Application of dynamic subgrid-scale concepts from large-eddy simulation to modeling landscape evolution

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1 Click Here for Full Article WATER RESOURCES RESEARCH, VOL. 4,, doi:10.109/006wr004879, 006 Application of dynamic subgrid-scale concepts from large-eddy simulation to modeling landscape evolution Paola Passalacqua, 1,,3 Fernando Porté-Agel, 1,,3 Efi Foufoula-Georgiou, 1,,3 and Cris Paola 1,,4 Received 10 January 006; revised 5 April 006; accepted 8 April 006; publised 30 June 006. [1] Landscapes sare important similarities wit turbulence: bot systems exibit scale invariance (self-similarity) over a wide range of scales, and teir beavior can be described using comparable dynamic equations. In particular, modified versions of te Kardar- Parisi-Zang (KPZ) equation (a low-dimensional analog to te Navier-Stokes equations) ave been sown to capture important features of landscape evolution. Tis suggests tat modeling tecniques developed for turbulence may also be adapted to landscape simulations. Using a toy landscape evolution model based on a modified -D KPZ equation, we find tat te simulated landscape evolution sows a clear dependence on grid resolution. In particular, mean longitudinal profiles of elevation at steady state and bulk erosion rates bot ave an undesirable dependence on grid resolution because te erosion rate increases wit resolution as increasingly small cannels are resolved. We propose a new subgrid-scale parameterization to account for te scale dependence of te sediment fluxes. Our approac is inspired by te dynamic procedure used in large-eddy simulation of turbulent flows. Te erosion coefficient, assumed exactly known at te finest resolution, is multiplied by a scale dependence coefficient, wic is computed dynamically at different time steps on te basis of te dynamics of te resolved scales. Tis is acieved by taking advantage of te self-similarity tat caracterizes landscapes over a wide range of scales. Te simulated landscapes obtained wit te new model sow very little dependence on grid resolution. Citation: Passalacqua, P., F. Porté-Agel, E. Foufoula-Georgiou, and C. Paola (006), Application of dynamic subgrid-scale concepts from large-eddy simulation to modeling landscape evolution, Water Resour. Res., 4,, doi:10.109/006wr Introduction [] Te fascinating self organized spatial patterns of natural landscapes ave long attracted te attention of researcers. Te most obvious and widespread of tese patterns are te tributary cannel networks generally caracteristic of erosional landscapes. Building on earlier landscape models suc as tose of Culling [1960, 1963], wic used a diffusion model of slope erosion, te 1990s saw a renaissance of landscape modeling [e.g., Willgoose et al., 1991a, 1991b; Case, 199; Rinaldo et al., 199; Howard, 1994; Rodriguez-Iturbe et al., 1994; Rodriguez-Iturbe and Rinaldo, 1997; Smit et al., 1997a, 1997b; Tucker et al., 001]. In general, tese models ave focused on reproducing wole-system properties of te landscape suc as fractal dimensions, network topology, and spatial statistics (e.g., slope distributions, slope-area relations). Landscape evolution models ave also been coupled to tectonic models 1 St. Antony Falls Laboratory, Minneapolis, Minnesota, USA. National Center for Eart-Surface Dynamics (NCED), Minneapolis, Minnesota, USA. 3 Department of Civil Engineering, University of Minnesota Twin Cities, Minneapolis, Minnesota, USA. 4 Department of Geology and Geopysics, University of Minnesota Twin Cities, Minneapolis, Minnesota, USA. Copyrigt 006 by te American Geopysical Union /06/006WR004879$09.00 to simulate te evolution of mountain belts on long timescales [Kooi and Beaumont, 1994; Tucker and Slingerland, 1994; Koons, 1995]. Compreensive reviews on landscape evolution modeling approaces are given on Dietric et al. [003], Peckam [003], and Willgoose [005]. [3] A fundamental problem arises in numerical modeling of systems wose dynamics spans a wide range of scales: selection of a computational grid (usually dictated by te size of te domain over wic a solution is sougt and te smallest grid tat can be afforded computationally) leaves out scales wose dynamics are not explicitly resolved. Yet, it is known tat even if te interest is not in resolving te smallest scales, teir effect on te dynamics of te larger scales (due to nonlinearities) is considerable. Tus ignoring te subgrid scales compromises te accuracy of te solution at te resolved scales and also makes te numerical simulation resolution-dependent. Tis problem presents itself in numerical modeling of many natural processes wic exibit multiscale variability, including flow and transport in porous media, atmosperic modeling from cloud resolving models to mesoscale to global circulation models, landatmospere interactions, atmosperic turbulence and, foremost, modeling of turbulent flows. Several metodologies ave been proposed to address tis problem and tese include derivation of effective parameters in coarse grained equations [e.g., Bear, 1988; Bou-Zeid et al., 004], statistical downscaling [e.g., Harris and Foufoula- 1of11

2 PASSALACQUA ET AL.: APPLICATION OF DYNAMIC SUBGRID-SCALE CONCEPTS Georgiou, 001], and analytical derivation of closure terms [e.g., Meneveau and Katz, 000; Sagaut, 00], among oters. [4] Te field in wic subgrid-scale parameterizations ave been te most advanced is tat of turbulence, were direct numerical simulation (DNS), i.e., numerical solution of te Navier-Stokes equations using a resolution as small as te dissipation (Kolmogorov) scale, is only feasible for relatively low Reynolds number flows. A tecnique tat as become popular to simulate iger Reynolds number turbulent flows is large-eddy simulation (LES), wic consists of solving te spatially filtered Navier-Stokes equations, using a spatial filter of size equal to or sligtly larger tan te grid size. Tis filtering operation applied to te nonlinear advection terms leads to te so-called subgrid-scale fluxes, wic represent te effect of te subgrid scales on te evolution of te resolved scales and need to be parameterized. As a result, LES explicitly resolves all scales of motion (eddies) larger tan te grid scale, wile te subgridscale fluxes are parameterized using a subgrid-scale model. Compreensive reviews on LES and subgrid-scale modeling are given by Meneveau and Katz [000], Pope [000, 004], and Geurts [004]. A particularly interesting development in subgrid-scale modeling of turbulent flows is te so-called dynamic modeling approac [Germano et al., 1991; Moin et al., 1991; Porté-Agel et al., 000; Porté- Agel, 004]. It takes advantage of te scale similarity of turbulence to optimize te value of te subgrid model coefficient(s) based on te dynamics of te resolved scales, tus not requiring any parameter tuning. [5] In te case of landscape evolution, it is well known tat landscapes present multiscale self-similar properties troug a wide range of scales, from te system scale (typically km) down to te spacing of te smallest cannels, wic is typically on te order of m and below wic diffusion processes dominate. As in ig Reynolds number turbulence, numerical solution of te entire range of scales is usually impractical. Instead, landscape models are run at relatively coarse resolution, i.e., one solves te so-called coarse-grained transport equations (Figure 1). However, te accuracy of tis metodology is unknown since cannels smaller tan te grid size are not taken into account. Tis suggests te possibility tat te calculated erosion rates and landscape evolution are likely affected by te grid resolution. Tis was pointed out by Stark and Stark [001] wo developed a subgrid-scale parameterization based on a parameterization measure called cannelization. Rodriguez-Iturbe and Rinaldo [1997] ave sown te effect of coarse graining a specific landscape on te scaling relationsips of elevation. [6] Landscapes sare important similarities wit turbulence: bot systems exibit scale invariance (self-similarity) over a wide range of scales and teir beavior can be described using comparable dynamic equations. Tis similarity can be seen, for example, in te beavior of power spectra: Turbulence velocity spectra exibit a well-known 5/3 slope in te inertial subrange [Kolmogorov, 1961], representing te energy cascade from large scales to small scales. In te case of landscapes, power spectra of linear transects in topograpy also exibit a log-log scaling range wit slope of. Anoter parallel between te two systems is te existence of a lower limit on te size of te turbulent structures (eddies): te Kolmogorov scale, te scale at wic viscous effects dominate and te effective Reynolds number approaces unity. In landscapes, te analogous fine scale would be te spacing of te smallest cannels, determined by te scale at wic (diffusive) illslope processes dominate [e.g., Dietric et al., 003]. Tis analogy between te viscous lengt scale of turbulence and te illslope scale in landscapes as also been discussed by oters [e.g., Peckam, 1995]. Moreover, turbulence as been used as a metapor for oter complex systems suc as eartquakes [Kagan, 199] and stream braiding [Paola, 1996; Paola et al., 1999]. [7] Te purpose of tis paper is to explore concepts of LES in te context of landscape evolution modeling. Using a minimum complexity model, used previously by several autors for landscape simulation [e.g., Sornette and Zang, 1993; Somfai and Sander, 1997; Banavar et al., 001], we demonstrate its scale dependence and propose a dynamic subgrid-scale model to take into account te effect of subgrid-scale processes in a landscape evolution model. [8] It is important to point out tat te goal of tis study is not to strictly apply te LES tecnique, as developed for turbulent flows, to landscape evolution simulations. Tere are some limitations to te direct extension of te LES tecnique to landscapes. Typical governing equations for landscape evolution, even toug often similar in form to te Navier-Stokes equations (including nonlinear terms tat generate fluctuations as well as diffusion terms), are not as well establised for te description of te system at all scales. For example, te nonlinear erosion flux term is already a parameterization containing tuning coefficient(s). Tis makes it callenging to formally define te subgridscale erosion fluxes and to develop subgrid-scale models for tem. Instead, our approac ere consists of developing a tuning-free dynamic procedure, inspired from te dynamic modeling approac used in LES, to optimize te value of te erosion coefficient (in te nonlinear erosion flux term) using te scale dependence of te coefficient quantified from te smallest resolved scales in te simulations.. Landscape Evolution Modeling and Effect of Grid Resolution.1. KPZ Model [9] As discussed above, a number of different models ave been proposed for landscape evolution. Here we use a modified version of te Kardar-Parisi-Zang (KPZ) equation, originally used in te context of growt of atomic interfaces by ion deposition [Kardar et al., 1986]. Te KPZ equation as applied to modeling te evolution of land surface elevation reads [Sornette and Zang, 1993; Somfai and Sander, 1997; Banavar et al., ð~x; ¼ Dr þ Cjrj þ ð~x; tþ ð1þ Te rigt and side of equation (1) includes, from left to rigt, a diffusion term, were D is te diffusion coefficient, a nonlinear term, were C is a constant and r is te slope, and a wite noise term. [10] It is important to note tat, wit a simple transformation of variables, te KPZ equation witout noise of11

3 PASSALACQUA ET AL.: APPLICATION OF DYNAMIC SUBGRID-SCALE CONCEPTS becomes te Burgers equation, a low-dimensional analog to te Navier-Stokes equations governing fluid flow and turbulence. [11] Te initial application of te KPZ equation to landscape evolution [Sornette and Zang, 1993] sows te importance of te nonlinear terms in te evolution of surface topograpy and te associated drainage network. At a coarse grained scale, te effect of diffusion is often neglected [Somfai and Sander, 1997; Banavar et al., 001] since tis mecanism is effective mainly at te small (subgrid) scales. Neglecting also te noise term, (1) as only te non linear term on te rigt and side. Furtermore, taking into account tat te evolution of te landscape is coupled wit te water flux q acting on te surface, te constant C in front of te nonlinear term in (1) can be written as an erosion coefficient a times te water flux q [Somfai and Sander, 1997; Banavar et al., 001]. Te governing equation ten ¼ j j : ðþ Under te assumption of uniform rainfall acting on te surface, te water flux at a given point is proportional to te area draining at tat location. We ave cosen a fairly simple landscape evolution model, best suited for bedrock landscape evolution modeling, wic does not allow bot erosion and deposition to occur. Tis coice as been motivated by te fact tat a simple type of equation would maximize clarity in deriving te subgrid model. Tis work will be extended in te future to more compreensive landscape evolution models. [1] Notice tat () is a special case of te general governing equation, widely used in landscape modeling [e.g., Rodriguez-Iturbe and ¼ aam r j j n ; ð3þ wit m n 0:5. [13] Te nature of te steady state reaced by te system depends on te external conditions applied in te problem. If te boundary condition at te output is a fixed elevation, wit constant rock uplift, te steady state is reaced wen te erosion rate balances te rock uplift rate over te wole system [Hack, 1960; Adams, 1980; Howard, 1994; Somfai and Sander, 1997; Willett and Brandon, 00]. If te uplift 3of11 Figure 1. Scematic of te separation between resolved and subgrid scales in turbulence (Figure 1a) and landscapes (Figures 1b and 1c). (a) Gray scale rendering of te vertical velocity component measured in a turbulent boundary layer at te St. Antony Falls Laboratory wind tunnel. (b) Bareeart LIDAR saded-relief image (1-m resolution) of a portion of te Angelo Coast Range Reserve, nortern California, grid spacing of 500 m. (c) Fourier spectrum of te topograpy sown in Figure 1b, sowing a power law dependence of spectral power on wave number. Separation between resolved and unresolved scales is 100 m. In bot turbulence and landscapes, only structures wit lengt scales larger tan te grid size are explicitly resolved.

4 PASSALACQUA ET AL.: APPLICATION OF DYNAMIC SUBGRID-SCALE CONCEPTS Figure. Boundary conditions and allowed flow directions. Te output of te system is located at te downstream end of te field were te elevation is kept fixed at zero. At te upstream end te boundary condition is an infinite wall, so tat no flow exits upstream of te field. At te east and west boundaries, te flow directions are likewise restricted to tose inside te field. Tus te only allowed output is at te downstream end of te field. rate is zero, te system reaces a steady state wen tere is no remaining material to erode [Inaoka and Takayasu, 1993]. [14] Te river networks obtained wit te modified KPZ equation ave been sown to satisfy scaling laws caracteristic of natural landscapes: te slope-area law, te power law of distribution of drainage area, and Horton s laws for brancing ratio and lengt ratio [Somfai and Sander, 1997]. In addition to tese laws, te simulations also yield realistic profiles for te average elevation along te mainstream direction [Banavar et al., 001]... Numerical Implementation [15] Te initial field is a sloping surface wit a small noise, obtained using te following expression [Somfai and Sander, 1997]: x; ð y; t ¼ 0Þ ¼ s 0 ðy þ dy randðx; yþþ; ð4þ were is te elevation, s 0 represents te initial slope, y is te nort-sout coordinate, dy is te grid constant and rand is a uniform random number in te range [0,1]. Tis initial configuration prevents te formation of lakes. [16] We study te evolution of te system at tree different resolutions: te same field is divided into grid cells, grid cells and grid cells. We focus initially on te simplest case (uniform rainfall, no groundwater, uniform and structureless substrate, no redeposition), applying te simplified erosion model discussed earlier (equation ()) wit te addition of a constant uplift ¼ u a q j j ; ð5þ Equation (5) represents an erosional model for an incisional process were te erosion rate depends linearly on water flux and nonlinearly on slope. [17] Water is routed using te steepest descent rule. In every site te elevation is compared wit te one of te eigt surrounding neigbors and te water is assumed to follow te steepest pat. Recently, Pelletier [004] as sown tat computing te slope using a multipledirection algoritm eliminates an undesirable consequence of te steepest descent rule: evolution to a frozen steady state of te river network in wic erosion exactly balances uplift at eac point [Hasbargen and Paola, 000, 003]. Once te flow direction is computed in every site, te slope and te water flux can be computed and used to update te elevation via (5). Because of te assumption of a uniform rainfall and no loss of water, te water flux is given by te drainage area times te unit rainfall. [18] Te boundary conditions are: an infinite wall at te upstream end of te field (nort boundary); an output boundary at fixed eigt equal zero located at te downstream end of te field (sout boundary); on te lateral sides (east and west boundaries) te flow is forced to drain into te system. Tis condition could be easily canged to periodic boundary conditions [Somfai and Sander, 1997; Banavar et al., 001]. A sketc of te computational domain wit te applied boundary conditions is sown in Figure. [19] Te simulations wit te tree resolutions are run independently until te systems reac steady state. Te steady state is reaced wen te erosion rate is in equilibrium wit te uplift. Te simulated evolution of te landscape sows two different timescales: a freezing time, at wic te river network reaces its final configuration but te elevation continues adjusting, and a relaxation time at wic te system reaces its equilibrium profile and te surface stops evolving [Sinclair and Ball, 1996; Banavar et al., 001]. Te time needed to freeze te system is usually smaller tan te time needed to reac te final profile. It sould be noted tat a freezing time, and a corresponding frozen configuration of te system, can be defined as we ave done only because te model allows for an (unrealistic) static steady state. Revised definitions would be needed for te more realistic dynamic-steadystate condition. [0] Te freezing time of te system is obtained by computing at every time step te number of unstable sites. An unstable site is defined as a point in te system were te flow direction canges in one time step [Inaoka and Takayasu, 1993]. Wen te number of unstable sites remains equal to zero for a sufficiently large number of time steps, te river network is considered at its final configuration. Before reacing te equilibrium profile, te number of unstable sites remains zero, wile te topograpy continues adjusting. Te relaxation time instead is given by te time at wic te topograpy also reaces a static steady state..3. Effect of Grid Resolution [1] We analyze te results of te numerical simulations using 56 56, and grid cells in terms of several statistics. Te systems obtained at steady state wit resolutions 56 56, and are 4of11

5 PASSALACQUA ET AL.: APPLICATION OF DYNAMIC SUBGRID-SCALE CONCEPTS sown in Figure 3. Te corresponding river networks were extracted wit River Tools (ttp:// and are sown in Figure 4 (cannels of Straler order greater tan only). Te modified KPZ model produces cannel networks wit te expected loss of detail as te resolution is decreased. [] Te power spectral density (spectrum) provides an estimate of te distribution of elevation variance across scales. Figure 5 sows a comparison of spectra from transects across te upstream part of te domain for te tree resolutions. Except at te largest scales, tere is a wide range of scales for wic te spectra obtained at all resolutions sow a slope of approximately, wic is in good agreement wit observations from linear transects in topograpy [Vening Meinesz, 1951; Mandelbrot, 1975; Sayles and Tomas, 1978; Newman and Turcotte, 1990]. However, te total variance in elevation observed at te tree resolutions does depend on scale. [3] Mean longitudinal profiles obtained at steady state wit te tree resolutions are sown in Figure 6a. Te results indicate tat te basin topograpy required to produce a balance between erosion and rock uplift in te simulation is strongly scale dependent. In particular, bot slope and curvature increase wit decreasing resolution, wic is not realistic. Tis beavior can be attributed to te fact tat erosion due to subgrid-scale cannel networks (occurring at scales smaller tan te grid scale) is not accounted for in te simulations. Since te subgrid-scale erosion flux is expected to be relatively larger in te case of coarser resolutions, landscapes simulated at tose resolutions experience less efficient erosion tan te ones obtained at iger resolutions. [4] Strong scale dependence is also sown by te volume of material eroded per time step. As Figure 6b sows, te iger te resolution, te iger te volume of eroded material per time step. Tis is consistent wit te observed beavior of te mean longitudinal profiles. Te area under te curves, wic gives te total amount of material eroded until steady state is reaced, increases wit resolution. Te rate of erosion gives also an idea of te timescale dependence of te erosion process at te tree resolutions: te iger te resolution, te iger te rate of erosion and te faster te process. Te lowest resolution needs more time to reac te steady state. Figure 3. Elevation fields obtained for tree grid resolutions at steady state using equation (5): (a) 56 56, (b) 18 18, and (c) A Dynamic Subgrid-Scale Model 3.1. Derivation of te Dynamic Subgrid-Scale Model [5] Te resolution dependence of te results obtained in te previous section using (5) igligts te need to account for te fact tat erosion rates depend on te grid size used in te simulations. In tis section, we develop a procedure to account for te scale dependence of te erosion coefficient a in (5). Te metodology is based in part on te so-called dynamic modeling approac used in LES of turbulent flows [Germano et al., 1991; Moin et al., 1991; Meneveau et al., 1996]. In te context of landscapes, we parameterize te effect of te subgrid-scale erosion rates by calculating a modified erosion coefficient a using information contained in te resolved elevation field and assuming scaling in te elevation statistics. [6] For te purpose of developing te tecnique, one needs to know te exact value of te erosion coefficient at some reference scale. For simplicity, witout loss of generality, ere we consider te igest resolution simulated (56 56) as te exact solution, and te value of te erosion coefficient at tis scale is assumed to be exactly known. However, te same approac can be extended to oter reference scales for wic erosion coefficients could be determined. Te grid size corresponding to resolution is taken as D/. 5of11

6 PASSALACQUA ET AL.: APPLICATION OF DYNAMIC SUBGRID-SCALE CONCEPTS [7] At coarser resolutions te model is now written in a filtered form, as it is done in LES. At resolution 18x18 (resolution D), (5) ¼ u a D ~q r ~ ; ð6þ were te tilde indicates quantities spatially filtered (wit an implicit filter imposed by te grid size) at scale D. At resolution (resolution D) equation (5) ¼ u a D ~q r ~ ; ð7þ were te overbar denotes spatial filtering at scale D. [8] Modeling erosion witout explicitly accounting for scale effects on te erosion coefficient, as we did in te previous section, implicitly amounts to assuming tat a D ¼ a D ¼ a D= ¼ a 0 ; ð8þ were a 0 is te erosion coefficient, wic we assume is known and independent on resolution. Te simulation results from section indicate instead tat a depends on te scale D. Our goal, ten, is to account for tis, dynamically computing a at eac time step as te simulation progresses. To do tat, one as to make some assumptions about te dependence of a on D. As a first approximation, we assume tat te ratio between te erosion coefficients at scales D and D/ is te same as te ratio between scales D and D, i.e., we assume a constant value for te scale dependence ratio b, defined as b ¼ a D a D= a D a D a 4D a D : ð9þ Note tat tis is a muc weaker assumption tan te original one tat te erosion coefficient a does not depend on scale. Te erosion coefficients at scales D (resolution Figure 4. River network extracted from elevation fields for tree grid resolutions at steady state. Only cannels of Straler order greater tan are sown: (a) 56 56, (b) 18 18, and (c) of11 Figure 5. Elevation power spectra at te tree resolutions averaged during te simulations. Te spectral slope is not affected by te grid resolution and is near, consistent wit observed spectra from natural topograpy.

7 PASSALACQUA ET AL.: APPLICATION OF DYNAMIC SUBGRID-SCALE CONCEPTS [30] On te basis of tese ideas, we derive te new model wit subgrid-scale parameterization in detail for simulations at resolution D. Applying te model at scales D and D leads to (6) and (7), given above. Spatially filtering (6) using a filter of size D (operation denoted by an overbar) and ten averaging equations (6) and (7) over te entire field (operation denoted by i) yields ~ + u a D ~q r ~ ð11þ ~ + u a D ~q r ~ ð1þ [31] Combining (11) and (1) leads to an expression for te ratio between te effective erosion coefficients at scales D and D: D a ~q r ~ E D ¼ a D ~q r ~ ð13þ Figure 6. Dependence on grid resolution of model results using equation (5). (a) Mean longitudinal profiles obtained at steady state. (b) Volumes of eroded material per time step during te simulations. Bot steady state profile and eroded volumes sow strong scale dependence ) and D (resolution 64 64) can be expressed as a function of te erosion coefficient at te finest scale (D/) as follows: a D ¼ b a D= a D ¼ b a D ¼ b a D= ð10þ Note tat we assume te coefficient at te smallest scale equal to te known value, i.e., a D/ = a 0. [9] On te basis of te expression for te scale dependence coefficient given by (9), b must be computed at scale D based only on information at te available scale D and larger, since during te simulation te beavior at finer scales is not known. Tus in our case, to compute b dynamically, we use information at scales D and D, togeter wit te assumption tat b is constant. Te variables corresponding to scale D can easily be computed by spatially filtering te simulated field (implicitly filtered at scale D) using a two-dimensional filter of size D. As mentioned above, tat operation is denoted by an overbar. Te rigt and side of (13) can be explicitly calculated using information contained in te simulated elevation field. [3] Taking advantage of te scale similarity assumption in (9), (13) can be used to define te scale dependence coefficient b: D E b ¼ a D a D= a D a D ¼ ~q r ~ ~q r ~ ð14þ Following te above procedure and using (14), we compute b dynamically at every time step, tus not requiring any a priori calibration or tuning. It is important to point out tat te averaging operation ( i) is needed to avoid unrealistic local fluctuations of b tat would be obtained witout averaging. [33] Te same approac can be followed to compute te coefficient b to be used in te simulations at oter resolutions. For example, in te case of a grid of size D, te information at scale 4D would be used, and te expression for te scale dependence coefficient b would become b ¼ a D a D a 4D a D ¼ d D q r E ; ð15þ ^q r^ were te at denotes a filtering operation using a twodimensional filter of size 4D over te simulated variables, obtained at a grid resolution of D. Just as for scale D, b is computed dynamically at every time step following (15), and again does not require calibration or tuning. [34] Te new value of b is used to define te erosion coefficient ad at te corresponding grid scale D in terms of 7of11

8 PASSALACQUA ET AL.: APPLICATION OF DYNAMIC SUBGRID-SCALE CONCEPTS Figure 7. Comparison of results from te original model (equation (5)) and te dynamic subgrid model (equations (16) and (17)). (a) Mean longitudinal profiles obtained at steady state. Te results obtained wit te subgrid-scale parameterization sow a relatively weak dependence on grid resolution. (b) Volumes of eroded material per time step obtained at steady state. Te results obtained wit te subgrid-scale parameterization again sow a relatively weak dependence on grid resolution. te erosion coefficient at scale D/, wic we assume is exactly known, according to (10). Tus te model at scale D is now given ¼ u b a D= ~q r ~ Furtermore, te model at scale D can be written ¼ u b a D= ~q r ~ ð16þ ð17þ [35] To test te efficacy of te proposed dynamic subgridscale sceme, we redo te simulations at te tree resolutions, using (5) at resolution 56 56, (16) at resolution and (17) at resolution Initial and boundary conditions are te same as used in te previous section. At every time step, te slope and te water flux are computed at eac location from te simulated field. Ten te field is filtered at a resolution double te grid size and slope and water flux are computed from te filtered field at eac location. Wit tese quantities te scale dependence coefficient b is computed from equation (14) (or (15) at 64 64) and te field is ten updated using equation (16) (or (17) at 64 64). 3.. Results and Discussion [36] Te elevation fields and te extracted river networks are qualitatively similar to te ones obtained in te previous section using equation (5) at all resolutions. However, te erosion rate is clearly affected by te new formulation of te model at te lower resolutions. Te mean longitudinal profiles obtained at steady state for te simulations wit dynamic subgrid-scale modeling are sown in Figure 7a. Te profiles at resolution and obtained wit te new model are close to te profile at 56 56, indicating tat te dynamic subgrid-scale metod accounts for most of te scale dependence. Te same beavior is found in te volume of eroded material per time step, sown in Figure 7b: te new model wit dynamic subgrid-scale parameterization yields muc more consistent erosion rates across te different grid resolutions tan te simulations using a constant erosion coefficient. [37] We stress tat in tis first stage of our investigation, we ave used te simplest plausible sceme for dynamic subgrid-scale modeling. Despite tis, te metod seems able to eliminate muc of te dependence of erosional landscape dynamics on grid resolution. Te dynamic procedure is now modified to allow for scale dependence of te coefficient b. A similar dynamic, tuning-free approac as recently been developed in te context of subgrid-scale models for LES of turbulent flows [Porté- Agel et al., 000; Porté-Agel, 004; Stoll and Porté-Agel, 006.]. 4. A Scale-Dependent Dynamic Subgrid-Scale Model 4.1. Derivation of te Scale-Dependent Dynamic Subgrid-Scale Model [38] Te dynamic procedure is now modified to allow for scale dependence of te coefficient b. Tis requires te use of an additional test filtering operation (e.g., at scale four times te grid scale), from wic te scale dependence of b can be determined dynamically. Te scale dependence coefficient b is now allowed to cange wit scale, and terefore b D 6¼ b D 6¼ b 4D : ð18þ [39] At tis point, an assumption as to be made about te functional form of te scale dependence of b. Assuming a simple power law dependence of b wit scale [Porté-Agel et al., 000], we can write b D b D b D b 4D b 4D b 8D : ð19þ Note tat tis is a muc weaker assumption tan te previous one of b constant across scales. Using equation (19), te 8of11

9 PASSALACQUA ET AL.: APPLICATION OF DYNAMIC SUBGRID-SCALE CONCEPTS Te same procedure can be applied at scale D to express te unknown parameters b D and b D as a function of b 4D and b 8D as b D b 4D b 4D ; b 8D b D b D b D ¼ b b 4D b 4D 1 ¼ b3 4D 4D b 8D b 4D b ; and 8D b D b D b5 4D b 3 8D ðþ Note tat b 4D can be obtained from te resolved elevation field using equation (1), and b 8D can also be computed dynamically using te identity * + _ b 8D ¼ a ^q r^ 8D ¼ * a + ; ð3þ 4D ^q _ r ^ _ were te curved overbar denotes a filtering operation using a two-dimensional filter of size 8D over te simulated variables, obtained at a grid resolution of D. [40] Wit te new definitions of b D and b D, a D and a D become a D ¼ b D a D= ¼ b D b 4D a D= a D ¼ b D b D a D= ¼ b5 4D b 3 a D= 8D and equations (16) and (17) become ð4þ Figure 8. Comparison of te results from te original model (equation (5)) and te new scale-dependent dynamic subgrid model (equations (5) and (6)). (a) Mean longitudinal profiles obtained at steady state. Te profiles obtained wit te subgrid-scale parameterization are almost indistinguisable. (b) Volumes of eroded material per time step obtained at steady state. Te results obtained wit te subgrid-scale parameterization again sow very little dependence on grid resolution and time. scale dependence coefficient at scale D, b D, can be expressed as b D b D b D b 4D ¼ b D b 4D ; ð0þ were b D and b 4D, recalling (14) and (15), can be computed dynamically from te resolved elevation field as b D ¼ a ~q r ~ D ¼ a D ~q r ~ D b 4D ¼ a q d r E ð1þ 4D ¼ ; a D ^q ¼ u b D a D= ~q r b ~ ¼ u b5 4D b 3 8D a D= ~q r ~ ð5þ ð6þ To test te proposed scale-dependent dynamic subgrid-scale sceme, we perform te same simulations using equation (5) at resolution 56 56, and equations (5) and (6) at resolutions and 64 64, respectively. [41] Initial and boundary conditions are te same used in te previous sections. At every time step, te slope and te water flux are computed at eac location from te simulated field. Ten te field is filtered at a resolution two and four times te grid size and slope and water flux are computed from te filtered field at eac location. Te scale dependence coefficients b corresponding to scales twice and four times te grid scale are computed dynamically using equations (1) for te resolution, or equations (1) and (3) for te resolution. Tese values are ten used in equations (5) and (6) to obtain te time evolution of te simulated elevation field. Te simulations are run until steady state is reaced. 4.. Results and Discussion [4] Similar to te case of te dynamic model presented in Section 3, te elevation fields and river networks obtained wit te scale-dependent dynamic model are 9of11

10 PASSALACQUA ET AL.: APPLICATION OF DYNAMIC SUBGRID-SCALE CONCEPTS quantitatively similar to te ones obtained witout subgridscale model. However, te mean longitudinal profile obtained at steady state and te volume of eroded material per time step sow an additional improvement compared to te scale-invariant dynamic model. As sown in Figures 8a and 8b, te simulation results obtained wit te tree resolutions are very similar, wic igligts te ability of te new model to systematically (and witout parameter tuning) account for te scale dependence of te erosion coefficient. Moreover, te previously observed time dependence of te results is substantially reduced (Figure 8b), indicating tat te new model is also able to minimize te effects of resolution on te time evolution of te simulated landscapes. [43] Note tat tere are a number of ways in wic te scale-dependent dynamic approac could be improved. For example, te coefficient b could be computed locally using alternative averaging metods, suc as te Lagrangian dynamic procedure introduced by Meneveau et al. [1996]. Hig-resolution digital elevation data could also be used to test some of te assumptions made in te dynamic models (e.g., power law scaling of te coefficients) and provide guidance for furter improvements, as done in a priori experimental studies of turbulent flows [e.g., Meneveau and Katz, 000]. 5. Conclusions [44] 1. Landscapes simulated using a modified -D KPZ equation sow a systematic dependence on grid resolution: increasing resolution allows for increased cannel density, and tus erosion rates and mean longitudinal profiles of elevation at steady state ave an undesirable dependence on grid resolution. [45]. A new subgrid-scale parameterization, inspired by te scale-dependent dynamic modeling approac used in turbulence simulations, is able to correct most of tis scale dependence. Te erosion coefficient, assumed exactly known at te finest resolution, is multiplied by a scale dependence coefficient, wic is computed dynamically as a function of time based on te landscape dynamics at te resolved scales. Te sceme takes advantage of te selfsimilarity tat caracterizes landscapes over a wide range of scales and produces landscapes tat sow very little dependence on grid resolution. [46] Tere is no reason te proposed approac could not be applied to oter landscape evolution models. Te applicability of te LES-inspired approac to modeling erosional landscapes suggests tat te tecnique may be generalizable to oter systems as well. Te basic requirement is tat te system be self-similar over a sufficiently wide range of lengt scales to justify te estimation of te effect of subgrid processes by comparison of te model beavior over scales coarser tan te resolved scale. 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