HESSHydrology and Earth System SciencesHESSHydrol. Earth Syst. Sci.1607-7938Copernicus PublicationsGöttingen, Germany10.5194/hess-20-479-2016Does the Budyko curve reflect a maximum-power state of hydrological systems? A backward analysisWesthoffM.martijn.westhoff@ulg.ac.behttps://orcid.org/0000-0002-8413-5572ZeheE.ArchambeauP.DewalsB.https://orcid.org/0000-0003-0960-1892Department of Hydraulics in Environmental and Civil Engineering
(HECE), University of Liege (ULg), Liege, BelgiumKarlsruhe
Institute of Technology (KIT), Karlsruhe, GermanyM. Westhoff (martijn.westhoff@ulg.ac.be)28January201620147948623July201511August20156November201511January2016This 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/20/479/2016/hess-20-479-2016.htmlThe full text article is available as a PDF file from https://hess.copernicus.org/articles/20/479/2016/hess-20-479-2016.pdf
Almost all catchments plot within a small envelope around the Budyko curve.
This apparent behaviour suggests that organizing principles may play a role
in the evolution of catchments. In this paper we applied the thermodynamic
principle of maximum power as the organizing principle.
In a top-down approach we derived mathematical formulations of the relation
between relative wetness and gradients driving run-off and evaporation for a
simple one-box model. We did this in an inverse manner such that, when the
conductances are optimized with the maximum-power principle, the steady-state
behaviour of the model leads exactly to a point on the asymptotes of the
Budyko curve. Subsequently, we added dynamics in forcing and actual
evaporation, causing the Budyko curve to deviate from the asymptotes.
Despite the simplicity of the model, catchment observations compare
reasonably well with the Budyko curves subject to observed dynamics in
rainfall and actual evaporation. Thus by constraining the model that has been optimized with the
maximum-power principle with the asymptotes of the Budyko curve, we
were able to derive more realistic values of the aridity and evaporation
index without any parameter calibration.
Future work should focus on better representing the boundary conditions of
real catchments and eventually adding more complexity to the model.
Introduction
In different climates, partitioning of rainwater into
evaporation and run-off is different as well. Yet, when plotting the
evaporation fraction against the aridity index (ratio of potential
evaporation to rainfall), almost all catchments plot in a small envelope
around a single empirical curve known as the Budyko curve .
The fact that almost all catchments worldwide plot within this small envelope
around this curve inspired several scientists to speculate whether this is
due to co-evolution of climate and terrestrial catchment characteristics
e.g.. Co-evolution between climate and the terrestrial
system could in turn be explained by an underlying organizing principle which
determines optimum system functioning
.
As hydrological processes are essentially dissipative, we suggest that
thermodynamic-optimality principles are deemed to be very interesting
candidates.
The most popular among these are the closely related principles of maximum
entropy production
and maximum power on the one
hand – both defining the optimum configuration between competing fluxes
across the system boundary – and, on the other hand, minimum energy
dissipation or maximum
free-energy dissipation , focusing on free-energy
dissipation associated with changes in internal state variables as a result
of boundary fluxes, i.e. soil moisture and capillary potential, and a related
optimum system configuration. In this research we focus on the maximum-power
principle.
The validity and the practical value of thermodynamic-optimality principles
are still debated e.g., and the partly promising results
reported in the above-listed studies might be just a matter of coincidence.
There is a vital search for defining rigorous tests to assess how far
thermodynamic-optimality principles apply. The Budyko curve
appears very well suited for such a test, as it condenses relative weights of
the steady-state water fluxes in most catchments around the world. It is thus
not astonishing that there have been several attempts to reconcile the Budyko
curve with thermodynamic-optimality principles. For example,
used the maximum-entropy-production principle to optimize
the run-off conductance and evaporation conductance of a bucket model being
forced with observed rainfall and potential evaporation of the 35 largest
catchments in the world. The resulting modelled fluxes were plotted in the
Budyko diagram and followed the curve with a similar scatter as real-world
catchments.
Another very interesting approach was presented by and
, using the perspective of the atmosphere. They maximized
power of the vertical convective motion transporting heat and moisture
upwards using the Carnot limit to constrain the sensible heat flux. This
motion is driven by the temperature differences between the surface and the
atmosphere, while at the same time depleting this temperature gradient,
leading to a maximum in power. Additionally, evaporation at the surface and
condensation in the atmosphere deplete this gradient even further at the
expense of more vertical moisture transport and thus more convective motion.
Their approach showed some more spreading around the Budyko curve for the
same 35 catchments as used in , but they used a simpler
model that has to be forced with far fewer observations, namely solar
radiation, precipitation, and surface temperature.
Very recently, used the maximum-entropy-production principle
to derive directly an expression for the Budyko curve. They started from the
expression of , and by maximizing the entropy production of
the whole system they reached the expression for the Budyko curve as
formulated by . This is an intriguing result that partly
contradicts the findings of , whose study revealed, in
simulations with an HBV type conceptual model, that joint optimization of
overall entropy production results in optimum conductances approaching zero.
The objective of this study is to define a model which, under constant
forcing, leads to a point on the asymptotes of the Budyko curve when flow
conductances are optimized by maximizing power. The model is comparable to
the one proposed by , but with different relations between
relative wetness of the subsurface store and driving gradients. We derived
the gradients driving evaporation and run-off in an inverse manner, with both
the asymptotes of the Budyko curve and the maximum-power principle as
constraints. Subsequently, we added dynamics in forcing or in actual
evaporation similar to to move away from these
asymptotes to more realistic values of the aridity and evaporation index,
without calibrating any parameter. Finally, these sensitivities were compared
to observations.
The maximum-power principle
The maximum-power principle implies that a system evolves in such a way that
steady-state fluxes across a systems boundary produce maximum power. It is
directly derived from the first and the second laws of thermodynamics, and it is
very well explained in e.g.. Here we give only a
short description: let us start by considering a warm and a cold reservoir,
which are connected to each other. The warm reservoir is forced by a constant
energy input Jin, and the cold reservoir is cooled by a heat flux
Jout. In steady state Jin=Jout and both
reservoirs have a constant temperature Th and Tc,
respectively, with Th>Tc. The heat flux between the
two reservoirs produces entropy, which is given by
σ=JoutTc-JinTh.
However, instead of transferring all incoming energy to the cold reservoir,
the heat gradient can also be used to perform work (to create other forms of
free energy). This means that, in steady state, the incoming energy flux
Jin equals the outgoing energy flux Jout plus the
rate of work P (which is power) performed by the system.
For given temperatures of both reservoirs, the theoretical maximum
rate of work is given by the Carnot limit:
PCarnot=JinTh-TcTh.
Now we introduce an extra flux cooling the hot reservoir as a function of its
temperature Jh.out=f(Th). This flux is in competition
with the flux Jh-c between both reservoirs, while both reduce the
temperature gradient between the two reservoirs. In Eq. ()
Jin should then be replaced by Jh-c, while
Th and Tc are not fixed anymore but are a function
of all fluxes. In this setting, there exists a flux Jh-c
maximizing power. In the extreme cases of Jh-c=0 and
Jh-c→∞, power is zero, while for intermediate
values power is larger than zero.
In hydrological systems, power is often generated by water fluxes and is
given as the product of a mass flux and the potential difference driving this
flux, note that several authors have divided this formulation by the
absolute temperature, while naming it maximum entropy production:
e.g..
Although these formulations are equivalent in isothermal circumstances, the
here-derived maximum-power principle is, in our opinion, more sound.
In the remainder of this article we use specific water fluxes
(LT-1) and potential differences μhigh-μlow in meter water column (L), where the flux is given
as the product of a specific conductance k (T-1) and the
potential difference. We recognize that, in order to come to the same units
as power, these formulations should be multiplied by the water density,
gravitational acceleration, and a cross-sectional area, but since we are
looking for a maximum, and these parameters are constant, we can leave them
out. We also use the word gradient for the potential difference
μhigh-μlow, where the length scale with which the
difference should be divided is incorporated into the conductance. With these
formulation, power is given by
P=kμhigh-μlow2,
where k is the free parameter we optimized to find a maximum in power.
Mathematical framework
Here we derive the model that, when conductances are optimized with the
maximum-power principle, always results in a point on the asymptotes of the
Budyko curve independent of the value of the given constant atmospheric
inputs (here rainfall and chemical potential of the atmosphere). To reach
this, proper relations between relative wetness and gradients driving run-off
and evaporation were derived, which is explained in the following.
Initial model setup
Our model consists of a simple reservoir being filled by rainfall
Qin and drained by evaporation Ea and run-off
Qr. Using the same expressions as in , the
steady-state mass balance and corresponding fluxes are expressed by
Qin=Ea+Qr,Ea=keμs-μatm,Qr=krμs-μr,
where μs, μr, and μatm are the
chemical potential of the soil, chemical potential of the free water surface
of the nearest river, and chemical potential of the atmosphere, respectively, while
ke and kr are the specific conductances of
evaporation and run-off. In these expressions, μs and
μs-μr are functions of the relative
saturation h in the reservoir:
Ge(h)=μs(h),Gr(h)=μs(h)-μr(h),
where Ge(h) and Gr(h) can have any form as long
as they are strictly monotonically increasing with increasing relative
saturation. For example, used the van Genuchten model
and gravitational potential to derive the chemical
potential of the soil. However, here we will derive them in such a way that,
under constant forcing, we end up exactly at the Budyko curve.
Backward analysis to determine the driving gradientsOptimum ke* matching the Budyko curve
Let us first find an optimum conductance ke* leading to a
point on the asymptotes of the Budyko curve B. An expression describing
these asymptotes exactly is given by adapted fromB=EaQin=1+Epot/Qin-Epot/Qin-122,
with Epot being the potential evaporation. Now we make an
important assumption to define Epot: we assume that evaporation
is purely described as the product of a gradient and conductance – ignoring
the influence of radiation. It is assumed to be maximum when in
Eqs. () and () μs=0, meaning that
the relative wetness is 1, implying no water limitation. With this
assumption, potential evaporation is given by
Epot=ke*(-μatm) (note that
μatm is always negative). Combining this equation with
Eqs. (), (), and () results in
ke*=QinGe(h*)-μatmB(ke*),
where h* is the steady-state relative wetness leading to a point on the
asymptotes of the Budyko curve (note that this is the relative wetness
occurring when ke=ke*).
Maximum power by evaporation
As mentioned above, ke* should also correspond to a maximum
in power by evaporation (Pe). We achieved this in a backward
analysis, implying that we start with defining a function
Pe(ke) which is always larger than zero for
ke∈(0,+∞) and where ∂Pe/∂ke=0 at ke=ke*. A function
satisfying these constraints is
We have also tested the function
Pe(ke)=P0exp-(ke-a)/k02, but this led to
two non-trivial solutions for ke* and is thus less
convenient to use than the expression in Eq. ()
Pe(ke)=keP0k0e-ke-ak02,
where P0 and k0 are the reference power (L2T-1) and
reference conductance (T-1), introduced to come to the correct
units. In all computations they have been set to unity. Setting the
derivative to zero for ke=ke* yields
∂Pe∂ke=2ke*a-2ke*2+k02P0k03e-ke*-ak02=0→a=ke*-k022ke*,
resulting in Pe(ke)=keP0/k0e-(ke-ke*)/k0+k0/(2ke*)2.
Combining this expression with Eqs. () and ()
(Pe=keGe-μatm2), Ge is expressed as
Ge(ke)=±P0k0e-ke-ke*k0+k02ke*2+μatm.
Since we neglect condensation (Ge(ke)-μatm≥0), only the positive solution remains. Inserting
Eq. () into Eq. () and setting
ke=ke* yields
ke*=QinP0k0e-k024ke*2B(ke*),
which can be solved iteratively for ke*.
Combining these results with the mass balance
(Eqs. –) yields the following expression for run-off
gradient Gr as a function of ke:
Gr(ke)=Qinkr-kekrP0k0e-ke-ke*k0+k02ke*2.
Note that any value of kr leads to a point on the Budyko
curve.
Maximum power by run-off
Although the Budyko curve does not depend on the value of kr,
an optimum kr* can still be found by maximizing power by
run-off. For this, steps similar to those for optimizing ke are
used, where in Eqs. ()–() ke is simply
replaced by kr, resulting in a gradient for run-off as a
function of kr:
Gr(kr)=P0k0e-kr-kr*k0+k02kr*2,
while from the mass balance (Eqs. –),
kr is given by
kr=Qin-Ge(h)-μatmGr(h).
Combining these two equations and setting kr to
kr* yields
kr*=Qin-ke*Ge(ke*)-μatmP0k0e-k024kr*2,
which can also be solved iteratively for kr*.
Forward analysis
To apply the maximum-power principle in any hydrological model, the model
should run until a (quasi-)steady state is reached. Within the above-presented
backward analysis the steady-state optimum gradients are simply
found by giving ke the value of ke* in
Eq. () and kr=kr* in
Eq. ().
However, when the relative wetness h evolves over time, the gradients
should be resolved as a function of the relative wetness
(Ge=Ge(h) and Gr=Gr(h)).
To do this, we assumed that h is a linear function of
Gr(ke) scaled between zero and unity (for
sensitivities to different initial relations between relative wetness and one
of the gradients see Supplement S1):
Gr(h)=minGr(ke)+maxGr(ke)-minGr(ke)h,
where the maximum in Gr(ke) occurs when the second
term on the right-hand side of Eq. () is zero
(maxGr(ke)=Qinkr) and the minimum value is derived when
this second term is maximum, occurring at ke=kemax=1/2ke*-k022ke*+ke*-k022ke*2+4.
Inserting this into Eq. () yields
minGr(ke)=Qinkr-kemaxkrP0k0e-kemax-ke*k0+k02ke*2.
If we now plot h vs. Ge, a unique relation between the two
exists (Fig. ).
The gradients driving evaporation
(Ge) and run-off (Gr) as a function of the
relative saturation (h) for different values of μatm with
kr=kr*. At h=0, the slope of the gradient
Ge is vertical, while the value of Gr is set to
zero to avoid run-off at zero saturation.
(a) Analytical Budyko curve
(Eq. ) and result from forward mode with constant forcing and
(b) time evolution of relative saturation and both gradients for
complete initial saturation (solid lines) and initial dry state (dashed
lines). μatm=-0.7.
Sensitivity to periodic dry spells in the
forward model. MOPEX catchments are filtered to have only those catchments
having at least 1 month with a median rainfall <2.5mmmonth-1 and a coefficient of variance <0.5 for all
months with a median rainfall >25mmmonth-1. The final number
of dry months was determined maximizing the difference between the mean
monthly precipitation of the X driest months minus the mean monthly
precipitation of the 1-X wettest months, where X=1,2…12. Error
bars indicate 1 standard deviation and are determined with bootstrap
sampling.
With the gradients as functions of h, the non-steady mass balance equation
is written as
Smaxdhdt=Qin-krGr(h)-keGe(h)-μatm,
where Smax is the maximum storage depth (L) and t is
time (T). Now, the time evolution of the relative wetness can be
simulated.
Results and discussion from forward analysisConstant forcing
With the known relations between relative wetness and gradients driving
evaporation and run-off, the forward model was run and ke was
optimized by maximizing power. With constant forcing, each value of
μatm resulted in a point on the asymptotes of the Budyko curve
(Fig. a). In Fig. b, the time
evolution of the relative wetness and both gradients are shown for an
initially saturated and an initially dry state, indicating that, irrespective
of the initial state, the forward model evolves to a steady state.
Sensitivity to dry spells
By introducing dynamics in forcing, we expected the resulting Budyko curve to
deviate from the asymptotes.
In the literature, the deviation from the asymptotes is often done by introducing
an empirical parameter e.g..
To move away from this empiricism, we started at the asymptotes of the Budyko
curve. Subsequently, we added dry spells and dynamics in evaporation (e.g.
when trees lose their leaves the evaporative conductance ke
goes to zero) and tested how this influenced the Budyko curve.
To test sensitivities to dry spells, simple block functions were used, with
either a predefined constant input or no input at all. For longer relative
lengths of the dry spell, the slope of the curves becomes smaller until a
maximum of Ea/Qin=0.98 (Fig. ).
The reason the asymptotes do not reach unity lies in the fact that already at
very short dry spells a second maximum in power evolves, while the first
maximum disappears quickly with increasing dry spells. This is in line with
results of , and in a second
optimum is also present. Although interesting, we leave a better exploration of
this transition zone where two maxima exist for future research.
These curves were compared with data of real catchments that have a
relatively stable wet period interspersed with a regular dry period. The
Mupfure catchment Zimbabwe,, with approximately 7
months without rain (Fig. S2.1 of Supplement), plots very close to the
theoretical curve with the same length of the dry spell. However, catchments
from the Model Parameter Estimation Experiment (MOPEX) database with clear, consistent dry spells
still plot far from the respective theoretical curves. This discrepancy can
be partly explained by the somewhat arbitrary way the number of dry months
is determined: the MOPEX catchments are filtered to have only those
catchments having at least 1 month with a median rainfall <2.5mmmonth-1 and a coefficient of variance <0.5 for all
months with a median rainfall >25mmmonth-1. The final number
of dry months was determined maximizing the difference between the mean
monthly precipitation of the X driest months minus the mean monthly
precipitation of the 1-X wettest months, where X=1,2…12.
For example, the MOPEX catchment with a 4-month dry spell could also be
argued to have a dry spell of 7 months (Fig. S2.1, MOPEX ID: 11222000);
similarly, the MOPEX catchment with a 5-month dry spell (Fig. S2.1, MOPEX ID:
11210500) could also be argued to have one of 6 months. If these
“corrections” are made, the variability within the MOPEX catchments is
consistent (with longer dry spells plotting more to the right), but there is
still a discrepancy with the simulated curves of 1 to 2 months, indicating
that the model should still be improved.
Sensitivity to on–off dynamics in actual
evaporation in the forward model. MOPEX catchments were filtered to have only
those catchments having a coefficient of variance <0.12 for monthly median
rainfall and with at least 1 month with a median maximum air temperature <0∘C; a month is considered to have no actual evaporation if
the monthly median maximum air temperature <0∘Cafter. Error bars indicate 1 standard deviation and
are determined with bootstrap sampling.
Sensitivity to dynamics in actual evaporation
We also tested the sensitivity of dynamics in actual evaporation by
periodically turning ke on and off, while keeping the rainfall
constant. This sensitivity analysis shows that the longer actual evaporation
is switched off, the smaller the slope of the Budyko curve and the smaller
the maximum value of the evaporation index (Fig. ).
Comparing the different curves with real catchments shows that data from the
Ourthe catchment (Belgium) are relatively close to its respective line
its months without actual evaporation are estimated from Fig. 6.1
of. Also the MOPEX catchments plot relatively close to their
respective lines. However, the way the MOPEX catchments were filtered is
somewhat arbitrary (only those having a coefficient of variance <0.12 for
monthly median rainfall and with at least 1 month with a monthly median
maximum ambient temperature <0∘C are taken into account; a
month is considered to have no actual evaporation if the monthly median
maximum air temperature <0∘C; after ,
Fig. S2.2 of Supplement).
At first sight the comparison with data looks better than in the case of dry
spells. However, all plotted catchments have an aridity index between 0.5 and
0.71, and within this range the different curves also plot close to each
other. Yet, it is still somewhat surprising that the comparison is relatively
good, since the modelled lines were created by assuming a constant
atmospheric demand (μatm) for each run, which is different from
real catchments that have a more-or-less sinusoidal potential evaporation
over the year. However, we consider it as future work to better represent the
real-world dynamics in the model.
Conclusions and outlook
The Budyko curve is empirical proof that only a subset of all possible
combinations of aridity index and evaporation index emerges in nature. It
belongs to the so-called Darwinian models , focusing on
emergent behaviour of a system as a whole. Since the maximum-power principle
links Newtonian models with the Darwinian models, it has indeed potential to
derive the Budyko curve with an, in essence, Newtonian model.
We presented a top-down approach in which we derived relations between
relative wetness and chemical potentials that lead, under constant forcing,
to a point on the asymptotes of the Budyko curve when the maximum-power
principle is applied. Subsequently sensitivities to dynamics in forcing and
actual evaporation were tested.
Since the Budyko curve is an empirical curve, a calibration parameter is
often linked to catchment-specific characteristics such as land use, soil
water storage, climate seasonality, or spatial scales
e.g..
Although correlations between characteristics and the calibration parameter
have been found, it remains a calibration parameter.
Here we presented a method to derive the Budyko curve without any
calibration parameter, but sensitive to temporal dynamics in boundary
conditions. Although we used simple block functions to test these
sensitivities, they compare reasonably well with observations. Nevertheless,
improvements could be made by modelling dynamics closer to reality, or even
by adding multiple parallel reservoirs to account for spatial variability
within a catchment.
Even though the model represents observations reasonably well (despite its
simplicity), the method used here is by no means proof that the
maximum-power principle applies for hydrological systems. This is due to the
top-down derivation of the gradients in which the maximum-power principle is
used explicitly. In principle, the method could also be used with respect to
any other optimization principle. However, the reasonable fits with
observations are grounds upon which to further explore this methodology – including the
maximum-power principle.
The Supplement related to this article is available online at doi:10.5194/hess-20-479-2016-supplement.
Acknowledgements
We would like to thank three anonymous reviewers for their fruitful comments.
Furthermore we would like to thank Miriam Coenders-Gerrits for providing data
of the Mupfure catchment, Wouter Berghuijs for his help with the MOPEX
data set, and Service Public de Wallonie for providing river flow data of the
Ourthe catchment. This research was supported by the University of Liege and
the EU in the context of the MSCA-COFUND-BeIPD project.Edited by:
D. Wang
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