Processbased modeling of tsunami inundation ... - Wiley Online Library

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 116, F01006, doi:10.1029/2010JF001797, 2011

Process‐based modeling of tsunami inundation and sediment transport Alex Apotsos,1 Guy Gelfenbaum,1 and Bruce Jaffe2 Received 8 June 2010; revised 15 November 2010; accepted 30 November 2010; published 10 February 2011.

[1] The infrequent and unpredictable nature of tsunamis precludes the use of field experiments to measure the hydrodynamic and sediment transport processes that occur. Instead, these processes are often approximated from laboratory, numerical, and theoretical studies or inferred from observations of the resultant sediment deposits. Here Delft3D, a three‐dimensional numerical model, is used to simulate the inundation and sediment transport of a tsunami similar in magnitude to the 26 December 2004 Indian Ocean tsunami over one measured and three idealized morphologies. The model is first shown to match well the observations taken at Kuala Meurisi, Sumatra, and then used to examine in detail the processes that occur during the tsunami. The model predicts that at a given cross‐shore location the onshore flow accelerates rapidly to a maximum as the wavefront passes, and then gradually decelerates before reversing direction and flowing offshore. The onshore flow does not tend to zero everywhere at maximum inundation, but instead flow reversal occurs near the shoreline even as the wavefront continues to inundate landward. While some sediment is eroded by the passing wavefront, the suspension of sandy sediment is dominated by the long‐duration, high‐velocity backwash that occurs along the beach face and offshore of the shoreline. Some of the sediment suspended during backwash is advected shoreward by the subsequent wave, creating large spatial gradients in the suspended sediment concentrations, which may not be in equilibrium with the local hydrodynamics. The inundation and transport of sediment during a tsunami can be affected by complexities in the morphological profile and interactions between multiple waves, and many of the hydrodynamic and sediment transport processes predicted here are similar to analogous processes previously observed in the swash zone. Citation: Apotsos, A., G. Gelfenbaum, and B. Jaffe (2011), Process‐based modeling of tsunami inundation and sediment transport, J. Geophys. Res., 116, F01006, doi:10.1029/2010JF001797.

1. Introduction [2] The 26 December 2004 tsunami in the Indian Ocean killed more than 237,000 people and left over a million homeless (USAID fact sheet, 7 July 2005, available at http:// www.usaid.gov/our_work/humanitarian_assistance/disaster_ assistance/countries/indian_ocean/fy2005/indianocean_ et_fs39_07‐07‐2005.pdf). While this was the most destructive tsunami in recent history, it is by no means unique. Over the past century, other tsunamis have resulted in a significant loss of life, including over 30,000 killed in Sanriku, Japan (1896) and 2,000 killed in Papua New Guinea (1998) and on Flores Island (1992) [Tappin, 2007]. Unfortunately, owing to the infrequent, unpredictable, and destructive nature of tsunamis, it is difficult to estimate accurately their historical 1 Coastal and Marine Geology Program, USGS, Menlo Park, California, USA. 2 Pacific Science Center, USGS, Santa Cruz, California, USA.

This paper is not subject to U.S. copyright. Published in 2011 by the American Geophysical Union.

recurrence, and to identify the factors that dictate their response in the nearshore. Without an adequate understanding of tsunami frequency and nearshore response, appropriate coastal management programs that properly plan for and mitigate losses owing to tsunamis cannot be developed. [3] Analytical and laboratory studies of tsunami generation, propagation, and inundation have been ongoing since the early 1900s (see Synolakis and Bernard [2006] for a detailed review). Many analytical studies have been based on the hodograph type transformation first used to solve the nonlinear shallow water equations by Carrier and Greenspan [1958] and now include closed solutions to tsunami‐like initial‐value problems [e.g., Carrier et al., 2003]. The number and complexity of laboratory experiments has grown to include both detailed studies in two‐dimensional flumes [e.g., Synolakis, 1987] and experiments using complex bathymetries in fully three‐dimensional basins [e.g., Briggs et al., 1995]. Recently, laboratory studies have been extended to include the suspension and transport of sediment under breaking solitary waves [Tonkin et al., 2003; Young et al., 2009].

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[4] While many early studies of tsunamis relied primarily on laboratory and analytical experiments, over the past several decades tsunami research has evolved to include numerical, geological, and field studies as well. Complex numerical models have been developed (e.g., MOST, Coulwave, Delft3D) that can simulate accurately tsunami propagation and inundation using realistic wave forms and bathymetries [Vatvani et al., 2005a, 2005b; Lynett and Liu, 2005; Titov et al., 2005; Arcas and Titov, 2006; Wang and Liu, 2006; Grilli et al., 2007; Ioualalen et al., 2007; Apotsos et al., 2010a]. Sediment transport, on the other hand, has typically only been studied using idealized wave forms and simplified bathymetries [e.g., Simpson and Castelltort, 2006; Pritchard and Dickinson, 2008], although some recent studies have used complex numerical models that more accurately depict the actual wave form [Gelfenbaum et al., 2007; Apotsos et al., 2009]. [5] Posttsunami field observations include eyewitness accounts, video recordings [Goff et al., 2006; Fritz et al., 2006; McAdoo et al., 2006], and field measurements of tsunami water levels and sediment deposits [e.g., Gelfenbaum and Jaffe, 2003; Borrero, 2005; Jaffe et al., 2006; Goff et al., 2006; Y. Tsuji et al., Distribution of the tsunami heights of the 2004 Sumatra tsunami in Banda Aceh measured by the tsunami survey team, http://www.eri.u‐tokyo.ac.jp/namegaya/ sumatera/surveylog/eindex.htm, 2005]. These field observations provide estimates of the inundation distance, flow depth, surface velocity, and onshore distribution of sediment that can be used both to identify specific characteristics of the tsunami as well as for model calibration and verification. [6] Geological studies have focused primarily on improving our understanding of the recurrence interval of and risk from tsunamis [e.g., Atwater, 1987; Minoura and Nakaya, 1991; Atwater and Hemphill‐Haley, 1996; Bourgeois and Minoura, 1997; Jaffe and Gelfenbaum, 2002; Burroughs and Tebbens, 2005; Geist and Parsons, 2006]. These studies are based on the fact that sediment deposited by large tsunamis can persist in the sedimentary record for hundreds, if not thousands, of years, and can offer clues to the magnitude and frequency of tsunamis that occurred before written records. Dating the soil layers surrounding tsunami deposits can lead to estimates of about when a tsunami occurred. Complex conceptual models derived from observations of the internal structure of modern sediment deposits [e.g., Paris et al., 2007; Morton et al., 2008; Nanayama, 2008; Choowong et al., 2008] have been formulated to identify the processes that dominate sediment deposition during a tsunami. Based on the processes identified, simplified mathematical models [e.g., Jaffe and Gelfenbaum, 2007; Smith et al., 2007; Soulsby et al., 2007; Moore et al., 2007] have been developed to estimate the magnitude and flow velocity of tsunamis from the characteristics of onshore sediment deposits. [7] A multifaceted approach to tsunami research, especially the transport of sediment, is necessary as the collection of detailed field measurements during a tsunami is currently unfeasible, and each type of analysis described above has inherent limitations. Their limitations notwithstanding, numerical models offer a way to describe the time‐varying hydrodynamic and sediment transport processes that occur. It is suggested here that models based on our current understanding of sediment transport processes can be used as part of an iterative process, whereby the models are used to suggest which processes may dominate during a tsunami. Appropriate

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laboratory experiments can then be conducted or targeted field observations taken to determine if the models are accurately replicating these processes. Any necessary modifications to the models can be made, and the iterative process continued until both numerical models that accurately predict tsunami‐ induced sediment transport and a number of benchmark‐like data sets for model validation are developed. This study is but an early step in this iterative process. [8] Here, a three‐dimensional numerical model (Delft3D) is used to examine the hydrodynamic and sediment transport processes of a tsunami similar in magnitude to the 26 December 2004 Indian Ocean tsunami. The model and simplified setup are outlined in sections 2 and 3, respectively. Model results are presented in section 4 and discussed in section 5. Conclusions are presented in section 6.

2. Model [9] Delft3D is a three‐dimensional coupled hydrodynamic/ sediment transport/morphological change numerical model. The hydrodynamic model solves the nonlinear shallow water equations (NLSWEs) on a three‐dimensional staggered grid using a finite difference scheme [Stelling and van Kester, 1994]. The numeric method used in this study to solve the NLSWEs is based on the conservation of mass, momentum (flow expansions), and energy head (flow contractions), and was specifically developed for rapidly varying flows with a wide range of Froude numbers, and the rapid wetting and drying of grid cells [Stelling and Duijmeijer, 2003]. The hydrodynamic model has been validated against numerical and laboratory data including several standard tsunami benchmarks [Apotsos et al., 2010a] and has been shown to model accurately the propagation and inundation of the 26 December 2004 Indian Ocean tsunami [Vatvani et al., 2005a, 2005b]. [10] Sediment transport is calculated by solving the conservation of mass equations within the hydrodynamic model. Bed load and suspended load transport are calculated following van Rijn [1993, 2007a, 2007b] combined with a k − " turbulence model (see Appendix A for model formulations). Sediment fluxes between the water column and the bed are calculated within each grid cell based on the vertical diffusion of sediment owing to turbulent mixing and deposition from settling, and the bottom morphology is updated accordingly at each time step [Lesser et al., 2004; van Rijn et al., 2004]. Sediment‐induced density stratification owing to high suspended sediment concentrations is inherently accounted for within the turbulence model by adjusting the fluid density to include the mass of the suspended sediment. The settling velocity is determined from the clear water settling velocity of van Rijn [1993] combined with the effects of hindered settling following Richardson and Zaki [1954]. [11] The combined sediment transport/morphological change model has been shown to model well many coastal and estuarine settings [Lesser et al., 2004; Gerritsen et al., 2007; van Rijn et al., 2007], and has been shown to qualitatively match the sediment deposition from the 2004 Indian Ocean tsunami (see section 4.1) [Gelfenbaum et al., 2007; Apotsos et al., 2010a]. However, many of the sediment transport formulations used were developed and calibrated for less extreme flow conditions (i.e., depth‐average velocities 1600 m). As the model simulations are based on morphology measured following the tsunami, the initial shoreline in the model is not the same as the shoreline prior to the tsunami owing to tsunami‐ induced shoreline recession. As the simulated tsunami erodes the shoreline further landward (i.e., ∼300 m), the predicted deposit starts further landward than the observations. It was not possible to reconstruct accurately the pretsunami morphology such that the two shorelines were the same. The thick deposit observed near the cliff is composed of fine sediment and may not be captured by the model owing to the small range of grain

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Figure 2. (a) Measured topography (solid black curve), flow depths (gray circles), and model predictions of the maximum water level (dashed black curve). (b) Measured deposit thickness (gray circles) and predicted change in bed level for erodable sand everywhere (dashed black curve) and deposit thickness for erodable sand just offshore (dashed gray curve) at Kuala Meurisi. sizes simulated, most of which are coarser than those observed near the cliff. Furthermore, when erodable sediment is assumed to be distributed over the entire profile, it was not always possible to determine which sediment within the bed was deposited by the tsunami, and therefore it was only possible to compare the observed deposit thickness with the predicted change in bed level at each location (Figure 2b; note negative values for dashed black curve). However, if erodable sediment is assumed to exist only offshore, the model still predicts the general thickness and overall patterns of the observations well (Figure 2b, dashed gray curve). The deposit from this simulation starts closer to the initial shoreline as the distance over which the shoreline can erode landward is restricted by the model setup (i.e., no erodable sediment is initially present onshore of x > 150 m). Owing to the large number of parameters on which the exact details of the deposit thickness depends [see Apotsos et al., 2009], it is difficult to quantitatively assess model performance. However, as the model predicts well the important characteristics of the flow and sediment observations at Kuala Meurisi, it is used to examine in more detail the processes that occur during a tsunami. 4.2. Hydrodynamics [20] The model predicts that the onshore flow depth associated with the two large, peaked waves reaches a maximum near the shoreline soon after the wavefront passes, and then begins to decrease before the wave reaches maximum inundation. The resulting depth‐averaged onshore flow velocities have an asymmetrical shape described by a rapid acceleration as the wavefront passes (Figures 3b and 3c, t ∼ 24–26 and 55– 57 min), followed by a gradual deceleration as the wave slows and then flows offshore. The simple saw‐tooth temporal

velocity time series becomes more complicated under less‐ peaked waves. For example, during inundation of the second wave (Figure 1c, t ∼ 24–40 min), the broad peak results in a roughly constant onshore flow velocity (e.g., Figure 3b, t ∼ 38– 45 min). Similar patterns in the flow velocity are observed a few hundred meters seaward of the initial shoreline (Figure 3a). [21] During both the runup and backwash, depth‐averaged velocities can exceed 10 m/s (Figures 3a and 3b, t ∼ 55– 65 min). These predicted velocities are similar to survivor estimates from close to the open ocean in Banda Aceh (i.e., 8– 10 m/s) [Lavigne et al., 2009]. Furthermore, velocities (2– 5 m/s) estimated from video recordings taken approximately 3 km from the open ocean [Fritz et al., 2006] are similar in magnitude to the model predictions at x = 800 m (Figure 3c). While the waves that inundated at Kuala Meurisi and Banda Aceh were likely somewhat different, the similarities between the model predictions and observations suggests the flow velocities predicted here are reasonable for the 2004 Indian Ocean tsunami. [22] The details, including the magnitude and timing, of the saw‐tooth temporal velocity profile associated with the peaked waves are partially dictated by the slope of the topography and the distance from the shoreline. For example, the onshore flow decelerates more rapidly over steeper topographies (Figures 3b and 3c; compare dashed and dotted curves at t ∼ 55–62 min) owing to the relatively stronger gravitation force, resulting in shorter inundation distances (i.e., 1675 and 4775 m for the linear and long‐transect profiles, respectively). Conversely, for the profiles examined, maximum runup increases with increasing onshore slope (i.e., 16.6 and 7.2 m for the linear and long‐transect profiles, respectively). Estimates of the maximum runup and inundation distance for the other two profiles (i.e., measured

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Figure 3. Time series of the depth‐averaged velocity approximately (a) −500 m, (b) 20 m, and (c) 800 m from the shoreline for the measured (solid curves), linear (dashed curves), short‐transect (dash‐dotted curves), and long‐transect (dotted curves) profiles. Positive velocities are directed onshore, and positive locations are onshore of the shoreline. The dashed gray line is 0 velocity. and short transect) are not necessarily comparable as they are predominately dictated by the presence of the onshore cliff. It is, therefore, unclear if it is possible to extrapolate results concerning the runup and inundation distance from linear profiles to more complex onshore topographies. [23] Steep onshore topographic features, such as coastal bluffs or nearshore cliffs, can affect the onshore flow. The sharp flow deceleration predicted for the measured and short‐transect profiles following the second peaked wave (e.g., Figure 3c, solid and dash‐dotted curves at t ∼ 59– 62 min) is caused by wave reflections off the cliff face. Model simulations show that as the reflected wave propagates seaward, it interacts with the shoreward propagating part of the wave, causing the sharp deceleration observed in the depth‐averaged velocity. Under certain circumstances the reflected wave can enhance the offshore‐directed flow (Figure 3; compare dotted and dash‐dotted curves at t ∼ 62– 68 min). The effect of the reflected wave is only noticeable following the second peaked wave as reflections following the first two waves are not large enough to significantly affect the onshore propagating wave. [24] At a given cross‐shore location the maximum onshore‐ and offshore‐directed velocities can have very different magnitudes (e.g., Figure 3c). However, the maximum onshore‐ and offshore‐directed velocities (i.e., ∼10– 12 m/s) predicted over the entire model domain are much more similar (i.e., typically within 30%) (e.g., Figure 4). For the simulations conducted here, the maximum onshore‐ directed velocity typically occurs offshore of the original shoreline, while the maximum offshore‐directed velocity occurs near the seaward edge of the backwash. For the onshore‐directed flow (Figure 4, curves above 0), the near‐

bed velocities are reduced due to bed friction, while the velocities higher in the water column are similar, suggesting the flow is roughly depth‐uniform above the bottom boundary layer. For the offshore‐directed flow (Figure 4, curves below 0), the magnitude of the velocities gets progressively larger higher in the water column, consistent with an accelerating flow. While the maximum onshore‐ and offshore‐directed velocities are generally larger for steeper onshore slopes (compare the maximum onshore‐ and offshore‐directed velocities in Figures 4b and 4d), the largest offshore‐directed flow velocities predicted occur for the short‐transect profile, owing to the contribution of the reflected wave. [25] The flow velocity at a location generally decreases with increasing distance onshore of the shoreline (compare Figures 3b and 3c). However, the presence of local topographic variations can affect this pattern (Figure 5a). The velocity over a topographic low (Figure 5a, x ∼ 800 m) can be smaller than that over a more landward topographic high (Figure 5a, x ∼ 1000 m) if the change in water depth is large enough relative to the distance between the two locations. In fact, the presence of topographic variations can dominate the onshore distribution of flow velocities and can have a significant effect on the deposition of sediment (i.e., increased deposition in topographic lows, i.e., Figure 2b). [26] As the predicted onshore flow velocity varies in time (e.g., Figure 3), the onshore flow does not tend to zero everywhere at maximum inundation. Instead, water can begin to flow offshore near the shoreline before the wave reaches maximum inundation (Figures 5b and 5c; note the opposite flow directions at the wavefront and near the shoreline). While the internal structure of the resulting

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Figure 4. Maximum onshore‐ and offshore‐directed velocities at each cross‐shore location for the first (solid black curves), third (dashed black curves), fifth (dash‐dotted black curves), eighth (dotted black curves), and tenth (solid gray curves) sigma layers for the (a) measured, (b) linear, (c) short‐transect, and (d) long‐transect profiles. The first layer is the top layer of the water column, and the vertical dashed gray line is the initial shoreline. The small discontinuity in the velocities near the shoreline is owing to the change in bed roughness. divergent flow varies with the onshore topographic slope (Figures 5b and 5c), flow divergence is predicted even on completely horizontal topographies (not shown). On the measured and the short‐transect profiles the reflected wave drives the flow offshore before flow divergence occurs. Onshore flow divergence and the associated flow reversal may affect the depositional patterns of sediment, and is likely to affect to a greater extent finer sediment, which settles more slowly. While including nonhydrostatic effects, and thereby the effects of the curvature of the free surface, in the model will likely affect the internal structure of the diverging flow, such a phenomena has been observed in other inundation processes (i.e., swash zones; see section 5.3) and is therefore not an artifact of the model formulation. [27] While the magnitude of the maximum onshore‐ and offshore‐directed velocities are similar, the duration of the high‐velocity backwash is often longer (e.g., Figure 3) and the flow depth is generally smaller (e.g., Figure 6). Here, the backwash lasts 30%–60% longer for the broad wave and 100%–400% longer for the second peaked wave as compared to the preceding runup. However, when multiple waves are simulated, the backwash can end abruptly with the arrival of the next wave, and the duration of the backwash becomes a function of the time between consecutive wave crests. Due to the shallow trough between the first peaked wave and the broad wave, the runup of the first peaked wave lasts approximately 30% longer than the subsequent backwash. If only a single wave is simulated, the backwash can last longer relative to the preceding runup than predicted for any of the

waves simulated here. Generally, the backwash not only lasts longer, but sustains larger velocities for a larger percentage of the time. The duration of the large offshore‐directed velocities in the backwash is important for sediment suspension (see section 4.3), while the small flow depths can result pffiffiffiffiffi in supercritical flow (i.e., Froude numbers (Fr = U/ gd , where U is the depth‐averaged velocity, g is gravitational acceleration, and d is the water depth) between 1 and 2.5). Conversely, the onshore‐directed flow remains subcritical (Fr < 1) except near (i.e., within a few tens of meters) the leading edge of the inundating wave. [28] Owing to the steeper slope (i.e., on the three composite profiles) and the increased duration of the flow (i.e., on all four profiles), offshore‐directed flow velocities in the backwash are larger offshore than onshore of the initial shoreline (compare maximum negative velocities in Figures 3a and 3c; Figure 4). The maximum seaward extent of the backwash is larger for the three idealized profiles, which have a linear offshore slope, than for the measured profile (compare Figure 6a with Figures 6b and 6c) owing to its nonlinear offshore slope. [29] As the steep wavefront inundates over land, turbulent energy (a measure of internal mixing) is generated near the bed (Figure 7). This turbulent energy does not mix from the bed all the way to the surface of the wave and persists for only a few hundred meters behind the passing wavefront. This is consistent with field observations of rip‐up clasts and unbroken shells in onshore deposits [Gelfenbaum and Jaffe, 2003] that have been used to suggest turbulence and shear

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Figure 5. Instantaneous snapshots of the horizontal velocity during inundation of the second peaked wave for the (a) measured (t = 56.5 min), (b) linear (t = 60 min), and (c) long‐transect (t = 68 min) profiles. Positive velocities are onshore. The time shown for the measured profile corresponds to just before the second peaked wave reflects off the steep cliff, while the time shown for the other two profiles is approximately the maximum in flow divergence. The color contours are in 0.5 m/s intervals and are different in Figure 5a versus Figures 5b and 5c, owing to the different stages of inundation displayed. within the flow can be small [Shiki et al., 2008]. On linear topographies little turbulent energy is generated away from the wavefront during inundation, while turbulent energy can be generated by onshore topographic variations as the flow adjusts to changes in the local water depth. The generation of turbulence over actual onshore landscapes, which include vegetation, manmade structures and natural obstacles, may differ significantly from that generated over the idealized topographies used here. 4.3. Sediment Transport [30] In all the simulations conducted here, a significant amount of sediment is suspended and transported within the tsunami. Some sediment is suspended locally by the passing wavefront (Figure 8; note areas of high suspended sediment concentrations immediately behind the wavefront). However, as the turbulent energy generated by the wavefront persists for only a few hundred meters, this sediment settles quickly back to the bed. The higher concentrations of suspended sediment that can occur far behind the wavefront (Figure 8, right) are owing to advection of sediment suspended offshore during backwash (as explained below). For all four profiles, the amount of sediment transported in suspension is more than an order of magnitude larger than in bed load owing to the large velocities induced by the tsunami, with the amount of sediment transported as bed load relative to suspended load varying from 0.25% (100 mm) to 2% (900 mm).

[31] The majority of sediment suspended by the tsunami is eroded during the long‐duration, high‐velocity backwash that occurs offshore of the shoreline (Figure 9). This backwash generated erosion is consistent with previous analytical and numerical studies that used idealized tsunami waves [Simpson and Castelltort, 2006; Pritchard and Dickinson, 2008]. Unfortunately, detailed measurements of tsunami‐ induced offshore erosion may be difficult to obtain as most of the erosion occurs in the nearshore, including within the surfzone. Accurate determination of tsunami‐induced erosion requires detailed knowledge of the bathymetry both before and after the tsunami. However, the bathymetric profile of the nearshore often varies over the course of a year due to changing forcing conditions, and surfzone processes (i.e., waves and currents) can quickly smooth out any erosional scarps or accerational bars formed during a tsunami. For example, using observations following the 26 December 2004 tsunami, Morton et al. [2008] suggested that the sand transported offshore by the tsunami was moving back toward the beach as a low bar within two weeks of the tsunami. Therefore, to accurately record tsunami‐induced erosion, measurements will have to be taken just prior to and soon after the tsunami. [32] The erosive capacity of the backwash is a function of both the bed slope and the amount of water onshore as the water level recedes. As backwash flow velocities typically increase with increasing bed steepness and flow duration, the erosive backwash generated over the steeper bathymetry just offshore of the shoreline should not be confused with the

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Figure 6. Instantaneous snapshots of the horizontal velocity during the backwash preceding the second peaked wave (t ∼ 50.5 min) for the (a) measured, (b) linear, and (c) short‐transect profiles. Negative velocities are offshore. The small discontinuity in the bed levels near the shoreline is owing to the change in bed roughness. Results for the long‐transect profile are similar to Figure 6c. Color contours are in 1 m/s intervals, and velocities >1 m/s are not resolved to show better the variations in the offshore‐directed velocities.

Figure 7. Instantaneous snapshots of the turbulent energy during inundation of the second peaked wave (t ∼ 56 min) for the (a) measured, (b) linear, and (c) short‐transect profiles. Color contours are in 0.1 m2/s2 intervals. 9 of 20

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Figure 8. Instantaneous snapshots of the sediment concentrations during inundation of (left) the first peaked wave (t ∼ 27 min) and (right) the seconded peaked wave (t ∼ 56 min) for the (a and b) measured, (c and d) linear, and (e and f) short‐transect profiles. Results for the long‐transect profile are similar to Figures 8e and 8f. Color contours are in 10 kg/m3 intervals. Concentrations greater than 200 kg/m3 are not resolved.

Figure 9. Instantaneous snapshots of the sediment concentrations during the backwash preceding the second peaked wave (t ∼ 50.5 min) for the (a) measured, (b) linear, and (c) short‐transect profiles. The small discontinuity in the bed levels near the shoreline is owing to the change in bed roughness. Results for the long‐transect profile are similar to Figure 9c. Color contours are in 50 kg/m3 intervals. 10 of 20

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Figure 10. Cumulative erosion (negative values) and deposition (positive values) for the simulations using the measured (solid curve), linear (dashed curve), short‐transect (dash‐dotted curve), and long‐transect (dotted curve) profiles over the entire profile (main plot) and just onshore (inset). relatively low velocity backwash generated on land that erodes little or no sediment. Furthermore, when no water is present onshore (e.g., when a leading depression tsunami first reaches shore), the nearshore water level simply draws down as a fast receding tide, and no backwash and little erosion occur. On linear offshore slopes, the largest velocities and highest suspended sediment concentrations generally occur at the seaward edge of the backwash (Figures 6b, 6c, 9b, and 9c), where a water‐sediment slurry can form. On nonlinear offshore slopes the location of the velocity/concentration maximum may occur more shoreward (e.g., Figures 6a and 9a) depending on the local gradients in the slope. For all cases the largest amount of erosion occurs near the original shoreline (Figure 10). [33] When the sediment‐laden backwash collides with the onrush of the next wave, some of the suspended sediment is deposited locally as the flow abruptly changes direction. In the simulations conducted here, the sediment deposited by the backwash/onshore rushing wave collision forms a large, offshore bar near the seaward extent of the backwash (Figure 10), consistent with bathymetric surveys conducted off Sumatra following the 2004 tsunami in which a large offshore bar was observed [Gelfenbaum et al., 2007]. Although the sediment‐laden backwash collides with the onrush of the next wave in all simulations conducted here, under other circumstances (e.g., longer waves or steeper bed slopes) the duration of the backwash may be limited by the volume of water onshore, and no interactions between consecutive waves may occur. [34] Some of the sediment suspended by the backwash is subsequently advected shoreward by the onrush of the next wave (Figure 8, right; note areas of high suspended sedi-

ment concentrations away from the wavefront) and deposited on land (Figure 10, note areas of deposition shoreward of the shoreline; Figure 11). As no backwash occurs prior to the first peaked wave, no backwash‐suspended sediment is observed during the inundation of that wave (Figure 8, left). As the backwash‐suspended sediment is suspended offshore before being advected onshore, the suspended sediment concentrations in the inundating wave may not be in equilibrium with the local hydrodynamics. Furthermore, the large lateral gradients predicted in the suspended sediment concentrations within the onshore flow (e.g., Figure 8, right) may affect sediment deposition. For example, Apotsos et al. [2009] suggested that if sediment eroded by the wavefront is of a finer composition than that suspended by the backwash inverse grading can occur at the base of the deposit. [35] The sediment deposited onshore generally fines both landward and upward in the deposit (Figure 11), consistent with previous field studies [e.g., Moore et al., 2006; Paris et al., 2007], although this pattern is complicated by onshore topographic variations. These fining trends are predicted for simulations conducted both with erodable sediment available everywhere, as well as simulations where erodable sediment is only available offshore, and are consistent with the majority of sediment settling from suspension. The discontinuity in the predicted deposit at approximately 1700 m and 2000 m onshore for the measured and short‐transect profiles, respectively, is owing to the sharp deceleration of the onshore propagating wave caused by interactions with the reflected wave, consistent with previous studies [e.g., Higman et al., 2007]. [36] The dominant processes of sediment suspension predicted here, particularly the highly erosive offshore

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Figure 11. Composition of the deposit for the (a) measured, (b) linear, (c) short‐transect, and (d) long‐ transect profiles. The color contours are in 50 mm intervals.

Figure 12. (a–c) Instantaneous snapshots of the sediment concentrations during inundation of the second peaked wave (t ∼ 56 min) and (d) final deposit thicknesses for simulations using the short‐transect profile and including stratification and hindered settling (Figure 12a and solid black curve in Figure 12d), hindered settling only (Figure 12b and dashed curve in Figure 12d), and stratification only (Figure 12c and dash‐dotted curve in Figure 12d). Color contours in Figures 12a–12c are in 20 kg/m3 intervals, and dotted gray line in Figure 12d is zero deposit thickness. Concentrations greater than 200 kg/m3 are not resolved. 12 of 20

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Figure 13. Instantaneous snapshots of the horizontal velocity for the short‐transect profile (a–c) as the onrush of the second peaked wave collides with the preceding backwash and (d–f) during inundation of the second peaked wave excluding (Figures 13a and 13d and black curves in Figures 13c and 13f) and including (Figures 13b and 13e and gray curves in Figures 13c and 13f) suspended‐sediment‐induced density stratification. The cross‐shore locations of the black and gray curves in Figures 13c and 13f are marked in Figures 13a, 13b, 13d, and 13e with vertical dashed lines of the same color. The color contours are in 1 m/s intervals. backwash, result in suspended sediment‐induced density stratification and hindered settling. Suspended sediment‐ induced density stratification occurs when large amounts of sediment are suspended in the water column, displacing less dense water. When sediment is suspended from the bed, a stably stratified water column is created as the near‐bed water‐sediment mixture is denser than the surface water. This stable density stratification inhibits near‐bed turbulence and the mixing of sediment into the upper water column [Gelfenbaum and Smith, 1986; Conley et al., 2008]. [37] Neglecting the effect of the sediment on the fluid density (and thus the effect of suspended sediment‐induced density stratification) increases the amount and modifies the vertical profile of the suspended sediment in the inundating wave (compare Figures 12a and 12b), resulting in a thicker onshore deposit (Figure 12d, compare dashed and solid curves). Much of the extra sediment deposited onshore when sediment‐induced density stratification is neglected is suspended offshore by the backwash where suspended sediment‐ induced density stratification has the largest effect. Suspended sediment‐induced density stratification also affects the vertical profile of the horizontal flow velocity by preventing the faster moving surface water from mixing down into the lower water column [Gelfenbaum and Smith, 1986]. Just after the backwash collides with the onrush of the next wave (i.e., when the sediment concentrations in the water column are large) the vertical profile of the horizontal velocity differs significantly from that predicted under clear water conditions

(compare Figures 13a and 13b and black and gray curves in Figure 13c). However, after the wave has inundated a few hundred meters onshore, the suspended sediment concentrations have decreased and the effect on the flow is less noticeable (compare Figures 13d and 13e and black and gray curves in Figure 13f). [38] Hindered settling results when suspended sediment concentrations increase to the point where the settling velocity of a particle is reduced compared to what it would have been in a clear fluid. Previous studies have shown that at suspended sediment concentrations of 20% a particle’s settling velocity can be reduced by up to 40%, while higher concentrations can reduce the settling velocity by more than 90% [Baldock et al., 2004; van Rijn, 2007b]. Here, the settling velocity is reduced by more than 90% offshore of the shoreline during the strongest backwash (Figure 14a), but is less affected onshore of the shoreline (Figures 14b and 14c) and offshore of the extent of the backwash (not shown). The reduction in the settling velocity is typically larger near the bed than in the upper water column (Figure 14, compare solid and dash‐dotted curves). Neglecting the effects of hindered settling reduces the amount of sediment suspended in the water column during inundation (compare Figures 12c and 12a) as more sediment settles out offshore, and results in a thinner deposit near the shoreline (Figure 12d, compare solid and dash‐dotted curves). Neglecting the effects of hindered settling does not have a large effect on the lateral extent of sediment deposition because this extent is dictated by

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Figure 14. Percent reduction in the settling velocity owing to high sediment concentrations (a) −500 m, (b) 20 m, and (c) 800 m from the shoreline for the first (solid curves), third (dashed curves), eighth (dash‐ dotted curves), and tenth (dotted curves) vertical sigma layers for the simulation using the short‐transect profile. sediment settling from the upper water column, where suspended sediment concentrations are typically small.

5. Discussion 5.1. Stratification and Hindered Settling [39] The high suspended sediment concentrations predicted here and the resulting sediment‐induced density stratification and hindered settling can affect the flow and sediment transport during a tsunami in several ways. First, as the backwash accelerates offshore, sediment is selectively suspended, with coarser sediment being eroded later than finer sediments. Under these conditions, sediment‐induced density stratification may prevent coarse sediments, which will be suspended into an already stably stratified water column, from mixing all the way to the surface of the wave. Second, suspended sediment‐induced density stratification influences how sediment eroded during backwash is distributed vertically within the subsequent wave as the highest sediment concentrations typically occur just before the two water masses collide. Both of these effects may influence the vertical distribution of sediment within the inundating wave and thus the vertical and horizontal composition of the onshore sediment deposit, which may affect interpretations of paleo‐tsunami deposits. [40] Third, the effect high suspended sediment concentrations have on the flow suggests uncoupling the flow and sediment transport may not be appropriate when modeling tsunami‐induced sediment transport. Modeling the hydrodynamics and sediment transport processes separately may result in overprediction of both the suspended sediment concentrations and the near‐bed flow velocities. Finally, the accuracy of simple advective models [e.g., Moore et al., 2007; Soulsby et al., 2007; Smith et al., 2007] could be reduced if density stratification is strong enough to prevent the coarsest

grains from mixing into the upper part of the water column, or if hindered settling reduces the settling velocity of sediment over part of the inundation period. Quantifying the relative magnitude of these effects depends on a variety of site specific parameters, and is beyond the scope of this paper. 5.2. Multiple Waves and Composite Profiles [41] Many previous laboratory, numerical, and analytical studies of tsunami inundation and sediment transport examined solitary or solitary‐like waves propagating over linear bathymetries [e.g., Synolakis, 1987; Tadepalli and Synolakis, 1994; Tonkin et al., 2003; Pritchard and Dickinson, 2008]. The simulations conducted here expand on those studies by examining the effects of multiple waves inundating over more complex morphologies. [42] The use of multiple waves demonstrates that interactions between consecutive waves can be important to accurately predicting the onshore transport and deposition of sediment. For example, sediment suspension is dominated by the long‐duration, high‐velocity backwash. As a result, the amount of sediment advected shoreward by the first peaked wave, which has no preceding backwash, is significantly less than that advected shoreward by the second peaked wave. Owing to the nonperiodic nature of tsunamis, the importance of interactions between consecutive waves will vary with the specific characteristics of the individual waves. For example, even though the broad wave has a similar maximum wave height as the second peaked wave, the broad wave transports almost no sediment onshore owing, at least partly, to the short duration and low velocity of the preceding backwash. Interactions between multiple waves can also influence the predicted inundation distance, runup height, and flow depth as later waves often inundate over water remaining onshore from previous waves [Apotsos et al., 2009].

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[43] While many of the hydrodynamic and sediment transport processes identified in this study are consistent across the range of morphological profiles examined, some differences exist. Steeper onshore slopes cause flow reversal to occur more quickly resulting in shorter periods of inundation, more sediment being deposited near the shoreline (Figures 10 and 11), and more water draining offshore before the next wave arrives. Conversely, the predicted runup increases with increasing onshore slope. [44] Steeper onshore slopes can generate stronger return flows (Figure 3c, compare dashed and dotted curves). These stronger offshore‐directed return flows may lead to re‐erosion of sediment deposited during inundation, reducing the thickness of the onshore sediment deposit. Offshore of the shoreline, the erosion of sediment is a function of both the duration of the backwash and the magnitude of the flow. For the simulations conducted here, the initial shoreline is eroded further shoreward for profiles with shallower topographic slopes (i.e., ∼150–300 m on the composite profiles as compared to ∼50 m on the linear profile; Figures 10 and 11). This is likely owing both to the decreased duration of the backwash and to the greater amount of sediment that needs to be eroded to shift the shoreline landward on the steeper linear profile. Erosion near and offshore of the shoreline tends to be greater and more peaked on the composite profiles than on the linear profile, likely owing to the prolonged duration of higher velocities in the backwash. The increased erosion just offshore of the shoreline for the short‐transect profile as compared to the long‐transect profile (Figure 10; compare dash‐dotted and dotted curves at x = −1000 m to −100 m) is caused by the increased offshore velocities generated following the second peaked wave owing to the contribution of the reflected wave. [45] The overall onshore deposit thickness is generally similar for the three composite profiles, but is significantly larger near the shoreline on the linear profile (Figures 10 and 11). On the three idealized profiles, the deposit thickness tends to increase rapidly before decreasing gradually away from the shoreline. Conversely, on the measured profile the deposit thickness is strongly controlled by the onshore topographic variations, with thicker deposits in topographic lows and thinner deposits on topographic highs. The large cross‐shore variation in sediment thickness, which does not thin monotonically with distance from the shoreline, on uneven topographies has implications for interpreting the tsunami flow characteristics from the observed deposits. Similarly, if deposits in lows are preferentially preserved in the geologic record, these deposits may offer a biased record of historical tsunamis. [46] Sediment is deposited over a greater extent of the inundation distance on steeper slopes (95% and 70% for the linear and long‐transect profiles, respectively). This is likely because an inundating wave slows less quickly on shallower slopes, allowing sediment to settle out before the wave reaches maximum inundation. Conversely, the total volume of sediment deposited onshore is similar for the three idealized profiles, but approximately 50% less on the measured profile. The large reduction in onshore deposition on the measured profile is likely owing to the nonlinear offshore slope which results in a shorter backwash extent and a steeper slope directly offshore of the shoreline. The former reduces the offshore‐directed velocities (i.e., Figure 6) and thus the backwash‐suspended sediment concentrations (i.e.,

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Figure 9), while the later causes more sediment to settle out as the subsequent wave inundates toward the shoreline. 5.3. Inundation and Sediment Transport During Tsunamis and in Swash Zones [47] Many of the hydrodynamic and sediment transport processes described above are similar to analogous processes previously observed in the swash zone, the area of the beach near the shoreline that is intermittently covered and uncovered by water. The similarity between these two natural phenomena is likely owing to the fact that both involve bore‐like structures inundating over a dry, or drying, bed. While differences exist (see discussion below), the similarities may suggest a better understanding of the physical processes that occur during a tsunami can be developed using detailed observations from the swash zone. [48] Analogous hydrodynamic processes observed or predicted in the swash zone that are similar to processes predicted here include (1) the nonsymmetric, saw‐tooth shaped profile of the onshore flow [Kemp, 1975; Hibberd and Peregrine, 1979; Cox et al., 1994; Raubenheimer et al., 1995; Hughes et al., 1997; Puleo et al., 2000]; (2) runup being characterized by decelerating flow, while backwash is characterized by accelerating flow [Elfrink and Baldock, 2002]; (3) an onshore diverging lens of flow and a longer‐duration backwash [Hibberd and Peregrine, 1979; Raubenheimer and Guza, 1996; Baldock and Holmes, 1998]; and (4) large backwash‐ generated velocities that can become supercritical [Butt and Russell, 1999]. Similarities concerning the suspension and transport of sediment include (1) the high suspended sediment concentrations that occur during backwash [Hughes et al., 1997]; (2) offshore to onshore velocity transitions that can lead to subsequent onshore advection of sediment [Butt and Russell, 1999]; (3) sediment being maintained in suspension close to the inundating bore, but settling quickly out behind the wavefront [Butt et al., 2004]; and (4) strong lateral gradients in the sediment concentrations within the inundating wave [Elfrink and Baldock, 2002]. [49] This type of comparison can be helpful in accurately depicting the evolution of the onshore hydrodynamics, which are difficult to observe during tsunamis. This is especially important as several models developed recently to estimate tsunami flow characteristics from sediment deposits [e.g., Jaffe and Gelfenbaum, 2007; Smith et al., 2007, Soulsby et al., 2007; Moore et al., 2007] are based on different simplifying assumptions concerning the onshore flow. Furthermore, similarities between the backwash characteristics of the two phenomena may offer opportunities to better understand some of the processes that occur. Of particular interest is the manner in which sediment is eroded during backwash and the vertical mixing of sediment during the collision of the sediment‐laden backwash and the onrush of the next wave. [50] Significant differences also exist between the two phenomena. Most obviously, the time and length scales of the individual waves differ by several orders of magnitude. The larger flow depths that occur during tsunamis allow a large amount of sediment to be transported in suspension, while it is still a matter of debate whether sediment is primarily transported as bed or suspended load in the swash zone [Hughes et al., 1997; Pritchard and Hogg, 2005]. The advection of surf‐zone generated turbulence and wave‐ breaking suspended sediment in to the swash zone can

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dominate turbulence generation and suspended sediment concentrations during runup [Elfrink and Baldock, 2002]. There does not appear to be an analogous process for nonbreaking tsunamis. Finally, the total sediment transport in the swash zone typically occurs over time scales that are long compared to individual wave periods, whereas sediment transport occurs over only a few waves during a tsunami. 5.4. Study Simplifications and Limitations [51] The application of the model as described in this paper involves several simplifications that may limit the applicability of the results. First, the model domains used exclude the effects of alongshore topographic and bathymetric variations, as well as waves that approach the shoreline at an angle, such as trapped edge waves. Previous studies have shown that large‐scale features such as headlands and bays can affect incoming tsunami waves [Liew et al., 2008; Paris et al., 2009], and that small‐scale alongshore topographic variations can concentrate return flows [Gelfenbaum and Jaffe, 2003; Paris et al., 2009]. Furthermore, Fagherazzi and Du [2008] suggest the backwash onshore can cut approximately evenly spaced channels in the alongshore. The concentration of flow within these channels could affect the offshore erosion and suspension of sediment that occurs during backwash and is predicted here to dominate sediment suspension. However, simulations conducted by the authors using the same wave forcing but more complicated morphologies, including a three‐dimensional morphology measured at Jantang, Sumatra, suggest the general conclusions (i.e., general characteristics of the onshore flow velocity profile, the predominance of backwash suspended sediment, the importance of interactions between multiple waves, and the influence of sediment‐induced density stratification and hindered settling) presented here are not unique to the morphologies used. However, the introduction of alongshore nonuniformity can affect significantly the magnitude of the erosion and deposition of sediment, as alongshore features can deflect or focus flow creating areas of preferential deposition or erosion. [52] Second, only a limited range of some of the important initial conditions and model parameters was examined in this study. For example, a single offshore slope and wave forcing were used. Previous studies [Madsen and Fuhrman, 2008; Apotsos et al., 2009] have suggested that the impact of a tsunami varies with both parameters through the pffiffiffiffiffiffiffiffiffi Iribarren number (x = b/ H=L, where b is the slope, and H and L are the wave height and length), and further research is needed to better quantify this variation. Furthermore, only a single value of the bed roughness and sediment source composition were examined. The inclusion of vegetation [Gelfenbaum et al., 2007] and variations in the initial sediment source [Apotsos et al., 2009] will affect the model predictions. For example, dense vegetation along the coast could increase local resistance to erosion, while vegetation further inland could create zones of preferential deposition or modify the near‐bed turbulence structure. Furthermore, in many coastal environments, sediment is available only in certain areas. For example, on American Samoa, erodable, sandy sediment is often only available along narrow beaches (∼20 m wide). This limited supply of sediment has been suggested as the reason for the thin, patchy nature of the tsunami deposit observed fol-

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lowing the 29 September 2009 South Pacific tsunami [Apotsos et al., 2010b]. However, while variations in these parameters affect the relative magnitude of the predicted processes and details of the onshore deposit, the general conclusions drawn here are not unique to the values chosen [see Apotsos et al., 2009]. [53] Third, the model, as used, does not include all the relevant processes that likely occur during a tsunami. For example, assuming an approximate wave period of 16 min, a wave height of 8 m, and an offshore slope of 1/100 the offshore Iribarren number can be estimated as 4.3. Based on nearshore studies [Galvin, 1968; Battjes, 1974], surfzone waves with x = 4.3 are on the verge of breaking. If the tsunami waves broke before inundating, breaking‐wave‐ generated turbulence, which is not accounted for here, could have affected the patterns of both turbulence generation and sediment suspension. Furthermore, pore pressure gradient‐ induced sediment liquefaction, which can be important during drawdown [Tonkin et al., 2003; Young et al., 2009], and infiltration of water into the soil onshore are not accounted for. There may also be other processes not yet identified that play an important role during tsunamis owing to the large flow velocities generated. Similarly, the appropriateness of using empirical sediment transport models calibrated using laboratory and field data of fluvial, tidal, and short wave phenomena to model tsunami transport is unclear, and needs to be explored further. However, most of the general processes identified above have been observed previously in numerical or laboratory studies or in other inundation processes (see section 5.3) and the model captures well the general characteristics of the observations at Kuala Meurisi (see section 4.1). [54] Finally, forcing numerical simulations of long waves from relatively close to the shore (i.e., h ≤ −35 m) can produce unrealistic reflections off the artificial offshore boundary. To reduce these reflections, Delft3D allows a so‐ called zero‐order weakly reflective boundary condition to be created by adding the Riemann invariants to the prescribed offshore water level [Verboom and Slob, 1984; Stelling, 1984]. Simulations conducted here indicate that neither artificial offshore reflections nor the technique used to reduce these reflections affect significantly the general conclusions drawn in this study.

6. Conclusions [55] Delft3D, a three‐dimensional numerical model based on the nonlinear shallow water equations, was used to simulate tsunami inundation and sediment transport over one measured and three idealized morphological profiles for an event similar in magnitude to the 26 December 2004 Indian Ocean tsunami. This study is an initial step to identify and quantify the physical processes that dominate during tsunamis. [56] The model was first compared with observations recorded at Kuala Meurisi, Sumatra following the 2004 tsunami where the tsunami flow depth exceeded 15 m. The model predictions compare well with the observed flow depths and inundation distance, as well as with the general characteristics of the sediment erosion and deposition. However, owing to large uncertainties in the model setup, a purely quantitative analysis was not appropriate. Instead,

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model results were analyzed qualitatively to identify general trends in and the characteristics of the tsunami‐induced hydrodynamics and sediment transport. [57] For peaked waves the temporal variation in the onshore flow is well described by a saw‐tooth pattern, with the flow accelerating rapidly behind the passing wavefront and then gradually decelerating. However, for more complicated wave forms, the onshore velocity profile can deviate significantly from this pattern. The onshore flow does not tend to zero everywhere at maximum inundation, but instead flow reversal can occur near the shoreline even as the wavefront continues to inundate landward. Even though the magnitudes of the maximum onshore‐ and offshore‐directed velocities are similar, the backwash generally lasts longer than the preceding runup, has a smaller flow depth, and can result in supercritical flow. Onshore topographic variations can have a leading‐order effect on the cross‐shore distribution of velocities. [58] While some sediment is eroded by the wavefront as it inundates shoreward, nearshore sediment suspension is dominated by the long‐duration, high‐velocity backwash that occurs offshore of the shoreline. Some of this backwash‐suspended sediment is subsequently advected onshore, creating large lateral gradients in the suspended sediment concentrations that may not be in equilibrium with the local hydrodynamics. The resulting onshore sediment deposit generally fines landward and upward within the deposit, though this pattern is affected by onshore topographic variations. Suspended‐sediment‐ induced density stratification and hindered settling owing to high suspended sediment concentrations may be important to the onshore deposition of sediment, but the local importance of these processes decreases with increasing distance from both the nearshore and the bed. [59] Deviations from linear profiles and interactions of multiple waves can be important to tsunami inundation and sediment transport. However, the exact effect of these variations likely depends on a variety of site specific parameters. Finally, many of the hydrodynamic and sediment transport processes that occur during a tsunami are similar to analogous processes observed in the swash zone.

Appendix A: Sediment Transport Formulations A1.

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Suspended Sediment Transport

The eddy diffusivities are estimated as "s ¼ "f ;

where "f, the vertical mixing coefficient, is calculated from the k − " turbulence closure model and b is the “beta factor,” 

ws  ¼1þ2 u*

2 ;

ðA4Þ

in which u* is the bed shear velocity. Following van Rijn [1993], b is limited to 1 to 1.5. [61] Suspended sediment includes all sediment transported above the reference height, a, found from the current‐ related effective roughness, ks, and limited to 20% of the local water depth [van Rijn, 2007a]. The reference concentration, ca, at a is [van Rijn, 2007b] d50 ðTa Þ1:5 ca ¼ 0:015s  0:3 ; a D*

ðA5Þ

where rs is the specific density of the sediment; d50 is the median sediment diameter; D* is the nondimensional particle diameter,  1 ðs  1Þg 3 D* ¼ d50 ; 2

ðA6Þ

and Ta is the nondimensional bed shear stress, Ta ¼

uc b;c  cr : cr

ðA7Þ

The efficiency factor, uc, is uc ¼

fc′ ; fc

ðA8Þ

and g is the gravitation acceleration, s is the relative density of the sediment (taken here to be 2.65), and u is the kinematic viscosity coefficient of water. The gain related ( fc′ ) and total current related ( fc) friction factors are

[60] Suspended sediment transport is calculated in Delft3D by solving the advection‐diffusion equation,   @c @uc @vc @ ðw  ws Þc @ @c þ þ þ  "s; x @t @x @y @z @x @x     @ @c @ @c "s; y  "s; z ¼ 0;  @y @y @z @z

ðA3Þ

   12d 2 fc′ ¼ 0:24 log10 3d90

ðA9Þ

   12d 2 fc ¼ 0:24 log10 ; ks

ðA10Þ

and ðA1Þ

where c is the mass concentration of sediment; u, v, and w are the flow velocity components; "s, x, "s, y, and "s, z are the eddy diffusivities; and ws is the settling velocity. For 2DV simulations, where alongshore variations are neglected @ ¼ 0), (A1) reduces to (i.e., @y     @c @uc @ ðw  ws Þc @ @c @ @c þ þ  "s;x  "s;z ¼ 0: @t @x @z @x @x @z @z

where d is the water depth and d90 is the sediment passing size (taken here as 1.5 d50). The bed shear stress, t b, and the critical shear stress, t cr, are b ¼ w u2*

ðA11Þ

cr ¼ ðs  w Þgd50 cr ;

ðA12Þ

and

ðA2Þ

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where rw is the specific density of the water and

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cr ¼ 0:24D1 * ; f or 1 < D*  4;

ðA13Þ

The cumulative flux of sediment between the water column and the bed (i.e., the total erosion and deposition of suspended sediment) within each grid cell during each time step can be found from (A22) and (A23).

cr ¼ 0:14D0:64 * ; f or 4 < D*  10;

ðA14Þ

A3.

Bed Load Transport

[63] Bed load transport, Sb, is [van Rijn et al., 2004] 0:1

cr ¼ 0:04D* ; f or 10 < D*  20;

ðA15Þ

jSb j ¼ 0:006s ws d50 M 0:5 Me0:7 ;

cr ¼ 0:013D0:29 * ; f or 20 < D*  150;

ðA16Þ

where M and Me, the sediment mobility number and excess sediment mobility number, respectively, are

and cr ¼ 0:055; f or 150 < D* :

ðA17Þ

 c ¼ ca

 aðd  zÞ A : zðd  aÞ

ðA18Þ



ckmx

ðA19Þ

the exponent, A, can be found as   ckmx ln ca : A¼  aðd  zkmx Þ ln zkmx ðd  aÞ

ðA21Þ

c  c

@c a kmx ;  "s Dz @z

ðA22Þ

where a is a correction factor that accounts for the difference in the concentrations in the middle and at the bottom of the kmx layer and Dz is the difference in elevation between the kmx layer and a. The downward flux of sediment owing to settling is D ¼ ws c:

Me ¼

ðvr  vcr Þ2 ; ðs  1Þgd50

ðA26Þ

[64] The effect of sediment on the fluid mixture is estimated by extending the empirical relation of Eckart [1958]. The density of the fluid is adjusted to account for both the mass of sediment in the water column as well as the mass of water displaced by the sediment particles, making the density of the fluid mixture, rmix, mix ¼ w þ

 X  w : c 1 s

ðA27Þ

As the mixing length (i.e., the level of mixing within the flow) is a property of the flow and thus the fluid density, density stratification is inherently accounted for in the k − " turbulence model when the fluid density is estimated from (A27). The effect of sediment‐induced density stratification is excluded in the model by neglecting the mass of sediment in the density of the fluid (i.e., rmix = rw). A5. Settling Velocity, Including Effects of Hindered Settling

as E ¼ "s

ðA25Þ

in which vr is the magnitude of an equivalent depth‐averaged velocity computed from the velocity in the bottom computational layer assuming a logarithmic velocity profile and vcr is the critical depth‐averaged velocity for the initiation of motion based on the Shields curve. While the magnitude of the bed load transport is adjusted to account for bed‐slope effects (i.e., effects owing to a nonhorizontal bed) within the model, the details of how these effects are computed are not presented here owing to the relatively small contribution of bed load to the overall transport of sediment.

ðA20Þ

The vertical diffusion of sediment through the kmx layer can be found from the concentration gradient of the Rouse profile,     @c aðd  zÞ A1 ad ; ¼ Aca @z zðd  aÞ z2 ðd  aÞ

v2r ðs  1Þgd50

A4. Fluid Density, Including the Effects of Sediment‐Induced Density Stratification

With the concentration in the kmx layer given by  aðd  zkmx Þ A ¼ ca ; zkmx ðd  aÞ

M¼

and

A2. Sediment Fluxes Between the Water Column and the Bed [62] Sediment fluxes between the water column and the bed are calculated based on the transfer of sediment through the bottom of the near bed, or kmx, layer, taken here as the lowest sigma layer that is completely above the reference height. The suspended sediment concentration between a and the center of the kmx layer is assumed to follow a Rouse profile,

ðA24Þ

[65] The clear water settling velocity, ws,0, of suspended sediment particles is [van Rijn, 1993] ws;0 ¼

ws;0

ðA23Þ

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10 ¼ ds

ðs  1Þgds2 ; f or 65 m < ds < 100 m; 18

ðA28Þ

" 0:5 # 0:01 ðs  1Þgds3 1þ 1 ; 2

f or 100 m < ds < 1000 m;

ðA29Þ

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and ws;0 ¼ 1:1½ ðs  1Þgds 0:5 ; f or 1000 m < ds ;

ðA30Þ

where ds is a representative diameter (taken here as d50). For the model simulations conducted here, only (A29) is used as all grain sizes are between 100 mm and 1000 mm. The settling velocity, ws, is estimated in such a way to include hindered settling effects owing to high suspended sediment concentrations by following Richardson and Zaki [1954], ws ¼

  c tot 5 1 s ws;0 ; cd

ðA31Þ

where ctot s is the total mass concentration of sediment in the water column and cd is a reference density, taken here to be 1600 kg/m3. The effect of hindered settling is neglected by setting the reference density to an artificially large number such that cstot  0; cd

ðA32Þ

ws ¼ ws;0 :

ðA33Þ

giving

[66] Acknowledgments. This research was funded by a USGS Mendenhall Postdoctoral Fellowship and the USGS Coastal and Marine Geology Program. Edwin Elias is thanked for his help and guidance concerning the details of the numerical model. Deepak Vatvani is thanked for providing the offshore water level time series. Peter Ruggiero is thanked for helping collect the bathymetry at Kuala Meurisi. Critical reviews from Britt Raubenheimer and Mark Buckley, as well as three anonymous reviewers, significantly improved this manuscript.

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