The present study confirms that a thermodynamic perspective on soil water is well suited to distinguishing the typical interplay of gravity and capillarity controls on soil water dynamics in different landscapes. To this end, we express the driving matric and gravity potentials by their energetic counterparts and characterize soil water by its free energy state. The latter is the key to defining a new system characteristic determining the possible range of energy states of soil water, reflecting the joint influences of soil physical properties and height over nearest drainage (HAND) in a stratified manner. As this characteristic defines the possible range of energy states of soil water in the root zone, it also allows an instructive comparison of top soil water dynamics observed in two distinctly different landscapes. This is because the local thermodynamic equilibrium at a given HAND and the related equilibrium storage allow a subdivision of the possible free energy states into two different regimes. Wetting of the soil in local equilibrium implies that free energy of soil water becomes positive, which in turn implies that the soil is in a state of storage excess, while further drying of the soil leads to a negative free energy and a state of storage deficit. We show that during 1 hydrological year the energy states of soil water visit distinctly different parts of their respective energy state spaces. The two study areas compared here exhibit furthermore a threshold-like relation between the observed free energy of soil water in the riparian zone and observed streamflow, while the tipping points coincide with the local equilibrium state of zero free energy. We found that the emergence of a potential energy excess/storage excess in the riparian zone coincides with the onset of storage-controlled direct streamflow generation. While such threshold behaviour is not unusual, it is remarkable that the tipping point is consistent with the underlying theoretical basis.
Only a minute amount of global water is stored in the root zone of the soil. Yet this tiny storage compartment crucially controls a variety of processes and ecosystem functions. The root zone soil water stock essentially supplies savannah vegetation (e.g. Tietjen et al., 2009, 2010) and more generally ecosystems during severe droughts (Gao et al., 2014). The soil water content controls infiltration, runoff formation and streamflow generation (Graeff et al., 2009; Zehe et al., 2010), it partly determines habitat quality of earthworms (e.g. Schneider et al., 2018) and it is of key importance for soil respiration and emission of greenhouse gases in mountain rainforests (e.g. Koehler et al., 2012). Soil water dynamics are controlled by the triple of infiltration, moisture retention and water release. These processes are driven by the intermittent rainfall and radiative forcing and controlled by multiple forces arising from capillarity, gravity, root water uptake and possibly osmosis. Steady-state hydraulic equilibrium conditions imply that the driving forces act in a balanced manner. In the simple case of absent vegetation and of a flat topography this force balance corresponds to the well-known hydraulic equilibrium, where the matric potential equals the negative of the gravity potential along the entire soil profile. The corresponding equilibrium soil water content profile, which is straightforward to calculate if depth to groundwater and the soil water retention curve are known, reflects thus a balance between the most prominent influences: the local capillary control and the non-local gravitational control. Although these two controls are sensitive to distinctly different systems properties, these properties are not necessarily independent. The climatological and geological setting constrains the co-development or co-evolution of soils, geomorphology and vegetation (as suggested by e.g. Troch et al., 2015; Sivapalan and Bloschl, 2015; Saco and Moreno-de las Heras, 2013). One might hence wonder whether this constrained co-development created a distinctly typical interplay of capillary and gravitational controls on soil moisture. In the present study we show that this interplay manifests through (a) distinct differences in soil water dynamics among different hydrological landscapes and (b) a thermodynamic perspective on soil water dynamics to discriminate typical differences that cannot be inferred from the usual comparison of soil moisture observations.
Thermodynamic reasoning has a long tradition in Earth science, ecology and hydrology, and one of its key advantages is a joint treatment of mass fluxes and the related conversions of energy, including dissipation and entropy production. In geomorphology it dates back to the early work of Leopold and Langbein (1962) on the role of entropy in the evolution of landforms. Howard (1971, cited in Howard, 1990) proposed that angles of river junctions are arranged in such a way that they minimize stream power. Bolt and Frissel (1960) related soil water potentials to Gibbs free energy of soil water (referring to the early pioneers Edlefson and Anderson, 1940) and established a link between soil physics and thermodynamics. In ecology Lotka (1922a, b) proposed that organisms that maximize their energy throughput have an advantage within the evolutionary selection process.
Thermodynamics have gained substantial attention in catchment hydrology since the work of Reggiani et al. (1998) and Kleidon and Schymanski (2008). Reggiani et al. (1998) employed thermodynamic reasoning and volume averaging to derive a model framework of intermediate complexity (Sivapalan, 2018). They introduced the idea of a representative elementary watershed, REW, which can be seen as the least spatial entity for building mesoscale hydrological models. This idea has been picked up and advanced by several follow-up studies dealing with the coding and successful application of REW-based hydrological models (Reggiani et al., 1998, 1999, 2000; Reggiani and Schellekens, 2003; Lee et al., 2005, 2007; Zhang et al., 2006; Tian et al., 2006) or the challenge to derive the necessary closure relations (Zehe et al., 2006; Beven, 2006).
Along a different avenue, Kleidon and Schymanski (2008) discussed the opportunity of using maximum entropy production (MEP, originally proposed by Paltridge, 1979) to predict steady-state, close-to-equilibrium functioning of hydrological systems and to infer model parameters based on thermodynamic optimality. This idea has motivated several efforts to predict the catchment water balance using thermodynamic optimality. For instance, Porada et al. (2011) simulated the water balance of the 35 largest basins on Earth using the SIMBA model and inferred parameters controlling root water uptake by maximizing entropy production. They tested the plausibility of their assessment within the Budyko framework (Budyko, 1958). Zehe et al. (2013) showed that a thermodynamic optimum density of macropores created by worm burrows which maximized dissipation of free energy during recharge events allowed an acceptable uncalibrated prediction of the rainfall–runoff response of a lower mesoscale catchment with a physically based hydrological model. While this finding is at least an interesting incidence, the explanation why the worms should create their burrows in such a way is not straightforward. Hildebrandt et al. (2016) proposed that plants optimize their root water uptake by minimizing the necessary energetic investment through a spatially uniform water abstraction from uniform soils. Along similar lines of thought but at much larger scales Gao et al. (2014) proposed that ecosystems optimize their rooting depth. This is deemed to balance the advantage of vegetation to endure droughts of increasing return periods with the necessary energetic investment to expand their root system to enlarge the water holding capacity.
Kleidon et al. (2013) tested whether the topology of connected river networks can be explained through a maximization of kinetic energy transfer to sediment flows. They showed that the depletion of topographic gradients by sediment transport can be linked to a minimization in frictional dissipation in streamflow networks, which in turn implies a maximization of sediment flows against the topographic gradient and thus of power in the sediment flows. The idea that the topology of river networks reflects an energetic optimum – more precisely a minimum – is in fact much older and was already suggested by Howard (1990) and picked up by Rinaldo et al. (1996) as the concept of minimum energy expenditure. Hergarten et al. (2014) transferred this idea to groundwater systems by analysing preferential flow paths that minimize the total energy dissipation at a given recharge under the constraint of a given total porosity and by verifying those against data sets for spring discharge in the Austrian Alps.
Kleidon et al. (2014) and Renner et al. (2016) tested whether a two-layer energy balance model based on maximum power in combination with Carnot efficiency is suited to predicting the partitioning of net shortwave radiation into longwave outgoing radiation and turbulent fluxes of latent and sensible heat. During convective conditions their predictions were in good accordance with flux tower data at three sites with different land use.
While some of us might find the search for thermodynamic optimality exciting and promising, it is certainly not the Philosopher's stone. Westhoff et al. (2013) found for instance that a conceptual model structure which was in accordance with MEP was not suited to predict the water balance in the HJ Andrews experimental watershed. Thermodynamic optimality should thus be seen as a testable and sometimes helpful constraint, but it should not be mixed with a first principle such as the first and second laws of thermodynamics (Westhoff et al., 2019). And thermodynamic optimality is restricted to explaining system steady-state, close-to-equilibrium functioning. The challenge is however to explain operation of hydrological systems under temporarily variable forcing (Westhoff et al., 2014) and far-from-equilibrium conditions.
In summary we think that there are four general arguments why a thermodynamic
perspective on soil water dynamics and hydrology in general has much to
offer. Firstly, surface runoff and particularly soil water fluxes dissipate a
very large amount of their driving energy differences. As the dissipation and
related entropy production rates depend on the soil material and on the
spatial organization of the material as well (Zehe et al., 2010), one may
quantify feedbacks between morphological/structural changes and hydrological
processes within the same current (joule). Secondly, energy is an extensive quantity;
as such it is additive when different systems are merged, it grows with
increasing system size and changes can be described through a balance. One
may hence apply volumetric averaging and upscaling to energy for instance to
derive macroscale effective constitutive relations and macroscale equations
as shown by de Rooij (2009, 2011). By contrast, the related gravity and
matric potentials are intensive state variables and as such not additive in
the sense specified above, nor can their changes be balanced. Thirdly, it can
be used to define and explain hydrological similarity based on a
thermodynamically meaningful combination of catchment characteristics (Zehe
et al., 2014; Seibert et al., 2017; Loritz et al., 2018). Last but not least,
one may test whether thermodynamic optimality provides, despite the fact that
it is controversial, a means to test the recent proposition of Savenije and
Hrachowitz (2017), stating that “Ecosystems control the hydrological
functioning of the root zone in a way that it
In the following, we show that the free energy state of soil water is well suited for characterizing distinct differences in soil water dynamics among different landscapes. Based on the free energy state we define a system characteristic called energy state function, which jointly accounts for the capillary and gravitational control of soil water dynamics, using height over the nearest drainage (HAND, Renno et al., 2008; Nobre et al., 2011) as a proxy for the gravity potential. These energy state functions are strongly sensitive to differences in topography and soil water characteristics of the study area and allow an instructive visualization of soil water dynamics in energetic terms. By comparing two different catchments we found that the soil water storage dynamics in both landscapes operate distinctly differently with respect to the local thermodynamic equilibrium state of minimum free energy. More specifically we provide evidence that the local thermodynamic equilibrium state separates two regimes of a storage deficit and storage excess. During a 1-year period the observed energy states of the soil water in the study areas operated distinctly differently with respect to these regimes and visited distinctly different ranges of their corresponding energetic state space. Last but not least, we provide evidence that the state of zero free energy not only separates regimes of a storage deficit and a storage excess, it is furthermore also a theoretically motivated threshold, explaining the onset of storage-controlled runoff generation, saturation excess overland flow or subsurface storm flow in our study areas.
In the following we express the matric and gravity potentials by their energetic counterparts, following largely the work of Bolt and Frissel (1960) and de Rooij (2009), to characterize soil water storage by its free energy state and derive the energy state function.
Following the micro approach of Bolt and Frissel (1960) we start our
derivation with the Gibbs free energy
In the next step, we express Eq. (1) as a change in volumetric energy
density. When recalling (a) that
When deriving Eq. (3) with respect to time (and neglecting changes in
The state of minimum Gibbs free energy corresponds to a state of maximum
entropy and thus to thermodynamic equilibrium. With respect to Eq. (3) this
is the case when gravity and matric potential are equal in absolute terms in
the entire profile of the unsaturated zone:
Soil water retention curves as a function of relative saturation
determined as explained in Sect. 3.1. The dashed black lines mark the
relative saturation at hydraulic equilibrium, assuming arbitrarily a depth to
groundwater of
Note that the different soils are at the same depth to groundwater, for instance at
The equilibrium storages shown in Fig. 1 separate ranges of
relative saturation where the corresponding free energy of the soil water is
either negative or positive. This becomes obvious when plotting the specific
free energy per unit volume
Weight-specific free energy state of the soil water storage, as defined in Eq. (8), plotted against the relative saturation of the three different soils, assuming a depth to groundwater of 10 m. The green lines mark the local equilibrium state where the absolute value of the specific free energy is zero and the corresponding equilibrium saturations. Free energy in the P- and C-regimes is plotted in solid blue and red, respectively; the arrows indicate the way back to equilibrium.
Relative saturations smaller than
Figure 2 shows that the three different soils, when arranged at the same
geopotential level, are characterized by distinctly different energy state
curves as a function of relative saturation. The P-regime is very prominent
for the Colpach soil – potential energy dominates over a wide range of
saturation, and its
As described in Eq. (8), the energy state functions shown in Fig. 2 depend on the soil
water retention curve and the depth above groundwater. While depth to
groundwater is usually not exactly known, height over the next drainage
(HAND, Renno et al., 2008; Nobre et al., 2011) provides an easy-to-measure
surrogate when taking the water level of the closest stream as a reference.
While depth to groundwater increases obviously proportionally to HAND, the
related proportionality factor
This family of curves characterizes how HAND and soil physical characteristics jointly control the free energy state of soil water as a function of the relative saturation. The presentation of the energy state functions for our study areas in the following Sect. 3 will reveal that all points in the root zone with the same soil water retention curve and which fall into the same bin of HAND are represented by the same energy state curve.
The derived energy state function introduced in the last section defines the possible energy states of the soil water storage, a thermodynamic state space of the root zone so to say. Due to the intermittent rainfall and radiative forcing, their respective annual cycles, the free energy state of soil water will be pushed and pulled through this state space. It appears thus straightforward to visualize these storage dynamics, either observed or modelled, as pseudo oscillations of the corresponding free energy state in the respective energy state functions. This will teach us (a) which part of the state space is actually visited by the system, and (b) whether the system predominantly operates in one of these regimes or within both of them. In the following, we briefly characterize the study areas and the data set we use for this purpose.
The Colpach and Wollefsbach catchments belong to the Attert experimental basin (Pfister et al., 2002, 2017), and are distinctly different with respect to soils, topography, geology and land use (Fig. 4). Both catchments have been extensively characterized in previous studies with respect to their physiographic characteristics, dominant runoff generation mechanisms and available data (Wrede et al., 2015; Martinez-Carreras et al., 2015; Loritz et al., 2017; Angermann et al., 2017). Hence, we focus here exclusively on those system characteristics which determine their respective energy state functions. The Colpach has an elevation range from 265 to 512 m. Soils are young silty haplic Cambisols that formed on schistose periglacial deposits. Despite their high silt and clay contents they are characterized by a high permeability and high porosity (Jackisch et al., 2017), because the fine silt aggregates embed a fast draining network of coarse inter-aggregate pores. In contrast, the Wollefsbach has a much more gentle topography, from 245 to 306 m a.s.l. Soils in this marl geological setting range from sandy loams to thick clay lenses.
For this study, we use data from a distributed network of 45 sensor clusters spread across the entire Attert experimental basin (Fig. 3) collected within the hydrological year 2013/2014. These sensor clusters measure, among other variables, soil moisture and matric potentials within three replicated profiles in 0.1, 0.3 and 0.5 m depths using Decagon 5TE capacitive soil moisture sensors and MPS1 matric potential sensors. In this application we focus on data collected at 0.1 m depth; the distributions of sensors along HAND in the Wollefsbach and the Colpach are shown in Fig. 3b and c, respectively. Note that we use HAND as an estimator for the depth to groundwater here. Soil water retention was in both catchments analysed by Jackisch (2015) using a set of 62 undisturbed soil cores from the Colpach and 25 undisturbed soil cores from the Wollefsbach (Fig. 4a and b).
Map of the Attert basin with the Colpach and Wollefsbach catchments
(
Here we do not use these point relations, but representative, macroscale soil
water retention functions to derive the energy state function of our study
areas (Fig. 4c and d). These were derived by Jackisch (2015) from the raw
data of all lab analyses as follows. He pooled the matching pairs of soil
water content and matric potential observed in each soil core experiment in the same landscape setting into a
single statistical sample (Fig. 5c and d). When using the tension (
We define the representative retention curve as the one that relates the
expected soil water storage to the matric
potential
Loritz et al. (2017, 2018) used these effective retention functions for setting up physically based hydrological models for both catchments, which yielded simulations of streamflow and soil moisture dynamics in good accordance with observations. Test simulations with randomly selected retention functions derived from individual soil samples (Fig. 4b) or based on the averages of the van Genuchten parameters of 62 experiments performed clearly worse.
Based on these representative retention functions and the frequency distributions of HAND (Fig. 5b and d), we compiled the energy state functions of both catchments (Fig. 5a and c) according to Eq. (8), using HAND as a surrogate for the depth to groundwater. As stated in the previous section, the energy state function consists of a family of curves, which characterize the free energy state of the soil water as a function of the relative saturation, stratified along the bin centroids of the corresponding frequency distributions of HAND. So each line corresponds to a certain HAND value.
Energy state functions of the Colpach
Note that the wider HAND range in the Colpach causes a clear dominance of the
P-regime over a large saturation range. More importantly, Fig. 5a reveals
that for relative saturations larger than
In a first step we inter-compare the free energy states of the soil moisture
storage (Fig. 6) which was observed at two arbitrarily selected sites in the
respective study catchments. Both sites are located 20 m above their
respective streams. The soil water content in the clay-rich top soil of the
Wollefsbach site is in the winter and autumn period rather uniform and on
average 0.15 m
Figure 6b and c provide the corresponding free energy states of both soil
water time series as a function of the soil saturation. Observations are
shown as black circles, and the related theoretical energy state curves
calculated following Eq. (8) are in blue. The first thing to note is that the
observed free energy states for both sites scatter nicely around the
theoretical curves. More interestingly one can see that the spreading of the
free energy state of the soil water stock is at both sites distinctly
different. The free energy state of soil water at the Colpach site is during
the entire hydrological year in the P-regime and hence subject to an
overshoot in potential energy (Fig. 6b). The site operates in the linear
range of the energy state curve and fluctuates around an average
weight-specific energy density of 3.2 m, which corresponds to an energy
density of
Top soil water content observed at cluster sites in the Colpach and
Wollefsbach catchments
In the Wollefsbach the weight-specific free energy density of soil water
spreads across a much wider range of almost 180 m, which corresponds to
We hence state that the free energy state of the soil water stock reveals a
distinctly different dynamic behaviour at both sites, which cannot be derived
from the comparison of the corresponding soil water moisture time series. The
Colpach site is characterized by permanent storage excess, though the
corresponding soil water content is always smaller than in the Wollefsbach.
Free energy of the soil water storage is in this range a linear function of
relative saturation. This implies that the energy difference which
predominantly drives soil water dynamics changes linearly with soil water
content, or in other words gravity potential dominates against matric
potential. The retention function in Fig. 1 shows that the matric potential
in the Colpach at the minimum observed saturation of
Figure 7 presents the free energy states of the soil moisture which was
observed at all cluster sites in the Colpach (Fig. 7a,
Free energy of all observations in the Colpach
Generally, the observed free energy states plot nicely around the
energy state curves of the corresponding HAND. The Colpach (except
for a few sites) operates most of the time in the linear range of the P-regime,
indicating that soil moisture dynamics is dominated by potential energy
differences. Observations in the Colpach generally spread across a wide range
of relative saturations, and the corresponding “amplitudes” of the free
energy deviations are clearly larger than at the single site shown in
Fig. 6b. This is because sensor clusters with the same HAND were pooled into
the same subsample regardless of their separating distance. For instance, at
Despite the large spreading, 80 % of the Colpach sites operated permanently in the P-regime (Fig. 7c). During the wet season it is more than 90 % of the sites, between days 250 and 400, and quite a few profiles switch to the C-regime and thus to a storage deficit. These profiles mostly have low HAND values, with only some having large values of 37 and 22 m.
In the Wollefsbach we find, consistent with Fig. 7b, a clear drop in free energy into the C-regime during the dry spell in the summer period. All sites drop clearly below the permanent wilting point, which corroborates the strong evaporative drying of the top soil in this landscape. In contrast to the Colpach, the fractions of profiles which operate in the different regimes are much more variable in time (Fig. 7d). During the observation period, on average 50 % of the profiles operate in the C-regime and thus in a storage deficit. The minimum is 30 % and the C-regime fraction peaks at 90 % at day 250. Note that more than 50 % of the sites are continuously in the C-regime during the second half of the observation period. These differences are consistent with the strongly different runoff generation behaviour of both systems, as further detailed in the next section.
An interesting question is whether the free energy state of the soil water content and particularly the separation of the C- and P-regimes is helpful for explaining the onset of storage-controlled streamflow generation in both landscapes. As storage-controlled runoff response to rainfall is not generated everywhere in the catchment but mostly in the riparian zone, the energy state of soil water at large values of HAND is unimportant in this respect. We thus plotted for the entire hydrological year the observed streamflow in both catchments against the energy state of soil water for sites at the smallest HAND values of 2 m, which are close to the riparian zone (Fig. 8a and b).
Observed streamflow in the Colpach (
Both scatter plots reveal distinct threshold-like dependence of streamflow on the free energy state of soil water and note that the threshold coincides with the state of zero free energy, which separates the C-regime from the P-regime. Streamflow in the Colpach is uniformly low if the riparian zone is in a storage deficit with respect to the local equilibrium (Fig. 8a), while streamflow shows a strong variability when the system switches to a storage excess in the P-regime. The transition to a state of storage excess, which implies that the system needs to release water to relax back to local equilibrium, coincides with the onset of storage-controlled streamflow generation. The variability of streamflow in the P-regime does of course also reflect the variability in the rainfall forcing. Streamflow in the C-regime likely feeds exclusively from groundwater. This behaviour is in line with our theoretical expectation.
In the Wollefsbach we observe a slightly different pattern. On the one hand
there is a similar sharp increase in streamflow when free energy of soil
water in the riparian zone switches from the C- to the P-regime. On the other
hand one can observe distinct values of streamflow for specific free energy
densities in the range between
The presented results provide evidence that a thermodynamic perspective on soil water storage offers holistic information for judging and inter-comparing soil water dynamics, which cannot be inferred from soil moisture observations alone. In the following we reflect on the general idea of using free energy as a state measure, and discuss its promises as well as its limiting assumptions. We then move on to the more specific differences in the storage dynamics in both studied catchments. And we close by reflecting on the apparent paradox between the known local non-linearity of soil physical characteristics and the frequent argumentation that hydrological systems often behave much more linearly.
Our results show that free energy as a function of relative soil
saturation holds the key to defining a meaningful state space of the root
zone of a hydrological landscape. This space of possible energy states
consists of a family of energy state curves, where each of those
characterizes how free energy density evolves at a specific height above groundwater, depending on the triad of the matric potential, HAND (as a
surrogate for the unknown gravity potential) and soil water content. The free
energy state of soil water reflects in fact the balance between its capillary
binding energy and geo-potential energy densities, and we showed that this
balance determines
whether a system is at a given elevation above groundwater locally in its
equilibrium storage state ( the regime of storage dynamics. Soil water dynamics in the C-regime
(
The energy level function turned out to be useful for inter-comparing
distributed soil moisture observations among different hydrological
landscapes, as it shows the trajectory of single sites or of the complete
set of observations in its energy state space. This teaches us which part of
the state space is actually “visited” by the system during the course of the
year, whether the system operates predominantly in a single regime, whether
it switches between both regimes during dry spells and how much water needs
to be released or recharged locally for relaxing back to local equilibrium
and how often it actually reaches its equilibrium.
Note that the usual comparison of soil water contents alone did not yield this information. On the contrary, from this we would conclude that the site in the Wollefsbach is, due to the higher soil water content, always “wetter” than the corresponding site in the Colpach. The free energy state reveals, however, the exact opposite: we have a storage excess at the Colpach site for the entire year, while the Wollefsbach site is in summer in a storage deficit. We thus propose that the terms “wet” and “dry” should only be used with respect to the equilibrium storage as a meaningful reference point.
The free energy state of soil water in the riparian zone of both study catchments has furthermore been proven to be rather helpful in explaining streamflow generation. We found a distinct threshold behaviour for storage-controlled runoff production in both catchments, and clear hints at overland flow contributions in the Wollefsbach. While we admit that a threshold-like dependence of runoff generation is frequently reported (Tromp-van and McDonnell, 2006a, b), we would like to stress that the tipping point we found here has a theoretical basis. In both catchments it coincides with the local equilibrium state of zero free energy – the onset of a potential energy excess of soil water in the riparian zone coincides with the onset of a strongly enlarged streamflow generation.
The apparent strong sensitivity of the free energy state of soil water to the estimated depth to groundwater offers new opportunities for data-based learning and an improved design of measurement campaigns, but it also determines the limits of the proposed approach. With respect to the first aspect, we could show that an underestimation of 2 m in the assumed depth to groundwater leads to a clear deviation of the observed free energy states from the theoretical energy level curve. This offers the opportunity to estimate depth to groundwater from joint observations of soil moisture and matric potential, in case the local retention function is known. This can, for instance, be done by minimizing the residuals between the observation and the theoretical curve as a function of depth to groundwater, or it allows for the derivation of a retention function based on the joint observations of soil moisture, matric potential and depth to groundwater. Here, we need to minimize the residuals between the observation and the theoretical curve, but this time as a function of the parameters of the soil water retention curve. Due to this strong sensitivity it is furthermore important to stratify soil moisture observations both according to the installed depth of the sensor and according to the elevation of the site above groundwater, or the height over the next stream. The latter is important because depth to groundwater determines the equilibrium storage the site will approach when relaxing from external forcing.
Despite all these opportunities for learning, the sensitivity of free energy to the estimated depth to groundwater implies that the site of the system is still in hydraulic contact with the aquifer. This key assumption is certainly violated if the soil gets so dry that the water phase becomes immobile while the air phase becomes the mobile phase. And it might get violated if depth to groundwater becomes too large. Last but not least, the groundwater surface may change either seasonally or in some systems more rapidly, and this might imply step changes in the energy state function and the storage equilibrium.
We nevertheless conclude that it is worth collecting joint data sets either of the triple of soil moisture, matric potential and the retention function at distributed locations (as we did in the CAOS research unit as explained in Zehe et al., 2014) or even preferably of the quadruple of soil moisture, matric potential, retention function and depth to groundwater. Soil moisture observations alone appear to be not very informative about the system state. This is because they neither tell anything about the binding state of water nor about how the system deviates from its equilibrium and which process is “needed” to relax.
In line with our proposition, we found a distinctly typical interplay between capillary and gravitational controls on soil water in our study areas, which were in the Colpach were substantially different compared to the Wollefsbach. The observations clearly revealed that the topsoil in the Colpach operates the entire hydrological year largely in a state of storage excess due to an overshoot in potential energy. Soil water dynamics is mainly driven by differences in potential energy, which means that the linear and non-local gravitational control dominates. Most interestingly we found that the free energy state of the soil operated for a considerable time of the year in the linear range of the P-regime, which implies that the storage dynamics is (multi-)linear. This means that the specific free energy density is at each HAND level a linear function of relative saturation, but the slope of the energy state curves does increase with increasing geopotential. We found furthermore that the annual variation of the averaged free energy of the soil water storage was rather small. Zehe et al. (2013) found a similar, almost steady-state behaviour for the averaged free energy of the soil water storage in the Mallalcahuello catchment in Chile, which also operated in the P-regime the entire year. Note that both landscapes are characterized by a pronounced topography, by well-drained highly porous soils (Blume et al., 2008a, b, 2009), and that both are predominantly forested. In both landscapes, subsurface storm flow and thus storage-controlled runoff generation are the dominant mechanisms of streamflow generation. This is consistent with our finding that gravity is the dominant control of soil water dynamics.
In contrast, the Wollefsbach was characterized by a seasonal change between both regimes: operation in the P-regime during the wet season and a drop to a C-regime and a storage deficit during the dry summer period. Free energy was at all sites on average negative, and a non-linear function of the relative saturation. Interestingly we found the same seasonality for the Weiherbach catchment in Germany, a dominance of potential energy during the wet season and a strong dominance of capillary surface energy in summer (Zehe et al., 2013). Note that both landscapes are characterized by cohesive soils, more silty in the Weiherbach and more clay-rich in the Wollefsbach, and a gentle topography. And both are used for agriculture. In both areas Hortonian overland flow would play the dominant role, but this process is actually strongly reduced due to a large number of worm burrows acting as macropores (Zehe and Blöschl, 2004; Schneider et al., 2018). Both landscapes are also controlled by tile drains. In both areas the soil water dynamics is dominated by capillarity during the summer period, which means that the local soil physical control dominates root zone soil moisture dynamics.
Overall we conclude that a thermodynamic perspective on hydrological systems provides valuable insights, helping us to better understand and characterize different landscapes. Given the strong relation between a potential energy excess of soil water in the riparian zone and strongly enlarged streamflow production, we found in our study areas that it seems promising to further explore the value of free energy for hydrological predictions. We also conclude that it makes sense to use the terms “wet” and “dry” only with respect to the equilibrium storage as a meaningful reference point, because the latter determines whether the soil is in a state of storage excess or a storage deficit with respect to the free energy state. Another key finding is that the energy level function, which can be seen as a straightforward generalization of the soil retention function, accounts jointly for capillary and gravitational control on soil moisture dynamics. With this we link the non-linear soil physical control and the topographical control on storage dynamics in a stratified manner and use HAND as a surrogate for the gravitational potential. A nice additional finding is that a linear dependence of free energy on soil saturation does not compromise the non-linearity of soil water characteristics. On the contrary, it may be explained by the dominance of potential energy in catchments with pronounced topography and during not too dry conditions, and this implies that at least the energy difference driving soil water dynamics is a linear function of the stored water amount. The latter is the basis of the linear reservoir, which is frequently used in conceptual modelling. The option for linear behaviour of the subsurface is hence inherent not only to Darcy's law of the saturated zone, as has been shown by de Rooij (2013) by deriving aquifer-scale flow equations for strip aquifers. Even in the top of the unsaturated zone a linear relation between storage and driving potential energy differences might emerge. This inherent option for linear behaviour is likely the reason why conceptual models, which usually do not account for soil physical characteristics, work very well in some catchments, while in others they do not. Based on the presented findings one could speculate that conceptual models work well in systems which are dominated by potential energy.
The code and the data underlying this study are freely available by email request to the contact author.
EZ wrote the paper, had the main idea for the underlying theoretical framework and carried out most of the analysis. RL and CJ contributed both to the theoretical framework, derived the distribution functions of HAND in both study catchments and measured the effective soil water retention curves. MW, AK and HHS contributed strongly to the idea of the energy state functions and their thermodynamic interpretation. SKH and TB collected the soil water content and matric potential data in the field and provided quality controlled data for the analysis.
The authors declare that they have no conflict of interest.
This article is part of the special issue “Thermodynamics and optimality in the Earth system and its subsystems (ESD/HESS inter-journal SI)”. It is not associated with a conference.
We sincerely thank both reviewers, particularly Gerrit de Rooij, for their thoughtful and valuable feedback. This study contributes to and greatly benefited from the “Catchments as Organized Systems” (CAOS) research unit. We thank the German Research Foundation (DFG) for funding (FOR 1598, ZE 533/11-1, ZE 533/12-1). The authors acknowledge support by the Deutsche Forschungsgemeinschaft and the Open Access Publishing Fund of the Karlsruhe Institute of Technology (KIT). The service charges for this open-access publication have been covered by a Research Centre of the Helmholtz Association. The article processing charges for this open-access publication were covered by a Research Centre of the Helmholtz Association. Edited by: Stefan Hergarten Reviewed by: Gerrit H. de Rooij and one anonymous referee