HESSHydrology and Earth System SciencesHESSHydrol. Earth Syst. Sci.1607-7938Copernicus PublicationsGöttingen, Germany10.5194/hess-20-3441-2016A thermodynamic formulation of root water uptakeHildebrandtAnkeanke.hildebrandt@uni-jena.dehttps://orcid.org/0000-0001-8643-1634KleidonAxelhttps://orcid.org/0000-0002-3798-0730BechmannMarcelFriedrich-Schiller-Universität, Jena, GermanyMax-Planck-Institut für Biogeochemie, Jena, GermanyAnke Hildebrandt (anke.hildebrandt@uni-jena.de)29August2016208344134549December201521December20152August20168August2016This work is licensed under a Creative Commons Attribution 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0/This article is available from https://hess.copernicus.org/articles/20/3441/2016/hess-20-3441-2016.htmlThe full text article is available as a PDF file from https://hess.copernicus.org/articles/20/3441/2016/hess-20-3441-2016.pdf
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.
Introduction
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 , how root length and
hydraulic properties affect uptake , how
plant communities exploit heterogeneously distributed soil water
, how to identify efficient
rooting depth , how soil water storage is shared between
plants , how plants may optimize water flow
in order to prevent cavitation , and about
relations between root water uptake and stomatal control , as well as crop yield .
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 . Much of our process understanding on
the spatial distribution of water uptake and its evolution in drying soil is
based on physically based models of the root system .
Relying on the electrical analogue of water flow and mass balance
, they mimic the flow of water over a
chain of resistances along continuously dropping water potentials from the
soil to the root, further up within the root xylem, and sometimes up the canopy
. At the same time, root water uptake depletes the soil
reservoir, leading to more negative soil hydraulic potentials which need to be
overcome in order to maintain the necessary gradient between soil and
atmosphere to allow for flow. Both processes (flow over a resistance network
and increasing soil water retention) impede transpiration, but comparing
their mutual contribution in form of resistances is not suitable, since the
change of soil water retention per water removed has no proper resistance
analogue.
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-1).
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 soil water movement and uptake
by plants involve changes in binding and gravitational energy, as expressed
by the respective matric and gravitational potentials, the fluxes of water in
the soil–vegetation–atmosphere system are associated with fluxes of energy,
and we can compare which one of the processes in the uptake chain requires
most energy, as well as quantifying the total energy expense of the uptake.
Thus, the thermodynamic perspective allows us to evaluate the efficiency of
different temporal dynamics of root water uptake and
differentiate more efficient from less
efficient root systems.
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 and soil hydrologyThermodynamic background
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
, i.e., defined as the change in
Gibbs free energy resulting from an incremental change in mass.
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 Vs,i of sufficiently large size
such that the porosity and the soil water retention curve are defined at that
scale. Soil properties are considered homogenous within but may vary between
soil compartments. We thus use sums to integrate over the soil space in the
following. The variable names used in this manuscript are summarized in
Table .
Forms of energy associated with soil water content
Two types of energy are relevant for describing soil water states. We will
refer to the sum of these as the total hydraulic energy
(Uw, J) contained in a soil volume, which consists of the binding
energy, Uwb (J), and the gravitational energy, Uwg
(J), defined in each soil compartment i (with i=1..n):
Uw,i=Uwb,i+Uwg,i.
For the total soil volume we have
Uw=∑i=1nUw,i.
The gravitational energy (Uwg,i, J) relates to the energy
necessary to lift the water from a reference level up to the point where it
is stored in the soil:
Uwg,i=Vs,i⋅ρw⋅g⋅θi⋅(zi-zr),
where ρw is the density of water (ρw=1000kgm-3), g is the gravitational acceleration (g=9.81 m s-1), zr is the elevation of the reference level, zi the elevation of
compartment i, and θi refers to the volumetric water content. This
formulation neglects variations
of gravitational energy within the compartment i. More rigorously, the
gravitational energy should be integrated over depth from the bottom to the
top of the compartment. This requires information on the vertical
distribution of soil water within the compartment, which is typically not
available in model applications. If compartments are chosen to be shallow,
the error associated with this simplification is small.
The binding energy (Uwb,i, J) relates to the capillary forces
in the soil pores. With the soil matric potential being the change of Gibbs
free energy per change of mass, the related energy can be found by
integration. We obtain it here by integration of soil water volume of each
compartment i:
Uwb,i=Uwb,i(θi)=Vs,i⋅ρw⋅g⋅∫θminθiψM(θ′)⋅dθ′.
Essentially, Uwb,i is the integral of the water retention curve
(ψM(θ)). The multiplication with ρw⋅g serves to convert the units of the matric potential (ψM)
from meter water to joule. An example for both ψM(θ) and
the related Uwb(θ) is depicted in Fig.
for a sandy loam using the van Genuchten parameterization with parameters
given in Table . The lower integration point
(θmin) should refer to completely dry soil. However, some analytic
formulations for the water retention function are not well defined in the
very dry range , and we therefore chose
θmin>0. This has a great influence on the absolute values of
binding energy in the soil, but, as will be shown below, the relative (i.e., temporal)
differences in Uwb,i are of relevance. Therefore, the exact
choice of θmin does not affect the results. We propose choosing a
value just below the water content at the permanent wilting point, or another
suitable value smaller than the water contents that will be reached in the
desired application.
Example of the
hydraulic and thermodynamic states of a sample soil (sandy loam;
): (a) water retention curve with logarithmic
y axis; parameters are given in Table .
(b) For the same soil, binding energy,
Uwb=∑i=1nUwb,i, as a function of soil water
content, for homogenous and heterogenous soil water distribution in a total
soil volume 1 m3. The ratio indicated in the legend corresponds to the
ratio of soil water contents in two compartments of equal size but different
soil water content. The blue arrow indicates how much energy is available for
driving fluxes to equalize the gradients in water potentials between
compartments.
Figure b shows the binding energy of the soil water as a
function of the volumetric soil water content both for homogeneously and
heterogeneously wetted soil. Like the soil matric potential, Uwb
is negative, reaching the lowest values at soil saturation. The negative sign
relates to the fact that energy is released (in form of a very small amount
of heat) when water attaches to the pore walls (“heat of immersion”;
). The same amount of energy has to be
transferred to the soil when water is removed from the pores, and hence the
bond between the water and the pore wall is broken. Thus, decreasing the
water content via root water uptake constitutes an export of negative energy
along with the mass export of water from the soil system to the plant (with
details described below).
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. b are steeper the greater the soil water
heterogeneity. This illustrates that more energy has to be invested per
decrease in total soil water content in heterogeneously compared to
homogeneously wetted soils if water is to be extracted at an equal rate from
the compartments.
Dissipation and energy export associated with soil water movement and root water uptake
Soil water fluxes lead to dissipation (D, J s-1) of total hydraulic
energy. Generally, dissipation of energy occurs when water flows over a
resistance, and it is expressed as the product of the driving gradient and the
water flow. Those fluxes may occur within the soil between compartments
during redistribution of bulk soil water or at the small scale due to root
water uptake, or further, within the plant tissue. The representation of
fluxes in models differs according to model complexity, yet the dissipative
nature of these fluxes should remain unaffected by how these are represented.
Energy dissipation due to soil water flow between compartments is written as
Df,k=ρw⋅g⋅(hi-hj)Jw,k
and for the entire soil volume as
Df=∑k=1lDf,k,
where h refers is the soil hydraulic potential
(hi=ψM+(zi-zr), m water) of the neighboring
compartments i and j; Jw,k refers to the water flux between
two neighboring compartments i and j; Df,k (J s-1)
refers to the respective dissipation of energy over the boundary k between
those compartments; and Df (J s-1) refers to the total
dissipation due to water fluxes within the total soil volume, i.e., over all
interfaces between compartments (l). The dissipation is always negative,
since it indicates a loss of hydraulic energy from the system, which is
released in the form of a very small quantity of thermal energy.
The same applies to the dissipation of energy due to the small-scale radial
root water uptake (Du,i, J s-1). The formulation depends
a great deal on the complexity of the applied flow model. The flow may be
represented over several steps, i.e., from bulk soil to the root surface,
from root surface to the xylem, and within the xylem. Here, we represent
the flow in a simple form, corresponding to
the water uptake (Jwr,i) of the water flow model presented in
the next section (Eq. ). Note that this may easily be adapted
to more comprehensive formulations of water uptake. We write dissipation due
to microscopic radial water uptake as
Du,i=ρw⋅g⋅(ψM,i-ψx)⋅Jwr,i,
with Du,i (J s-1) being the dissipation due to root water
uptake in each reservoir i and ψx being the root xylem potential (m). The total dissipation
Du becomes
Du=∑i=1nDu,i.
Lastly, the root water uptake constitutes an export of energy (JE,exp, J s-1) from the soil root system which is generally defined as
the product of the potential at and the flow across the boundary. In our case, water leaves at the top of the
root collar as total transpiration at the xylem water potential:
JE,exp=ρw⋅g⋅ψx⋅Jwu.
The sign of JE,exp is positive since in our case water leaves
the system (a negative flux) over the root collar at negative hydraulic
potential. Correspondingly, this increases the total hydraulic energy as the
soil dries (compare Fig. ). JE,exp would be
negative should water enter the system via the roots.
Although the dissipation terms (Df, Du) and energy fluxes (JE,exp) carry the same units, their difference is noteworthy. Dissipative
fluxes refer to internal processes within the thermodynamic system. They are
irreversible. In our example they reflect the heat dissipated when water
fluxes degrade the gradients in soil water potential. On the other hand,
JE,exp is an energy flux (energy transported) across the system
boundary. Note, however, that in general this flux also depletes a gradient
(between the soil and the atmosphere), but this gradient is not described in
our simple soil–root model explicitly.
Energy balance equation
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:
∑i=1ndUw,idt=∑k=1lDf,k+∑i=1nDu,i+JE,exp.
Some properties of this equation are noteworthy. First, re-arranging
Eq. () yields an expression that relates the characteristics
at the outlet of the system to a series of internal processes:
JE,exp=ρw⋅g⋅ψx⋅Jwu=∑i=1ndUw,idt-∑k=1lDf,k-∑i=1nDu,i.
The units in all terms of Eq. () are joules per second
(J s-1), as they all indicate rates of energy flux and changes of
energy content with time. More practically, JE,exp, as the
product of root collar xylem potential and transpirational flux, is
influenced by several processes, and Eq. () shows that they
act as a sum (remember that all dissipative terms have negative signs). For a given water flux,
Eq. () shows that the collar xylem potential has to be more
negative when water has to be moved within the soil (thus decreasing
Df), when water is taken up in drier soil at more negative soil
water potentials, and also when soil water potentials are more
heterogeneously distributed (both increasing Uwb, as shown in
Fig. ). For model applications, comparison of the
magnitude of the separate terms of the sum in Eq. () provides
a tool to assess which of the successive pathways involved in root water
uptake most strongly impedes water flow.
Parameters and initial conditions applied for each of the scenarios
in the conceptual model for the compartments (i=1…4). Given are the
differences between scenarios in words and the corresponding manipulations in
initial states and parameters.
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.
SymbolVariableUnitsDfDissipation due to soil water flowJ s-1 or WDuDissipation due to root water uptakeJ s-1 or WgGravitational acceleration9.81 m s-2hSoil hydraulic potentialmi, jSoil compartment indices–JE,expExport of energy from the soil–rootJ s-1 or Wsystem through the root collarJwuTotal root water uptakem3 s-1Jw,kSoil water redistribution betweenm3 s-1compartments over the interface kJwr,iWater flux from bulk soil into the rootm3 s-1kIndex for interfaces between–compartmentslNumber of interfaces between–compartmentsKijSoil hydraulic conductivity betweenm2 s-1compartments i and jKr,iEffective radial conductivity of them2 s-1active roots in compartment inNumber of soil compartments4tTimesUwTotal hydraulic energyJUwbBinding energyJUwgGravitational energyJWiTotal soil water storagem3VVolumem3Vs,iVolume of the model compartmentsm3zElevationmzrElevation of the reference levelmψMSoil matric potentialmψxXylem water potentialmρwDensity of water1000 kg m-3θVolumetric soil water content–θaveAverage volumetric soil water content–θminLower integration boundary for Uwb–Conceptual root water uptake model
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 system
consists of four soil water reservoirs, from which water is extracted by root
uptake. No water is added during the simulation. All soil reservoirs are
assumed to be of equal volume, Vs,i(m3), and their
water storage is described by the variables Wi=θi⋅Vs,i (in m3), with θi being the volumetric
soil water content (–) of the reservoir i. Soil water is extracted from
the soil reservoirs by root water uptake. In the model scenarios presented
below, we will assume that soil compartments are isolated; hence no flow
between compartments takes place. However, in order to formulate all terms in
the energy balance, we also state the corresponding flow equations for bulk
water flow between compartments for reference.
The mass balances of the reservoirs describing the temporal changes in Wi
in terms of the root water uptake fluxes, Jwr,i,
(m3 s-1) and soil water flow between the reservoirs,
Jw,k (m3 s-1), are expressed as
dWidt=Jwr,i+∑k=1mJw,k,
with both fluxes carrying negative signs when directed outward of the
reservoir. Here, m describes the total number of neighbors of cell i. The
water flux (Jw,k, m3 s-1) from the neighboring
reservoirs is expressed by Darcy's law, being proportional to the difference
in hydraulic potentials (hi=ψM+(zi-zr), m water).
For any two neighboring cells i and j, it is given as
Jw,k=-Kij(hi-hj),
where Kij (m2 s-1) is the effective unsaturated soil hydraulic
conductivity between adjacent compartments i and j. For convenience the
spatial scale is factored into the effective conductivity Kij.
The total root water uptake (Jwu, m3 s-1) is the sum of
the uptake fluxes from each compartment:
Jwu=∑iJwr,i,
which are described in analogy to Darcy's law:
Jwr,i=-Kr,i(ψM,i-ψx),
where ψx (m) is the xylem water
potential which is taken to be equal throughout the entire root system and
the index i runs over the number of compartments. The conductivities
Kr,i (m2 s-1) are effective radial conductivities
of active roots in compartment i. They encompass the notion of active root
length and hydraulic conductivity of the flow path from the bulk soil into
the root xylem, all of which are positively related to Kr. In our
conceptual model, we will change the proportion of
Kr,i/Kr,j to create heterogenous root water uptake
from the different reservoirs (see below). As mentioned above, root water
uptake may be represented more comprehensively. We keep it simple here to
better support the purpose of demonstrating the thermodynamic diagnostics.
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. () and () with a prescribed boundary
condition (total transpiration, Jwu) for the unknown xylem water
potential ψx as follows:
ψx=Jwu+∑i=1n(ψM,i⋅Kr,i)∑i=1n(Kr,i).
For each time step (Δt), Eq. () yields the xylem
water potential, based on the current water contents in the compartments.
Next, we obtain the water uptake for the same time step using
Eqs. () and (). Water contents are then
updated for the next time step based on the root water uptake. Initial soil
water contents and root conductivities are applied as shown in
Table . For purposes of simplicity, we run the model
until soil water limits uptake. We somewhat arbitrarily assume that soil
water becomes limiting when the root xylem potential falls somewhat below the
permanent wilting point (-150 m). When this point is reached, we fix
ψx=-150 m.
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.
Scenarios
We run the model for four scenarios, as shown in
Table . In the scenarios we vary the distribution of
initial soil water content and the implied root length by changing to
compartment root conductivity to impose increasingly heterogenous conditions
while keeping the average constant.
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 (Kr,i) were either homogeneously distributed or
heterogenous with two compartments having more roots and two less than
average. Working with four compartments allows us to combine the manipulation
such that average root conductivity is equal between the dry and wet
compartments and between all scenarios (see Table ,
Fig. ).
The model is representative of a plant having access to a soil volume of
0.5 m2, consisting of a soil monolith of 0.5 m depth and a surface area
of 1 m2. Each of the compartments is the same volume (0.125 m3). The
transpiration rate is indicative of a hot summer day in Germany, with
6 mm d-1. Effective root conductivities correspond to roots with
radial conductivity of 3×10-6 m s-1 MPa-1, which is
on the upper end of the values summarized in , and total
root length densities in the compartments varying between 1 cm cm-3
(most densely rooted), 0.5 cm cm-3 (average), and 0.1 cm cm-3
(least densely rooted). This is within the range of observed root length
densities for maize .
The soil hydraulic properties are equal in all compartments and derived using
with parameters for a sandy loam
given in Table . Starting from the initial condition,
we model a dry-down event until limiting soil water xylem potential is
reached. In each simulation, first root water uptake and the resulting
temporal evolution of soil water content were explicitly solved for each
compartment. Second, the thermodynamic evaluation was applied a posteriori
based on the results of the root water uptake model.
Results
Figure shows the evolution of the root collar
potential obtained from the process model over the course of the drying cycle
and the associated creation of heterogeneity of soil water contents (as
reflected in the coefficient of variation). Shown are the results of all
scenarios given in Table . Remember that the difference
between scenarios is only with regard to the prescribed heterogeneity. The
average initial water contents, root conductivities, and root water uptake
are the same in all simulations. The scenario called “optimal” is one where
both initial soil water content and root distribution are homogenous. It can
be seen as the optimal scenario from a plant's point of view, as it minimizes
dissipation, which keeps the xylem potentials less negative and delays time to water stress. It is obvious from the
evolution of the root collar potential that, despite everything relating to
the overall water balance being the same in all scenarios, the homogenous
(optimal) scenario is the one where limiting xylem potentials are reached at
the lowest average soil water content and longest time after the beginning of
the experiment. The limiting xylem potential is reached earlier the more
heterogenous the distribution of root water uptake and soil water contents
is. Also, Appendix shows analytically that
uptake from homogeneously distributed soil water minimizes (i.e., optimizes
from the plant's point of view) the dissipative losses due to root water
uptake in a situation in which soil hydraulic properties are homogeneous.
Model results
of the simple model: (a) evolution of xylem potential over the
course of root water uptake, (b) evolution of the coefficient of
variation of soil water content during the simulation. Average initial soil
water content is the same in all simulations. Only the unstressed uptake
is shown. The time axis has been replaced by average volumetric soil water,
which evolves parallel with time in this constant-flux
experiment.
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. , the time axis
was replaced by the average soil water content. (a) Total energy
exported from the system at the root collar. It is the sum of the two
components given in the other subplots. (b) Component due to
decrease of soil binding energy, which is due to both soil drying and
enhanced heterogeneity (compare Fig. ).
(c) Component due to energy dissipation by water flow from the soil
into the root.
Based on the output of the root water uptake model, we applied
Eq. () to diagnose the impediments to root water uptake. The
individual terms of Eq. () (except dissipation to soil water
flow, which was not modeled) are plotted separately in
Fig. : on the left the total export of energy
(JE,exp), which proves to be composed of the change of binding
energy in the soil (dUwb/dt, middle) and dissipation due
to root water uptake (-Du, right). All individual terms
(dUwb/dt, Du and JE,exp) were
calculated separately, applying Eqs. (), (), and
(). Thus, Fig. provides a proof
of concept for the correct derivation of Eq. () because all
the terms balance.
The energy export (JE,exp) corresponds closely to the evolution
of the root xylem potential (Fig. , left), because
the transpirational flux is prescribed as constant. JE,exp
increases continuously as the soil dries in order to maintain the constant
rate of uptake. The decomposition of the energy export provides information about the
impediments to root water uptake in the different scenarios. For example, the
greatest contribution to JE,exp in wet soil in our setup
originates from the dissipation when water flows from the soil into the root,
which constitutes about 97 % of the energy export. When the soil dries out,
it becomes increasingly more costly to detach water from the soil matrix, and
the change of the binding energy makes up a somewhat more substantial
proportion of the total energy exported from the system (17–22 %,
depending on the scenario).
The optimal case (grey solid line) is the one with the least possible
expenditure in dUwb/dt, and the difference between
the solid grey curve and the other curves illustrates the impact of soil
water heterogeneity on the water uptake at each time step. At the same
average soil water content, differences in dUwb/dt
between our scenarios are entirely due to heterogenous soil water
distribution. When comparing the optimal scenario and the one with strongly
heterogenous roots at θave=0.15, we observe that less than
half of the investment in detaching water is due to soil drying and the
remaining part is due to the heterogenous distribution of the soil water. The
effect of soil heterogeneity increases further after this point.
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.
Discussion
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 dUwb/dt.
Generally, hysteresis can be included in the framework to investigate this
effect further in the future.
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
. At the same time heterogeneous soil water retention
properties may induce root growth that alleviates water stress, and root
systems are likely adapted to such conditions. This reasoning may be further
extended to understanding horizontal and vertical distribution of root
systems and uptake in adapting to their environment also in terms of reducing their dissipative losses.
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 – which
include more process details, particularly more complex description of water
flow within the root system – or any other process models describing root
water uptake. Application of thermodynamics as proposed in this paper may
help to identify and understand the effects of heterogeneity in more
realistic models of root water uptake. Furthermore, a thermodynamic
evaluation may be applied to investigate the effects of dynamic root growth,
aquaporin and stomatal regulation , or mucilage
on the impediments of the whole plant root water
uptake. have applied thermodynamics to root water uptake
studies in order to differentiate efficient root parameterizations from less
efficient ones by minimizing the time average of JE,exp. More
practically, measurements of leaf water potential and transpiration are used
to assess plant water relations, and Eq. ()
elucidates the processes
involved. Thus, when information on potentials and flux along the flow path
are available, the formulations can also be implemented in experimental
studies, while imposed system boundaries can be adapted to fit the specific
setup.
At the more general level, this study adds to the thermodynamic formulation
of hydrologic processes and the application of thermodynamic optimality
approaches
. What
we described here is more targeted towards reduction in dissipative losses,
rather than the maximization of dissipation, or entropy production, as
suggested by some previous studies
. This is, however, not a
contradiction. A reduction of dissipative losses in a system allows greater
fluxes for the same forcing gradient to be maintained, which may then result
in a greater depletion of the driving gradients, thus maintaining low xylem
potentials and delaying the onset of water stress or cavitation. In our
study, we did not consider this effect on the driving gradients, which in the
case of root systems are the difference in chemical potential between soil
moisture and the water vapor in the near-surface atmosphere. The minimization
of internal dissipation has already been applied in hydrology for characterization
of river network structure and
vertical root water uptake . Notably, it has also been used in
vascular networks as the starting point to derive scaling laws and the
fractal nature of plant branching systems . It would seem
that our study fits very well into the scope of this previous study and
extends it to include the transport of soil water towards the vascular
network of the rooting system. In a further step, this transport would need
to be linked into the whole soil–vegetation–atmosphere system along with its
driving gradient to fully explore the thermodynamic implications of an
optimized root system. Such extensions could form the scope of future
research. The thermodynamic formulation of root water uptake as described
here provides the necessary basis upon which to test the applicability of thermodynamic
optimality approaches to root system functioning.
Summary and conclusions
Systems approaches and modeling will certainly be tools to investigate plant
water relations and efficient rooting strategies in the future
. In this paper we give a description of how root water
uptake relates to changes of total energy in the system, which can be used to
quantify the contribution of individual processes to impeding root water
uptake. It also sheds new light on some impediments not yet accounted for,
like heterogeneity in soil water. This is a slightly different and
potentially complementary approach to describing flow resistances over
potential gradients. Our derivation shows that the product of xylem water
potential and transpiration flux carries a great deal of information, as it
can be partitioned into the sum of individual processes impeding water flow
in the soil–plant system. Particularly in process models on root water uptake
, the changes of total
hydraulic energy and energy dissipation provide the opportunity to evaluate
which processes dominate the impedance to root water uptake at given times
and shed light on whether those are of biotic (within the plant) or abiotic
(within the soil) origin.
Analytical derivations
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. c). Such a minimization of dissipation
then results in a lower decrease dUw/dt and/or in a
lower export JE,exp, as expressed in Eq. (). To
show this minimum analytically, we consider a simplified setup of only two
reservoirs, a and b, yet use the same formulations as in the main text and
the same boundary condition of a prescribed flux of root water uptake,
Jwu.
We consider the case of a uniform root system (i.e., Kr,a=Kr,b=Kr) that takes up water from the two soil
reservoirs. The distribution of soil water is described by matric potentials
ψM,a=ψM-Δψ and
ψM,b=ψM+Δψ. When Δψ is
relatively small, then the water retention curve is approximately linear with
the soil water content, so that this formulation represents a case in which
the total soil water of the two reservoirs is the same, and it is only the
distribution across the two reservoirs that differs, as described by Δψ. With this formulation, the prescribed boundary condition in terms of
the root water uptake Jwu results in a constraint of the form
Jwu=-Kr(ψM-ΔψM-ψx)-Kr(ψM+ΔψM-ψx)=-2Kr(ψM-ψx)
so that ψx=ψM+Jwu/(2Kr).
The dissipation, Du, associated with root water uptake then
becomes
Du=Du,a+Du,b=-ρwg⋅Kr2Jwu2Kr2+2ΔψM2.
It is easy to see in this expression that the minimum is reached when ΔψM=0 (which can also be derived analytically by ∂Du/∂ΔψM=0). In other words, for a
uniform root system, dissipation associated with root water uptake is at a
minimum when moisture is distributed homogeneously in the soil.
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.
Soil parameters
Parameters used for calculation of soil
hydraulic properties using .
SymbolDescriptionValuenShape parameter1.38mShape parameter, m=1+1n0.275αShape parameter0.068 cm-1θminLower integration boundary in Eq. ()0.07θrResidual soil water content0.041θsPorosity0.453
The Supplement related to this article is available online at doi:10.5194/hess-20-3441-2016-supplement.
Acknowledgements
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
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