ecohydrological modelling

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Prelude

Information for students

These notes are used in the module Ecohydrological Modelling that takes place each winter term. In this module, we lecture for six weeks and the second part of this module is taken over by our colleagues at the Leichtweiß-Institute.

A rough schedule of our six weeks together is as follows:

Week 1 Introduction to ecohydrology
Week 2 Plant physiology and soil physics
Week 3 Soil water budget and Richards(on) equation
Week 4 Soil–plant–atmosphere continuum modelling
Week 5 Modelling plant–water relations
Week 6 Terrestrial biogeochemistry

Introduction

Rüsvai de aşk-ı emraz
Bin yıl etse methimi az
Aşıkların elinde saz
Dahi kaval kemanım ben

— Ali Cem Akbulut, Ağaçname

What is ecohydrology?

Ecohydrology is a subdiscipline of hydrology that focuses on the bidirectional interactions among the water cycle, the biosphere, and the microclimate.

Hydrology is the study of the circulation of water between the Earth and its atmosphere—the hydrological cycle, shown in Fig. 1. This cycle has no beginning nor end. Water evaporates from the oceans and the land surface to become part of the atmosphere, where it is transported. After some time, water precipitates back on the land surface and the oceans, where it is partitioned into overland flow and subsurface flow, and groundwater. From here, water eventually discharges into streams as surface runoff, where it either evaporates or flows until reaching the ocean and thus, restarting the hydrological cycle.

ecohydrology_c.png

Figure 1: A conceptual sketch of the water cycle with green and blue water flows.

Following Falkenmark and Rockstom (2004)°, we can categorise water flow in terrestrial ecosystems as blue and green water flows. Blue water flow is the liquid water flow moving above and below the ground into the oceans or groundwater aquifers—that is to say, surface and subsurface runoff. Green water flow is the vapor water flow to the atmosphere—evapotranspiration. Green water flow outweighs blue water flow in terrestrial systems by 61% to 39% of the total precipitation. In this context, recent field observations have given rise to the so-called two water worlds hypothesis: plants and streams return distinct water pools to the hydrosphere, that is to say that there are two distinct water masses involved in blue and green water flows that do not exchange (Goldsmith et al., 2011)°.

Ecological processes play a crucial role in the partitioning of blue and green water flows and thus, the hydrological cycle. For example, plants play an important role in runoff partitioning and generation through leaf interception, evapotranspiration, and root water uptake. While plants affect the hydrological cycle, hydrological processes also affect plants—water availability is crucial in determining vegetation patterns and plant growth. This strong interlinkage between plants and water makes it probably impossible to consider interactions in a unidirectional manner (Baird and Wilby, 1999)°. We shold acknowledge that animals also affect hydrological processes (Müller et al., 2014)°. For example, earthworms create preferential flow paths that route the water faster through the subsurface and beavers build dams that block and divert water inside catchments.

The soil–plant–atmosphere continuum

A fundamental unit of terrestrial ecohydrology is the soil–plant–atmosphere continuum (SPAC), which describes the path taken by the water from the soil to the leaf, where it evaporates into the atmosphere. The SPAC controls two key ingredients of the hydrological cycle: (i) root water uptake and (ii) transpiration. These processes are heavily influenced by the plant itself, the soil, and the hydrological conditions of its surroundings. At the same time, the transpiration from the plants affects the properties of the atmospheric boundary layer, which then feedbacks into the local microclimate. Thus, to accurately model these processes, a bidirectional interaction between plant and water needs to be considered.

The SPAC is typically modelled through a series of resistances that represent different compartments of the plant, for example, the root, the xylem (and sometimes the phloem), and the stomata (Ruffault et al., 2022)°.

Julia

The tool doesn't matter, but we have to choose one. Here we choose Julia°, a programming language that specialises in scientific computing. If you have previous knowledge of the programming languages Matlab or Python, you will be able to read Julia without any problems.

Recommended reads

There are several good text books on ecohydrology available:

Text books on related topics:

Fundamentals

Water potential

Plant physiology

We will now briefly review fundamental plant physiology°. We put our focus on vascular plants—a large fraction of terrestrial vegetation that feature lignified tissues for transporting water and minerals, called the xylem.

The plant cell

A generalised plant cell° consists of a cell wall that surrounds the protoplast. The cell wall is composed of cellulose and contains insterstices that allow water and solutes to enter the cell. Behind the cell wall, the plasma membrane is found, which regulates the exchange of the cell with its environment. The plasma membrane surrounds the cytoplasm, which contains chloroplasts and mitochondria. The chloroplast is the site for photosynthesis. The mitochondria is the site for respiration. The centre of the cell is occupied by the central vacuole, a large aqueous compartment surrounded by a membrane called the tonoplast.

The leaf

Plants must navigate through a range of potentially adverse environmental conditions. The leaf° intercepts sunlight and uptakes CO2 during photosynthesis to grow. During photosynthesis, the plant loses water due to evaporation. The plant aims to optimise this process by maximising CO2-uptake while minimising water loss. In nature, we observe different shapes and sizes of leaves. This means that different strategies exist to approach these trade-offs.

In general, leaves are 4–10 cells thick. The upper and lower sides of the leaf feature a colorless epidermis layer with a thickness of one cell. Epidermal cells have a waterproof cuticle on the outwards facing side, which minimises water loss through the epidermis. The lower epidermis contains guard cells that can open and close the stomatal pore to regulate photosynthesis. These cells are of upmost importance to the field of ecohydrology and we will discuss them in more detail later. Between the epidermal layers, the leaf consists of chloroplast-containing mesophyll tissue. Palisade mesophyll cells and spongy mesophyll cells are loosely packed and provide intercellular air spaces to enable fast diffusion of gases into and out of the cells.

The vascular tissue

Vascular systems transport water, nutrients, and organic compounds through the plant. They span from the roots through the stem to the leaves. The vascular tissues that build these systems are the xylem° and the phloem°. The xylem is the woody part of the plant that conducts water. The phloem is usually located external to the xylem, for example, in trees, phloem builds the bark. Xylem and phloem differentiate from the cambium, which is the region of meristematic activity.

In the xylem, water and nutrients are transported from the soil to the upper parts of the plant. The conducting cells of the xylem tissue are narrow, elongated, hollow dead cells. In lower plants, they are called tracheids. In higher plants, they are called vessel members, which have adjoining end walls with small holes, called perforation plate. Water and nutrients move through these vessel members in the direction of lower hydrostatic pressure.

In the phloem, organic compounds are distributed throughout the plant in both directions. The flow rate in the phloem (about \(0.6~\mathrm{mh^{-1}}\)) is smaller than the xylem flow rate (\(1\) – \(6~\mathrm{mh^{-1}}\) in general with up to \(16~\mathrm{mh^{-1}}\) in wide vessel members). The conducting cells of the phloem tissue—referred to as sieve cells—form a transport system through the whole plant. In contrast to the xylem, these sieve cells contain cytoplasm.

The root

The root° uptakes water and nutrients from the soil towards the stem. The tip of the root is referred to as the root cap, which is the part of the root that grows into the soil (with up to \(0.002~\mathrm{mh^{-1}}\)). The apical meristem is located above the root cap, followed by a cell elongation zone, where cells elongate along the direction of the root axis to push the root cap further. The cell elongation zone is followed by a region of cell differentiation, where cells differentiate to form root hair. Root hair increases the root surface area and increases nutrient and water uptake. As the root approaches the stem, it becomes thicker. The root now specialises on conducting the water to the stem and its surface gets less permeable to water.

In the region of cell differentiation, moving from the root epidermis inside the root, we first encounter a number of cell layers called the cortex. These layers feature air spaces that enable the diffusion of O2 and CO2 in this tissue. Inside the cortex, the endodermis is a layer of cells with cell walls that feature a waxy surface. These walls form a band with low permeability, called the Casparian strip, which hinders water and solutes to move directly inside the root. Instead, water must move through lateral walls and enter the cytoplasm of endodermal cells. This gives the plant a means to regulate water and nutrient uptake. After passing the endodermal cells, water moves in the so-called apoplast, a region consisting of cell walls and xylem vessel members.

Plant growth

Vascular plants grow from a single cell called the zygote. As the plant grows longitudinally, it develops a polar structure that lead the growth. The poles are called apical meristems. Meristematic cells are undifferentiated cells that are capable of cell division to develop into other tissues in the plant. Cell division continues until they get differentiated and lose their ability to divide. From these poles, the shoot and the root systems emerge. Plants with extended lifespans feature additional meristem layers that are called cambium in their roots and stems, which increase the girth of the plant along its longitudinal axis.

Plant growth analysis distinguishes between absolute growth rate (AGR [\(\mathrm{gd^{-1}}\)]) and relative growth rate (RGR [\(\mathrm{d^{-1}}\)]). These are defined as functions of the total biomass (M [\(\mathrm{gg^{-1}}\)]) as

\begin{equation*} \mathrm{AGR} = d_t \mathrm{M}, \end{equation*}

and

\begin{equation*} \mathrm{RGR} = \frac{1}{M} d_t \mathrm{M}. \end{equation*}

Typical values of RGR for tree seedlings range between 10–100 \(\mathrm{mg g^{-1} d^{-1}}\).

Conceptually, we often link the plant's net growth to its net CO2 assimilation rate (An [\(\mathrm{gm^{-2}d^{-1}}\)]), which is controlled by leaf area (LA [\(\mathrm{m^2}\)]). An can be calculated as

\begin{equation*} \mathrm{A_n} = \frac{1}{\mathrm{LA}} d_t \mathrm{M}. \end{equation*}

The RGR can be expressed in terms of An as

\begin{equation*} \mathrm{RGR} = \mathrm{A_n} \frac{\mathrm{LA}}{\mathrm{LM}} \mathrm{LMF}, \end{equation*}

where LM [\(\mathrm{g}\)] denotes the mass of leaves and LMF [\(\mathrm{gg^{-1}}\)] is the leaf mass fraction, denoting the ratio of LM to total mass.

As trees increase in size, RGR and AGR decline, because large allocation of biomass to the trunk instead of leaves, which decreases the photosynthesis rate, and thus, reduces An. For decidious trees, these rates become seasonal because most of the biomass growth is in the leaves.

A simple software for plant growth analysis is provided by Hunt et al. (2002)°.

Tree allometry

Depending on environmental conditions and available C to produce sugar, plants allocate biomass to different compartments—leaves, roots, and stems—to gain advantage in their environment. The proportions of between these compartments is the subject of tree allometry. The balance in biomass allocation° between plant compartments is referred to as functional equilibrium. The biomass allocation is typically expressed in fractions of total biomass: the leaf mass fraction (LMF), the stem mass fraction (SMF), and the root mass fraction (RMF). These denote dry mass and have the unit [\(\mathrm{gg^{-1}}\)].

Tree allometry° considers proportions between characteristic dimensions of trees and relations to other properties. The aim is to predict tree measurements that are difficult to obtain from more convenient tree measurements such as diameter at breast height (DBH) or tree height. For example, a typical application is to estimate tree biomass from DBH (Picard et al., 2012)°. The relations used are rather empirical and as such, rely on regression analyses of field data. Such allometric equations for common trees have been put in databases, for example GlobAllomeTree°. An additional challenge when formulating these allometric equations is that environmental factors and tree age affect these relations.

Cohesion–tension theory for sap flow

The cohesion–tension theory explains how water travels over long distances from the root to the leaf. The theory goes back to (Böhm, 1893)° and plays a key role in our understanding of plant functioning. Simply put, the cohesion–tension theory for sap flow postulates that the evaporation of water through the stomata lowers the water potential in the leaves and causes water to move from the xylem to the stomatal cavities. This in turn causes the water potential in the xylem to drop. Now, some key assumptions are made for this theory to work: firstly, it is assumed that the cohesion due to hydrogen bonds between water molecules is strong enough to create a continuous water ''chain'', and that this chain—supported by adhesion to the xylem walls and the surface tension—is able to withhold the large tension created by the aforementioned evaporation. Secondly, it is assumed that this water chain forms a continuous system from the leaf through the stem to the roots and into the soil—the SPAC. Thus, the tension created in the leaves travels through this water chain to the root, lowering the water potential along the way. This is sufficient to lower the water potential in the root below the water potential in the soil, such that water moves from the soil into the root. Once it reaches the xylem, it then travels along the gradients in water potential to the leaves.

Soil physics

Soil properties

Soil° is a complex biomaterial that is often idealised as a mixture of mineral, liquid, and gaseous components. It stores water and nutrients that are crucial for life on Earth. It also controls the partitioning of water and energy fluxes in terrestrial systems. Soils form from the weathering of bedrock. The weathering is driven by climate conditions that trigger biogeochemical processes. This is why we observe that soil develops in horizontal layers that we call soil horizons.

Soil horizons° are horizontal layers in the subsurface with distinct characteristics. The “O” horizon is rich with organic material and develops from recent accumulation of plant litter. The underlying “A” horizon is a mixture of minerals and accumulated organic matter. Most roots are found in the “A” horizon. Subsequent B and C horizons have undergone less and less weathering and feature less and less organic matter until unweathered bedrock is reached. The soil water budget typically considers water stored in the O and A horizons. See Fig. 2, for a sketch of the soil horizons.

soil_horizons.png

Figure 2: Soil horizons, reproduced from wikipedia.

Soil textural classes° are a helpful taxonomy to describe soil properties relevant to ecohydrology, especially regarding their permeability and water storage properties. The distinction is made on the basis of particle size. Here, we recognise three “true” soils. Sand particles are between 2 mm and 0.05 mm, silt particles between 0.05 mm and 0.005 mm, and clay particles are less than 0.05 mm. Particles above 2 mm are considered to be gravel. All other soils are considered to be a mixture of these true soils and can be classified according to the U.S. Department of Agriculture's soil texture triangle°.

The soil water budget

The soil water budget quantitatively describes how the water input is partitioned into outputs and storage. The primary water input to the soil is precipitation and agricultural sites may also have irrigation water input. Outputs from the soil are evaporation losses, drainage, overland runoff, and subsurface runoff. The difference between inputs and outputs is the storage component of the soil water budget. These components are sketched in Fig. 3 below.

swb.png

Figure 3: Components of the soil water budget. The balance considers the change of soil moisture in the root zone (brown area).

Balancing these inputs and outputs gives a simple mass balance that tracks the changes in soil moisture as

\begin{equation*} w_0 \frac{ds}{dt} = R + J - (C_i + E_v + E_t + L + Q), \end{equation*}

where the definitions of all terms are summarised in Tab. 1. We calculate the soil storage capacity as \(w_0 = n Z_r\). Typical values for the rooting depth \(Z_r\) range between 1 and 2 m.

Table 1: Components of the soil water budget equation.
Symbol Unit Definition
\(n\) \(-\) Porosity of the soil
\(Z_r\) \(\mathrm{mm}\) Rooting depth
\(w_0\) \(\mathrm{mm}\) Soil storage capacity per unit area
\(t\) \(\mathrm{h}\) Time
\(s\) \(-\) Vertically averaged relative soil moisture (\(s \in [0,1]\))
\(R\) \(\mathrm{mm/h}\) Precipitation rate
\(J\) \(\mathrm{mm/h}\) Agricultural irrigation rate
\(C_i\) \(\mathrm{mm/h}\) Canopy interception rate
\(E_v\) \(\mathrm{mm/h}\) Soil evaporation rate
\(E_t\) \(\mathrm{mm/h}\) Transpiration rate
\(L\) \(\mathrm{mm/h}\) Percolation rate
\(Q\) \(\mathrm{mm/h}\) Surface runoff

Terrestrial biogeochemistry

Biogeochemistry is concerned with biogeochemical cycles, particularly the pathways of carbon and nitrogen, and their interactions with biological activity on Earth. Other elements of interest are phosphorus, sulfur, and iron. Here, we will review the carbon, nitrogen, and phosphorus cycles. Carbon and nitrogen dynamics are coupled through the soil organic matter, thus, we discuss these cycles together.

Soil organic matter

The biogeochemical cycles are linked to the hydrological cycle through the soil organic matter° (SOM), a complex mixture of organic substances—plant residues, microorganism biomass, and humus. SOM controls the chemical reaction rates of these elements in the soil. SOM and hydrology are deeply interlinked. SOM increases the soil surface protection, its water holding capacity, and nutrient availability. It further causes soil darkening. This affects the hydrology by reducing evaporation, increasing infiltration capacity and water availability to plants, thus, also increasing transpiration, and further moderates soil temperature.

In return, hydrology triggers mineralisation and immobilisation processes, denitrification, and biological fixation. Increase of water fluxes may cause leaching of SOM. This modulates soil respiration, nutrient availability, and emissions of nitrogen gases. Leaching may cause stream and groundwater pollution.

Carbon cycle

Carbon is crucial for life on Earth. In the atmosphere, carbon is mostly found in the form of CO2. Plants take in CO2 by means of photosynthesis, separate carbon (C) and oxygen (O2), and incorporate C into their biomass.

A simplified sketch of the C-cycle is shown in the figure below. The C in the plants eventually turns into plant litter (residues) and moves as part of SOM into the soil. Each SOM has a specific carbon-to-nitrogen (C/N) ratio, which affects its decomposition° rate. SOM decomposition produces mineral compounds and CO2 that is returned to the atmosphere through soil respiration. Part of the mineral compounds are metabolised by soil microbes and the rest is turned into humic substances°. Such humic substances play an important role in maintaining high organic levels and nutrients. SOM decomposition is closely related to immobilisation°, the conversion of inorganic compounds to organic compounds by microorganisms and plants. The immobilisation is regulated by the C/N ratio of the SOM as well as the soil water potential, because they influence the bacterial colonies.

carbon-cycle-c.png

Figure 4: Simplified soil–plant–atmosphere carbon cycle. Adapted from Porporato et al. (2003)°.

Nitrogen cycle

Nitrogen (N) is a crucial nutrient for plants. In soil, N is usually found in the form of organic compounds, which prevents the plants from directly accessing it. Plants uptake mineral N only in the form of ammonium (NH4) and nitrate (NO3), which are produced by SOM decomposition. Because SOM decomposition is linked to the C-cycle, it is directly coupled to the N-cycle. SOM contains about 5% N. While the atmosphere is composed of 78% dinitrogen (N2), N in this form is not usable by most plants.

The figure below shows a simplified soil–plant–atmosphere nitrogen cycle. Note that N cycles from the soil into the plants, which then returns as SOM into the soil. SOM is decomposed by bacteria to produce NH4 and NO3. The bacterial activity is dependent on pH, temperature, and soil moisture. Part of these products is mineralised. The rate of this mineralisation depends on the C/N ratio, because the C/N ratio in a microbial colony remains constant as the colony grows and shrinks. This means, that the mineralisation rate is either limited either by C or by N.

nitrogen-cycle-c.png

Figure 5: Simplified soil–plant–atmosphere nitrogen cycle. Modified from Porporato et al. (2003)°

A simple model for soil organic matter dynamics

We follow the predator–prey model by Manzoni and Porporato (2007)°, where the following system of equations is solved:

\begin{aligned} d_t C_s = \mathrm{add} - \mathrm{dec} + \mathrm{bd}\\ d_t C_b = (1 - r) \mathrm{dec} - \mathrm{bd} \end{aligned}

Here, \(C_s~[\mathrm{gm^{-3}}]\) and \(C_b~[\mathrm{gm^{-3}}]\) are the carbon masses stored in the soil system and the microbial pool, respectively. \(r\) is the respired C fraction. The term \(\mathrm{bd}~[\mathrm{gm^{-3}h^{-1}}]\) denotes microbial decay and is modelled through a first order reaction equation as

\begin{equation*} \mathrm{bd} = k_b C_b, \end{equation*}

with \(k_b~[\mathrm{h^{-1}}]\) is the coefficient of proportionality, representing a decay rate of the microbial pool. \(\mathrm{dec}~[\mathrm{gm^{-3}h^{-1}}]\) is the unlimited flux of carbon decomposition, if we assume that sufficient nitrogen is always available. It can be calculated by a Michaelis–Menten kinetic reaction as

\begin{equation*} \mathrm{dec} = k_s C_b \frac{C_s}{k_m + C_s}, \end{equation*}

\(k_m\) is the Michaelis constant. The actual decomposition \(k_s~[\mathrm{h^{-1}}]\) is calculated from the imposed potential decomposition rate \(k_s^*~[\mathrm{h^{-1}}]\) as

\begin{aligned} k_s = k_s^* f_s (s, s_b, s_{fc}) f_T(T, T_{\mathrm{min}}, T_{\mathrm{max}})\\ f_s (s, s_b, s_{fc}) = \begin{cases} 0 & s \leq s_b,\\ \frac{s - s_b}{s_{fc} - s_b} & s_b < s \leq s_{fc} \\ \frac{s_{fc}}{s} & s_{fc} < s \leq 1 \end{cases}\\ f_T (T, T_{\mathrm{min}}, T_{\mathrm{max}}) = \frac{(T - T_{\mathrm{min}})^2}{(T_{\mathrm{max}} - T_{\mathrm{min}})^2} \end{aligned}

where \(f_s\) and \(f_t\) are reduction functions due to soil moisture \(s\) and temperature \(T~[\mathrm{^\circ C}]\), respectively. \(s_b\) is the species-specific microbial biomass stress point and \(s_{fc}\) is the field capacity. \(T_{\mathrm{min}}~[\mathrm{^\circ C}]\) is the critical temperature at which \(f_T\) approaches nil and \(T_{\mathrm{max}}~[\mathrm{^\circ C}]\) is the maximum temperature observed in the field. Finally, \(\mathrm{add}~[\mathrm{gm^{-3}h^{-1}}]\) is a constant rate of litter input.

With this simple dynamical system, we can model the C dynamics in the soil. The figures below show exemplary output of this model.

c-cycle-1c.png c-cycle-2c.png

The C content in the soil increases at the beginning, because the microbial biomass is not sufficient to decompose the litter input at the same rate. However, increased C input leads to a fast growth of microbial biomass, reflected in an increase in the carbon decomposition flux \(\mathrm{dec}\). As \(C_b\) grows, the microbial decay also increases. At a saddle point, microbial activity overpowers the litter input rate and C content in the soil starts to decrease. The system reaches an equilibrium where decomposition, litter input, and microbial decay balance each other out. This can also be observed in the phase space portrait, where the dynamical system features an attraction point.

Acknowledgements

Thanks to Dr. Mikael Gillefalk, who helped me develop these notes for the course Ecohydrological Modelling at Technische Universität Braunschweig. These notes were written as part of the Juniorprofessorship programme funded by the state of Lower Saxony.


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Author: ilhan özgen xian

Created: 2024-01-15 Mon 23:07

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