Longitudinal Profile Development Into Bedrock: an Analysis of Hawaiian Channels1

Longitudinal Profile Development into Bedrock: An Analysis of Hawaiian Channels1

Michele A. Seidl,2 William E. Dietrich, and fames W. Kirchner Department of Geology and Geophysics, University of California at Berkeley, CA 94720

ABSTRACT Analysis of topographic maps of rivers incised into dated Hawaiian lava flows shows that the long term average bedrock erosion rate along certain reaches is linearly related to stream power. Field observations suggest that two processes may control Hawaiian channel downcutting: (1) stream power-dependent erosion, including abrasion of the channel bed by transported particles, and (2) step-wise lowering caused by knickpoint propagation. Modeling results indicate that a simple stream power-dependent erosion law predicts the straight to weakly convex longitudinal profiles characteristic of Kauai channels but is insufficient to predict two other characteristic features: the upslope propagation of knickpoints and the straight 5-8° channel slopes below the knickpointS; thus more than this single transport law is apparently required to model bedrock channel incision. Field surveys also indicate that significant portions of the channel lengths below the knickpoints are mantled with large boulders. We propose that the boulder mantling of long channel reaches inhibits channel incision, reducing downcutting to a rate set by boulder weathering, breakdown and transport of the material, and perhaps by knickpoint propagation sweeping under the boulder armor. Partial boulder mantling of bedrock-dominated channels is common in mountainous regions, and a theory which takes boulder armoring into account will have broader applications than one which ignores these limiting effects.

Introduction Bedrock channels are common in mountainous have recognized that river longitudinal profiles are landscapes, yet no theory, supported by field data, generally concave upward, and in predictions of exists to predict the development of river longitu profile evolution, profile concavity is often inter dinal profiles into bedrock. This lack of knowledge preted as confirmation that the simulations are inhibits our ability to model landscape evolution. realistic (e.g., Snow and Slingerland 1987; Slinger- Diverse topography generally is formed by the ac land and Snow 1988). However, close inspection of tion of water cutting channels and forming a hier long profiles reveals that profile concavity is often archical network of progressively larger valleys interrupted by steep or convex reaches, commonly downstream. The intriguing hypothesis that late referred to as knickpoints, and that some profiles Cenozoic uplift is linked to climatically acceler are convex through their entire length. Many pub ated erosion (Molnar and England 1990; Burbank lished profiles appear to be made up of several rela 1992) emphasizes the importance of understanding tively flat reaches, linked by steeper or more con how the evolution of the rate and depth of river vex reaches. Profiles of large and small rivers in incision and subsequent canyon formation control Europe, the U.S., and Africa are characterized by the evolution of terrain. overall convex shapes, straight and convex reaches, Most studies of river longitudinal profile form and/or by sharp, local convexities that outcrop start with the assumption that the profile in ques along their respective profiles (e.g., Woodford 1951; tion is in an equilibrium condition, or at grade. McKeown et al. 1988; Pavich et al. 1989; Brice Since the work of Gilbert (1877), geomorphologists 1964; Peters 1978; L. B. Leopold pers. comm.). Re cent work by Young and McDougall (1993) docu

1 ments the persistence of steep reaches through 20 Manuscript received September 1, 1993; accepted March m.y. of profile evolution in southeastern Australia. 15, 1994. 2 Present address, Lamont-Doherty Earth Observatory of Local reaches of steep gradient usually are as Columbia University, Palisades, NY 10964. sumed to correlate either with (1) areas of more

457 458 MICHELE A. SEIDL ET AL. resistant strata in bedrock-bedded streams (e.g., include solution weathering and seepage erosion Lewis 1945; Wolman 1955; Hack 1957; Brush (e.g., Laity and Malin 1985; Baker 1990) and cavita 1961), (2) sites where coarser load is introduced tion at the base of waterfalls and the related retreat (e.g., Woodford 1951; Hack 1973), (3) loci of tec of waterfall walls (e.g., Barnes 1956). Little is tonic activity (e.g., Reed 1981; Burnett and known, however, about the relative importance of Schumm 1983; McKeown et al. 1988), or (4) the each of these mechanisms, and few field data exist upslope propagation of base level change (e.g., that either fully support or contradict the hypothe Penck 1924; Foley 1980a; Seidl and Dietrich 1992). sis that propagating knickpoints may shape stream Of these four knickpoint-forming mechanisms, the profiles. most controversial, and perhaps the most impor Here we investigate longitudinal profile devel tant, is propagation of base level change via knick opment on the island of Kauai where dated lava point migration. Penck (1924), who introduced the flows, well-preserved initial surfaces, and dis term knickpoint, recognized that propagation of tinctly varying base level conditions provide clues steep reaches up a river system would profoundly to rates and controls on profile evolution. We start change landscape-forming processes and valley by assessing whether a stream power erosion law evolution. The controversy revolves around for these boulder-bedrock channels adequately rep whether (1) knickpoints can migrate very far before resents the long-term average erosion rate of the they become indistinct and (2) whether or not channels. We then use a finite-difference model to knickpoint formation and preservation require the assess the effects of stream power dependent ero presence of a more resistant rock unit at the top of sion on profile evolution and knickpoint migra the knickpoint. Gilbert (1896) discussed the impor tion. tance of a resistant caprock to the maintenance and retreat of Niagara Falls, while Young (1985) described retreating waterfalls which lacked a re Stream Power Dependent Erosion sistant caprock and discussed waterfall retreat Most theories for longitudinal profile development mechanisms. are based on sediment continuity and alluvial sedi In alluvial channels, knickpoints have been ment transport equations and therefore implicitly shown both to propagate long distances and reju assume bedrock to be eroded as if it were composed venate fluvial transport, and to affect channel of cohesionless grains (e.g., Smith and Bretherton processes only a short distance upstream (e.g., 1972; Pickup 1977; Begin et al. 1980; Ahnert 1987; Schumm and Hadley 1957; Leopold et al. 1964; Snow and Slingerland 1987, 1990). (An exception is Leopold and Bull 1979; Begin et al. 1980, 1981; the work of Foley [1980b], who coupled sediment Gardner 1983). Foley (1980a) identified several transport equations with abrasion models derived knickpoints formed in glacial outwash materials from the engineering literature to model bedrock on tributaries to the Dearborn River, Montana, erosion.) Many channels in hilly and mountainous and connected them to episodic incision on the regions, however, cut into bedrock and lack a thick Dearborn. alluvial mantle. The sediment transport capacities Profiles incised into bedrock are often character of channels in these regions can be much greater ized by. prominent knickpoints, the locations of than the supply derived from either the canyon which appear to be related neither to lithology, walls or upstream channel delivery. In these cases tributary confluence, nor uplift (Woodford 1951). the bedrock's resistance to erosion presumably Knickpoint migration in bedrock remains poorly plays a dominant role in controlling long profile understood, although it has often been inferred evolution. A field-tested theory that models bed (e.g., Jones 1924; Gardner 1983). If knickpoints rock erosion, rather than cohesionless sediment propagate, they could be very important in trans transport, may contribute to our understanding of lating base level change into bedrock channel inci profile development in mountainous regions. sion (e.g., Reed 1981). Howard and Kerby (1983) described the parallels The detailed mechanisms by which rivers cut between erosion of cohesive soils in flume experi through bedrock remain poorly understood. Recent ments (e.g., Akky and Shen 1973; Croad 1981) and work highlights the importance of three processes erosion of bedrock in rivers. They argued that inci shaping bedrock channels: (1) erosion of channel sion is proportional to shear stress, specifically to beds by flowing water and sediment, (2) scour by the shear stress exerted on the bed by the dominant debris flow, and (3) knickpoint propagation (Seidl discharge. To test this argument, Howard and and Dietrich 1992). Other proposed mechanisms Kerby (1983) analyzed profiles developed in a 0.13 Journal of Geology LONGITUDINAL PR FILE DEVELOPMENT 459 km2 badlands landscape. By using Manning's equa Field Site tion and arguments from channel geometry, they inferred that the erosion rate of a bedrock channel Our analyses rely on observations made on rivers is proportional to the product of drainage area and located on the Hawaiian islands of Kauai, Molokai, slope raised to the 0.4 and 0.8 power, respectively. Maui, and Hawaii, with the focus placed on Kauai While Howard and Kerby's model is mechanisti river longitudinal profiles incised into gently dip cally based, it requires information about channel ping basalt (figure 1). The Kauai channels have geometry and roughness that is difficult to obtain. three features particularly useful for profile evo This motivated us to take a simpler approach based lution studies, including: (1) they are runoff- on stream power. Although our approach cannot dominated channels downcutting through uniform easily account for important roughness and chan bedrock lithology of well-constrained age, (2) they nel scale effects, it is simpler, and thus potentially have preserved initial surfaces above the present more general. Because Howard and Kerby studied day channels, enabling us to calculate long-term a microlandscape, their measured exponents and downcutting rates, and (3) they have well-char coefficients are site-specific, and the coefficients acterized and varied base level histories. Channels and exponents required by Howard and Kerby's on both the Napali Coast and Mana Plain of west model are more readily measured in their scaled- ern Kauai (figure 2) provide tests of the erosion law down system than in stream basins. and examples of river profiles developed under Following our previous analysis (Seidl and Die varying base level conditions. The Napali Coast is trich 1992), we hypothesize that the rate of river located on northwestern Kauai. The Mana Plain is incision into bedrock is proportional to the power an apron of sediments that forms a broad flat sur expended by the stream flow, i.e., face along the southern edge of the Napali Coast. Deposition of the Plain has caused the sea cliffs that were once at the coast to be isolated by the -~ = k{Qsr (i) sea (Stearns 1985). These cliffs now ring the inland boundary of the Plain. Basin Morphology. Several authors have argued where z is the elevation of the channel above a that particular Hawaiian valleys form primarily by datum, t is time, Q is river discharge, S is the local groundwater sapping and have contended that slope of the river, and k and n are empirical con morphologically similar Martian valleys formed stants. Bedrock erosion is probably an infrequent by similar processes (e.g., Baker 1988; Kochel 1988; event that usually occurs during high runoff. If we Baker 1990). This basin morphology, termed "am assume that on average these peak runoff events phitheater headed valley" by Hinds (1925), occurs scale with drainage area, A, then equation (1) can on the windward sides of the Hawaiian Islands, be written in the more convenient form, most notably along the Kohala Coast of Hawaii, and is often located in close proximity to calderas - — = kAmS" (2) and their associated dike complexes, which may serve as catastrophic seepage sources when breached. These amphitheater headed valleys are where m is an empirical constant. defined on the basis of morphology and assumed Here we examine this proposed stream power process of formation. The valleys typically have erosion law through empirical analysis and numer u-shaped cross-sections, steep walls that often ical modeling of Hawaiian river channels. We first meet the valley floor at nearly right angles, plan show that long-term erosion rates of western Kauai views that widen to a blunt shape in the upstream rivers correlate well with stream power as repre direction, non-dendritic drainage patterns, and sented by equation (2), despite a nearly continuous short tributaries located far apart from each other boulder mantle in some of these channels which (e.g., Baker 1990). Dunne (1990), however, chal probably inhibits local bedrock erosion. We then lenges the notion that particular basin morpholo use equation (2) to simulate the long-term evolu gies are related uniquely to seepage erosion. tion of stream profiles on Kauai and test whether In contrast, the channels we have chosen to the profile forms, predicted using equation (2) and study do not have either the morphology or struc reasonable initial and boundary conditions for the tural controls proposed by Baker and his col Kauai channels, are consistent with the rivers' con leagues, nor have we seen any evidence in the field temporary forms. of seepage processes. The Napali Coast and Mana 460 MICHELE A. SEIDL ET AL.

Kauai• 159° 158» Explanation . 22° Hawaiian Islands Sedimentary Deposits (Holocene) Niha Molokai 155° Koloa Volcanics (Pleistocene and Pliocene) Waimea Canyon Basalt Maui (Pliocene and Miocene (?)) Makaweli Member (Pliocene) Olokele Member 0 40 Miles ocene) Bathymetric Contour Interval Napali Member 1000 Feet (Pliocene and Miocene (?))

Contact — — • Fault Contact KAUAI 159°40' 159°30' 159°20'

Figure 1. Location map of the Hawaiian Islands (modified from Stearns 1985) and map of Kauai showing generalized geology (modified from Langenheim and Clague 1987 and Sterans 1985).

Plain channels are located on the leeward, drier widths that taper regularly toward the channel side of Kauai, with yearly precipitation ranging head, and elongate tributaries. from 100 cm at the divide to 50 cm at the coast (Macdonald et al. 1960), and are not located near Lithology. Located on the oldest portion of the dike complexes. These channels have dendritic Kauai shield, channels of the Napali Coast flow drainage patterns, v-shaped cross-sections, valley over the remarkably uniform and continuous Na- Journal of Geology LONGITUDINAL PROFILE DEVELOPMENT 461

shield volcano" (Macdonald et al. 1960, p. 27). Py- !59°4."i' roclastic rocks comprise <1% of the mass of the

t 22° 10' formation (Macdonald et al. 1960). On western N Kauai, the Napali member flows are not overlain by other volcanic rocks and the thickness of the formation is on the order of 6500 m when mea sured from the seafloor (Stearns 1985). The Napali

Napali Coast Kaawciki Valley member flows are approximately 5.1 m.y. old and Hikimoe Valley are the oldest rocks on Kauai (Clague and Dalrym- Haeleele Valley ple 1987; Langenheim and Clague 1987). Kaulaula Valley JP Paleosurfaces. On western Kauai the geometry 22°05' of the shield volcano allows the ridges above the

Ohaiula Valley channels to be identified as the initial basaltic sur face into which the channels have incised. The sur

Nahomalu Valley faces can be connected across valleys and the resul tant form is representative of the original shield

Kaawaloa Valley volcano. These paleosurfaces are considered by us to be approximately equivalent in slope and rela Niu Valley 22canyons. The paleosurfaces traverse the distance from the Pacific Ocean valley headwaters to the sea, where their seaward edges are truncated by the sea cliffs. Kauai Base Level Histories. Past and present fluctua tions in base level must be determined clearly be fore we can predict longitudinal profile evolution from an erosion law. It is now recognized that the Figure 2. Location of digitized river channels on Kauai. Hawaiian volcanoes subside 2-4 km, with the Dashed channels were not digitized. The Mana Plain is the surface to the south bounded by the dashed line and greater part of this subsidence occurring in the first the coast. The Napali Coast stretches to the north of the million years after volcanism on the sea floor be Plain. Ancient sea cliffs demarcate the inward (eastern) gan (Moore 1987). The southernmost Hawaiian is edge of the Mana Plain. Outline of box around figure lands currently are subsiding actively, whereas the corresponds to outline on figure 2. northwestern islands, including Kauai, Molokai, and Oahu, effectively have been stable for a long period (Moore 1987). pali member of the Waimea Canyon Basalt of Comparing the Napali Coast and Mana Plain Pliocene age (McDougall 1964; Langenheim and channels illustrates how boundary conditions af Clague 1987} (figure 1). Channels that flow over fect longitudinal profile evolution. The Napali the Mana Plain and into the sea also originate in Coast is an area of active cliff erosion, with sea the Napali member (figure 1). The member con cliffs reaching upward of 600 m in height. These sists of thin-bedded tholeiitic and pahoehoe flows cliffs may have been formed by a major landslide of predominantly olivine basalt erupted during the while the island was subsiding (Moore et al. 1989) shield-building stage on the island (Stearns 1985). and have been modified by wave erosion over the The flows exposed in the valley walls are uni last several million years. Similar large-scale fea formly thin, averaging several meters in thickness tures have been described on other volcanic land- and dip, on average, 6° (Macdonald et al, 1960). In forms, and the prevalence of massive landsliding dividual flow units can be traced for long distances, events over the course of volcanic evolution has and intercalated ash and soil beds are rare, as are been documented recently (Borgia et al. 1992). erosional unconformities between flows (Macdon Whereas the Napali Coast channels have been af ald et al. 1960). This continuity evidenced by the fected by cliff retreat and sea level change through lava flows is "the characteristic result of the rapid their entire history, the Mana Plain channels have accumulation of lavas of uniform basaltic composi been buffered from sea level changes during the tion on the unobstructed flanks of a highly active Pleistocene. The Plain is an uplifted coral reef 462 MICHELE A. SEIDL ET AL.

across which streams have built a large alluvial culated. Similarly, a curve was fit to the paleosur fan. The Mana Plain may have been uplifted as a face to determine the elevation of the ridge for a result of tilting during Pleistocene volcanism that given distance. The distance between the paleosur affected eastern Kauai and may be as old as 2 m.y. face and channel was then calculated for every con (D. Clague, pers. comm.). Therefore, channels in tour interval along the profile. both areas flow over the same rock formation and Profiles of the main channels draining the Na have experienced similar early histories, yet chan pali Coast and Mana Plain were digitized from U.S. nels of the Napali Coast have been subjected to Geological Survey 7 1/2' topographic maps. Eleva base level changes due to glacio-eustatic fluctua tion and distance coordinates were determined for tions whereas channels of the Mana Plain appear every contour interval along river traces. Paleosur- to have been protected from sea level fluctuations faces for each channel were similarly digitized and cliff erosion during the Quaternary. from the 7 1/2' topographic maps. Elevations and distances of the paleosurfaces were digitized ap proximately every 200 ft. Nineteen channels that Methods drain the Napali Coast and Mana Plain were digi Longitudinal Profile Analyses. The analysis pre tized (figure 2 and table 1). Three channels that sented here relies on field observations and de flow to the Mana Plain were not digitized because tailed interpretation of topographic maps. We vis they terminate far from the Waimea Canyon di ited representative channels from both the Napali vide. Three of the 12 digitized Mana Plain channels Coast and Mana Plain to confirm the accuracy of were not used in the erosion rate analysis because the maps and the assumptions made in the theory. several nearby cinder cones obscured the paleosur To test the hypothesis that bedrock channel ero faces above these streams. sion rates are proportional to stream power, we cal Drainage areas were determined directly from culated erosion rates, drainage areas, and channel topographic maps with a digital planimeter. Be slopes from topographic maps and compared the tween five and 18 separate determinations of drain empirical relationship between these factors to the age area were made along each of the longitudinal relationship predicted by equation (2). Erosion profiles. These contributing drainage areas and as rates were calculated from the difference in eleva sociated channel lengths were used to determine tion between the paleosurface and the adjacent drainage area as a function of distance from the channel. A polynomial was fit to the channel's dig divide for each of the catchments. itized profile so that, for any distance along the Channel slope values were calculated using the channel, the corresponding elevation could be cal digitized profile data. Measurements of local slopes

Table 1. Summary of Analyses of Kauai Channels

Channel Drainage Area (A) Length as a function of Erosion Rateb Area-Slopeb Basin (km) Relief Channel Length (L)a r2 (ER) + (AS) r2 n

Makaha 7.2 1.1 20.7 X LlA 1.00 (4.4 X io-* + 7.5 x 10-11) .81 66 5 11 Kauhao 7.4 1.1 4.5 X L1.6 .99 (3.7 X 10" + 3.7 x IO" ) .23 53 Kaaweiki 7.3 1.0 2.1 X I17 1.00 (-1.1 X 10"5 + 1.1 x 10"10) .95 55 Hikimoe 7.5 1.1 24.3 X L1-4 .97 (1.4 X 10"5 + 3.6 x IO"11) .86 42 Haeleele 7.9 1.0 133.2 X L11 . .99 (3.6 X 10"* + 7.7 x IO"11) .97 72 Kaulaula 9.4 1.0 116.5 X L12 .99 (7.3 X 10"* + 4.7 x IO"11) .59 70 14 Kahoaloha 10.3 1.1 25.6 X L 1.00 (6.5 X 10"* + 4.9 x IO"11) .89 75 6 Ohaiula 7.3 .7 11.4 X I1-5 .91 (4.5 X 10" + 8.6 x 10"11) .85 54 5 11 Kahoana 10.9 .7 9.3 X L1.5 .97 (1.2 X 10~ + 3.8 x 10" ) .87 57 14 5 Waipao 9.7 .7 12.7 X L .99 (1.1 X 10" + 3.5 x 10"11) .84 52 11 Paua 5.5 .4 190.8 X L 1.00 (5.9 X 10"* + 3.6 x IO"11) .96 31 11 Waiaka 7.7 .6 36.2 X L1.3 .99 (8.3 X io-* + 5.0 x IO" ) .68 37 Kapilimao 6.4 .5 3.0 X L1-7 .99 (5.1 X 10"* + 78.7 x 10-11) .90 36 E.Kapilimao 5.4 .5 .1 X L2-1 .92 (6.9 X 10"* + 9.0 x 10"12) .21 24 11 Hukipo 4.6 .4 5.1 X L1.6 .98 (7.0 X 10"* + 4.3 x 10" ) .71 29 Kaleinamanu 2.8 .3 19.8 X L14 .94 (1.4 X 10"* + 1.9 x 10"10) .97 19

' This column is the empirically determined relationship between Drainage Area (A) in km2 and Channel Length (L) and should be read as: A = 20.7 x L and so on down the column. b This column represents the empirically determined relationships between Erosion Rate (ER) and the Drainage Area (A)-Slope (S) product and should be read as: ER = [(4.4 x IO-6) + (7.5 x 10"n) (AS)]. Journal of Geology LONGITUDINAL PROFILE DEVELOPMENT 463 made in the field compared favorably with those Kauai Basins calculated from the topographic maps. The slope 1.0 10' T. . T .J_ used in this analysis was a spatially averaged value, because we are calculating long-term average ero sion rates. For any point on a profile, the slope of the present channel bed was determined by averag ing slope values over five contour intervals: two contours above and two contours below the ele vation in question. Several channel profiles had short, notably steep reaches that correlated with waterfalls in the field. These reaches of nearly ver tical gradient were excluded from the data set. The 100 m directly downstream from the channels' headwaters were similarly excluded from the an alysis because erosion rates along these upper reaches approached zero. The slope of the paleosurface above the channel was similarly averaged, 2.0 4.0 6.0 8.0 10.0 12.0 over 15 contour intervals, to generate a value repre 5 2 sentative of the initial surface. These channel and Area x Slope (10 m ) ridge slopes were combined to calculate the slope Figure 3. Calculated erosion rate as a function of the value used in the analysis. product of drainage area and slope for channels on both Erosion rates were calculated assuming a 5.1 Ma the Napali Coast and Mana Plain, Kauai. age for the basalt and were plotted against the prod uct of the measured drainage area and the com should manifest itself in differing area-length rela puted slope for points on 16 profiles (table 1). tionships in basins of disparate ages- We-chose channels flowing over uniform rocks of younger age than the Napali member flows, in areas with Empirical Analysis precipitation gradients, paleosurf ace gradients, and Empirical Test of Erosion Law. Clear relation channel lengths similar to those of western Kauai. ships exist between erosion rate, drainage area, and Channels on East Maui, which flow over the rela slope on both on an inter- and an intra-basin basis. tively uniform Kula Volcanics of Pleistocene age, Local river incision rates into basalt on Kauai range and channels on northwestern Hawaii, which are from 5 to 80 m/m.y., as compared to the 5 to 30 incising into the Hualalai Volcanics of Pleistocene m/m.y. incision rates reported in Young and and Holocene age (Langenheim and Clague 1987), McDougall (1993) for Australian rivers carved into were analyzed in the same manner as on Kauai. basalt in a similarly humid environment. Regres Only drainage basins that extended from the divide sion of the data for both the Napali Coast and to the sea were examined. This analysis indicated Mana Plain basins indicates that that area-length relationships do not vary system atically from island to island; there is enough scat ter in the data that no systematic variation in expo d 6 U - ft = (6.1 x 10- ) + [(6.3 x 10- )[AS)\, (3) nent-length relationships can be ascertained (table 2, figure 4). Although there is a 4.7 m.y. difference with an i of 0.88 (figure 3). The erosion rate was in their ages, drainage basins on the island of Maui measured in meters/year and the drainage area in appear to be well developed and similar in shape square meters. The measured drainage areas vary to the older Kauai valleys (figure 5). This result was over four orders of magnitude. Regression of the unexpected, and we speculate that the nonuniform data within each basin indicated that the exponent initial topography created by successive surface of length varied from 1.1 to 2.1 and averaged 1.5 flows contributes to the early definition of a char on Kauai (table 1). acteristic, but spatially variable, catchment area- It should be noted at this point that we are com draining distance relationship. Subsequent dissec paring a time-averaged erosion rate with measure tion leads to the creation of locally steep valley ments of current drainage area. Because of this dis walls, much steeper than the initial surface, but crepancy, we analyzed several channels on two not to an increase in drainage area. We assume that other Hawaiian islands to determine if changes in the catchment relationships given in table 1 apply drainage area could be related to increases in age. throughout the development of the profile. A systematic variation in drainage area with time In contrast to the range in drainage area values, 464 MICHELE A. SEIDL ET AL.

Table 2. Summary of Area-Length Data for Hawaii and convexities, or knickpoints (figure 6). The Hawai East Maui ian knickpoints occur as both small, local features tens of meters in length and as much larger fea Drainage Area (A) in tures hundreds of meters of length. Each knick- km2 as a function of b point is either a waterfall, from 20 to 100 m in Basin Channel Length (L ) height, or a series of 2 to 10 m bedrock steps. Hawaii: Woodford (1951) distinguishes between sharp Waiaha 1.7 X I17 1.00 nicks, or knickpoints, and longer convexities, and 2 North Waiaha .1 X L > .97 asserts that the nicks evolve into the convex Unnamed 118.9 X L" .98 reaches. For simplicity, we will call both very local East Maui: Waiopai 1.5 X L17 .93 and larger-scale, strongly convex regions knick Wailaulau 341.9 X L" .93 points. A plot of local slope against relative dis 21 Pahihi .1 X L .99 tance toward the sea along the profile reveals 17 Manawainui 1.5 X L .90 where these generally straight profiles are punctu ated with convex reaches (figure 7) and shows the greater frequency and magnitude of knickpoints in the profile shapes are often straight along some the Napali Coast channels. The effects of the lower portion of their length, and because the paleosur- boundary condition are shown clearly as well. faces are also straight, the range in calculated slope Whereas the Mana Plain slopes rapidly decline at values is small, ignoring the nearly vertical sec about 0.7(x/L) (where x is distance from the divide, tions made up of waterfalls. As such, the exponent and L is total length to the sea) due to fan construc on slope is not as tightly constrained as it would tion across the Plain, the Napali Coast channels be if the slopes of the present day river profiles steepen at about 0.4(x/L) and maintain an average varied more. slope of 0.15 to the sea. The Mana Plain channels Longitudinal Profiles. For the empirically de all have about the same mean slope (0.10), and this rived erosion law to be applicable to longitudinal is less steep than the 0.15 slopes of the Napali profile evolution, it should predict at least the gen Coast channels. eral properties of the observed Kauai channels. Most casual impressions of waterfalls formed in Here we describe the digitized profiles to compare volcanic terrain and elsewhere are that they origi them with subsequent model predictions. In sharp nate due to some resistant unit, typically an apha- contrast to the classic concept of a graded profile, nitic dike, and greatly retard channel incision. longitudinal profiles of channels on the Mana Plain Many waterfalls in Hawaii originate this way, but have straight to weakly convex profiles (figure 6), in our field area this does not appear to be the case. whereas the Napali Coast channels are character Map traces of Mana Plain and Napali Coast chan ized by straight reaches punctuated by distinct nels are generally free of mapped dike localities (Macdonald et al. 1960). Six valleys do contain such mapped intrusions, including Milolii Valley, 2.2 with nine dikes mapped in the lower reaches, Ma- kaha Valley, with five dikes mapped crossing the lower valley, Hikimoe Valley, with one mapped intrusion. Nahomalu Valley, Kaawaloa Valley, and 1.8 — Kahoana Valley, all with several mapped locations (Macdonald et al. 1960). The accuracy of the intru -E 1.6 — sion locations has been called into question; in deed, the mapped dikes probably do not exist (D. 1.4 — Clague, pers. comm.) On Kahoana Valley a mapped dike locality corresponds to a channel knickpoint. 1.2 — This site was visited in the field, however, and no sign of an intrusion was noted. The rock constitut ing the knickpoints that we observed in the field 0.1 1 10 is indistinguishable from the surrounding bedrock. Age of Rocks (Million years) Dikes may have been mapped on the basis of waterfalls rather than from independent data; thus Figure 4. Drainage area-length relationships for the three Hawaiian Islands of Kauai (5.1 m.y.—dots), Maui dike mapping may assume that dikes cause knick (0.4 m.y.—triangles), and Hawaii (0.1 m.y.—squares). points. On channels with both pronounced knick- Journal of Geology LONGITUDINAL PROFILE DEVELOPMENT 465

KAUAI

Pacific Ocean Figure 5. Outlines of drainage basins on Kauai and East Maui. Note that both drawings are at the same scale. points and mapped intrusions, the two features oc channels in the two areas evolved, when con cur in the same place on only one valley, Milolii trasted with their similarities, suggests that the Valley, which was inaccessible in the field. In two knickpoints may be formed by base level changes, of the valleys containing mapped dikes, Makaha but may, without continued base level fluctua Valley and Hikimoe Valley, the mapped intrusions tions, die out over time. are located kilometers downstream of the digitized Dramatic morphologic changes occur in the val knickpoints. In general, the locations of the Kauai leys above and below the knickpoints. Upstream knickpoints do not correspond to sites of more re of the knickpoints the adjacent hillslopes are soil- sistant strata. mantled and have relatively gentle gradients. The Channels crossing the Mana Plain have been bedrock outcropping in the channel bed and banks buffered from sea level change for a long time and is extremely weathered. Downstream of the knick display singularly straight to slightly convex pro points, valley walls have steep, often nearly verti files (figure 6). Conversely, Napali Coast channels cal gradients three to five times greater than those may have been subjected to sea level fluctuations above the knickpoint and lack substantial soil and cliff retreat for several more million years and cover. The digitized channel reaches below the exhibit profiles characterized by marked steps knickpoints have remarkably uniform, straight (figure 6). Although knickpoints presumably slopes of approximately 8°. Channel incision is a formed on the Mana Plain channels at the time of boundary condition to hillslope development, and sea cliff formation, knickpoints are generally not a change in the process of incision across the wa present on the profiles; channels bounded by the terfalls, from propagating knickpoints downstream Mana Plain would have ceased generating knick to abrasion and dissolution upstream, could ex points at the onset of uplift of the coral reef. These plain the distinctive landscapes. different boundary conditions under which the Our field surveys also revealed that significant 1000 - —•—1—.—1—.—1—.—1—.— 1200 - , 1 . 1 . 1 . 1 .

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Figure 6. Representative profiles from Kauai with elevation and distance plotted in meters on the y- and x-axis, respectively. The straight to weakly convex Mana Plain profiles are shown on the left; the flat channel segments beginning at the sea correspond to the Mana Plain. Napali Coast channel profiles are shown on the right. Note the sharp knickpoints and longer convex reaches that punctuate the profiles. Open squares represent the channel profiles and solid squares represent the paleosurfaces above the channels. Note that the amount of vertical exaggeration differs from profile to profile. Journal of Geology LONGITUDINAL PROFILE DEVELOPMENT 467

1.0 fining of sediment either in the channel or on the

s Hukipo boulder terraces in the downstream direction. We X Kapilimao D Waiaka saw no clear evidence of flood transport of large 0.8- + Paua boulders. Debris flow mobilizaiton of material A Waipao I Kahoana probably predominates on the tributaries, but the O Kaawaloa 0.6-- o Nahomalu main channel gradient is close to depositional o Kahelenui - Ohaiula grade and so debris flow scour is less likely. The large, apparently immobile boulders impose 0.4 another constraint on the river incision process. Rivers can overcome the resistance to erosion of fered by boulders and cut into the underlying bed rock by (1) moving them and breaking the boulders down during large flood flows, especially in narrow canyons where flows can be deep, or during mobili zation of debris flows (e.g., Baker 1977; Pierson 1980; Baker and Pickup 1987; Benda 1990; Miller 1990; Wohl 1992), (2) eroding decay products of weathered boulders (Schumm 1977), and (3) un dermining the boulder mantle by propagating knickpoints that are significantly thicker than the boulder pile size (Seidl and Dietrich 1992). None of these processes fit into a conventional cohesionless sediment transport law normally used in pre dicting longitudinal profile evolution. In a river that has cut progressively through bedrock, even when receiving untransportable boulders, either

' 0.4 the boulders must be removed through weathering and breakdown, or they must be undercut by prop agating knickpoints. The former process may still be crudely accounted for in a simple stream power like erosion law if breakdown can be thought of as slowing the rate of bedrock erosion. This presumes that the weathering rate is spatially constant and that the boulder influx rate is not sufficient to in hibit eventual bedrock erosion. Cosmogenic iso Figure 7. Plot of slope against normalized distance (x/ tope analyses of boulder olivines on one of the Na- L) for representative Mana Plain channels (top) and Na- pali Coast channels, Kaulaula Valley, indicate that pali Coast channels (bottom). larger boulders may lie in the valley for at elast 180,000 yr (Seidl 1993), suggesting that the rate of breakdown is relatively slow. In the modeling portions of the channel lengths above and below to follow, we will account for both the progres knickpoints are mantled with boulders. Below the sive downwearing and the propagating knickpoint knickpoints, the boulders are very large with an hypotheses. average diameter of several meters, often angular, interlocked in a fashion that shows no evidence of Theoretical Analysis reworking by the stream and appear to be weather ing in situ. Most boulders are stored in debris ter Model Framework. A finite-difference model races adjacent to the channel. These terraces are was developed to predict the present general form angular ramps of material rising 2-8 m above the of river profiles in western Kauai from the inferred present channel. The channel itself is mantled by initial conditions and our empirical erosion law this material over much of its length, but the peri (equation 2). If equation (2) is valid, it should at odic outcropping of resistant bedrock in the chan least predict the gross features of the longitudinal nel bed and banks indicates the valley is not filled profiles from which it was derived. In our one- by a deep layer of boulders. There is no noticeable dimensional model, drainage area was replaced 468 MICHELE A. SEIDL ET AL. with distance from the divide using an empirical sea level rise and fall are incorporated in (a). If the correlation and our erosion law thus becomes, offshore gradient is steep, a drop in sea level will be accompanied by a steepening of the distal profile. 4| Similarly, sea level rise accompanied by erosive -if--*K£)' ' wave action at the river-sea confluence will result in an increase in channel gradient at the conflu in which x is distance from the divide, (— dz/dx) is ence. The steep slope imposed on the profile in (a) slope, c is 6.5 x 10 ~4, b equals 1.5, and n equals acts as a proxy for both these phenomena. 1.0, where lengths are in meters and time is in Straight Initial Condition. This model represents millions of years. A value of b equal to 1.5 is used the case of profile evolution with a fixed base level because the A a Lb relationships average roughly from the initial, essentially straight paleosurface A a Lls (table 1). created by successive lava deposition. If knick- The initial condition in most modeled cases was points formed and propagated, it is assumed to be similar to observed paleosurfaces and consisted of of secondary importance. This model simulation a constant slope of 0.08, relief of 1 km, and a divide would apply most closely to the profiles on the positioned 12 km from the sea. The length and re Mana Plain which have had a fixed base level with lief values correspond to linear projections of the out cliff retreat. Figure 8a shows the predicted pro digitized ridge topography downstream from the file evolution. Note that the profile consists of two Waimea Canyon divide, across and beyond the sea distinct reaches: a straight to weakly convex por cliffs, to present sea level. Because the extensions tion that becomes progressively steeper through parallel the straight ridge surfaces and the offshore time and a reach of zero slope that extends toward gradient is low, a different sea level would affect the divide through time. No erosion occurs at the the length and relief of the initial profiles, but divide because there x equals zero, hence the pro would have little effect on the slope. A steep off file rotates about this hinge point. shore gradient would make such a topographic Qualitatively this result is in good agreement reconstruction less straightforward. Island subsi with the Mana Plain profiles (figure 6), which have dence is considered to have occurred during the approximately straight upper reaches showing lit shield building stage and prior to this recon tle or no erosion at the divide and increasing inci structed stage. This simple projection is therefore sion along the profile in the downstream direction. assumed to be a reasonable proxy for the initial The flat lower reaches shown in figure 8a corre volcanic surfaces. For each profile, 100 nodes were spond to the Mana Plain depositional surface and used, which is similar to the number of map con should not be equated with the reaches of zero tour intervals used to measure the actual channel slope predicted by the model. The upper reaches of profile. Starting from this initial condition, each Napali Coast channels also display notably straight node was iteratively lowered according to (4), with profiles with little or no erosion at the divide and the coefficients specified above. This erosion pro successively more incision downstream along the cess was repeated for the equivalent of 4 m.y. profile. In all simulations, the divide elevation and the The reaches of zero slope near the sea occur in sea level elevation remained fixed. The two domi the model because base level remains fixed during nant base level effects on the Kauai channels are simulation. Clearly such a zero slope does not oc the sudden formation of sea cliffs due to massive cur in nature. Instead, deposition of boulder-sized landslides and the slower shoreline retreat and sea material along the lower channel reaches may both cliff formation through wave action, chemical restrict incision and be instrumental in setting the weathering, and mechanical failure. These two 5-8° channel slopes found on the profiles. The de processes might be expected to affect the profile position of fairly uniformly sized material is likely differently. Sea cliffs were therefore simulated in to yield a straight slope. This process is not in two different ways: (a) by altering the initial condi cluded in our simple model, but may be added in tions and imposing a steep slope over a long reach subsequent analyses. of the paleosurface and (b) by imposing a constant In the model, the approximately straight to rate of cliff retreat, which formed steep sea cliff weakly convex reaches result from the relatively like features at the coast. Initial condition (a) ef low value of the exponent m in the distance term fectively modeled a massive landsliding event, of equation (4). whereas (b) simulated the modifications wrought Steepened Reach Imposed on Initial Profile. This by more gradual erosional processes. The effects of model represents profile development after a mas- Journal of Geology LONGITUDINAL PROFILE DEVELOPMENT 469

1000 Massive landslide 800 + n = 1

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Distance (m) Distance (m) Figure 8. Longitudinal profile development predicted by numerical solutions of stream power dependent erosion law assuming m and n equal to 1.0, unless otherwise stated, and (a top left) a stable base level, and changing boundary conditions simulated by [b top right) imposition of a steep reach, (c bottom left) cliff retreat continuing for 4 m.y. and [d bottom right) imposition of a steepened reach with slope exponent changed to 2.0. sive landslide. A steepened reach 300 m in height initially straight slope. This rate corresponds to with a slope of 0.6 was imposed on the profile. that calculated for the western Kauai channels as The height and length of the step approximate the suming an initial shield volcano geometry, pro Napali Coast cliffs. jecting the present day paleosurfaces out to sea, The initial profile looks very similar to the Na and assuming a 4 m.y. time interval. In figure 8c, pali Coast channels. We have hypothesized that the cliff retreat function runs for the full 4 m.y. such a change in base level will result in the forma The resultant profiles are characterized by no ero tion and propagation of knickpoints. The main sion at the divide, straight to weakly convex pro purpose of this model run was to see if the cata files, and a steep reach that gains in height through strophic generation of a 300 m high step resulted time at the advancing shoreline. As in the case of in the migration of the step headward. As shown the landsliding event, the steep reaches do not in figure 8b, the height and slope of the steep fea propagate upslope as knickpoints but instead gain ture are preserved on each successive profile; the in elevation over time, while retreating in a paral area of steeper gradient does not appear to either lel fashion. Although the increase in height of the wear back or migrate upstream. feature might appear at first glance to be mimick Cliff Retreat Rate Imposed on Straight Initial Pro ing a propagating feature, the formation of the file. We modeled the effect of progressive cliff re cliffs does not result in the formation of a knick- treat on profile development by shifting base level point that migrates faster upslope than the cliff re toward the divide at a rate of 1 m/1000 yr on an treat rate. 472 MICHELE A. SEIDL ET AL.

tween different Hawaiian watersheds with similar modeling profile evolution requires both a theory age, rock, climate, and boundary conditions. These for boulder production, breakdown, and removal consistencies suggest that the same erosion pro and a theory for knickpoint evolution. The advent cesses are operating in different locales. We have of surface exposure age dating using cosmogenic shown that bedrock incision on some channel isotopes may allow a quantitative investigation of reaches correlates with a drainage area-slope prod the relative rates of incision by both the proposed uct, a surrogate for stream power. Analyses of the processes, as well as assessment of the residence straight and convex longitudinal profiles of Kauai time of boulder-sized material in the basins (Seidl streams suggest that base level history may affect 1993). Models incorporating these processes should the evolution of channel profiles. Stable base level yield important insight into how landscapes re results in the development of straight profiles, spond to climatic fluctuations and uplift. such as are observed in channels draining to the Mana Plain. Changes in base level may form knickpoints which propagate upstream, perhaps re sulting in the steep reaches found on the Napali ACKNOWLEDGMENT Coast channels. Knickpoint propagation may be a We gratefully acknowledge field support and criti primary process of bedrock lowering along many cal discussions with K. Loague, N. Shubin, R. Tor channels and reflects base level history, as well as res, A. Howard, and M. Power. We also thank local tectonic events. V. Baker for raising important questions with us. A simple stream power-dependent erosion law This work was partially funded by the National cannot fully account for the evolution of stream Science Foundation, the Lawrence Livermore Na channel profiles on western Kauai. Longitudinal tional Laboratory Institute of Geophysics and Plan profile development is complicated by many fac etary Physics, the American Association of Uni tors, including sediment supply, storage effects, versity Women, and the Geological Society of and weathering. We believe that comprehensively America.

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