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## 1 Ecohydrology

The study of the interaction among hydro- and biosphere. In general, three spheres are studied (see, McClain et al., 2012):

1. plants and water in atmosphere, soil, vadose zone, and groundwater
2. hyporheic zone processes
3. lake, estuary, and coastal zone processes

Water—especially soil water—is a control on vegetation (Eagleson, 2002; Rodriguez-Iturbe et al., 1999). Vegetation impacts hydrology (Berry et al., 2017; Chen et al., 2015).

### 1.1 Hydrological connectivity

Hydrological connectivity describes how compartments in a system connect through mass and energy fluxes.

#### 1.1.1 Topographic Wetness Index

The Topographic Wetness Index (TWI) quantifies topographic controls on hydrological processes.

$$TWI = \ln \frac{a}{\tan \beta}$$

$$a$$ is the local upslope area and $$\beta$$ is the local slope.

TWI is linked to DOC export (Musolff et al., 2018); iron export (Tittel et al., 2022); and groundwater level (Soransen et al., 2006).

References: Beven & Kirkby (1979)

#### 1.1.2 Flux re-scaling

Hydrodynamic processes occur at different spatio-temporal scales. But hydrodynamic fluxes aggregate along the landscape and manifest as observable behaviour at hillslope to catchment scales — see Caviedes-Voullieme et al. (2021). This process is still not fully understood (Reis et al., 2017).

### 1.2 Disturbance ecology and disturbance ecohydrology

Imagine the amazing good fortune of the generation that gets to see the end of the world. This is as marvelous as being there at the beginning. How could one not wish for that with all one’s heart? How could one not lend one’s feeble resources to bringing it about? – Jean Baudrillard, Fragments

The study of disturbances in ecosystems to understand causes, consequences, and interactions. For example:

• insect plague (bark beetles 1.2.2)
• pathogens 1.2.1
• climate change impacts, for example drought 1.2.4

As noted in Newman (2019), we distinguish between:

• natural disturbances — re-occuring events in the natural regime
• unnatural disturbances — disturbances out of the natural regime

See also Jentsch et al. (2007; 2011) for the distinction between event-based and trend-based studies.

Disturbance studies often make use of systems theory 2. Collapse and regime shifts occur at so-called tipping points 2.2.

In order to cope with disturbances, the best property a system could have is anti-fragility 2.3 sensu Taleb (2014).

Some pathogens spread with water availability. For example, the Malaria vector spreads along water gradients, see Jiang et al. (2021); Wilkening et al. (2021).

Rinaldo and Rodriguez-Iturbe (2022) discuss catchment models of water-related diseases.

#### 1.2.2 Bark beetles

Emergence of bark beetles can be linked to drought and plant hydraulic failure. This could be studied in terms of a tipping cascade 2.2.

#### 1.2.3 Wildfire

Some thoughts on modelling:

There exist field experiments that can be used to validate wildfire models.

• coupled advection–diffusion–reaction equation with Euler equation
• Sink term for bottom cells for the Euler solver:
• shear velocity can be related to flow variables
• IMEX scheme

#### 1.2.4 Drought

Drought is a time interval when the supply of moisture does not match the demand. Commonly, we distinguish between:

1. meteorological drought
2. agricultural drought
3. hydrological drought
4. socio-economic drought

Auft (2020) gives a review of tools for drought analysis.

Deep adaptation is a concept for climate adaptation, put forward in Bendell (2018), a preprint that did not go through peer review. The main message is that climate change makes societal collapse 1.2.6 inevitable. It is therefore necessary to take deep, as in radical, adaptation measures to address it. Bendell (2018) and Carr & Bendell (2019) suggest a framework with four key ingredients: resilience (surviving), relinquishment (letting go), restoration (rediscovering), and reconciliation (making peace). The claims in Bendell (2018) have been both criticised (Nicholas et al., 2020) and supported (Servigne et al., 2020) in the academic community. Bendell (2020) provides a response to the criticisms.

#### 1.2.6 Collapse science

Collapse denotes the process of a decrease in the complexity of a society. While collapse has occured throughout the history of humankind, in our current times, it stands for the unraveling of the globalized, interconnected, and materially abundant societies.

There are several different reasons for collapse. Internally, collapse can occur if the society reaches a level of organizational complexity that it cannot sustain. Collapse of a society can also occur due to external stressors, for example, economical or political instabilities.

There is an emerging anxiety that the end is nigh, that climate change, financial instabilities, and peak oil might eventually lead to a collapse of our global society. However, not all academics agree with this prediction (Butzer & Endfield, 2012; Haldon et al., 2020; Spinney, 2020).

While valuable data points, historical collapse scenarios, especially ancient ones, suffer from temporal inaccuracies, related to current dating technologies and missing, incomplete, or wrong records. This makes it difficult to infer environmental circumstances at the point of collapse, and therefore limits the predictive capability of extrapolations from specific case studies to current times.

Predicting collapse is also challenging. Computational sociology, the branch of sociology that provides computer models of societal collapse, relies on (multi) agent-based modelling techniques that are coupled to climate models. The capability of agent-based models to simulate complex human societies sufficiently accurate is an open debate (Crooks et al., 2008). This is not to say that collapse is impossible, but that it might not be as inevitable as some might suggest.

Historical case studies show that societies with sufficient environmental, political, or socio-cultural resilience can overcome breakdown and avoid collapse (Butzer & Endfield, 2012). Thus, it is important to study not only the stressors and disruptions, but also the resilience of the systems they impact. See also anti-fragility 2.3.

In cases with high societal resilience, the societal system displays thresholding behaviour in form of regime changes or ideological shifts. In such societies, military leaders with the support of new elites may rise to prevent the breakdown of the social order, civil wars, and the eventual collapse (Butzer & Endfield, 2012).

### 1.3 Hillslopes

Hillslopes are fundamental units for watershed hydrology and element cycling (Wainwright et al., 2022; Fan et al., 2019; Loritz et al., 2017; 2018).

#### 1.3.1 Representative hillslopes

Parsimonious representations of a catchment in a physically-based model (Loritz et al., 2017).

#### 1.3.2 Trait-based approaches

A concept from ecology that may or may not be useful to analyse hillslope scale processes. See the fourth-corner problem (Brown et al., 2014).

There's a paper by McDonnell that discusses trait-based hillslopes, referenced in Wainwright et al. (2022).

1. Watershed traits

Watershed functional traits are spatiotemporal landscape and process patterns that arise as a result of the watershed. Watershed traits could be used as proxies for watershed functions—capture, storage, and release of water.

For ecohydrology, we can include biogeochemical reactions and habitat for flora and fauna to the functions (Black, 1996; Schulz et al., 2006).

See: McDonnell et al. (2007)

#### 1.3.3 Elementary functional units

Defined in Zehe et al. (2014) to denote sub-hillslope scale domains with homogeneous functions. Conceptually relates to representative hillslopes 1.3.1, but one scale below.

See: Zehe et al. (2014; 2021)

#### 1.3.4 Watershed zonation

Delineating watersheds to identify hillslopes and unsupervised clustering to classify them into hillslope scale functional units (Wainwright et al., 2022).

### 1.4 Darwinian hydrology

Darwinian hydrology applies Darwinian approaches to hydrology (Harman and Troch, 2014).

Darwinian approaches have following properties:

1. anatomy vs. biogeography in the type of questions asked: instead of studying properties, study explanations.
2. creating explanatory hypotheses — given an outcome B, generate a prior condition A, such that if A were found, B would necessarily follow
1. measure and extrapolate observable processes of change
2. classification and space for time substitution — classifying by form to obtain time series in space
3. testing explanatory hypotheses using a hypothetico-deductive approach:
• collect extensive and detailed observations of patterns
• conceive hypothesis
• derive a set of circumstances that must hold if hypothesis is correct
• verify these consequences

#### 1.4.1 Pushback against Newtonian models

The pushback against Newtonian models is related to a shift in hydrological research. For example, Sivapalan (2018) argues that we need to move away from prediction-based studies to explanatory studies (Darwinian approaches 1.4) that might lead to universal laws of hydrology. The existence of the Budyko curve suggests the existence of universal laws that can be discovered. In the same vein, Kleinhans et al. (2010) state that rather than making specific predictions as accurately as possible, the aim of hydrological science should be to explain general hydrological phenomena through explanatory models.

### 1.5 Isotope ecohydrology

Using stable and radioactive isotope methods to study age and origin of water in ecosystems.

Stable water isotopes can be used to trace water. Lighter oxygen (16O) tends to evaporate at higher rate than heavier oxygen (18O). Thus, seawater contains more heavy oxygen than rain and snow. Rainfall and groundwater have also different isotopic signatures.

In stable water isotope transport modelling, we consider two aspects (Kuppel et al., 2018):

• isotope mixing
• isotope fractionation

#### 1.5.1 Isotope fractionation in plants

Oxygen is fractionated in plants during the creation of plant organic material. Photosynthesis fractionates oxygen and hydrogen depending on the photosynthesis type — C3 and C4 plants behave differently (White, 2015).

#### 1.5.2 ech2o-iso

Ech2o-iso is an ecohydrological model with isotope transport capability 1.5 (Kuppel et al., 2018).

Isotope mixing in ech2o-iso is assumed to be instantaneous. There is research that shows that this assumption has limiatations, for example, Zhao et al. (2016); Vargas et al. (2017).

Isotope fractionation is considered only during soil evaporation, using the Craig–Gordon model (Craig and Gordon, 1965; Gat, 1995).

### 1.6 Terrestrial ecohydrology

Mama said, "Boy, stay clear of the water
When you're blue and still catching fire."
And I've learned from my father
To listen to my mother.
So, I'm cooling, waiting by the shore.
— Jacob Banks, Coolin

#### 1.6.1 Soil–plant–atmosphere continuum

Models of soil–plant–atmosphere continuum (SPAC) are often based on Ohm's analogy, which results in the van den Honert equation (van den Honert, 1948). Flow through the SPAC are driven by gradients in the total water potential in the soil, root, and leaf (Porporato and Yin, 2022). The plant conductivity comprises root, xylem/stem, and leaf conductivity. The conductivity is dependent on the water content in the plant, which gives rise to vulnerability curves.

SPAC models can be used for isotope ecohydrology 1.5.

#### 1.6.2 Integrated hydrology

Especially during low flow, the interactions between surface and becomes significant—see, Staudinger et al. (2019).

Runoff generation 1.6.2.1 is an emergent property (see 2) of integrated hydrological systems. According to Staudinger et al. (2019), feedback between surface and subsurface is important when:

• analysing exchange fluxes and processes they control:
• ecology (Woenner, 2017)
• sedimentation (Partington et al. 2017)
• soil hydrology and runoff generation where the interfaces control the partitioning (Weiler and McDonnell, 2001; Bachmair and Weiler, 2011)
• water quality
• low flow periods (Smakthin, 2001; Garner et al., 2015; Lauber et al., 2015; Stoelzle et al., 2015)

Kleinhans et al. (2010) state that rather than making specific predictions as accurately as possible, the aim of hydrology should be to explain general hydrological phenomena through explanatory models.

1. Runoff generation

Information on runoff generation can help to increase the realism of spatial and temporal recharge patterns (Staudinger et al, 2019). Runoff generation is a non-linear process that may feature thresholding behaviour (Meerveld & Weiler, 2008; Tromp-van Meerveld & McDonnell, 2006) and is vulnerable to climate change.

The characteristics of the catchment's response, that is to say the composition of streamflow depends on the runoff generation, the temporal water sources (event or pre-event) and water flow paths (surface or subsurface).

Runoff generation mechanisms:

• Hortonian overland flow
• Dunne overland flow
• Subsurface stormflow 1.6.2.2

See: Tetzlaff & Soulsby (2008); Wandzell et al. (2007)

2. Subsurface stormflow

Subsurface flow can transmit at rates that are high enough that it contributes to stormflow (Freeze, 1972; Fiori et al., 2007). The contribution can be quite significant, for example, Wenninger et al. (2004) report that 80% of the streamflow consisted of subsurface water. Different mechanisms that try to explain this high and fast contribution of subsurface flow to streamflow have been proposed—see Rawitz et al. (1970); Hills (1971) for early studies on the subject.

Further studies include:

1. Abdul & Gilham (1984; 1989) propose the capillary-fringe-induced groundwater-ridging, also explored in McDonnell & Buttle (1998); Park et al. (2011). However, this mechanism is not sufficient to explain the fast response of subsurface contribution.
2. An alternative mechanism proposed in Becker (2005); Vidon (2011) is the pressure wave translatory flow. It states that as soon as the subsurface connects hydraulically, high pressure can push groundwater into the stream.
3. Another mechanism is the transmissivity feedback flow: saturated soil permeability decreases with increasing depth, which deplaces shallow groundwater and pushes it into the stream (Detty & McGuire, 2010; Laudon et al., 2004). Bachmair & Weiler (2014) and Weiler & McDonnell (2004) investigate these mechanisms and propose a theory of drainable porosity.

Important controls on subsurface stormflow are:

• Macropores (Gerke & van Genuchten, 1993; Park et al., 2011)
• Mechanical dispersion (Ghasemizade & Schirmer, 2013)

• Ghasemizade & Schirmer (2013) for a review of the current understanding of subsurface contribution to streamflow.
• Beven & Germann (1982) on macropores and water flow in soils.

#### 1.6.3 Ecohydrological interfaces

River networks and landscapes are the fundamental compartments and the interfaces across them control ecohydrological and biogeochemical processes (Rinaldo and Rodriguez-Iturbe, 2022; Krause et al., 2017).

#### 1.6.4 Vegetation self-organisation

The ability of plants to organise in spatial patterns without environmental complexity. The most common mathematical model 3 is Rietkerk's model — a phenological model describing a Turing instability.

See Crompton et al. (2021) and Caviedes-Voullième & Hinz (2020) for recent modifications. See also Nes & Scheffer (2005) and Dakos et al. (2010).

An interesting link might be the understanding of invasive species dynamics, see disturbance ecology 1.2.

### 1.7 Urban ecohydrology

It is difficult for anyone born and raised in human infrastructure to truly internalize the fact that your view of the world is backward. Even if you fully know that you live in a natural world that existed before you and will continue long after, even if you know that the wilderness is the default state of things, and that nature is not something that only happens in carefully curated enclaves between towns, something that pops up in empty spaces if you ignore them for a while, even if you spend your whole life believing yourself to be deeply in touch with the ebb and flow, the cycle, the ecosystem as it actually is, you will still have trouble picturing an untouched world. You will still struggle to understand that human constructs are carved out and overlaid, that these are the places that are the in-between, not the other way around. — Becky Chambers, A Psalm for the Wild-Built

Urbanisation leads to rapid (dis)connectivity among environmental compartments. Thus, urban systems function differently than undisturbed natural systems. Most significantly, the co-evolution of preferential flow paths is disturbed.

#### 1.7.1 Socio-hydrology and coupled human–environment system

• coupled human and natural system (CHANS)

Dynamical two-way interactions between human systems (economic, social) and natural (hydrological, atmospheric, biological, geological) systems. The evolution of these systems cannot be treated individually, they form a complex system 2.1

See: Sivapalan and Blöschl (2015); Turner et al. (2003); Sheppard and McMaster (2004)

#### 1.7.2 Crown transparency

Crown transparency is an indicator of tree health and can be determined by two-dimensional images. While the approach is quite heuristic and relies on expert knowledge, there exist efforts to automate and standardise the process—see, for example, Borianne et al. (2017).

P. Borianne and G. Subsol and Y. Caraglio (2017). "Automated efficient computation of crown transparency from tree silhouette images". Computers and Electronics in Agriculture, 133, 108–118.

### 1.8 Richardson–Richards equation

After Richardson (1922) and Richards (1931):

$$\partial_t \theta + \nabla K(\psi) \nabla \psi - \sigma = 0$$

A similar form of this equation is used in plant hydraulics 1.8.3.

#### 1.8.1 Dynamic non-equilibrium RRE

Describes the phase of wetting and drying when the gas–water phases have not equilibriated yet. This may lead to hysteresis in the soil water-retention curve.

#### 1.8.2 Physics-informed neural networks for RRE

PINNs solve PDEs through automatic differentiation and thus, physical constraints and relations can be encoded in their structure.

Bandai and Ghezzehei (2021; 2022) solve the RRE through PINNs.

#### 1.8.3 Plant hydraulics

Flow through plants is—in general—assumed to be passive, see Baird & Wilby (1999) and Yin & Porporato (2022). This passive flow is driven by the gradient in total water potential between soil and leaf. The gradient between soil and leaf is caused by photosynthesis; as the stomata open to capture carbon, water evaporates through their openings, causing the leaf water potential to drop below the soil water potential. Following Ruffault et al. (2022), the flow rate at any location within the soil–plant–atmosphere continuum is expressed as a general mass balance as

\left\{ \begin{aligned} \frac{d\Theta}{dt} = \nabla \cdot \mathscr{J} + s,\\ \mathscr{J} = \left( k \nabla \psi \right). \end{aligned} \right.

Here, $$\mathscr{J}$$ is the water flux, $$\Theta~[MV^{-3}]$$ is the water content, $$t~[T]$$ is time, $$k~[MT^{-1}]$$ is a conveyance parameter, $$\psi~[L]$$ is the total water potential, and $$s~[MV^{-3}T^{-1}]$$ is a sink or source term. Note the similarity with the Richardson–Richards equation 1.8, because both of these equations describe potential flows.

J. Ruffault and F. Pimont and H. Cochard and J.-L. Dupuy and N. Martin-StPaul (2022). "SurEau-Ecos v2.0: a trait-based plant hydraulics model for simulations of plant water status and drought-induced mortality at the ecosystem level". Geoscientific Model Development, 15, 5593–5626.

### 1.9 Metabolic Theory of Ecology

E.A. Johnson and Y.E. Martin, Introduction, in: A Biogeoscience Approach to Ecosystems (ed: Johnson & Martin), Cambridge University Press, Cambridge, UK, 2016:

One of the most interesting developments in ecology has been the Metabolic Theory of Ecology (MTE). This theory (West et al., 1997; 1999; Brown et al., 2004; Enquist et al., 2003; 2007) argues that mass conservation, biological mechanics, hydraulics, heat budgets, and thermodynamics can be used to explain the flux of energy, water, and nutrients from cells to ecosystems.

## 2 Dynamical systems theory

A system is a group of elements that interact among each other with a governing set of rules that determines their behaviour. The aggregated actions of all elements lead to the so-called emergent behaviour of the system. All systems are characterised by their structure, function, behaviour, and interconnectivity.

Systems theory aims to predict a system's dynamics, constraints, and relations, and to infer transferable principles.

### 2.1 Complex system

A complex system has behaviour that is intrinsically difficult to predict and understand, because a high degree of non-linear relationships among its elements. Consequently, complex systems have distinct properties such as non-linearity, emergence, spontaneous order, adaptation, and feedbacks.

#### 2.1.1 Hysteresis

Hysteresis is the dependence of a system on its history. This can result from a lag between input and output (rate-dependent hysteresis), which is often found in ecohydrology.

Rate-independent hysteresis also exists.

Hysteretic models:

• Preisach model
• Lapchin model
• Bouc–Wen model
• Jiles-Atherton model

May connect to flux re-scaling 1.1.2 in the sense of: Does the re-scaling necessarily cause a hysteresis?

If two subsystems are coupled, a tipping cascade can emerge—see, Klose et al. (2020).

Let $$S1 \leftarrow S2$$ be unidirectionally coupled systems. Then we distinguish between:

1. facilitated tipping — $$S1$$ reduces the threshold of $$S2$$ and facilitates transition into an alternative state
2. impeded tipping — the opposite of facilitated tipping
3. back tipping — $$S1$$ is not strong enough to cause a transition, but pushes $$S2$$ towards its threshold

Tipping cascades, according to Klose et al. (2020) are

1. domino cascades — see Lenton et al. (2019); Brummet et al. (2015)
2. joint cascades — $$S1$$ and $$S2$$ tip jointly
3. two-phase cascades — see Dikker et al. (2018)

See further: Wunderling et al. (2020)

### 2.3 Anti-fragility

Anti-fragility is a concept by Taleb (2012) to describe systems that benefit from disturbances 1.2. Taleb (2014) gives an order of fragility as: fragile — robust — anti-fragile.

On page 59, Tab. II, Taleb (2012) summarises fragility of mechanical vs. organic and non-complex vs. complex systems, whereby the latter are considered anti-fragile. An example for anti-fragility might be fire ecology— see also wildfire modeling 1.2.3.

Further, Taleb (2012) points out that antifragile systems often feature redundancy.

#### 2.3.1 Anti-fragile (dis)connectivity

If we take a step back and more generally consider the issue of partitioned versus connected systems, partitioned systems are more stable, and connected systems are both more vulnerable and have more opportunities for collective action. Vulnerability (fragility) is connectivity without responsiveness. Responsiveness enables connectivity to lead to opportunity. If collective action can be employed to address threats, or to take advantage of opportunities, then the vulnerability can be mitigated and outweighed by the benefits. This is the basic relationship between the idea of sensitivity as we described it and your [N.N. Taleb's] concept of antifragility. — Y. Bar-Yam, quoted in Taleb (2012), page 458.

Connects to disturbances 1.2, hydrological connectivity 1.1.

## 3 Numerical analysis

Numerical analysis studies algorithms to numerically approximate continuous mathematical problems. Common applications include the numerical resolution of partial differential equations, for example, the Richardson–Richards equation 1.8

### 3.1 Mathematical modelling

Baird & Wilby (1999, p. 339) point out that despite their shortcomings, mathematical models are formal encodings of our theories and hypotheses. If coded correctly, mathematical models allow for testing and exploring these theories.

A.J. Baird & R.L. Wilby (1999). Eco-hydrology: Plants and Water in Terrestrial and Aquatic Environments, Routledge, Abingdon-on-Thames, UK.

Created: 2023-02-03 Fr 17:10