Geomorphic threshold conditions for ephemeral gully incision

IORPHOiOGY ELSEVIER

Geomorphology 16 (1996) 161-173

Geomorphic threshold conditions for ephemeral gully incision Karel Vandaele a, ,, Jean Poesen a,b, Gerard Govers a,b, Bas van W e s e m a e l a a Laboratory for Experimental Geomorphology, K.U. Leuuen, Redingenstraat 16b, B-3000 Leuven, Belgium b National Fund for Scientific Research, Leuven, Belgium Received 4 May 1995; accepted 26 September 1995

Abstract

Ephemeral gully erosion in cultivated land is an important source of sediment that is frequently being overlooked and not accounted for in soil erosion studies. Furthermore, little is known about the factors controlling ephemeral gully erosion. In this paper the available information on the initiation and location of (ephemeral) gullies is summarised, focusing on the relationship between the upslope drainage area (A) and the critical slope gradient (Scr) for ephemeral gully initiation. By plotting the non-linear relationship between critical slope gradient (measured immediately upstream of the incision head) and drainage area (at the incision head) for gullied sites it was possible to draw a straight line on log-log paper through the lower-most points for each of the available datasets, representing a critical slope-area relationship for incision. Consequently, below this line no incision occurs. This line or critical relation can also be written as a power function between critical slope and area. Although many factors vary between the different datasets, the exponent of the drainage area relationship ( - 0 . 4 ) showed very little variation. The observed critical slope-area relationship can be related to a simple model of channel initiation by overland flow. Furthermore, this relationship can be used to identify potentially unstable sites where control measures should be undertaken.

1. Introduction

The term ephemeral gullies is used to describe linear erosion forms (1) which result from hydraulic erosion by overland flow, (2) with a cross sectional area larger than a square foot (Hauge, 1977), (3) occurring where ow~dand flow concentrates in the landscape, i.e., either in natural drainage ways (thalweg) as or along linear landscape elements, e.g. parcel borders, field roads, plough furrows, etc. The term ephemeral is u:~ed because these linear erosion forms are temporary features, removed by tillage

* Corresponding author. Tel: +32 16 22 69 20, Fax: +32 16 29 33 07, E-mail: [email protected]

(i.e. ploughing) and recurring in the same place (Foster, 1986; Laflen et al., 1985; Poesen, 1989). Ephemeral gullies are larger than fills but usually smaller than permanent gullies. The recognition of ephemeral gully erosion as a separate erosion class is relatively recent (Foster, 1986). The most important differences between ephemeral gullies, fills and permanent gullies are given in Table 1. In the loess region of Western Europe many researchers have already pointed to the importance of ephemeral gully erosion in natural drainage ways (Evans and Cook, 1987; De Ploey, 1989, 1990; Poesen and Govers, 1990; Papy and Douyer, 1991) and in linear elements (Poesen, 1989). However, little information is available about the relative im-

0169-555X/96/$15.00 ~ 1996 Elsevier Science B.V. All rights reserved SSDI 0 1 6 9 - 5 5 5 X ( 9 5 ) 0 0 1 4 1 - 7

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K. Vandaele et al. / Geomorphology 16 (1996) 16I 173

portance of ephemeral gully erosion in the total sediment budget. Volumetric measurements of rill and ephemeral gully erosion in cultivated catchments carried out by Auzet et al. (1993) in northern France, Baade et al. (1993) in Germany, and Vandaele (1993) and Vandaele and Poesen (1995) in central Belgium indicate that ephemeral gully erosion is a dominant sediment source. Consequently, the incision of ephemeral gullies into valley floors can cause a real threat for the productivity of the loessial soil. Nevertheless, ephemeral gully erosion is frequently overlooked in many soil erosion estimates. Most soil loss equations do not include the soil loss due to ephemeral gully erosion (Beasly et al., 1980; Foster, 1986; Merkel et al., 1988; Borah, 1989; Bingner et al., 1989). The failure to include this sediment source in the equations and models can lead to significant underestimates of the severity of soil loss. Furthermore, flooding of off-site houses by

soil-laden water is often associated with the formation of ephemeral gullies developed in normally dry valley bottoms (Papy and Douyer, 1991; Vandaele, 1993). Costs of such events may be considerable and are largely borne by individual house occupants (Boardman et al., 1994). Efficient control measures to reduce soil erosion and the risk of massive flooding are needed. Therefore, it is important to identify those valleys which are prone to ephemeral gullying. Schumm and Hadley (1957) already indicated the importance of the geomorphic characteristics of the drainage basin upon discontinuous gully development. They found that discontinuous gullies are formed on locally oversteepened segments of the valley floor (Patton and Schumm, 1975). Upslope drainage area and slope gradient are also used in more recent models predicting the initiation, location and soil loss caused by gully erosion (Patton and Schumm, 1975; Begin and Schumm, 1979; Foster,

Table 1 Characteristics of different types of erosion processes. After Laflen et al. (1985) Sheet and rill erosion

Ephemeral gully erosion

Gully erosion

slopes above

Occurs along shallow drainagelines upstream from incised channels or gullies.

Generally occur in well defined drainagelines.

May be of any size but are usually smaller than concentrated flow channels.

May be of any size but are usually larger than rills and smaller than permanent gullies.

Usually larger than concentrated flow channels and rills.

Flow pattern develops many small disconnected parallel channels which end at concentrated flow channels, terrace channels or in depositional areas.

Usually forms a dendritic pattern along water courses beginning where overland flow, including rills, converge. Flow patterns influenced by tillage, rows, terraces, man made features.

Dendritic pattern along natural water courses. May occur in non-dendritic patterns in road ditches, terrace or diversion channels, etc.

Rill cross-sections usually are narrow relative to depth.

Cross-sections usually are wide relative to depth. Sidewalls not wel defined. Headcuts not readily; do not become prominent because of tillage.

Cross-sections usually narrow relative to depth. Sidewalls are steep. Headcut prominent. Eroding channel advances upstream.

Rills normally removed by tillage, usually do not reoccur in the same place.

Temporary feature, usually removed by tillage; reoccur in same place.

Not removed by tillage.

Soil removed in thin layers or shallow channels. Soil profile becomes thinner over entire slope.

Soil removed along narrow flow path, to tillage depth if untilled layer is resistant to erosion, or deeper if untilled layer is less resistant.

Soil may erode to depth of profile, and can erode into soft bedrock.

Low erosion rates not readily visible.

Area may or may not be visibly eroding.

Erosion readily visible

Detachment and transport by raindrops and flowing water.

Detachment and transport by flowing water only.

Detachment by flowing water, slumping of unstable banks and headcut retreat; transport by flowing water.

Occurs on smooth side drainageline.

K. Vandaele et al. / Geomorphology 16 (1996) 161-173

1986; Merkel et al., 1988; Thorne et al., 1986; Auzet et al., 1993; Montgomery and Dietrich, 1994). This is based on the fact that the location and size of gullies is essentially ,controlled by the generation of concentrated surface runoff of sufficient magnitude and duration to initiate and sustain erosion. In most of these studies upslope contributing area is used as a surrogate for the volume of runoff involved because in most cases no runoff discharge data are available. It is believed that in landscapes where Hortonian overland t]ow dominates, runoff volume increases proportional to catchment area (Leopold et al., 1964). The objective of this paper is to summarise available information on the initiation and location of ephemeral gullies. Attention is particularly focused on the relationship between the upslope drainagebasin area (A) and critical slope gradient (Scr) of ephemeral gully initiation. This is primarily because in most studies only data on slope gradient and

163

drainage area at the head of the ephemeral gully are available. The findings from the available data are compared and discussed. Furthermore, these results are compared with the findings obtained from rills, discontinuous gullies and gullying by landslide scarring.

2. Methods and materials

The critical slope angle and drainage area for gully initiation can be obtained by various methods. Poesen et al. (in prep.), Govers (1991) and Montgomery and Dietrich (1988) measured the maximum slope gradient where gully head formation started in the field by using an optical clinometer. Boardman (1992), the Institut Gtographique National de France (I.G.N., 1983) and Patton and Schumm (1975) measured the slope gradient on topographic maps. Recently, Digital Elevation Models can also be con-

Table 2 Overview of the data sour,zes used in the analysis: techniques used and the erosional processes Region

Reference

Observation method

Calculation of slope gradient

Calculation of drainage area

Incision by

Central Belgium

Vandaele et al. (1995)

Aerial photographs

G.I.S., topographic map, scale 1:10,000

G.I.S.

Ephemeral gullying

Poe~;en et al. (19c~5) Govers (1991)

Field survey

Field measurements Field measurements

Field measurements Field measurements

Ephemeral gullying Rilling

France

I.G.N. (1983)

Aerial photographs

Topographic map

Ephemeral gullying

UK - South Downs

Boardman ( 1992)

Field survey

Topographic map, scale 1:10 000 Top~graphic map

Topographic map

Ephemeral gullying

Portugal

Vandaele et al. (1995)

Aerial photographs

G.I.S., topographic map, scale 1:25,000

G.I.S.

Ephemeral gullying

California and Orego (U.S)

Moutgornery and Dietrich (1988)

Field survey

Field measurements

Topographic map (planimeter)

Small scale landsliding and overland flow

Colorado (U.S.)

Patton and Schumm (1975)

Aerial photographs

Topographic map, scale 1:50,000

Topographic map

Discontinuous gullying

Field survey

164

K. Vandaele et al. / Geomorphology 16 (1996) 161-173

structed by digitizing the contour lines from largescale topographic maps to obtain the topographically derived attributes (upslope drainage area and slope gradient). After converting this vector information into raster information several algorithms are available to calculate slope energy (Zevenbergen and Thorne, 1987) and upslope drainage area (Desmet and Govers, 1995). Vandaele et al. (1995) used this technique to assess the slope gradient and the upslope drainage area at the gully head. The upslope drainage area can also be assessed in the field (Poesen et al., in prep.; Govers, 1991) or on topographic maps (Montgomery and Dietrich, 1988). An overview of the techniques used in this study is given in Table 2. The measurement of drainage area and slope gradient at the gully head can cause a considerable bias, especially for old gullies. This is based on the

fact that once a gully has developed, it can migrate upslope due to headcut retreat. Therefore, Patton and Schumm (1975) measured the slope gradient and the upslope drainage area at the steepest valley slope along the discontinuous gullies. Govers (1991) conducted his field survey in central Belgium in order to assess rill erosion rates on arable land during winter. During this field survey the critical length (L) for rill initiation on the fields was measured at the most upslope rill headcut. To obtain the critical drainage area (A) the critical length (L) was multiplied with the mean spacing between rills. Based on field evidence (Govers, 1991) the spacing between rills is approximately constant and equals 2.5 m. Until now different methods have been used to study erosional systems (Table 2). In order to cover a longer period of observation aerial photographs can

Table 3 General information on the available data sets in the literature Reference

Region

Annual rainfall (mm)

Soils

Land-use

Slope angles

Vandaele et al. (1995)

Central Belgium

700-800

Loam

Cultivated

< 25%

Poesen et al. (1995)

Central Belgium

700-800

Loam

Cultivated

< 25%

Govers (1991)

Central Belgium

700-800

Loam

Cultivated

< 25%

I.G.N. (1983)

Northwestern France

700-1000

Loam

Cultivated

< 20%

Boardman (1992)

South Downs

750-1000

Loam

Cultivated

< 50%

(U.K.) Vandaele et al. (1995)

South Portugal

500-600

Red schist soil

Cultivated

< 35%

Montgomery and Dietrich (1988)

Oregon (U.S.A.)

1500

Soil on sandstones

Logged forest

9-100%

Montgomery and Dietrich (1988)

Southern Sierra Nevada (U.S.A)

260

Soil on deeply weathered granitic rocks

Open oak woodland and grasslands

9-100%

Montgomery and Dietrich (1988)

California (U.S.A.)

760

Soil on greywacke

Coastal prairie

9-100%

Patton and Scbumm (1975)

Colorada (U.S.A.)

300-500

Soil on sandstone

Sagebrush and scattered trees

-

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K. Vandaele et al. / Geomorphology 16 (1996) 161-173

be more useful than field surveys. On the other hand, field surveys are more precise. The available information in this study deals mostly with ephemeral gullying, rilling and discontinuous gullying due to overland flow (Table 2). However, limited information about gullying due to small scale land-sliding is also included. Information for ephemeral gullying was obtained for central Belgium (lPoesen et al., in prep.; Vandaele et al., 1995), the South Downs (Boardman, 1992), northern France (I.G.N., 1983) and southern Portugal. Govers (1991) studied rill initiation in central Belgium while discontinuous gullying in Colorado was studied by Patton and Schumm (1975). The gullies in California, Oregon and the Sierra Nevada were dominantly the result of small land-sliding (Montgomery and D,ietrich, 1988). The data available in the literature ,:over a broad range of climates, i.e. from semi-arid to humid-temperate. Also morphology, land-use, soil characteristics and the observed erosion processes differ markedly between the available data ('['able 3). The land in the north-

western part of France and also in central Belgium is dominantly used for conventionally tilled winter and summer crops. The study area in Portugal is characterised by a coarse weathering mantle over schists resulting in a soil with a high rock fragment cover (-t-30%). The land in this region is dominantly used for winter wheat. Erosion of agricultural land occurs mainly on fields with a low crop cover (i.e. after seeding) during autumn and winter. Mainly winter wheat and barley are grown in the South Downs. Therefore, most erosion problems occur during autumn and winter and are situated in the valley bottoms. In this region frost-shattered chalk underlies the silty-loamy A horizon (at 0.20 m). Consequently, many soils on the South Downs have a rather high rock fragment cover ( + 39%). In many areas of the loess belt of northwestern Europe (i.e. central Belgium and northwestern France) soil erosion occurs mainly during two periods, winter and late spring (Vandaele and Poesen, 1995; Auzet et al., 1993). Poesen et al. (in prep.) and Vandaele (1993) recorded ephemeral gully erosion after rather ex-

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Fig. 1. Critical slope gradient versus upslope drainage area for the initiation of ephemeral gullies and straight lines fitted through the lower-most points of the slope gradient - upslope drainage area. Central Belgium field (Poesen et al., 1995), central Belgium photo

(Vandaele et al., 1995), South Downs (Boardman, 1992), northernFrance(I.G.N., 1983), southernPortugal(Vandaeleet al., 1995).

K. Vandaele et al. / Geomorphology 16 (1996) 161-173

166

treme events. The data from the I.G.N. are related to high-intensity rainfall events in late spring. On the contrary, there is very little bare ground in spring and summer on the South Downs and soil erosion is mainly a winter phenomenon. The study sites in the United States are characterized by much steeper slope gradients (Table 3).

points are plotted. The slope gradient is plotted on the ordinate and drainage area on the abcissa. Following the method of Patton and Schumm (1975) a straight line was fitted through the lower-most points for sites which were incised (Fig. 1). This was done for each data set and was based on Figs. 1 and 2. This line, i.e. the lower limit of scatter of the data, represents an approximation to the critical slope-area relationship for incision. Consequently, below this line no incision occurs. This line or critical relation can also be represented by a power function between critical slope and area (Eq. 1):

3. Results

For each data set the critical slope gradient (Scr) and upslope drainage area ( A ) for incision were replotted on l o g - l o g paper. The results for ephemeral gully initiation are given in Fig. 1. Fig. 2 represents the results for all the data sources together (for filling, ephemeral gullying, gullying, and gullying due to small landslide scarring). For the data of Montgomery and Dietrich (1994) only the lowermost

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Fig. 2. Critical slope gradient versus upslope drainage area for all the different datasets and straight lines fitted through the lower-most points of the slope gradient - upslope drainage area. California (Montgomery and Dietrich, 1988), Oregon (Montgomery and Dietrich, 1988), Sierra Nevada (Montgomeryand Dietrich, 1988), central Belgium rill (Covers. 1991), Colorado (Patton and Schumm, 1975), central Belgium field (Poesen et al., 1995), central Belgium photo (Vandaele et al., 1995), South Downs (Boardman, 1992), northern France (I.G.N., 1983), southern Portugal (Vandaele et al., 1995).

K. Vandaele et aL / Geomorphology 16 (1996) 161-173

Table 4 Coefficients in Eq. (1) corresponding to the different data sets Reference a - b Vandaele et al. (1995), 0.025 -0.40 to -0.35 central Belgium Govers (1991) 0.0035 - 0.40 Poesen et al. (1995) 0.08 -0.40 to -0.30 Boardman (1992) 0.09 - 0.25 I.G.N. (1983) 0.06 - 0.40 Vandaele et al. (1995), 0.02 - 0.35 Portugal Montgomery and Dietrich 0.25 - 0.40 (1988), Oregon Montgomery and Dietrich 0.27 - 0.40 (1988), California Montgomery and Dietrich 0.35 -0.60 (1988), Sierra Nevada Patton and Schurnm (1975) 0.16 -0.26

more or less constant and equals - 0 . 4 0 . Only a few data have slightly different values for the exponent - b ranging between - 0 . 2 6 and - 0 . 6 , namely the Colorado data of Patton and Schumm (1975) and the Sierra Nevada data of Montgomery and Dietrich (1988). Despite the important differences between the study areas in terms of morphology, climate, land-use and erosional systems (i.e. filling, discontinuous gullying,...), the exponent ( - b ) of the critical slope-area relations are almost identical. The influence of the methods used and the specific characteristics of the different data sets on the critical slope-area relation is believed to be reflected in the value of the constant a. Therefore, the constant a shows important variation and ranges over several orders of magnitude. For study areas Characterized by more or less the same range of slope gradients, drainage areas and erosional processes the results indicate that the critical drainage area needed to start incision (for a given slope gradient) was largest for the South Downs study area (i.e. a = 0.09) and lowest for the central Belgium study area (i.e. a = 0.025). Although there are important differences in climate and soil characteristics, the results obtained from studying ephemeral gullying on aerial photographs in central Belgium and Portugal are more or less equal. For the Oregon and California study areas, which have significantly steeper slopes we fi3und the highest a value ( a =

167

0.35). Looking at the different erosional processes we can conclude that lowest values for a are found for filling (i.e. a = 0.0035) and highest for gully initiation by small land-sliding (a = 0.35). This indicates that the drainage area above the headcut for a given slope is systematically larger for ephemeral gullies than for fills. Even within the same region (for example central Belgium) the critical drainage area to initiate ephemeral gullying for a given slope shows important variation (i.e. 0.025 to 0.08). The largest drainage areas at the gully head for a given slope are obtained from the field survey conducted by Poesen et al. (in prep.).

4. Discussion As already pointed out, this study is essentially based on the concept of geomorphic thresholds, defined by Schumm (cited by Patton and Schumm, 1975). The inverse relation between Scr and A is assumed to represent a critical threshold for incision. This threshold condition plots as a straight line on a l o g - l o g paper. For points below this line gully erosion does not occur. However, as stated by Patton and Schumm (1975) this relation does not imply that gullying necessarily takes place. Begin and Schumm (1979) tried to relate this critical slope-area relation to a parameter which has a physical meaning. They chose the average shear stress exerted by the flow on the valley bottom. Rilling and gullying is often considered to be associated with a critical shear stress (Foster, 1982; Rauws, 1987; Govers, 1990). Using empirical relations between the hydraulic radius of the flow (R) and discharge (Q), and similar relationships between discharge and drainage area (Leopold et al., 1964; Dunne and Leopold, 1978) they substituted the hydraulic radius of the flow (R) in the original shear stress formula. As a result a relationship between a critical shear stress indicator, drainage area and valley slope was established: Fcr= (c'y) a r f s

(2)

where Fcr is the critical shear stress indicator, A is the drainage area (ha), S is the valley slope gradient ( m / m ) , rf is an exponent, c is a constant and 3' is the weight per unit volume of the water

168

K. Vandaele et al. / Geomorphology 16 (1996) 161-173

According to this equation, a line of equal values of the critical shear stress indicator plots a straight line on a log-log paper. If valley slope S is plotted on the ordinate and drainage area on the abcissa, then the slope of such line is equal to - rf: S = da

rf

(3)

Based on the foregoing theoretical considerations, Begin and Schumm (1979) found that the exponent - r f should range between - 0 . 4 and - 0 . 2 . The values obtained for the exponent - b (Eq. 1) in this study (Table 4) agree well with the theoretical predicted values (Eq. 3). The observed data are consistent with the predicted form of the critical drainage area-slope gradient relation for overland flow. This means that the inverse critical relation between S and A can be related to a measure of shear stress which initiates rilling and gullying. However, the different values of the constant a are expected to vary with geology, soils, climate and vegetation (Begin and Schumm, 1979). An important variation in the constant (a) may of course be caused by the different methods used. The assessment of slope gradient and upstream area in the field is more accurate and precise than when obtained from topographic maps or from Geographical Information Systems. On a topographic map the slope gradient is measured between two contourlines resulting in a mean critical slope gradient. Furthermore, the size of the grid cells (in the G.I.S. approach) will affect the accuracy of the slope measurements. In general, slope gradients measured in the field will normally be higher. Also, the upstream area is most accurately defined and measured in the field. The different methods used may partly explain the differences observed between the data for ephemeral gullying in central Belgium collected by Vandaele et al. (1995) and by Poesen et al. (in prep.). Although these two studies are conducted in the same study area and deal with ephemeral gullying in the growing and dormant season, the value of the constant a differs markedly (Table 4). During field surveys (immediately after a rainstorm) the critical slope gradient can be measured exactly at the ephemeral gully head or where ephemeral gully formation started. Because permanent gullies may have developed over a longer period through headward extension it is very difficult

to identify the exact location where incision started. This headward extension will not only affect the size of the drainage area but also the slope gradient. Therefore, Patton and Schumm (1975) and Poesen et al. (in prep.) measured the drainage area and slope gradient at the steepest valley slope along the gully. Also the magnitude of the rainfall event which caused gullying may influence the erosion thresholds, i.e. rainfall events with a high return period might be expected to give rise to gullying for a given drainage area in valley bottoms with lower gradients (Boardman, 1992; Vandaele, 1993). This is based on the fact that these events will generate a higher runoff discharge for a given drainage area. Consequently, during high-intensity, low-frequency rainstorms, a smaller drainage area is needed to reach the critical shear stress for incision. Also Patton and Schumm (1975) already stressed the importance of the magnitude of runoff on the critical slope of gully incision. The observed ephemeral gullies in the loam belt of central Belgium (Vandaele et al., 1995) can be surely associated with extreme events in late spring an early summer (Table 5). These events had return periods of at least 50 to 100 years (Demarte, 1985). Also the ephemeral gullying at the I.G.N. site was due to high-intensity low-frequency rainfall events during the spring and summer of 1982, while the data of Poesen et al. (in prep.) refer to ephemeral gullying due to both low-intensity high-frequency rainstorms (dormant season) and high-intensity lowfrequency rainstorms (growing season). On the South Downs however, there is very little bare ground in spring or summer and severe erosion by water on hillslopes and in dry valley bottoms is exclusively a winter phenomenon (Boardman et al., 1994). The formation of ephemeral gullies in this period is mainly due to rainfall events with high rainfall amounts but rather low rainfall intensities (Boardman, 1993). Laboratory experiments (Govers et al., 1990) and field observations (Vandaele and Poesen, Table 5 Rainfall data related to extremeevents in Belgium Date of aerial Date of stormevent Rainfallamount photographs (mm) 29/07/1963 12-13/06/1963 94.0 25/06/1986 6-7/06/1986 37.3

K. Vandaele et aL / Geomorphology 16 (1996) 161-173

1995) illustrated how erodibility of a loamy material is affected by antecedent soil-structural and moisture conditions. Loamy soils appear to be very sensitive to soil erosion, particularly after a long period of drying. Due to the occurrence of high-intensity rainstorms and the higher erodibility of the soil top layer, erosion rates are higher in spring and summer (Vandaele and Poesen, 1995). Therefore, the a value for incision due to high-intensity low-frequency rainfall events in late spring and summer will be lower than for incision due to low-intensity high-frequency rainstorms in autumn and early winter. This may partly explain why the value of a is so high for the South Downs (U.K.) region. Furthermore, Montgomery and Dietrich (1988) stated that the drainage area required to initiate a channel, for the sarae gradient, increases with increasing aridity. However, this trend is not observed in our results. The resistance of the valley floor to incision is increased by the biomass (Graf, 1979). Therefore, the critical shear stress to start incision for a given slope is higher for more densely vegetated sites. The steep valley bottoms studied by Montgomery and Dietrich (1988) and Patton and Schumm (1975) are densely vegetated. Also rock fragments will have a pronounced effect on the overland flow velocity (Poesen, 1992) increasing flow resistance and reducing the detaching and transporting capacity (Govers, 1990). On the other hand, stony soils are much less erodible (Poesen et al., 1994). Boardman (1990) and Evans (1990) found that stony silt loams, such as those of the South Downs, are less erodible than soil in many other areas. Donker and Damen (1984) found for the region near Daroca (Spain) that gully sites have a considerably lower gravel content than non-gully sites. Taking these findings into account we can conclude that the upslope contributing area for a given slope will be systematically larger for more vegetated valleys and for valleys with a high rock fragment cover. Even for rill erosion by overland flow on slopes (Govers, 1991) the exponent - b equals + - 0.40. In our study we found that in central Belgium, for a given slope gradient, rills tend to have a smaller drainage area at the most upward rill headcut than do ephemeral gullies. This can be attributed to the fact that rills are smaller features than ephemeral gullies

169

(Table 1). Indeed, the volume of material eroded is often considered to be associated with shear stress (Foster, 1982; Govers, 1991). On the other hand, shear stress is dominantly determined by the discharge of the flow and the slope gradient (Rauws and Govers, 1988; Torri et al., 1987). Unfortunately, information about discharges of overland flow is very sparse, nevertheless most studies use drainage area (or slope length) as a surrogate for flow discharge. This means that small erosion features can be related to small flow discharges. Assuming all the other factors constant, smaller drainage areas will be sufficient to initiate rills. Consequently, larger erosion forms (i.e. ephemeral gullies, permanent gullies) will have larger drainage areas at the their most upward incision head. The critical slope-area relationship for incision can also be used for gully formation due to landslide scarring. The value of the - b exponent for the data of Montgomery and Dietrich (1988) ranges between - 0 . 4 and - 0 . 6 . All these data indicate that for a given slope gradient larger erosion features tend to have a larger drainage area at the most upward incision head than smaller features. These results also suggest that the slope of the line through the lower limit of scatter of the data (representing a critical slope-area relation for incision) is independent of the erosional system (Figs. 1 and 2) and the specific properties of the individual sites. Therefore, we may assume that the observed critical slope-area threshold relations can be related to a simple method or model of channel initiation by overland flow. This model can also be used in steep humid landscapes where channel initiation occurs by small land-sliding. Based on the data of Montgomery and Dietrich, Willgoose et al. (1991) already stated that the general inverse trend indicates a single underlying channel initiation mechanism. On the other hand, Montgomery and Dietrich (1994) found that simple models of processes for gully initiation by overland flow and land-sliding were able to predict the observed slope-area relations. These models predict that S c~ A -°5 for erosion by unconfined overland flow on low-gradient slopes and a more rapid decrease in drainage area with increasing slope for small land-sliding. For channel initiation by turbulent overland flow the model predicts that S ~ A - 6 / 7 . Furthermore, Thorne et al. (1986) and Moore et al.

K. Vandaele et al. / Geomorphology 16 (1996) 161-173

170

(1988) basically used drainage area (per unit contour length; A b) and slope gradient (S) to predict the location of ephemeral gullies. Their models predict that Scx A-1 for ephemeral gully initiation by concentrated overland flow. The most frequently used relationship between fill erosion rates and slope gradient and slope length has the form E r (x 3 1 4 5 L 0"75 (Govers, 1991). Assuming that the drainage area increases linearly with distance from the divide (L) this length can be used as a surrogate for drainage area. This is the case if the hillslopes are tilled perpendicular to the contourlines like on most fields in central Belgium. The ratio between the length exponent and the slope exponent for filling, which illustrates the relative importance of length and slope in the rill process, equals 0.5. Most of these results

I

3

3

I

I

agree well with the ratio obtained for ephemeral gully incision in our study (0.3-0.5).

5. Application The observed critical relation between slope gradient and upslope drainage area can help the soil conservationist to identify those areas within a catchment which are prone to ephemeral gullying in order to establish anti-erosion measures. This approach is illustrated with a small example for a cultivated catchment in central Belgium (Kinderveld catchment, Korbeek-Dijle). We compared the actual slope gradient (Sac) of each grid cell (5 m × 5 m) within

3 I

I

170370 mN

170370 mN

m

Ephemeral gully location.

m

Zone prone to gullying (Sac - Scr > 0).

8rid

~North meter~ 470

168170 m N

168170 m N

i

i

~

Idri-~i

Fig. 3. Location of the ephemeral gullies observed on the aerial photographs and the zones which are prone to gullying (Sac - Scr > 0) superimposed on the topographic map. Thin lines refer to contour lines. The difference in height between two contour intervals equals 2.5 m.

K. Vandaele et al. / Geomorphology 16 (1996) 161-173

the catchment with the critical slope gradient (Scr). To obtain the topographically derived attributes (drainage area, A, and valley slope at valley head, Sac) a Digital Elevation Model was constructed by digitizing the contour lines from the Belgian topographic map (N.G.I.) on a 1:10,000 scale. For this purpose, we used the TOSCA program of the IDRISI geographic information system (Jones, 1991; Eastman, 1992). The vector information was converted into a raster Digital Elevation Model by using the IDRISI procedures LINERAS and INTERCON. These procedures use a linear interpolation method. As already mentioned above, a grid resolution of 5 m was taken. The energy slope was calculated from the digital elevation model by using the Zevenbergen and Thorue (1987) formula which is employed in the IDRISI software. For each grid element the upslope drainage area is cah:ulated by the FORTRAN program written by Desmet and Govers (1995). Critical slope gradient was calculated by using the threshold relation obtained from the dataset of Vandaele et al. (1995): Scr :

0.025 A-°4°

(4)

The upslope drainzLge area (A) is expressed in hectares and Sot in m / m . In this first approach land-use was not taken into account in the calculation of the upslope drainage area. For each grid cell the Scr was subtracted from the Sac (OVERLAY procedure). Consequently, the following conclusions can be drawn; 1. if Sac - S , > 0 then ephemeral gullying is likely to occur in this grid cel, 2. if Sac- Scr < 0 then ephemeral gullying is not likely to occur in this grid cel. Fig. 3 shows the location of all ephemeral gullies observed on the aerial photographs superimposed on the topographic map and the map of the grid cells which are prone to gullying and where soils are cultivated. Using this technique we found a good agreement between the predicted and observed locations of the ephemeral gullies. Further research is needed to refine this method, perhaps by taking into account the land-use within the gully catchments. Therefore, we believe that this method opens new perspectives and can be successfully applied to other regions using the intormation collected in Table 4.

171

6. Conclusions By plotting on log-log paper critical slope gradient S (measured immediately upstream of the incision head) versus drainage area A (at the incision head) for gullied sites it was possible to draw a straight line through the lower-most points for each of the datasets. It is assumed that this line represents a threshold condition for incised and non-incised sites. This line can also be written as a power function between slope gradient and drainage area. Although many factors vary between the different datasets, the exponent of the drainage area obtained for the different datasets in Eq. (1) showed very little variation and equalled - 0 . 4 . However, the constant in this function ranges over several orders of magnitude reflecting the site specific conditions. Although other factors also vary, the critical drainage area for a given slope to start incision is smaller for small erosion features (i.e. small rills and ephemeral gullies). The results suggest that the drainage area needed to initiate ephemeral gullies for a given slope will be higher for well vegetated soils. Also, the magnitude of the rainfall event will affect gullying, i.e. rainfall events with a high return period falling on a dry soil might be expected to give rise to gullying for a given drainage area in valley bottoms with lower gradients. These results also suggest that the observed critical slope-area threshold relations can be related to a simple method or model of channel initiation by overland flow. The critical slope gradient-drainage area relation for filling, ephemeral gullying and small-land sliding, observed from the available data is consistent with the threshold theory of incision by Hortonian overland flow. The results from a cultivated catchment in central Belgium suggest that this critical slope-area relation can be used to identify potentially unstable sites.

Acknowledgements This research was conducted as part of a collaborative research project funded by the European Commission (D.G. XII) whose support is gratefully acknowledged. We would like to thank R. Geeraerts for the drawings. This paper was much improved thanks to the comments of two anonymous referees.

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