Katul et al AnnRevEcolSys 2007

28 Pages·2012·1.13 MB·English

ANRV328-ES38-30 ARI 24September2007 10:5
Stochastic Dynamics of
Plant-Water Interactions
g
s.or Gabriel Katul,1,2 Amilcare Porporato,1,2
w
vie and Ram Oren1
e
alr
nuy. 1NicholasSchooloftheEnvironmentandEarthSciences,DukeUniversity,Durham,
m www.annal use onl 2NDDuoekrptehaUrCtnmairveoenlrtisnoiatfy2,C7Di7vu0ilr8ha-na0md32,E8NnvoirrtohnCmaernotlailnEan2g7i7n0e8e-r0in3g2,8P;ermattaiSl:[email protected],
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791. n 07/ Annu.Rev.Ecol.Evol.Syst.2007.38:767–91 KeyWords
007.38:767-University o FSTSeyihrsptsetetemApmanubtnbieculsirasilh5sRe,od2env0oli0iennw7leinoafetEacsoalogRye,vEievwoluintioAnd,vaanndceon daAtimbmsoetnsrpsaihcoetnreresdysutcetmion,nonlinearity,scaleeffects,soil-plant
yst. 2State http://ecolsys.annualreviews.org Describingwaterflowfromsoilthroughplantstotheatmosphere
Evol. SKansas T10h.1is1a4r6t/icalnen’sudroevi:.ecolsys.38.091206.095748 rTehmisaicnhsaallefonrgmeiidsanboletssucirepnrtiisfiincgchgiavlelenngtheedheisgphitediymeaernssioofnraelisteyaracnhd.
Ecol. by CAlolpryigrhigthstre(cid:2)cser2v0e0d7byAnnualReviews. degree of nonlinearity of the soil-plant system, which evolves in
v. space and time according to complex internal physical, chemical,
e 1543-592X/07/1201-0767$20.00
R andbiologicallawsforcedbyexternalhydroclimaticvariability.Al-
u.
n thoughrigorousmicroscopiclawsforthissystemstillawaitdevelop-
n
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ment,someprogresscanbemadeontheformulationofmacroscopic
lawsthatupscaleknownsubmacroscopicprocessesandusesurrogate
stochasticitytopreservetheprobabilisticandspectralinformation
contentofthehighdimensionalsystem.Theexternalhydroclimatic
forcing is inherently intermittent with variability across all scales,
therebyprecludingtheuseofstandardapproximationsemployedin
analysisofstochasticprocesses(e.g.,smallnoiseperturbations).Ex-
amplesareprovidedtoshowhowsuperpositionofstochasticityat
multiplespace-timescalesshapesplant-waterinteractions.
767
ANRV328-ES38-30 ARI 24September2007 10:5
ItissurelyoneofthetriumphsofevolutionthatNaturediscoveredhowtomakehighly
accuratemachinesinsuchanoisyenvironment.
(Phillips&Quake2006)
1.INTRODUCTION
We explore the stochasticity in plant-water dynamics with emphasis on its conse-
quencesforderivinglawsforplant-waterinteractionsatlargerscales(referredtoas
macroscopiclaws)startingfromwhatisknownatthesmallerscales.Thethemeof
this review purposely resembles a centuries-old problem whose answer led to the
development of statistical mechanics and linkages to thermodynamics (see Nelson
2004foraclearintroductiontotheapplicationsofthermodynamicsandstatistical
g
s.or mechanicstobiologicalsystems;Dewar2003,Martyushev&Seleznev2006,Ozawa
w
e etal.2003,Roderick2001,andWhitfield2005,amongothers,advocatethermody-
vi
alre namicprinciplestoexplainmacroscopicbehaviorsincomplexnaturalsystems).The
nuy. originalproblemconsideredwhetherknowledgeofmolecularmatterviamicroscopic
w.ane onl lawspermitteddescriptionofmacroscopicbehavior.Naturally,thelargenumberof
m wwnal us mtisotilceaclutlreesatthmatenmt.akAeltuhpoumgahcrpolasncot-pwicatseyrstienmtesraacntdiotnhseisrhsatroecahtatsrtibicumteostwioitnhinmvoitleescustlaa-r
d froperso systems,includingtheirhighdimensionality(becauseofthelargenumberofinter-
nloade2. For afucntidnagmpernotcaelsdseifsfeimrepnaccetsinthgawtpatreervemnotvimemmeendtiawtiethaipnptlihceatsiooinl-spolafnsttastyissttiecmal)m,tehcehraenairces
w1
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791. n 07/ 1. Thefundamentalsubmacroscopiclawsdescribingwatermovementwithinthe
yst. 2007.38:767-State University o 2. piAvinnilvdatetenehrrtaaeagscmxyityniseotlgaeennmmtsihn,aeagnarifndneuddllnaidiovcneitksdtechounrfeaitplsilrctpeeiaarollofyen)p;kfseoenrprtoiatwehrsanetoni(foeetn.xhgt(.eh,is.wieee.asr,utaperbrrcmehflsiaeoccnwarcloelasetcovotfehplsebiigcernlcoaiaofiwutcs-saesmnootiaflvynianonrtnoiealtrbifpnialrecioatey-r,
Evol. SKansas 3. aTthaelldsrciavleerss);oafnmdanymacroscopiclawsremainstochasticbecausethesoil-plant
col. by systemisopentoexternalenvironmentalforcingsuchasrainfall,temperature,
v. E and radiation. In other words, the soil-plant system is open and far from
Re equilibrium.
u.
n Our main objective is to present what is known about plant-water interactions
n
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within a stochastic framework, building on recent advances in dynamical systems
theory,complexsystemsscience,andstochasticprocesses.Long-termfieldmeasure-
mentsatvarioustimeandspatialscalesarebecomingincreasinglyavailablethereby
providingunprecedenteddetailsofthespatialandtemporalstatisticsofplant-water
interactions (e.g., long-term ecosystem water vapor flux measurements via eddy-
covariance methods, gas-exchange measurements, sapflow and soil moisture mea-
surements, tree-ring reconstruction of net primary productivity, and space-borne
leafareaindexmeasurements,tonameafew).Intheabsenceofatheoryformicro-
scopic dynamics, such a stochastic framework may be a first step that inspires the
futurestatisticalmechanicstheoryinecologyandplant-waterdynamics.
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ANRV328-ES38-30 ARI 24September2007 10:5
2.REVIEWOFREVIEWS
Previousreviewsofplant-waterinteractionshavebeenconductedbyNoyMeir(1973)
for desert ecosystems and Lathwell & Grove (1986) for tropical ecosystems. The
formerprimarilydealtwithspace-timestochasticityinrainfall,whereasthelatterdealt
withthephysicalandchemicalpropertiesofsoilsthatimpactecosystemproductivity.
Attheplantscale,awealthofreviewarticleshasalsodealtwithphysical,chemical,
andphysiologicalaspectsofplant-waterinteractions,primarilytheeffectsofwater
stressonphysiologicalprocessesinvariousorgans,gasexchangeprocesses,andplant
hydrodynamics.WementioninparticularthereviewsofHsiao(1973)onplant-water
stress, Sack & Holbrook (2006) on leaf hydraulics, and Sperry (2000) and Sperry
etal.(2002)onhydraulicconstraintstoleafwatersupply,aswellasseveralbooksthat
g
or summarizethestateofknowledgeuptothepasttwodecadesorso(e.g.,Jones1992,
s.
w
e Kramer&Boyer1995,Larcher1995,Nobel2005,Steffen&Denmead1988).
vi
e Atthewatershedscale,anumberofreviewspresentedtheconsequencesofplant-
alr
nuy. water interactions on the hydrologic cycle with emphasis on potential feedbacks
w.ane onl ontheclimatesystem(Brutsaert1982,Eagleson2002)andbiogeochemicalcycling
m wwnal us (eEaragdlyensoanm2ic0s0i2n,Seccohsleyssitnemgesrh1a9s97al)s.oTrheeceinivteedrpsliagynbifiectwanetenatstteoncthioanst—iceitvyidaenndcnedonblyina-
d froperso recentspecialissueofOecologia(in2005;seealsoAustinetal.2004,Schwinning&
nloade2. For Stiaolna2w0it0h4i)n.Fwiantaelrly-,litmheitestdocehcoasstyiscteamspsecwtseroefahlysdortohcelismuabtjieccftoorfcisnegvearnadlrtheceeirnptrroepviaegwa-s
Dow20/1 (Kull & Jarvis 1995, Porporato & Rodriguez-Iturbe 2002, Porporato et al. 2002,
791. n 07/ Reaordlireirgureevzi-eIwtusrbbye&focPuosirnpgoroantost2o0c0h4a)s.tiTcihtyeascnodpheoowftihtiasrriseevsieiwnpdliavnetr-gwesatferromdynthaemse-
007.38:767-University o iscasm.Wpleecaolsnotedmispcuosrsarwyheactotlhoegiccoanlsperqoubelnemcess.areofincorporatingsuchstochasticityin
yst. 2State 3.STOCHASTICPROCESSES:ANALEATORYORIENTATION
Evol. SKansas Tevhoelvsionigl-ipnlaspnat-caetmanodsptihmeereacsycostredmingcatnobitesicnotnesrindaelrpehdyasisctaolc,hchasetmicicdayln,aanmdicbailosloysgtiecmal
col. by laws and subject to external variability. We speak of stochasticity in a broad sense,
E
v. referring to the practical impossibility of precisely modeling and predicting tem-
e
R poral evolution and spatial configuration of the system at all spatial and temporal
u.
n scales.Weavoidthephilosophicaldiscussiononthetruenatureofthisunpredictable
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andirregularbehavior(Ford1983)andwhetheritshouldbeattributedtohighdi-
mensionality,instabilities,andsensitivitiestoinitialconditions(Boffettaetal.2002,
Cross&Hohenberg1993,Eckmann&Ruelle1985),ortosimpleignoranceofthe
detailedfunctioningofthesystemthatbyitselfrequiresastatisticalapproach(Clark
&Gelfand2006,Jaynes2003,Kass&Raftery1995,Sivia&Skilling2006).Tosome
extent,allthesearepresentandsomewhatinterlinkedineachcompartmentofthe
soil-plant-atmospheresystem.
Stochasticfluctuationsareoftenreferredtoasrandomnoise,andsolutionsofequa-
tionswithrandomparameters(orrandomdynamicalsystems)arecalledstochastic
processes or random functions. The novice may be easily disoriented and daunted
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ANRV328-ES38-30 ARI 24September2007 10:5
bythevastliteratureinthisbranchofmathematics,whichwasinspiredbydifferent
disciplines such as physics, chemistry, theoretical finance, communication and in-
formationtheory,hydrology,etc.,andwhichbenefitedfromfundamentalcontribu-
tionsbymathematiciansandscientistssuchasLaplace,Poisson,Maxwell,Bolzmann,
Einstein,andKolmogorov,tonameafew.Thefollowingisanincompleteorienta-
tionlistofreferencesforapplicationsofrandomfunctions(Bendat&Piersol1971,
Papoulis1991,Vanmarke1983),stochasticprocesses(Cox&Miller1977,Gardiner
2004,Horsthemke&Lefever1984,Larson&Shubert1979,vanKampen1992)and
theirapplicationsinthenaturalsciences(Sornette2004).
Themathematicalmodelingofnoisydynamicstypicallyrequiresaconsiderable
degreeofabstractionandsimplification.Ingeneral,whentheanalysisislimitedtothe
org temporaldomain,theexistingliteratureissufficientlydevelopedforsomeclassesof
ws. noise.Ifthegoverning(orstate)variablesareorcanbeapproximatedasdiscrete,one
e
vi canemploythewell-establishedtheoryofMarkovchains(Cox&Miller1977,Norris
e
alr 1998);moretypically,thestatevariablesaretimecontinuousandonemustresortto
nuy.
w.ane onl soof-ccoanllteindusotoucsh-satsattiecsdtoifcfehraesntitciaplreoqcueassteiosnasre(SeDithEesr).GTahuessimanosntotiyspesic(aBlrbouwilndiianngmboloticokns
m wwnal us orWienerprocess),usedwhentheirregularfluctuationsdonotappeartobepulsing
d froperso athnedpinutlseirnmgitctoemntp,oonrejnutmspansdcotmhepsopuancdee-dtimweithintseurimtabitlteenpcoyinbtepcoromceesdseosm,iunsaendtw(eh.gen.,
wnloade12. For hinoduerplyenodrednatiliynrcarienmfaelln).tsTshoetbhaastitchmeyoddeolsnjoutstamddennteiwonmedemaroercyhtaoratchteersiyzsetdemby.Braencdauosme
Do20/ oftheirflatpowerspectrum,theyarealsocalledwhitenoises.Theirusealwaysentails
791. n 07/ somesubtlemathematicalaberrationsatveryfinescalessuchasdiscontinuitiesinthe
007.38:767-University o scTtawasteoefmvoararniBaifbreolsewtsa,ntaiioasnnissmothofettihocenasseeanafodbreirtjrusamtriepolnaptsirvoaepcseps(esCaeros,xwoh&reinnMatinhllaeelryfiz1risn9t7gd7te,hrGeivaatrrtaidvjieenc,etarosr2yi0so0tf4h)ea.
yst. 2State Beqruoawtnioiann.Tpharetfiicrlsetamtainnfiifneisttealtyiosnmiasliltstedmiscpoonratilnsucoaluessvueslioncgitoiense,wsthoicchhaisstiinctdeirfpfereretendtiaasl
Evol. SKansas bcreoinssgaprgoidvuecnedlebveylainnufinnriteeallyistmicainnyfintiimteefsoricne,aanndintfihneitseesciomnadlitsimtheatitnhteertvraalj.ecTtohreisees
col. by pathologiesarethenecessarypricepaidforsimplicity,andmaybeacceptablesolong
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v. asnospecialphysicalmeaningisattributedtothefluctuationsatsuchshorttimescales.
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u.
n whichfuturestatesdonotdirectlydependonpaststates.Theycanbedescribedmath-
n
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ematicallyaseitherSDEsforthestatevariables,emphasizingaspecificrealization
(e.g.,onepossibleoutcome)ofthestochasticprocess,ortheensembleofrealizations
(e.g.,allpossibleoutcomes)intermsofpartialdifferentialequations(PDEs)forthe
probabilitydensityfunctionsofthestatevariablesdescribedbytheSDEs(Gardiner
2004,Horsthemke&Lefever1984,vanKampen1992).
Theuseofnon-Markovian(orcolorednoises),fractionalBrownianmotion,and
Levy-type processes has gained recent popularity because of their ability to de-
scribefinetemporaldynamics,presenceofscaling(e.g.,power-laws),andlong-term
memoryeffectssignifiedbyslowlydecayingtemporalautocorrelations(Horsthemke
& Lefever 1984, van Kampen 1992). Unfortunately, this standard mathematical
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770 Katul Porporato Oren
ANRV328-ES38-30 ARI 24September2007 10:5
machinery becomes difficult to implement analytically for nonlinear systems with
twoormoredegreesoffreedom(Arnold2003),andevenmoredifficultwhenconsid-
eringspace-timeprocesses(Bouchaud&Georges1990,Cross&Hohenberg1993,
Durrett&Levin1994,Falkovichetal.2001,Sagues&Sancho2004).
Weconcludethisorientationbynotingthatthepresenceofnoiseinthecontextof
plant-waterdynamicsmaynotbenecessarilyasourceofdisorderwithinthesystem
butcanbeastabilizingfactor(e.g.,promotingcoexistenceamongplants),apattern
generator, or a mechanism responsible for phase transitions (e.g., abrupt changes
suchasextinctionsandcatastrophicshifts),allofwhicharenowgainingattentionin
anumberoffields(D’Odoricoetal.2005,Horsthemke&Lefever1984,Porporato
&D’Odorico2004,Rietkerketal.2004,Sagues&Sancho2004,Schefferetal.2001,
org Sornette2004,Vandenbroecketal.1994).
s.
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alre 4.STOCHASTICDYNAMICS
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m wwnal us considertheconceptualdiagraminFigure1showingontheabscissathedimension-
d froperso adleitsycroibfinthgethsyesitnetmeraacntdioonnsatmheoonrgdiitnsavtaeritahbeledse.gArseweeodfinscounslsinbeealroiwty,wofattehremeoqvueamtioennst
wnloade12. For igniotnheofstohili-spfilagnutres.yWstehmentythpiecdalylnyarmesiicdseosfitnhethseeihnitgehra-cdtiimonesnsisioanpaplraonxidmnaotendlinbeyaarfreew-
Do20/ coupled equations, the high dimensionality is replaced by stochasticity to preserve
791. n 07/ theprobabilisticandspectralinformationcontentpresentinthenaturalsystem.As
007.38:767-University o ahatyradelrspuarulotlpitchecreotisnetdso,ucehcttacisv.)tiitacysitowyfeathlplepaessoairinlsptihonermeesxoatdneedrlnppallaarvnaatmrcioaebnteidlriustiyotsfo,ftfhothelieasgihmeydpdilrsiofitrceilbdimustyaisotitncesmf,osrt(coei.mngg.-,
yst. 2State (e.gT.,hriasinsftaolcl,hwasitnicd,rteepmrepseernattautrioe,nraodfipatliaonnt-).water dynamics is logical in the context
Evol. SKansas oofnpercoopsyasgtaetmingprporcoejsescetse.dVcitloimusaetkic’sfl(1u9c9tu2a)trioenvise—wa“nGeloxabmalpElenvoifroenxmteernnatallfCorhcainngg—e”
col. by pointedoutsomeoftheproblemsanddifficultiesinpropagatingclimateprojections
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v. toecosystemprocesseswiththecurrentmodelingframework.Thisdifficultyisdriven
e
R bythefactthatecosystemsrespondtothefullrangeofhydroclimatevariability(par-
u.
n ticularly the extremes). Hence, there are two practical benefits in considering the
n
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plant-watersysteminastochasticbutsimplifiedframework:(a)reducedcomplexity
and(b)properaccountingofhydroclimaticvariability.
4.1.OriginofInternalStochasticity
Movementofwaterinthesoil-plant-atmospheresystem(Figure2)beginswithwater
migratingfromwettertodriersoilporesadjacenttotherootingsystemfollowingpo-
tentialenergygradients.Onceithasreachedandenteredtherootingsystemthrough
a patchy and heterogeneous root membrane, water flows through a tortuous and
complexnetworkwithinthexylem.Itexperiencesphasetransitionwithintheleaves,
www.annualreviews.org • StochasticityandPlant-Water 771
ANRV328-ES38-30 ARI 24September2007 10:5
Number of variables (n)
(surrogate for dimensionality)
n=1 n=2 n=3 n>>3 ?
Lumped linear
Linear
soil moisture dynamics
Lumped soil Vertically averaged Soil moisture + Competition for water Reaction diffusion
dynamics + soil-water balance plant and atmospheric among plants (biomass and
quasi-steady + plant capacitance boundary layer nutrient patterns)
transpiration dynamics Root growth dynamics
Soil aggregates Richards equations
Non-
Logistic (Lumped) Lumped water, (spatially explicit
linear
g biomass growth soil moisture nutrients, and Dendritic tree soil-water dynamics)
s.or + average + biomass carbon balance and river network
w hydroclimate dynamics equations structures
vie Navier-Stokes
nualrey. (turbeuqlueantcioen) swith
w.ane onl Dimension reduction Plant coTmhpee ftritoionnti eforr water; CeOn2e, rwgya tfelur,x aensd
m wwnal us sHuigrrho gdaimteedn sbiyo ninatleitryn aisl muplltaisnct amleic mroofldueidliincgs ;of
d froperso Degree of stochasticity aspnadt itaelm hpeoterarol gneoniseeit.y..
nloade2. For nonlinearity
w1
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791. n 07/ (e.g., rSaitnofcahlla, sratidci aetxitoenr,n aailr f toermcipnegrature)
007.38:767-University o FCdiiomgnuecrneespit1ounaallidtiya/gnroanmlinsheoarwitiyngpltahneec.oSmtopclheaxsittiyciotfytahreisseosiwl-phleanntsismysptleimfiedviemwoeddelisnatthteemptto
Evol. Syst. 2Kansas State cssdiyaimssptttpieunmlrigfie.ueWtidshhiemsirtcoeofdfrmeeorlmpstltoeotxhiinpetrtyeeex(rsrnteeeravrvlniessaettlodhfceofhrrpaocrsmitonibcgSia,ttbwyriolhaigssiactritheczliaa1snt9ped9dr4estsp)o.eenScthtutreraartlpoairgnlelafstsoeecrnamsclteeoastcoiahofnanrdsatcniisocdinsotttymoeicnshptaaaodsrftdaitmechdieenttoeonrratisghttuieonrael.
col. by The“frontier”representsopenproblemslackingrigorousmathematicalformulations.
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v.
Re andexitstotheatmosphereintheformofwatervaporthroughleafstomata.Thevapor
nu. moleculesarethentransportedbyturbulenteddiesfromwithinthecanopyintothe
n
A freeatmosphere.Thetransportingenergyandsizesoftheseeddiesarepartiallysetby
complexinteractionsamongcanopyattributes(e.g.,leafareaandheight),mesoscale
forcing(e.g.,geostrophicwindsandweatherpatterns),landscapeheterogeneity,and
theairflowabovethecanopy.
Resolvingallspatialscalesneededtodescribethetrajectoryofwaterinthesoil-
plant-atmospheresystemnecessitatesathree-dimensionalsimulationdomainspan-
ning0.1μmtotensofkilometers,equivalenttorequiring∼(1010)3 nodespertime
step.Thetimestepmustbesufficientlyfinetoresolvethefastestprocess,whichisthe
actionofviscousdissipationonturbulentfluctuationsintheatmosphere(∼0.001s).
This high dimensionality in space and time is well beyond the capacity of any
· ·
772 Katul Porporato Oren
ANRV328-ES38-30 ARI 24September2007 10:5
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yst. 2State k(cn)oPwlannitn-aptmlaonstp-wheatreergdaysneaxmchicasngmeoidsecloinngtr(oslclaeldeb∼y1s0t0omμmata;a(fstcearleZ∼im1m0μermm;afnro1m98G3)r.ant&
col. Evol. Sby Kansas Vhateamttenorisocpkghe2en0ree0i4t(ys).c(a(sdlcea)l∼Teu1∼0r1bm0ul)ke.mn(et)).eAdNdtmoietoessttprhhaanetsrspitcoosrctthawatesatstiecairrteyvamipsoopdrrueflrsaoetnmetdsabttoyamltlhasetpalaattoniadtlshscecaapflreeese.
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v. brute-forcecomputationatpresentandintheforeseeablefuture.Furthermore,there
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u. areinsurmountablescaleissuesinattemptingtorelatewaterflowinthesoil-plant
n
n systemwithitsdrivingforces.Forone,theconstitutivelawscurrentlyusedtodescribe
A
water movement in the soil, root, plant, and atmosphere systems do not share the
samerepresentativeelementaryvolume(REV),definedastheminimalspatialscale
ofrepresentation.
Considertheconstitutivelawsineachofthethreecompartmentsofthesoil-plant-
atmospheresystem:
1.Soil.Darcy’slawandthewatercontinuityequationaretypicallycombinedin
the Richards equation, which describes water movement in unsaturated soils near
the rooting zone at an REV scale containing a sufficiently large number of pore
spaces. This is a nonlinear partial differential equation that provides a space-time
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ANRV328-ES38-30 ARI 24September2007 10:5
descriptionofwatermovementbutaveragesoutthevariabilityofthesoilandroot
matrix at scales smaller than the REV (Hillel 2004, Jury et al. 1991). It is a major
theoreticalchallengetoup-scalethisequationbeyondtheREVtoincludetheeffects
ofspatialheterogeneitiesinsoilproperties,macroporosity,andpreferentialflowsof
waterandnutrientsatdiscontinuities(e.g.,largeroots,rocks,etc.).Eventheapplica-
tionofDarcy’slawwithinaREVthatincludessuchrandomnessandheterogeneities
isquestionable(Steffen&Denmead1988).Inparticular,theoriginofpreferential
flowsandtheirimpactsontheplant-waterrelationshipremainopenareasofresearch
(deRooij2000).
2.Plant.Similarproblemsariseattheplantlevel.Laminarflowequations(e.g.,
Hagen-Poiseulle’slawforcapillarytubes)basedonthecontinuumassumptioninfluid
org mechanicsaretypicallyusedtodescriberootandxylemwatermovement.Theseas-
ws. sumptions are now being challenged by recent research. For example, predictions
e
vi oftheonsetofembolismintheplantxylemrequiremicroscalethermodynamicde-
e
alr scriptionofairandwatermicrofluiddynamicsnotcapturedbyPoiseulle’slaw;the
nuy.
w.ane onl dfiersrtivpartiinoncipolfesobhsaesrnveodtyveutlnbeereanbitlaictyklceudr,vaensd(Sthpeercroyn2n0e0c0ti,oSnpbeertrwyeeetnasl.to2m00at2a)lfcroonm-
m wwnal us ductanceandtheplant-xylemsystemremainsasubjectofresearch(Katuletal.2003,
d froperso Ore3n. eAttmal.o1sp9h99er,eS.aMcka&ss,Hmoolbmroeonktu2m0,06a,nSdpeenrreyrg&yHexacchkaen2g0e0s2b,eStpweerernyetthael.c2a0n0o2p)y.
wnloade12. For adnedtaitlheedlodwesecrriapttmioonspohfetrheeabreoudnesdcarriybecdonbdyitthioenNs aatvitehr-eSptolaknets-aetqmuaotsipohnesraenidntreerqfuaciree.
Do20/ Describingtheboundaryconditionsfortheseequationsiscomplicatedbystochas-
791. n 07/ ticityatmultiplescales.Randomness,beginningwithpatchinessatthestomatallevel
007.38:767-University o alfeonardfadpsirdsotygrnirbaeumsstiiioncngb,otloueanpfdaatacrrheyaincdoeensnsdsiiinttiyos,tnoasnmtdoaottanhlwecoaNrndadvutioectrta-hSnetcoaekt(meBsouescqpkuhleaeytrieoetnmasul(.sA1tl9bb9ee7ra)tc,scoroannuedntoteamld.
yst. 2State 2eq0u0a1t)i.oEnsvecnanfnoorttbheessoimlvpedleactaasleltohfeansetcaetsicsabryouscnadlaersy(Fcionnndigitaionn2,0t0h0e).Navier-Stokes
Evol. SKansas Theseconstitutivelaws(theRichardsequation,Poiseuillelaw,andNavier-Stokes
col. by equations) often provide reasonable approximations at a particular scale, typically
E wheremicroscopicheterogeneitiescanbeaveragedout.However,amajorchallenge
ev. is the derivation of effective parameters in simplified models at larger scales (e.g.,
R
u. Bohreretal.2005,Chuangetal.2006,Ewersetal.2007,Katuletal.1997).Clearly,
n
An novel tactics are needed to further the development of these constitutive laws for
applicationsintheplantsystematmuchfinerscales(<10μm),andofwaystoproperly
scalethemtocoarserlevels.Someanswersmaybefoundinmicrofluiddynamics,a
fieldthatisrapidlygainingattention(Squires&Quake2005).
Formalapplicationsofhomogenizationandaveragingtechniques(Torquato2002)
may guide the derivation of effective parameters for the current constitutive laws.
Applicationsofsuchtechniquestothesetofequationsdescribedabovearefurther
complicatedbythreefactors:(a)theinherentnonlinearitiesintheresistancetowater
flow,soilhydraulics,canopyturbulence,andplantresponsestotemperatureandlight,
amongothers;(b)thefactthattheexternalforcingcanbehighlyintermittent(e.g.,
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774 Katul Porporato Oren
ANRV328-ES38-30 ARI 24September2007 10:5
a b
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ali
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w.annue only. Espxaacmepalnedsotifmsem.o(ao)thSipnagceo:fdnifofnerliennecaersitiinesleaapfp-eanardi-ncganinopmya-cprhoostcoospyinctehqeusiastiroenspsobnyseavteoraliggihntgin
m wwnal us wphhoetnosvyenrttihceaslilsymavoedraegli(nFgarfqroumharleeatfatol.e1n9t8i0re)icsaunsoepdy.foArsliemafpplihfioetdosvyenrstihoensios,fwthheerFeaarsqtuhhear
wnloaded fro12. For perso Cel(iatgAsahsNlut.ma2Vt0itEne0gGn2u)m;matethoiaoednneCa,lilAiresNatuefsVmpehEdpyGetsroiaomstlcuooargdleeye,,lirlteiesulaapaftiemfvoneurehlrttghuilymeaybciedaarnliatoryne,pcpwyer,ie(nBasdenandlsdtpatoeutceircodbhn,uialo1enf9ndt9thC2ter,OaLcna2asnipcoeooptnrytaclept.hnr2aot0trc0aeit0nsi,scoeSlnusidqaeubseoivrea
791. Don 07/20/ ttshhoeeilcdmaanoiloiysptluyervaeerlem,wuenahnaelrt(cid:3)eesar(cid:4)estdt(hL).ea(ibdo)aTeshtimeadl.ea:2ne0qd0u2di)ov.atTtleehndetlcisnooenilstwirneaufteoerrutsloolsilnsoefnuigns-ctthetireomnnoafnv(el(cid:3)irsna(cid:4)egta)erfsolforostrshdfeuiftnfeecmrteiponontraatl
yst. 2007.38:767-State University o nrfSeao∗ri,inessenfoaitllleslvm,teasoltusiesnrsteufl;parerenceadksets(n,tcht)eientthcgo.en)r,sareiapntrrfeooaflpdlwruveaactsrieeinarngbscitljeriuetoyms.sf;Spaahscn,ltdtehhaSaerftchs,ayctbgahrrleeuofispsceetollpdypaicrcfaoapptroaicocieninttty.;h.ASewss,aytchsoteenwmsieltqtionuwgenaprcodein,dtt;hife-
Evol. SKansas utrpa-tsecdaslipnagtiasmllyofootrhpehsoatnodsyrnetdhuecseissannodntleinmeparoirtiaellsybfuotrsnoeivlewraetelirmininFaitgesurthee3m).F(ausritlhluesr--
col. by more,theclassicsimplifyingtechniques,suchas“small-noiseandsystem-sizeexpan-
v. E sions,”arenotdirectlyapplicableinthiscontext(Gardiner2004,vanKampen1992).
e
R
u.
n
n
A 4.2.OriginofExternalStochasticity
One of the main challenges in modeling the temporal dynamics of the soil-plant
systemisthenoisestructureofthehydroclimaticforcing.Dependingonthetemporal
scale,somehydroclimaticcomponentsmaybemodeledascontinuousGaussiannoise
orprocessesdrivenbyGaussiannoise(e.g.,annualrainfall,dailytemperature,and
hourlywindvelocity)andothersmaybemodeledasintermittentjumpprocesses(e.g.,
dailyrainfall,subhourlyphotonfluxdensityinthecanopyunderstory).Thisstrong
intermittencyalonesufficestopreventdirectapplicationsoftraditionaltechniques
usedtosimplifystochasticprocesses,suchassmall-noiseexpansions(Gardiner2004,
vanKampen1992).
www.annualreviews.org • StochasticityandPlant-Water 775
ANRV328-ES38-30 ARI 24September2007 10:5
Next we show examples of the complex noise structure at different scales of
temperature,lightandrainfall,whichareamongthemostimportanthydroclimatic
drivers of the soil-plant system. We start in Figure 4 with measurements of air
temperature above the canopy at scales ranging from interannual to fractions of
seconds. Note the large and organized turbulent fluctuations around the diurnal
cycle in mean air temperature (∼5◦C). The figure shows that these excursions are
intermittent and highly nonstationary during daytime as evidenced by progressive
changes in the variance of air temperature. Even at sufficiently small timescales
(∼10s),largeramp-liketemperatureexcursionsareoftendetected(Figure4).The
latteronesareconnectedwithburstsofairhavinghighvelocityandlowtemperature
(so-calledsweepevents)thatpenetratethecanopyfromabove,exchangeheatwiththe
org vegetation,andthenareejectedbacktotheatmosphere(ejectionevents)withlower
ws. velocityandhighertemperature(e.g.,Katuletal.2006).Suchrapidchangesinair
e
vi temperaturecansignificantlyaffecttemperature-dependentkineticconstantsinplant
e
alr processes.
nuy.
w.ane onl of tLhieghptaassbaogveeothfeclcoaundospy(Kmnaayppals1o9b9e3c).omFueritnhteerrmmoitrtee,nwtaitthsihnotrhtetimcaensocpalyesthbeeclaiguhset
m wwnal us regimecanbehighlystochasticwithintermittentpulses(Figure5)atscalesofsec-
d froperso opnhdostotsoynmtihneustiessoofwthinegutnodrearnstdoormy,wshhaidcihngisbayntohneloinveearrlyfiunngcctiaonnopoyf.lTighhitsimnteeannssittyh,aist
wnloade12. For dsereivneinnbFyigraunrdeo5m. sunflecks(seeNaumburgetal.2001,Pearcyetal.1997),ascanbe
Do20/ The effects of these intermittent and high-frequency variations in light levels
791. n 07/ onthespatialandtemporalwatermovementwithinvariousplantorganshavenot
007.38:767-University o birneegaedncilasynuaffirvecadiieluancbtelleytrtsaotnusdsupiepirdpaot(isroetnehEiogfwhseeorrsmweeattlaeelra.v2fle0us0xo7es,rOobfrraetnhnceehtbeeastl.bte1ur9t-9iml8lu)o.mrIeinntwartaaectderrobwbrneancsochmhaedes-s.
yst. 2State Swohmereeesvtiedmen-lceeveolftflhuixsecsomdopennostatsiiognnimficaaynbtelygvraapryhiicnallsypadceemaonndstraaptpeedairnbFeitgteurrebe5-,
Evol. SKansas hthaevebdrainncfhoelslodwoiwngnstthreeaomv.erTahlledciuormnbalincaytciolenoofflrigechetnvtaaridavtaionncseswihnepnlacnotmhpyadrreoddtyo-
col. by namic models of water movement in trees and stochastic models of light attenu-
E
v. ation within canopy has been employed to successfully represent such variability
e
R anditsintegratedeffectsonstomatalconductanceandbranchandtreetranspiration
u.
n (Ewers et al. 2007), and may be employed to a similar end in representing carbon
n
A
uptake.
Finally, Figure 6 shows rainfall variability from daily to century timescale.
Theannualrainfallcanbedecomposedintoseveralembeddedstochasticprocesses
(D’Odorico et al. 2000, Porporato et al. 2006). Similar to temperature, at annual
timescalestherainfallprocessislessintermittent(andnear-Gaussian),whileatthe
dailytimescaleitishighlyintermittent.Thepropagationofsuchrainfallfluctuations
toplantdynamicsisgainingincreasingusageforpredictinganecosystem’sresponse
tofutureclimaticshiftsorforexplaininghistoricalresponsestoclimaticvariations.It
isnowrecognizedthatchangesintotalrainfallamountsarenotsufficienttoexplain
changesinnetprimaryproductivity(Knappetal.2002).Shiftsinrainfalldistributions,
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776 Katul Porporato Oren