Applicability of the local inertial approximation of the shallow water equations to flood modeling

Transcription

1 WATER RESOURCES RESEARCH, VOL. 49, , doi: /wrcr.20366, 2013 Applicability of the local inertial approximation of the shallow water equations to flood modeling Gustavo A. M. de Almeida 1 and Paul Bates 1 Received 30 July 2012; revised 13 June 2013; accepted 13 June 2013; published 7 August [1] Recent studies have demonstrated the improved computational performance of a computer algorithm based on a simplification of the shallow water equations the so-called local inertial approximation which has been observed to provide results comparable to the full set of equations in a range of flood flow problems. This study presents an extended view on the local inertial system, shedding light on those key elements necessary to understand its applicability to flows of practical interest. First, the properties of the simplified system with potential impact on the accuracy of the solutions are described and compared to the corresponding full-dynamic counterparts. In light of this discussion, the behavior of the solutions is then analyzed through a set of rigorously designed test cases in which analytical solutions to the shallow water system are available. Results show a general good agreement between the local inertial and full-dynamic models, especially in the lower range of subcritical flows (Fr < 0.5). In terms of steady nonuniform flow water profiles, the error introduced by the local inertial approximation leads to milder water depth gradients, which results in attenuated spatial changes in depth. In unsteady problems, the local inertial approximation leads to slower flood propagation speeds than those predicted by the fulldynamic equations. Even though our results suggest that the magnitude of these errors is small in a range of floodplain and lowland channels, it becomes increasingly relevant with increasing Fr and depth gradients. Citation: de Almeida, G. A. M., and P. Bates (2013), Applicability of the local inertial approximation of the shallow water equations to flood modeling, Water Resour. Res., 49, , doi: /wrcr Introduction 1 University of Bristol, Bristol, UK. Corresponding author: G. A. M. de Almeida, University of Bristol, University Rd., Bristol BS8 1SS, UK. (gustavo.dealmeida@bristol.ac.uk) American Geophysical Union. All Rights Reserved /13/ /wrcr [2] The past few decades have witnessed exceptional progress in the development of computer models to simulate the propagation of floods, much of which has been driven by the need for tools to evaluate flood risk. Most of these methods are based on the shallow water equations, a nonlinear hyperbolic system of equations describing the conservation of mass and momentum of water in two horizontal dimensions. In spite of the substantial progress achieved in key areas of numerical methods for the solution of this system and the unprecedented evolution of computational power, computations over large space and time scales and at high resolution are still hampered by the unfeasibly high computational cost involved. With the growing availability of fine resolution remotely sensed data, this computational limitation represents a clear barrier for a range of applications aimed at understanding and predicting large-scale flood problems. [3] The demand for computationally efficient methodologies to solve flood propagation problems has led to the development of simplified formulations that provide accurate solutions within the range of their applicability at a considerably high performance. The two most popular examples are the kinematic and diffusion wave approximations. The first is derived by neglecting the two Eulerian inertial terms in the momentum equation (i.e., local and convective acceleration) along with the pressure gradient term, while the second neglects inertia terms only. The properties and limitations of these models as well as of other models based on simplifications of the Saint Venant equations (i.e., quasi-steady dynamic wave and gravity wave) have been widely explored over the last decades [e.g., Lighthill and Whitham, 1955; Woolhiser and Liggett, 1967; Di Silvio, 1969; Ponce and Simons, 1977; Ponce et al., 1978; Morris and Woolhiser, 1980; Vieira, 1983; Ferrick et al., 1984; Ferrick, 1985; Ponce, 1986; Hromadka and Yen, 1986; Dooge and Napiorkowski, 1987; Govindaraju et al., 1988a, 1988b, 1990; Parlange et al., 1990; Singh and Aravamuthan, 1996; Lamberti and Pilati, 1996; Cappelaere, 1997; Tsai, 2003; Tsai and Yen, 2004; Moramarco et al., 2008; Philipp et al., 2012]. Much less attention has been given to a third class of shallow water approximate solvers the so-called local inertial approximation which neglects only the convective part of the inertial terms [e.g., Xia, 1995, Aronica et al., 1998, Bates et al., 2010, de Almeida et al., 2012]. [4] From a conceptual point of view, the local inertial formulation provides a better physical description of the problem than the diffusive model, as the local rate of change of the fluid momentum is now represented by the 4833

2 additional term. In discrete terms, this means that the fluid momentum at any given time step is used to update the next time step, so that the flow has to be accelerated from a previous state. Therefore, in terms of the physical representation of shallow water flows, the local inertial formulation lies between the diffusion wave approximation and the fulldynamic equations. From the computational perspective, the main advantage of the local inertial formulation (compared to the diffusion wave model) is related to the stability condition in explicit finite difference schemes. Namely, the maximum stable time step in the explicit diffusion wave model decreases quadratically with grid refinements [see Hunter et al., 2006], while a linear relation is obtained with the numerical schemes based on the solution of the hyperbolic local inertial system (which is governed by the Courant-Friedrichs-Lewy condition, hereafter referred to as CFL). This key difference substantially enhances the computational performance at fine resolution. In addition, the stable time step for the diffusion wave model also depends on the water surface gradient [see Hunter et al., 2006], so that its computational performance is dramatically reduced in zones of near horizontal water surface. This is particularly important for large-scale simulations that unavoidably include flat water surface areas (e.g., lakes and lowland rivers). The local inertial model therefore provides an excellent alternative for the simulation of flow problems that would become unfeasibly slow with an explicit diffusive model. [5] Bates et al. [2010] proposed a simple finite difference numerical scheme for the solution of the local inertial system that provides accurate solutions to a range of subcritical flow conditions at a very attractive computational performance. A number of subsequent studies have benchmarked this model against field data, analytical solutions, as well as the solutions of a range of industry numerical models that solve the full-dynamic and simplified versions of the shallow water equations [e.g., Bates et al., 2010, Neal et al., 2012, Falter et al., 2012]. Recently, an alternative numerical scheme was proposed by de Almeida et al. [2012] that solved some stability problems reported by Bates et al. [2010], providing a robust and efficient model to simulate a range of subcritical flood propagation problems. [6] Neal et al. [2012] compared the performance of the Bates et al. [2010] local inertial model against a diffusion wave and a full-dynamic model (a Godunov-type finite volume scheme that uses the approximate Riemann solver of Roe). They observed that the local inertial model performed up to seven times faster than the full-dynamic model, and more than 2 orders of magnitude faster than the diffusion wave model. The speed-up of model performance compared to a full-dynamic model obviously comes at the cost of neglecting one of the terms in the governing equations, and it is the objective of this paper to analyze how this assumption affects the accuracy of the model. [7] The relative simplicity of the resulting set of algebraic finite difference equations [e.g., Bates et al., 2010; de Almeida et al., 2012] not only reduces the computational cost but also significantly increases its potential to be implemented as part of a range of models that would benefit from a simple, robust, and efficient two-dimensional (2-D) hydraulic module. This is reflected in recent efforts to implement the Bates et al. [2010] scheme as the hydrodynamic engine of the CAESAR landscape evolution model (e.g., Coulthard et al., Integrating the LISFLOOD- FP 2-D hydrodynamic model with the CAESAR model: Implications for modelling landscape evolution, submitted to Earth Surface Processes and Landforms, 2013) and JULES, the UK Land Environment Simulator. [8] With an increasing interest in this formulation, it becomes of paramount importance to understand its main capabilities and limits of applicability to simulate flood propagation problems. This paper is organized as follows. First, the governing equations and the corresponding numerical scheme used to solve the system are briefly presented. This is followed by a discussion comparing the main differences between the full-dynamic and local inertial systems, which reveal some of the properties of the latter with a potential influence on practical problems. The analysis is structured around two main aspects of modeling capabilities: (i) the ability of the model to accurately simulate steady flow problems and (ii) flood propagation (onedimensional (1-D) unsteady flow). The mathematical insights developed in these sections are supported by and further analyzed through a sequence of test cases with analytical solutions to the full-dynamic equations. These test cases are used to illustrate the most relevant differences in the behavior of the solutions obtained with the two systems. The tests were specifically designed to explore the same aspects of the local inertial model previously discussed and cover a wide range of flow conditions. Finally, the results of the local inertial and full-dynamic models are discussed in light of the properties of the two systems. These results illustrate the accuracy obtained with the local inertial model at a range of subcritical flow conditions, as well as the main flow characteristics influencing this accuracy. 2. Governing Equations [9] The model examined in this paper is derived from the two-dimensional shallow water equations, a hyperbolic, nonlinear system of PDEs describing the conservation of mass and momentum of shallow water flow. In two horizontal dimensions, the system can be written as {z} local {z} ¼ 0 uq vq xþþ ðh þ zþ fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} convective pressure þ acceleration bed y þ fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} convective uq y þ ðh þ fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} pressure þ bed gradients ð1þ þ gn2 jjqjjq x ¼ 0 fflfflfflfflffl{zfflfflfflfflffl} h 7=3 friction ð2þ þ gn2 jjqjjq y ¼ 0 fflfflfflfflffl{zfflfflfflfflffl} h 7=3 friction where x and y are the two Cartesian directions, t is the time, h is the water depth, q x and q y are the x and y components of the discharge per unit width vector q (the magnitude of ð3þ 4834

3 which is jjqjj), u and v are the x and y components of the flow velocity, z is the bed elevation, g is the acceleration due to gravity, and n is the Manning s friction coefficient. The key simplification introduced by the local inertial approximation is based on the assumption that the convective acceleration terms are negligible compared with the other terms in equations (2) and (3) and, therefore, can be ignored in the numerical scheme. This leads to the following system of simplified local ¼ 0 þ ðh þ zþ þ gn2 jjqjjq x ¼ 0 h þ ðh þ zþ þ gn2 jjqjjq y ¼ 0 h 7=3 [10] The numerical model investigated in this paper uses these simplified momentum equations (i.e., equations (5) and (6)) to estimate flow between rectangular computational cells, while equation (4) is used to update the water depths inside the cells. 3. Numerical Scheme [11] The computational model tested in this paper uses the numerical scheme proposed by de Almeida et al. [2012], which provides a more robust solution than the previous scheme proposed by Bates et al. [2010]. The numerical solution uses a finite difference scheme applied to a staggered grid (Figure 1) and was implemented as part of the LISFLOOD-FP set of numerical solutions to different forms of the shallow water equations [see Bates and De Roo, 2000; Villanueva and Wright, 2006; Hunter et al., 2008; Bates et al., 2010]. The LISFLOOD-FP package was developed from conception to directly make use of the expanding wealth of raster terrain data that are now available and therefore uses only structured, rectangular grids. In the numerical method used here, the local inertial approximation of the momentum equations (i.e., equations (5) and (6)) is evaluated in the interface between two adjacent computational cells using the finite difference scheme proposed by de Almeida et al. [2012], which yields an explicit algebraic equation to estimate the water discharge between cells: h q n q nþ1 i 1=2 ¼ i 1=2 þ ð 1 Þ 2 q n i 3=2 þ qn iþ1=2 i gh n f 1 þ gdtn 2 jjq n i 1=2 jj=h7=3 f Dt Dx n i n i 1 ð7þ Figure 1. Grid and variables used in the numerical scheme. Only the x spatial dimension is shown here for simplicity. Note the staggered characteristic of the grid: is evaluated in the center of cells, while q is evaluated at cells interfaces. where ¼ h þ z is the water surface elevation, h n f is the depth at the interface between cells (estimated as the difference between max n i ;n i 1 and max ð zi ; z i 1 Þ), subscripts i 1/2, i 3/2 and i þ 1/2 denote, respectively, points located at the cell interface and the two adjacent cell interfaces, subscripts i and i 1 denote points at the center of two neighboring cells (Figure 1), the parameter is a weighting factor that adjusts the amount of artificial numerical diffusion, Dt is the time step, and Dx is the grid spacing. The value of can be set between zero and unity, and no artificial diffusion is added when ¼ 1, while increasing amounts of numerical diffusion are added as is reduced below unity. In general, artificial diffusion is needed to counteract the formation of discontinuities in the solution that typically arise as a result of the nonlinearity in the system (i.e., shocks). In practical problems, this becomes particularly necessary for low friction surfaces or problems involving fast changes of flow, in which case values between 0.7 and unity have been found to perform well. The value of jjq n i 1=2jj is estimated as: r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 jjq n i 1=2 jj ¼ þ q n q n x;i 1=2 y;i 1=2 [12] The presence of jjqjj in equation (7) poses an inconvenience to the discretization of the friction term in two dimensions using the staggered grid adopted by the model, where the q x and q y components are evaluated at different points of the domain. This difficulty can be overcome by estimating the missing variable (i.e., q y or q x ) as its average value at four neighboring points: q iþ1=2;j y ¼ 1 4 qi;j 1=2 y q i;jþ1=2 x ¼ 1 4 qi 1=2;j x þ q i;jþ1=2 y þ qy iþ1;j 1=2 þ q iþ1=2;j x þ q i 1=2;jþ1 x þ q iþ1;jþ1=2 y þ q iþ1=2;jþ1 x ð8þ ð9þ ð10þ where i and j superscripts represent x and y indexes of the cell center. [13] The flow discharges estimated at the four faces of the rectangular cell are then used to update water depth inside the cells using the following discrete form of equation (4): nþ1 i;j ¼ n i;j þ Dt Dx 2 q x j nþ1 i 1=2;j q xj nþ1 iþ1=2;j þ q yj nþ1 i;j 1=2 q yj nþ1 i;jþ1=2 ð11þ where the subscript j is used here to denote the second horizontal dimension, and Dx 2 assumes a square cell. The main characteristics of this numerical scheme are described in 4835

4 detail in Bates et al. [2010] and de Almeida et al. [2012]. [14] The value of Dt follows the standard CFL condition for the stability of hyperbolic systems, which for the local inertial model yields: Dt ¼ pffiffiffiffiffiffiffiffiffiffiffi Dx gh max ð12þ where h max is the maximum water depth over the domain. The CFL is a necessary but not sufficient condition for the stability of nonlinear hyperbolic systems. In equation (12), the coefficient 0 < 1 is used to further reduce the time step, which significantly enhances the stability of the model ( ¼ 0.7 is used in this paper). Flow discharge boundary conditions are implemented by simply setting the value of q x or q y at the edges of boundary cells, while water level boundary conditions are set at ghost cells contiguous to the boundary cells. 4. Relevant Properties of the Local Inertial Approximation to the Shallow Water Equations [15] This section compares some relevant differences between the local inertial and full-dynamic representations of the shallow water equations for both steady nonuniform and unsteady flow regimes. At this point it is important to highlight that under steady flow conditions, the local inertial system is in fact equivalent to the diffusion wave model, as the local acceleration term in the momentum equation is zero. The analysis of steady state solutions (i.e., water surface profiles) to the diffusion wave system has been the object of a number of previous works [see Govindaraju et al., 1988b; Parlange et al., 1990; Singh and Aravamuthan, 1996; Moramarco et al., 2008]. However, these previous studies focused only on the idealized situation of flow over a plane (i.e., constant bed slope), which considerably restricts the analysis of the influence of the convective acceleration term on the water surface profiles. This is because the magnitude of this term clearly depends on flow gradients, which usually result from, or are accentuated by, irregularities in the bed profile. This dependency will be illustrated in sections 5.1 and 5.2 by exploring scenarios with increasing depth gradients linked to variations in the bed profile. As a result of the equivalence between the local inertial and diffusion wave models, the new conclusions obtained from our analysis of steady flow regime are also extensible to the diffusion wave model Steady Flow Problems [16] One question of high relevance for flood modeling is how the results obtained with the inertial formulation compare with those of a full-dynamic model under different scenarios of steady flow. Classically, the equation describing steady, gradually varied flow is derived from the full-dynamic model (e.g., equations (1) and (2)) by setting all time derivatives to zero, which yields [e.g., Henderson, 1966]: dh dx ¼ S o S f 1 Fr 2 ð13þ where S o and Sp f are, respectively, the bed and friction slopes, Fr ¼ u= ffiffiffiffiffi gh is the Froude number, and u is the flow velocity. Equation (13) is known as the backwater equation and is widely used in 1-D open-channel hydraulics to describe the behavior of water surface profiles at different flow conditions. To compare steady flow water surface profiles predicted by the local inertial and full-dynamic models, we derive a similar ODE by neglecting not only the time derivatives but also the convective acceleration term in the Saint Venant equations, which yields. dh dx ¼ S o S f ð14þ [17] Equations (13) and (14) are, respectively, equivalent to the full-dynamic and local inertial models for the specific case of a steady flow regime. The comparison between these equations helps to understand the expected behavior of the local inertial model at steady flow. At very low Fr regimes, the denominator of equation (13) tends to unity, and the inertial model provides an accurate approximation to the full-dynamic model. With increasing Fr (under subcritical flow conditions), the absolute values of the water depth gradients predicted by the inertial model are increasingly smaller than those corresponding to the full-dynamic model. A relative measure of the error in the estimation of the depth gradient can be written as: E h 0 ¼ h0 IN h0 FD h 0 FD ¼ Fr 2 ð15þ where h 0 IN ðþand x h0 FDðÞare, x respectively, the depth gradients given by equations (13) and (14). For instance, at Fr ¼ 0.5, the local inertial model predicts a depth gradient approximately 25% smaller than the full-dynamic counterpart. This indicates that the integration of equation (14) will, in general, lead to profiles with less pronounced depth changes than those obtained with equation (13). In addition, these depth deviations will lead to different estimations of friction. For flows approaching the critical condition (i.e., transcritical flow), it is well known that the full-dynamic model will in general lead to a near vertical depth profile as a result of the denominator in equation (13) tending to zero. Oppositely, in the local inertial model (equation (14)), finite depth gradients are predicted for Fr ¼ 1. [18] At subcritical flow, the difference between the two formulations is reflected only in the magnitude of the depth slope, leading to water surface profiles with milder depth changes predicted by the local inertial model, compared with those corresponding to the full-dynamic equations (i.e., gradients predicted by equation (14) are smaller than those of equation (13)). When Fr exceeds unity (i.e., supercritical flow conditions), the two models start to exhibit an important qualitative difference, with equations (13) and (14) predicting depth gradients not only with different magnitudes but also with opposite signs. To illustrate this disparity, we examine the case of a supercritical bed slope. In this situation, the full-dynamic model predicts water surface profiles approaching the normal depth (i.e., uniform flow) as we move toward downstream (dh/dx is negative if h is higher than the uniform depth, and positive if h is smaller than this depth). On the other hand, the local inertial model predicts water surface profiles which depart 4836

5 from the normal depth in the downstream direction, or in other words, which approach the uniform depth toward the upstream direction. This is the same pattern observed for subcritical flow and highlights the need for a downstream depth boundary condition in the inertial model even under supercritical flow conditions. This exposes the unsuitability of the model at Fr > 1, as supercritical flows are controlled exclusively by the upstream flow conditions Unsteady Flow Problems [19] Neglecting the convective acceleration term eliminates part of the system s nonlinearity, but the resulting system is still nonlinear because of the pressure gradient term. A key property of this local inertial model is that the eigen values p of the Jacobian matrix of this system are 1;2 LI ¼ 6 ffiffiffiffiffi gh [de Almeida et al., 2012], while the full-dynamic counterparts are 1;2 FD ¼ u6 p ffiffiffiffiffi gh. Understanding how this fundamental difference affects the structure of solutions to a local Riemann problem provides very useful insights into the behavior of the model. In particular, the local inertial formulation admits only solutions containing one shock wave traveling in one direction and an opposite rarefaction wave, while the full-dynamic equations also admit more complex solutions (e.g., involving two shocks, two rarefactions, as well as transcritical rarefactions; see LeVeque [2002] and Toro [2001]). If the water depth is higher on the left than on the right, then the local inertial system will exhibit a shock wave propagating to the right and a rarefaction wave toward the left direction. Conversely, if the depth on the right is highest, the solution will contain a left-going shock and a right-going rarefaction. This pattern of two waves always propagating in opposite directions again exposes the need for one (and only one) boundary condition at each boundary (i.e., one characteristic curve entering each side of the domain), an important difference compared with the full-dynamic model, in which the number of boundary conditions depends on the flow regime. The requirement of a downstream boundary condition under supercritical regimes is inconsistent with the physics of the problem and reaffirms the model s inappropriateness to simulate supercritical conditions, as discussed in section 4.1. [20] From the viewpoint of unsteady flow problems, the difference between and FD is expected to affect the speed of wave propagation predicted with the local inertial model, as briefly reported in de Almeida et al. [2012]. Understanding the flow conditions under which this difference becomes relevant is of paramount importance for practical applications. A relative measure of the error in introduced by the local inertial assumption can be obtained as: E ¼ FD LI ¼ p u ffiffiffiffiffi ¼ 6Fr ð16þ LI gh [21] That is, the relative error E increases with Fr, so that the local inertial formulation is expected to produce more accurate wave speeds at low Fr conditions. However, the model will in general propagate waves slower than the full-dynamic model, and the gap between the two solutions increases with Fr. This problem will be examined using a series of wave propagation test cases in sections 5.3 and 5.4. [22] The model s dependency on Fr highlighted here (both in steady and unsteady problems) is not surprising, as it measures the relative importance of inertial and gravity forces, and therefore represents the relevance of the neglected term at particular flow conditions. 5. Test Cases and Results [23] This section describes the set of test cases used to support and illustrate the discussion presented in section 4. Each individual subsection begins with a description of the particular problem tested and the corresponding analytical solution to the Saint Venant equations, which is then compared to the results of the local inertial model. Test cases 1 and 2 are based on the methodology proposed by MacDonald [1996], which describes an ingenious technique to derive analytical solutions for a variety of steady open-channel flow problems. The method is based on the solution of an inverse problem, where given a particular water surface profile, flow discharge, and channel characteristics, the bed profile is obtained by integrating the channel slope analytically derived from the steady flow form of the Saint Venant equations. The particular sequence of problems analyzed here provides a rigorous test of the model s ability to accurately simulate a range of steady state flow conditions. A range of subcritical Fr values were tested to analyze the influence of the differences in equations (13) and (14) on the model results. In addition, the convective acceleration term neglected in the local inertial approximation clearly depends on flow gradients, which indicates that these gradients will also influence the accuracy of the model. To analyze this influence, the analytical solutions were manipulated to generate flow scenarios with increasingly steeper gradients. [24] The second set of tests (test cases 3 4) was taken from Hunter et al. [2005] and explores the propagation of flood waves over both horizontal and sloping beds that are initially dry. These tests provide important insights into the main capabilities of the model to simulate flood propagation problems, including situations where the results are expected to deviate from those of the full-dynamic system. [25] All the analytical solutions have been compared to a full-dynamic shallow water numerical model implemented as part of the LISFLOOD-FP package. This model solves the 2-D shallow water equations using the first-order finite volume Godunov method with fluxes computed with the approximate Riemann solver of Roe [Roe, 1981; Villanueva and Wright, 2006]. The results of this comparison have shown an almost perfect match between the analytical solutions and the full-dynamic numerical model and are not shown here to avoid redundancy. [26] The set of tests presented here has been restricted to 1- D problems with analytical solutions because previous 2-D testing in practical applications proved largely successful. [27] Convergence checks were made to ensure that grid truncation errors were negligible in all test cases in comparison with the errors associated with the local inertial approximation. Figure 2 shows examples of the convergence analysis for test cases 2 (steady nonuniform flow) and 3 (unsteady flow) Test Case 1: Near Critical Steady Subcritical Flow in a Rectangular Channel [28] The first test case proposed by MacDonald et al. [1997] explores a steady flow regime in a 1 km long and 4837

6 Figure 2. Model convergence check to grid resolution. (a) Test case 2. (b) Test case 3 at t ¼ 3600 s. The results show the root mean square error (RMSE) relative to the model results at 10 cm resolution. 10 m wide rectangular channel with a Manning coefficient of 0.03 and a discharge of 20 m 3 s 1. The flow is subcritical throughout the reach (approximately 0:54 < Fr < 0:99), reaching near critical conditions close to both the downstream and upstream boundaries. This first test case aims to analyze the accuracy and stability of the simplified inertial formulation for subcritical flows approaching Fr ¼ 1. The solution originally developed by MacDonald et al. [1997] for 1-D channels accounted for the lateral friction resulting from the walls of the channel (wetted perimeter P ¼ b þ 2h, where b is the width of the channel). On the other hand, the LISFLOOD-FP formulation does not include lateral friction between cells, which represents an inconvenience for the comparison with the above-mentioned solution. To produce a more consistent comparison, the MacDonald et al. [1997] solution was slightly modified to eliminate the effect of lateral friction. The resulting solution for the channel slope reads: " # S o ðþ¼ x 1 4 ghðþ x 3 h 0 ðþþ x 0:0036 ð17þ hx ðþ 10=3 which can be easily integrated using high-resolution quadrature methods to obtain the channel profile z(x). In equation (17), h(x) is the water depth profile and h 0 ðþ¼dh=dx x its spatial derivative, as proposed in MacDonald et al. [1997]: hx ðþ¼ 4 ( " 1=3 1 þ 1 g 2 exp 16 x L 1 #) 2 2 ð18þ h 0 ðþ¼ x 4 1=3 16 g L x L 1 2 " exp 16 x L 1 # 2 2 ð19þ where L ¼ 1000 m, which is the length of the channel. As the problem is essentially one dimensional, the numerical model was set up using a grid with 1 m resolution. Boundary conditions are set as q ¼ 2 m 2 s 1 (upstream) and the depth given equation (18) at the downstream end of the channel. The value of ¼ 1 was used in all the experiments described in this section. The analytical solution is used as the initial condition, and the model is run until a new steady nonuniform water surface profile is obtained. [29] Figure 3 shows the water surface profile predicted by the numerical model along with the corresponding Froude number and the analytical solution. The results show a very good agreement between the numerical model and the analytical solution, even with Fr values approaching unity close to the boundaries. As expected, the local inertial formulation predicts a water surface profile with slightly milder depth variations than that obtained with the full-dynamic equations, but in this example the differences are virtually negligible (almost imperceptible in Figure 3). The close agreement observed in this test is, at least in part, due to the very gradually varied depth profile used in this example (water depth changes of 36 cm in 500 m), which translates into a negligible convective acceleration term in equation (2). As the spatial changes in the water depth profile increase, the neglected term becomes increasingly relevant, and the results obtained with the local inertial formulation are expected to deviate from the full-dynamic 4838

7 Figure 4. Water depths predicted by the local inertial model for different channel lengths in test case 1, along with the corresponding full-dynamic analytical solution. Figure 3. Water surface profile predicted by the local inertial model in test case 1 (L ¼ 1000 m), along with the corresponding analytical solution of the full-dynamic equations. The Froude number variation over the reach is presented on the right axis. counterparts. To analyze this effect, a set of similar analytical solutions were derived by compressing the x axis in equations (17) (19). This is obtained by reducing the channel length L in equations (18) and (19) and integrating equation (17) numerically to obtain the corresponding bed elevations. The channel lengths tested here are L ¼ 1000, 500, 250, 125, and 62.5 m. To ensure grid independence, the L ¼ 62.5 m test used a 50 cm grid resolution, while all others were run with Dx ¼ 1 m. Figure 4 shows the water depth profiles obtained with the local inertial numerical model for different channel lengths, along with the corresponding analytical solution (all analytical solutions collapse into a single curve when the x/l axis is used). As expected, the error introduced by the local inertial approximation increases with decreasing L. In this example, the reach averaged absolute error varies from 1.9 cm (2.1% of the analytical depth at corresponding points) for the L ¼ 1000 m experiment up to 7.5 cm (7.8%) for L ¼ 62.5 m, while the respective maximum values range from 3.5 cm (3.9%, L ¼ 1000 m) to 15.3 cm (13.8%, L ¼ 62.5 m). The L ¼ 62.5 m analytical depth profile (i.e., equation (18)) exhibits a 36 cm depth change in m (i.e., 1.15% average depth slope between the boundaries and the central position) and h 0 (x) reaches a maximum value of Even though these gradients are 16 times higher than those corresponding to the L ¼ 1000 m experiment, they still represent a gradually varied flow condition. [30] The above results explored the upper range of subcritical flow ð0:54 < Fr < 0:99Þ. In order to analyze the influence of Fr on the errors introduced by the local inertial approximation, a new set of tests was carried out by increasing the depth profile (i.e., equation (18)) by a constant. Figure 5 plots the ratio between the depth predicted by the local inertial model and the corresponding analytical solution ðhðþ=h x ðþ x analyt Þ for 0:54 < Fr < 0:99 (Figure 5a) and 0:35 < Fr < 0:54 (Figure 5b, obtained by adding 36 cm to equation (18)) for different values of L. Note that the depth gradients are unchanged for each pair of curves with the same channel length L in Figures 5a and 5b, so that the influence of Fr is analyzed separately from that of the depth gradient. Figure 5b shows the same pattern of a gradual increase in the error with decreasing channel lengths (i.e., increasing depth gradients), as observed in Figures 4 and 5a. However, results in the lower Fr range (Figure 5b) exhibit a considerable reduction of the error introduced by the local inertial approximation, as expected from equation (15). The maximum error observed in this set of low Fr simulations is 6.8 cm (i.e., 4.6%, for the L ¼ 62.5 m test) Test Case 2: Subcritical Flow Over an Undulating Bed Elevation in a Rectangular Channel [31] This test case is similar to problem 3 in MacDonald et al. [1997], which analyzes an undulating water depth profile (sinusoidal) in a trapezoidal channel. However, in order to avoid inherent errors arising from the geometric representation of a trapezoidal section in a 2-D domain, the problem was simplified by using a rectangular channel. The lateral friction was also neglected by assuming P ¼ b. Apart from further testing the model s ability to capture the nonuniform flow spatial variations in subcritical flows, this test is used to assess larger scale effects. This is because local differences in the depth predicted by the local inertial and full-dynamic models will lead to differences in the friction estimation, the spatial integration of which may result in a large-scale side-effect. The reach is 5 km long, the flow discharge per unit width was set to q ¼2m 2 s 1 and a Manning s coefficient n ¼ 0.03 was used. The value of was set 4839

8 Figure 5. Ratio between the water depths predicted by the local inertial model and that corresponding to the fulldynamic analytical solution for different channel lengths in test case 1. (a) Upper range of Fr; (b) Lower range of Fr. to unity in all experiments discussed in this section. The analytical solution was derived using the same guidelines outlined in MacDonald et al. [1997], which yielded the following equations for S o, h(x), h 0 (x): " # S o ¼ 1 4 ghðþ x 3 h 0 ðþþ x 0:0036 hx ðþ 10=3 hx ðþ¼ 9 8 þ 1 4 sin x 500 h 0 ðþ¼ x 2000 cos x 500 ð20þ ð21þ ð22þ [32] Figure 6 shows the water profile predicted by the local inertial model, along with the corresponding analytical solution and Froude number. The results show an excellent agreement between the local inertial model and the analytical solution of the full-dynamic problem. As in the previous test case, a sequence of increasingly more stringent tests was derived from equations (20) (22) by compressing the x axis, which is achieved by reducing the channel and the wave lengths in equation (21) by the same Figure 6. Water surface profile predicted by the local inertial model in test case 2 (L ¼ 5000 m), along with the corresponding analytical solution of the full-dynamic equations. The Froude number variation over the reach is presented on the right axis. factor, differentiating equation (21), and integrating equation (20) numerically. This leads to an increase in water depth gradients and, as a result, an increased departure from the solution obtained with the local inertial model from that of the full-dynamic equations. In addition, three groups of experiments were run to illustrate the influence of Fr on the results. In these scenarios, Fr was varied by subtracting 16 cm from and adding 47 cm to equation (21) (as well as the corresponding x-compressed depth profiles). As this last manipulation does not alter depth gradients, the effect of Fr is analyzed separately from that of depth gradients. Figure 7 compares the depth profiles predicted by the model for L ¼ 5000, 1000, and 250 m against the corresponding analytical solutions for 0:44 < Fr < 0:95. Figure 8 shows the ratio between the water depth predicted by the local inertial model and that corresponding to the analytical solution for the three channel lengths and three ranges of Fr. These figures illustrate some of the patterns already observed in the previous test case. First, the disparity between the local inertial model and the full-dynamic solution increases with the depth gradient, as expected from the main assumption adopted in the derivation of the local inertial model. In particular, the local inertial model attenuates the oscillations in depth predicted by the full-dynamic model (i.e., analytical solution in Figures 7 and 8) as a result of the underestimation of depth gradients discussed in section 4.1. This error is as large as 15 cm (20.8% of the analytical depth) for the L ¼ 250 m test (Figure 8a), which 4840

9 Figure 7. Water depths predicted by the local inertial model for different channel lengths in test case 2 ð0:44 < Fr < 0:95Þ, along with the corresponding fulldynamic analytical solution. exhibits a maximum depth gradient h 0 ðþ¼3:1%. x This occurs in the upper range of Fr ð0:44 < Fr < 0:95Þ studied but is significantly reduced at lower values of Fr (e.g., Figures 8b and 8c, which show maximum errors of 10.9% and 2.4%, respectively). Apart from the dampening of water depth oscillations, Figures 7 and 8 show a gradual reduction of the depth predicted by the inertial model (compared to the full-dynamic) from the boundary condition toward upstream. This suggests an influence of the nonlinearity in the friction term, which will lead to higher energy losses at a highly oscillatory depth profile than in the relatively smoother counterpart. In other words, the attenuation of the spatial variations of depth induces a reduction of the integral of the friction term, the side effect of which is an underestimation of flow depths. However, the results shown in Figures 7 and 8 indicate that the error introduced by this effect is relatively small, even in the upper range of subcritical flows tested Test Case 3: Nonbreaking Wave Propagation Over a Horizontal Plane [33] This test case explores the ability of the model to accurately simulate the propagation of a flood wave over a horizontal plane that is initially dry. For a very specific set of boundary conditions, this problem admits an analytical solution to the full-dynamic shallow water equations, which can be used to thoroughly benchmark the accuracy of the model in unsteady flow problems. This analytical solution and the corresponding boundary conditions were derived by Hunter et al. [2005] by imposing a constant velocity over the whole mass of fluid propagating in one space dimension, ux; ð tþ ¼ constant. Rewriting equations (1) and (2) as a function of velocity and depth and imposing the constant velocity condition yields: Figure 8. Ratio between the water depths predicted by the local inertial model and that corresponding to the fulldynamic analytical solution for different channel lengths in test case 2. Three ranges of Fr representing subcritical flow conditions are depicted in Figures @x þ n2 u 2 h 4=3 ð23þ ð24þ [34] Equation (23) shows that water depths are simply advected at constant velocity u. The analytical solution to the water depth profile h(x,t) is obtained by ¼ 0 in equation (24) (horizontal plane), integrating and imposing the moving boundary condition hut; ð tþ ¼ 0, yielding: hx; ð tþ ¼ 7 3 n2 u 2 3=7 ðx utþ ð25þ 4841

10 [35] At x ¼ 0, the upstream boundary condition then becomes hð0; tþ ¼ 7 3=7 3 n2 u 3 t : ð26þ [36] A set of tests were run to analyze the influence of u (and therefore Fr) on the performance of the model. In these tests, both u and n were gradually varied, but the product n 2 u 3 was held constant, so that the boundary condition (equation (26)) is the same in each of these runs. The specific values of n tested are 0.005, 0.01, 0.02, and 0.03, and the respective velocities are approximately 0.95, 0.60, 0.38, and 0.29 ms 1. The problem is again essentially one dimensional, and the model was set up using a 1 N computational grid of square cells with Dx ¼ Dy ¼ 1 m, where N is the number of cells needed to prevent the wave from reaching the downstream boundary during the simulation. The n ¼ 0.02 and n ¼ 0.03 runs were stable with ¼ 1, while the n ¼ and 0.01 required the addition of numerical diffusion to suppress numerical oscillations ( ¼ 0.8 and 0.9 were used respectively). [37] Figure 9 presents the model results at t ¼ 3600 s. The effect of Fr (which at corresponding depths is proportional to the value of u used in each particular experiment) on the accuracy of the model is patent in this figure. At relatively low velocities, the local inertial model provides an excellent approximation to this unsteady flow problem. For instance, in the experiment with u ¼ 0.29 ms 1 an almost perfect match is obtained between the local inertial model and the full-dynamic analytical solution. On the other hand, the gap between the local inertial and the full-dynamic solutions gradually widens with the flow velocity, resulting in a nonnegligible underestimation of the wave front position. The error in the position of the wave front is 10, 29, 134, and 529 m (1.0, 2.1, 6.2, and 15.4% of the analytical solution), respectively, for u ¼ 0.29, 0.38, 0.60, 0.95 m s Test Case 4: Nonbreaking Wave Propagation Over an Adverse Slope [38] The second unsteady flow test explores the propagation of a flood wave over a nonhorizontal plane with an adverse slope. Even though equations (23) and (24) are also valid in this case (under the assumption of a constant velocity profile), the analytical integration of equation (24) is only available ¼ 0. In order to derive accurate benchmark solutions for the full-dynamic system, numerical solutions of equation (24) were obtained with the fourth-order Runge-Kutta method. Four different tests were run using the same computational grid and values of n and u used in test 4, and the results are shown in Figure 10. The values of used in these tests were ¼ 1 for n ¼ 0.02 and 0.03, and ¼ 0.9 for the low friction scenarios (n ¼ and 0.01). The influence of Fr on the speed of the wave propagation is again confirmed by comparing the local inertial water surface profiles with the corresponding fulldynamic solution for different values of u. While the low velocity local inertial numerical results show an excellent agreement with the full-dynamic solutions, the former increasingly underestimates the wave speed with increasing values of u. The error in the wave front position is 12.4, 32.7, 112.0, and m (1.2, 2.4, 5.2, and 9.9% of the Figure 9. Water depths predicted by the local inertial model (solid lines) along with the corresponding analytical solutions (dashed lines) for the experiments with Manning s values of 0.005, 0.01, 0.02, 0.03, and velocities of approximately 0.95, 0.60, 0.38, and 0.29 m s 1, respectively. Figure 10. Water surface elevations predicted by the local inertial model (solid lines) along with the corresponding full-dynamic solutions (dashed lines) for the experiments with Manning s values of 0.005, 0.01, 0.02, 0.03, and velocities of approximately 0.95, 0.60, 0.38, and 0.29 ms 1, respectively. 4842

11 analytical solution), respectively, for u ¼ 0.29, 0.38, 0.60, and 0.95 ms Discussion [39] Results from our test cases shed light onto important characteristics of the local inertial approximation to the shallow water equations. In particular, these examples illustrate and confirm the issues discussed in section 4, concerning the relevant properties of the system and the resulting behavior of the solutions. [40] The set of steady flow problems (test cases 1 3) indicates that the model is capable of providing highly accurate solutions to the shallow water system, particularly in the lower range of subcritical flows (e.g., Fr < 0.5). This covers a wide range of floodplain and lowland river flows of practical interest. On the other hand, it has been observed that the accuracy is gradually degraded with increasing values of Fr. In the upper range of subcritical flows (e.g., 0.5 < Fr < 1), the model s accuracy is considerably influenced by both Fr and flow spatial gradients. In general, the error introduced by the local inertial approximation will lead to milder spatial variations of depth than those obtained with the full-dynamic system; and this becomes more relevant in situations with steep flow gradients and at high Fr. This effect is highest at Fr values close to unity, in which case only scenarios involving very low water depth gradients can be accurately predicted by the simplified model. In summary, the local inertial model provides a relatively accurate and efficient model to flows in the lower range of subcritical flow, as well as the upper range involving only mild depth gradients. Applications requiring a highly accurate depth estimation at Fr close to unity and with steep depth changes (e.g., mountain rivers) need to be based on the full-dynamic set of equations. [41] The set of flood propagation test cases (i.e., test cases 4 and 5) confirms the influence of Fr also on the accuracy of the speed of the wave propagation predicted by the local inertial model. This is intimately related to the differences between the characteristic speeds of the local inertial and full-dynamic systems ( and FD ), as discussed in section 4.2. While the wave propagation speeds obtained with both models show an excellent agreement at low Fr, the former gradually underestimates the wave front position with increasing Fr. This may prove particularly problematic for practical applications involving high Fr values, as the timing of wave propagation is very often used to calibrate channel or flood plain Manning s n. In this case, lower values of n will be needed to compensate the slower propagation speed intrinsic to the local inertial equations. This means that n is to be recognized as an aggregate (effective) parameter, rather than one with a more generic and extrapolable physical interpretation (i.e., a representation of bed roughness). [42] It is important to highlight that from the perspective of practical applications, the accuracy of model predictions not only depends on the model s structure, as they are significantly influenced by inherent errors in the input data (e.g., discharge estimated from gauging stations and topography discretization). These input errors very often overshadow structural errors and constitute an important factor influencing the selection of a particular model. 7. Conclusions [43] This study presents an extended view on the applicability of the local inertial approximation of the shallow water equations to a range of flow problems of practical interest. First, the system of PDEs was thoroughly analyzed and its main characteristics were compared to those corresponding to the full-dynamic system. The discussion is divided in two different parts, which separately look at the ability of the model to accurately simulate (i) steady flow problems and (ii) flood propagation (unsteady flows). A series of test cases with analytical solutions to the fulldynamic system was then used to illustrate each aspect of the inertial model discussed in the paper. [44] The analysis of the equations representing steady nonuniform flow showed that the error introduced by the local inertial assumption leads to an underestimation of depth gradients. In addition, it has been demonstrated that the relative error in the depth gradients predicted by the local inertial model scales with Fr 2. The underestimation of depth gradients suggests that the integration of the local inertial equations will result in an attenuation of oscillations in the water depth profile. This behavior was illustrated and confirmed by a series of test cases considering steady flow regimes, in which the results of the numerical model have been compared against analytical solutions to the full-dynamic system. At flow conditions involving relatively low Fr values (e.g., Fr < 0.5), the errors introduced by the local inertial approximation were relatively low (the maximum error in the estimated depth in these tests is around 2 4% of the analytical depth for depth slopes up to 3%, approximately), and the model results showed an excellent agreement with the full-dynamic solution. A relatively good agreement was also obtained at the upper range of subcritical flows where only mild depth spatial changes take place (e.g., a maximum depth error of 4.5% was observed in the test with Fr in the range and maximum depth slope of 0.16%). In situations involving 0.5 < Fr < 1.0 combined with more pronounced depth variations, the disparity between the two models becomes more relevant; the local inertial model in general exhibiting profiles with milder water depth changes than the fulldynamic equations. [45] The analysis of unsteady flow problems has confirmed the influence of Fr also on the model s ability to accurately simulate flow propagation. This is because the elimination of the convective acceleration p leads to simplified characteristics speed ¼ 6 ffiffiffiffiffi gh, which ignores the flow velocity, and clearly p contrasts with the full-dynamic counterparts FD ¼ u6 ffiffiffiffiffi gh. Our test cases have confirmed that at low Fr this effect is of a relatively small significance but becomes relevant at high Fr. This translates into a slower propagation of flood waves by the local inertial model, compared with the full-dynamic system. [46] Acknowledgments. This research was funded under UK Natural Environment Research Council grant NE/I005366/1 awarded to the DEMON project. References Aronica, G., T. Tucciarelli, and C. Nasello (1998), 2D Multilevel model for flood wave propagation in flood-affected areas, J. Water Resour. Plan. Manage., 124, , doi: /(asce) (1998)124:4(210). 4843