The effects of storm movement on rainfall-runoff processes through a V-shaped watershed are investigated using a dynamic wave model based on shallow water equations. The governing equations are solved by a fourth-order accurate finite volume method to obtain accurate computational results. Once the numerical model is verified, a series of rainfall-runoff events on a V-shaped watershed are simulated. To investigate the contribution of each term in the depth-averaged momentum equation, a scale analysis, and quantitative assessment are conducted on the basis of the computational results. As a result, it is found that the speed and the direction of storm movement can play an important role in determining peak discharge and peak arrival time. In addition, the storm movement can generate loop in the stage-discharge relationship curve. Finally, it is revealed that the rating curve depends on the watershed characteristics rather than the rainfall conditions, at least in the idealized V-shaped watershed.
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 The effects of rainstorm movement on runoff hydrographs and peak flows were pioneered by Maksimov . Yen and Chow [1968, 1969] demonstrated the importance of the rainstorm movement to the shape of the runoff hydrograph through a series of laboratory experiments using the watershed experimentation system (WES). Black  studied watershed models to investigate the effect of watershed characteristics on hydrograph parameters. From the study, it was found that most of the contribution to the peak discharge was dependent on the lower half of the drainage. Thus, the direction of storm movement had little effect on the time of concentration. However, the direction change significantly affected the peak discharge and lag time at the outlet of the watersheds.
 By using a numerical kinematic wave model, Ogden et al.  found a specific relation between the speeds of equivalent moving storms and the flow velocities during runoff processes. In particular, the equivalent moving storms had a common storm length but the moving storms with different speeds had different rainfall intensities in proportion to the storm movement speeds. Previously, equivalent moving storms were defined by Yen and Chow  as storms moving at different speeds but having the same rainfall duration at every point on the watershed and having the same total rainfall volume. Singh  proposed analytical solutions of kinematic wave equations to model the flow on a plane when storms moved upward or downward. Lee and Huang  used a nonlinear numerical kinematic wave model to examine the criteria for equilibrium discharge by nonequivalent moving storms. Based on the numerical simulations for an overland plane and a V-shaped watershed, it was found that runoff can reach an equilibrium discharge even if the storm length is shorter than the watershed length and the rainfall duration is less than the time to equilibrium of the watershed for stationary uniform storms. Recently, consideration of the effect of rainstorm movement has not been constrained to flow hydrographs but extended to other interests such as the interaction between rainstorm movement and building environments [Isidoro et al., 2012], erosion [Nunes et al., 2006; de Lima et al., 2009; Heng et al., 2009, 2011], or sediment transport [de Lima et al., 2008, 2011].
 To obtain results more physical than those of kinematic wave approaches, dynamic wave models based on the Saint Venant equations, or shallow water equations (SWE), can be used. The SWE can be derived by depth averaging the Navier-Stokes equations when the horizontal length scale is much greater than the vertical length scale. In addition, the SWE assume that the flow is under hydrostatic pressure and the horizontal velocity field is constant throughout the water depth. Liang  compared a one-dimensional (1-D) kinematic wave model and a 1-D dynamic wave model. The results showed that due to the backwater effects the peak flows estimated at the watershed outlet by the dynamic wave model were different from the results estimated by the kinematic wave model. Recently, two-dimensional (2-D) dynamic wave models have been developed and applied for rainfall-runoff predictions [Ajayi et al., 2008; Yeh et al., 2011; Costabile et al., 2012a, 2012b; Caviedes-Voullième et al., 2012]. Yeh et al.  and Costabile et al. [2012a], similar to Liang , reported that a full dynamic wave model should be used to obtain accurate results for complex flows. Only under simple flow conditions will simplified versions of full dynamic wave models yield acceptable computational results.
 However, it should be noted that few studies have incorporated dynamic wave models to evaluate the effect of rainstorm movement on the resulting hydrographs. Although various researches have been conducted to investigate the effects of moving storms on hydrographs, most of the interest has been focused on what happens at the watershed outlet. In fact, it is difficult to find thorough investigations based on dynamic wave modeling approaches of the effects of storm movement on runoff processes throughout the interiors of watersheds. Hence, this paper presents a detailed quantitative analysis based on a 2-D dynamic wave model of the rainfall-runoff processes caused by moving storms throughout an idealized V-shaped watershed. In turn, we precisely examine the justification of various routing models simplified on the basis of scales that have traditionally been assumed. In addition, this paper provides the mechanism by which the storm movement affects the characteristics of loop-rating curves.
2. Dynamic Wave Model
2.1. Governing Equations
 The SWE are used to describe overland flows as follows [Yoon, 2011]:
where t is time, are the horizontal coordinates, and are the depth-averaged horizontal velocities in the x and y directions, respectively. The subscripts are . H is the water depth (from bottom to water surface), d is the distance from a datum to the bottom elevation, i is the rainfall intensity of a storm, g is the gravitational acceleration, and ρ is the density of water. The bottom shear stress in the xy plane is defined as follows:
 The bottom friction coefficient cf is determined using the Manning friction formula, , where n is Manning's roughness coefficient.
2.2. Numerical Methods
 In this paper, for research purposes rather than forecasting, the SWE are solved by higher-order accurate numerical schemes. A fourth-order accurate finite volume method based on a compact monotone upstream-centered scheme for conservation laws (MUSCL) with total variation diminishing (TVD) [Yamamoto and Daiguji, 1993] is used to calculate the flux terms. Using the values of the interface constructed, the numerical fluxes are calculated with the Harten-Lax-van Leer-contact (HLLC) approximate Riemann solver [Toro, 2002]. A well-balanced scheme that can be applied for arbitrarily varying topography, including steps, is employed in order to prevent the numerical oscillation that arises from the spatial variation of topography. More details of this well-balanced scheme are given in Kim et al. . A moving boundary scheme that can be applied for continuous or discontinuous slopes, including surface-piercing and submerged topography, is used to simulate changes in wet and dry beds. The details of the moving boundary scheme are given in Kim  and Lynett et al. . For the time integration of the conserved variables, the third-order accurate Adams-Bashforth predictor scheme and the fourth-order accurate Adams-Moulton corrector scheme are used. An implicit property of the time integration scheme results in minimization of the numerical instability that can arise from the friction terms. In addition, the water depth that appears in the friction term of the SWE is taken as H = 10−4 when H < 10−4, and the flow velocity is set to zero when H = 10−5. More details of the numerical schemes can be found in Kim and Lee , where the density change and the temporal bottom variation are considered. Thus, by neglecting two variables, the numerical schemes in Kim and Lee  can be applied directly to the flows in this paper. No other modification is implemented for the numerical simulations.
2.3. Verification of Numerical Model
 Following previous studies on rainstorm movements, this study uses experimental results from the WES [Yen and Chow, 1968, 1969] to verify the numerical model. In the WES laboratory, the dimensions of the V-shaped watershed are . On the overland planes, long bars that have cross section are added in the lateral direction (y direction) to guide lateral inflow to the main stream.
 In numerical simulations, the reflective boundary condition is imposed at the side boundaries (at and ) and the upstream boundary . To minimize the downstream boundary effects, the longitudinal dimension of the watershed is extended to as shown in Figure 1 and the transmissive boundary condition is imposed at the downstream boundary. Although not reported here, the backwater effects of the downstream boundary were tested. When the longitudinal dimension was greater than , only negligible differences were observed. As initial conditions, and are given on the entire computational domain. The width of the storm is always and the rain falls only on , like in the WES experiment. The grid size is and the time step is . In the case of a downstream-moving storm, , the bottom slope in the longitudinal direction (x direction) is , , the storm length is , and the storm movement speed is . In the verification of an upstream-moving storm, the case tested is , , , and . It is difficult to determine a value of n that considers unsteady nonuniform flow under rainfall. Thus, after many trials, the value , which gives the best agreement with the WES data, is adopted.
 Figure 2 depicts the comparisons between the computed results and the WES data. The result shows that the runoff hydrographs of the moving storms can be reasonably reproduced using the proposed numerical model.
3. Effects of Storm Movement on Hydrographs
 This study examines two types of moving storms: equivalent and nonequivalent storms. An equivalent storm is an equivalent moving storm to a stationary storm, where rainfall intensity is uniform throughout a watershed. Therefore, it is useful to investigate the effect of spatiotemporal rainfall variation on the resulting hydrographs due to rainstorm movement compared with a uniform and stationary rainfall. In this study, an equivalent storm preserves the total rainfall volume on the watershed [Ogden et al., 1995]. In contrast, nonequivalent storms do not preserve the total rainfall volume but rainfall intensity, which is more typical in nature. Therefore, the total rainfall volume depends on duration, that is, storm speed.
 To investigate the effects of rainstorm movement on runoff hydrographs of a 2-D watershed, numerical tests are conducted for a V-shaped catchment with a relatively mild bottom slope of . Thus, the same geometry as shown in Figure 1 is used for all the numerical simulations and the analysis in this paper, because the laboratory data collected from the WES [Yen and Chow, 1968, 1969] have been the basis for many studies on the effects of rainstorm movement [Shen et al., 1974; Ben-Zvi, 1970; Marcus, 1968; Lee and Huang, 2007; Liang, 2010]. In addition, for all the numerical simulations, , , , and the storm width is .
 When a storm moves downstream, a surge that flows in the same direction as the storm can be accumulated with the sequential inflow from the overland in the main channel as shown in Figure 3. Conversely, when a storm moves upstream, the flow in the main channel is not accumulated with the lateral inflow from the overland planes. Therefore, it is expected that runoff hydrographs will be affected by whether the storms move upstream or downstream. Thus, the two directions of movement are investigated and the results are discussed in the sections below.
3.1. Equivalent Moving Storms
 Rainfall-runoff from the equivalent moving storm proposed by Ogden et al.  is simulated: (longitudinal extent of catchment), and moving storms with different Vs have different i in proportion to Vs, such as .
 As shown in Figure 4, Qp (the peak discharge at the outlet during an event) is significantly affected by and the direction of storm movement. Specifically, Qp is proportional to for the equivalent moving cases, and the downstream-moving storm results in larger Qp than the upstream-moving storm when these have the same . The reason for this can be deduced from Figure 5. As shown in Figure 5a, the considerable amount of water in the main channel overtakes the storm front when the storm moves slowly , so the water is not accumulated. However, as shown in Figure 5b, when a storm moves rapidly , only a small part of the water in the main channel can exceed the storm front. Consequently, the sequential lateral inflow from overland can be accumulated successively with the flow in the main channel. Therefore, the Qp of the fast moving equivalent storm becomes larger than that of the slow moving storm, as shown in Figure 4. As can be seen in Figure 5c, when a storm moves upstream , the water in the basin is released through the outlet from the beginning of the rainfall event. Hence, the amount of water accumulated in the main channel becomes less than in the corresponding case of the storm moving downstream .
 Moreover, tp (the time to peak discharge of an event) is also affected by . Specifically, tp is proportional to . However, the computed results shown in Figure 4 indicate that the differences in tp between a pair of storms with the same and opposite directions are always very small, unlike the differences in Qp. That is, tp is almost independent of the direction of storm movement, as reported for an experiment [Black, 1972] and a conceptual approach [Seo and Schmidt, 2012]. The directional independence of tp results from the fact that Qp can be produced only after the entire storm area (with ) enters the basin (e.g., at ), and the travel times of the two events with opposite directions are similar after the instant to for the simulated cases.
 In addition, another type of equivalent moving storms is investigated in this study. Equivalent moving storms were defined by Yen and Chow  as storms moving with different Vs but having the same rainfall duration T at every point on the watershed and having the same total rainfall volume. Therefore, to maintain the volume with a fixed value of i, Ls varies in proportion to Vs. Although not presented here due to space limitations, the results of the dynamic wave simulations for the moving storm cases defined by Yen and Chow  were consistent with those for the case of Ogden et al. .
3.2. Nonequivalent Moving Storms
 The runoff hydrographs of nonequivalent moving storms are generated and analyzed in this section. Here the total volumes of rainfall during the events are not equivalent, because Vs varies but and are fixed. The effect of the direction of storm movement on tp is negligible for nonequivalent moving storms as well as equivalent moving storms. Furthermore, Qp is affected significantly by the direction of storm movement when the storm movement speed is high, as shown in Figure 6.
 There is, however, a fundamental distinction from the cases of equivalent moving storms. As shown in Figure 6, the slower nonequivalent moving storm produces a larger Qp than the faster one, because the total rainfall amount is greater when a nonequivalent storm moves slowly than it moves fast.
3.3. Stationary Equivalent Storms
 Equivalent stationary storms are defined here as storms that do not move and have the same total rainfall volume, , largest i(t), and T. The spatial rainfall distribution is uniform but the hyetograph can have temporal variations as shown by cases 1–5 in Figure 7. Yen and Chow  used a different definition of equivalent stationary storms in which the rainfall distribution is uniform in space and time over the entire watershed as shown by case 6 in Figure 7. For comparison, a rainstorm that is uniform in space and time but has the same T as the moving storms is shown by case 7 in Figure 7.
 Figure 8a presents the runoff hydrographs at the outlet of the V-shaped watershed by the stationary storm cases 1–5. The time variation of the hyetographs of the stationary storms considerably affects tp but not Qp. That is, in terms of Qp, the movement of storms is a more important factor than the temporal variation of i under the modeling conditions considered in this paper.
 In terms of a local point of the test basin, the stationary storm in case 6 and the two moving storms in Figure 8b have the same hyetographs, as noted by Yen and Chow . However, as shown in Figure 8b, the Qp generated by the upstream-moving storm is only half of that in case 6 and similar to that in case 7, while the i in case 7 is only half of that in case 6. It is also worthwhile to observe that the downstream-moving storm results in larger Qp than the stationary one. Of course, if an equivalent storm with moves downstream very slowly, then the Qp of the moving storm is smaller than that of a stationary storm. That is, the equivalent stationary storms defined by Yen and Chow  do not always produce greater peak discharges than moving storms.
 In terms of the entire basin, the stationary storm in case 3 and the two moving storms have the same precipitation volume with respect to time, but the hydrograph of case 3 has appreciable differences from those of the moving storms. In other words, the effect of movement can be critically important in flood forecasting.
3.4. Relationship Between Storm Location and Maximum Discharge Location
 Following on the above sections, the relationship between Qp and Vs is investigated by looking at the relationship between the locations of moving storms and the locations of Qm, where Qm is defined as the maximum discharge between the outlet and the upstream end at a given instant. Note that Qm should not be confused with the peak discharge Qp, which is the greatest discharge at the outlet during a rainfall-runoff event. For simplicity, the location of Qm will be expressed as x(Qm) hereinafter.
 When a storm moves in the upstream direction from the downstream end, x(Qm) always occurs at the outlet of the basin, as shown in Figure 9. Note that if the Qm occurs beyond , then x(Qm) is assumed to coincide with the outlet of the watershed because we are interested in the watershed area. The downstream-moving storms produce x(Qm) curves with different shapes in the xt plane from the upstream-moving storms, as shown in Figures 10 and 11. The x(Qm) starts from the upstream end and progresses downstream, and the difference between the x(Qm) and the location of the storm becomes larger as the storm rapidly moves.
 Because we can interpret the time rate of x(Qm) as the speed of movement in the location of maximum discharge, the results of resonance condition can be deduced from the figures. The resonance condition is that the storm front and x(Qm) nearly coincide. By considering Figures 6 and 11, we can see that if a nonequivalent storm moves slowly enough, the resonance condition results in a perfect storm because of continuous lateral flow into the main channel until Qp reaches Qe, which is consistent with what has traditionally been believed. However, the resonance condition of equivalent moving storms in Figure 10 results in smaller Qp and does not lead to Qe as shown in Figure 10, which is contrary to our traditional expectation.
 The lateral inflow is affected by the backwater from the main channel, especially in the downstream areas, which makes it more complex to estimate the time lag between the overland plane and the main channel. Therefore, a resonance condition based on a quantitative relationship between the speeds of main stream flow and storm movement is not given in this paper.
4. Effects of Storm Movement on Loop-Rating Curve
 To study the effects of storm movement on the rating curves, the stage-discharge curves of the main channel are generated at , where the flow is more stable and parallel than at the outlet. The water surface profile is wavy at the outlet due to the effect of the lateral strips installed on the overland.
4.1. Stationary Storms
 The stage-discharge relationship curves generated with the stationary storms in Figure 7 become a single rating curve as shown in Figure 12. The reason can be deduced from Figure 13, in which are plotted all the profiles of the components of the depth-averaged momentum equation along the thalweg in the main channel. As can be seen from the profiles, the friction term and the bottom-slope term overwhelm the others, which have very small magnitudes during the rainfall event. Thus, the flow in the main stream is dominated by the balance between the bottom-slope and friction terms. Therefore, regardless of the steadiness or unsteadiness of the hyetograph in Figure 7, if stationary storms have a uniform rainfall distribution, , we can expect that a kinematic wave model will predict the runoff as accurately as a dynamic wave model, at least for the simple geometry of the watershed modeled in this paper.
4.2. Moving Storms
 Based on the scale parameters in 1-D space, the 1-D depth-averaged dimensionless momentum equation can be expressed as
where is the Froude number and
with u*, l*, h*, and d* as the characteristic scales of the horizontal velocity, horizontal length, water depth, and bottom elevation, respectively. The magnitude of the friction coefficient cf is typically greater than 10−2 for an open channel flow at Reynolds number Re < 106 [French, 1985], but l*/h* will be larger by several orders of 10, which makes the typical magnitude of the friction term greater than those of the terms and . However, the relative importance of the terms in and depends on Fr.
 For overland flows, has traditionally been assumed, which resulted in diffusion wave models by ignoring the contributions of the terms and . However, as can be seen from Figure 14, the value of Fr is between 0.5 and 0.8 in the overland planes and the main channel. Thus, , and so the scale analysis indicates that the terms and are comparable in magnitude to the term . Therefore, all of the terms in equation (2) should be included in the surface runoff modeling, which is inconsistent with the assumption made in the usual scale analysis for runoff modeling. As expected from the scale analysis considering the Fr value, the results computed for a downstream-moving storm reveal considerable magnitudes not only for the terms , , and but also for the terms in and around the surge front, as shown in Figure 15.
 Considering the contributions of all the terms, the stage-discharge relationship can be obtained by rearranging equation (2) as follows:
where the Manning friction formula is used and is assumed for simplicity. Although, as stated above, the terms , , and should be included in equation (6) by considering the magnitudes plotted in Figure 15, we observe that becomes very small, even around the surge front, as shown in Figure 16. Therefore, equation (6) can be simplified like a diffusion wave model:
 Here we should note that the simplification from a dynamic wave model to a diffusion wave model does not result from the scale analysis. Thus, to obtain accurate results, the use of a dynamic wave model needs to be considered as reported by Liang , Yeh et al. , and Costabile et al. [2012a].
 Finally, by considering the magnitude and sign change of the term in equation (7), from a negative value to a positive value as shown in Figure 15, we can see that Hu can have multiple values for the corresponding to a single value of H in a loop-rating curve for a downstream-moving storm, as shown in Figure 17.
 Similar to the cases of downstream-moving storms, the cases of storm movement in the upstream direction generate loops in the stage-discharge curves, as shown in Figure 18. However, the loop-rating curves of the upstream-moving storms have an inverse sense of rotation with respect to those of the downstream-moving storms.
 In the loop generation processes, the only difference from the stationary case is the movement of the storm. Thus, we conclude that storm movement can be a mechanism to generate loops in stage-discharge relationship curves and that the direction of storm movement affects the sense of rotation in the loop-rating curves.
4.3. Dependence of Stage-Discharge Relationship on Watershed
 All the loop-rating curves shown in the previous sections are plotted together in Figure 19. Although each event has different rainfall conditions, all the loops exhibit a single tendency of stage-discharge relationship, with little variation over the V-shaped watershed. Therefore, it can be concluded that for a V-shaped watershed that has idealized topography, a uniform land cover property, and nonerodible bottom the stage-discharge relationship is a function of basin characteristics rather than rainfall conditions.
5. Concluding Remarks
 In this paper, we used a dynamic wave model to investigate the effects of storm movement on rainfall-runoff processes over an idealized V-shaped watershed. For research purposes, the SWE were solved by a fourth-order accurate finite volume method coupled with HLLC approximate Riemann solver. As shown by the computed results, very shallow flows generated by precipitation could be simulated stably and accurately on the given watershed, even where vertical steps existed. After verifications, the dynamic wave model was applied to various rainfall-runoff events on the V-shaped basin and the findings below were obtained. Because all the computations and analysis were performed for a single fixed-bed watershed and under the shallow water assumptions, with , it should be noted that the proposed conclusions are limited by the stated conditions in this paper.
 First, the storm movement can generate a loop in the stage-discharge relationship curve, and changes in the direction of storm movement can invert the sense of rotation in the loop-rating curve.
 Second, through a quantitative examination of the each term in the depth-averaged momentum equation, it was found that the Fr of the overland flows could be during the rainfall-runoff processes. Thus, we conclude that relying on the traditional assumption to remove the local and advective acceleration terms from dynamic wave models is not valid, which means that a dynamic wave model is recommended to obtain accurate results for complex flows as reported by Liang , Yeh et al. , and Costabile et al. [2012a].
 In addition, the stage-discharge relationship curves generated by various rainfall conditions for the simple V-shaped watershed (fixed sx, sy, and n) showed an identical tendency. That is, we conclude that the parameters α and β appearing in are functions of watershed characteristics rather than functions of rainfall conditions, at least, in the idealized V-shaped watershed.
 The effects of on hydrographs depend on the equivalence. For equivalent moving storms with , the results computed from the SWE showed that high produces large Qp. Contrary to our usual expectation, the equivalent moving storms resulted in smaller Qm or Qp under the resonance condition. However, the nonequivalent moving storms with resulted in smaller Qp as was increased, and under the resonance condition larger Qm or Qe could be produced, which is consistent with our intuition. These results from the dynamic wave model are consistent with previous studies in that the traditional resonance effect should be carefully reexamined in terms of the moving storm effects on peak flow increases. The results of dynamic wave model also show that the backwater interaction between the lateral and the main channel flow implies complexity of estimating the lag time.
 The direction of storm movement also greatly affected the rainfall-runoff process with respect to the magnitude of Qp. However, tp was revealed to be almost independent of the direction of storm movement when , regardless of whether the storms were equivalent or not.
 The future study will be focused on the diversity of basin characteristics and spatial rainfall distribution. In this paper, for example, the total acceleration was very small due to the regular shape of the main channel. Therefore, nonuniformity in the slope and rainfall distribution may lead to different hydrologic results. In addition, it will be more difficult to find a concrete relationship between the movement of the storm and the movement of x(Qm) because the latter will exhibit more complicated behavior for an irregularly shaped basin.
 The authors thank anonymous reviewers for their efforts in making this a more comprehensive and complete paper. The research presented here was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2011-0013032), and by a grant from Construction Technology Innovation Program (11-TI-C06) initiated by Ministry of Land, Transportation and Maritime Affairs (MLTM) of Korean government.