentially a delay term. The harvest residue term is given by

HC (CS,CR, t) = fH (t) · p(1 − hfrac)CS + CRq, (6.8)

where fH is a time-dependent term that describes the harvest timing and hfrac is

the fraction of CS that is harvested. CR is the crop root biomass (further explained

in Section 6.3). In this way, (1-hfrac) is the fraction of CS which remains in the soil system, while the entirety of CR is assumed to enter the soil system after harvesting occurs. The decomposition term is given by

DECC (s, CNL, t) = kC fD(s, t)CNL, (6.9)

where kC,DEC is the ﬁrst order decomposition rate of non-labile carbon and fDEC (s) gives the sensitivity of decomposition to soil moisture, for which we assume the following form (Porporato et al., 2003)

( s if s ≤ sfc(t), f (s, t) = sfc(t) (6.10) D sfc(t) s if s > sfc(t),

where sfc(t) is the ﬁeld capacity (Rodr´ıguez-Iturbe and Porporato, 2004a). The labile

soil carbon (CL) balance, which is taken to be the component of the soil carbon pool which drives the dynamic PSD model of Chapter5, is

dC L = DEC (s, C , t) − RESP (s, C , t), (6.11) dt C NL L where RESP is the soil carbon respiration, given by

RESP (s, CL, t) = kRfD(s, t)CL, (6.12)

where kR is a respiration coeﬃcient and fD(s, t) is the respiration dependence on soil moisture, taken to be the same as the decomposition dependence.

98 6.2.3 Soil nitrogen balance

The non-labile soil nitrogen (NNL) balance is

dN NL = H (N ,N , t) − DEC (s, N , t), (6.13) dt N S R N NL where HN is the nitrogen addition to the soil due to crop residue and DECN is decomposition of non-labile soil nitrogen. The concentration of the nitrogen in the residue is assumed to be the same as the concentration of nitrogen in the shoots and roots which make it up, and so the nitrogen addition from this source is

HN (NS,NR, t) = fH (t) · p(1 − hfrac)NS + NRq, (6.14)

where NS and NR are the shoot and root nitrogen, which will be discussed in Section

6.4. The decomposition of non-labile nitrogen is similar in form to DECC and is given by

DECN (s, NNL, t) = kN fD(s, t)NNL, (6.15) where kN is the decomposition rate of non-labile soil nitrogen. The labile soil nitrogen balance is

dN L = DEP (t) + FERT (N ,N ,C , t) + DEC (s, N , t) dt L S S N NL (6.16) − UP (s, NL,CS,NS, t) − L(s, NL, t),

Where NL is labile soil nitrogen, DEP is deposition, FERT is fertilization, UP is

plant nitrogen uptake, and L is nitrogen leaching. The presence of FERT in the NL

but not the NNL equation assumes that the applied fertilizer is composed entirely of labile nitrogen, though it could also include a non-labile component to account for e.g. organic fertilizer. FERT may in general depend on the current state of

labile nitrogen in the soil (NL), the nitrogen status of the crop (NS/CS), or the time,

99 as fertilization is generally applied at speciﬁc points in the growing season. The nitrogen leaching is given by

aDNL L(s, NL, t) = Q(s, t), (6.17) ˜sφ(t)Zr(t)¸

where aD is the fraction of mineral nitrogen dissolved in the soil moisture. With a value between 0 and 1, it is approximately equal to 1 for nitrate but less for ammonium (Porporato et al., 2003). The nitrogen uptake ﬂux UP will be discussed in Section 6.4.

6.3 Plant carbon dynamics

The crop component of the model presented here is based in part on the AquaCrop model (Steduto et al., 2007; Raes et al., 2009; Hsiao et al., 2009) as well as other works on crop modeling and nutrient dynamics (Thornley and Johnson, 1990; Pelak

et al., 2017). We begin with the carbon component, for which CP = CS + CR, where

CP is the total carbon in the plant, CS is the carbon content of the shoots, and CR is the carbon content of the roots. Taking the time derivative gives

dC dC dC P = S + R . (6.18) dt dt dt

To model the increase in biomass, we make use of a normalized above ground water productivity, WP , which is then multiplied by the actual transpiration ﬂux divided by the reference evapotranspiration (Steduto et al., 2007, 2009). With the addition of a negative harvesting term, we have the following expression for the balance of above ground carbon

dC T r(s, C , t) S = WP S − f (t)C dt ET (t) H S 0 (6.19) CS = WP · Kcb(t) 1 − exp − β − fH (t)CS, ˜ ˜ CS,max ¸¸ 100 1.6

1.4

1.2 ] 2 1

0.8 [kg/m s

C 0.6

0.4

0.2

0 0 20 40 60 80 100 120 140 t [d] Figure 6.1: The fully irrigated above ground carbon data from Hsiao et al.(2009), converted to above ground carbon using the conversion factor cfrac = 0.43, compared to the model equation for the above ground carbon, CS, under no stress conditions.

where Kcb(t) is the time-dependent crop coeﬃcient, and the expression with β and

CS,max determines the eﬀect of CS on crop growth (in Section 6.4 we also additionally modify Equation (6.19) with a plant nitrogen dependence). A comparison of this equation with data from Hsiao et al.(2009) for maize grown in unstressed conditions can be seen in Figure 6.1. In order to convert the dry weight biomass values which are typically reported in the literature to carbon, we have assumed that dry weight

biomass has a constant fraction of carbon, cfrac (a value of cfrac = 0.43 for maize was taken from Latshaw and Miller(1924)). Crop coeﬃcients (Allen et al., 1998) are a useful empirical method to account for the evapotanspiration of a crop over the course of a growing season. They are tabulated for a variety of crops and growing situations, and consist of an initial crop

coefficient, Kcb,ini which accounts for the ET at the beginning of the growing season, when its ability to transpire is minimal. From a specified time t1 until t2 the crop coefficient rises linearly to the mid-season value, Kcb,mid, which is constant until a time t3. At this point, it declines linearly until it reaches an end value of Kcb,end at

101 K cb,mid 1 [-] cb K

K cb,end K K cb,ini cb C /C 0 S S,max Tr/ET 0

t t t t t 0 1 2 3 4 t [d] Figure 6.2: Plot comparing the crop coeﬃcient Kcb(t) during the growing season to the above-ground plant carbon CS (normalized by dividing by CS,max) and the transpiration ﬂux T r (normalized by dividing by ET0). The water productivity WP has been parameterized to allow the crop to reach CS,max at t = t4 in the absence of stress.

time t4, accounting for crop senescence and corresponding decreased transpiration as the crop approaches maturity. This form of the crop coeﬃcient is shown by the dashed grey line in Figure 6.2.

In our model, instead of relying on Kcb,ini to account for reduced crop transpiration at the beginning of the season, we relate the potential transpiration of the crop

to the above ground carbon, CS, via an empirical equation

C CC = 1 − exp − β S , (6.20) ˜ CS,max ¸

where CC is the canopy cover fraction, with the assumption that the ratio between transpiration and the maximum potential transpiration is proportional to canopy cover (Steduto et al., 2007), and β is an empirical ﬁtting parameter. A comparison of this equation to data from fully irrigated maize (Hsiao et al., 2009) can be seen in Figure 6.3. Equation (6.20) is similar to the empirical relationship between canopy cover and leaf area index (LAI) given by Equation (1) of Hsiao et al.(2009). Here,

102 1

0.8

0.6

0.4

Canopy Cover [-] Fitted Data 0.2 Remaining Data fit, =15.55 0 0 0.2 0.4 0.6 0.8 1 C /C S S,max Figure 6.3: Canopy cover (CC) and above ground carbon CS (converted from biomass by assuming a constant carbon concentration and normalized by the maximum value for each dataset) from Hsiao et al.(2009). The data in black was included in the fitting of Equation 6.20, while the data in grey was excluded because the canopy cover had begun to decline for those data points. a transpiration reaches a maximum as CC approaches 1. While we do not use an explicitly time-dependent crop coefficient during the earlier stages of the growing season, we do employ one during the late stages of crop development (i.e. between times t3 and t4) when crop transpiration declines. The time-dependent crop coefficient Kcb is then given by ( Kcb,m t ≤ t3, Kcb(t) = (6.21) mkt + bk t3 < t < t4,

Kcb,m−Kcb,e Kcb,mt4−Kcb,mt4 where mk = and bk = − . In Figure 6.2, Kcb(t) can be seen t3−t4 t3−t4 as the solid grey line which merges with the previously described ‘full’ crop coeﬃcient from Allen et al.(1998). In the same ﬁgure, the dashed black line shows the ratio of the potential transpiration to the reference evapotranspiration, including the decline induced by Kcb(t) after t = t3, and the dash-dotted black line shows the value of CS normalized by CS,max. Now that we have described the aboveground carbon dynamics, we now discuss

103 the root:shoot relationship, which we have also incorporated into this model. The root:shoot carbon relationship can be approximated with a power law equation (An- derson, 1988; Bonifas et al., 2005), such as

β2 CR = β1CS , (6.22)

in which β1 and β2 are ﬁtting parameters. A ﬁt of this root:shoot relationship to maize data from Anderson(1988) can be found in the top panel of Figure 6.4. In using a power law relationship, we are assuming allometric allocation rather than the balanced growth hypothesis (Shipley and Meziane, 2002). Taking the time derivative of Equation (6.22) yields

dC dC R = β β Cβ2−1 S , (6.23) dt 1 2 S dt ´ ¯ which is the balance equation for CR.

6.4 Plant nitrogen dynamics

We now turn to the plant nitrogen dynamics. As with carbon, the total plant nitrogen can be written as a sum of the root and shoot components, NP = NR + NS, and we again take the time derivative to obtain

dN dN dN P = S + R . (6.24) dt dt dt

The rate of change of the total plant nitrogen is equivalent to the nitrogen uptake

dNP minus the nitrogen lost to harvest residues, dt = UP − fH (t)(NS + NR) (for the moment we do not include the functional dependencies of the nitrogen uptake UP ). Anderson(1988) reports root:shoot ratios of maize for both carbon and nitrogen, for fertilized and unfertilized treatments, which suggest that root nitrogen concentration can be taken to be approximately constant throughout the growing season, especially

104 0.14

0.12

0.1 ] 2 0.08

0.06 [kg C/m R C 0.04 F=0 180 kg N/ha F=180 kg N/ha 0.02 C = C 2 R 1 S 0 0 0.2 0.4 0.6 0.8 1 C [kg C/m 2] S

0.04 0.035 0.03

[-] 0.025 R 0.02 (N/C) 0.015 F=0 180 kg N/ha 0.01 F=180 kg N/ha 0.005 C = C 2 R 1 S 0 0.07 0.08 0.09 0.1 0.11 0.12 C [kg C/m 2] R 0.08 F=0 180 kg N/ha 0.07 F=180 kg N/ha C = C 2 0.06 R 1 S

[-] 0.05 S

0.04 (N/C)

0.03

0.02

0.01 0 0.2 0.4 0.6 0.8 1 C [kg C/m 2] S Figure 6.4: Data and ﬁtted relationships for root and shoot carbon, CR and CS (top), the root nitrogen ratio NR/CR to the root carbon CR (middle), and the shoot nitrogen ratio NS/CS to the shoot carbon105CS (bottom). Data from Anderson(1988). relative to the aboveground component (see the middle panel of Figure 6.4). If root

NR nitrogen concentration is constant, then = ηC , or NR = ηC CR, and taking the CR time derivative gives dN dC R = η R . (6.25) dt C dt

dCR We now have an expression for NR in terms of dt , which was deﬁned in Equation

dNS (6.23). We can then combine the previous equations to obtain an expression for dt

dN dN dN dC S = P − R = UP − η R − f (t)(N + N ). (6.26) dt dt dt C dt H S R

We must now account for the reduced crop growth when nitrogen is limiting, and deﬁne the nitrogen uptake term, including the ability of the plant to restrict uptake when it has a suﬃcient supply of nitrogen. This is done by making use of nitrogen dilution curves, which are commonly used in agronomy to assess the nitrogen status of a crop, among other applications (Lemaire et al., 2008). The critical dilution curve is a power-law relationship (Lemaire and Gastal, 1997) which gives a critical above ground nitrogen content below which the growth is nitrogen-limited, as a function of above ground crop biomass (which we have converted to carbon in this study).

We multiply the dilution curve by CS to obtain the critical (above ground) nitrogen level as

1−bc NC = acCS , (6.27) where ac is the critical aboveground plant nitrogen when CS is equal to cfrac·1 ton/ha, and 1−bc is the power law exponent for this relationship. A comparison of the critical curve to maize data can be seen in Figure 6.5. While in Parolari and Porporato

(2016) the plant C:N ratio (C/N)p was assumed to be constant, here we will allow it to vary in such a way that it is equal to the dilution curve, unless the plant experiences nitrogen stress, in which case the nitrogen concentration drops below the value speciﬁed by the dilution curve. The bottom panel of Figure 6.4 shows the

106 Figure 6.5: Figure from Lemaire and Gastal(1997), showing the nitrogen dilution curves for a maize crop. Nitrogen levels represented by closed circles correspond to nitrogen levels which are non-limiting, while open circles represent. data from Anderson(1988) alongside the critical curve from our model. Figure 6.6 compares maize timeseries data of CS, CR, NS, and NR from Anderson(1988) for two fertilization treatments (no fertilization and high fertilization) to the model results for unstressed conditions. Because the results of our model (solid black line) are for unstressed conditions, they are a closer ﬁt to the dashed lines, which represent an adequately fertilized crop, than to the dotted lines, which are from an unfertilized crop.

When the shoot nitrogen content, NS, is below the critical level NC , the shoot carbon growth is reduced by a function C∗, which is deﬁned as

 βC  NS ∗ N if NS < NC , C (NS,NC ) = C (6.28) ´ 1¯ if NS ≥ NC ,

∗ where βC is a parameter which controls the shape of the C function, shown in Figure

∗ 6.7. If βC = 1, then C (when NS < NC ) is equivalent to the nitrogen nutrition index

107 1.4 0.14 F=0 kg N/ha 1.2 F=180 kg N/ha 0.12 model 1 ] ]

2 2 0.1 0.8 0.08

[kg C/m 0.6 [kg C/m S R

C C 0.06 0.4 F=0 kg N/ha 0.2 0.04 F=180 kg N/ha model 0 0.02 0 20 40 60 80 100 120 140 0 20 40 60 80 100 120 140 t [d] t [d] (a) (b)

10-3 0.03 5 F=0 kg N/ha 0.025 F=180 kg N/ha model 4 ] ]

2 0.02 2 3 0.015 [kg C/m [kg C/m S 2 R

N 0.01 N

0.005 1 F=0 kg N/ha F=180 kg N/ha model 0 0 20 40 60 80 100 120 140 0 0 20 40 60 80 100 120 140 t [d] t [d] (c) (d)

Figure 6.6: A comparison of model predictions under unstressed growing conditions and data from Anderson(1988), for a) above ground carbon CS, (b) root carbon CR, (c) above ground nitrogen NS, (d) root nitrogen NR.

(NNI) (e.g. Lemaire and Meynard, 1997). Lemaire and Meynard(1997) present data suggesting that the crop carbon accumulation rate decreases linearly with NNI, and

∗ therefore we set βC = 1. C appears in the dynamical system as

dCS ∗ T r(s, CS, t) = C (NS,NC ) · WP · Kcb(t) · − fH (t)CS. (6.29) dt ET0(t)

The rate of nitrogen demand DEM(t) is equal to the sum of the rates of increase

108 1

0.8

0.6 [-] * C 0.4

0.2

0 0 0.5 1 1.5 2 N /N [-] S C ∗ Figure 6.7: The growth limitation term, C , as a function of NS/NC , the ratio of the actual shoot nitrogen to the critical shoot nitrogen content.

of N in the shoots and roots

dN dN DEM(t) = S + R . (6.30) dt dt

If nutrients such as nitrogen are available in the rooting zone, crops are generally able to meet their nutrient needs. Therefore we assume that the nitrogen uptake UP is equal to DEM, unless there are exceptionally low levels of nitrogen, as would occur in a depleted soil. If NL falls below a given value, NL,C , it is reduced by a factor N ∗

( NL NL < NL,C , ∗ NL,C N (NL) = (6.31) 1 NL ≥ NL,C .

The expression for uptake is then

∗ UP = N (NL) · DEM(t). (6.32)

We also assume that if the nitrogen uptake is insuﬃcient to meet the demand from both roots and shoots, the demand for the roots is met before that of the shoots.

109 6.4.1 Critical curve: pre- and post- vegetative stage

The power law critical dilution curve, Equation (6.27), is valid only during the vegetative stage of crop growth, when the aboveground dry weight biomass is approximately 1 t/ha to 22 t/ha for maize, which normally occurs at approximately 600 degree-days or 25-30 days after ﬂowering (Lemaire and Gastal, 1997). We therefore require a diﬀerent critical curve relationship when biomass is outside of this range. When is it below, Lemaire and Gastal(1997) and others propose a constant N con-

centration which is equal to ac, which we also adopt here. When the biomass is above this range, the slope in a log-log scale of the NS concentration to CS curve

(i.e. the parameter bc) decreases, so we likewise adopt a reduced bc value once the

0 crop has passed the vegetative stage (bc). At the end of the vegetative growth stage, the crop has taken up approximately 95% of the total amount that will be taken up, as during the later, grain-filling stages, the crop remobilizes nitrogen from leaves and other parts of the plant to put into the grain. This means that the post-flowering nitrogen concentration will not, in any case, have a large influence on the overall crop nitrogen balance. With this in mind, the equation for the critical nitrogen content

NC over the entire range of aboveground crop biomass is

 a C if C < c · 1 [t/ha],  c s S frac 1−bc NC (CS) = acCS if cfrac · 1 [t/ha] ≤ CS ≤ cfrac · 22 [t/ha], (6.33) 0  0 1−bc acCS if CS > cfrac · 22 [t/ha].

0 0 0 bc−bc Note that bc is the new slope and ac = acCf is a parameter chosen such that the critical curves above and below Cf = cfrac·22 t/ha meet.

6.5 Evolution of soil properties

The results presented in this section are intended as initial explorations of the coupling between the agroecosystem crop-soil model developed in this chapter and the

110 dynamic PSD model of Chapter5. Here we focus the impact of contrasting management strategies on soil properties, and do so by varying the irrigation (rainfed vs. stress avoidance), tillage (till vs. no-till), and crop residue management strategies (returning a low, medium, and high amount of crop residue at the end of each growing season). The same fertilization treatment (a single amount applied at the beginning of the growing season) was used for all simulations. Because the main purpose of these results is to demonstrate the impact of each management strategy on the soil properties, rather than to represent a particular climate in detail, we assume constant climatic conditions (i.e. constant reference evapotranspiration and rainfall parameters). The simulations are run for a period of 25 years, which is sufficient time for large differences in soil properties to develop between the different treatments. We highlight four key variables, which are intended to represent a variety of important changes in the soil biogeochemistry and hydraulic properties. The first is

the labile soil carbon, CL, as we use it for the soil organic matter (C) variable which in Chapter5 was used to encapsulate the eﬀect of soil biogeochemical changes on

the soil properties. We use CL rather than the non-labile soil carbon CNL because the latter experiences jumps (in particular at the end of the growing season when crop residues are added to the soil) which would result in corresponding unrealistic jumps in the parameters of the PSD. Figure 6.8 shows the change in CL for different combinations of strategies. Here, unsurprisingly, the main driver of differences is the amount of crop residue which was added (Naveed et al., 2014), while the tillage strategy (grey vs. black) did not make a substantial difference in the amount of soil carbon. The top panel is for rainfed irrigation while the bottom panel is for stress avoidance irrigation, with the latter having somewhat higher levels of CL due to the increased crop yields (and thus increased residue amounts) of irrigated vs. rainfed crops. The second variable which we present is porosity, φ, which is shown in Figure

111 32

30 ]

2 28

26 [kg C/m L

C 24 lowC-NT midC-NT highC-NT 22 lowC-T midC-T highC-T 20 0 5 10 15 20 25 t [yr] 34

30 ] 2 28

26 [kg C/m L C 24 lowC-NT midC-NT highC-NT 22 lowC-T midC-T highC-T 20 0 5 10 15 20 25 t [yr] Figure 6.8: This ﬁgure shows the eﬀects on the labile soil carbon CL of no till (grey) vs. tillage (black) strategies and crop residue management strategies: no residue returned (dotted line), half of residue returned (dashed line), and all residue returned (solid line). The top panel is for rainfed agriculture and the bottom is for stress avoidance irrigation.

112 6.9. The largest porosity values occurred with the high crop residue input and tilled strategies, while the smallest were with the low residue and no till strategies. There were not large diﬀerences between the irrigated and rainfed cases, and we note that at the end of the simulation period the porosity under low residue input and tillage was approximately the same as under high residue input and no tillage, highlighting that diﬀerent combinations of management strategies may result in the same soil

properties. In Figure 6.9, once the value of CL moves above or below the limits of the soil carbon in the data which were used to parameterize the model, additional changes in CL were not considered to impact the soil properties, and this is the reason for the change in slope which can be seen in the trajectories of porosity as well as the other soil properties presented in this section. This was done so that the relationships which were obtained, in this case between porosity an soil organic matter, were not extended past the range over which they are supported by data (Naveed et al., 2014). As mentioned previously, the water retention curve is an important soil hydraulic property which can be strongly impacted by management practices (Dexter, 2004; Vereecken et al., 2010), and which also has implications for plant growth and response to water stress (Rodr´ıguez-Iturbe and Porporato, 2004a). Figure 6.10 shows a timeseries for the soil wilting point sw for each of the management strategies. The wilting point for the tilled cases is much lower than for the untilled case, which reflects the general purpose of tillage in improving soil hydraulic properties for the promotion of plant growth (Hillel, 1980). Smaller differences come about as a result of the residue strategy, with higher residue inputs leading to lower values of sw, which is consistent with many studies which relate organic matter to soil quality (e.g. Dexter, 2004). There are again not large differences between the rainfed and irrigation strategies. Another key soil hydraulic property is the hydraulic conductivity curve. To illustrate the changes in the HCC, Figure 6.11 shows the change in the saturated

113 0.46

0.44

0.42

0.4 [-] 0.38

0.36 lowC-NT midC-NT highC-NT 0.34 lowC-T midC-T highC-T 0.32 0 5 10 15 20 25 t [yr] 0.46

0.44

0.42

0.4 [-] 0.38

0.36 lowC-NT midC-NT highC-NT 0.34 lowC-T midC-T highC-T 0.32 0 5 10 15 20 25 t [yr] Figure 6.9: This ﬁgure shows the eﬀects on the porosity φ of no till (grey) vs. tillage (black) strategies and crop residue management strategies: no residue returned (dotted line), half of residue returned (dashed line), and all residue returned (solid line). The top panel is for rainfed agriculture and the bottom is for stress avoidance irrigation.

114 0.4 lowC-NT midC-NT 0.38 highC-NT lowC-T midC-T 0.36 highC-T

0.34 [-] w

s 0.32

0.3

0.28

0.26 0 5 10 15 20 25 t [yr] 0.4 lowC-NT midC-NT 0.38 highC-NT lowC-T midC-T 0.36 highC-T

0.34 [-] w

s 0.32

0.3

0.28

0.26 0 5 10 15 20 25 t [yr] Figure 6.10: This ﬁgure shows the eﬀects on the wilting point of no till (grey) vs. tillage (black) strategies and crop residue management strategies: no residue returned (dotted line), half of residue returned (dashed line), and all residue returned (solid line). The top panel is for rainfed agriculture and the bottom is for stress avoidance irrigation.

115 hydraulic conductivity under different management strategies. The tilled soils exhibit much larger values of Ksat than in the no-till case, resulting from several factors, including an increase in the porosity and the maximum effective pore radius (see the expression for K(s) in Chapter5). Again, little difference was observed between the rainfed and irrigated cases.

6.6 Conclusion

We have developed a novel agroecosystem model for soil-crop interactions which takes into account soil biogechemistry, crop carbon assimilation, and nitrogen dynamics and builds on previous work in these areas (Porporato et al., 2003; Steduto et al., 2009; Pelak et al., 2017). This model was then coupled with a dynamic pore size distribution evolution model (presented in detail in Chapter5), which allowed for the incorporation of soil properties which change dynamically as a result of temporal and biogeochemical processes. The resulting coupled agroecosystem model was then used to make an initial exploration of the impact of contrasting management strategies, such as irrigation, fertilization, tillage, and crop residue management, on the soil properties. The results focused on the eﬀect of each strategy on the labile soil carbon, porosity, soil wilting point, and saturated hydraulic conductivity, which provided a representative sample of key soil variables and properties which drive the feedbacks between soil properties and biogeochemical and ecohydrological processes in agroecosystems.

116 6 lowC-NT midC-NT 5 highC-NT lowC-T midC-T highC-T 4

[m/d] 3 sat K 2

0 0 5 10 15 20 25 t [yr] 6 lowC-NT midC-NT 5 highC-NT lowC-T midC-T highC-T 4

[m/d] 3 sat K 2

0 0 5 10 15 20 25 t [yr] Figure 6.11: This ﬁgure shows the eﬀects on the saturated hydraulic conductivity of no till (grey) vs. tillage (black) strategies and crop residue management strategies: no residue returned (dotted line), half of residue returned (dashed line), and all residue returned (solid line). The top panel is for rainfed agriculture and the bottom is for stress avoidance irrigation.

117 7

Conclusion

In Chapter2 we developed a minimal model of soil-vegetation interaction to study the feedbacks between soil production, erosion, and vegetation. Bistable conditions were found for certain regimes of soil and vegetation dynamics and harvesting pressures. When the system is bistable, an increase in harvesting pressure beyond a certain threshold leads to soil and vegetation loss and an eventual collapse to a degraded soil state. Rather than soil resources, in Chapter3 we focused on the optimal use of water resources. We extended results which were originally used to characterize the probability distribution of the soil water balance to obtain an analytic solution for the optimal volume of a rainwater harvesting (RWH) cistern as a function of the collection area, water demand, rainfall characteristics, and costs of utility water and cistern construction. We also analyzed the eﬃciency of particular RWH systems based on their conﬁguration. The work of Chapter4 considers the optimal management of water and nutrient cycling in an agroecosystem context. We developed a minimal crop model which captures the key elements of the soil-crop system and which was based on more

118 complex crop models. This model was then used to explore how changing climate parameters and irrigation and fertilization strategies impact crop yield, proﬁtability, and other important metrics. In Chapter5, we extended our studies of agroecosystem dynamics to also account for changing soil properties. Soil properties change in time as a result of management activities and hydrological and biogeochemical processes, but such changes are often not taken into account in ecohydrological models. We developed an evolution equation for the soil pore size distribution (PSD), which has as one of its solutions the power law distribution. The parameters of the soil PSD were connected to soil management and biogeochemical processes, key soil properties were derived from the PSD, and their temporal evolution was decribed. Finally, in Chapter6, we applied the dynamic PSD model from Chapter5 in an agroecosystem model. To do so, it was coupled to a novel crop model, based in part on the dynamic crop model of Chapter4, which was capable of modeling crop growth with water and nutrient stress, and to a soil biogeochemical cycling model. An initial exploration of the the eﬀect of this coupling on soil properties was made. Taken together, the chapters in this dissertation provide insights into several important topics relating to the optimal use and management of soil and water resources in ecohydrological systems. These topics range from soil-vegetation interactions and ecosystem stability, to rainwater harvesting, to agroecosystem modeling and the evolution of soil properties. A variety of models and tools have been used, but especially minimal ecohydrological models, stochastic modeling, and dynamical systems theory.

119 Appendix A

Analytical results for the stochastic water balance

A.1 Chapman-Kolmogorov equation and derivation of steady-state PDF

The Chapman-Kolmogorov forward equation for the probability of the normalized variable c is (Cox and Miller, 1977)

∂ ∂ Z c p(c, t) = [p(c, t)ρ(c)] − λp(c, t) + λ p(u, t)fY (c − u, u)du + λp0(t)fY (c, 0). ∂t ∂c 0 (A.1) The atom equation is

d p (t) = −λp (t) + ρ(0)p (0, t). (A.2) dt 0 0 0

The loss function ρ is constant and equal to the constant water demand rate h

−γy R ∞ −γu normalized by dividing by V . Note that fY (y, s) = γe + δ(y − 1 + c) 1−c γe du. The steady state solutions are given in Equations (3.6) and (3.7).

120 A.2 Crossing Time Analysis

The mean length of time, T¯, spent by the system under a generic threshold ξ is

D 1 T¯(ξ) = c eξ(Dcγ−γ) − . (A.3) λDc − 1 λDc − 1

For the case of ξ = 0, this reduces to

1 T¯(0) = . (A.4) λ

The frequency of downcrossings (or upcrossings) is

ξ(Dcγ−γ) ν(ξ) = λp0e , (A.5)

which reduces to ν(0) = λp0 if the threshold ξ is equal to 0. The expected number

of occurrences of an empty cistern over a given period of time T is therefore λp0T . A detailed derivation of the preceding equations can be found in Rodr´ıguez-Iturbe and Porporato(2004b).

121 Appendix B

Method of characteristics

The method of characteristics (Logan, 2013) can be used to analytically solve the evolution equation for the soil PSD, which is

∂f(r, t) ∂ = ρ(r, t)f(r, t) − m(r, t)f(r, t), (B.1) ∂t ∂r ” ı where the drift term ρ(r, t) is

r ρ(r, t) = a0(t) − a(t)b0(t) ln(r) , (B.2) a(t)b(t) ´ ¯ and the source/sink term m(r, t) is

a0(t) b0(t) m(r, t) = − 1 + ln(r) . (B.3) a(t)b(t) b(t) ´ ¯ Equation (B.1) simpliﬁes to

∂f(r, t) r ∂f(r, t) = a0(t) − a(t)b0(t) ln(r) (B.4) ∂t a(t)b(t) ∂r ´ ¯ with the initial condition

−b(0) f(r, 0) = f0(r) = a(0)r , 0 < r < Rm(0), t = 0. (B.5) 122 In the method of characteristics, the coordinate system in (r, t) is changed to a new

one in (r0, s) in such a way that the PDE becomes an ODE along certain curves, (r(s), t(s)). These curves are referred to as the characteristic curves. To ﬁnd r(s) and t(s), we solve the characteristic equations, which are

dr dr r = = a0(t) − a(t)b0(t) ln(r) , (B.6) ds dt a(t)b(t) ´ ¯ and dt = 1. (B.7) ds

From the second characteristic curve, s = t. From the ﬁrst, using the initial condition

r(0) = r0, we obtain

b(0) 1 r a(t) b(t) r(t) = 0 , (B.8) a(0) ´ ¯

and solving for r0,

b(t) 1 a(0)r b(0) r (r, t) = . (B.9) 0 a(t) ´ ¯ Inserting this into the initial condition yield the power law as the solution,

−b(t) f(r, t) = f0(r0) = a(t)r , 0 < r < Rm(0), t = 0. (B.10)

123 Appendix C

Water retention curve

The relative soil moisture s is deﬁned as

θ s = . (C.1) φ

Following Brutsaert(2005), we assume that soil moisture will occupy smaller pores first, allowing us to set the relative soil moisture s(r) equal to the P (r), the cumulative distribution function (CDF) of the pore size distribution (where r is the effective radius of the largest pore which is filled at the specified soil moisture level),

s(r) = P (r). (C.2)

We can invert this expression and solve for r as

r = P −1(s). (C.3)

We then apply the Young-Laplace equation, which gives the pressure diﬀerence across the interface between two immiscible ﬂuids (Brutsaert, 2005)

1 1 ∆p = σ ± , (C.4) R1 R2 ´ ¯ 124 in which R1 and R2 are the radii of curvature of the interface and σ is the surface tension. Assuming that the capillary tube is sufficiently small that the interface is approximately spherical, we define the radii of curvature as R = r/cos β, where r is the pore radius and β is the contact angle of the air-water interface. For a sphere, the radii of curvature are also equal, so that we can now write the previous equation as 2σ cos β ∆p = . (C.5) r The contact angle for water and typical soil materials is generally small, so we let β → 0, and define the pressure difference across the interface as the matric pressure,

∆p = −Ψs. We then obtain the following expression for the matric pressure as a function of r C Ψ (r) = − s , (C.6) s r where Cs = 2σ. By substituting the previous expression for r(s), we can obtain the matric pressure as a function of s (i.e. the soil water retention curve)

C Ψ (s) = − s . (C.7) s P −1(s)

With the power law PSD as our starting point, we obtain the following expression for the WRC

Cs −1/(1−b) Ψs(s) = − s , (C.8) ˜Rm ¸

At the extreme values of s, this function is no longer dependent on the power law exponent b:

Cs Ψs(s = 1) = , (C.9) Rm which is known as the air-entry pressure, and

Ψs(s = 0) → ∞. (C.10)

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140 Biography

Norman Frank Pelak III attended the University of Alabama, where he earned a Bachelor of Science Degree in Civil Engineering with a minor in Environmental Engineering. He then moved to Durham, North Carolina to begin his PhD at Duke University in the Department of Civil and Environmental Engineering, during the course of which he earned his Masters Degree in Environmental Engineering in 2016. In 2017, he relocated to Princeton University with his advisor, Amilcare Porporato, while remaining a student at Duke University. During his time in graduate school, he was awarded the National Defense Science and Engineering Graduate Fellowship and received the 2016 Utku Award from the Department of Civil and Environmental Engineering for his paper, Bistable Plant-Soil Dynamics and Biogenic Controls on the Soil Production Function. In the summer of 2019 he will begin a postdoctoral position in the Sierra Nevada Institute at the University of California, Merced, where he will work on a project funded by the National Institute for Food and Agriculture that aims to model the ground and surface water in California’s Central Valley and to assess strategies for enhancing groundwater recharge and storage for the forest and valley ﬂoor.

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