Optimality approaches to describe characteristic fluvial patterns on landscapes

Autho

One coproductand imp

Phil Trans R Soc B (2010) 365 1387ndash1395

doi101098rstb20090303

Optimality approaches to describecharacteristic fluvial patterns

on landscapesKyungrock Paik1 and Praveen Kumar2

1School of Civil Environmental and Architectural Engineering Korea University Anam-dong 5 gaSeongbuk-gu Seoul 136 713 South Korea

2Department of Civil and Environmental Engineering University of Illinois Urbana IL 61801 USA

Mother Nature has left amazingly regular geomorphic patterns on the Earthrsquos surface These pat-terns are often explained as having arisen as a result of some optimal behaviour of naturalprocesses However there is little agreement on what is being optimized As a result a numberof alternatives have been proposed often with little a priori justification with the argument that suc-cessful predictions will lend a posteriori support to the hypothesized optimality principle Given thatmaximum entropy production is an optimality principle attempting to predict the microscopic be-haviour from a macroscopic characterization this paper provides a review of similar approaches withthe goal of providing a comparison and contrast between them to enable synthesis While assump-tions of optimal behaviour approach a system from a macroscopic viewpoint process-basedformulations attempt to resolve the mechanistic details whose interactions lead to the systemlevel functions Using observed optimality trends may help simplify problem formulation at appro-priate levels of scale of interest However for such an approach to be successful we suggest thatoptimality approaches should be formulated at a broader level of environmental systemsrsquo viewpointie incorporating the dynamic nature of environmental variables and complex feedback mechanismsbetween fluvial and non-fluvial processes

Keywords landscape evolution hydrology fluvial geomorphology

1 INTRODUCTIONOver a range of time scales the flow of water has leftcharacteristic patterns on Earthrsquos terrestrial surfacesuch as self-similar binary tree organization of rivernetworks (eg Peckham 1995) meandering streams(eg Tal amp Paola 2007) regular stream bed profiles(eg Leopold 1953) systematic downstream grain-size sorting (eg Yatsu 1955) and power-law hydraulicgeometry relationships (Leopold amp Maddock 1953)(figure 1) These self-organized patterns are signaturesof past hydrological processes over a number of scalesMany interesting statistical relationships have beenreported on the regularity of geomorphic patternsFor example the number of streams and averagelength of streams of given stream order exhibit log-linear relationships with the stream order (Horton1945) As another example relationships betweenupstream area and main channel length ubiquitouslyexhibit a power law indicating self-similarity of rivernetwork organization with their scaling exponents ina narrow range (Hack 1957)

Identifying and quantifying the signatures of geo-morphic patterns and attributing them to thecausative mechanism is of significant interest in a

r for correspondence (paikkoreaackr)

ntribution of 17 to a Theme Issue lsquoMaximum entropyion in ecological and environmental systems applicationslicationsrsquo

1387

variety of contexts ranging from enabling a betterunderstanding of complex feedback mechanisms onEarthrsquos surface processes to predicting the impact ofdisturbance on future landform as well as its influenceon broader environmental systems Understandingmechanisms underlying the formation of these pat-terns also helps investigate the existence of water onother planets such as Mars Characteristic geo-morphic patterns similar to what we observe onEarth such as self-similar tree network and meander-ing have been found on Marsrsquo surface (eg Stepinskiet al 2002 Wood 2006) indicating the possibilitythat a substantial flow of water existed in the pastand has carved such patterns on the Marsrsquo surface

One viewpoint for the formation of these regularpatterns is that there is a certain principle that dictatesthe landform evolution and observed regular patternscan be explained by this principle as the destination oflandform evolution Significant efforts have been madeto identify this principle Since Leopold amp Langbein(1962) introduced the concept of entropy in landformevolution a number of metrics have been proposed torepresent entropy production in landscape formationThis has given birth to several alternative hypotheses

A contrasting viewpoint holds that the observedregularity of patterns arises as emergent characteristicsfrom local interactions of physical biological andchemical processes occurring at a number of scalesFor example we have shown (Paik amp Kumar 2008)that inherent randomness is a sufficient condition for

This journal is q 2010 The Royal Society

(a)

(b)

400 km

Figure 1 Some patterns on Earthrsquos surface (a) Meandering patterns of Songhua River (Image courtesy of NASAGSFCMETIERSDACJAROS and USJapan ASTER Science Team) (b) Self-similar tree network of Upper Susquehanna Riverbasin (Image modified from Saco (2003))

1388 K Paik amp P Kumar Landscape patterns

the generation of self-similar tree organization drivenby the erosionmdashdeposition process under a gravita-tional gradient and demonstrated that the decreasein total energy expenditure the optimality conditionsuch as that used in the optimal channel networktheory (Rodrıguez-Iturbe et al 1992) is not thecause but a consequent signature

Both these differing perspectives are valuable andcan contribute to improving our predictive capabilityPrediction of Earthrsquos surface processes is vital for a

Phil Trans R Soc B (2010)

variety of environmental studies There are emergingsocial issues as consequences of artificially modifiedenvironments such as channelization dams miningurbanization and their restoration or removal as wellas complex environmental changes over longer timescales such as climate change However our abilityto predict the dynamics of systems that evolve intime such as Earthrsquos surface processes is limited(Kumar 2007) Better prediction of their dynamicsdemands the integration of a greater number of

landscapeevolution studies

fieldobservations

modelling

physical modelsmathematical

models

process-basedmodels

objective-basedmodels

Figure 2 Classification of different approaches in investi-gating landscape evolution problems The arrow indicatesthat field observations provide references for all modellingapproaches

Landscape patterns K Paik amp P Kumar 1389

processes and feedback mechanisms When usingprocess-based models this means increasingly compli-cated formulation and generation of numerousparameters that are subject to uncertainty Howeveridentification of macroscopic governing laws such asoptimality principles can provide a simpler approachfor reproducing the observed patterns The value ofmany existing optimality hypotheses including themaximum entropy production principle has to berecognized in this context

The purpose of this review paper is to reconcile anumber of different notions that are prevalent in thiscontext We provide a comprehensive overview ofexisting optimality hypotheses a detailed comparisonof these ie the scale issues and philosophical back-ground and synthesis of their implication towards abetter understanding of characteristic patterns in land-form dynamics In sect2 we define fluvial landscapeevolution and discuss issues on its modelling Thenexisting optimality hypotheses are reviewed in sect3In sect4 an insightful discussion is given for thesehypotheses Conclusions are provided in sect5

2 NATURE OF FLUVIAL LANDSCAPEEVOLUTIONLand surfaces including both channels and hillslopesserve as corridors for transporting a variety of constitu-ents such as biomass carbon nutrient pollutantsediment water etc Spatial and temporal variabilityin these components continuously reshapes the flowpath by lifting materials that form the land surfaceand depositing them at other places Fluvial landscapeevolution refers to the continuous variation oflandform as a result of the relocation of land-forming materials powered by water flow Whilesurface water flow is at the core of landscape evolutionlandscape evolution is also driven by complex feedbackmechanisms among ecological geomorphological andhydrological functions where different climaticgeographical and geological conditions play key roles

Fluvial landscape evolution has been studied byeither fieldwork-based observations laboratory exper-iments or mathematical modelling Observationsprovide key references for modelling studies but areavailable only for limited time spans simply becausenobody can watch landscape evolution over its vastscales (spatially ranging from a definable minimumgrain size to the entire Earthrsquos surface and temporallyfrom an instance to geological time scale and beyond)

Models for fluvial landscape evolution can belargely divided into two types (i) physical modelsie models physically built at scales usually smallerthan the prototype where physical experiments canbe conducted under controlled flow characteristics(eg Schumm amp Khan 1972) and (ii) mathematicalmodels which enable simulations based on specifiedformulations Based on the model formulation weclassify the mathematical models into process-basedmodels where a set of physically based equations isused (eg Willgoose et al 1991) and objective-basedmodels which use hypothesized governing principleswhere each hypothesis defines an objective or a goalthat the evolutionary process attempts to achieve


(eg Leopold amp Langbein 1962) (figure 2) Anumber of optimality hypotheses have been proposedas the governing principle for objective-basedmodels The subject of this paper is the mathematicalmodels with a focus on optimality hypotheses used forobjective-based models

Process-based models simulate landscape evolutionusing physically formulated governing equations suchas conservation of mass and momentum (for bothsediment and water) under various scenarios ofatmospheric and tectonic activities over a range oftime scales They are targeted towards engineeringgoals eg to estimate variation of stream bed profilesunder given conditions (Wu et al 2004) or scientificexplorations eg to reproduce the observed tree net-work patterns on the Earthrsquos surface (Paik amp Kumar2008) For detailed reviews on existing process-basedlandscape evolution models for the scale of the wholelandscape readers may refer to Willgoose (2005)and Codilean et al (2006)

Although process-based models are rooted in firstprinciples albeit with a variety of approximationsand parametrizations their practical implementationsare rife with challenges For example current modelshave large uncertainty in their predictions of sedimenttransport through natural streams Sediment flux isinfluenced by not only the distribution of the sizebut also the shape of each grain and their interactionsuch as hiding effect (eg Egiazaroff 1965) Particledetachment mechanism for cohesive soils differsfrom that for non-cohesive soils which addsuncertainty in the estimation of sediment transport

Vegetation also plays a significant role by increasingthe resistance to water and sediment flux with its leavesand stems Further existence of roots substantiallyaffects particlesrsquo aggregate stability Evaluating theeffect of vegetation is also crucial in estimating thewater balance of a catchment system since transpira-tion constitutes a significant portion (as high as 90in certain regions) of a catchmentrsquos water budget Sur-facendashsubsurface interaction is also an importantelement in that this governs water balance that directlyregulates surface flow Infiltration losses reduce streampower in overland flow Therefore an appropriate esti-mate of infiltration is important as well However


existing models for whole landscape evolution havevery unrealistic settings ie bare soil with zero infiltra-tion capacity and no groundwater flow Thisassumption definitely limits modelsrsquo capability to cap-ture complex feedbacks Other limitations includeaccommodating stream bank failure and sensitiveprocesses at channel confluences Besides these phys-ical processes existing models have limitedimplementation of biological and chemical processesHowever as we increase model complexity weincrease the number of model parameters As eachmodel parameter value is subject to uncertainty com-plicated model structure does not necessarily helpobtain better scientific insights

Objective-based models can often provide a closurecondition for the solution of process-based modelsFor example finding hydraulic geometry exponentshas been treated as a problem that has more unknownsthan known relationships (deterministic conditions)and optimality hypotheses have been used as theadditional constraint to solve the problem (egLeopold amp Langbein 1962 Yang 1976) In objective-based models the prediction of landform evolution isconsidered as an optimization problem where theproposed optimality principles become objective func-tions (or cost functions) while the deterministicconditions (either biological chemical or physical)serve as constraints In sect3 we discuss details ofexisting optimality hypotheses

3 OPTIMALITY HYPOTHESESOptimality approaches proclaim that landscapes evolvetowards a state that is characterized by an optimal con-dition The exact nature of this optimality remainsmoot and several alternative hypotheses often invol-ving statistical measures such as entropy have beenproposed They were called lsquostochasticrsquo approaches(eg Yang 1971a) However the term stochastic ismisleading since many process-based models alsohave stochastic properties in their model parametersboundary or initial conditions Alternatively optimal-ity hypotheses were often called extremal hypotheses(Davies amp Sutherland 1983) since proposed objectivefunctions of most hypotheses pursue extremal statesie minimization or maximization of a definedmetric However hypotheses in non-extremal formdo exist such as the hypotheses of lsquoconstant Froudenumberrsquo (Lacey 1958) and lsquoequal energy expenditureper unit area of channelrsquo (Rodrıguez-Iturbe et al1992) Here we use the term optimality hypothesesto group such approaches This terminology is generalenough to encompass hypotheses in non-extremalform and clear enough to be distinct from process-based approaches In this section we provide areview on existing optimality hypotheses Throughthis review it will be shown that most existinghypotheses focus on flow efficiency

The attempt to describe landform patterns as pre-determined by an optimality principle dates back tothe study on channel geometry by British engineersin India Kennedy (1895) proposed the empiricalregime theory motivated by stable irrigation canals inIndia The regime theory proposed that there is a


regime or optimal range of flow and channel geome-try that enables the irrigation canals to be stable Theregime theory developed for artificial waterways ieirrigation canals was later expanded to similar empiri-cal relationships called hydraulic geometryrelationships (Leopold amp Maddock 1953) applicableto natural rivers that cover a wider range of flow dis-charge and channel slope The basic principle thatguides landform evolution towards the stable regimehas been an elusive question

Leopold amp Langbein (1962) were the first to use theconcept of entropy in landscape evolution This workwas restated using a statistical term as the lsquominimumvariancersquo hypothesis (Langbein 1964) since it assumesthat naturally evolved channels follow the lsquomost prob-ablersquo geometry Here the definition of the mostprobable state depends on the choice of variablesunder consideration Hence the minimum variancehypothesis yields multiple solutions depending on theconstraints used and the choice of dependent variableswhose variance would be minimized (eg see casesshown in Williams (1978)) Since the minimum var-iance hypothesis explains the evolution only througha probabilistic perspective it could be applied regard-less of physical attributes (eg cohesiveness of channelboundaries) (Williams 1978) However this implies alack of physical basis to support the minimum variancehypothesis (eg Kennedy et al 1964)

Most hypotheses proposed after the minimum var-iance hypothesis have therefore emphasized physicalinterpretation by adopting physical terms such asenergy stream power Froude number sedimenttransport and friction factor instead of the statisticalmeasure lsquovariancersquo Brebner amp Wilson (1967) appliedthe principle of lsquominimum energy degradation ratersquo(von Helmholtz 1868) to the channel geometry pro-blem and showed that this principle yields channelgeometry equations close to the empirical regimetheory This early optimality hypothesis on energydissipation rate was tested for pressurized conduits ina laboratory (Brebner amp Wilson 1967) This test hasbrought an argument that the comparison of theresults obtained from the pressurized conduits withthe observed regime equations for open channels thathave free water surface exposed to atmosphericpressure is inappropriate (eg Barr amp Herbertson1967 Lacey 1967)

Yang (1971a) proposed two hypotheses of lsquoaveragestream fallrsquo and lsquoleast rate of energy expenditurersquo toexplain meandering (Yang 1971ab) and the formationof riffles and pools (Yang 1971c) Then Yang (1972)found that the rate of energy expenditure can beexpressed using the term lsquostream powerrsquo originallyproposed by Bagnold (1960) Yang (1972) proposeda sediment transport equation where the sedimenttransport is expressed as a function of the unitstream power In his successive work Yang (19731976) proposed the hypothesis of lsquominimum unitstream powerrsquo (MUSP) stating that lsquoan alluvial chan-nel with subcritical flow in the lower flow regimetends to adjust its velocity depth slope and channelroughness in such a manner that given water dischargeand sediment concentration can be transported withthe minimum amount of unit stream power under

Table 1 List of selected existing optimality hypotheses V is

the mean flow velocity S is the channel slope r is thedensity of water g is the gravitational acceleration constantQ is the flow discharge L is the reach length rs is thedensity of sediment Qs is the sediment transport rate W isthe channel width H is the hydraulic depth and h is a

constant Qj and Lj are the discharge and the length of link j(channel between two confluences) respectively Inapplication of these hypotheses Q and Qs are treated asgiven constants and channel or network geometry isadjusted to obtain the optimal combination of variables

such as V S H W and L to satisfy a given objectivefunction

hypotheses mathematical form references

MUSP min VS Brebner ampWilson (1967)Yang (19731976)

MSP frac14MSTC

min rgQS Chang amp Hill(1977) Chang(1979a) Whiteet al (1982)

MEDR min (rgQ thorn rsgQs)LS Yang et al (1981)

MFF max W2H3SQ2 Davies ampSutherland(1980 1983)

minimumFroude

number

minethV=ffiffiffiffiffiffiffigHp

THORN Jia (1990)

minimumtotal energyexpenditure

min hP

j Q05j Lj Rodrıguez-Iturbe

et al (1992)


given geologic and climatic constraintsrsquo (Yang 1976)However it should be noted that the formulation ofthe lsquoMUSPrsquo is identical to that of an earlier hypothesisof minimum energy degradation rate (Brebner ampWilson 1967) which was derived from a differentapproach (table 1)

Stimulated by the MUSP hypothesis Chang amp Hill(1977) proposed a similar lsquominimum stream powerrsquo(MSP) hypothesis This hypothesis was used to pro-vide a boundary condition for numerical calculationfor delta streams (Chang amp Hill 1977) and to explainhydraulic geometry (Chang 1979a 1980) as well asmeandering (Chang 1979b) The measure minimizedin this hypothesis is rgQS where r g Q and S arethe density of water the gravitational accelerationconstant the flow discharge (volumetime) and thechannel slope respectively This measure was calledthe cross-sectional stream power by Rhoads (1987)

Later Yang amp Song (1979) proposed the lsquominimumenergy dissipation ratersquo (MEDR) hypothesis statingthat lsquoit appears that all natural rivers have a tendencyto adjust whatever possible under the given constraintsto achieve an objective of transporting water and sedi-ment with a minimum rate of energy dissipationrsquoBecause of the general nature of this principle the for-mulation of this principle can be derived in variousways For example MEDR can be achieved by adjust-ing channel geometry (the formulation derived for thiscase is shown in the table 1) If channel boundary isfixed MEDR can be still accomplished by adjusting


the flow depth the velocity distribution and other tur-bulent characteristics Song amp Yang (1980) showedthat the total rate of energy loss is equal to the totalstream power when the river boundary is fixed andclaimed MEDR as a general hypothesis for thisthread of reasoning that has the previous hypothesesof MUSP and MSP as its special cases Differencesamong MUSP MSP and MEDR are shown intable 1 and further described by Griffiths (1984)Based on a similar idea Rodrıguez-Iturbe et al(1992) proposed principles of lsquominimum energyexpenditure in any link of the networkrsquo and lsquominimumtotal energy expenditurersquo in a river network andapplied them for the river network organization pro-blem These two later unified as the lsquoglobal optimalenergy expenditurersquo hypothesis by Molnar amp Ramırez(1998) can also be grouped with the MEDRhypothesis

The Froude number has also been popularly usedfor building optimality hypotheses Through ananalytical study on the regime relationships Lacey(1958) showed that the Froude number should beconstant for the constant slope and sediment loadHowever Barr amp Herbertson (1968) argued whetherinsights gained from two-dimensional formulationwhich Laceyrsquos hypothesis is based on can be extendedto real three-dimensional relationships This constantFroude number hypothesis seems to be connectedwith the concept of minimum energy degradationrate according to Brebner amp Wilson (1967) whostated lsquoFor the minimization of energy degradationrate the hydraulic radius of the waterway will havebeen adjusted so that a certain value of the Froudenumber is obtained This value is a function of the par-ticle properties and concentration but is completelyindependent of the dischargersquo Jia (1990) furtherargued that the maximum channel stability is achievedwhen the Froude number is not only a constant butalso a minimum value This lsquominimum Froudenumberrsquo hypothesis is related to the earlier MUSPhypothesis since Yang (1978) already found that theprediction accuracy of the MUSP hypothesis improvesas the sediment concentration and the Froude numberdecreases Later Grant (1997) proposed that mobile-bed channels adapt to prevent supercritical flow (thestate defined as its Froude number is greater thanthe unity)

Another approach has focused on the sedimenttransport capacity (Pickup 1976 Kirkby 1977Ramette 1980 White et al 1982) White et al (1982)proposed the hypothesis of lsquomaximum sediment trans-porting capacityrsquo (MSTC) This hypothesis wasrecognized as being equivalent to the MSP hypothesis(White et al 1982 Griffiths 1984) This can be con-ceptually explained as follows under a fixeddischarge if the flow can reduce energy loss streamhas more ability to transport sediments fromupstream which helps prevent deposition This con-cept in turn agrees with the lsquomaximum flowefficiencyrsquo (MFE) hypothesis (Huang amp Nanson2000) proposed as a more general principle offormer MEDR and MSTC Here the flow efficiencyis defined as the MSTC per unit available streampower The MFE hypothesis was derived for a straight


single-thread channel Later Huang et al (2004) pro-posed the lsquomaximum energyrsquo as the principle toillustrate the condition of MFE for a more generalopen channel flow not limited to straight channelsAll optimality hypotheses discussed by far have acommon tenet geomorphological features adjusttowards the state that enables the most efficient flowor least energy loss in turn the maximum capacity totransport sediment particles This state is also under-stood as the state of greatest stability where netdeformation is the least (eg Jia 1990 Eaton et al2004)

These hypotheses have shown a wide range of appli-cations Many early optimality hypotheses wereapplied to estimate the scaling exponents of hydraulicgeometry relationships (eg Leopold amp Langbein1962 Langbein 1964 1965 Williams 1978 Chang1980 Yang et al 1981 Huang amp Nanson 2000)Many of them were also used to explain the meander-ing (Yang 1971ab Chang 1979b Huang et al 2004)the formation of riffles and pools (Yang 1971c) andself-similar river network formation (Rodrıguez-Iturbe et al 1992) Although existing optimalityhypotheses have achieved insightful results in severalapplications there have been criticisms as explainedin the next section

4 DEBATES OVER EXISTING OPTIMALITYHYPOTHESESThere have been continuous debates over existingoptimality hypotheses mostly pertaining to their phys-ical justification (eg Griffiths 1984) It has beenargued that they mimic entropy production of linearthermodynamics with highly nonlinear energy trans-formations in river flows (Davy amp Davies 1979) anduse assumptions only applicable to laminar flows forriver flows that are turbulent in nature (eg Davies ampSutherland 1983) To fortify their physical justifica-tion additional components have been incorporatedin the formulation of optimality hypotheses Forexample bank stability relationship was added toWhite et alrsquos (1982) formulation for non-cohesive(Millar amp Quick 1993 Eaton et al 2004 Millar2005) and cohesive (Millar amp Quick 1998) banksNevertheless existing optimality hypotheses stillsuffer from significant limitations such as (i) theyhave little consideration on the dynamic nature ofenvironmental systems and (ii) feedbacks from closelyrelated non-fluvial components such as tectonic activi-ties and ecosystem functions are rarely incorporated

First the dynamic nature of environmental vari-ables is rarely accounted for in the formulation ofexisting optimality hypotheses For example existinghypotheses predict the channel formation under a con-stant flow discharge (mostly dominant discharge)However stream discharge is directly dependent onhydrological variation and consequently time-variantin nature If discharge is not a fixed constant but a vari-able quantity MSP would be satisfied for dischargeQ frac14 0 while MSTC would be satisfied as Q divergesto infinity Based on this argument under varying QMSP and MSTC may no longer be equivalent con-flicting with earlier studies on these hypotheses


(White et al 1982) Consequently existing hypothesesare only applicable to the ideal steady flow conditions(eg Yang 1978 Jia 1990) Existing hypotheses alsoadopt the constant sediment load condition Sincesediment load continuously varies along with flow dis-charge this assumption is also problematic

Existing hypotheses have little consideration onfeedbacks between fluvial and non-fluvial processesLandscape evolution cannot be isolated from atmos-pheric processes geological activities and ecosystemfunctions (eg Dietrich amp Perron 2006) For examplevegetation coevolves with landform and this coevolu-tion is found to be critical in forming meanderingpatterns (Tal amp Paola 2007) However no existingoptimality hypothesis integrates coevolutionaryprocesses

These two points ie the dynamic nature ofenvironmental systems and feedbacks between fluvialand non-fluvial processes are related to each otherAs a result of this complexity landform evolutionmay never reach a stable state but is always subjectto dynamic changes Phillips (1990) demonstratedthat at-a-station hydraulic geometry is inevitablyunstable questioning the validity of equilibrium con-straints Simon amp Thorne (1996) observed rapidadjustment of the Toutle River to the debris avalancheaccompanied by the eruption of Mount St Helens andshowed that existing optimality hypotheses are onlypartially applicable

Sometimes conflicts between existing optimalityhypotheses have been reported in a general contextHoward (1972 p 477) stated

Since plane Couette flow has the least energy dissipa-

tion among all solenoidal fields satisfying the boundary

conditions and is in fact the motion that will occur

when R (Reynoldrsquos number) is small one might be

tempted to consider as a sort of metaphysical principle

the statement that lsquonature chooses that motion which

minimizes energy dissipationrsquo Such a statement

while true for R Rc (critical Reynolds number)

could not always be true since plane Couette flow

does not in fact occur if R is large But even for

small R such a statement is misleading for one

should not compare the flow that occurs with all sole-

noidal vector fields but only with those motions that

are possible ie the real question is lsquoAmong all sol-

utions (with steady averages) of the NavierndashStokes

equations which one (or ones) actually occur under

the given boundary conditionsrsquo Since when R Rc

there is actually only one competitor the exactly oppo-

site metaphysical principle lsquoNature chooses (from

among the possibilities available) that motion which

maximizes the energy dissipationrsquo is equally true

Any selection principle at all will be lsquocorrectrsquo when

there is no choice

Motivated by Howard (1972) Davies amp Sutherland(1980 1983) investigated the case that the evolutionproceeds towards the state of lsquomaximum frictionfactorrsquo (MFF) which is conceptually opposite to theminimum energy dissipation Ironically Davies ampSutherland (1983) found that the MFF hypothesispursues the same extrema as the MEDR grouphypotheses depending on circumstances The MFF


hypothesis was refined as the hypothesis of lsquomaximumresistance to flow in the fluvial system as a wholersquowhich considers the friction factor as well as otherresistance terms (Eaton et al 2004) The MFFhypothesis was also criticized in that no MFF is avail-able when channel width depth and slope aredependent variables (Griffiths 1984)

There have been insufficient field observations tosupport existing optimality hypotheses To make mat-ters worse field observations are often inconsistentwith the formulation of these hypotheses For severalmajor rivers in the USA Leopold amp Maddock(1953) reported that the rate of flow velocity increasedownstream is very little Based on this the formu-lation of the optimal channel network theory wasmade assuming no velocity increase downstream(Rodrıguez-Iturbe et al 1992) However observationsmade later in New Zealand showed that the flow vel-ocity increases downstream at a rate much greaterthan was widely accepted putting in question the val-idity of the formulation of the optimal channelnetwork theory (Ibbitt 1997)

5 CONCLUSIONSThis paper provides a review on the mathematical for-mulation of landscape evolution that has leftcharacteristic patterns with a focus on optimalityapproaches There have been a number of optimalityhypotheses proposed and large efforts to find theirbackgrounds from a physical basis Although existingoptimality hypotheses have achieved insightful resultsin several applications there have been continuous cri-ticisms on their physical basis This indicates thedifficulty of interpreting the entropy production ofthe thermodynamics principle for the complex anddynamic geomorphological system

While the early study of Leopold amp Langbein(1962) used the entropy concept most hypothesesproposed thereafter have chosen different approachesin formulating their optimality principles as describedin this review paper This contrasts to other disciplineswhere thermodynamics-based optimality approachessuch as the maximum entropy production hypothesishave been established based on theoretical grounds(eg Lorenz 1960 Dewar 2003) Treating landscapeevolution with entropy aspects is difficult in that it isthe mechanical energy that plays the main role in driv-ing landform evolution while the main theme of theclassical thermodynamics is heat energy As a resultthe maximum entropy production principle has notbeen used in landform evolution problems While theformulation of landscape evolution into the contextof maximum entropy production is challenging it isworth investigating the maximum entropy productionprinciple in this research context

We suggest that the formulation of optimality prin-ciples should accommodate the dynamic nature of thesystem such as hydrological variability and complexfeedbacks from non-fluvial processes eg atmosphericprocesses geological activities and ecosystem func-tions Future interpretation of the thermodynamicsprinciple for landscape evolution problems should bemade in this context of coevolution with a broader


environmental system It would be also worth consid-ering different dissipation regimes in a natural riversystem in relation to the necessity of structuralpatterns

Nevertheless it should be noted that the scope ofoptimality approaches is to find the final or optimalstate of evolution In other words a well-formulatedoptimality hypothesis can give an acceptable predic-tion of geomorphological variables for aninstantaneous snapshot of a continuous movie of land-scape evolution ie the final state expected undergiven ideal conditions but basically may not be agood tool for predicting time-dependent evolution Ifcarefully used with a good understanding on theselimitations optimality principles can contribute tosimplifying the formulation of process-based modelsOptimality approaches can also provide insightful find-ings on the macroscopic behaviour of environmentalsystems

Some of the first authorrsquos ideas stated in this paper developedfrom personal communications with Gavan McGrath andMurugesu Sivapalan We thank Erwin Zehe and StanSchymanski for their helpful comments during the reviewThis research was supported by the International AridLands Consortium grant no AG AZ Y702424-01R-02 theNational Science Foundation (NSF) grant no EAR 02-08009 and a Korea University Grant Any opinionsfindings and conclusions or recommendations expressed inthis publication are those of the authors and do notnecessarily reflect the views of these funding agencies

REFERENCESBagnold R A 1960 Sediment discharge and stream power

a preliminary announcement US Geological Survey Cir-cular 421

Barr D I H amp Herbertson J G 1967 Discussion on

derivation of the regime equations from relationships forpressurized flow by use of the principle of minimumenergy-degradation rate Proc Inst Civil Eng 37777ndash778

Barr D I H amp Herbertson J G 1968 A similitude frame-

work of regime theory Proc Inst Civil Eng 41 761ndash781(doi101680iicep19687818)

Brebner A amp Wilson K C 1967 Derivation of the regimeequations from relationships for pressurized flow by use of

the principle of minimum energy-degradation rate ProcInst Civil Eng 36 47ndash62

Chang H H 1979a Geometry of rivers in regime J HydrDiv (ASCE) 105 691ndash706

Chang H H 1979b Minimum stream power and river

channel patterns J Hydrol 41 303ndash327 (doi1010160022-1694(79)90068-4)

Chang H H 1980 Geometry of gravel streams J HydrDiv (ASCE) 106 1443ndash1456

Chang H H amp Hill J C 1977 Minimum stream power for

rivers and deltas J Hydr Div (ASCE) 103 1375ndash1389Codilean A T Bishop P amp Hoey T B 2006 Surface pro-

cess models and the links between tectonics andtopography Prog Phys Geogr 30 307ndash333 (doi1011910309133306pp480ra)

Davies T R H amp Sutherland A J 1980 Resistance toflow past deformable boundaries Earth Surf Processes 5175ndash179 (doi101002esp3760050207)

Davies T R H amp Sutherland A J 1983 Extremal hypoth-

eses for river behavior Water Resour Res 19 141ndash148(doi101029WR019i001p00141)


Davy B W amp Davies T R H 1979 Entropy concepts influvial geomorphology a reevaluation Water Resour Res15 103ndash106 (doi101029WR015i001p00103)

Dewar R C 2003 Information theory explanation of thefluctuation theorem maximum entropy production andself-organized criticality in non-equilibrium stationarystates J Phys A 36 631ndash641 (doi1010880305-4470363303)

Dietrich W E amp Perron J T 2006 The search for a topo-graphic signature of life Nature (London) 439 411ndash418(doi101038nature04452)

Eaton B C Church M amp Millar R G 2004 Rational

regime model of alluvial channel morphology andresponse Earth Surf Processes Land 29 511ndash529(doi101002esp1062)

Egiazaroff I V 1965 Calculation of nonuniform sedimentconcentration J Hydr Div (ASCE) 91 225ndash247

Grant G E 1997 Critical flow constrains flow hydraulics inmobile-bed streams a new hypothesis Water Resour Res33 349ndash358 (doi10102996WR03134)

Griffiths G A 1984 Extremal hypotheses for river regimean illusion of progress Water Resour Res 20 113ndash118

(doi101029WR020i001p00113)Hack J T 1957 Studies of longitudinal stream profiles in

Virginia and Maryland US Geological Survey Pro-fessional Paper 294B

Horton R E 1945 Erosional development of streams and

their drainage basins hydrophysical approach to quanti-tative morphology Bull Geol Soc Am 56 275ndash370(doi1011300016-7606(1945)56[275EDOSAT]20CO2)

Howard L N 1972 Bounds on flow quantities Annu RevFluid Mech 4 473ndash494 (doi101146annurevfl04010172002353)

Huang H Q amp Nanson G C 2000 Hydraulic geometryand maximum flow efficiency as products of the principle

of least action Earth Surf Processes Land 25 1ndash16(doi101002(SICI)1096-9837(200001)2511AID-ESP6830CO2-2)

Huang H Q Chang H H amp Nanson G C 2004 Mini-mum energy as the general form of critical flow and

maximum flow efficiency and for explaining variationsin river channel pattern Water Resour Res 40 W04502(doi1010292003WR002539)

Ibbitt R P 1997 Evaluation of optimal channel network andriver basin heterogeneity concepts using measured flow

and channel properties J Hydrol 196 119ndash138(doi101016S0022-1694(96)03293-3)

Jia Y 1990 Minimum Froude number and the equilibriumof alluvial sand rivers Earth Surf Processes Land 15

199ndash209 (doi101002esp3290150303)Kennedy R G 1895 The prevention of silting in irrigation

canals Min Proc Inst Civil Eng 119 281ndash290Kennedy J F Richardson P D amp Sutera S P 1964 Dis-

cussion on geometry of river channels J Hydr Div(ASCE) 90 332ndash341

Kirkby M J 1977 Maximum sediment efficiency asa criterion for alluvial channels In River channel changes(ed K J Gregory) pp 429ndash442 New York NYJohn Wiley

Kumar P 2007 Variability feedback and cooperative pro-cess dynamics elements of a unifying hydrologic theoryGeogr Compass 1 1338ndash1360 (doi101111j1749-8198200700068x)

Lacey G 1958 Flow in alluvial channels with sandy mobile

beds Proc Inst Civil Eng 9 145ndash164Lacey G 1967 Discussion on derivation of the regime

equations from relationships for pressurized flow by useof the principle of minimum energy-degradation rateProc Inst Civil Eng 37 775ndash777


Langbein W B 1964 Geometry of river channels J HydrDiv (ASCE) 90 301ndash312

Langbein W B 1965 Closure to geometry of river channels

J Hydr Div (ASCE) 91 297ndash313Leopold L B 1953 Downstream change of velocity in

rivers Am J Sci 251 606ndash624Leopold L B amp Langbein W B 1962 The concept of

entropy in landscape evolution US Geological Survey

Professional Paper 500ALeopold L B amp Maddock T J 1953 The hydraulic geome-

try of stream channels and some physiographicimplications US Geological Survey Professional Paper

252Lorenz E N 1960 Generation of available potential energy

and the intensity of the general circulation In Dynamics ofclimate (ed R C Pfeffer) pp 86ndash92 Oxford UK Per-gamon Press

Millar R G 2005 Theoretical regime equations for mobilegravel-bed rivers with stable banks Geomorphology 64207ndash220 (doi101016jgeomorph200407001)

Millar R G amp Quick M C 1993 Effect of bank stability ongeometry of gravel rivers J Hydr Eng 119 1343ndash1363

(doi101061(ASCE)0733-9429(1993)11912(1343))Millar R G amp Quick M C 1998 Stable width and depth

of gravel-bed rivers with cohesive banks J Hydr Eng124 1005ndash1013 (doi101061(ASCE)0733-9429(1998)12410(1005))

Molnar P amp Ramırez J A 1998 Energy dissipation theoriesand optimal channel characteristics of river networksWater Resour Res 34 1809ndash1818 (doi10102998WR00983)

Paik K amp Kumar P 2008 Emergence of self-similar treenetwork organization Complexity 13 30ndash37 (doi101002cplx20214)

Peckham S D 1995 New results for self-similar trees withapplications to river networks Water Resour Res 31

1023ndash1029 (doi10102994WR03155)Phillips J D 1990 The instability of hydraulic geometry

Water Resour Res 26 739ndash744Pickup G 1976 Adjustment of stream-channel shape to

hydrologic regime J Hydrol 30 365ndash373 (doi10

10160022-1694(76)90119-0)Ramette M A 1980 A theoretical approach on fluvial pro-

cesses In International Symposium on River SedimentationBeijing China Chinese Society of HydraulicEngineering

Rhoads B L 1987 Stream power terminology Prof Geogr39 189ndash195 (doi101111j0033-0124198700189x)

Rodrıguez-Iturbe I Rinaldo A Rigon R Bras R LMarani A amp Ijjasz-Vasquez E J 1992 Energy dissipa-

tion runoff production and the three-dimensionalstructure of river basins Water Resour Res 28 1095ndash1103 (doi10102991WR03034)

Saco P 2003 Flow dynamics in large river basins self-simi-lar network structure and scale effects PhD dissertation

University of Illinois Urbana-ChampaignSchumm S A amp Khan H R 1972 Experimental study of

channel patterns Geol Soc Am Bull 83 1755ndash1770(doi1011300016-7606(1972)83[1755ESOCP]20CO2)

Simon A amp Thorne C R 1996 Channel adjustment of anunstable coarse-grained stream opposing trends ofboundary and critical shear stress and the applicabilityof extremal hypotheses Earth Surf Processes Land 21155ndash180 (doi101002(SICI)1096-9837(199602)21

2155AID-ESP61030CO2-5)Song C C S amp Yang C T 1980 Minimum stream power

theory J Hydr Div (ASCE) 106 1477ndash1487Stepinski T F Marinova M M McGovern P J amp

Clifford S M 2002 Fractal analysis of drainage basins


on Mars Geophys Res Lett 29 1189 (doi1010292002GL014666)

Tal M amp Paola C 2007 Dynamic single-thread channels

maintained by the interaction of flow and vegetationGeology 35 347ndash350 (doi101130G23260A1)

von Helmholtz H 1868 Zur Theorie der stationaren Stromein reibenden Flussigkeiten Verh Naturh-Med VerHeidelb 11 223

White W R Bettess R amp Paris E 1982 Analyticalapproach to river regime J Hydr Div (ASCE) 1081179ndash1193

Willgoose G 2005 Mathematical modeling of whole

landscape evolution Annu Rev Earth Planet Sci 33443ndash459 (doi101146annurevearth33092203122610)

Willgoose G Bras R L amp Rodrıguez-Iturbe I 1991 Acoupled channel network growth and hillslope evolutionmodel 1 Theory Water Resour Res 27 1671ndash1684

(doi10102991WR00935)Williams G P 1978 Hydraulic geometry of river cross

sectionsmdashtheory of minimum variance US GeologicalSurvey Professional Paper 1029

Wood L J 2006 Quantitative geomorphology of the Mars

Eberswalde delta Geol Soc Am Bull 118 557ndash566(doi101130B258221)

Wu W Vieira D A amp Wang S S Y 2004 One-dimen-sional numerical model for nonuniform sedimenttransport under unsteady flows in channel networks


J Hydr Eng 130 914ndash923 (doi101061(ASCE)0733-9429(2004)1309(914))

Yang C T 1971a Potential energy and stream morphology

Water Resour Res 7 311ndash322 (doi101029WR007i002p00311)

Yang C T 1971b On river meanders J Hydrol 13231ndash253 (doi1010160022-1694(71)90226-5)

Yang C T 1971c Formation of riffles and pools WaterResour Res 7 1567ndash1574 (doi101029WR007i006p01567)

Yang C T 1972 Unit stream power and sediment transportJ Hydr Div (ASCE) 98 1805ndash1826

Yang C T 1973 Incipient motion and sediment transportJ Hydr Div (ASCE) 99 1679ndash1704

Yang C T 1976 Minimum unit stream power and fluvialhydraulics J Hydr Div (ASCE) 102 919ndash934

Yang C T 1978 Closure to minimum unit stream power and

fluvial hydraulics J Hydr Div (ASCE) 104 122ndash125Yang C T amp Song C C S 1979 Theory of minimum rate

of energy dissipation J Hydr Div (ASCE) 105769ndash784

Yang C T Song C C S amp Woldenberg M J 1981

Hydraulic geometry and minimum rate of energy dissipa-tion Water Resour Res 17 1014ndash1018 (doi101029WR017i004p01014)

Yatsu E 1955 On the longitudinal profile of the gradedriver Trans Am Geophys Union 36 655ndash663

(a)

(b)

400 km

Figure 1 Some patterns on Earthrsquos surface (a) Meandering patterns of Songhua River (Image courtesy of NASAGSFCMETIERSDACJAROS and USJapan ASTER Science Team) (b) Self-similar tree network of Upper Susquehanna Riverbasin (Image modified from Saco (2003))


the generation of self-similar tree organization drivenby the erosionmdashdeposition process under a gravita-tional gradient and demonstrated that the decreasein total energy expenditure the optimality conditionsuch as that used in the optimal channel networktheory (Rodrıguez-Iturbe et al 1992) is not thecause but a consequent signature

Both these differing perspectives are valuable andcan contribute to improving our predictive capabilityPrediction of Earthrsquos surface processes is vital for a


variety of environmental studies There are emergingsocial issues as consequences of artificially modifiedenvironments such as channelization dams miningurbanization and their restoration or removal as wellas complex environmental changes over longer timescales such as climate change However our abilityto predict the dynamics of systems that evolve intime such as Earthrsquos surface processes is limited(Kumar 2007) Better prediction of their dynamicsdemands the integration of a greater number of


fieldobservations

modelling


models

process-basedmodels






























MSP frac14MSTC




minimumFroude

number


THORN Jia (1990)


min hP


et al (1992)









































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fieldobservations

modelling


models

process-basedmodels






























MSP frac14MSTC




minimumFroude

number


THORN Jia (1990)


min hP


et al (1992)









































there is no choice








































































































































MSP frac14MSTC




minimumFroude

number


THORN Jia (1990)


min hP


et al (1992)









































there is no choice






























































































































MSP frac14MSTC




minimumFroude

number


THORN Jia (1990)


min hP


et al (1992)









































there is no choice
























































































































































there is no choice





































































































































































































































































































































































Optimality approaches to describe characteristic fluvial patterns on landscapes

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