JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 99, NO. B7, PAGES 13,971-13,986,JULY 10, 1994
Modeling fluvial erosion on regional to continental scales
Alan D. Howard
Departmentof Environmental
Sciences,Universityof Virginia, Charlottesville
William E. Dietrich and Michele A. Seidl•
Departmentof Geologyand Geophysics,Universityof California, Berkeley
Abstract. The fluvial systemis a majorconcernin modelinglandformevolutionin
response
to tectonicdeformation.Three streambedtypes(bedrock,coarse-bed
alluvial,
and fine-bedalluvial) differ in factorscontrollingtheir occurrenceand evolutionand in
appropriatemodelingapproaches.Spatialandtemporaltransitionsamongbed typesoccur
in responseto changesin sedimentcharacteristics
and tectonicdeformation. Erosionin
bedrockchannelsdependsuponthe ability to scouror pluck bed material;this detachment
capacityis oftena powerfunctionof drainageareaandgradient. Exposureof bedrockin
channelbeds, due to rapid downcuttingor resistantrock, slowsthe responseof headwater
catchmentsto downstreambaselevelchanges. Sedimentroutingthroughalluvial channels
mustaccountfor supplyfrom slopeerosion,transportrates,abrasion,and sorting. In
regionallandformmodeling,implicitrate laws mustbe developedfor sedimentproduction
from erosionof sub-grid-scaleslopesand small channels.
Introduction
in the fluvial system. This paperreviewsfluvial processes
withinthecontextof modelinglong-termlandscape
evolution
Sincethe days of Huttonand Playfair, we have recog- overregionalscales.The initialdiscussion
(generalframenized that landscapes
are createdby erosional/depositionalwork) focuses
uponthe majorchannelbedtypesoccurring
processesacting upon tectonicallycreatedsurfaces. Late in naturalchannels,the factorscontrollingtheir occurrence,
19th century studiesrecognizedthe interactionsbetween andtheirimplications
for long-termlandformevoh•tion.We
tectonicsand erosionthat occurvia isostasy.Until recently
expandon the work of Howard [1980, 1987]andidentify
this interactionhas largely been deeoupledin geologic severalchanneltypes,all of which couldoccurin a large
studies,withgeomorphologists
considering
tectonicdeforma- river system. We arguethata singletransportor erosion
tion as an imposedconstraintandgeodynamicists
specifying law cannotsuffice. Several field examplesillustratethe
asboundaryconditionsthe erosionalunloadingin mountains primaryrole thatbedmaterialtypeplaysin channelevoluand the sedimentaryloadingin basins. However, apprecia- tion. A secondsectionreviews •antitative models for
tion of the strong coupling of tectonic and geomorphic channelevolutionand suggestsan approachto modeling
processes
in the evolutionof landscapes
at regionalto conti- regionalscalelandformevolutionthroughexplicittreatment
nentalscaleshas been growing. For example,erosionof of transportand erosionin larger streamscoupledwith
passive continentalmargin scarps induces lithospheric implicitparameterization
of slopeand low-orderchannel
flexure that affectsrelief and drainagepatternsover a wide erosion.Finally,we conclude
witha synopsis
of remaining
belt [Gilchrist and $ummerfieM, 1990]. The style of uncertainties
in modelingof long-termfluvialevolutionand
deformationin orogenicbelts may be influencedby the crucial research needs.
amount and spatial distributionof erosion [Dahlen and
In nearly every terrestriallandscape,fluvial processes
$uppe,1988;Koons,1990;Isacks,1992l. In recognitionof dominateremovalof weatheringproducts,their transport,
the stronginteractionsbetweentectonicsand topography, andstabsequent
deposition
at locations
thatmaybe separated
severalprocess-based
modelsof regionalerosion[Willgoose fromthesourceby thousands
of kilometers.By connecting
et al., 1991a,b; Koons, 1989; Chase, 1992; Slingerlandet landscapes
to their boundaries,
riversprovidethe primary
al., 1994] and basin sedimentation
[Paola, 1989; Flemings linkagebetweentectonicdeformationand landscapereandJordan, 1989;Jordanand Flemings,1990; Paola et al.,
sponse.Adequatemodelingof the fluvial systemrequires
1992a;Slingerlandet al., 1994] havebeendeveloped.
use of calibratable, mechanistic,transport/erosionlaws.
However,mostmodelingapproaches
haveoversimplified Unlikecrustalprocesses
thatare at leastperceivedasbeing
erosion,transpert,and depositionalprocesses,particularly driven by large-scaleand relativelycontinuously
acting
forces, the erosionof landscapes
occursepisodicallyby
INow at Lamont-Doherty
EarthObservatory
of Columbia spatially variable processesthat neverthelesscreate a
University, Palisades,New York.
Copyright1994 by the AmericanGeophysical
Union.
Papernumber94JB00744.
0148-0227/94/94JB-00744505.00
coherent,integratednetworkof avenuesof transportand
concentrated
erosion(valleys). Althoughit is temptingto
modelthe fiver systemwith a singlerule of transportor
incision(e.g., a diffi•sionequation),
suchoversimplifications
havelittle bearinguponwhat is observed.
In modelingsediment
deposition
in thedistalportionsof
13,971
13,972
HOWARD
ET AL.'
MODELING
FLUVIAL
Erosion
drainage
basinsasfans,deltas,or floodplains
(thusleaving
a sedimentary
record,redistributing
mass,andaffecting
the
EROSION
of Bedrock
Channels
By contrast,bedrockchannels,whichlackan alluvialbed,
overalltectonicforcebalance),treatmentof thetransport
of
occur when stream flow has excesstransportingcapacity
sediment derived from hillslope erosion is essential.
comparedto supplyrate for all size rangessuppliedfrom
However,modelingof erosionintotheunderlying
bedrock
upstreamand from local slope erosion. Therefore the
(particularly
in headwater
areas)is moreproblematic.In sedimentflux divergencein (1) is relatedto flow detachment
large-scale
modelsit has almostalwaysbeentreatedas capacity•rather thanto transportcapacity. The bed scour
resulting
fromtheunsatisfied
sediment
transport
capacity
in Oyb/Ot
depends
uponintrinsicbedrockerodibilityKr, specific
the conservation
of massequation,which is inappropriate
dischargeq, channelgradientS, bed sedimentflux qs, and
becausebedrockincisionis not equivalentto erosionof
loose sediment. Yet incisionis essentialto any large-scale possiblythe grain size of sedimentin transportd:
0Yb
--•(Kr,
q,qs,
$,d).
Ot
landscape
model,as it createsreliefandis thecriticallink
betweentectonicsand erosionalprocesses
via base level
(2)
control.
If theplanetconsisted
of onegrainsizeof cohesionlessThe functionalform of the detachmentcapacityis discussed
sediment,the rulescommonlyusedto createfluvialsystems later. Variable aspectsof channelmorphology,such as
in landscape
modelswouldbe acceptable.Inspection
of width-depthratio, may alsobe important.
rivers revealsa different picture. Those in steepland Controls on Channel Bed Types
typicallyhavesignificant
portions
of theirbedsin exposed
A crucial, but poorly understood,issue in long-term
bedrock. Even in actively uplifting land and rapidly
downcutting
rivers,thebedsof riversmaybe mantiedwith streamresponseis the factors determiningwhich type of
setting. This is
boulders
or gravel. Riverprofilesusuallychange
abruptly channelwill occurin a givenphysiographic
addressed below for both alluvial and bedrock channels.
wherethe gravelmantlegivesway to sand. Thesegrain
size changes
exert a primarycontrolon river incision, Role of transport mechanicsin determining bed type
in alluvial channels. Althoughstreamchannelstransporta
transport
rate, andprofileevolution.
widerangeof grainsizes,onlya narrowproportion
of this
rangepredominates
onthebedanddetermines
theequilibriGeneral Framework
um channelgradientrequiredto transportthe imposed
sediment load. Natural channels tend to be dominated either
Quantitativeanalysis of long-term stream responseto
by relativelyfine bedload(live bed conditions)
or by a
tectonic,lithologic,andclimaticvariationsis complicated
by
coarse,relativelyimmobilebed (threshold
conditions).As
the occurrenceof severaltypesof streamchannelsdiffering
we now discuss,this dichotomyis a consequence
of the
in morphology,dominantprocesses,
and timescales
of adfunctional
formof sediment
transport
relationships.
justment. Howard [1980, 1987] suggeststhree major
In alluvialchannels,
sediment
transport
ratein (1) is often
channeltypes:bedrock, frae-bedalluvial, and coarse-bed
expressedas a functionalrelationship•' betweentwo
threshold. Knighton [1987] makes a similar distinction.
dimensionless
parameters,
•I, (transport
number)and 1/•
However, many coarse-bedchannels carry appreciable
(Shield'sparameter)[Einstein,1950]:
sedimentloads, so that identificationof an additionaltype,
1
live bed gravel, is warranted. For purposesof initial
discussion,these four channeltypes can be merged into
bedrock
and alluvial.
ß - ,.•'(-•-)
(3a)
where
• =
Conservation Equation for Alluvial Channels
qs
todps
--1 =
?
x
.
(Os-p/)gd
(3b)
In alluvial channels, conservationof bed sediment mass In (3a) and (3b), w is the fall velocityof the sediment
relateschangesin the channelbed surfacealtitudey to uplift grains,pf andPsare the fiuid andsediment
graindensities,
U, the spatialdivergenceof bed sedimenttransportrate qs, r is thebedshearstress,andg is thegravitational
acceleraandthe influx of sedimentfrom adjacentslopesqh:
tion. Bedloador total load formulasare commonly
ex-
pressed
asa powerfunctionrelationship
[e.g., Yalin,1977]:
Oy•
Oy
U= I {0q,
qa}(1)
at
at
p,(1-•l)
ax
W
{1 1}•'
(4)
where x is the downstreamdirection,Ps is bed sediment where 1/• c is the thresholdfor transport. In this and
density,•/is sedimentporosity,qsis expressed
in massflux subsequent
formulasconstants
of proportionality
areindicatperunit channelwidth W, qhis massinfluxperunit channel ed by a subscripted
K and exponents
are indicatedby
length of bedload-sizedsedimentfrom adjacentslopeson lowercaseletters;unlessotherwisenoted, suchconstantsand
bothsidesof the channel,U is the tectonicuplift rate, which exponentsare assumedto be spatially and temporally
maybe a functionof locationandtime, andYbis bedaltitude invariant. Becausesedimenttransportrate increases
as
referencedto a matehal coordinatesystem. In this relation- channelgradientincreases(for a given grain size and
shipbothqsandqh measuresedimentonly in the grainsize discharge),
channel
gradient
tendsto adjusttoanequilibrium
rangeconstitutingthe channelbed. Thusin alluvial channels valuejust steepenoughto transport
sediment
loadsupplied
the importantissuesare quantifyingbed sedimenttransport from upstream(in the long-termcontextconsidered
here,
rate and sediment contribution from local erosion.
thiswouldbe the rate of supplyfrom slopeerosionwithin
HOWARD
ET AL.'
MODELING
the drainagebasin). This conceptof equilibrium,or grade,
wasexplicatedby Mackin [1948]. The equilibriumgradient
canbe illustratedvia expressing(4) as a functionaldependenceof gradienton discharge,sedimentlead, and sediment
caliberthroughtheuseof equationsof steady,uniformflow:
x = p/gas
v=
13,973
• =
2Psdv s
(9)
their concentrationon the bed, but this is overbalanced in
unityinmeterseconds
and1.5in feetseconds,
andKpisa
form factor closeto unity. Substituting(5)-(7) into (3) and
solvingfor gradient, $, gives
naturalstreamsby the rapidlydecliningsupplyratesfrom
slopeerosionfor largersizes. For grain sizeslargerthan
somelimitingsizede, the supplyof grainswill be insufficient to make an appreciablecontributionto the sedhnent
bed in comparisonto more abundantfreer bedload.
In light of the above arguments,Howard [1980, 1987]
suggeststhat observedchannel gradientsand their correspondinggrain sizescorrespondto a narrowgrain size range
within the spectrumof suppliedlead which requiresthe
steepestgradientin transportequationsof the form of (8).
$s1)dK,Kp
6•7 (8) Even in mountainous terrain,
where$sis the sedimentspecificgraviS. For a givenr•ch
•d hence f•
"do•t"
water disc•rge, q = Q•,
• the s•ond braces •e cons•t,
.
The slow rate of motion of large grains tendsto increase
s !
whereR is the hydraulicradius, V is meanvelocity, Q is
te•
EROSION
d is givenby themassflux qsof grainsized timestheratio
of grain cross-sectional
area dividedby the grain voh•me,
the averageparticlevelocityvs, andthe particledensityPs'
For sphericalgrainsthis gives
3q,
(5)
discharge,Nm is Manning'sresistance
coefficient,Kn is
$= • •lp+
I
FLUVIAL
•d
the first bracket
conm• two te•s, onerelat• to •s•
lead, (•/K•, •d
the s•o• to the t•eshold of motion (1/•c). As shown
below, the two ma• b• ty•s corres•nd to do••ce
by
one of the two
• s•d b• strew, 1/• cis generallysmallcompar• to
the tr••
par•eter ß due to the •sy mobili• of s•d,
sot•t the gradientof suchfme-b• str•
is largelydeter••
by the s•ent
lead suppli• by slo• erosion,•d
live b• conditio• prevail. • mostgravel•d boulderbed
c•els,
•gh dischargesbarely exc• the critical shear
stress,so t•t tr•s•a
ratesare ve• low, •d the 1/• c
te• largelycontrolsch•el gradient(coarse-b• t•eshold
c•els).
For coarse, u•fo•
siz• gra•s, 1/• c is
essentiallya cornrot, so that gradient is nearly l•early
relat• to gra• size.
However, the above•alysis begsthe questionof whch
c•el
W• will occur• a stre• system•to w•ch a ve•
wide r•ge of gra• sizes is suppli• by slo• erosion.
•tiom
(4) •d (8) are bas• u•n ex•r•ents
•d
obse•ation h ch•els with a narrow size r•ge of b•
s•ent;
tr•z•a
relationshpsfor •x•
sizesof sed•ent
are more complicat• due to •teractions betw•n sizes
trina,
as is discuss• •aher below. Neveaheless,
trina
relations•ps of the above fo• provide a firstorderexpiation of factorscontrollhgbedty•. • natural
c•els,
theb• is com•s• of gra• of a muchnarrower
s•e r•ge t•
t•t of the sed•ent suppli• from slo•
much of the sediment dis-
chargedby a river is relativelyfme; hencethe concentration
term is high for free sediment,althoughthe critical shear
stressterm is low. If sedimentsupplyhasa sizedistribution
which is lognormallydistributedaboutfree sandthen a plot
of solutionsof (8) as a function of grain size showstwo
regions,eachdominatedby one of the two termsdiscussed
above (Figure 1). In the free size range there is a peak
"required"gradientat a grain size somewhatcoarserthan
the median grain size (usually in the sand size range for
natural streams)due to the high supplyrate, whereasthe
critical shearstressterm producesa slopingline increasing
indefinitelyas size increases. If the limiting coarsegrain
sizedeis small,thenthepeakgradientin thefree sizerange
will determinethe channelgradient,and the coarsergrain
sizes, although present and participatingin downstream
transport,will be diluted in their representation
in the bed
due to the small value of the areal density•. On the other
hand,if dc is large,thenthe gradientwill be determined
by
0.1
........
Median
Grain
Size
Threshold
ofSus13ension
/
•luired Gra
0.01
._e
0.001
erosion;a 2-4 + r•ge generallyencompasses
the majority
of b• s•ent
s•es, whereasthe r•ge of supplymay be
20 • or more [e.g., Howarda• Dohn, 1981, Figure9].
•e reasonfor lack of sizessig•fic•tly freer th• the b•
is cl•r: it is tr••a•
as washlead.
Few gra•s muchcoarserth• the do••t
size occur on
theb• b•ause of l•t•
supplyrate. •s c• be illustrat• by assu•g that the layer of activetr•s•a
of the bed
is one gr• t•ck, whch will be appropriateo•y for the
coarsestb•load. •en the ar•l concentration• (fraction
of the b• surfacecover• by the gra•) of s•ent
of size
0.0001
o.o ......
........ i ....... i'0 ......
Grain
Size
..... i0o
(rnm)
Figure 1. Theoreticalcurve of requiredgradientversus
mediangrain size. Median grain size0.3 mm and gradients
are calculatedusing the Einstein-Browntransportformula
[from Howard 1987, Figure 4.2].
13,974
HOWARD ET AL.:
MODELING
FLUVIAL EROSION
coarsegravelor bouldersnearthreshold,
and sand-sizedtectonics,the importantissueis that such channelsare
relativelysteep,therebyaffectingoverall relief, and the
gradients
arecontrolled
by a minorbutcoarse
component
of
represented
on thebed.
material will be carried as wash lead and will be poorly
yield, the limitinggrainsizedediscussed
Distribution of alluvial channeltypes. Fine-bedalluvial thetotalsediment
Theslowratesof transport
andcomminution
of
channels
occurprimarilyin lowlandsand are favoredby previously.
low-relief,abundant
supplyof fme sediment,
andabsence
of the coarsedetritusmeansthat channelprofile evohltionis
influenceof major floods.
coarsedetritus(eitherdueto little production
in headwater slow, despitethe occasional
Mountainstreams
areasor due to abrasionor sortingout of coarsedebris Muchof thecoarsedebrisin Appalachian
may
have
been
produced
by
periglacial
physical
weathering
duringtransport).Deposits
fromfrae-bed
alhlvialchannels
arethosemostcommonly
represented
in fluvialdepositional duringglacialmaxima[Pizzuto,1992].
environments. The characteristictimescaleof responseof
Channelgradientin thresholdgravelstreamsmay be a
frae-bedalluvial channelsto changesin sedimentsupply, dependent
or independent
variable.Forlong-term
equilibridischarge,
baselevel,or tectonic
deformation
istheshortest um betweensupplyof coarsedebrisfrom slopesand its
comminutionand selectivetransport within the fluvial
of all the bedtypes[Howard, 1982].
Coarse-bedrivers are common in mountainotisregions, system,
gradient
becomes
a dependent
variable.However,
wherephysical
weathering
processes
produce
coarse
detritus. tectonicdeformationor constractionallandscapeprocesses
may imposeor modifyvalleygradients,
Such streamsrange from those carryingan abundant suchas glaciation
distribution
of grainsizes
sediment
lead ("live bed" channels
with dominance
of the sothatthe restiltingdownstream
•/K etermin (8))to those
withverylowtransport
ratesof reflectsortingeffectson the local alhlvium. The rapid
fining observedby Fergusonand Ashworth
bedsediment
andgradients
juststeepenough
to permitsome downstream
canalsoalterthe
transport
atveryhighflowstages
("threshold"
channels
with [1991]is an example.Climaticchanges
rates
of
production
and
removal
of
coarse
debrisin the
dominance
of 1/• cin (8)). Channels
mayalternate
between
live bed and thresholdconditionsif sedimentsupplyratesare fluvialsystem,
sothatgradients
andgrainsizesmayberelict
episodic.
features.
Livebedgravelchannels
aresimilarto sandbedchannels Occurrence of bedrock channels. Channel incision into
in thatthe gradientis affectedby bothsediment
sizeand bedrockoccurswhenthe supplyof sedimentto the channel
mantledwithan alhlvialcover,
sedimentsupplyrate. Live bed gravelmay occurin cannotkeepit continuously
or to meagersexliment
mountainous,
alpine,arid, andarcticareaswheresediment usuallydueeitherto steepgradients
yieldsrelative
todischarge
arehighandphysical
weatheringsupply.Thusbedrockchannels
arefavoredby oneor more
predominates
over chemical. Suchchannels
generally of the followingfactors:highrelief, highuplift rates,local
conveya widerangeof grainsizeson thebed, andmost upwarpingor faulting, resistantbedrock,low sediment
of debrisflow transport.
grainsizesaremobilized
at aboutthesameflowstage(the yields,andpossiblya dominance
"equalmobility"concept
[ParkerandKlingeman,
1982]). Theheadwatertributariesof manyriversdrainingmountainThe majordifference
from fme-bedchannels
is thatboth ous areas are bedrock floored. Becauseof scoutingand
thatoccursduringhighflow stages,
channels
with
downstream
sortingandabrasion
playan importantrole, so plucking
that grainsizesdiminishfairly rapidlydownstrean•.In a thin alluvial cover can slowly erode the underlying
channels
wherethe gravelonlythinlymantles
thebedrock bedrockwhilemaintainingan alhlvialcoverduringlow flow
andtherefore
provides
littlestorage
of sediment,
theprimary conditions[Howard and Kerby, 1983]; but the bedrock
capacityof alhlvialchannels
is limited,so thatif
causesof downstream
finingmustbe particlebreakdown
or erosional
downstreamreductionin size of locally contributedsedi- downstreamerosionratesexceedthis capacity,local gradiment. However, even in such channelsthere may be entssteepenand bedrockbecomesexposed[Merrittsand
appreciable
temporary
sortingeffectsif sediment
supply Vincent,1989], often when channelgradientsexceedabout
fromslopes
is episodic
[Dietrichet al., 1993]. In deposi- 3-10%. Thismay occurin particularlyresistantrock, as a
tional environments(alluvial piedmontsand fans), most resultof differentialuplift along a river profile, or as a
modeling
studies
ascribea leadingroleto sorting[Parker, resultof relativeland-seaelevationchanges(suchasthe Fall
Line in the Appalachian
Mid-Atlanticregion[Reed,1981;
1991a,b; Paolaet al., 1992b;vanNiekerket al., 1992].
Threshold
gravelchannels
generally
occurin areaswhere Hack, 1982]). The importantcharacteristicof bedrock
presentor pastphysicalweathering
has supplied
coarse channelsis their resistanceto erosion;expostireof resistant
gravel,butoverallsediment
yieldsarelow, suchasin the bedrock isolates headwater areas from short-term effects of
Appalachian
Mountains.In suchsituations,
reworking
of base level fluctuations. The timescalefor tipstreammigrathegravelby floodsmaintains
gradients
closeto threshold tion of baselevel control in bedrock channelsis much longer
conditions.But the supplyratesand sizesof graveland
boulders
oftenvariesspatiallyin a complicated
way, sothat
thereis no uniquepatternof downstream
change
in grain
sizeor channel
gradient[Hack,1957;Brush,1961]. Large
changes
in grainsize and gradientcan occurover short
distances[Fergusonand Ashworth, 1991]. However,
Pizzuto[1992]has shownthat gravelstreamgradients
in
thanthat of regradingof alluvial channels.
Mixed
alluvial-bedrock
channels.
Channels in which
bedrockexposures
alternatewith shortalluvialsections
are
common[Miller, 1991; Seidl and Dietrich, 1992; Wohl,
1992a, 1993]. Suchmixedchannelsarisefrom at leastthree
scenarios. The first is where regional rates of strean•
downcuttingare such that, on the average,the gradient
largebasins
intheAppalachians
canbeclosely
estimated
by requiredfor bedrockincisionis marginallysteeperthanfor
fidly alluvialchannel.Undersuchconditions,
a routingmodelcombining
a modelof hydraulic
geometry anequivalent
andtransport
withtheassumption
thatgravelisproduced
in an alluvial channelmight require episodicexpostireof
headwaterbasins and is abraded systematicallyduring bedrockin order to erode bedrock as well as lowering bed
transport.In thecontextof regional
response
to long-term elevationthroughsedimentflux divergence.Also, in such
HOWARD
ET AL.-
MODELING
FLUVIAL
EROSION
13,975
channels,bedrock exposurescommonlyoccur due to
small-scale
areal
variations
in bedrock
resistance.
The
secondcase occurs when changesin sedimentload and
dischargeoccasionedby climatic oscillationscause the
channelto alternatebetweenbedrock(during times of low
sedimentyield) and alhwial cover. Seasonalvariationsof
this type betweensand-bedalhwial and bedrockchannels
have been documentedin badlands [Howard amt Kerby,
1983]. In larger temperate-climate
fiver systemsbedrock
exposuremight be causedby alternatingglaciopluvialand
interglacialclimates, with low sexlimentyields at present
exposingbedrock. Catastrophicflooding with headwater
debris avalanchingmight also discontinuouslymantle a
bedrock channel with immobile
coarse debris.
The third
caseoccurswhensuddendrop of baselevelcausesdissection
of a former alluvial channelsystem. Althoughknickpoint
migration causesthe greatestdissection,channelsections
well upstreamfrom the knickpointexperiencesomesteeponing and incision,as is discussedfurther below. This case
may accountfor the sparse,incompletealluvial coveralong
sectionsof bedrockchannelbetweenknickpointsalongElder
Creek, northern California studiedby Seidl and Dietrich
[1992]. The extensiveoccurrenceof nfixexlbedrock/alluvial
channelstreamsof the AppalachianMountains,United States
[e.g., Brush, 1961], may be due either to dissectionabove
the fall line knickpointor to postglacialdecreasein sediment
loading.
Effects of Bed Type on Long-Term Stream Evolution
The role of bed type in controllingfluvial evolutionis
illustratedbelow with topical discussions
of channelgradientsin the Grand Canyon,the role of bedrockknickpoints
in evolutionof streamprofiles, and the behaviorof water-
Figure 3. A rapidsof the ColoradoRiver in the Grand
Canyon. Coarsesedimentdeliveredby debrisflow from the
coarse debris.
Coarse-bed reaches at local sources of
tributaryhas createda fan, narrowedthe river, and forced
boulderydetrituscan have appreciableinfluenceon river the ColoradoRiver into a rapids. Repeateddebrisflows
gradientsand thus on the connectionbetweenbase level over a long time period has causedthe Colorado River to
changesand uplift on headwatererosionrates. A prenfier preferentially
erodethe oppositebank. The tributaryfan is
partiallymantiedby thin sandterracesof the ColoradoRiver
(photoA. Howard).
falls.
The Grand Canyon, an exampleof gradient control by
2500
exampleis theColoradoRiverin theGrandCanyon. The
profileof the ColoradoRiveris veryirregular(Figure2).
Portionsof the river, suchas the now-submerged
Glen
Colorado
2000
Canyonsection,are low-gradient,frae-bedalluvial. Howev-
15oo
er, in theGrandCanyon
thegradient
is controlled
primarily
by rapids
developed
in coarse
boulders
contributed
bydebris
J
I lyon
Green
River
flows debauchinginto the river from small tributaries
drainingthe steepcanyonwalls[HowardatulDolan, 1981;
Kieffer,1985](Figure3). Theserapidshavegradients
near
5OO
the threshold of motion for the coarsestfraction of the debris
• Grand
Canyon,
,
0
500
1000
1500
Distancefrom Gulf of Califomia(kin)
2000
flow sediment.Betweenthe rapidsare longsections
with
sandybed and low gradient;thesesandsectionscontribute
2500
littletotheoverallelevation
dropthrough
thecanyon.
The overallgradientthroughtheGrandCanyonis deterFigure 2. Longitudinal
profileof the ColoradoandGreen minedby the balancebetweenproduction
andmobilization
Rivers,showing
convexities
andsections
of variablegradient of coarsedebris and its removal after abrasion and weather(basedon work by the U.S. Bureauof Reclamation[1946, ing withinthe rapids[HowardandDolan, 1981;Kieffer,
Figure 16]). The sectionlabeled"GrandCanyon"includes 1985;Webbetal., 1989]. Thisbalance
is poorlycharacterthe nominalGrand Canyonas well as Marble and Boulder ized,butthedeliveryof coarse
debrisis certainlyrelatedto
Canyons.
the physicalcharacteristics
of the exposed
rocksas well as
13,976
HOWARD
ET AL.:
MODELING
the relief and width of the canyonandthusindirectlyto past
rates of base level lowering. The balanceof additionand
removal have probably also been strongly influencedby
climaticchangesthroughoutthe late Cenozoic.
Exposuresof bedrockon the bed of the ColoradoRiver
in the Grand Canyon are very rare. Thicknessof bed
sediment near the Glen Canyon and Hoover dam sites
averaged20 m and locally exceeds60 m [Howard and
Dolan, 1981]. Scourholesbelow rapidscommonlyexceed
20 m in depth. These observationssuggestthat bedrock
control is minimal and that the river has excessscouring
capacityeven in the metamorphicand igneousrock portions
of the canyon. Were the relativelysmallquantitiesof coarse
bouldersnot locally producedwithin the canyon,thinningof
the alluvial cover would probablyhaveacceleratedbedrock
erosionand loweringof the fiver profile, producingconsiderably deeper dissectionof the Colorado Plateau and its
bordering mountainrangesthan has occurred. Thus the
productionof coarsedebrisin steeplands
createdby relative
uplift,has a negativefeedbackon erosionratesthroughits
slow transportand commintationwithin the fluvial system
and its role in maintainingsteepchannelgradients.
Knickpoints. We define a knickpointto be a relatively
steep gradient sectionof channelbetweenlower-gradient
sections,no matter whether it is prodracedby tectonic
deformation,baselevel changes,or variablerock resistance.
In alluvial streams,sedimenttransportactsrapidlyto smooth
perturbations
in streamprofilesby a combination
of erosion
of steep channel sectionsand redepositiondownstream,
includingsteepgradientsintroducedby fatalting,baselevel
change,or tectonicdeformation[e.g., Brush and Wolman,
1960]. This mode of knickpoint decay has been called
"inclination"or "rotation"by Gardner [1983].
FLUVIAL
EROSION
Whenbedrockexposures
occuralonga river, loweringof
baselevel elicits erosionalresponseupstreamonly to the
degreethat the bedrockexpostares
can be eroded. If, as
suggestedbelow, erosion rate can be characterizedas a
powerfunctionwith positiveexponents
on drainageareaand
gradient,thenupstreampropagation
of baselevelchangesis
assured. In general, channelerosioninto uniform bedrock
following such a power function tends to smooth out
irregularitiesand perturbations
of the bed profile due, for
example,to suddenchangesin baselevel,sothatknickpoints
migrateupstreambut graduallyflatten,as observedexperimentally by Gardner [1983] and termed "replacement".
However, upstreammigration of knickpointswith nearly
constantdrop and steepness
can occurin two circumstances.
The first is where gently dipping resistantbedrock is
sandwichedbetweenless resistantlayers. Miller [1991]
discusses
naturaloccurrences
of small,migratingknickpoints
in horizontalsedimentary
strata;Hollandand Pickup[1976]
performedflume experhnentsof knickpoint migration in
layered sediments;and Howard [1971a, 1988] provides
simulationsof parallelknickpointmigrationin layeredrock.
The sex5ondcase occurs where steep, bedrock-floored
channelsalternatedownstreamwith low gradient,generally
alluvialchannels.Bedrock-floored
knickpointsoccurin the
profiles of many rivers. The fall-line steepeningis a
prominentfeature of Piedmontstreamsin the Mid-Atlantic
United States(Figure 4). Detailed terracemapping[Dunford-Jackson, 1978; Reed, 1981] in Virginia shows that
alluvialterracesgenerallyextenddownstream
fromtheupper
lip of knickpoints,with the terraces"runningaground"onto
the flatter channelsectionsbetweenknickpoints(Figtare4).
A similar situationoccursin Elder Creek, California [$eidl
and Dietrich, 1992]. The simplestinterpretationis that
300.
E25O
Goas/a/ PIo/h
P/gdmon/
2OO
150
FALL
L IN•-
IO0
50
0
-75
-5O
-25
0
I
25
i
50
7•5
I00 Km
Figure 4. Profileandterracesof the Rappahannock
River on theVirginiaCoastalPlainantiPiedmont.
ProfileextendsupstreamfrombelowtheFall ZonenearFredericksburg,
Virginia(Piedmont
terracesfrom
Dunford-Jackson
[1978, Figure4-0] andCoastalPlainterracesfrom Pavicher al. [1989, Figure liD.
Based
uponrelative
weathering
andaltitude,
Markewich
etal. [1987]suggest
anageof about
5x105
years
for terrace
7 andlx106yearsfor terrace
5. Correlations
forterraces
1-3aretancertain
nearthefall line
and may be faultedor tilted.
HOWARD
ET AL.:
MODELING
FLUVIAL
EROSION
13,977
rapid downstream incision is transmitted tapstreamas
discreteknickpoints,suchthatknickpointsfarthesttapstream
reflect the earliestdowncuttingevents. Headwardmigration
and long-term maintenanceof knickpointsin bedrock have
also been demonstratedby Youngand McDougall [1993],
where, for example, a knickpointabout 250 m high has
migratedabout 15 km in 20 m.y. without diminishmentof
gradientand without obviouslithologiccontrol.
Rapid, episodicdrop of baselevelproducingmigrating
knickpoints not only can occur as a result of relative
land-oceanelevation changesbut also occurs in tributary
streams can also occur due to master stream incision due to
climatic change,expostareof weak rocks, fatalting,stream
piracy, etc. [Seidland Dietrich, 1992]. Knickpointshave
alsooriginatedon portionsof the Hawaiianislandcoastlines
as a result of massive submarinelandslidescreating high
scarpsat the edgeof the islands[$eidl, et al., 1994].
Thus it appearsthat knickpointsin bedrockchannelsin
homogeneous
rockscan migratelongdistancestapstream
in
contrastto the experimentsand conclusionsof Gardtmr
[1983] that knickpointsdeclineby replacement. In a later
section we introduce a simulation model that illustrates such
knickpointmigration.
Waterfalls. The classic model of developmentand
migrationof waterfallsilhastrated
in nearlyevery elementary
geologytextbookis exemplifiedby the NiagaraFalls, where
plungepool erosionof underlyingshalepurportedlyunderminesthe superjacentlimestone[e.g., Gilbert, 1907]. As
pointed out by Horton [1945], shear stressexerted on
channel beds by flowing water reachesa maximtamat
inclinations
of
45 ø and diminishes
to
zero
as vertical
free-fall conditionsare approached. Retreat of waterfalls Figure 5. Waterfallsandplungepoolsat a canyonheadin
requires other processes,with plunge pool undermining the Kohalaregionof the islandof Hawaii. Note the minimal
being the most widely recognized. However, the general evidencefor bedrockunderminingat thephangepools(photo
applicabilityof this model is questionable,
even in the case A. Howard).
of Niagara Falls [Tinkler, 1993]. Weathering,stressrelief
fracturing of the caprock, and its underminingdue to
poolis inhibitexl
throughdevelopment
of an
gravitationalcreepof the subjacentshalemay play a role in fromthephange
addition to hydraulic erosion. In arid landscapeswith alhavialstreamgradedto a downstreambaselevel. Occurpermeablecaprocksoverlyingimpermeableshales,ground- fencesof steepbedrocksectionsbetweencoarse-bedalluvial
water sappingis locally importantin canyondevelopment sectionsare commonin the Hawaiian Islands[$eidl, et al.,
1994]. Also tandersuch conditions,waterfalls, rather than
[Laity and Malin, 1985; Howardand Kochel, 1988].
locationsof rapidbackwasting,
may insteadbe
Backcuttingvia phangepool underminingmay have representing
limitedapplicabilityfor waterfallsin massiverock. Water- sites of inhibited backcutting,above which streamsare
falls with basalphangepoolsaboundon the basalticHawai- relativelyisolateAfrom baselevel control. Backwasting
ian Islands(Figtare5). However, backcuttingand overlying ratesat suchwaterfallsmay be as much relateAto ratesof
rockcollapseconcentrated
at plungepoolsare notconunonly rock weathering,inchadingstressrelief fracturing, as to
observed.Kocheland Piper [1986] andBakeret al. [1990] hydraulicprocesses. However, when channelsegments
attributecanyonbackcutting
at waterfallsto sappingby basal above and below a waterfall are boulder covered, erosion
groundwater
emergingat the footof the waterfalls. Howev- rateson thesesegmentsare muchdiminished,and headward
er, well-developed
alcoves,secondary
porosity,or obviously retreatof the waterfall, even thoughslow, may dominate
weathered rocks are rare.
channelincision[$eidl et al., 1994].
Hawaiian waterfallsare conunonlystepped,with several
phange
poolsinterrupting
the cascading
falls (Figtare5). It
seemslikely thatdowncuttingat the phangepoolsdue to the
Quantitative Modeling of Channel Evolution
momentumof falling water and debris is more prevalent
than backwasting. In this interpretation,the waterfalls
expanddownwardthroughtimewithrelativelylittlebackwasting;in fact, many waterfallsare but little incisedinto the
surroundingslopes (Figtare5), suggestingthat rates of
headwardretreatare not much greaterthan erosionrateson
adjacentslopes. The vertical limit to such downcutting
would be the elevationat which transportof debris away
The discussionnow turns to processdescriptionof
bedrockerosionand alhavialsedimenttransportwithin the
contextof large-scalemodeling.
Erosion
in Bedrock
Channels
In streams with bedrock beAs the critical concern is the
rate of bed scour. Erosionmay occurby severalmechanisms,inchadingphmking,abrasionby sediment,sohation,
13,978
HOWARD
ET AL.:
MODELING
and weathering. The relativeimportanceof theseprocesses
dependsupon rock type, channelhydraulics,water chenfistry, sedimenttype and load, and climate. Thus there is no
universallaw of bed erosion, and due to the generalslownessof bed erosionin resistantrocks, few processobservations have been madeß
As a first-orderapproximation,the functionalrelationship
in (2) might be approximatedby a power law:
0y/,
sd½]$
Ot__-X•qa[1.Xxq
k,
FLUVIAL
EROSION
the ratio m/n. They assumethat erosionrates of main
streamsand their tributariesshouMbe nearly equal if the
junctionsare accordantand channelprofilesare smooth. If
the subscript1 refers to the main stream and 2 to the
tributary,thenequalityof erosionratesimpliesthe following
relationship:
m _-log(S2
/ Sl)/ log(A2
/ A•).
(17)
They f'md an estimatedvalue for the ratio close to unity,
(10)which
is consonantwith a streampower rate law for bed
where the bracketedterm reflectsthe possibilitythat some erosionif dischargeis directlyproportionalto drainagearea:
bed erosionmay occureven in clear waterdischargedue to
plucking and weathering. The constantsKr and Kl, and
OYb
_possiblythe exponents,would vary amongrock types. In
areasof uniform lithology,climate,and relief q, qs, and d
shouldvary systematicallywith drainagearea A, so that However,the $eidl and Dietrich [1992] analysiscannot
distinguish
theabsolute
vah•eof rn andn, onlytheirratio,
further simplificationmay be appropriate:
sothatvaluesof, say,m=0.5 andn=0.5 are alsoconsonant
at - Kp
p/A$.
0Yb
Ot
(18)
is addressed
in theirstudy
(11) withtheirdata. Thisdeficiency
___KzAmSn,
of fluvial dissectionof Hawaiian volcanos, where areal
variations
in amountof dissection
correlate
reasonably
with
where the exponentscan be expectedto be positive.
the area-slope
product[$eidl,et al., 1994]. Youngand
Howard and Kerby [1983] and Howard [1994] suggest
McDougall[1993]andWohl[1992b,1993]alsosuggest
that
thatbedrockerosionin somerock typesmaybe proportional
bedrock
stream
erosion
in
southeastern
Australia
may
occur
to a dominant bed shear stress r:
OYb
=
Ot -K, .
(12)
in proportionto streampower.
Foley [1980] has developeda modelfor circumstances
whereerosionby bedloadis dominant. He usestheoretical
treatments
by Bitter [1963a,b],Nedsonaml Gilchrist[1968]
whereKt is bedrockerodibility. Theyalsoassume
thatmost andFinney[1960]to parameterize
erosionrateby particles
dueto cuttinganddeformation
(surfacedistortion)abrasion,
providingproceduresfor estimatingparametersfrom
measurements
of Moh's hardness. His analysisgives
abrasion
ratesas complicated
fimctions
of particlemassM,
relativevelocityVr, angleof impact0, and impactorand
surface
properties.A simplerempiricalformulation
report-
erosionoccursas a resultof high (flood) discharges
andthat
erosionby any given flood is small compareAto overall
basinrelief. They thereforeassumethata characteristic,
or
dominant discharge can be defmed that representsthe
averageeffect of the natural sequenceof flows. Furthermore, the erodibilityKt in (12) is assumed
to be adjustedfor
theflow durationof the dominantdischarge.The powerlaw
equationsof hydraulicgeometryintroducedabove((5)-(7))
are assumedto be valid, togetherwith expressions
relating
dominantdischargeQ andchannelwidth W to drainagearea:
E o•v,.d0I,
(19)
(13) wheretheexponents
areempirically
evaluated.
Foley[1980]
Q -- K,,A•,
(14)
Combining
(12) with (•), (6), (7), (12t)and(14) gives
OYb
S0.?
at ---K,KzAO.•O-b)
,
edbyHeadandHarr [1970]andEngel[1976,pp. 149-157]
relatessingle-particle
wear erosionE for brittlematerialsto
vr and 0:
(15)
shows
thaterosion
bybedload
should
beproportional
to the
sedimentload qs times the single-particle
erosionrate
divided
bythecharacteristic
saltation
pathlength
X. Combiningthiswith(19) yieldsthefollowing:
OYbo, _ qsvra 0!
Ot
ß
(20)
•
where
In bedrock
channels
qsislessthantheequilibrium
rategiven
by
sediment
transport
relationships
for
equivalent
discharge,
(16)
gradient,andbedloadgrainsize. Regression
analysis
of
data from simulations
of bedloadsaltation
reportedby
WibergandSmith[1985]and Wiberg[1987]suggest
the
Thusfor thismodel,n=0.7 andm•0.25 in (11)fortypical following
dependencies
for Vr, X, and0 at hightransport
exponent
valuesb•0.5 ande•0.8 [e.g., Knighton1987].
(v./v.c > 1.7),wherev. is shear
velocity
andv.c is
HowardandKerby[1983]measured
erosionratesin bedrock stages
thecriticalshearvelocityfor movement
of grainsof sized:
channels
in shalebadlands
andfoundthatspatialvariations
of erosionratesare well approximated
by (15) withestimat-
Vro•v,•'• d-ø'3,
(21)
0 o•v,-0-•d0.3'
(22)
•. ,• v.2'
(23)
ed valuesof n=0.7 and m=0.45, valueswhichare reason-
ably closeto the modelprediction.
$eidlandDietrich[1992]lookatgradient
relationships
at
streamjunctionsin bedrockchannelsto estimatea valuefor
HOWARD
ET AL.:
MODELING
FLUVIAL
EROSION
13,979
Althoughthesesimulations
are for bedloadgrain motion models the interactionof weatheringand detachmentby
above alluvial beds, bedload grain trajectoriesabove a debrisavalanchesin formingbedrockchuteson mountainous
bedrockbedshouldshowsimilardependencies
uponv. and slopes.
The previous discussionof knickpoint retreat indicates
d. Use of (21)-(23) togetherwith equationsof hydraulic
geometry(5)-(7) permit(20) to be recastinto the form of that knickpointsformed by rapid incisionin bedrockchannels can migrateupstreamfor long distancesunder suitable
OYb
• -qsqO.Sa-o.•s/-o.6
S0.•-0.2/-0.7
circumstances.
This scenario is tested in a simulation model
of streamprofile evolution. Drainageareaanddischargeare
Ot
to increaseas the squareof distancedownstream.
.d0.3/_0.3•. (24) assumed
Streamsare assumedinitially to be bedrockwith a concave
Empiricalcorrelations[Head and Harr, 1970] for brittle profile in equilibriumwith a slow, constantrateof baselevel
materialsindicatevaluesof about3.0 for the exponentd and lowering (negativerelative times in Figure 6a), so that all
2.7 forf, suggesting
a negligibledependency
of erosionrate partsof the profile are erodingat an equalrate (time zero in
Figures6a-6e). After time zero, baselevel is assumedto be
on grain size.
Althoughthe relationshipfollows the form of (10), characterizedby longperiodsof stabilityinterruptedby brief
appropriate
valuesfor the exponents
are not as well con- episodesof uplift (Figure 6a). Figures6b-6e show stream
strainedas the analysiswould suggest. Studiesof wear profiles developedin responseto the base level lowering
usuallyconcernimpactvelocitiesin the rangeof 20 to 300 scenario,with profiles keyed to the times shownin Figure
m/s [Engel,1976],well outsidetheprobableupperlimit of 6a. Bedrock erosion is assumedto be governedby (11);
about 5 m/s in streams. The above relationshipmay Figures6b and 6d assumeunity rn and n for erosionratesE
approximately
characterize
cuttingandsurfacedeformation proportionalto streampower [$eidlandDietrich, 1992], and
dueto bedloadimpacton a flat bed,butotherprocesses
may the 0.3 and 0.7 valuescorrespondto erosionin proportion
alsobe involved,suchasweathering,particlefragmentation, to shearstress[Howard and Kerby, 1983]. The heavy lines
andscourby suspended
sediment
or bedmatehalin vortices, are alluvial channelsections(discussedbelow); light lines
resulting
in potholes
andlongitudinal
grooves[Wohl,1992b,
are bedrock
channels.
A power law relationshipfor bedrockerosionhnpliesthat
1993].
gradients
will graduallydiminish if base level is fixed and
Thus three different models of bexlrock channel erosion
that
the
lowest
gradientswill occur at the downstreamend.
(Howardand Kerby's[ 1983]shearstressmodel;Seidl
However,
as
gradients
decline, a minimum gradientwill be
Dietrich's [19921 stream power model; and the above
sedimentscourmodel)resultin equations
(i.e., (15), (18), reachedwhere the gradient will only be just sufficient to
and (24)) that canbe expressed
in the form of (10) or (11), transportsedimentsuppliedfrom upstream,and that section
of
the
stream
will
be
converted
to fme-
or coarse-bed
butwithdifferentexponents
andconstants
of proportionality.
At present,our knowledgeof spatialvariationsin bedrock alluvial, dependingupon sedimentsupply characteristics.
This conversion will occur first at the downstream end and
channelerosionratesand of inherenterodibility of bedrock
migrate
upstream,graduallyreplacingthe bedrockchannel
is insufficientto permitadequateevaluationof the relative
(time
1
in
Figure 6).
merits of these models.
The
minimum
alluvialchannelgradient$mcanbe evaluatBedrockstreamshave no simpledownstreamhydraulic
ed
from
(8).
Hydraulic
geometryrelationships
(13)-(14) can
geometrybecausegradientis a semi-independent
variable.
However, if bedrock is homogeneous
and the rate of base be usedto express$mas a fi•nctionof drainageareaA:
level lowering has been constantfor a long time, then
$. -- K•,A•.
(26)
erosionrates shouldbe uniform throughoutthe basin and
In the simulationsu equals-0.25, consistent
with a sandbed
(11) implies
channel
at
high
sediment
transport
rates
with
areallyuniform
s., A .
(25)
and temporallyconstantsedimentyield [Howard, 1980]. If
the alluvial channelsare coarsebed with gradientscloseto
the thresholdof motionthe valuesof/qn and u will differ,
for uniform grain size [Howard, 1980;
steep hollows of the Appalachians,erosionby debris with u •-0.4
avalanches
may dominatethroughremovingaccumulated Knighton,1987], andKmwill vary nearlylinearlywith bed
In someareas,suchasthe U.S. Pacificcoastalrangesand
sediment and weathered bedrock from channels [Dietrich grain size (8). In channelswith sedimenttransportdominatand Dunne, 1978; Dietrich et al., 1982; Pierson, 1980; ed by debrisflows, there may be little areadependenceupon
Benda, 1990; Wohl and Pearthree, 1991]. Due to the high transportcapacity,so that u •0 [$eidl and Dietrich, 1992].
sedimentconcentrations,debris flows have fiandamentally So long as base level remains constant,the alluvial section
different mechanicsof transportand erosionthan normal of the channelis assumednot to lower but does gradually
river flows, andthey will requirea separateanalysiswhose extendheadwardin responseto continuederosionof the
long-termspatialpatternsof erosionmay not be described bedrockchannel(e.g., times 1, 4, and 5 in Figure 6). In a
very well by (11). $eidl and Dietrich [1992] suggestthat more realistic simulation, continued erosion of headwater
debrisflows in mountainousregionsare the primary agent areaswould causesomedeclinein suppliedload and grain
of bed scour in headwaterchannelswith gradientsgreater size, requiring some regrading of the alluvial channel
than 0.2.
section.
Immediatelyafter a rapid base level drop, the farthest
In some casesdetachmentin bedrock channelsmay be
limited by the rate of bed weathering. Howard [1994] downstreamsection of the channel is steepenedto values
discusses
howweatheringandsheardetachment
may interact well above$m(e.g., time 2), so that this sectionis reconin bedrock channelsin shale badlands,and Howard [1990] verted into a bedrock-flooredknickpoint,which gradually
13,980
HOWARD ET AL.'
MODELING FLUVIAL EROSION
c-
-2O
-4O
-6O
-8O
-lOC
o
-20
20
40
60
leo
80
Time
150
100
5O
0
-50
-leo
150
100
i
0
10
20
30
40
50
60
Distance Downstream
70
80
90
100
0
10
20
30
40
50
60
70
80
90
Distance Downstream
Figure6. Simulations
of knickpoint
evolution.Timeanddistance
scales
arein arbitraryunits. Drainage
areais assumed
to be proportional
to thesquare
of distance
downstream,
withunitdrainage
areaat unit
distance.(a) Temporal
change
in downstream
baselevel(matehal
coordinates
areusedin thissimulation);
(b) successive
streamprofilesproduced
for the erosional
historyin Figure6a with bedrockerosion
proportional
to streampower(m=1, n= 1, andK2=0.0004in (11);Kin=1 in (26)). Heavylinesare
alluvialchannel
segments
thatformduringperiodsof baselevelstability;
(c) conditions
similarto Figure
6bexcept
bedrock
erosion
proportional
to shearstress
andK2=0.096;(d) conditions
similarto Figure6b
exceptfor steeper
assumed
alluvialchannel
gradients
(Kin=6.5); notethatat timezerothereis a
downstreamtransitionfrom bedrockto alluvial channel;(e) conditionssimilarto Figure 6c exceptfor
steeperassumedalluvialchannelgradient.
leo
HOWARD ET AL.: MODELING FLUVIAL EROSION
erodesheadward,replacingthe upstreamalluvial channel
section(times 3 and 4).
Formeralluvialchannelsections
whichlie upstreamfrom
a bedrockknickpointalso graduallysteeponin advanceof
thearrivalof the knickpoint.Thisgradualsteepening
is an
effectof the baselevel loweringwhichis smallerin magnitudebutmorerapidlytransmitted
tipstream
thantheknickpoint. Thissteepening
canoccurin the modelbecause
during
each time incrementa potentialbedrock erosion rate is
13,981
downstreamtrend in transportrate, which in turn depends
upon sediment supply rate and grain size distribution,
channelwidth, and discharge. Treatmentof downstremn
changesin transportrate alsomustalsoaccountfor abrasion
and sorting. Complicatingthe issuefor longterm drainage
basin evolution is the necessityfor developingtransport
relationships
that integratethe effectsof thenaturaltemporal
spectrumof dischargesand sedimentsupply.
Equilibriumtransportrelationships
for narrow grain size
calculated, even in alluvial channel sections. That erosion rangesof suppliedsedimentunder constantdischargeare
is permittedto occurtinlessthe gradientwould be reduced reasonablywell developed,althoughthereare a plethoraof
belowthe criticalgradientSmas a resultof the erosion. empiricaltransportlaws (see summariesby Vanoni [1975],
Becausethe gradient is steepened,the alluvial channel Chang, [1988], and Gomezand Church [1989]). Most of
sections
tipstreamfrom a knickpointare rapidlyreconverted these can be formulated into relationshipssinfilar to (4),
to bedrock,althoughthe gradients
are onlyslightlysteeper althoughsome are valid for only a limited range of grain
thanan equivalentalluvial channel. In naturalstreams,one sizes (e.g., gravel or sand) or for high or low transport
mightf'mdpatchyalluvialcoveror shortalternating
sections rates. Someare for bedloadtransportonly, whereassome
of alluvial and bedrock channels in such circumstances.
inchidesuspended
loadtransportof bedmatehalat highflow
Gardner[1983] noteda similarsteepening
tipstreamfrom stagesin sandbed streams(total load formulas). In channels
the knickpointsin his experimentsbut attributedit to with a narrow grain size range of bed matehal, hysteretic
drawdown effects.
effects in transport rates as a function of bed shear are
The combinationof upstreammigrationof knickpoints minor. However, stage-related
changesin form dragdue to
with slightsteepening
and loweringof the formertipstream bedformdevelopmentcan introducetemporalvariationsand
alluvial sectionswouldtypicallyresultin creationof alluvial time lags in the proportionof total perimetershearthat is
terracesfrom renmantsof the former alluvial floodplain. availablefor transportingsediment. Nonetheless,for longSuchterracesshouldoccurhigh abovethe fiver below the term, large-scaleevaluationof sedimenttransportrates,the
knickpoint
but be onlyslightlydissected
abovetheknickpo- useof an appropriatetransportrelationshiptogetherwith an
int. Suchterracesoccur in the Rappahannock
River in empiricalresistance
relationshipgenerallypermitsevaluation
Virginia (Figure4) andhavebeeninterpretedto havearisen of transportrateswith fair accuracyif a representative,or
from knickpoint migration of the type discussedhere dominant discharge is used which is an average of the
[Dunford-Jackson,1978].
spectrumof natural dischargesweightedby their transport
The shape,height,andgradientof the knickpoint,andits capacity,but only so long as the grainsize rangeof supplied
temporalperseverance
duringupstreammigration,depend sediment is small.
uponthe value of the exponentsrn and n, the magnitudeof
Transientmodelingof streamprofile evolutionin response
base level drop, and the steepness
of the alluvial channel to changesin hydraulic regime or base level is possible
sections.For high valuesof rn and n (erosionproportional [Howard, 1982; Snow and Slingerland, 1987, 1990; Willto streampower),knickpoints
remainsteepas theymigrate gooseet al., 1991a,b;Bonneauand Snow, 1992]. Transient
upstreamand former alluvial sectionsare little erodeduntil responsesof sand bed channelsto differential uplift have
theyare engulfedby themigratingknickpointbecauseof the been documented [Burnett and Schumm, 1983; Ouchi,
strongdependencyof erosionratesupon gradient(Figtire 1985]. Sand or sand-siltbed channelsare most suitablefor
6b). On the otherhand,for the caseof fractionalpowersof modeling using the approachdiscussedabove. In such
rn and n, knickpointsare convexlyrounded,they maintain channelsthe rangeof grain sizeson the bed is usuallysmall,
a generallyconstantgradientbut decreasingheightduring and gravel, althoughpresent, is transportedin generally
upstream
migration,andtheformeralluvialsections
undergo negligible quantities. In addition, downstreamchangesin
appreciableerosionduring subsequent
dissectionalthough grain size due to comminutionor sorting are appreciable
theirgradientis onlymodestlyincreased
untiltheknickpoint only over distancesgreaterthan severalhundredkilometers,
migratesthrough(Figure 6c). If the gradientof the alluvial althoughmore rapid downstreamfining sometimesoccurs
channelsthat form duringtimesof baselevel stabilityare [Pickup, 1984]. Furthermore,becauseof the low gradients
steeperthan the previoussimulations,then erosionof the of sandbed channelstheir contributionto overallrelief may
former alluvial sectionsafter baselevel loweringis more be small, so that in large-scalemodeling,errors in estinmtpronouncedand knickpointslower and gradually loose ing gradients may be inconsequential. However, two
individuality(Figtires6d and 6e). The relevantcriterionis complications
arisein large-scalemodeling. Becauseof the
the ratio of the alluvial channel gradientto that of the wide rangeof grain sizesfed into headwaterareas,a model
knickpoint. The knickpointgradientis a functionof the must be capableof predictingwhen and where sand bed
exponentsrn and n, the magnitudeof the baselevel drop, channelswill occur. Furthermore, changesin hydraulic
andthe distanceof knickpointmigration.
regime or tectonic warping can causetransitionsto other
Althoughthe simulationsassumea downstreamincrease channel types. Diminishment of sedimentload (e.g., by
in discharge,
development
andmigrationof knickpoints
also upstreamreservoirs)can lead to developmentof coarse-bed
occurwhendischarge
is constant
or decreasing
downstream, armoring. Similartransitionsto coarse-bed
conditionsmight
althoughthe patternof upstreammigrationis different.
accompanytectonicsteepeningof fiver profiles. A transition to bedrockchannelsmay accompanyrapiddowncutting
SedimentTransport in Alluvial Channels
and removalof the alluvial bed (Figure 6).
Modeling of bed elevationchangesin alluvial channels
In channelscarrying a wide range of sedimentsizesthe
usingthe conservation
equation(1) requiresestimationof the simpletransportrelationshipsof (3) and (4) are not satisfac-
13,982
HOWARD
ET AL.:
MODELING
FLUVIAL
EROSION
tory. In recentyearsa numberof transportrelationships
for developed
to adequately
coupletectonicsandgeomorpholomixed grain sizes have been proposed[e.g., Shih and gy. In a large-scale
erosionalmodel,thefocusshouldbe on
Komar, 1990; Parker, 1990;Bridgeand Bennett,1992; van the evolutionof the river longitudinalprofile and erosion,
of sediment.Numerical,coupled
Niekirket al., 1992]. Someof thesemodelsare computatio- transport,anddeposition
nally very demanding
andmay not be suitablefor the type slope-channel
modelsof landformevolutionhave been
of large-scale,long-termmodelingdiscussed
here. None of developedin the last several years [e.g., Ahnert, 1976;
thesemodelspredictsthe transitionsbetweensandbed and Koons,1989; Willgooseetal., 1991a,b;Chase, 1992;Lifton
gravelbedchannels
discussed
above;suchabrupttransitions and Chase, 1992; Howard, 1994]. However, it is impractimaybe dueto changein transportefficiencyassociated
with cal (and unnecessary)to extend this type of model to
of
the differencebetweengravel-on-graveland gravel-on-sand regionalor continentalscales. Adequaterepresentation
slopemorphologywouldrequireordersof magnitudemore
conditions(Y. Kodama, personalconununication,1993).
Effects of downstreamabrasionupon grain sizes have memory and computationalresourcesthan are presently
generallybeen treated empirically, with mean grain size available. On the other hand, the strongfeeAbackbetween
decreasingas either an exponentialor power function of channelprofile evolutionand slopeprocessesrequiresthat
travel distance (see reviews by Kodama [1992], Pizzuto suchinteractionsnot be ignored. The approachsuggested
[1992], andMikos [1993]). Parker [1991a,b]hasdeveloped here is to explicitlymodelprofileevolutionof high-order
a more mechanisticapproach. Models of sedimenttrans- channelswhile implicitly treating the interactionbetween
response(floodhydrolport, sorting,and abrasionin suchstreamsare reasonably channelevolutionand sub-grid-scale
well advanced [e.g., Parker, 1990, 1991a,b]). In sire ogy and the delivery rates and grain size distributionof
abrasion within streams [Schummand Stevens, 1973] and sedimentfrom slopesand low-order channels). This takes
duringtemporarystoragein floodplains[Bradley, 1970]may an oppositeapproachto that usedby Koons [1989], which
help to account for very rapid downstreamgrain size modeledslopeerosionusing a diffusionequationbut simply
diminishmentin coarse-bedstreams[e.g., Pizzuto, 1992; specifiedthe profile of high-orderchannels.
A large-scalefluvial modelmightbe basedon a matrix of
Kodama, 1992].
cells,
with eachcell representinga high-orderchannelplus
Downstreamfming as a result of sorting should only
occur in areas in which sedimentis actively depositing. surroundingsub-grid-scalecontributingarea, such as the
Paola et al. [1992a] use the simplestapproach,modeling networkmodelsof Howard [1971b, 1991]. Typical cell
sortingas a successive
depletionof grainsin transportfrom dimensionsmight range from 1 to 10 km square. The
of sub-grid-scale
coarsestto finest. The models of Parker [1991a,bl and van importantissueswouldbe characterization
erosionand sedimentcontribution,routing of fluvial sediNiekerket al. [1992] offer more detailexlmodeling.
mentdownstream,rate of channelbederosionor deposition,
tectonicdeformation,temporaland spatialchangesbetween
Discussion- A SuggestedModeling Approach
channel types, flow directions, and initial and boundary
For coupling of erosionalprocesseswith tectonicand
climatic forcing on large spatialscalesand over long time
spans,a criticalconcernis predictionof spatialandtemporal
ratesof erosionand deposition. If erosionof the landscape
were everywherein balancewith a long-termconstantrate
of uplift, then correlation studiesof erosion rates as a
function of relief [e.g., Ahnert, 1970, 1984; Ruxton and
McDougall, 1967; Pinet and Souriau, 1987; Milliman and
Syvitski, 1992], climate [Langbein and Schumm, 1958;
but the remainingissuesare beyondthe scopeof this paper.
However, the treatment of sub-grid-scaleprocessesis
particularlycrucial, so that we suggestone possibleapproach.
Sedimentyieldsfrom local slopesand channelsmightbe
modeledas a convolutionfunctionof presentand pastrates
of incisionof the high-orderchannelflowing througheach
grid cell:
Wilson, 1973; Ohmori, 1983; Schmidt, 1985; Pinet and
Souriau, 1987; Milliman amt Syvitski,1992] anduplift rates
[Schumm, 1963; Adams, 1985; Yoshikawa, 1985] would
sufficeto coupletectonicsand erosion. Hack [1960, 1975]
makesan argumentfor suchan equilibriumin the Appalachians, Suppe [1981] and Dahlen and Suppe [1988] for
Taiwan, and Adams [1985] for the New Zealand Alps.
However, seriousdiscrepancies
betweenlong-termuplift or
deformationrates and denudationexist in many tectonic
regions(e.g., the TibetanPlateauandthe Altiplanoof South
America [Dahlen and Suppe, 1988; Isacks, 1992] and the
fall line knickpoints of the Appalachians). On shorter
timescalesand smaller spatial scales, river profiles are
commonlydirectly affectedby tectonicdeformation[e.g.,
Burnett and Schumm, 1983; Ouchi, 1985; Gregory and
Schumm, 1987; Merritts and Vincent, 1989].
Thusmechanistic,predictivemodelsof relief development
and associatedweathering, erosion and deposition with
parameterizationappropriate for regional or continental
spatialscalesand > 105year temporal
scalesmustbe
conditions. Some of these issueshave been discussedabove,
q•(O
= ),+1ß
(t-i)
,
(27)
whereqhis the sedimentinflux (per unit length)to the
channelfrom slopeerosionat timet, b is thedin•ension
of
a unit cell in the simulation,3, is a characteristicrelaxation
timescale
(3, >_0) measured
in iterations,Oyb/Ot
is thelocal
channelerosionrate at time t-i in the past, and the summa-
tion goesover all iterationtime stepsi. If the channel
erosionrate is temporallyconstant
or if 3, = 0, then
q• = Ps150Yb
.
ot
(28)
The weightingby •5comesfrom considering
thecell areato
be•52,thelength
of thechannel
through
thecelltobe•5,and
the width of the channelto be small comparexlto the cell
dimensions.Equation(27) canbe expressed
as an equivalent difference formula which is easier to compute:
qh(t)__
Ps•
OYb(t)+qh(t_l)[
)•] (29)
HOWARD ET AL.'
MODELING FLUVIAL EROSION
The use of large scalesimulationmodelsthat explicitly treat
slope erosion [e.g., Ahnert, 1976; Willgoose, 1991a,b;
Howard, 1994) can help to scale3, as a functionof parameters governingmasswastingand fluvial erosion. Figure 7
showsan exampleof temporalvariationin qhas a function
of changesin Oyb/Ot
using(29).
The grain size distributionof sedimentsuppliedfrom
13,983
bedrockslopeswouldrespondvery slowly(verylong3,)to
erosionof masterstreams.Andersonand Humphrey[1990]
suggest
weathering
andstreamdowncutting
andtransportare
essentiallydecoupledin suchcircumstances
(indefmite3,).
However,mostphysicalweatheringprocesses,
suchas frost
actionand developmentof sheetingfracturing,work from
the surfaceinwardsand diminishin intensitywith depth.
On steeperslopesmasswastingrequireslessweatheringto
remove the partially weatheredbedrock than on gentler
slopes,so that there is a positivefeedbackbetweenslope
steepness
and erosionrates. Furthermore,somephysical
weatheringprocesses,
suchas fracturingdueto gravitational
local slopes is also important in determining bedload
transport rates. A reasonableassumptionwould be a
lognormaldistributionwith logarithmicmean and standard
deviations/x and a, respectively. Both /x and a would be
functionsof bedrockcharacteristics,
climate,andpresentand
pastchannelerosionrates(expressedthrougha convolution stresses,are directly relatedto slopesteepness.Thus there
functionas above), under the assumptionthat the steeper is an indirectpositivecouplingbetweenrelief generation
by
relief associatedwith more rapid erosion would produce streamincisionanderosiononbedrockslopes.The contrast
coarserand possiblymore variable debris.
in timescalesbetween regolith-mantledslopes (low to
The primary issuein applicationto naturallandscapes
is moderate3,) and bedrockslopes(long 3,) is somewhat
providingreasonableestimatesof X,/z, and a. The charac- analogous
to the differencein responsetimes between
teristicrelaxationtime 3,scalesthe time requirodfor changes alluvial and bedrock channels.
in stream erosion rates to be transmitted upslope and
Evaluation
of a and/z (or othersuitable
descriptors
of
upstreamwithin sub-grid-scale
tributariesasperturbations
of grainsizes
supplied
fromslope
erosion)
require
sampling
of
channelgradient,slopesteepness,
and drainagedensity. In sediment
transportin headwaterbasins. Because
of difficulthe limiting case of 3,=0, slope and low-order channel tiesof accurate
measurement
of thefull spectrum
of sizesin
responseto changein erosionrate is instantaneous,
and the transport,the most reliablemethodis bulk analysisof
entire cell erodesat the samerate. Two broadcategoriesof sediment
deposited
in headwater
reservoirs
or lakes[e.g.,
slopeshavebeen identified:regolith-mantled(transport-limi- Smith et al., 1960]. However, suchdata for mountainous
ted) slopeswhoserate of erosiondependsuponthe capabili- areas are rare, and measurement in downstream reservoirs
ty of erosional processesto remove the regolith, and is confounded
by downstream
finingandsediment
storage.
bedrock (weathering-limited) slopes whose erosion rate
Glaciated
areaswill requirea separate
parameterization
of
dependsuponthe rate of weathering[Culling, 1960; Carson the relationship
betweenuplift ratesand sub-grid-scale
and Kirkby, 1972]. In low-relief landscapeswith thick erosion rates.
regolith,low drainagedensity,anda dominanceby creepike
A numberof the otherissuesthatmustbe addressed
in
processes,3, may be fairly long. In some mountainous large-scale
erosionmodelscanbe onlybrieflymentioned.
areas, efficient frost weatheringproducesabundantcoarse Sediment
mustbe routedthroughthe fluvialsystem,with
debrisandslopesare closeto thresholdof stability[Carson, appropriate
attention
to prediction
of bedtype,sortingand
1971; Carson and Petley, 1970]. In such cases, slope abrasionusing the quantitativeapproaches
summarized
steepness
varieslittle with erosionrate, andchanneldowncu- earlier. Localbedtypein alluvialchannels
depends
upon
tting provokesan immediateresponsein masswasting, so which term is dominantin (8), and transitionsbetween
that 3, would be close to zero. High-relief areas with bedrock and alluvial channels could be treatexl as in the
profilesimulations
in Figure6. In addition,
boundary
and
initial conditions
mustbe specified,includingtectonic
deformation.
25
-20
Conclusions
15
10
'•
'5
,1• 25
0
Q- 20-
thecontrolling
processes.
Severaltypesof channels
occur
in nature, with bedrock,frae-bedalhwial, and threshold
coarse-bed
alluvialbeingthe end members.Eachtype
requiresa differentapproach
to modeling
andprediction.
Mechanisms and rates of erosion in bexlrockchannels are
15-
poorlycharacterized,
butan approach
relatingerosionrate
to a powerfunctionof drainageareaandchannelgradient
10.
5-
0
0
Theemphasis
in thispaperis on ourinadequate
understandingof long-termevoh•tionof fluvial channelsand of
10
20
30
40
50
60
70
80
90
100
may be sufficientfor many situations. Fine-bedalluvial
channelsare the best understood,with the critical issue
ment yield is assumedto be in equilibriumwith an uplift
beingcharacterization
of sediment
yieldfromslopes
andthe
interactions
between
channel
entrenchment/aggradation
and
ratesof slopeerosion.Gravelchannels
oftenhavegradients
near the threshold
of motion,and the importantissueis
determining
deliveryratesandsizedistributions
of gravel
fromslopesandtherolesof sortingandabrasion
in down-
rate of 10.
stream transport of this debris.
Time
Figure 7. Sedimentyield for two differentvaluesof X as a
functionof time anduplift rateusing(29). Time anduplift
rate scalesare arbitrarywith ps= 1 and •5=1. Initial sedi-
13,984
HOWARD
ET AL.:
MODELING
FLUVIAL
EROSION
The most critical uncertaintiesin predictionof long-term Bonneau, P.R., and R.S. Snow, Character of headwaters
adjustment
to baselevel drop, investigated
by digitalmodelevolutionof fluvial systemsare (1) determinationof what
type of channelwill occurin a giventopographic-geologic- ing, Geomorphology,5, 475-487, 1992.
hydraulic-climatologicsetting, (2) parameterizationof Bradley, W.C, Effect of weatheringon abrasionof granitic
gravel, ColoradoRiver, (Texas), Geol. Soc.Am. Bull., 81,
slope-channelinteractions(inch•dingsize distributionand
61-80,
1970.
amount of sediment shed to channels), (3) quantitative
characterizationof erosionratesin bedrockchannels,and (4)
the role of debrisproduction,sorting,and comminutionin
evolutionof gravel bed channels.
Not discussedin this paper, but of considerableimportancein long-termchannelevolutionare possiblechangesin
Bridge, J.S., and S.J. Bennett,A modelof the entrainmentand
transportof sedimentgrainsof mixed sizes, shapes,and
densities,Water Resour. Res., 28, 337-363, 1992.
Brush,L.M., Jr., Drainagebasins,channelsandflow characteristicsof selected
streamsin CentralPennsylvania,
U.S. Geol.
SurveyProfi Pap. 282-F, 181 pp., 1961.
channel pattern (meandering,braided or straigh0 and Brush, L.M. and M.G. Wolman, Knickpoint behavior in
drainagenetworkpatternthroughdividemigrationor stream
noncohesive
material:A laboratorystudy, Geol. Soc. Am.
Bull., 71, 59-74, 1960.
capture. In addition,climateandclimaticchangeinfluences
often outweighbaseleveleffects in mountainous
regions Burnett,A.W., and S.A. Schumm,Alluvial river responseto
neotectonicdeformationin Louisiana and Mississippi,
(e.g., altiplanation,glaciation,high-elevation
deserts).
Science, 222, 49-50, 1983.
Ourrecommendations
for futurestudyaddressing
regional
Carson,
M.A., An applicationof the conceptof threshold
scalelandformevolutionare to proceedon severalfronts.
slopesto the Laramie Mountains, Wyoming, Inst. Br.
The first is development
andtestingof simulation
modelsof
the type proposedabove in order to assessthe nature of
channel
typeinteractions
andthetypesof simplifications
that
canbe madein suchmodeling. At the sametime, regional
studiesof channelgeomorphology
are needed,particularly
in highrelief areas,with an emphasis
on channeltypeand
bed sedimentsize, and processes
of transportand erosion.
Characterization
of sedimentsupplyratesand grain sizes
from headwater slopes and channelsis also essential,
particularlyin mountainous
areas. Finally, applicationof
absoluteage datingtechniquesto estimationof erosionrates
is necessaryfor model calibration.
Geogr. Spec.Publ., 3, 31-47, 1971.
Carson,M.A., andM.J. Kirkby, HillslopeForm and Process,
475 pp., CambridgeUniversityPress,New York, 1972.
Carson, M.A., and D.J. PetIcy, The existenceof threshold
slopesin the denudationof the landscape,Trans. Inst. Br.
Geogr., 49, 71-95, 1970.
Chang,H.H., FluvialProcesses
in RiverEngineering,432 pp.,
JohnWiley, New York, 1988.
Chase, C.G., Fluvial landsculptingand the fractal dimension
of topography,Geomorphology,
5, 39-57, 1992.
Culling, W.E.H., Analytical theoryof erosion,J. Geol., 68,
336-344, 1960.
Dahlen,F.A., andJ. Suppe,Mechanics,growthanderosionof
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Acknowledgments.
The manuscript
wassubstantially
improved
1988.
as a resultof reviewsby E. Wohl and G. Willgoose. This Dietrich, W.E., and T. Dunne, Sedimentbudgetfor a small
workhasbeenpartiallysupported
by NASA Planetary
Geology
catchmentin mountainous
terrain, Z. Geomo•phol.Suppl.,
and GeophysicsProgramgrant NAGW- 1926.
29, 191-206, 1978.
Dietrich, W.E., T. Dunne, N. Humphrey, and L. Reid,
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