## 1. Introduction

[2] Hydraulic resistance to open-channel and overland flows is an important characteristic that needs to be represented properly in modeling runoff, flood routing and inundation, and soil erosion. Resistance estimation affects not only the accurate calculation of flow variables, such as the water depth, velocity, and shear stress, but also the prediction of their derivative outcomes, such as the time of concentration, flow distribution in a basin, the transport capacity, the total sediment yield, etc. The resistance of a surface can be characterized with several hydraulic roughness coefficients. The most widely used are the Manning roughness coefficient (*n*), the Chezy resistance factor (*C*), and the Darcy-Weisbach friction factor (*f*). Manning's *n* is most popular in hydrological and soil erosion models, while using the Darcy-Weisbach *f* is more common than the other resistance formulations in experimental studies [*Hessel et al.*, 2003]. Theoretically, hydraulic resistance can be divided into five components: surface (grain) resistance, form resistance, wave resistance, rain resistance, and bed-mobility resistance [*Abrahams and Parsons*, 1994; *Hu and Abrahams*, 2006; *Smith et al.*, 2007].

[3] Numerous studies have performed field or laboratory experiments and theoretical analyses seeking ways to relate hydraulic characterization of flow to roughness coefficients. These studies tried to investigate a number of dimensionless variables in an attempt to find suitable relationships using various metrics such as the Reynolds number (*Re*), the Froude number (*Fr*), the characteristic roughness length (e.g., the ratio of depth to roughness element), domain slope (*S*), and vegetation or obstacle cover fractions. Since early studies of overland flow, resistance was described by a roughness coefficient, in analogy to the resistance relations used to characterize flows in pipes. A relationship between the roughness coefficients and the Reynolds number (e.g., *f-Re*) has been well established for shallow overland flows as well as for flows in pipes and smooth channels [*Chow*, 1959; *Emmett*, 1970; *Li and Shen*, 1973; *Phelps*, 1975; *Savat*, 1980]. The *f-Re* relationship has a negative slope of 1.0 in the laminar flow regime [*Blasius*, 1913]; in turbulent flow, different *f-Re* relationships are obtained, depending on the value of the relative roughness [*Nikuradse*, 1933]. These findings indicated that among several possible dimensionless variables, *Re* has a predominant effect in quantifying the flow resistance in conditions where the flow completely submerges a plane bed with either a smooth or a rough surface. In such conditions, the roughness height is significantly smaller than the flow depth and the hydraulic resistance is dominated by the surface resistance component arising due to the presence of roughness elements beneath the flow surface.

[4] However, in conditions where the surface is covered by stones, organic litter, or stems of vegetation that protrude through the flow, the aforementioned *f-Re* relationships are not applicable. Other dimensionless variables (e.g., *Fr*, relative roughness height, vegetation cover, etc.) may become more dominant, reflecting that the form and wave resistance can become the primary components of the total flow resistance [*Emmett*, 1970; *Roels*, 1984; *Abrahams et al.*, 1986; *Gilley and Finkner*, 1991; *Gilley et al.*, 1992b; *Hirsch*, 1996; *Lawrence*, 1997; 2000; *Takken and Govers*, 2000; *Hu and Abrahams*, 2006]. For example, *Emmett* [1970] was the first to emphasize the importance of form resistance caused by microtopography, which can significantly exceed the surface resistance. *Roels* [1984] and *Abrahams et al.* [1986] stated that the standard *f-Re* relationship is not ubiquitous: the *f-Re* relationship can have a convex upward or a negatively sloping power law relation. These relationships can be attributed to the progressive inundation of roughness elements, implying that the surface configuration of the elements, and not just the flow state, becomes dominant in quantifying the resistance. Further, *Gilley and Finkner* [1991] presented a regression equation for predicting *f* and *n* by including the characteristic length scale, i.e., a “random roughness index” as the primary variable. *Gilley et al.* [1992b] suggested that *f* is largely controlled by a measure of the gravel cover fraction. *Hirsch* [1996] developed a flow resistance model that explained flow conditions when the fraction of roughness elements was greater than 10% and *Fr* was greater than 0.5.

[5] Recently, *Lawrence* [1997] further demonstrated the importance of other dimensionless variables in conditions of emerging vegetation and other types of obstacles protruding through the flow. Rather than using the Reynolds number, *Lawrence* [1997] advocated the use of the inundation ratio, *h/k*, as the ratio of the flow depth *h* to the characteristic height of roughness elements *k*. *Lawrence* [1997] identified distinct flow regimes, such as partial and marginal inundation, and well-inundated flows, with various fractions of obstacles (hemispheres) placed in the flow. Depending on whether the flow depth *h* was greater/smaller than the characteristic height *k*, *Lawrence* [1997] estimated *f* as a function of the inundation ratio by using a drag model for the partial inundation, a mixing length model for the marginal inundation, and a rough turbulent flow formula for well-inundated flows. Since the estimation of the drag model showed an underestimation of flow resistance for the partial inundation case, *Lawrence* [2000] later modified the form drag model to obtain higher *f* values by increasing the drag coefficient, which was negatively correlated with *h/k*.

[6] The modified model of *Lawrence* [2000] was successfully applied for the estimation of flow resistance for the case of marginal inundation, where roughness elements were randomly distributed and relatively uniform in size. However, when this model was applied under conditions differing from the setting under which the model was developed, such as complex flow geometries, the performance was not always satisfactory. *Ferro* [2003] tested the model using laboratory measurements and showed that the modified mixing length formulation provided accurate estimates, while the modified drag model resulted in a limited accuracy in estimating *f*. *Takken and Govers* [2000] also tested the partial inundation case of *Lawrence* [1997] and concluded that for situations with the complex configurations of roughness elements, a single independent variable (i.e., *h/k*) was insufficient to predict *f*. Thus, other variables, such as the flowrate, *Fr*, and *Re* need to be considered to fully characterize the flow resistance [*Takken and Govers*, 2000; *Smart et al.*, 2002; *Hu and Abrahams*, 2006].

[7] In shrubland or forested hillslopes, typical flow depths are much smaller than the height of roughness elements such as vegetation stems and thus inundation ratios are very small. Such flow conditions generally prevail in hillslope hydrological dynamics. Characterization of flow for partially inundated conditions with a nonuniform distribution of roughness elements is therefore significant for modeling runoff routing and soil erosion. However, these flow conditions remain poorly characterized by empirical observations. For example, experimental data from previous studies (see *Lawrence* [1997, Figure 4] reporting data from eleven studies) are limited to partial inundation cases, i.e., most of the observed inundation ratios were between 0.1 and 1.

[8] In order to establish a general relationship applicable to a wide range of conditions, numerical modeling based on the two-dimensional shallow-water equations was carried out in this study. The numerical simulations corresponded to overland flow on hillslopes covered with shrubby or woody vegetation. An application of a numerical model, as compared to field or experimental manipulations, provides several advantages. Specifically, in the case of small depths of overland flow (few mm to cm length scale), the minimum requirement of water depth for measuring the velocity with Acoustic Doppler velocimeters or electromagnetic current meters is not satisfied [*Lawless and Robert*, 2001]. When the requirements are satisfied, small depths still represent an issue in terms of measurement accuracy [*Biron et al.*, 1998]. These difficulties result in large measurement errors in laboratory or field experiments. Furthermore, the determination of the friction (or energy) slope used in the calculation of roughness coefficients is also cumbersome. It can be normally substituted with the bed slope under uniform flow conditions, but it may not represent truthfully a spatially varying friction slope in situations with many protruding obstacles. Lastly, the difficulty of controlling conditions for high-flow rates prevents empirical observations in field and laboratory studies [e.g., *Takken and Govers*, 2000; *Hessel et al.*, 2003]. For example, *Takken and Govers* [2000] used discharges ranging from 4.2 × 10^{−6} to 2.7 × 10^{−4} m^{3} s^{−1}.

[9] High-resolution, hydrodynamic numerical simulations can overcome all of the above problems by specifying arbitrary flow conditions, including both high and low-flow rates that occur in real world situations. Using detailed simulations performed at fine space-time scales, the properties of the resistance coefficient at larger spatial scales can be investigated. In order to represent a system with tree/shrub stems, a sloped plane populated with “obstacle cells” that have infinitely long vertical dimension was designed. A number of scenarios with different domain slopes (*S*), inflow rates (*Q*), bed substrate roughness conditions (*n _{b}*), and vegetation cover fractions ( ) were considered. Based on the simulation results, two methods were developed to obtain the upscaled Manning coefficient. A predictive equation was developed using multiple regression and dimensional analyses and verified with five different experimental data sets and a proposed wave resistance equation. Finally, the characteristic controls of several independent variables on the roughness coefficient are described and evaluated.