By extracting bound water from the soil and lifting it to the canopy, root systems of vegetation perform work. Here we describe how root water uptake can be evaluated thermodynamically and demonstrate that this evaluation provides additional insights into the factors that impede root water uptake. We derive an expression that relates the energy export at the base of the root system to a sum of terms that reflect all fluxes and storage changes along the flow path in thermodynamic terms. We illustrate this thermodynamic formulation using an idealized setup of scenarios with a simple model. In these scenarios, we demonstrate why heterogeneity in soil water distribution and rooting properties affect the impediment of water flow even though the mean soil water content and rooting properties are the same across the scenarios. The effects of heterogeneity can clearly be identified in the thermodynamics of the system in terms of differences in dissipative losses and hydraulic energy, resulting in an earlier start of water limitation in the drying cycle. We conclude that this thermodynamic evaluation of root water uptake conveniently provides insights into the impediments of different processes along the entire flow path, which goes beyond resistances and also accounts for the role of heterogeneity in soil water distribution.

Root water uptake is an important process, determining the transport of water
between soil and atmosphere and influencing plant productivity and crop
yield. A wealth of studies using both models and observations deal therefore
with understanding root water uptake, that is, to learn where plants take up
water

In order to evaluate the efficiency of root water uptake and learning how
plants may regulate it, we require some understanding of the impediment for
water flow and how it is distributed along the soil–plant–atmosphere
continuum, especially whether it lies within the plant or the soil
compartment

In this paper we show that additional information about the system can be
obtained from a thermodynamic perspective, specifically by combining the
hydraulic potentials with mass fluxes, yielding fluxes of energy. This
approach has the advantage that different processes, such as the change of
soil water potential with decreasing soil water content as well as the
transport of water over a resistance, can be expressed in the same currency of
energy fluxes and dissipation, with units of joules per second (J s

While thermodynamics is most commonly associated with heat, its formulation
is much more general and can be used to express the constraints and
directions of energy conversions of any form

As will be shown in this paper, the thermodynamic formulations are comparatively simple and straightforward to implement in models. Since the hydraulic potential is just the specific energy per mass (or volume), that is, the derivative of the Gibbs free energy to mass (or volume), the related soil energy content can be obtained by integration. The thermodynamic representation has, however, several advantages that are currently not well explored by the hydrological community. One of these advantages is, for example, related to describing the effects of soil heterogeneity. While soil water potential is an intensive property (i.e., a property that does not depend on the size of the system) that cannot meaningfully be averaged, the associated energy content is an extensive property (i.e., a property that depends on the size of the system) and therefore is additive, and the total energy content in heterogenous soil can be calculated. As will be shown, the total energy content offers insights into the role of soil heterogeneity that cannot be derived when focussing only on the potential or the soil water content alone.

In the following, we will derive formulations for the energy contained in unsaturated soil as well as for the dissipation of energy for fluxes in unsaturated soil and along the root system. In order to illustrate how these fluxes can be interpreted to evaluate impediments to root water uptake and the role of soil water heterogeneity, we illustrate them in a simplified process model, which is a conceptual four-box model for root water uptake.

Thermodynamics is a general theory of physics that describes the rules for energy conversions. The first law of thermodynamics ensures energy conservation and formulates that the internal changes in energy are balanced with external additions or removals and internal conversions between different forms. The second law describes that with every conversion of energy, energy is increasingly dispersed, which is described by entropy as a physical quantity. It is the second law that sets the natural direction of processes to deplete their driving gradients, and that is, for instance, reflected in soil water movement depleting gradients in soil water potential. The state of thermodynamic equilibrium is then described as a state of maximum entropy and represents a state in which no driving gradients are present within the system.

To describe soil water movement in thermodynamic terms, it needs to be
formulated in terms of the energies involved, and it needs to be associated
with entropy. The energies involved consist of the binding energies
associated with capillary and adhesive forces, gravitational energy, and
heat. The first two forms of energy are directly relevant to soil water
movement. Their formulation in energetic terms is straightforward as these
are directly related to the matric and gravitational potentials. These
potentials are formally defined as chemical potentials

The use of heat is important as it is required to ensure energy conservation within the soil when the other forms of energy change, and because heat is directly linked to the entropy of the system. When water is redistributed within the soil due to gradients in soil water potential, this results in a reduction of the binding and gravitational energy, with the reduced energy being released as heat of immersion (see also below). The state of thermodynamic equilibrium is reached when there is no gradient in soil water potential. This state corresponds to a state of minimum Gibbs free energy; i.e., the binding and gravitational energy is minimized for a given amount of stored water. As the remaining energy is converted into heat, this reduction to a minimum of Gibbs free energy corresponds to a maximum conversion into heat and thus a maximization of entropy that can be achieved by soil water redistribution. This is despite the fact that the actual amounts of heat involved are rather small compared to the heat fluxes involved in heat diffusion in the soil.

Next, we describe how these forms of energies are determined quantitatively
from their respective potentials and how these forms of energy change during
root water uptake and soil water redistribution. We state equations for
discrete bulk soil compartments

Two types of energy are relevant for describing soil water states. We will
refer to the sum of these as the total hydraulic energy
(

The gravitational energy (

The binding energy (

Essentially,

Example of the
hydraulic and thermodynamic states of a sample soil (sandy loam;

Figure

When soil water potential is distributed heterogeneously, the binding energy increases (is less negative). Technically, this results from the strongly nonlinear water retention function. From a process perspective, this additional energy will drive water fluxes for equalizing the soil water gradients between compartments and will during this process eventually dissipate this amount of energy by conversion into heat.

During root water uptake, a given amount of energy has to be invested to take
up a certain volume of water over time. Hence differential changes of binding
energy per change in water content are relevant. Note that the slopes on the
curves in Fig.

Soil water fluxes lead to dissipation (

Energy dissipation due to soil water flow between compartments is written as

The same applies to the dissipation of energy due to the small-scale radial
root water uptake (

Lastly, the root water uptake constitutes an export of energy (

Although the dissipation terms (

The energy balance for the soil–root system can be written as the sum of the
changes in total hydraulic energy over all compartments, the dissipation
terms, and the energy export:

Some properties of this equation are noteworthy. First, re-arranging
Eq. (

Parameters and initial conditions applied for each of the scenarios
in the conceptual model for the compartments (

Schematic of the numerical split-root experiment. One plant has access to four soil compartments, two densely rooted (left) and two sparely rooted (right). Color shading of the containers indicates high (dark color) and low (bright color) initial soil water content. The average initial soil water content is the same in all simulations. In the same way, the average water content over the two left (densely rooted) and two right (sparsely rooted) containers is the same in all simulations.

Variables used in this study.

The thermodynamic evaluation introduced in the last section is meant to be
applied to a water flow model (process model). We illustrate the application
using a simple model system as shown in Fig.

The mass balances of the reservoirs describing the temporal changes in

The total root water uptake (

For the purpose of demonstration, we keep the model simulations deliberately
simple. All soil compartments are arranged horizontally, so that differences
in gravitational energy do not play a
role and all changes in hydraulic energy will be due to changes in water
content. Also, we model a split-root experiment, where no water flow between
compartments is possible and all changes in soil water content are due to
root water uptake. This enables us to increase the heterogeneity in the soil
water content and demonstrate its effect on the energy balance, xylem
potential, and uptake dynamics. In this simplified setup we solve
Eqs. (

For each time step (

Additional scenarios with diurnal fluctuations of transpirational forcing and other soil hydraulic properties are given as a reference in the Supplement. They yield similar results.

We run the model for four scenarios, as shown in
Table

The first scenario is completely homogenous with a uniform initial soil water
content and root conductivity across compartments. Three additional scenarios
are initialized with heterogenous initial soil water and differ with regard
to the heterogeneity of root conductivity. In all simulations the average
initial soil water content is the same. In the same way, the effective root
conductivities (

The model is representative of a plant having access to a soil volume of
0.5 m

The soil hydraulic properties are equal in all compartments and derived using

Figure

Model results
of the simple model:

Components of the energy balance (Eq. 11) for the
soil–plant system over the course of a drying experiment and different root
water uptake scenarios. As in Fig.

Based on the output of the root water uptake model, we applied
Eq. (

The energy export (

The optimal case (grey solid line) is the one with the least possible
expenditure in d

At the same time, in heterogenous soils the impediment to uptake due to water flow over the root resistance increases, since uptake occurs preferentially in a limited part of the root system (the compartment with greatest root length that was initialized as wet, data not shown). However, this dissipation effect is less dynamic over time than the one related to soil drying in this modeling exercise.

We used thermodynamics to evaluate the dynamics of a very simple process model for root water uptake to demonstrate that besides fluxes and potentials there is more relevant information in the system that relates to change of hydraulic energy and dissipative losses of water uptake. The main contribution of the paper lies in providing a tool for assessing where the impediments to root water uptake lie along the flow path between soil and atmosphere. For this the thermodynamic formulations are applied a posteriori to water fluxes and changes of soil water contents calculated with the hydrological model. The relative contribution of each of the impediments can then be quantified, by evaluation of the relative contribution of each process to the total energy export. At the same time, the calculations with the simple model serve as a proof of concept: the energy balance is closed; i.e., the sum of change in hydraulic energy and dissipation equal the energy export.

In our thermodynamic description of the soil–plant system, we have not considered the changes of soil temperature, which should be induced particularly when heat is generated as water attaches to the soil. We have done this because the related changes of temperature are so small that they would not affect the water flow and generally small compared to changes of temperature due to radiative soil heating.

Also, we have assumed in this derivation that the soil water retention
function is known and is non-hysteretic. The latter may have considerable
influence on the resulting trajectory of d

Finally, we have also deliberately limited our model scenarios to situations where roots do not grow and where root length does not depend on water availability, and we have not allowed for redistribution of water between compartments. This way, we artificially maintained heterogeneity, which was done in order to demonstrate in the separate scenarios how heterogeneity alone affects uptake and its thermodynamics.

An important advantage of evaluating the process model output in the energy
domain lies in the possibility for evaluating the role of heterogenous soil
water potentials. The water potentials, the derivative of the Gibbs free
energy per mass, are an intensive property of the system, and in heterogenous
systems they cannot be meaningfully averaged. The Gibbs free energy itself is
an extensive property, can be averaged, and hence allows states in
heterogenous systems to be efficiently described. An additional advantage of
working in the “energy domain” is the possibility to consider the influence
of the water retention function, the heterogeneous soil water distribution,
and the various resistances along the flow path in the same realm and using
the same units. In particular, heterogeneity of soil water increases the
total hydraulic energy, which necessarily implies that xylem water potentials
have to be more negative to transpire at the same rate and same average soil
water content if root systems are equally distributed. Thus, with everything
else being equal and independent of soil water potential distribution, plants
rooted in heterogeneously wetted soils are expected to reach water limitation
earlier. This phenomenon has already been observed in models dealing with
spatially heterogenous infiltration patterns caused by forest canopies

We have given equations for our simple system, but the concept can easily be
extended to more complex systems, for example three-dimensional models of
root water uptake

At the more general level, this study adds to the thermodynamic formulation
of hydrologic processes and the application of thermodynamic optimality
approaches

Systems approaches and modeling will certainly be tools to investigate plant
water relations and efficient rooting strategies in the future

It can be shown analytically that a homogeneous soil water distribution
results in the least dissipation associated with root water uptake (as shown
in Fig.

We consider the case of a uniform root system (i.e.,

In principle, one can also show that a uniform root system results in a minimum of dissipation. This requires an integration over time, which makes an analytical treatment more complex so that it is more easily illustrated by the numerical simulations done in the main text.

Parameters used for calculation of soil
hydraulic properties using

Anke Hildebrandt acknowledges support from the Collaborative Research Centre “AquaDiva” (SFB 1076/1, B02) funded by the German Science Foundation (DFG). Marcel Bechmann has been supported by the Jena School for Microbial Communication (JSMC). Axel Kleidon acknowledges financial support from the “Catchments As Organized Systems (CAOS)” research group funded by DFG through grant KL 2168/2-1. We thank Stefan Kollet for discussion and helpful references; Thomas Wutzler; and the reviewers Gerrit de Rooij and Uwe Ehret as well as the editors Erwin Zehe and Alberto Guadagnini for critical comments during the manuscript discussion. The article processing charges for this open-access publication were covered by the Max Planck Society. Edited by: A. Guadagnini Reviewed by: U. Ehret and G. H. de Rooij