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Agriculture, Ecosystems and Environment 142 (2011) 95– 101

Contents lists available at ScienceDirect

Agriculture, Ecosystems and Environment

jo ur n al homepage: www.elsev ier .com/ lo cate /agee

cale effects in Hortonian surface runoff on agricultural slopes in West Africa:ield data and models

ick Van de Giesena,∗, Tjeerd-Jan Stomphb, Ayodele Ebenezer Ajayi c,d, Fafré Bagayokoe

Delft University of Technology, Stevinweg 1, 2628 CN Delft, NetherlandsWageningen University, Haarweg 333, 6709 RZ Wageningen, NetherlandsFederal University of Technology, PMB 704 Akure, Ondo State, NigeriaFederal University of Lavras, CEP 37200-000 Lavras MG, BrazilGTZ, Bamako, Mali

r t i c l e i n f o

rticle history:eceived 12 October 2009eceived in revised form 3 June 2010ccepted 8 June 2010vailable online 7 July 2010

eywords:urface runofforton flowcale effectsest Africa

a b s t r a c t

This article provides an overview of both experimental and modeling research carried out over the past15 years by the authors addressing scaling effects in Hortonian surface runoff. Hortonian surface runoffoccurs when rainfall intensity exceeds infiltration capacity of the soil. At three sites in West Africa (Côted’Ivoire, Ghana, and Burkina Faso) runoff was measured from plots of different lengths to assess scaleeffects. Consistently, longer plots showed much lower runoff percentages than shorter plots. There werelarge variations in runoff percentages from one rainstorm to the next but there were very good correla-tions between plots of equal length for each single event. This strongly suggests that temporal dynamicsare the cause behind the observed scale effects. In the literature, spatial variability is often proffered asexplanation for such scale effects without providing a mechanism that would cause consistent reductionin runoff percentages with increasing slope length. To further examine whether temporal dynamics can

indeed provide the explanation, Hortonian runoff was simulated using models with increasing levelsof complexity. The simplest model was already able to reproduce the observed scale effects. Also morecomplex models were used that accounted explicitly for spatial variability. The conclusions remained thesame regarding the role of temporal dynamics. Finally, a dimensional analysis was developed that helpspredict under which circumstances one can expect scale effects similar to the ones observed in WestAfrica.

© 2010 Elsevier B.V. All rights reserved.

. Introduction

Water stress is an important constraint on agricultural pro-uctivity in arid and semi-arid regions, such as West Africa. It

s important to understand how much water is available overime in the root zone in order to make optimal use of availableater resources. When rainfall is intensive, as is typical for tropical

egions, part of the rain falling on sloping fields will not infiltratento the root zone but runoff over the surface. This surface runoff

oes not only constitute a local water loss but may also cause ero-ion and loss of nutrients. The main objective of this paper is toescribe the spatial and temporal dynamics of the surface runoff

∗ Corresponding author.E-mail addresses: n.c.vandegiesen@tudelft.nl (N. Van de Giesen),

jeerdjan.stomph@wur.nl (T.-J. Stomph), ayo.ajayi@gmail.com (A.E. Ajayi),bagayoko@yahoo.fr (F. Bagayoko).

167-8809/$ – see front matter © 2010 Elsevier B.V. All rights reserved.oi:10.1016/j.agee.2010.06.006

process by presenting field measurements and models. Water thatruns off over the surface does not necessarily keep running off untilthe bottom of the slope has been reached. Water may infiltratea bit further downhill once the rain has stopped or has becomeless intensive. Such redistribution makes that the total runoff froma slope tends to be less than the average point runoff times theslope length. Longer slopes or plots produce less runoff per meterlength than shorter slopes or plots. Quantitative understanding ofthis scale effect is essential not only for the development of cor-rect hydrological models but also the design and implementation offield-scale measures that aim to capture surface runoff and reduceerosion.

Surface runoff that occurs when the infiltration capacity of thesoil is exceeded by rainfall intensity is called Hortonian runoff

(Horton, 1933). When this happens, water will accumulate on thesurface and start to run downhill. Hortonian flow, or infiltrationcapacity excess flow, is usually contrasted to so-called Dunne flow(Dunne, 1978). Dunne flow, or saturation excess flow, occurs when

9 system

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6 N. Van de Giesen et al. / Agriculture, Eco

ater storage capacity of soil is exceeded and rain can no longer betored. Dunne flow typically occurs in bottom lands while Hortonow can mainly be found on slopes.

Once Horton flow occurs, local (micro) depressions fill up. Afteruch depressions are filled, they will be overtopped and water willtart to flow downhill. The flow pattern looks like “a shallow sheetf water with threads of deeper, faster flow, diverging and con-erging around surface protuberances” with mixed occurrence ofurbulent and laminar flow (Abrahams et al., 1986). This type of flowncludes a braiding pattern of water threads, without the completelope being covered by water. As long as the threads of water do notonverge into rivulets that may start rills and gullies, we speak ofheet flow. All observations and analysis here concern sheet flow.nce gullies start to form, very different processes become relevantnd the concepts put forward here are no longer relevant (Bryannd Poesen, 1989). It should be pointed out that, at watershed level,ther processes, such as Dunne flow and groundwater flow, becomeelevant as well. Our analysis concerns Hortonian flow only andtops at the bottom of the slope or when rills are formed, whicheveromes first.

Within the environmental sciences, there are many types ofcaling issues and phenomena. It is, therefore, important to definexactly the type of scaling we are considering. For the purposes ofhis paper, we are only interested in Hortonian sheet flow. A com-

on way of measuring this type of runoff in the field is by isolating small plot on a slope by inserting metal sheets at the top and sidesnd a gutter at the bottom. The water collected in the gutter is theneasured, either as total runoff after an event or as a flux during

rainstorm. It has been observed in many studies that the size,nd especially the length, of the plot is important when it comes tohe relative amount of rain that is captured in the gutter (Bagarellond Ferro, 2004; Blöschl and Sivapalan, 1995; Esteves and Lapetite,003; Gomi et al., 2008a; Lal, 1983, 1997; Sidle et al., 2006; Vane Giesen et al., 2000; Wilcox et al., 1997; Yair and Lavee, 1985).etween formation of runoff at a point and runoff reaching the bot-om of the plot, more water infiltrates than would be expected onhe basis of observations at scales smaller than the plot (see alsobrahams et al., 1995). The type of scaling issue addressed here isoncerned with understanding the processes of runoff generation,nfiltration, and surface flow.

The relevance of considering this type of scale effect is, at least,wofold. First, it is very common in so-called distributed modelso treat pixels as points and to simply route whatever point runoffhere may be to a lower lying pixel. As we will see, even whenelatively small pixels of, say, 10–100 m are used, this approach isften not valid, and sub-pixel scale effects will need to be takennto account. Second, the scaling effect is relevant for the designf management options within watersheds. To ensure optimal usef water and associated nutrient and sediment fluxes, one oftenants to ensure that surface water does not move over long dis-

ances along the slope. At what scale measures need to be taken,ill depend on the scale effects described and analyzed here.

The geographical scope of this paper is West Africa, where weid all our experimental work at sites that are typical for large partsf this region. The geography of West Africa is such that slopes areelatively moderate (1–8%). The agricultural land-use was repre-entative for the region with homogenous vegetation cover for eachite (upland rice, maize, and sorghum). Strictly speaking, the directbservations are limited to plots and do not include complete slopeslthough watershed scale studies point towards similar resultsMasiyandima et al., 2003). We focused on plot studies becausehis allowed us to vary, at each site, only one variable, namely slope

ength, while keeping all other variables the same. In addition, plottudies allowed for better parameterization of the experimentalurfaces and eliminated effects of breaks in slopes and upstreammpervious surfaces. More complex hillslope processes such as

s and Environment 142 (2011) 95– 101

hydrophobic soils, preferential sub-surface flow paths, and exfiltra-tion zones, were not observed and are not included in the modelingapproaches. Probably the most defining feature of the systems stud-ied is the high intensity and short duration of runoff generatingstorms, which greatly reduces complicating factors such as mois-ture redistribution effects.

This paper presents an overview of field work and modelingactivities over the past fifteen years. Emphasis lies on activities bythe authors with new work presented on recent field experimentsin Ghana and Burkina Faso and dimensional analysis. The next twoparts, “2 Field experiments” and “3 Modeling procedures” are thecore of the remainder of this paper. Under the field experiments, wedescribe the main results of measurements of scale effects at threesites in West Africa, in Côte d’Ivoire, Ghana, and Burkina Faso. Cli-mate and topography vary between the three sites but all threeshowed very clearly that longer plots do not produce commensu-rately more runoff than shorter plots. After the presentation of theempirical basis, we present an overview of different models wedeveloped to investigate the mechanisms underlying the observedscale effects. The models show different levels of sophisticationbut they all show that temporal dynamics of the rainfall-runoff-infiltration pattern can explain the observed scale effects. The paperends with conclusions and discussion.

2. Field experiments

Scale effects have been observed at several field sites. Here, wedescribe in some detail the experiments conducted by the authorsat three sites in West Africa. All sites fall within the Koppen class Aw‘equatorial, winter dry’, but have different average rainfall amountsand distributions of rainfall within the wet period.

All three experimental sites had a similar basic set-up withplots of different lengths on slopes with similar characteristics.The slopes were representative for their respective agro-ecologicalzones (Windmeijer and Andriesse, 1993). Land-use was agriculturalwith maize (Zea mays) in the more humid zone, upland rice (Oryzasativa) in the intermediate zone, and sorghum (Sorghum bicolor) inthe driest zone. These three species are important crops through-out the region. All slopes were relatively smooth and did not exhibitcomplex hillslope processes such as exfiltration. At all sites, rain-fall and total plot runoff was measured after each event. The ratiobetween total runoff and total rainfall is the runoff coefficient. Thecomparison of runoff coefficients between plots of different lengthswithin each site shows the occurrence of scale effects.

At each site, measurements were made regarding rainfall inten-sity distributions during the events. Only in Ghana was it possibleto also measure plot runoff over time. At all sites, all runoff pro-ducing events consisted of so-called rain squalls, which producevery heavy rainfall (150–250 mm h−1) over relatively short periods(20–40 min).

2.1. Bouaké, Côte d’Ivoire

The first experimental site was located at the M’bé experimentalfarm of the West Africa Rice Development Association (WARDA).The location (7◦42′N, 5◦06′W) is a few kilometers north of the townof Bouaké. The climate is sub-humid with an average rainfall of1200 mm/year. The climate zone is the Guinea savanna (KoppenAw). The distribution of the rainfall is pseudo-bimodal with a smallpeak in March–May and the main rains in August–October. Mostrain is associated with squall lines moving over the sub-continent

from East to West. During such storms, rainfall intensities of up to200 mm h−1 were measured.

The landscape consists of lightly undulating plateaus, valleybottoms, and slopes connecting the plateaus to the valley. The

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xperiments were undertaken on the slopes, which slant at about%. The soils on the slopes were sandy Orthi–Luvic Arenosols. Thenderlying geology is granitic, which is typical for large parts ofest Africa (Windmeijer and Andriesse, 1993).The experimental set-up consisted of three sets of paired plots,

ll with a width of 0.8 m. Each pair consisted of a short (1.25 m) andong (12.0 m) plot. The plots were surrounded by metal sheets and

ere equipped with a gutter at the bottom that collected runoff intoil drums. After each rainfall event, the drums were emptied andotal runoff was measured. Rainfall was measured with totaling andipping bucket raingauges. Measurements were taken during the996 rainy season. Two pairs of runoff plots were treated with her-icide during the fallow period preceding the experiments, causing

ess biological activity and an associated reduction in infiltrationapacity. All three pairs were planted with upland rice (Oryza sativa. cv Bouake 189) at the start of the experiments. Plant density was6 plants m−2, planted in a regular 0.25 m by 0.25 m grid. Plots wereand weeded four weeks after planting.

The results showed a clear reduction in relative runoff withncreasing slope length for all treatments (Van de Giesen et al.,000). Runoff percentages varied widely between rainfall eventsut showed a good correlation between the two herbicide treatedairs, for which the average runoff coefficient of the short plotsas 42% and for the long plots 24%, a reduction of more than 40%.

or the untreated pair, with higher infiltration capacity, the runoffoefficient was 24% for the short plot and 6% for the long plot.

.2. Ejura, Ghana

The second experimental site was located in the Kotokosuatershed (7◦20′N, 1◦16′W) near Ejura, Ghana (Ajayi, 2004). The

limate zone here is the transition zone between forest and savannaKoppen Aw) and can be characterized as humid with a long-termverage rainfall of 1445 mm/year. The distribution of the rainfalls bimodal with the main season running from April through Julynd a minor rainy season from September to mid-November. Ashroughout non-coastal West Africa, also here most rain falls duringhe passing of line squalls. During the observation period, maxi-

um intensities of 240 mm h−1 were measured.The landscape is more hilly than the landscape at the site in

ôte d’Ivoire with heights from 60 m to 300 m ASL. On the higherarts, the soils were acrisols and at the bottom of the slopes, mainlylinthosols were found. The underlying geology consists of Voltaianandstones. Strong bio-turbation through earthworm activity wasbserved throughout the site. The experiments were undertakenn the lower slopes, with a slope of 7%.

The experimental set-up consisted of two sets (A and B) of fourlots each. Metal sheets of 50 cm height were driven 20 cm into theoils to ensure that no outside water could enter the plots. Eachet consisted of twin long plots of 2 m × 18 m, one medium plot of

m × 6 m, and one short plot of 2 m × 2 m. The plots were orienteduch that the longer sides were perpendicular to the contour lines.unoff was collected in an aluminum trough at the bottom, fromhere it was lead to tipping buckets to measure runoff over time.

otal runoff was measured as well.The plots were well characterized with detailed elevation and

nfiltration measurements, and vegetation characterization (Ajayi,004). A full set of micro-meteorological observations was collected

ncluding rainfall intensities. Also soil moisture was monitored atifferent depths through the use of Delta-T Theta probes (Delta-T,ambridge, UK). These additional measurements later allowed forhe incorporation of spatial variability in runoff modeling.

Also the results in Ejura showed important variations in runoffoefficients between events. The correlation between runoff totalsor individual events was good for the twin long plots of set A withorrelation coefficient r = 0.94 and somewhat weaker for the twin

s and Environment 142 (2011) 95– 101 97

long plots in set B with r = 0.87. The average runoff coefficient forthe four long plots was 5.4%. For the two medium length plots, theaverage runoff coefficient was 14.3%, and for the short plots theaverage runoff coefficient was 27.0%.

2.3. Kompienga, Burkina Faso

The third West African experimental site was located near Kom-pienga (11◦05′N, 0◦48′E) in eastern Burkina Faso (Bagayoko, 2006).The climate zone here is drier than the previous two sites and iscalled the Sudan savanna (Koppen Aw). The rainfall regime is sub-humid or semi-arid with an average rainfall of 950 mm/year anda potential evaporation of 2200 mm/year. The distribution of therainfall is monomodal with a rainy season starting in May/June,peaking in August/September, and ending in October. Rainfallevents are again brief, less than 1 h, and very intense, with intensi-ties well over 200 mm h−1.

The landscape is relatively flat with slopes of 2–3%. Depend-ing on land-use intensity, the landscape is naturally covered withgrasses, sparse trees, shrubs, or woody savanna. The soils aremainly Lexisols with sandy-loam to loam textures. Granites formthe underlying geology, as witnessed by regular rock outcrops.

Two sites were used differing in land-use intensities. The lowland-use intensity site was located in a national park. At each site,two runoff plots were installed by surrounding the plots with metalsheeting that was partially driven into the soil. At each site, thesmaller plot was 0.8 m wide and 1.25 m long and the larger plot was2 m wide and 5 m long. At the bottom side of the plots, a gutter col-lected the surface runoff that was subsequently collected in buriedoil drums. During the 2004 rainy season, total runoff amounts weremeasured after each runoff producing rainfall event. The plots hadslopes of 2%.

The general results at the Kompienga site were comparable tothat of the other two sites; large variation in runoff coefficientbetween events and a consistent reduction in runoff coefficientbetween short and long plots. For the site with the higher land-use intensity, the runoff coefficient was 35% for the short plot and5% for the long plot. For the one with the lower land-use inten-sity, the runoff coefficients were 14% for the short plot and 2% forthe long plot. In both cases, the reduction in runoff coefficient is86% between the short and long plots. The differences in runoffbetween the sites follow a logical pattern with increasing runoffwith increasing land-use intensity. The very large reduction inrunoff coefficients is mainly due to the fact that the slopes werevery gentle.

The above cases show that on relatively smooth slopes in WestAfrica, significant scale effects are found over a range of settings.Slopes varied from 2% to 7% and rainfall regimes from 900 mm/yearto 1445 mm/year. The variation in runoff coefficients found fromone event to the next, combined with the high correlation betweenplots, could be regarded as an indication that temporal dynamics,and not spatial variability, are the cause of this phenomenon. Forfurther analysis of underlying causes, modeling is needed.

3. Modeling procedures

Throughout West Africa, we found that short plots had signif-icantly higher runoff coefficients than long plots. To determinewhether temporal dynamics are sufficient and necessary to explainthis reduction in runoff coefficients, model studies were under-taken. Hortonian runoff was simulated with three increasingly

complex models, Runoff01, LeapFrog, and LISEM. The first model,Runoff01, consisted of a simple infiltration model coupled to akinematic wave hydraulic routing model. The function of Runoff01was to determine if the temporal variation in measured rainfall

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Fig. 1. Runoff ratios. Ratio of the runoff coefficient of the total slope and the short-

8 N. Van de Giesen et al. / Agriculture, Eco

nd the associated changes in surface runoff over time, can indeedause a reduction in runoff with increasing slope length. Runoff01id indeed prove that the temporal dynamics could explain thebserved scale effects (sufficiency). To investigate if temporalynamics are the only process necessary to explain the observedcale effect, and to ascertain that other processes may not also giveise to scale effects, more complex models were built (necessity).pecifically, the more complex models (LeapFrog and LISEM) weresed to examine the effects of spatial changes in roughness andegetation. The more complex models confirmed that spatial vari-bility in surface characteristics does not play a significant role withespect to scale effects and that temporal dynamics are needed.inally, with sufficiency proven and necessity made plausible, weeveloped a dimensionless analytical solution to the basic equa-ions. The reason for developing a dimensionless solution is that itrovides direct qualitative insight in the relevant parameters andheir role in causing scale effects. Such insight cannot directly bebtained from the earlier numerical models. Here, we present withmphasis on results first the simple model Runoff01, followed byhe results from the more complex models, LeapFrog and LISEM.his section finishes with the dimensionless analytical solution.

.1. Runoff01

To explain the experimental results, a first simple model waseveloped called Runoff01 (Van de Giesen and Stomph, 2003; Vane Giesen et al., 2005). Runoff01 is simple but does fully accom-odate variations in rainfall over time and has realistic infiltration

ynamics.Runoff01 uses two very simple sub-models. The first sub-model

alculates the runoff at a point as the difference between the cal-ulated infiltration capacity calculated with the Philip-Two-Termquation (PTT) (Philip, 1957) and the measured or given rainfallntensity. The second sub-model routes the point runoff as a kine-

atic wave along the slope until the bottom of the slope is reached.ffects of micro-topography or surface roughness are not mod-led explicitly but are to a large extent implicitly captured in theoughness parameter and flow exponent of the kinematic waveAbrahams et al., 1986).

At the start of a rainfall event, all rain will infiltrate and itill take some time before the infiltration capacity is exceeded.

he moment infiltration capacity is lower than rainfall intensity isetermined using the Time Compression Algorithm (TCA) (Reevesnd Miller, 1975; Mls, 1980). Underlying assumptions are that infil-ration capacity is fully dependent on accumulated infiltration andhat no water redistribution occurs over the soil profile over thehort period to reach saturation of infiltration capacity. The lat-er is particularly appropriate for the typical high intensity shortainfall events in arid, semi-arid and sub-humid tropics for whichhe model was developed. Once rainfall intensity exceeds the infil-ration capacity equivalent to a cumulative amount of infiltratedainfall, ponding starts and thus water becomes available for runoff.nfiltration capacity after ponding is calculated with the PTT:

(t) = St−1/2 + A (1)

here I is the instantaneous infiltration rate (m/s), t is time (s), Ss the sorptivity (m s1/2), and A the effective hydraulic conductivity

hich is set here equal to the saturated hydraulic conductivity, Ksat

m/s). The PTT is a second-order development of the Richards’ equa-ion and effectively models vertical moisture redistribution. Theorptivity accounts for antecedent moisture status (Philip, 1957).

The ponded water is subsequently routed downhill by solving

he kinematic wave equation with the method of characteristics.he model allows for temporary drying up of the slope during theourse of an event. The validity of the combination of PTT and kine-atic wave for relatively smooth slopes such as those found in West

s and Environment 142 (2011) 95– 101

Africa has been tested by the authors under both laboratory condi-tions (Stomph et al., 2001, 2002) and field conditions (Van de Giesenet al., 2000). The model is capable of reproducing scale effects asobserved in field and laboratory. The simplicity of the sub-modelslead to a low parameter requirement, while they seem the sim-plest combination to capture all relevant processes to understandthe here analyzed scale effects in Hortonian runoff. This simplemodel does not account for spatial differences in roughness andinfiltration or more complex redistribution patterns of rainfall dur-ing rainfall events. Runoff01 does, however, allow us to concludethat temporal dynamics can explain the observed scale effects. Tosee if neglected processes also are important, we used two morecomplex models.

3.2. LeapFrog and LISEM

The simplicity of Runoff01 comes at the cost of not explicitlytaking certain processes into account. The most obvious miss-ing function in the present context is spatial variability. Becausereductions in flow speed through extra infiltration or roughnesscause shock waves when using Eq. (4), the hydraulic routingmodeling becomes more demanding. With a leapfrog scheme, atwo-dimensional surface runoff model was developed that couldaccommodate spatial variability, micro-topography, and vegeta-tion interception (Ajayi, 2004; Ajayi et al., 2008). With the detailedtopography and infiltration data, which were collected on therunoff plots in Ghana, this model was able to reproduce observedrunoff events, including associated scale effects.

Another example of a more comprehensive model is the LIm-burg Soil Erosion Model-LISEM (De Roo et al., 1996). In short, thisGIS incorporated distributed model contains the following featuresand routines; spatially and temporarily heterogeneous rainfall canbe entered, vegetation interception is accounted for, infiltrationand vertical soil water transport is modeled using Richards’ equa-tion, while infiltration excess is stored in micro-depressions beforerunoff starts, and overland flow is routed using a four-point finite-difference solution of a kinematic wave (Manning’s equation). TheLISEM model also includes a channel flow procedure and routinesrelative to erosion which are all beyond the scope of this paper.The model has been calibrated and used for analysis in a number ofdifferent settings (e.g. De Roo and Jetten, 1999; Hessel et al., 2003,2006).

We used LISEM to find that the same scale effects hold true when

est simulated slope at slope lengths between 100 m and 500 m tested with theLISEM model for a slope of homogenously medium roughness (1) or homogenouslylow roughness (2) or a mixture of 50% medium and 50% medium roughness witheither the bottom half of each slope segment with medium roughness (3) or lowroughness (4).

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In the case when the scale effect is relevant, the recession phasebecomes very important. To quantify the combined effects of build-up and recession, the “build-up–runoff” time, tbr, is defined. Wedefine tbr as the duration of a rainstorm such that water that starts

N. Van de Giesen et al. / Agriculture, Eco

ainfall event of 120 mm h−1 intensity during 2 min on a typicaloess soil. Runoff was simulated for slopes of 1–5 slope segmentsf 100 m length, which were in turn composed of 10 by 10 m gridells. The scenarios were: a homogenous relatively high surfaceoughness and micro-depression storage (1) or a homogenous rel-tively low soil roughness and micro-depression storage (2), a slopehere the bottom half of each segment had the characteristics of

lope 1 and the top half the characteristics of slope 2 (3), and a slopehere the bottom half of each segment had the characteristics of

lope 2 and the top half those of slope 1 (4). The simulated trends inunoff coefficient with increasing slope length (Fig. 1) were in lineith those that were obtained with the Runoff01 model, confirm-

ng the effect of recession infiltration also when more complexityas explicitly included in the overland flow modeling.

One could continue to add complexity to the simulations whenhe experimental set-up so requires. All models, however, do showhe importance of temporal variability. Only through inclusion ofainfall-runoff dynamics can the scale effects observed at our exper-mental sites be reproduced.

.3. Dimensional analysis of scale effects

It is difficult to obtain qualitative insight from numerical mod-ling. For this reason, we develop here a dimensionless solutionf the governing equations. The cost of the analytical approachs a loss of complexity. We assume constant infiltration and alock shaped rainstorm. The gain of the analytical solution isirect insight in the relative importance of different factors suchs slope, roughness, and intensities of rainfall and infiltration. Theharacteristic dimensionless numbers can be used to make coarseredictions on the relevance of the described scale phenomenalsewhere.

Immediately after the on-set of surface runoff, the water reach-ng the bottom of the slope will come from the lowest parts of thelope. If the rain continues, also water from higher parts will reachhe bottom. Eventually, when it has been raining long enough, theomplete slope will contribute to the runoff and a state of equi-ibrium will be reached. Once the rain stops, a recession phasetarts. Not all water found on the surface will make it all the wayo the bottom, unless the slope is completely impermeable. Forhort rains, we distinguish a build-up and a recession phase. Forong rains, we have an equilibrium phase between build-up andecession.

To quantify the scale effect, we define the scaling ratio = (RO/ROpnt*L),whereby RO [L2/T] is the observed runoff from thelope per unit width, ROpnt [L/T] the average point runoff alonghe slope, and L [L] the length of the slope. When the runoff fromhe slope is simply the point runoff times the slope length, such asould be the case from a glass roof, there is no scale effect and � = 1.

or strong scale effects, � � 1. It should be noted that under con-itions of saturation excess overland flow, we may have � > 1, buthis type of runoff generation is not under consideration here. Whenomparing runoff from plots of different lengths, as was done forhe field measurements, one does not necessarily know the pointunoff, so � cannot be used. In these cases, one can only compareunoff coefficients (cf. Fig. 1).

If the rainfall is of a duration that only briefly leads to runoff,nly a small part of the slope will contribute to the total runoff. Inuch cases, � will be small. In cases where the rainfall lasts for aong time after equilibrium has been reached, the complete slope

ill have contributed and � will approach one.

Clearly, what “short” and “long” rains are will depend on how

ast the water moves down the slope. Steep and smooth slopes willome into equilibrium quicker than gentle and rough slopes. Thentensity of the rainfall and the rate of infiltration will also play a

s and Environment 142 (2011) 95– 101 99

role. To simplify the analysis, we will look at what happens whenthe rainfall, P (t), has a block shape over time, t:

P(t) ={

P 0 < t ≤ tr

0 t > tr(2)

and the rate of infiltration is constant, I(t) = I, t > 0. In this case, wecan gather all relevant parameters in two dimensionless numbersthat characterize the system.

To route the infiltration excess down the slope, we use a kine-matic wave approach in which the average flow velocity, v (x,t) [L/T],depends as a power function on the water depth, h (x,t):

v(x, t) = ˛h(x, t)ˇ (3)

where x indicates the location along the slope, with x = 0 denot-ing the top of the slope and x = L the bottom. The constant ˛incorporates roughness and slope angle of the surface. The con-stant ̌ depends on the flow regime, with ̌ = 2/3 for turbulentflow and ̌ = 2 for laminar flow. Over natural surfaces, often ̌ = 1(Eagleson, 1970). The flow dimensions are such that the kinematicwave is a good approximation (see Daluz-Vieira, 1983, for validityof different approximations to the full flow equations). The two-dimensional flow, or the volumetric flow per unit of slope width,q(x,t), is given by:

q(x, t) = v(x, t)h(x, t) (4)

The governing partial differential equation can be found by usingabove equations and the mass balance:

˛ˇh(x, t)ˇ−1 ∂h(x, t)∂x

+ ∂h(x, t)∂t

= P(t) − I(t) (5)

This equation can be solved analytically for simple slopes andinputs such as the block input defined by Eq. (2) (Parlange et al.,1981). These representations, as well as the analytical solutionmethods, explicitly assume spatial homogeneity with respect toinputs, infiltration, and surface parameters.

Fig. 2 shows conceptually the different phases, build-up, equi-librium, and recession. The first characteristic time is te, the time ittakes for the plot to come into equilibrium. By solving Eq. (5) withthe input described by Eq. (2), we can calculate te (Stone et al., 1993;Van de Giesen et al., 2005):

te =(

L

˛

)1/ˇ

(P − I)(1−ˇ)/ˇ (6)

When tr is the duration of rainfall, the first dimensionless num-ber now becomes tr/te. When tr/te � 1, the scale effect will benegligible and the runoff from the slope will be close to the averagepoint runoff times the length of the slope.

Fig. 2. Conceptual “short” and “long” rainfall events. Conceptual difference betweenthe runoff from long rains (tr2 � te) and short rains (tr1 < te).

1 system

tic

t

mlispno

twasnpb

stohtt

4

oapvmtntr

enaddfl

eoata3crt2bti

Em

00 N. Van de Giesen et al. / Agriculture, Eco

o runoff from the top of the slope at the start of the rain, just makest to the bottom before the slope falls dry. The characteristic time tbran be calculated as (Stone et al., 1993; Van de Giesen et al. 2005):

br = 1P − I

(LI(P − I)

˛P

)1/ˇ

or tbr =(

I

P

)1/ˇ

te (7)

Both the ratio I/P, and the difference P − I are important deter-inants of the scale effect, which makes sense intuitively. Slope

ength, L, and slope angle/roughness, ˛, also determine te and tbrn a logical way. The second, and probably more relevant, dimen-ionless number now becomes tr/tbr. When tr/tbr < 1, only the lowerart of the slope will contribute to the runoff. In such cases, it doesot make sense to work with average infiltration capacity becausenly the characteristics of the lower slope matter.

The importance of the dimensionless numbers tr/te and tr/tbr ishat they define what long and short rains are for a given slope andhen scale effects may be expected. There is an intermediate situ-

tion, in which tr/te < 1 and tr/tbr > 1. In such intermediate cases, thecale effect will still be significant, but in practice this situation willot occur often. Eqs. (6) and (7) combine the most relevant slopearameters and allow prediction of the strength of scale effects toe expected.

When we go back to the experimental sites in West Africa, weee that almost all factors point towards strong scale effects withr/tbr < 1. First, we have very brief rainstorms so tr is small. Sec-nd, we have gentle slopes, leading to long tbr. Finally, also theigh rainfall intensity leads to a long tbr. The combined effect ishat wherever infiltration is significant, we will find with tr/tbr < 1hroughout West Africa.

. Discussion

The field experiments showed that scale effects consistentlyccur under the given circumstances of moderate slopes (2–7%)nd short duration high intensity rainfall. The results from pairedlots clearly point out that temporal dynamics, rather than spatialariability, is the cause of the observed scale effects. The differentodeling approaches, from simple to complex, further confirm that

emporal dynamics cause the scale effects. This confirms the earlyumerical results of Julien and Moglen (1990), who already showedhat the average effect of spatial variability does not account for theeduction in runoff coefficients with increasing plot lengths.

In the present paper we have extended earlier analyses of theffects of temporal variability in rainfall on scale effects in Horto-ian overland flow (Van de Giesen et al., 2000, 2005; Wainwrightnd Parsons, 2002). We present a dimensional analysis that usesimensionless numbers to predict if a given combination of rainfalluration, infiltration excess, slope, surface roughness, and overlandow regime will show minor or major scale effects.

In the literature, there is an abundance of observations on scaleffects in runoff. For the reader’s convenience, we provide a shortverview here. In the forest zone of Nigeria, earlier studies hadlready shown that shorter agricultural plots produced less runoffhan longer plots (Lal, 1983, 1997). Plots with a length of 10 m hadn average runoff coefficient of 5.2%, plots of 20 m had 4.7% runoff,0 m plots 3.0%, 40 m plots 2.3%, and 60 m long plots had a coeffi-ient of 1.9%. Also in arid Niger, consistent reduction in measuredunoff was found when runoff of large (42–130 m2) was comparedo runoff from small plots (0.8 m × 1.25 m) (Esteves and Lapetite,003). The effects were found under tiger bush, on fallow fields, andare piedmont surfaces. The scale effects were less pronounced inhis case, perhaps due to low infiltration capacity caused by crust-

ng.

Also outside West Africa, clear scale effects have been observed.arly work in Israel showed clear effects in a desert environ-ent (Yair and Lavee, 1985). In Chile, on the grassland slopes of

s and Environment 142 (2011) 95– 101

around 12.5% in the Andean piedmont, runoff coefficients fromlong plots (5 m × 10 m) were only 40% of the runoff coefficients ofsmall (0.5 m × 0.5 m) plots (Joel et al., 2002). In the Chilean case,spatial variability was again put forward as underlying cause butno process-based insight was provided on how this would func-tion. In southern Italy, significant reduction in runoff was foundduring an erosion study between microplots of 0.4 m × 0.4 m and0.2 m × 0.2 m (Bagarello and Ferro, 2004).

A careful study of the processes involved in the production ofsurface runoff at different scales was undertaken on steep (35–45◦)forested slopes in Japan (Gomi et al., 2008a,b; Sidle et al., 2006,2007). Even on these steep slopes, the reinfiltration and the tem-poral dynamics were put forward as explanation for the observedreduction in runoff with increasing slope/plot length. Here, how-ever, the process was complicated by the occurrence of connectedpathways along the slope, for example as pipe flow through thebiomat. The dynamics of connectivity always play a role as depres-sions will have to be filled with water before downhill flow can start.On the relatively smooth slopes of West Africa this phenomenoncould adequately be captured by adjusting the kinematic waveparameters, but this need not always be the case. As shown by themodeling work of Reaney et al. (2007), the exact temporal dynamicsbecome even more critical when connectivity plays a role.

Also Cerdan et al. (2004) reported a strong scale effect, but intheir study in Normandy France the plot size difference were suchthat clearly other processes could have equally contributed to thedifferences in observed runoff coefficients.

Overall, scale effects in Hortonian flow have been observed overa large range of climatic and topographic conditions. One has tobe careful to lump all these observations under one process butin all cases temporal variability in rainfall played a crucial role.Whether the full processes can be captured with a simple model likeRunoff01 or if one needs to include the effect of micro-topography(Wainwright and Parsons, 2002) and connectivity (Reaney et al.,2007) depends on the exact local conditions.

5. Conclusions

In the literature, roughly two categories of reasons for the expla-nation of the observed scale effects can be found, namely spatialvariability in soil characteristics and temporal and spatial variabil-ity in rainfall. We have repeated in this paper the point raised byJulien and Moglen (1990) and Wainwright and Parsons (2002) thatspatial variability can at best give a random variability in runoffcoefficients when slopes are compared. Spatial variability, in otherwords, is an issue of sampling. Mismatch between outcomes of pre-diction based on sampled soil characteristics are thus by definitionsampling error related under- or over-estimations. Temporal vari-ability in rainfall is due to occur in any rainfall event and we havereiterated the model analysis brought forward by Van de Giesen etal. (2000, 2005) and by Wainwright and Parsons (2002) that thisvery temporal variability in rainfall is the only true cause of scaleeffects in runoff.

Any slope region that would regularly produce surface runoffover its full length, would probably erode so fast that soon therewould no longer be a slope, or at least no soil. Natural slopes, there-fore, do normally not produce runoff over their full length. Whennatural slopes are disturbed, for example through agriculture orlogging, a number of processes will change. The kinetic impact ofraindrops on the soil will increase, leading to reduced infiltrationand particle dislodging. Biological activity, roughness and aggre-

gate stability will be reduced leading to reduced infiltration andincreased flow velocity of overland flow leading to larger volumesof runoff water and sediment load. Understanding the scale effectsin runoff in terms of the dimensionless numbers presented here

system

cppi

R

A

A

A

A

B

B

B

B

C

D

D

D

D

EE

G

G

H

H

H

Windmeijer, P.N., Andriesse, W. (Eds.), 1993. Inland Valleys in West Africa: An

N. Van de Giesen et al. / Agriculture, Eco

an help to decide on the management options. The frameworkresented here could help to make a quick scan of the order oflot sizes that would allow management of runoff for given rainfall

ntensities, slopes, rainfall durations, soil types and land uses.

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