Raindrop and flow interactions for interrill erosion with wind-driven rain

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Raindrop and flow interactions for interrill erosionwith wind-driven rainGunay Erpul a , Donald Gabriels b , L. Darrell Norton c , Dennis C. Flanagan c , Chi HuaHuang c & Saskia Visser da Department of Soil Science and Plant Nutrition, Faculty of Agriculture, University ofAnkara, 06110, Diskapi-Ankara, Turkeyb Department of Soil Management and UNESCO Chair on Eremology, Ghent University,Coupure Links 653, B 9000, Ghent, Belgium E-mail:c USDA-ARS National Soil Erosion Research Laboratory, 275 S. Russell St., PurdueUniversity, West Lafayette, IN, 47907-2077, USAd Land Degradation and Development Group, Department of Environmental Sciences,Droevendaalsesteeg 4, 6708 PB, Wageningen UR, The Netherlands E-mail:Published online: 24 Apr 2013.

To cite this article: Journal of Hydraulic Research (2013): Raindrop and flow interactions for interrill erosion with wind-driven rain, Journal of Hydraulic Research, DOI: 10.1080/00221686.2013.778339

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Journal of Hydraulic Research, iFirst, 2013, 1–10http://dx.doi.org/10.1080/00221686.2013.778339© 2013 International Association for Hydro-Environment Engineering and Research

Research paper

Raindrop and flow interactions for interrill erosion with wind-driven rainGUNAY ERPUL, Professor, Department of Soil Science and Plant Nutrition, Faculty of Agriculture, University of Ankara, 06110Diskapi-Ankara, Turkey.Email: [email protected] (author for correspondence)

DONALD GABRIELS, Emeritus Professor, Department of Soil Management and UNESCO Chair on Eremology, Ghent University,Coupure Links 653, B 9000 Ghent, Belgium.Email: [email protected]

L. DARRELL NORTON, PhD, Research Soil Scientist (Retired), USDA-ARS National Soil Erosion Research Laboratory, 275 S.Russell St., Purdue University, West Lafayette, IN 47907-2077, USA.Email: [email protected]

DENNIS C. FLANAGAN, PhD, Research Agricultural Engineer, USDA-ARS National Soil Erosion Research Laboratory, 275 S.Russell St., Purdue University, West Lafayette, IN 47907-2077, USA.Email: [email protected]

CHI HUA HUANG, PhD, Research Soil Scientist, USDA-ARS National Soil Erosion Research Laboratory, 275 S. Russell St.,Purdue University, West Lafayette, IN 47907-2077, USA.Email: [email protected]

SASKIA VISSER, PhD, Researcher, Land Degradation and Development Group, Department of Environmental Sciences,Droevendaalsesteeg 4, 6708 PB, Wageningen UR, The Netherlands.Email: [email protected]

ABSTRACTWind-driven rain (WDR) experiments were conducted to evaluate the interrill component of the Water Erosion Prediction Project model with a two-dimensional experimental set-up in a wind tunnel. Synchronized wind and rain simulations were applied to soil surfaces on windward and leewardslopes of 7, 15 and 20%. Since WDR fall trajectory varied with horizontal wind velocities of 6, 10, and 14 m s−1, magnitude of raindrop normal andlateral stresses on flow at the impact-flow boundary also changed and differentially directed lateral jets of raindrop splashes with respect to downwardflows occurred. To account for interactions between raindrop impact and interrill shallow flow, a vector approach with kinetic energy fluxes of bothraindrop splashes and flow were used and this resulted in greater correlations in predicting sediment delivery rates.

Keywords: Flow kinetic energy flux; interrill erosion; raindrop impact velocity vector; Water Erosion Prediction Project; wind-drivenrain

1 Introduction

Interrill sediment delivery mechanics includes the integratedaction of raindrop detachment and raindrop-impacted flow trans-port (RIFT) (Julien and Simons 1985, Guy et al.1987, Guy 1990,Kinnell 1993, Flanagan and Nearing 1995, Zhang et al.1998,Kinnell 2005). Given its thin depth, the detachment by overland

flow is often of minor importance for interrill erosion (Foster1982). Therefore, interrill detachment is considered to be mainlydue to raindrop impact, and detachment by runoff is entirelyascribed to rill flow. In fact, studies of modelling the inter-rill erosion mechanisms have mostly characterized the sedimenttransport capacity of rain-impacted thin flow (qs) as a functionof rainfall intensity (I ), unit discharge (qu) and channel bottom

Revision received 14 February 2013/Currently open for discussion.

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slope (So), and models with either linear or power terms havebeen developed (Eq. 1):

qs = f (I , qu, So) (1)

The USDA-Water Erosion Prediction Project (WEPP) modelalso uses similar approaches to explain the interrill sedimentdelivery (Flanagan and Nearing 1995). The general equationmerely formulates the integrated action of raindrop detachmentand RIFT (Eq. 2) (Kinnell 2005):

Di = kiIq (2)

where Di is the interrill sediment delivery rate to a rill(kg m−2 s−1), ki is the interrill erodibility (kg s m−4), I is therainfall intensity (m s−1), and q is the interrill runoff rate (m s−1).

There have been some attempts recently towards a combinedwind and water process-based erosion prediction model. Flana-gan and Visser (2004) give the details of the commonalitiesin WEPP (Nearing et al.1989, Flanagan et al.1995, Fosteret al.1995) and Wind Erosion Prediction System (WEPS) (Hagen1991, Hagen et al.1999). Fox et al.(2011) describe the inte-gration of the WEPP hydrology and water erosion equationsinto the WEPS model code. Flanagan et al.(2007) and Ascoughet al.(2011) also presented information on the status of com-bining wind and water erosion models within the ARS ObjectModelling System.

During the last decade, significant studies have been con-ducted to develop the fundamentals of wind-driven rain (WDR)erosion processes at the wind tunnel/rainfall simulator facilityat the International Centre for Eremology (ICE), Ghent Uni-versity, Belgium (Erpul 1996, 2001, Gabriels et al.1997, Erpulet al. 1998, 2000, 2002, 2003a, 2003b, 2004a, 2004b, 2005,2008, 2009a, 2009b, Cornelis et al. 2004a, 2004b, 2004c). Thesestudies have shown that the interrill erosion processes of WDRrequire directional or vectoral physical parameters to deal withthe oblique raindrop impact trajectories under the effects ofboth gravitational and drag forces. The raindrop impact param-eters emerge as a direct function of the driving wind currents.Erpul et al.(2008) used the kinetic energy flux term (KEr inJ m−2 s−1), which integrates the frequency, velocity and direc-tion of the wind-driven raindrop impacts with a varying angle offall incidence (�), to explain the sand detachment (Eq. 3):

KEr = f (ηwd , vr , �) (3)

where ηwd is the number of wind-driven raindrops per unit areaper unit time (m3 m−3 m−2 s−1) and vr is the resultant impactvelocity of raindrops (m s−1).

In terms of the mechanics of interrill erosion with WDR, thewind itself and the oblique raindrop impact trajectories also affectthe shallow flow hydraulics by changing the roughness inducedby both the growth of wind-induced waves (Kawai 1979) andthe raindrop impacts with an angle on the flow (De Lima 1989a,

1989b, 1989c, Erpul et al. 2004b). The surface drift velocity isassumed to be mostly proportional to the wind shear velocity(u∗, m s−1) (Kranenburg 1987), and the interaction between theair and the water motion is reflected largely in the along-surfacewind shear stress, τw (N m−2) (Eq. 4):

τw = ρau2∗ (4)

where ρa is the air density (kg m−3). Nevertheless, disturbancesby the wind shear stress on the flow will be much less than thoseby the shear stress of the wind-driven raindrops (τs, N m−2),given the density of water and the frequency of the impingingdrops. The latter is proportional to the along-surface velocitycomponent of the raindrop impact velocity (Eq. 5):

τs = ρwv2s (5)

where ρw is the water density (kg m−3) and vs is the along-surfaceraindrop impact velocity (m s−1).

Under the two-dimensional oblique impact conditions ofWDR, the partitioning of the normal and lateral shear stressesof the wind-driven raindrops, determined by the normal andalong-surface raindrop impact velocities (vn and vs, respec-tively), significantly varies with wind speed (Erpul et al. 2008).Obviously, WDR has enhanced lateral jetting of raindrops whencompared with those of wind-free rainfall (WFR) on a slopingsurface, which may exhibit a reverse situation of the raindropshaving greater normal stresses. Furthermore, under the WDRconditions the lateral jets directionally induce opposite effectson overland flow (De Lima 1989a, 1989b, 1989c). In perpendic-ular impacts, the lateral jets as a function of only slope gradient(θ ) are always in the direction of flow irrespective of the slopeaspect. However, as a function of rain inclination (α), slope gra-dient and aspect (which are all three integrated into the angle ofrain incidence (�)), the lateral jets of inclined raindrops drivenby wind are in the direction of the flow on the leeward slopes(� = α + θ) and in the opposite direction on the windwardslopes (� = α − θ) (Erpul et al. 2004b). De Lima et al. (2003)showed that soil loss caused by downstream-moving rainstormson the leeward slopes was greater than that caused by identicalupstream-moving rainfall storms on the windward slopes.

In the current study, sediment transport rates by raindrop-impacted shallow flow were measured under WDR, and a vectorapproach was introduced to deal with raindrop and shallow flowinteractions for the mechanics of the interrill erosion process,elucidating the differences between WFR and WDR physics.

2 Materials and methods

The WDR experiments were conducted in the wind tunnel rain-fall simulator facility of the ICE, Ghent University, Belgium.Three loess-derived agricultural soils, Kemmel1 sandy loam(57.6% sand, 31.1% silt, and 11.3% clay) and Kemmel2 loam

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Journal of Hydraulic Research, iFirst (2013) Raindrop and flow interactions 3

(37.8% sand, 44.5% silt, and 17.7% clay) from the Kemmelbeekwatershed (Heuvelland, West Flanders, Belgium) and Nukerkesilt loam (32.1% sand, 52.3% silt, and 15.6% clay) from theMaarkebeek watershed (Flemish Ardennes, East Flanders, Bel-gium) were used in this study. The soil samples were collectedfrom the Ap horizon and air-dried prior to the experiment. Soilwas sieved into three aggregate fractions: 1.00–2.75, 2.75–4.80,and 4.80–8.00 mm, and proportionality factors assigned to eachfraction were 28, 32, and 40%, of each size class, respectively.A 5-kg soil sample of this mix was then packed loosely intoa 55-cm-long by 20-cm-wide pan. The simulated rainstormswere driven by horizontal wind velocities of 6, 10, and 14 m s−1

with the soil pan placed on either windward and leeward slopegradients of 7% (4◦), 15% (9◦), or 20% (11◦).

The details of the wind tunnel were given by Gabriels et al.(1997) and Cornelis et al. (2004a). Additional information werereported by Erpul et al. (2002, 2003b, 2004a, 2004b, 2008,2009a, 2009b) on the experimental set-up, which was especiallydesigned for WDR erosion studies in the ICE wind tunnel rainfall

simulation facility (Fig. 1). The parameters of the wind-drivenraindrop impact vectors are given in Table 1.

The diverse WDR fall trajectories with the angle of rain inci-dences (�) between the wind vector and the plane of the testsurface were obtained by changing slope aspect from wind-ward (Fig. 1a) to leeward (Fig. 1b). The WDR intensities (Iwd ,m3 m−2 s−1) were directly measured on the inclined plane andare reported as a mean of nine measurements in Table 1. UsingWDR intensities and median drop sizes (d50, mm), ηwd was cal-culated by Eq. (6) (Table 1). The raindrop sizes of WDR in theICE facility, determined by the stain method (Hall 1970) havea narrow range around the median raindrop diameter, and thedominant size is 1.5 mm (Erpul 1996, Erpul et al. 1998, 2000).

ηwd = Iwd

∀ (6)

where ∀ (m3) is the volume of a raindrop, considered as a spherewith diameter d50 (m). Since the WEPP model assumes the

Sediment and runoff sampler

u

N2

N1

Flowdepth

s

rn

uf

Flow vector

Wind-driven raindrop fall vector

Wind-driven raindrop impact vector

u

Horizontal plane

Plane of the sloping test surface

Wind-driven raindrop fall vector

Wind-driven raindrop impact vector

Sediment and runoff sampler

Horizontal plane

Plane of the sloping test surface

Flow vectorFlowdepth

-

(a)

(b)

qa

qa

F F = a -q

N2

N1

uf

r

s

n F = a +q

F

Figure 1 Two-dimensional experimental set-up with different wind-driven raindrop impact vectors in the ICE wind tunnel for measuring interrillsediment delivery rates (Di, kg m−2 s−1): (a) windward set-up and (b) leeward set-up. Notes: α, rain inclination; θ , slope gradient; �, angle of rainincidence; vr , resultant raindrop impact velocity; vs, along-surface raindrop impact velocity; vn, normal raindrop impact velocity; uf , shallow flowvelocity; u, horizontal wind velocity; N1, normal to the horizontal plane; N2, normal to the sloping test surface

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Table 1 The parameters of the wind-driven raindrop impact vectors along with the measured WDR intensities (Iwd ) for rains with the referencehorizontal wind velocities of 6, 10, and 14 m s−1

u d50 α degree θ degree �a Mean (Iwd) CV Mean ηwd(m s−1) (mm) (◦) (◦) (◦) m s−1× 10−5 (Iwd)b (I ) m s−1× 10−5 m3 m−3 m−2 s−1× 104

6 – ww 1.63 53.0 ± 11.5d 4 49 2.53 0.03 3.86 1.121.38 ≤ d50 ≤ 1.84c 9 44 2.91 0.02 4.08 1.28

11 42 3.15 0.02 4.21 1.3910 – ww 1.53 68.0 ± 7.6 4 64 3.50 0.05 8.04 1.87

1.50 ≤ d50 ≤ 1.57 9 59 3.83 0.04 7.59 2.0411 57 3.94 0.03 7.21 2.10

14 – ww 1.54 74.0 ± 6.6 4 70 2.62 0.03 7.48 1.371.51 ≤ d50 ≤ 1.57 9 65 3.17 0.02 7.51 1.66

11 63 3.33 0.03 7.14 1.746 – lw 1.63 53.0 ± 11.5c 4 57 3.45 0.03 6.33 1.52

1.38 ≤ d50 ≤ 1.84 9 62 2.96 0.04 6.20 1.3111 64 2.71 0.03 6.25 1.20

10 – lw 1.53 68.0 ± 7.6 4 72 2.51 0.04 8.20 1.341.50 ≤ d50 ≤ 1.57 9 77 1.70 0.03 7.23 0.91

11 79 1.44 0.06 7.88 0.7714 – lw 1.54 74.0 ± 6.6 4 78 1.81 0.03 8.37 0.95

1.51 ≤ d50 ≤ 1.57 9 83 1.21 0.06 8.67 0.6311 85 1.15 0.05 12.70 0.60

Notes: u, horizontal wind velocity (ww, windward; lw, leeward); d50, median drop size; α, rain inclination from vertical; θ , slope gradient; �, angleof rainfall incidence; ηwd , the number of wind-driven raindrops per unit area per unit time.aThe angle of rain incidence is � = α − θ on the windward slopes and � = α + θ on the leeward slopes.bThe values of coefficient of variation (CV) were calculated as the ratio of the standard deviation to the mean of the related parameter.c95% confidence interval on mean values of d50.dStandard deviations of the rainfall inclination (α) are given next to the mean value with ± sign.

vertical maximum intensity (I , m s−1) for the interrill componentof the hillslope erosion model (Eq. 2), this was calculated by thecosine law of spherical trigonometry (Eq. 7):

I = Iwd

cos �(7)

The resultant impact velocities (vr , m s−1) (Eq. 9) were measuredby a piezoelectric ceramic kinetic energy sensor (SensitTM 2000,Erpul 2001), and are reported in Table 2.

Under WDR conditions, because only the component of rain-drop velocity normal to the soil surface (vn, m s−1) can resultin compressive stress (Erpul 2001, Erpul et al. 2002, 2003a,2003b), the flux of rainfall energy based on the normal velocitycomponent of raindrop impacts (KErn, J m−2 s−1) was calculatedby Eq. (8) (Table 2) to replace the rainfall intensity term of Eq. (2)for a comparative analysis:

KErn = 0.5Mv2nηwd (8)

vn = vr cos � (9)

where M is the mass of the raindrop (kg).Flow kinetic energy flux or stream power (�Flow, J m−2 s−1)

was used as a substitute for the interrill runoff rate (Eq. 2), andwas estimated by (Eq. 10) (Table 3):

�Flow = ρwgquSo (10)

where g is the gravitational acceleration (m s−2), qu is the unitdischarge (m2 s−1), and So is the bed slope (m m−1). Simulatedrainfall experiments were conducted under freely drained condi-tions and during each rainfall application and after runoff started,sediment and runoff samples were collected at 5-min intervals atthe bottom edge of the pan using wide-mouth bottles. Runoff unitdischarge was then calculated by Eq. (11):

qu = 1B

n∑

i=1

Qi (11)

where Qi is the runoff discharge (m3 s−1) measured at the ith5-min interval and B is the top width of the soil pan (B = 0.20 m).

Additionally, the effects of the splash bursts of the wind-drivenraindrops on the shallow flow were related to their impact fre-quencies and sizes, and the kinetic energy fluxes exerted by thesebursts (�Drop, J m−2 s−1) were estimated by Eq. (12):

�Drop = ρwgIwdd50 (12)

Under WDR, a vector field forms at the impact-flow boundary(Fig. 1) such that the lateral jets of raindrop splashes with respectto the downward flow vary with the slope aspect. Because of this,�Drop was partitioned into its normal (�Dropn) and along-surfacecomponents (�Drops), using the angle of rain incidence

�Dropn = �Drop cos � (13)

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Journal of Hydraulic Research, iFirst (2013) Raindrop and flow interactions 5

Table 2 The resultant impact velocities for the rains with the referencehorizontal wind velocities of 6, 10, and 14 m s−1 along with the normalraindrop impact velocities and kinetic energy fluxes

KErn × 10−2

(J m−2 s−1)

u (m s−1) θ degree (◦) vr (m s−1) vn (m s−1) Mean CVa

6 – ww 4 4.64 ± 0.56b 3.04 11.75 0.039 3.31 19.62 0.05

11 3.46 16.84 0.0310 – ww 4 7.64 ± 0.60 3.32 16.21 0.02

9 3.85 29.64 0.0411 4.17 31.11 0.02

14 – ww 4 10.48 ± 0.57 3.67 18.70 0.029 4.43 34.10 0.03

11 4.89 37.71 0.036 – lw 4 4.64 ± 0.56 2.53 11.01 0.03

9 2.21 6.99 0.0411 2.01 4.30 0.03

10 – lw 4 7.64 ± 0.60 2.34 7.03 0.049 1.80 2.51 0.03

11 1.39 0.98 0.0614 – lw 4 10.48 ± 0.57 2.27 5.61 0.03

9 1.46 1.53 0.0611 0.95 0.48 0.05

Notes: u, horizontal wind velocity (ww: windward; lw: leeward); θ ,slope gradient; vr , resultant raindrop impact velocity; vn, normal rain-drop impact velocity; KErn, normal kinetic energy flux of raindrops.aThe values of CV were calculated as the ratio of the standard deviationto the mean of the related parameter.bStandard deviations of the resultant raindrop impact velocity (vr) aregiven next to the mean value with ± sign.

�Drops = �Drop sin � (14)

Interactions between the raindrop impact velocity vectors andflow vectors were dealt with using a vector solution in the two-dimensional case (Fig. 1), considering the fluxes that act onthe shallow flow in both normal and along-surface directions(Table 3). Vector additions of normal (�n) and along-surface(�s) kinetic energy fluxes at the raindrop impact-flow boundaryare as follows:

�n = �Dropn (15)

�s = �Flow ± �Drops (16)

A plus-minus sign is used in Eq. (16) to show that�Flow and�Drop

are in the opposite directions in the windward slope (−) and inthe same direction in the leeward slope (+) as the along-surfaceplane (Figs. 1 and 2). The resultant vector of kinetic energy fluxesat the raindrop impact-flow boundary (�r) was then calculatedusing Eq. (17) (Table 3):

�r = (�2n + �2

s )1/2 (17)

In this paper, a prediction equation was developed for the interrillsediment delivery rate based on the WDR vector physics. The

linear terms of both energy flux related to the normal componentof resultant raindrop velocity (KErn, Eq. 7) (Erpul 2001, Erpulet al. 2002, 2003a, 2003b) and resultant vector of energy fluxesat the raindrop impact-flow boundary (�r , Eq. 17) were used toexplain the variations in Di (Eq. 18):

Di = kwdKErn�r (18)

where kwd is the interrill erodibility with WDR, with units of(kg m2 s J−2). The results obtained from Eq. (18) were finallycompared with those from Eq. (2).

3 Results and discussion

Table 4 provides the mean values for the measured interrill sed-iment delivery rates (Di) and the interrill runoff rates (q) for thethree soils, and Fig. 2 shows the scatter plots for all individualdata points between the measured Di and predicted Di valuesfrom both Eq. (2) (Fig. 2a) and Eq. (18) (Fig. 2b) for each soiland the combined data for all soils.

Equation (2) fitted the measured WDR interrill erosion poorly(r2 = 0.63 for all data) (Fig. 2a). The main reason for this wasthe fact that the WEPP model was not intended to deal withvariations that occur due to the vector physics of WDR, whichrequires a component-wise or vector-wise solution when theinterrill erosion processes occur in a two-dimensional case. Therainfall intensity term of the WEPP model (Eq. 2), as an impactparameter, cannot represent the significant differences in raindropfrequency (ηwd) (Sharon 1970, De Lima 1990), raindrop veloc-ity (vr) (Pedersen and Hasholt 1995) and direction (Erpul et al.2003b, 2005, 2008) under WDR. The soil detachment processoccurs differently due to the inclined raindrops having improvedtangential jetting during impact and after the impact inducedby wind. As the wind-driven raindrop impact angle decreases(Cruse et al. 2000, Erpul 2001), the soil surface experiencesa lesser impact force (Ellison 1947, Heymann 1967, Springer1976) when compared with that under the perpendicular fall tra-jectory of raindrops, which is explicitly assumed in WEPP forinterrill erosion mechanics. Erpul et al. (2003a) reported that theflux of rainfall energy based on the normal velocity of raindropimpacts (KErn) explained the variations in the sediment transportby rain-impacted shallow flow better than the resultant rainfallenergy flux (KEr), which could not account for directional effectsof the raindrop impact velocity vector on the process. One reasonwhy Eq. (18) explained the variation in the experimental data hereso successfully (r2 = 0.96 for all data) (Fig. 2b) was the fact thatKErn was able to distinguish variations not only in the raindropimpact frequency and velocity but also in the impact directionin the two-dimensional impact cases. This resulted in substan-tial partitioning of the normal and lateral (along-surface) stresseswhich varied depending upon the horizontal wind velocity anddirection.

The vector-free runoff term of the WEPP model (q,m3 m−2 s−1) (Eq. 2) could not differentiate the interactions

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Table 3 The parameters for performing a vector solution for interactions between raindrop splashes and flow in the two-dimensional set-up of theWDR simulations

�Flow × 10−3 (J m−2 s−1) �Dropn × 10−4 (J m−2 s−1) �Drops × 10−4 (J m−2 s−1) �r × 10−3 (J m−2 s−1)

u (ms−1) Sao (mm−1) Mean CVb Mean CV Mean CV Mean CV

6 – ww 0.07 3.92 0.30 2.65 0.03 3.05 0.03 3.63 0.320.15 5.83 0.35 2.29 0.05 4.71 0.05 5.37 0.370.20 4.71 0.29 1.35 0.03 3.71 0.03 4.34 0.31

10 – ww 0.07 10.09 0.41 3.33 0.02 3.23 0.02 9.77 0.420.15 15.48 0.32 2.94 0.03 4.92 0.04 14.99 0.330.20 11.81 0.29 2.01 0.02 4.34 0.02 11.37 0.30

14 – ww 0.07 15.43 0.36 3.72 0.02 3.36 0.02 15.10 0.360.15 22.52 0.36 3.20 0.03 4.95 0.03 22.03 0.360.20 18.17 0.27 2.27 0.03 4.48 0.03 17.73 0.28

6 – lw 0.07 5.25 0.27 2.99 0.03 4.62 0.03 5.72 0.250.15 3.24 0.39 1.16 0.03 3.57 0.04 3.60 0.350.20 1.61 0.29 0.57 0.03 2.67 0.03 1.87 0.25

10 – lw 0.07 7.90 0.38 2.21 0.04 4.17 0.04 8.32 0.360.15 3.10 0.34 0.57 0.04 2.48 0.03 3.35 0.310.20 1.78 0.20 0.22 0.06 1.81 0.06 1.96 0.19

14 – lw 0.07 8.61 0.50 1.89 0.03 3.89 0.03 9.00 0.470.15 3.00 0.49 0.41 0.06 2.11 0.06 3.21 0.450.20 1.81 0.26 0.15 0.05 1.73 0.05 1.99 0.23

Notes: u, horizontal wind velocity (ww, windward; lw, leeward); So, channel bottom slope; �Flow , kinetic energy flux of flow; �Dropn , normal kineticenergy flux of raindrop splashes; �Drops , along-surface kinetic energy flux of raindrop splashes; �r , resultant vector of energy fluxes at the raindropimpact-flow boundary.a0.07, 0.15 and 0.20 mm−1 correspond to the values of 4◦, 9◦ and 11◦ respectively.bThe values of CV were cslculated as the ratio of the standard deviation to the mean of the related parameter.

Table 4 Measured interill runoff and sediment delivery rates for the three soils of the WDR experiments

Nukerke Kemmel1 Kemmel2

Di × 10−4 Di × 10−4 Di × 10−4

(kg m−2 s−1) (kg m−2 s−1) (kg m−2 s−1)

u (m s−1) Sao (m m−1) q × 10−5 (m s−1) Mean CVb q × 10−5 (m s−1) Mean CV q × 10−5 (m s−1) Mean CV n

6 – ww 0.07 1.44 3.78 0.04 0.90 1.78 0.02 0.78 2.67 0.10 30.15 1.82 9.98 0.10 1.26 6.28 0.05 0.67 4.20 0.16 30.20 1.92 15.13 0.08 1.59 11.06 0.06 0.80 7.45 0.21 3

10 – ww 0.07 1.96 8.76 0.05 1.83 5.72 0.10 0.85 6.88 0.22 30.15 2.46 22.30 0.17 2.16 20.64 0.14 1.14 12.84 0.28 30.20 2.64 34.82 0.08 2.48 40.65 0.05 1.16 16.63 0.05 3

14 – ww 0.07 1.60 7.13 0.08 1.36 4.99 0.26 0.79 3.83 0.04 30.15 1.75 18.14 0.28 1.72 15.33 0.13 0.92 12.10 0.16 30.20 2.02 27.03 0.27 1.96 30.73 0.07 1.10 18.58 0.06 3

6 – lw 0.07 1.84 4.79 0.06 0.99 3.27 0.05 1.36 4.81 0.04 30.15 1.46 5.67 0.05 0.74 3.45 0.06 0.74 2.74 0.03 30.20 1.32 6.57 0.05 0.60 2.75 0.06 0.48 1.86 0.07 3

10 – lw 0.07 1.30 4.92 0.02 0.63 2.33 0.06 0.65 3.89 0.06 30.15 0.55 2.77 0.16 0.33 1.08 0.06 0.27 1.51 0.14 30.20 0.45 1.90 0.06 0.23 0.63 0.07 0.15 0.62 0.15 3

14 – lw 0.07 0.59 3.31 0.07 0.37 1.92 0.05 0.32 2.72 0.07 30.15 0.28 2.27 0.15 0.18 0.88 0.05 0.21 1.00 0.08 30.20 0.22 1.66 0.04 0.14 0.34 0.05 0.14 0.60 0.11 3

Notes: u, horizontal wind velocity (ww, windward; lw, leeward); So, channel bottom slope; q, interrill runoff rate (runoff unit discharge is calculatedusing channel length and was calculated by (qu = q × 0.55); Di, interrill sediment delivery rate; and n, number of replicates.a0.07, 0.15 and 0.20 mm−1 correspond to the values of 4◦, 9◦ and 11◦ respectively.bThe values of CV were cslculated as the ratio of the standard deviation to the mean of the related parameter.

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0.0 5.0e-10 1.0e-9 1.5e-9 2.0e-9 2.5e-90.000

0.001

0.002

0.003

0.004

0.005

Iq (m2 s–2)

0.0 5.0e–10 1.0e–9 1.5e–9 2.0e–9 2.5e–9

Di(k

g m

–2 s

–1)

Di(k

g m

–2 s

–1)

0.000

0.001

0.002

0.003

0.004Di = 1E + 06 (Iq) Di = 0.33 KErnWr

Di = 0.41 KErnWr

Di = 0.40 KErnWr

r2 = 0.61

0.000 0.002 0.004 0.006 0.008 0.010 0.0120.000

0.001

0.002

0.003

0.004

0.005

r2 = 0.97

r2 = 0.97

r2 = 0.95

r2 = 0.96

Nukerke Nukerke

0.000 0.002 0.004 0.006 0.008 0.0100.000

0.001

0.002

0.003

0.004

0.005

Kemmel1 Kemmel1

0 2e-5 4e-5 6e-5 8e-5 1e-4 1e-4 1e-4

Di (

kg m

-2 s

-1)

0.0000

0.0005

0.0010

0.0015

0.0020

0.0025Di = 1.7E+06 (Iq)

r2 = 0.63

0.000 0.001 0.002 0.003 0.004 0.0050.0000

0.0005

0.0010

0.0015

0.0020

0.0025

Kemmel2 Kemmel2

0.0 5.0e-10 1.0e-9 1.5e-9 2.0e-9 2.5e-9

Di (

kg m

-2 s

-1)

0.000

0.001

0.002

0.003

0.004

0.005Di = 1.5E+06 (Iq)

r2 = 0.63

0.000 0.002 0.004 0.006 0.008 0.010 0.0120.000

0.001

0.002

0.003

0.004

0.005

All data All data

(a) (b)

Di = 0.37 KErnWr

KErnWr(J m–2 s–1)2

Di = 17E + 06 (Iq)

r2 = 0.67

Figure 2 Interrill sediment delivery rates (Di, kg m−2 s−1) from wind-driven rainfall predicted by: (a) kiIq (Eq. 2); and (b) kwd Krn�r (Eq. 18).Results shown for each soil and for all data combined

between the raindrop impact velocity vector and the shallowflow vector under WDR. When compared with the maximumperpendicular impact of the windless raindrops, it was clear thatgreater stress components acted parallel to the surface flow andwith lesser stress components perpendicular to the surface flowin our experiments. Also, given the differences with respect to theslope aspect, the normal stresses decreased further as the WDRincidence angle increased on the leeward slopes compared withthose on the windward slopes (Fig. 1b).

For the case of the one-dimensional vertical impact of WFRon the surface shallow flow, the hydraulic friction caused bythe raindrop effect can be explained by the Darcy–Weisbach

friction factor and the Reynolds number (Gilley et al. 1985,Julien and Simons 1985, Katz et al. 1995). However, in situ-ations where increased lateral jetting by the wind existed, thepartitioning of the normal and lateral stresses of the raindropimpacts determined the shallow flow hydraulics by changing theroughness induced by the raindrop impacts at an angle with theflow and the unidirectional splashing in the wind direction (Erpulet al. 2004b).

Together with the discrepancies in the stress partitioning underthe impacts of wind-driven raindrops, a vector field also formedat the impact-flow boundary (Fig. 1), such that the lateral jets ofraindrop splashes with respect to the downward flows changed

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8 G. Erpul et al. Journal of Hydraulic Research, iFirst (2013)

with the slope aspect. This is greatly different from the case ofwindless rains where the lateral raindrop jets are always in thesame direction as the shallow flow direction (Erpul et al. 2004b).In fact, raindrop-induced flow resistance empirically estimatedby Shen and Li (1973) and Katz et al. (1995) represents thiscondition. Nevertheless, in our case, the lateral jets of raindropsplashes were in the opposite direction on the windward slopeand in the same direction on the leeward slope as the along-surface plane (Fig. 1). Therefore, the model represented by Eq.(18) functioned convincingly well with the experimental datafor WDR interrill erosion (r2 = 0.96 for all data) (Fig. 2b) sincethe resultant energy flux at the raindrop impact-flow boundary(�r , Eq. 17) worked vector-wise and discriminated well both thevariations in the stress partitioning of the raindrop impacts andthe interactions between the raindrop impact velocity vector andflow vector.

4 Conclusions

In this evaluational analysis of the performance of the interrillcomponent of the WEPP model for the WDR events, we foundthat sub-processes occurred in WDR which closely dependedupon the raindrop impact velocity vector. Since the terms forthe rainfall intensity and interrill runoff rate in the WEPP model(Eq. 2) do not include the vector physics of WDR, they could notadequately represent this sub-process in these experimental data.Introducing the kinetic energy fluxes including vector-dependentparameters for rainfall and shallow flow in WDR provided agreatly improved prediction equation, accounting for ≥95% ofthe variation in the observed sediment delivery rates. While themodel used in this study was WEPP, these same sets of physics-based equations could be utilized within any other current orfuture physical process-based erosion model, to simulate wind,raindrop, and flow interactions on interrill soil erosion. Withincreasing computational power of personal computers as wellas better observed and/or generated climate data for storm inten-sity patterns and wind speed and direction, future assessments ofsoil erosion from combined rain and wind storm events have thepotential to be greatly improved.

Acknowledgements

G.E. thanks the Abdus Salam International Center for TheoreticalPhysics (ICTP), Trieste, Italy, for providing financial support ofhis visits (as a Regular Associate) to both the ICTP and the part-ner institute, UNESCO Chair on Eremology, Ghent University,Belgium.

Notation

qs = sediment transport capacity of rain-impacted thin flow(kg m−1 s−1)

Di = interrill sediment delivery rate to a rill (kg m−2 s−1)

I = maximum vertical rainfall intensity (m3 m−2 s−1 orm s−1)

Iwd = WDR intensity (m3 m−2 s−1 or m s−1)

So = channel bottom slope (m m−1)

ki = interrill erodibility of soil (kg s m−4)

kwd = interrill erodibility of soil with WDR (kg m2 s J−2)

θ = slope gradient (◦)α = rain inclination (◦)� = angle of rain incidence (◦)N1 = normal to the horizontal planeN2 = normal to the sloping test surfaceρa = air density (kg m−3)

ρw = water density (kg m−3)

uf = shallow flow velocity (m s−1)

u = horizontal wind velocity (m s−1)

u∗ = wind shear velocity (m s−1)

τw = along-surface wind shear stress (N m−2)

τs = shear stress of a wind-driven raindrop (N m−2)

d50 = median raindrop size (mm)∀ = volume of a raindrop (m3)

M = mass of a raindrop (kg)g = gravitational acceleration (m s−2)

ηwd = number of wind-driven raindrops per unit area perunit time (m3 m−3 m−2 s−1)

vr = resultant impact velocity of a raindrop (m s−1)

vs = along-surface component of raindrop impact velocity(m s−1).

vn = normal component of raindrop impact velocity(m s−1)

KEr = resultant kinetic energy flux of rainfall (J m−2 s−1)

KErn = flux of rainfall kinetic energy based on the normalvelocity component of raindrop impacts (J m−2 s−1)

Qi = runoff discharge (m3 s−1)

B = top width of soil pan (0.20 m)qu = runoff unit discharge (m2 s−1)

q = interrill runoff rate (m s−1)

�Flow = kinetic energy flux of flow or stream power(J m−2 s−1)

�Drop = resultant kinetic energy flux of raindrop splash bursts(J m−2 s−1)

�Dropn = normal kinetic energy flux of raindrop splash bursts(J m−2 s−1)

�Drops = along-surface kinetic energy flux of raindrop splashbursts (J m−2 s−1)

�n = vector addition of normal components at the raindropimpact-flow boundary (= �Dropn) (J −2 s−1)

�s = vector addition of along-surface components at theraindrop impact-flow boundary (= WFlow ± �Drops)

(J m−2 s−1)

�r = resultant vector of kinetic energy fluxes at theraindrop impact-flow boundary (= �2

n + �2s )

1/2

(J m−2 s−1)

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