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Dynamics of the stream-power river incision model: Implications for height

limits of mountain ranges, landscape response timescales, and research needs

Article  in  Journal of Geophysical Research Atmospheres · August 1999

DOI: 10.1029/1999JB900120

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 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 104, NO. B8, PAGES 17,661-17,674, AUGUST 10, 1999

Dynamics of the stream-power river incision model:

Implications for height limits of mountain ranges,landscape
response timescales, and research needs
Kelin X. Whipple
Department
of Earth,Atmospheric
andPlanetarySciences,
Massachusetts
Instituteof Technology,
Cambridge

Gregory E. Tucker
Department
of Civil andEnvironmental
Engineering,
Massachusetts
Instituteof Technology,
Cambridge

Abstract. The longitudinalprofilesof bedrockchannelsarea majorcomponent of the relief struc-

tureof mountainous drainagebasinsandthereforelimit theelevationof peaksandridges.Further,
bedrockchannels communicate tectonicandclimaticsignalsacrossthe landscape, thusdictating,
to firstorder,the dynamicresponse of mountainous landscapes to externalforcings.We review
andexplorethe stream-power erosionmodelin an effortto (1) elucidateits consequences in terms
of large-scaletopographic (fluvial)reliefandits sensitivity
to tectonicandclimaticforcing,(2)
derivea relationshipfor systemresponse time to tectonicperturbations,(3) determinethe sensi-
tivity of modelbehaviorto variousmodelparameters, and(4) integratethe aboveto suggest use-
ful guidelines for furtherstudyof bedrockchannelsystems andfor futurerefinementof the stream-
powererosionlaw. Dimensionalanalysisrevealsthatthe dynamicbehaviorof the stream-power
erosionmodelis governedby a singlenondimensional groupthatwe termthe uplift-erosion
number,greatlyreducingthenumberof variablesthatneedto be considered in the sensitivity
analysis.The degreeof nonlinearityin the relationship betweenstreamincisionrateandchannel
gradient(slopeexponentn) emergesasa fundamental unknown.The physicsof the activeero-
sionprocesses directlyinfluencethisnonlinearity,which is shownto dictatethe relationship
betweenthe uplift-erosionnumber,the equilibriumstreamchannelgradient,andthetotal fluvial
reliefofmountain
ranges.
Similar[y,
thepredicted
response
timetochanges
inrockupliftrateis
shownto dependon climate,rockstrength,andthemagnitudeof tectonicperturbation, with the
slopeexponentn controllingthe degreeof dependence on thesevariousfactors.For typicaldrain-
agebasingeometriesthe response time is relativelyinsensitiveto the sizeof the system.Work
on thephysicsof bedrockerosionprocesses, their sensitivityto extremefloods,theirtransient
responsesto suddenchangesin climateor upliftrate,andthe scalingof localrockerosionstudies
to reach-scale
modelingstudiesaremostsorelyneeded.

1. Introduction determinemuchof the relief structureof the landscape, (3)

1.1. Motivation
riverstransmittectonicand/orclimaticsignalsthroughout
the landscape,and (4) bedrockchannelsset the boundary
Recentrecognition of potential global-scale interactions conditionsfor hillslope processes
(e.g.,soil creepand land-
between climate, surface processes, and tectonics [e.g., slides) responsible for denudation of the land surface. Thus
Adams, 1985; Molnar and England, 1990; Isacks, 1992; bedrockchannelssignificantlyinfluenceboth the rates and
Raytoo and Ruddiman, 1992] has sparked the field of tec- patternsof erosionalunloading in fluvial landscapesand,
tonic geomorphologyand broughtthe problem of the dynam- consequently,long-term sedimentfluxes to basins.
ics of bedrockchannelfluvial systemsto the forefront of theo- Significant
progress
hasbeenmadein developing
physi-
retical geomorphology[e.g., Seidl and Dietrich, 1992; Wohl, callybasedformalismsfor modelingthe dynamicsof bedrock
1993; Howard et al., 1994; Seidl et al., 1994; Wohl et al., channelsystems[Howard and Kerby, 1983; Seidl and
1994; Zen and Prestegaard, 1994; Montgomeryet al., 1996; Dietrich, 1992;Anderson,1994; Howard, 1994; Howard et
Tuckerand Slingerland, 1996]. Knowledge of the dynamics al., 1994; Kooi and Beaumont, 1994; Rosenbloomand
of bedrock channels is of profound importance for under- Anderson,
1994;Seidlet al., 1994; Goldrickand Bishop,
standingthe interaction of tectonicsand surficial processes 1996;Stock,1996; Tuckerand Slingerland,1996;Stockand
because(1) the channel network definesthe texture (plan- Montgomery, 1999]. Of themodels thathavebeenproposed,
view) of the landscape,(2)channel longitudinal profiles thestream-power (orshear-stress)modelis mostsatisfying
as
it is castdirectlyin termsof the physicsof erosion[Howard
Copyright1999by the AmericanGeophysical
Union.
andKerby,1983]. Thestream-power modelis quitegeneral
andhasbeenprofitablyusedin a diversityof modelingstud-
Papernumber1999JB900120. ies [Anderson, 1994; Howard, 1994; Rosenbloom and
0148-0227/99/1999JB900120509.00 Anderson,1994;Humphreyand Heller, 1995;Moglenand
17,661
17,662 WHIPPLE AND TUCKER: DYNAMICS OF THE STREAM-POWERLAW

Bras, 1995a,b; Goldrickand Bishop,1996; Tucker,1996'

Tucker
andSlingerland,
1996]. Thisgenerality,
however, l ........ 2 (•e.•h_c,•)•,mi•n
.•.,•T(e hc
)max
brings with it a numberof poorly understoodparameters
whoseeffective
values
represent,
intheworst
cases,
adiver-
sityofcomplex
interactions
amongasuite
ofphysical
proc- i• BedrOck
Channel
PrOfile •
esses.
Giventhesweeping
generality
ofthemodel,
itswide-
spread
use,andthecomplex
nature
ofthephysics
ofriverin-
cision
intobedrock,
anexploration
andsensitivity
analysis
of the stream-power modelseemsusefulto furtherdevelop- ?<xmax, ,
mentof landscapeevolution modelingand as a guide for 0 10 20 30 40

futurefield investigationsof river incisionprocesses. Distance from Divide (km)

1.2. Approach and Scope

In this paper we review and explorethe general stream-
powermodelfor bedrockchannelprofile evolution. Our first (S
c)max_,=L
=•
aim is to reveal the strengths,weaknesses,and limitations of ---(•'
j=)min
-----
the currentmodel [e.g.,Howard et al., 1994] in a thorough
review of its formulation. Further, we aim to illuminate the
aspectsof the model most critical to modeling the dynamic
0.1
Landslides/
responseof riversto tectonicor climaticforcing. That is, we
explorethe role of model parametersin dictating (1) equilib-
rium channelform, (2) sensitivityof equilibrium channelgra-
Cø11uvialChannels
I !! ck'•e
(S - constant)
Fluvial Bedro

dientto rock uplift rate, (3) equilibriumfluvial relief in active (A c ),rnin!,A,,,,?

)max
0.01
orogens,and (4)the timescaleof landscaperesponseto tec-
1E+2 1 E+3 1 E+4 1 E+5 1 E+6 1 E+7 1 E+8 1 E+9
tonic forcing. In this analysis we attemptto develop a rela-
tively completepictureof the dynamicbehaviorof the stream- b Drainage
Area(m2)
power erosionmodel and, in doing so, develop new insights
into the relative importanceof the various parametersin the Figure 1. Relief structureof active nonglacial orogensbased
on data fromfour drainagesin the northernCentralRangeof
model equations. A dimensional analysis of the bedrock
Taiwan(seeTable 1)..(a) Fluvial bedrockchannelrelief Rf
channelevolution equation is used to guide the sensitivity
and hillslope/colluvial channel relief Rhc shown along a
analysis. Limiting assumptionsthat restrictthe "complete- characteristicdivide-to-outlet channellongitudinal profile.
ness"of our analysis are clearly statedwhereverappropriate The boundarybetweenfluvially dominatedbedrockchannels
in the text and are outlined briefly in the paragraphbelow. anddebrisflow-dominatedcolluvialchannels,demarcated by
Finally we discussthe need for coupled field and modeling Xc is inferred from a break in scaling of the relationship
studies of bedrock channel systemsto place quantitative betweenchannelgradient $ and drainageareaA (see Figure
constraintson thoseparametersmostcritical to modelbehav- lb). Here and elsewhere(Table 1) fluvial relief represents80-
ior, and we discusssome of the physical processesand proc- 90% of the total relief. (b) Lines schematicallyrepresent
ess feedbacks that set the value of "effective" model slope-areadata along longitudinal profiles of four Taiwanese
rivers. Data for one river are shown for comparison(shaded
parameters.
dots;dataare smoothedby log-bin averagingafterTarboton
Howard et al. [1994], Montgomery et al. [1996], and
et al. [1991]). Note that concavity estimatesreported in
$klar and Dietrich [1998] have discussed at length the Table 1 were derived from regressionof raw (unsmoothed)
occurrenceof bedrockchannelsin the landscape,the poten- data. The transition from fluvially dominatedto colluvial
tial controlson bedrock-alluvial transitions, and approaches channelsis inferredto occurat the break in scalingobserved
to modeling these transitions at regional to continental to occurbetween
105and106m2 in the drainage
area[after
scales.In general,bedrock and mixedbedrock-alluvial chan- Montgomery and Foufoula-Georgiou, 1993]. Minimum and
nels dominatein headwaterregionsand in the uplandsof tec- maximum((Sc)maxand (Sc)min)colluvial channelgradients
tonically active orogenicbelts. We restrictthe focusof this andcriticaldrainageareas((Ac)maxand(Ac)min)representthe
paper to the exploration and discussionof fluvially domi- rangeof variability amongthe four drainagesexaminedand
nated bedrock channel erosion exclusively. Figure 1 is
definetherangeof Rhcindicatedin Figure la. Note the log-
log scale.
drawn on the basis of streamprofile data fromthe Central
Rangeof Taiwan and servesto definethe range and limits of
applicabilityof the stream-powererosionlaw and henceour
analysis. In the Central Range of Taiwan and in the King nels (typically 80-90% of total relief; Table 1). Here we
Range of northern California [Merritts and Vincent, 1989; make the interpretation, following Montgomery and
Snyder and Whipple, 1998; N. Snyder et al., Landscape Foufoula-Georgiou [1993], that the break in scaling ob-
responseto tectonicforcing:DEM analysisof streamprofiles served
ata drainage
areaof 105- 106m2represents
thetransi-
in the Mendocino triple junction region, northernCalifornia, tion from debris flow-dominated "colluvial" channels to flu-
submittedto Geological Society of America Bulletin, 1999, vially dominatedbedrock channels,herein describedby a
hereinafterreferredto as Snyder et al., submittedmanuscript, criticaldownstreamdistance(Xc;seeFigure la). Field obser-
1999], the two best examplesof active, fluvially sculpted vations confirm that the bedrock-alluvial transition is near
mountainrangesfor which we have data, relief is dominated the rangefront in both Taiwan [Hovius et al., 1999] and the
by the elevation drop on fluvially dominatedbedrockchan- King Range [Snyder and Whipple, 1998; Snyderet al., sub-
 WHIPPLE AND TUCKER: DYNAMICS OF THE STREAM-POWER LAW 17,663

Table 1. Fluvial Relief Statisticsin Active Orogens

FieldArea CriticalDrainageArea AverageColluvialSlope % FluvialRelief Concavity SampleSize

A½,a 10•m2 & Ri/Rt N

King Range,California 0.59 + .20 0.54 + .11 79 + 7 0.40 + .10 14

(highuplift rate)
King Range,California 0.72 + .24 0.36 + .05 80 + 5 0.49 + .10 7
(low uplift rate)
CentralRange,Taiwan 1.40+ .48 0.63 + .26 89 + 6 0.41 + .10 4

All uncertainties
indicate1-sigmaerrorbars.
aAcdeftnedby breakin slope-area
scalingin longitudinal
profiledata(only).
bThe
condition
0 = m/nholds
if andonlyif channels
areinequilibrium
andbothUandK areconstants.
Reported
values
werefit to longprofiledatabetweenAcandthebedrock-alluvial
transitiononly.

mitted manuscript,1999]. In other landscapes,fluvially erosionlaws havebeenincorporated into modelingstudies

dominated bedrock channels constitute a considerably thathaveyieldedusefulinsightsaboutlandscape evolution,
smaller fraction of the total relief [e.g., Montgomery and we do not pursuetheir dynamics here. In the paragraphs
Foufoula-Georgiou, 1993; Sklar and Dietrich, 1998], but belowwe reviewthe derivationof (1) in orderto highlight
we believe the two examplesin Table 1 are typical of non- the underlyingassumptions and to emphasize relationsbe-
glacialtectonicallyactivemountainranges. tweeneffectiveparametersk, n, and m and physicalvariables
Sklar and Dietrich [1998] give a cautionarynoteon range suchasprocess (e.g.,pluckingversusabrasion), lithology,
of applicability of the stream-power erosion law that high- climate,sedimentloading,anddrainagebasinshape.Under-
lightssomeaspectsof the dynamicsof bedrockchannelsthat standingtheseprocess-parameter linkagesand how they
are beyondthe scopeof our analysis.We restrictour analysis influencemodel predictionsis essentialto critical evaluation
to the fluvially dominatedpartof thebedrockchannelsystem, and furtherrefinementof landscapeevolution simulationsas
specificallyavoidingdebrisflow-dominatedchanneltips. In well as to the formulation
of effectivefield and laboratory
addition, we do not addressthe retreatof large-scaleknick- research efforts in the area of bedrock channel erosion
pointsthatSeidl et el. [1996] arguedmaybe limited by rock processes.
mechanicsand weatheringratherthan fluvial erosion. Fur- Derivationof (1) startswith the reasonable
postulatethat
ther, we intentionallyrestrict our discussionto fluvial land- erosionrate is a power law functionof shearstressxb or,
scapes,makingno mentionof the role of glacialerosion. alternatively,
stream
powerperunit areaof channelbed(the
productof shearstressandmeanvelocity V, henceforth"unit
2. The Shear-Stress/Stream-Power streampower"). Thesebasicpostulatesarewritten:
Erosion Model

The detachment-limited rate of bedrock channel erosion e

Shear
stress œ= kbxb
a (2a)
is often modeledas a power law function of drainageareaA Unit
stream
power •;=kb('lJbV)
a (2b)
and streamgradientS:
Where kb and a are positive constants. Note that ko is a
E=KAmS
n (1) dimensionalconstantwith dimensionsthat dependon both
where m and n are positive constantsand K is a dimensional the exponenta and whether the shear-stressor unit stream-
coefficientof erosion(dimensionsof all variablesare listed in powerformulationis used(seenotation section).Both forms
the notation section). The drainage area term appearsas a of (2) implicitlyassume that the threshold(e.g.,criticalshear
proxyfor discharge.Howard and Kerby [1983] showedthat stress) is negligible for the flows of interest. An erosion
local erosion rates derived by differencingchannel profiles thresholdterm can easily be incorporatedinto numerical
resurveyedover a 7-year interval in rapidly eroding badlands solutionsand has someinterestingeffects[Howard, 1997;
were well explainedby a formulationof (1) assumingincision Tuckerand Slingerland,1997] but is omittedherein keep-
rateslinearly proportionalto bed shearstress(m -- 1/3 and n ingwith standard formulationof thestream-power law and in
-- 2/3). the interestof obtaining analyticalsolutions. Coefficientko
Sincethat time, variousformulationsof (1) have been used dependson rock massquality (lithology, jointing, and
extensivelyin modelingstudiesof bedrockprofile evolution weathering),sedimentloading,and process.Similarly,the
[ Seidl and Dietrich, 1992; Anderson, 1994; Howard, 1994; exponenta likely dependson the dominantprocessand has
Rosenbloom and Anderson, 1994; Seidl et al., 1994; beenarguedto rangefrom 1 to asmuchas7/2 [Hancocket al.,
Humphrey and Heller, 1995; Moglen and Bras, 1995a, b; 1998; K. Whipple, et el., River incision into bedrock:
Tucker and Slingerland, 1996, 1997; Sklar and Dietrich, Mechanicsand relative efficacyof plucking, abrasion,and
1998]. Other bedrock channelerosionlaws have been formu- cavitation, submitted to Geological Society of America
lated [e.g., Beaumont et al., 1992; Kooi and Beaumont, Bulletin,1999, hereinafter
referredto as Whipple et el., sub-
1994] but are less readily cast in termsof the physics of ob- mittedmanuscript,1999]. Thus,for therangeof erosionproc-
served erosion processes. Therefore,although these other essesadequatelydescribed by (2) the exponenta, in particu-
 17,664 WHIPPLE AND TUCKER: DYNAMICS OF THE STREAM-POWER LAW

lar, carries information about the physics of the erosion [Hack,1973],a valueconsistentwith approximately
loga-
process. rithmicchannelprofilesobservedin nature[Hack,1957].
Several researchershave recently argued that erosion rate However,besidesthe pioneering
workof Suzuki[1982],no
is a function of the ratio of sedimentflux qsto sedimenttrans- comprehensivestudy of the controls on bedrock channel
port capacity qc [Sklar and Dietrich, 1997; Slingerland et width has been done.
al., 1997; Sklar and Dietrich, 1998], which likely varies Combining(2) -(7), the bedrockerosionrate for shear-
with uplift rate, climate,and position in a catchment.A sim- stressdependenterosioncan be written as
ple way to denote this dependencewithin the frameworkof
the stream-powererosionlaw is to write
œ
=rA2aC(1-b)/3S2a/3
(8a)
kb=kef(qs,qc) (3) K= k•k•
2432•(1-o)/3f(q,)C?p•g2•/3
kq (8b)
where ke depends on rock massquality and erosion process
Comparing(8) with (1), it canbe seenthatexponents
rn and n
andj(qs,qc) is an unspecifiedfunction. As argued by Sklar
arerelatedto erosionprocess,hydraulicgeometry,and basin
and Dietrich [1998], the role of sedimentflux (here denoted
hydrologyaccordingto
asj(qs,qc))encapsulates
at leasttwo competing
effects:
(1) ac-
celerated erosion due to an increased number of tools in the
flowand(2) reduced
erosion
asa resultof partialshieldingof m=2ac(1-
b)/3 (9a)
thebedfrom particleimpactandotherprocesses. For the sake n= 2a/3 (9b)
of simplicity and in keepingwith the standardformulationof
the stream-power law we restrictour analysisto the condi- •C --
(9c)
tion of constantkb(for a given lithology), an assumption
incorporatedinto mostlandscapeevolutionmodels. Similarresultsarereadilyfoundfor the unit stream-power
Coupling either (2a) or (2b) and (3) with relations case
describingflow hydraulics,channelgeometry,and basin
hydrologyresultsin a simpleexpressionfor channelerosion K- kek•ak•(1-t•)f(qs)paga (lea)
ratein termsof streamgradientand drainagearea,in the form
of(1). In this analysis,hydrologicand hydraulic variables m=ac(1-b) (10b)
(dischargeQ, flow depthD, flow width W, flow velocity V, (10c)
shearstress'•b,and erosionrateœ)aretakenastime-averaged
quantities,suchthat dischargecanbe takenas a simplefunc- (led)
tion of drainageareaA. Thus it is implicitly assumedthat an
effectivedischargecan be definedthat adequatelyrepresents Thusthe shear-stress andunit stream-power versionsof the
the integratedeffectsof the full-time history of flood dis- erosionlaw differin detail butare notfundamentally
different.
charges [Wolman and Miller, 1960; Willgoose, 1989; Moreover,giventhat the exponenta in (2) is unknown,it
Tuckerand Bras, 1997]. Given this assumption, the internal wouldbe difficult at presentto discriminatebetweenthe unit
relationsare conservationof mass(water) stream-power and shear-stressmodels on the basis of field
data.

Q = VDW (4) Equations(8b) and (lea) emphasize

the multivariatecon-
conservation
of momentum(steadyanduniformflow) in wide trols on the effectivecoefficientof erosionK in (1) and
channels conveytherelativesensitivityof K to lithology,climate,and
sediment load. Within the broad subset of fluvial erosion
'c!•
=pgDS
- pC
fV2 (5) processesadequatelydescribedby (2) -(7), them/n ratio dis-
cussedby Seidl and Dietrich [1992], Moglen and Bras
hydraulicgeometry
[1995b],Dietrichet al. [1996], Tucker[1996], andothersis
W= Ic•Q• (6)
shownby (9) and (le) to be independentof the dominant
and a relationfor basinhydrology
erosionprocess(e.g.,pluckingversusabrasion),depending
onlyontherelativeratesof increase
of discharge
with drain-
Q = kqAc (7)ageareaand of channelwidth with dischargeregardless of
whetheroneaccepts a shear-stress
or unit stream-power for-
In the above,p is densityof water,g is gravitationalaccelera- mulation.Fortypicalvaluesoftheexponents in (6) and(7)
tion,CSis a dimensionless frictionfactor,k• andkqaredimen- (0.7_<c _<1 andb -- 0.4 - 0.6)them/nratio is predicted to fall
sionalconstants,and b andc are positivedimensionless con- into a narrowrangenear0.5 (0.35 < m/n < e.6), consistent
stants.For convenience, the small-angleapproximation (sintz with empiricalvaluesderivedfi'omfield data [Howardand
-- tantz)hasbeenexploitedto write shearstressin termsof the Kerby,1983]andmanyderivedfi'ommapdatarelatingchan-
streamwise gradientS. Constantskwandb dependon rock nel gradientand drainagearea(seeTable 1) [Flint, 1974;
massquality,erosionprocess, sedimentloading,andhydrau- Tarbotonet al., 1989; Willgooseet al., 1990; Tarboton et
lic resistance
Cs. Constants kqandc area functionof climate, al., 1991; Willgoose, 1994; Moglen and Bras, 1995b;
runoffprocesses, the returnperiodof the effectivedischarge, Slingerlandet al., 1998]. Althoughthe m/n ratio is known
and basin topology. to stronglyinfluencethe concavityof equilibriumchannels
Equation(6) is well known empiricallyfor alluvial chan- [e.g.,Moglen and Bras, 1995a],manyadditionalfactorscan
nels fi'omthe hydraulic geometryliterature[Leopoldand affectprofile concavity. Thus the restrictionthat the m/n
Maddock,1953] (b-- 1/2). Similarvaluesof the exponentb ratioshould fall in a narrowrangedoesnotnecessarily imply
appear to apply to partially alluviated bedrock channels thatchannelconcavities arelikewiserestricted.Indeed,em-
 WHIPPLE AND TUCKER: DYNAMICS OF THE STREAM-POWER LAW 17,665

pirical estimatesof channel concavity (often equated to the z x U tU o

m/n ratio) outside the expectedrangehave been reported in z,=--H x,---L U,= U o t,=• H (15)
somelandscapes[e.g.,Sklar and Dietrich, 1998] and proba-
bly reflect some combination of disequilibrium conditions, Uo=U'(x,t) (16)
systematicdownstreamvariationsin either rock uplift rate or whereH and L are the representativevertical and horizontal
erodibility K, or regressionof data that cross the bedrock- lengthscales,respectively,and an asteriskis used to denote
alluvial transition. all dimensionless variables. The choice of H/Uo as a charac-
teristic timescale is convenient and assures that the dimen-

3. Nondimensionalization sionless rate of bed elevation change(dz,/dt,) is of order

unity:
It is usefulto expressthe governingequationfor the evo- dz, 1 dz
lution of bedrock channel profiles in termsof dimensionless dt,= U'•'•' (17)
variables. This allows a preliminary scaling analysis of the
evolutionequationand simplifiesanalysisof the dynamicsof The length scaleH need not be equal to the bedrockfluvial
river profile responseto external forcings (e.g., tectonics or relief Rf, thoughthis makesa convenientchoiceif known.
climate). Willgoose et al. [1991] give a nondimensionaliza- The fourth and final dimensionlessgroup on the right-
tion schemefor their transport-limitedequation set, which is hand side must involve the variables U, K, and k,, and can be
similar in spirit to the one presentedhere. In addition, determinedreadily by rewriting (13) in terms of the dimen-
Fernandes and Dietrich [1997] present a similar dimen- sionless variables defined above:
sional analysisof equationsdescribinghillslope evolution
by diffusiveprocesses.
For a detachment-limitedsystem(i.e., bedrock channels),
dz,
=S,_NE-lx,
hml
dz,
)n
conservationof mass(rock) dictatesthe form of the channel
where the dimensionlessuplift-erosionnumberN• is given
profile evolutionequation:
by

dt Uok•-mLn-hmH-n (19)
N/r=-•-
where z is the elevation of the river bed, x is the distance
downstream,and U is the rock uplift rate definedrelative to Note thatby definition,if the rockuplift rate U is steadyand
the erosionalbaselevel. Combining(11) with (1) and em- uniform,the dimensionless uplift rate U, is unity (equations
ployingHack's law [Hack, 1957], (15) and (16)).
The uplift-erosionnumbercanbe immediatelyidentifiedas
A= k•x• (12) the critical dimensionlessgroup governingthe dynamicsof
the bedrockchannelprofile evolution equation(18). More-
wherek, is a dimensionalconstantand h is the reciprocalof over, as with the familiar Reynolds and Froude numbersin
the Hack exponent,showsthatriver profilesare governedby fluid mechanics,dynamicresponses associated
with perturba-
a nonlinearkinematicwave equation[Whitham, 1974]: tions of the suite of variables Uo, K, ka,L, m, and H can be
fully capturedby simply consideringresponsesto perturba-
dt -(x,O
-amxhmn-'l-
I Xc
-<
X-< r tions in the uplift-erosionnumberNE.For instance,changes
in the rock uplift relative to baselevel U are dynamically
equivalentto changesin the coefficientof erosionK. In addi-
withwavespeed-Kk,mXhmS
•4, whereS = I(az/ax)l,
L is the tion, covarianceof empirically determinedK values and the
bedrockchannelstreamlength measuredfromthe divide, and
exponentm (the dimensionsof K depend on m) [Sklar and
the area-length exponent h is seen to vary over a narrow
Dietrich, 1998; Stock and Montgomery, 1999] does not
rangefi'om1.67 to 1.92 [Hack, 1957; Marltan et al., 1996;
complicatethe dynamic behavior of the profile evolution
Rigon et al., 1996].
equation as this effectis encapsulatedwithin the uplift-
Nondimensionalizationof the bedrock river profile evolu-
erosion number. Consideration of steady state conditions
tion equation first requiresthat we write (13) in general
terms:
will reveal the roles of the exponentsh, m, and n in the form
and dynamicsof modeledriver profiles.

z F(U,K,k
=
a,x,dx
dz' ) • t (14)
4. Steady State River Profiles
The right-handsideof (14) hassix independent variablesin In this section we explore the behavior of bedrock chan-
two dimensions(length and time), which thereforecan be nels as predictedby the shear-stress/stream-power model in
writtenasfour independentnondimensional groups. Note orderto draw out the significanceof the issuesoutlined ear-
thattheexponents h, m, and n do not appearas variableson lier in regard to channel profile form,the relationship be-
the right-handsideasthesearepartof the unspecified func- tween equilibrium channel gradient and environmental
tion F. Similarly,the variableXcintroduced earlier(seeFig- controls(climate,lithology,and uplift rate),and the equilib-
ure 1) doesnotappearasthisentersonly as a boundarycon- rium height of mountainranges. In the analysisthese envi-
ditionto theunspecified functionF. In orderto proceedwith ronmentalcontrolsare all representedby the uplift-erosion
thenondimensionalization we introducethreerepresentative numberNE introducedin section3, which can be quantita-
scales(H, L, and Uo)to definethe following dimensionless tively interpreted as either reflecting tectonic forcing
variables:
(throughUo)or climaticandlithologicforcing(throughK).
17,666 WHIPPLE AND TUCKER: DYNAMICS OF THE STREAM-POWER LAW

4.1. Equilibrium ChannelGradient Table 2. ParameterValuesUsedin Examples

At steadystatetheriverprofileis by definitioninvariant
in time(dz/dt= 0 anddz,/dt,= 0). Localerosionrateœmust Parametera Value Dimensions Notesb
everywherebalancetherockupliftrate. Settingdz,/dt,equal
to zeroandsolvingfordimensionless stream
gradientS,, the œ 2],500 ILl ---
steadystatesolutionof (18) isreadilyobtained: H 2,900 [L] ---
xc 320 [L] 105m2source
area
s,= (dZ*)NEl/ns,l/nx,-hm/n
= (20)
k,
h
m/n
6.69
1.67
0.50
[L2'h]
[]
[]
Hack[1957]
Hack [1957]
---
Equation(20) showsthat the slopeexponentn largely
dictatesthe sensitivityof streamgradientto changes
in rock K (n= 1) 2.00x 10-s [m]'2m
yr'•] SS
upliftrate,lithology,
andclimate(Figure2). Equilibrium
flu- K (n= 2/3) 9.28x 10's [m•'2m
yr'•] SS
vial relief will be shownlater (equation (22)) to scalepre- K (n= 2) 2.00x 10-s [m•'2•
yr'•] SS
ciselywith thestreamgradientandtherefore
is includedin
Figure2 forconvenience.Becauseonlythesensitivity
ofthe aAllin mksunits,exceptK asnoted.
equilibriumgradient
to differences
in climate
orupliftrateis bSSdenotes
Kschosen
toyieldequivalent
steady
state
profiles
of interesthere,channelgradientsare reportedrelativeto a for U = 2 mm yr-1 for all n (for convenience
only).
referenceconditionNErcalculatedusingthe parameters listed
in Table 2, reportedherefor completeness,
althoughthe
This convenientarti- ear erosionprocess(a = 1, equation(2) and n = 2/3, equation
actualvaluesusedare inconsequential.
fice is usedthroughoutthe paperto normalizeillustrative (9)), equilibrium channel gradient is very sensitive to
plotswithno lossofgenerality.Fortherestricted
caseof changesin the uplift-erosionnumber. For a slightly nonlin-
uniformblock uplift (U is constant),uniformcoefficient
of ear erosionprocess(n = 1, equation(9)), equilibriumchannel
erosion(K is constant),andno downstream changesin ero- gradient is linearly related to the uplift-erosion number.
sionprocess (n andm areconstant),thedimensionless uplift Finally,for a highly nonlinear process(n > 1, equation(9)),
rateU, is unity,andthe uplift-erosionnumbercaptures all equilibriumchannelgradientis only weakly dependenton
thedependencies of channelgradientontherockupliftrate, the uplift-erosionnumber(Figure2).
lithology, and climate. Thus landscaperesponseto tectonic regimeis critically
Asexemplified
by (19) and(20), theequilibrium
gradient dependent on the slopeexponentn. The directdependence of
of a bedrock channel reflects a balance between the rate of the slope exponentn on the physics of fluvial bedrockero-
rockupliftU andtherateof channelincisionperunit slope sion (equations(2), (9), and (10)) is powerful testimonyto
and areaK. Importantly,the steepness of a river profile the need for field studies of these processes. Moreover, the
depends ontheuplift-erosion
number raisedto a powergiven m/n ratio plays no direct role in the sensitivity of channel
bythereciprocal oftheslopeexponent(l/n) (Figure2). The gradientandrelief to the uplift-erosionnumberNE.
significance
of thisfundamental
predictionof(1) hasnotyet
beenwidelyappreciated [TuckerandBras,1998]. Fora lin- 4.2. Equilibrium Longitudinal River Profiles
4.2.1. River profile concavity. Assuming simple block
uplift (dU/dx = 0) and uniformlithology, precipitation,and
'- 100 / 100 erosionprocess(dK/dx= 0; m and n are constants),(18) can
readily be integratedto derive an expressionfor dimension-
..... n : 2/3 / less streambed elevation z, as a function of dimensionless
distance downstream x,:
• 10 n=l / 10
o n=2 / ...

--½ 1 (21a)
E 0.1 o.1
::] / / ill:
ß-
ß
__-- /
/
z,(x,)-z,(1)-NE1/nu,1/nln(x,)
hm-1(2lb)
'-•
O'
0 ' 01 ' ' ' •'"1 ........ ', ........ ', ....... 0.01
where L is total bedrockstreamlength and z,(1) is the dimen-
0.01 0.1 1 10 100
sionless elevation at the basin outlet (or at the bedrock-
Uplift/Erosion Ratio (N E/[N E] r ) alluvial transition). Equations (21a) and (2lb) arevalid for
X,c _<x, _<1 only, where X,c is the dimensionlessdistance
Figure 2. Sensitivityof dimensionless equilibriumchannel downstream fi'om the divide at which fluvial processes
gradientand dimensionless equilibriumfluvial relief to the become dominant [Montgomery and Foufoula-Georgiou,
uplift-erosion
numberNE as a functionof the slopeexponent
1993] (See Figure 1).
n. In orderto emphasizethe sensitivity to changesin the
uplift-erosionnumberboth dimensionless channelgradient Although calculationsusing the restrictive assumptions
and dimensionless fluvial relief are shown relative to a refer- incorporatedinto (2 la) and (2 lb) are illustrative (Figure3),
encevalue notedwith the subscriptr. Referencevalues in all we stress that nonuniform uplift rates [i.e., Adams, 1985;
figuresare computedwith the parameters
listed in Table 2. Koons, 1989], orographic precipitation [Beaumont et al.,
Note the log-log scale. 1992; Masek et al., 1994b], and systematic downstream
 WHIPPLE AND TUCKER: DYNAMICS OF THE STREAM-POWER LAW 17,667

1
variations in sediment loading in streams [Sklar and
Dietrich, 1997; Slingerland et al., 1997; Sklar and Dietrich,
1998] all play potentially important roles in even the sim-
plest realistic scenario. For steady,uniform uplift, constant
• '"•0.6 =
coefficientof erosion,and constantareaand slope exponents
(m and n) the ratio hm/n dictatesthe equilibrium formof river
o., profiles(Figure 3a). For typical valuesof h (1.67 < h < 1.92)
and the rn/n ratio (0.35 < rn/n < 0.6) the ratio hrn/n ranges
fi'om0.58 to 1,15, and predicted river profiles are approxi-
• 0.2
Z mately logarithmic,as documentedby Hack [1957] and illus-
trated in Figure 3b.
0
Althoughthe steadystateform (concavity)of river profiles
0 0.2 0.4 0.6 0.8 1
subjectedto the constraintsoutlined above is predictedto be
relatively insensitiveto the multivariate controls on bedrock
DimensionlessDistance (x I L )
erosion processes,any deviations fi'omsteady state, or any
systematicdownstreamchangesin uplift rate (e.g., tilting) or
3 1\.. erodibility (e.g., change in dominant process, sediment
supply, and cover)will complicatethe interpretation of river
•"•..... ß...... schematic
ridge profile data in terms of the m/n ratio. For instance, where
•' I!\. --- '-.. / profile uplift rate increasesmonotonically downstream(dU/dx > 0;
• 2 ul-:"•
• ..... \ '-,•-'---_1•
I', %
.... x •,R hc --",. back-tilting), profile concavity will be diminished and vice
versawhere uplift rate decreasesdownstream(dU/dx < 0). In
addition, spatially variable controls on erodibility (K)may
play an important role in channel profile form [Sklar and
o
Dietrich, 1998]. Because of such difficulties, Seidl and
0x c 5 10 15 20 25 Dietrich [1992] proposeda methodfor extractingrn/n ratios
from differencesof channelandtributarygradientsat tributary
Distance from Divide (km) junctions. Although their analysis did not accountfor pos-
sible differences in alluvial cover or channel width between
tributary channels, their method requires no assumptions
4
regarding steady state or equilibrium conditions. However,
their finding of m/n = 1 for streams in the Oregon Coast
• n =1
Range is at odds with other data (i.e., reasonable values for
mm' 3
exponentsin (6) and (7) and logarithmic channel profiles)
and hasnot yet been explained.
4.2.2. Equilibrium fluvial relief and the height of
mountain ranges. Over long timescalesthe height of moun-
• 'g 1 tain rangesis limited by either crustal strength[e.g.,Molnar
and Lyon-Caen, 1988; Bird, 1991; Masek et al., 1994a] or by
a balancebetweenrock uplift and erosion[e.g.,Adams, 1985;
0
Koons, 1989], whichever is more restrictive. In the case
0 25 50 75 100
where crustalstrengthis not limiting, the equilibrium height
c Range Half Width (L) (km) of a fluvially sculptedmountainrangeis dictatedby four fun-
dmental geomorphiccontrols: (1) rangewidth, (2) longitu-
Figure3. Equilibriumchannelprofilesand fluvial relief dinal profiles of transversebedrock streams,(3) the length
(assuming spatiallyconstantU, K, m, andn). (a) Longitudi- and gradientof colluvial channelsabovethe fluvial network,
nalprofileconcavity is controlledbythehm/nratio(concav- and (4)the length (= drainagedensity) and gradient of hill-
ity index). Exponenth is held constantat the observedvalue slopes(see Figure 1).
of 1.67 [Hack, 1957]. Natural channelsare approximately Equilibrium fluvial bedrockchannelreliefRf is given by
logarithmic in form: consistentwith h = 1.67- 1.92 and m/n = the difference between the elevation at the headwater of the
1/2. (b) At steadystatethe topographic envelopeof moun-
tain rangesis setby the longitudinalprofilesof the stream fluvial channel(i.e., at x = Xc)and the elevation of the basin
outlet or the bedrock-alluvial channel transition (i.e., at x =
networkplus the relief on hillslopes and colluvial channels
(Rhc).A theoreticalprofile,in dimensional form(trunk chan- L). In termsof dimensionless
variables,from(21), fluvial bed-
nel' solid black line; tributary channels(projectedinto rock channelrelief is given by
plane)'dashed blacklines),computed with U = 5xl 0-3ma-], K
= 1.2x10
-s,andn= 1(other
divide-to-outlet
parameters
longitudinal
aslistedin Table2) is
shown(forXc_(x _(L) for directcomparison againstdatafor a
profile(trunkchannelonly) for
g,f
:NE1/ns,1/nll-•!-l(1
-X,c
1-hm/n) hrn
an intermediate-sized basinin northernTaiwan (Table 1). ½1
(22a)
Stairstepsin theprofilefromTaiwanprobablyreflectnoisein n

the digital elevation data. (c)Dimensionless fluvial relief

increasesslowly with rangehalf width L for all values of n
anduplift-erosionnumberNE(shownfor n = 1). R,f=-NE1/nu,
1In
lnx,
c hm
=1 n
(22b)
17,668 WHIPPLE AND TUCKER: DYNAMICS OF THE STREAM-POWER LAW

Unsurprisingly, equilibrium fluvial relief varies in exactly in the responseof hillslopes to channel incision [e.g.,
the sameway with the uplift-erosion numberNE as does the Fernandes and Dietrich, 1997], nor any feedbacksdue to
equilibrium channel gradient (see Figure 2). Thus the slope eitherinteractionbetweenchannelsand hillslopes [Schumm
exponentn in (1) emergesonce again as a critical unknown. and Parker, 1973; Schumm,1979] or coupling with down-
Recall that in the caseof uniform rock uplift the dimension- streamdepositional systems [e.g., Humphrey and Heller,
less uplift rate U, is unity, and the uplift-erosion number 1995].
fully capturesthe dependenceof fluvial relief on environ- Before deriving results for a dimensionlessresponse
mental controls(lithology, climate, tectonics,and basin size). timescaleT, it is necessaryto return to the dimensional
Thus fluvial relief depends on rock uplift rate to the 1/n analysispresentedearlier. We arguedthat the representative
power,the coefficientof erosionto the -1/n power, and stream rock uplift rate Uo was the appropriatescaling term for the
length to a small power (-0.15 - 0.42 for typical values of rate of changeof channelbed elevation(dz/dt). However, we
hm/n), as illustratedin Figures 2 and 3c. In plotting Figure could equally well have introduceda characteristictimescale
3c it is assumedthat the largest transversedrainages(length T such that
L) scalewith rangehalf width.
Equation (22) makesthe direct prediction that all else
being equal, greaterrelief is expectedfor lower values of the
'•7=-•(dt,• =Uødt, (23a)
coefficientof erosionK. That is, greaterrelief is expectedfor where

more resistant lithologies and lower precipitation rates. H

r •
Although this result is not surprising,it runs counterto fre- Uo (23b)
quent argumentsin the literature that greaterprecipitation
Thus a consistentdimensionlessresponsetimescalecan be
leadsto greaterrelief [e.g.,Fielding et at., 1994; Masek et at., defined as
1994b].
Theoretical mainstreamand tributary profiles calculated
with (21a) are plotted in Figure 3b in dimensional form for T*=UøTH (24)
direct comparisonwith data froma channel typical of those
draining the northern Central Range of Taiwan. In this and 5.1. Sudden Base Level Fall
other tectonically active, fluvially sculpted landscapes(see
Table 1 and Figure 1)the total relief is dominatedby the ele- The dimensionlessresponsetimescalefor a sudden,finite
vation drop of bedrockchannels,and equilibrium range crest baselevelfall T,bcanbe readilyderivedfromthe profileevo-
elevation can be expectedto vary strongly with the fluvial lution equation (18), which, as first recognizedby
relief. Moreover, Schmidt and Montgomery [1995] have Rosenbloomand Anderson [1994], has the form of a non-
argued that hillslope relief rapidly attains a maximumin linearkinematicwaveequation. The kinematicwavespeedis
actively incising landscapes. Where this condition holds, the upstreamrate of knickpoint migrationand governsthe
the relationship between uplift rate, climate,and range crest rate at which changesin boundaryconditionscan be commu-
elevation above base level can be describedto first order by nicatedacrossa landscape. From (13) the kinematicwave
(22). However, drainage density may decreasewith rock speed,in general,varieswith both drainageareaand stream
gradient:
uplift rate [e.g., Howard, 1997; Tucker and Bras, 1998],
resulting in longer hillslopes and possibly greaterhillslope
relief (and proportionately reduced fluvial relief). In addi-
re---gka
mXhmSn-1 (25)
tion, little is at presentknown about the controls on either Equation(25) can be written in the equivalentdimensionless
the length (representedby Xc)or gradient of debris flow- form(seeequation(18)):
dominated channels [Howard, 1998]. Thus, although equi-
librium range crest elevation in fluvially dominated land-
lOOO
scapescan be describedto first orderby (22), this relation-
ship strictly relatesonly to the fluvial relief.
A 800
E
5. Transient Response to Tectonic Forcing
c: 600 stateprofile
Thus far we have considered only equilibrium (steady .o_
state) channelprofiles. Here we addressthe questions: (1)
under what conditions can we expect river profiles to be in
equilibrium with the imposed tectonic, lithologic, and
! 400]. ••_
l

U.I ,':•
!

fo(U,K,x,n)
sudden
climatic setting?, and (2) over what timescaleswill a river 200/"
|',
.... ""••..b.a.s.e
knickpoint ......
level
drop
systemreturn to equilibrium following a changein tectonic 0/; I I I
or climatic conditions? We focus our discussion on tectonic
0x c 1000 2000 3000 L 4000
perturbations, but the analysis is not significantly different
for suddenchangesin climate. Distance (m)
We considertwo typesof tectonic perturbation away from
Figure4. Schematic
illustrationof knickpointmigrationin
a base steadystate:(1)a single, suddenfall in base level and
responseto a sudden base level fall (coseismic,eustatic,
(2) a step function increase in uplift rate. In all cases we streamcapture,etc.)on a channelotherwisein a steadystate
assumethat K, U, m, and n do not vary along the river profile. condition.KnickpointmigrationspeedC, is givenby (26).
We do not consider any time lags associated with an Total knickpoint propagationtime fromL to x½definesthe
isostatic responseto denudationalunloading, any time lags systemresponsetime.
 WHIPPLE AND TUCKER: DYNAMICS OF THE STREAM-POWER LAW 17,669

100.00

Ce,
=_NE-lx,
hm(
dg*)ln-1
•,•,-, (26)
Thus(18) dictatesthat to firstordera suddendrop in base
level will producea knickpointthat propagatesupstreamat
• I...lO.OO n=i/3
n --

o E
aneverdecreasing ratefromthepointof disturbance,without '• i: •.oo
attenuation (Figure4). However,thedependence of the wave
speed onchannel gradientforn ½1 maycausechanges to the E =
shapeof the kinematicwave, possiblyproducingshock i5o o.o
wavesthat alter the rate of translation [Whitham, 1974]. For
this reasonthe derivationpresentedhereshouldbe consid-
eredpreliminaryin natureand only valid for infinitesimal, 0.01

stepfunctionperturbations.However,the derivationpre- 1.00E-01 1.00E+00 1.00 E+01

sented in the next section for the responseto a sudden
increasein uplift rate is not subjectto this limitationand Normalized Uplift-Erosion Number
yieldsa similarexpression forthe response time,suggesting
that the kinematic wave solution is, indeed, robust.
Bothdrainage areaandstreamgradientvary with position
alongthestream,so in the mostgeneralcaseit is difficultto 10.00
compute the response time,whichequatesto the integrated ....... n = 2/3
timerequired to carrya signalfromthebasinoutlet(x, - 1) to •n=l
theupperlimit of fluvialchannels(x - X,c). However,forthe .... n--2 ooO"
restrictedcaseof a smallperturbationaway fioom steadystate o
o

the form of dz,/dx, is known. Substituting (20) for the equi-

librium dimensionlesschannel gradient, the dimensionless 1.oo
kinematicwave speed(26) can be written as a functionof
dimensionless distance downstream: o
o
o
o
o

(Ce*
)s
s=-NE
-1/nU*l-1/nx*hm/n
(27)
To a first approximation,
dimensionless response
time T,b 0.10

is foundby integrating thetransittime of the kinematicwave 1.00E-01 1.00E+00 1.00E+01

alongthelengthof thestream(1 > x, > X,c):
Normalized Uplift Rate (U I[U ]r )
ßX* c

Figure 5. Sensitivityof responsetime to a suddenbase level

fall (T,•,;assuming
spatiallyconstant
U, K, m, andn). Model
1
parametersusedto determinereferenceuplift-erosionnumber
NL- and T,b are reported in Table 2. (a) Dimensionless
responsetime increasesmonotonicallywith uplift-erosion
number NE,withgreatersensitivityfor lowerslopeexponent
hm n. Note the log-log scale. (b) Responsetime Tbvariably
½1 (28)
n
increases,remainsconstant,or decreases with uplift rate U
dependingon whetherthe slope exponentn is less than,
T,b = _NE1/nU,1/n-1
lnx,c •hm
=1 n
(28c) equalto, or greaterthanunity. Note the log-logscale.

Equation (28) shows that dimensionless response time

increasesmonotonically with the uplift-erosion number NE
for all n (Figure 5a). However, it is important to recognize responsetime increasesrapidly with uplift rate for n < 1, is
that although dimensionlessrock uplift rate U, is unity for independent of uplift rate for n = 1, and decreasesrapidly
the uniform uplift scenario considered here, response time with uplift rate for n > 1 (Figure 5b). The reasonfor the sensi-
does in fact vary differently with rock uplift rate U and the tivity of the relationship between responsetime and uplift
coefficientof erosion K, as indicated by the different expo- rate is clear fi'om(26): for n < 1 the kinematic wave speed is
nentsin the uplift-erosion numberNe and the dimensionless inversely related to channel gradient; for n = 1 the wave
rock uplift U, terms: speed is independent of gradient; and for n > 1 the wave
speed increases with gradient. Finally, response time is
Tb
=Uo
HT*b
• mNœ1/nWo
-1w*l/n-1
• mK-1/nwl/n-1
(29) shownto be only weakly a function of basin size (the expo-
nent on streamlength (1 - hrn/n) typically ranges from -0.15
Responsetime is plottedin dimensionalform as a function to 0.42). This latter finding follows because downstream
of rock uplift rate in Figure 5b to emphasizethis differencein channelsegmentswith large drainage areasrespond quickly;
scaling. Becauseonly the sensitivity of predicted response the time requiredfor upstreamheadwaterchannel segmentsto
time to differencesin uplift rate is of interest here, response adjusteffectively limits the responsetime of the entire basin.
times are reported relative to a referencecondition Ur calcu- Thus tectonic disequilibrium in the landscapeis most likely
lated using the parameterslisted in Table 2. Interestingly, recorded in small headwater catchmentsand on hillslopes
 17,670 WHIPPLE AND TUCKER: DYNAMICS OF THE STREAM-POWER LAW

that have not yet respondedto rapid adjustmentsof the chan- Since the upper reach of the fluvial channel does not
nel system. Further, a field-testableprediction of this model respondto the changein uplift rate until the wave of erosion
is the location of knickpoints correlatedwith known (co- reachesthispoint,from(11) thetimerateof changeof the ele-
seismicor eustatic) base level drops [e.g., see Rosenbloom vationof the fluvial channeltips Z(Xc)is given by the differ-
and Anderson, 1994]. encebetweenthenewlyimposed
uplift rate Uf andthe previ-
ouslyestablishederosionrate (ci(Xc)):
5.2. Sudden Increase in Uplift Rate
Responsetime for a step function increasein uplift rate Tt•
is derivedin a differentmanner. Application of the kinematic Rearrangingand noting that for perturbationaway froman
wave solution is not straightforwardin this casebecausethis initialsteadystatetheerosionrateat the fluvialchanneltips
problementails a significantdeviation fromthe base steady is equalto the initial uplift rate(c•(Xc)=UO, we obtain the
state and becausethe channel is subjectedto a sustainedrate simple result:
of base level fall, rather than a discrete perturbation. In fact,
the derivation presentedhere is more robust than the kine-
matic wave solution above as it is not subject to any uncer-
tainties related to knickpoint shape evolution during
Tu= Uf - Ui (32)
upstreammigration. Using(32), the systemresponse
timescale
canbe directly
An increasein uplift rate initiates a wave of erosion that estimatedfrom field observations, providedcertainrestrictive
propagatesupstream(Figure 6). Equilibrium within the flu-
conditionsare met: (1) adjacentterranesof similarclimateand
vial system is reached when the wave of erosion reachesthe lithologyare experiencing different,
knownrockuplift rates,
fluvial channeltips (x = Xc),as argued above. We reasonthat and(2) channelprofilesappearto haveadjustedto imposed
arrival of this migratingknickzoneat the fluvial channeltips uplift rates;profileshavesmoothlogarithmicformswith no
coincides with the time that they attain their final elevation indicationof an active,propagating knickzone[Snyderand
(ZAXc)).In other words, the migratingknickzonedefinesthe Whipple,1998; Snyderet al., submittedmanuscript, 1999].
Theseconditionswill rarelybe met, however,and it is useful
boundary between a downstreamsegmentof the profile that
has achieved its final equilibrium gradient and an upstream to write (32) in termsof nondimensionalrock uplift rate,
segmentthat has not yet steepenedin responseto the change systemlength,and rock erosionparameters,
using(24) for
dimensionless timescale:
in rock uplift rate. Barring the effectsof numericaldiffusion,
this is precisely the behavior observed in numerical solu-
tions of the profile evolution equation (18). Thus response
time is set by the time required to elevate the fluvial channel U,f(]_$) (33a)
tips from an initial equilibrium position (zi(xc)) to a final
equilibrium position (Figure 6): Si
•f=f (33b)
(xc)- + (30)whereUo/ischosenasthereferencerockupliftrate.
Using (21) to determine
z,i(X,c)andz,•(X,c),substituting
into(33), andrearranging
givesan algebraicrelationshipfor
dimensionless
responsetimescaleT,t•:
4

1-hrn/
n
E

•
zt(x
c)'••final
steady
state
profiledz/dt =Uf '•i =Uf -Ui
E ta,f
o

n
• 1 (34a)
• of erosion

z,(xc)
•,.j .•..•..œ,• -"ßrE1/nrr
=
l/n-1
•'*f c(1_fl/n)--=
lnx, hm1 (34b)
T*u (l-f) n
0 5 10 15 20 25 Note that for the case of uniform rock uplift rate treated here
the dimensionlessuplift rate term U,/is unity. As in the
Distance from Divide (kin)
derivation of Tb,we have assumedblock uplift (dU/dx = O)
Figure6. Schematic illustrationof the transientresponse andspatiallyuniform coefficientof erosion and erosion proc-
(solidline)of anequilibrium channel(shaded line)to a sud- ess(dK/dx = 0; n and m are constants).For Ui = 0 the initial
den increasein rock uplift rate used in the derivation of
condition is a horizontal plane, and the erosion rate at the
response timescaleTs. Rockuplffi rates•e indicatedby fluvial channel headwater is zero until the wave of erosion
solid arrows;headwatererosionrates•e indicatedby
shadedarrows.An increasein uplffirateinstigates has swept throughthe entire fluvial system. In this casethe
a waveof
erosion that propagates upstream. Channel headwater responsetime is simply the quotientof the equilibrium eleva-
reaches•e uplffiedbut do not responduntil the wave of ero- tion above the base level of the channel at x = Xc and the
sionreaches them. Accordingly,z(xO increasesat a constant uplift rate Uo/. Interestingly, (34) reducesto the kinematic
rate (dz/dt = U•- UO until a new equilibriumis reached wave solution (equation(28)) in this case (Ui = 0; f= 0). In
(dashedshadedline). otherwords, it takesas long for a discretebaselevel fall to be
 WHIPPLE AND TUCKER: DYNAMICS OF THE STREAM-POWER LAW 17,671

Fixed Final Uplift Rate (U f ) than with the coefficientof erosionK. For this reason,
responsetime is plotted in dimensionalformas a function of
10
E ....... n = 2/3 therockupliftraterelativeto a reference
condition
(Figure7).
Additionally, the appearanceof both initial and final rock
•n=l
uplift rates in both the numerator and the denominator
.... n=2
(throughfand NE) indicatesthat there is an additional level
of complexityin the dependence of responsetime on rock
uplift rates. Accordingly,Figure7a is plotted to illustrate
therelationship
between
response
timeandfinalupliftrateUf
andFigure7b illustratesthe relationship
betweenresponse
o time andinitial uplift rate Ui.
Somenonintuitiveeffects
arerevealed
in the relationship
0.1 i i i i i iiI I i i i i i i •1 betweenthe magnitudeof the changein uplift rate and
0.01 0.1 responsetime (Figure7). As seenin the caseof a suddenbase
levelfall,forn = 1, responsetimeis independent of both the
a Uplift Rate Ratio (f =U / IU f ) finalupliftrateUf andthemagnitude of thechange in uplift
rate(Figures 7a and7b). Interestingly,
forthecasen > 1, sys-
temresponse time decreasesfor smallerchanges in uplift rate
Fixed Initial Uplift Rate (U • ) (i.e.,asUdUfapproaches 1)whereUfisheldconstant (Figure
lO 7a) but actuallyincreases for smallerchanges in uplift rate
2/3 whereU• is held constant(Figure7b). Conversely, for the
•n=l
caseof n < 1, response timeincreases forsmallerchanges in
upliftratewhereUf is heldconstant(equation(34); Figure
7a) anddecreases for smallerchangesin uplift ratewhereU• is
held constant(equation(34); Figure 7b). This yields the
_
-
seeminglyodd result that for n < 1 it takes significantly
longerto adjustto a minor changein uplift rate than it does
to raisetheentirerangestartingfrom a horizontalplane. The
reasonfor thisis twofold:(1) the relationshipbetweenuplift
rate and equilibrium channel headwater elevation is non-
0.1 i i i , i i ii I i i i i i i i i linearin the slopeexponentn (equation(21)), and(2) the rate
0.01 0.1
at which the channelheadwateris elevateddependson the
initial slopeat x = Xc.
Uplift Rate Ratio (f =U / I U f ) Simplifying (34) and writing it in dimensionalform, we
seethat the changein channelheadwaterelevationrequired
Figure 7. Timescaleof responseto a step function increasein with an increasein uplift rate scalesas
uplift rate (Tu; assumingspatially constant U, K, m, and n).
Referencevaluesof rock uplift rate Ur and the responsetime-
scaleTurusedto normalizethe plots are given by or derived
fromparametervalues reported in Table 2. Note that if the while the rate of changeof channelheadwaterelevation
initial uplift rate is zero (Ui = 0), the responsetime is given scales
withthedifference
between
finalandinitial rockuplift
by the base level fall solution (Figure 5). (a) Sensitivity of rates'
responsetime to the magnitudeof the increasein uplift rate
(UdUf) for the casewherethe final uplift rate Uf is held con-
stant. Responsetime increaseswith U• (increasingUdUf) for dt oc
Uf - Ui (36)
n < 1, is independentof Ui for n = 1, and decreaseswith U• for
n > 1. (b) Sensitivityof responsetime to the magnitudeof the This findinghas importantimplicationsfor the differing
increasein uplift rate (UdUf) for the casewhere the initial dynamic responseof landscapes (both real and simulated)
uplift rate U• is held ccnstant. Responsetime increaseswith etchedby differentsetsof erosional
processes (e.g.,abrasion
Uf (decreasingUdUf)for n < 1, is independent of Uf for n =1, by suspended load,abrasion by saltationload,andpluck-
anddecreases with Uf for n > 1. ing),throughtheircontrolof theslopeexponent n.

6. Conclusions: Research Needs

and Approaches
translatedthe lengthof a channelsystemat equilibriumwith
the prevailinguplift rateas it wouldto uplift the entirerange Reviewof the underlyingassumptions and approxima-
frombaselevel, a somewhatsurprisingresult with poten- tionsof theshear-stress/stream-powererosionmodel,consid-
tially interestingfield applications. erationof steadystateriver profiles,and explorationof the
Equation (34) reveals a rather complex relationship controlson bedrockchannelresponsetimesestablishin no
between the system responsetimescale,the initial condi- uncertain termsthat resolvingquestions regardingthe non-
tions, and the dominanterosionprocessesthat govern the linearityof the dominantbedrockerosionprocess(es) are
slopeexponentn. As with the responsetime to suddenbase paramount to furtherfundamentalprogressin understanding
levelfall, because
T,s is normalized
by averageuplift rateUof, landscaperesponse to tectonicandclimaticchange.Dimen-
the actualresponsetime scalesdifferentlywith uplift rate U sional analysisdemonstratesthat a singlenondimensional
 17,672 WHIPPLE AND TUCKER: DYNAMICS OF THE STREAM-POWERLAW

group, here termedthe uplift-erosionnumber,encapsulates Which processis dominant under what conditions? What
the dependenceof predicted erosion rates on tectonic, are the appropriateforms of (2) and (3)? Over what distance
lithologic,andclimaticvariables.Our sensitivityanalysisof is an appropriategradient measured? Similarly, the need to
the dimensionlessriver profile evolution equation (18) integrateover the full spectrumof flood dischargesto derive a
revealsthat boththe magnitudeand timescaleof the bedrock meaningfullong-term average rate is a problem [Willgoose,
channelresponseto an imposedtectonicor climaticforcing 1989; Tuckerand Bras, 1997]. One approachthat may help
arelargelygovernedby the uplift-erosionnumberraisedto a to bridge this gap is to pursue small-scaleprocess-oriented
power determinedby the slope exponentn in the stream- field studies in conjunction with reach-scalemodeling
powererosionlaw (equation(1)). Also, the muchdiscussed studies, pursuing top-down and bottom-up approachesin
m/n ratio is shown to influencesignificantly the shape(and concert. Field areas encompassing known differences in
therebyresponsetimescale)ofriver profiles. However,the either climate, lithology, or uplift histories but similar in
m/n ratio neither influencesthe sensitivity of channel gradi- other respectswill be critical to such studies [e.g.,Merritts
ent, relief, or responsetimescaleto changesin the uplift- and Vincent, 1989; Snyder and Whipple, 1998; Snyder et al.,
erosionnumbernorthe dependence of this sensitivity on the submitted manuscript,1999]. Moreover, field areas where a
slopeexponent n. Furthermore,
for thebroadsubsetof fluvial transientresponseto a recent climatic or tectonic perturba-
erosionprocesses adequatelydescribedby (2)- (7)the m/n tion from a known initial condition can be studied would be
ratio is shownto be restrictedto a narrow range(0.35 - 0.6). most advantageousbecauseo?the sensitivity of predicted
The slopeexponentn, whichemerges
fromtheanalysisas a transientresponsesto critical model parameters.
critical unknown, has been shown above to be directly
related to the degree of nonlinearity in the relationship Notation
betweenerosionrate and shearstress(or streampower). Thus
the dominant fluvial erosion processesdirectly and pro- Variables
foundly influencethe dynamicbehaviorof fluvial bedrock vertical
erosion
rate[LT-•].
streamchannels. Clearly, it is not satisfactorysimply to density
ofwater[ML-3].
assumethat erosion is linearly related to shear stress (or basalshear
stress
[ML'•T'2].
streampower); the relationship between channel gradient upstreamdrainage
area[L2].
and erosionrate and its potentialvariation betweenfield set- critical upstreamdrainagearea for fluvial erosion
tingsarefirst-orderproblemsin tectonicgeomorphology. processes
[L2].
As the slopeexponentn is directly relatedto the physics re kinematic
wavespeed
[LT'•].
of the activeerosionprocesses,directedsmall-scalefield and hydraulic friction factor.
modeling studies of these processesare greatly needed. averageflow depth[L].
Potential erosion processesinclude plucking, bashing by f(qs) erodibility scalingfactor for sedimentloading.
bedload,abrasion by suspendedload, cavitation, solution, f rock uplift rate ratio.
and weathering [Alexander, 1932; Maxson and Campbell, g gravitational
acceleration
[LT'2].
1935; Barnes, 1956; Foley, 1980; Wohl, 1993; Zen and H representative
verticallengthscale[L].
Prestegaard, 1994; Hancock et al., 1998; Sklar and L total bedrockstreamlength [L].
Dietrich, 1998; Whipple et al., submittedmanuscript,1999]. Q discharge
[L3T-•].
A numberof field and laboratory studies are underway [e.g., fluvial bedrockchannelrelief [L].
Slingerland et al., 1997; Sklar and Dietrich, 1998; Snyder R•c hillslopeand colluvial channelrelief [L].
and Whipple, 1998; Snyder et al., submittedmanuscript, s streamwisechannelbed gradient.
1999],but manyquestionsremainunanswered. Sc average gradient of hillslope/colluvial channel
Studyof the variationin the effectiveerosioncoefficientK profile.
accompanying adjustments to imposedboundaryconditions t time [7].
is also greatly needed. In most modeling studiesto date, responsetime for suddenbaselevelfall [7].
including our analysispresentedabove, the coefficientof responsetimefor stepincreasein upliftrate[T].
erosionhas been treated as a constant(dK/dt = 0; dK/dx = 0). rock uplift rate definedrelative to erosional base
In addition to the obvious assumptionthat lithology and level[LT4].
precipitationbe held constantin spaceandtime,holding K averagerockuplift rate definedrelative to erosional
constantcarriesthe implicit assumptionthat slope is the baselevel[LT-•].
only morphologicvariable that may adjust in responseto a v meanvelocity[LT'•].
changein boundaryconditions. Even in the simplestcases, w channelwidth [L].
this assumptionwill often be violated, with either the x streamwisedistancefrom divide [L].
amountof alluvial/colluvial cover [e.g., Howard, 1998; Sklar Xc critical distancefor transitionto fluvial erosion [L].
and Dietrich, 1998] or the channelwidth changingin con- elevationof streambed [L].
certwith the gradient. Spatialand temporalvariationsin the
coefficientof erosionmay importantlyinfluencethe dynamics
of river responseto tectonicandclimaticforcing. Exponents
The classicproblemof scaling observationsof local ero- a shear-stress
or stream-power
exponent.
sion rates and processesup to the reachscalerelevant in b exponentin channelwidth-dischargerelation.
landscape evolutionmodelsis, of course,a difficultyfacedby c exponentin discharge-arearelation.
small-scaleprocessstudies.A majorhurdlein this effortwill h exponent in area-lengthrelation (reciprocal
be finding an effectiveway to constrainthe reach-averaged Hack's exponent).
competitionand interactionof the variouserosionprocesses area exponent,erosionrule.
active at the bed. Answers to several questionsare needed. slope exponent,erosionrule.
 WHIPPLEAND TUCKER:DYNAMICS OF THE STREAM-POWER
LAW 17,673

Dimensional constants ference,

editedbyE.E.WohlandK. Tinkler,pp.23,Colorado
State
K coefficient
oferosion[L •-2m Univ., Fort Collins,1996.
ka area-length
coefficient
[L2-h]. Hack,J.T., Studies
of longitudinal
streamprofilesin Virginiaand
Maryland,U.S.Geol.Surv.Prof.Pap.,294-B,97, 1957.
kb totalerodibility
[M-•La+lr2a-1(shear-stress)
orM Hack,J.T., Streamprofileanalysis
andstream-gradient
index,J. Res.
LT3a-i(stream-power)]. U.S. Geol. Surv.,1,421-429, 1973.
ke intrinsic
erodibility
[M-aLa+lT2a-i(shear-stress)
or Hancock,
G.S.,R.S.Anderson,
andK.X. Whipple,Beyondpower:
M -•LT3a-1(stream-power)]. Bedrockriver incisionprocessand form, in RiversOver Rock:
kq discharge-area
coefficient(effectiveprecipitation) FluvialProcesses in BedrockChannels, Geephys.
Monogr.Set.,
vol. 107,editedby K. Tinklerand E.E. Wohl,pp. 35-60,AGU,
[L3-2½
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channelwidth-discharge
coefficient
[L •-3b
T •]. Hovius,
N., C.P.Stark,H.T. Chu,andJ.C.Lin,Supply
andremovalof
sedimentin a landslide-dominated
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Dimensionless variables Taiwan,J. Geol.,in press,1999.
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uplift-erosionnumber. Howard,A.D.,Badland
morphology andevolution:
Interpretation
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S, dimensionlesschannelgradient. Howard,A.D., Longprofiledevelopment
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t, dimensionless time.
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Geology,23, 499-502, 1995.
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landsurface
processes:
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and
climatehietnry
inmn,,ntaln
holtein T....t. Understanding
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Acknowledgments. Fundingfor thisresearch wasprovidedby NSF restrialEnvironment,
editedbyP.M. Mather,pp.21-36,Taylorand
grantsEAR-9614970andEAR-9725723with additionalsupportfrom Francis,Bristol,Pa., 1992.
the Army Construction EngineeringResearchLab (DACA-88--95-R- Kooi,H., andC. Beaumont,Escarpmentevolutionon high-elevation
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Insights
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sionsof bedrockchannelprocesses in the field andJ. Pelletierfor a combines diffusion,
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