## 1. Introduction

[2] In the western United States, forestry management decisions affect a large portion of public lands and many different uses and activities. When assessing the environmental impacts of various decisions, potential soil loss through erosion is an important consideration. Since empirical erosion data are scarce and time- and cost- intensive to collect, managers often depend on erosion models to make these assessments. To be fully effective these models must be designed for various management scenarios and diverse environments. Most of the currently available erosion models used for forest environments evolved from models that were developed from plot studies on agricultural soils with low slopes [*Bryan*, 2000]. Experiments in forest environments where soils are shallow and slopes are steep are necessary if erosion models are to be usable for forest conditions.

[3] Most physically based water erosion models divide erosion into inter-rill or splash and sheet flow erosion, and rill or concentrated flow erosion [*Foster and Meyer*, 1972; *Foster*, 1982]. Sediment delivery may be limited either by the ability of the erosive agents to detach the sediment (detachment or source limited), or by the ability of the runoff to transport the sediment (transport limited) [*Ellison*, 1946; *Foster and Meyer*, 1972; *Foster*, 1982]. As concentrated flow further increases in amount or duration, channel processes such as bed and bank scour and sediment transport begin to dominate sediment movement [*Hairsine and Rose*, 1992b]. The study described in this paper is aimed at better understanding and modeling the upland rill erosion processes on steep disturbed forested hillslopes.

[4] Early modeling efforts focused on transport capacity, detachment capacity, and flow mechanics such as hydraulic shear stress and stream power as drivers for erosion. *Foster and Meyer* [1972] proposed a rill and inter-rill erosion model where the rill erosion component of that model was a function of hydraulic shear stress. This approach was subsequently incorporated into the Water Erosion Prediction Project (WEPP) model [*Nearing et al.*, 1989].

[5] The erosion component of the WEPP model [*Nearing et al.*, 1989] was one of the first physically based models to account for soil erosion and sediment transport from both inter-rill and rill areas, and is described by

where *x* is the distance down the slope (m), *G* is the sediment load per unit width (kg s^{−1} m^{−1}), *D*_{r} is the rill erosion rate (kg s^{−1} m^{−2}) and *D*_{i} is the inter-rill erosion rate (kg s^{−1} m^{−2}).

[6] The erosion from inter-rill areas, *D*_{i}, is calculated using an inter-rill erodibility parameter, *K*_{i} (kg s^{−1} m^{−4}), and is a function of rainfall intensity and runoff rate. The rill sediment detachment capacity, *D*_{c} (kg s^{−1} m^{−2}), is the amount of soil that can be dislodged by the water per unit area and time and is a product of the rill erodibility parameter, *K*_{r} (s m^{−1}), which is a soil property, and the excess soil shear stress, *τ*_{s} (kg s^{−2} m^{−1} or Pa). Some stress must be applied to the soil before detachment occurs, and this has been termed the critical shear stress, *τ*_{c} (kg s^{−2} m^{−1}), leading to an equation in the form of [*Foster et al.*, 1995]

[7] *Foster and Meyer* [1972] noted in field observations that as the amount of sediment in transport increases, the detachment decreases. They proposed a model to describe this process as

where *q*_{s} is the sediment flux rate (kg s^{−1}) and *T*_{c} is the sediment transport capacity of clear water (kg s^{−1}). If the detachment rate is assumed to be the differential of *G* (equation (1)), equation (3) becomes [*Elliot et al.*, 1989]

[8] The above equations demonstrate how *Foster and Meyer* [1972] assumed that once sediment was entrained in the rill flow, the sediment was either delivered to the outlet of the rill or, when the rill sediment flux equaled the flow's sediment transport capacity, it was deposited in the rill. When the sediment in transport exceeded the transport capacity, sediment deposition would begin as a function of particle fall velocity, with larger sand particles depositing first and smaller or less dense aggregates and clay particles depositing last. The transport limiting condition may occur in agricultural settings due to relatively high availability of detachable soil particles and relatively low transport capacities achieved on low slopes. The transport capacity in mountainous forests often is much greater than the values reported for agricultural settings [*Elliot et al.*, 1989; *Hairsine and Rose*, 1992a, 1992b; *McIsaac et al.*, 1992; *Giménez and Govers*, 2002]. *Foster* [1982] indicated that if the sediment flux rate was less than the flow's transport capacity, additional detachment would occur in the rill to either 1) achieve the transport capacity, or 2) achieve the rill detachment capacity.

[9] *Hairsine and Rose* [1992a, 1992b] also developed a rill erosion model that results in less sediment detachment as the amount of sediment in transport increases. In their model, sediment entrainment and deposition can occur simultaneously [*Hairsine and Rose*, 1992a, 1992b]. In rill segments where the sediment flux rate is low, the deposition also will be low so the dominant process will be detachment of cohesive soil from the sides and bottoms of the rill. In rill segments where the sediment flux rate is high, deposition will become more pronounced, and the entrainment process will be dominated by entrainment of non-cohesive sediment—sediment that had been previously detached and subsequently deposited—from the rill bottom, with some detachment of cohesive sediment from the rill sides. Based on these *assumptions*, *Hairsine and Rose* [1992b] developed a stream power rill erosion model with the form

where *Q* is the volumetric flow rate per rill (m^{3} s^{−1}), *c*_{i} is the sediment concentration of particle size class *i* (kg m^{−3}), *H* is the fraction of the rill wetted perimeter base covered by deposited sediment, *W*_{b} is the width of the base of a trapezoidal rill (m), *W*_{s} is the horizontal width of the sides of a trapezoidal rill (m), *F* is the fraction of excess stream power (Ω − Ω_{0}) used in entraining or re-entraining sediment in class size *i*, Ω_{0} is the stream power below which no entrainment occurs (kg s^{−3}), *I* is the number of settling velocity classes, *J* is the specific energy of entrainment (J kg^{−1} or m^{2} s^{−2}), and *q*_{syi} is the inter-rill contribution to sediment in the rill (kg s^{−1} m^{−1}). *Hairsine and Rose* [1992b] concluded that the *J* term can only be derived from erosion experiments, suggesting it may be related to fall cone or shear device measurements. Other researchers, however, did not find field- or lab-measured soil strength properties useful for predicting rill soil erodibility [*Elliot et al.*, 1990].

[10] The work of *Hairsine and Rose* [1992a, 1992b] has been supported by more recent studies that found stream power to be a better predictor of rill detachment rates than shear stress [*Elliot and Laflen*, 1993; *Nearing et al.*, 1997; *Pannkuk and Robichaud*, 2003]. *Bryan* [2000] suggested that different erosion predictors work better for different experimental designs. In an extensive laboratory and field study on agricultural soils, stream power was found to be the best predictor of unit sediment load (sediment per unit time per unit width) [*Nearing et al.*, 1997]. *Nearing et al.* [1999] found that sediment detachment rates were better correlated in a power function of either shear stress (*R*^{2} = 0.51) or stream power (*R*^{2} = 0.59) than in a linear function of shear stress. In a laboratory study on burned soils, stream power was shown to be a better predictor of rill erosion than shear stress (*R*^{2} = 0.56 and 0.24, respectively) [*Pannkuk and Robichaud*, 2003].

[11] The unit stream power, Ω_{u} (m s^{−1}), has also been shown to effectively predict rill erosion rates [*McIsaac et al.*, 1992; *Morgan et al.*, 1998]. The unit stream power rill erosion model takes the form

where *q*_{s} is the sediment flux rate (kg s^{−1}), *K*_{Ωu} is the unit stream power rill erodibility (kg m^{−1}) and Ω_{u0} is the (critical) unit stream power below which no erosion occurs. *McIsaac et al.* [1992] used sediment particle diameter, sediment particle fall velocity, and fluid viscosity to estimate the erodibility term.

[12] *Govers et al.* [2007] presented a critical review of the *Foster and Meyer* [1972] and *Hairsine and Rose* [1992a, 1992b] models as well as a number of similar concentrated flow erosion models that had been proposed in recent decades. They presented data to suggest that soil detachment is not limited by the amount of sediment in transport until the amount of sediment in transport approaches the sediment transport capacity [*Govers et al.*, 2007]. *Giménez and Govers* [2002] suggested a simplified rill erosion model based on the unit length shear force, Γ (kg s^{−2})

where *D*_{L} is the sediment flux per unit length of rill (kg s^{−1} m^{−1}) and *α* (s m^{−1}) is a constant.

[13] Each of these models initially was used to predict erosion from agricultural lands. The WEPP model was intended for use on agricultural, range and forest lands [*Laflen et al.*, 1997]; initial parameterization of the model, however, focused on predicting erosion from heavily disturbed agricultural and range lands [*Elliot et al.*, 1989; *Laflen et al.*, 1991]. Data presented by *Hairsine and Rose* [1992a, 1992b], *McIsaac et al.* [1992], and *Giménez and Govers* [2002] were from either tilled or highly disturbed agricultural soils. *Bryan* [2000] lists some of the limitations of research in agricultural soils, including the homogeneity caused by plowing, changes in soil structure and organic matter content, and lack of macropores. The frequent tillage applied to agricultural plots alters the soil structure, disperses aggregates, increases porosity, and decreases compaction. As a consequence, the tilled layer is more erodible and the rill erodibility would likely be relatively high compared to a forest soil which had never been mechanically disturbed. In comparison, soils in undisturbed forests generally have greater cohesion, a more developed structure, greater aggregate stability, and protection from erosive forces by vegetation, litter, duff, and roots. Forest soils therefore have fewer particles available for detachment unless some disturbance occurs to disrupt this stability.

[14] Recently, use of the WEPP model has expanded into non-agricultural applications, including predictions of erosion from natural and disturbed range and forest hillslopes [*Elliot*, 2004; *Robichaud*, 1996]. Some of the hydrologic [*Robichaud*, 2000] and inter-rill [*Burroughs et al.*, 1992] modeling parameters for forest conditions have been presented but rigorous evaluations of the rill erodibility parameters for forest conditions have not yet been conducted.

[15] The soil parameters in the rill erosion models presented are fixed for all computations once a soil type and disturbance are selected. A recent study on forest roads suggested that rill erosion rates are much higher in the early part of a runoff event than in the latter part of the event [*Foltz et al.*, 2008]. Similarly, *Pierson et al.* [2008] reported changes in sediment concentration during a constant flow, short duration simulated runoff experiment on burned range land. These changes in rill erosion over short time periods may be caused by the winnowing of fine or easily detachable soil particles during the early stages of a runoff event. As the supply of easily erodible particles is smaller in forest soils than in agricultural soils, a constant erodibility model may not apply as well to forest soils as it does to tilled agricultural soils.

[16] In part 1 of this study [*Robichaud et al.*, 2010] we reported differences in runoff rates, runoff velocities, and sediment flux rates among natural forested sites and sites with three types of forest disturbance. The experiment was designed to measure the effects of rill flow on erosion rates, independent of inter-rill flow and sediment contribution. Although this would not occur in any natural system, this control allowed clear identification of the rill erodibility parameters and influences on those parameters. The objectives for the current paper were to 1) determine if the erosion response is detachment-limited or transport-limited; 2) calculate and compare the rill erodibility parameters for five classes of forest disturbance (natural, 10-month old low soil burn severity, 2-week old low soil burn severity, high soil burn severity, and skid trails); 3) determine if the erodibility for a given site changes between initial and late (steady state) runoff periods during a constant inflow; and 4) compare prediction capability among the four hydraulic parameters described above (soil shear stress, stream power, unit stream power and unit length shear force). Implications for erosion modeling are also discussed.