# Formulating Fine to Medium Sand Erosion for Suspended Sediment Transport Models

^{1}

^{2}

^{3}

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## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. 1DV Model

#### 2.1.1. Advection-Diffusion Equation

_{z}the vertical turbulent diffusion determined from the Prandtl mixing length formulation and W

_{s}the settling velocity.

_{d}are respectively the source and sink terms (erosion and deposition) at the bottom boundary. E and F

_{d}are described hereafter.

#### 2.1.2. Deposition Flux

_{s}is the settling velocity deduced from the sand grain characteristics following Soulsby [1]:

_{s}/ρ), ρ

_{s}the sediment density, ρ the water density and g the acceleration of gravity.

_{bot}is assumed to be similar to the concentration C

_{k}

_{= 1}modelled in the bottom layer (whose index k is 1) (Figure 1), which is reasonable when the settling velocity is small. When dealing with sand and bottom layer thicknesses in the range 0.1 m to a few metres, this assumption is no longer valid, and c

_{bot}has to be extrapolated from C

_{k}

_{= 1}. This can be achieved by coupling an analytical model for the concentration profile to the numerical model. Assuming a configuration near equilibrium conditions, an analytical Rouse-like profile can be used in the bottom boundary layer (e.g., [14,17,34]). However, the near-bed concentration generally becomes infinite when the distance from the bed to the location where it is computed tends to zero. Consequently, it is necessary to set a distance a

_{ref}from the bed where c

_{bot}is being calculated (Figure 1). The reference height is often considered to be twice the sediment diameter (e.g., [25,31]). In our case, a

_{ref}is chosen to be equal to the grain size roughness ${k}_{ss}=2.5D$ [35]. This corresponds to an equivalent Nikuradse roughness length for a flat sandy bed, considering that under this level, any concentration profile has probably no significance for suspended sediment transport.

_{bot}is dependent on the choice of the analytical models chosen for both velocity and concentration profiles in the boundary layer. Those analytical models are presented below.

**Figure 1.**Bottom boundary layer and coupling between the numerical and the analytical model. The principal notations defined within the text or the appendices are presented. OBL, outer boundary layer; IBL, internal boundary layer.

#### Bottom Boundary Layer Model and Near-Bed Concentration

_{0s}the skin roughness (z

_{0s}= k

_{ss}/30 deduced from the grain size; see Appendix A), z

_{0f}the total roughness (z

_{0f}= k

_{sf}/30 deduced from bedform geometry; see Appendix B) and κ the Von Karman constant (= 0.4). It can be noticed that this velocity profile is in agreement with the turbulence closure of the 1DV numerical model (mixing length), but would also have been compatible with other turbulence closures, as most of them are similar in the bottom boundary layer, provided the Reynolds number remains high.

_{0s}is assumed to only depend on the sediment size, while the form roughness is likely to change in response to the forcing, as the bedform itself depends on the forcing. A bedform predictor is used (Appendix B). When sheet flow is predicted (cf., Appendix B), IBL and OBL merge.

_{i}) by the form roughness k

_{sf}(Figure 1). When wave action is important for sediment motion, we assume that the wave boundary layer concerns the whole bottom layer in which the extrapolation is applied. Thus, following Soulsby [1], vertical mixing of sand induced by turbulent wave motion is accounted for by considering the maximum wave + current shear stress during a wave period in the Rouse number expression in both the IBL and the OBL.

_{i}) computed with Equation (8) for the sake of profile continuity and the Rouse number within the OBL defined as [1]:

_{bot}from the concentration C

_{k}

_{= 1}computed by the numerical model (Equation (1)) in its bottom layer (k = 1), it is assumed that this concentration is representative of the average concentration within the bottom layer of thickness δz

_{k}

_{= 1}. The average concentration obtained from the integration of the analytical concentration profile (Equations (8) and (10)) over the bottom layer should then match this representative concentration C

_{k}

_{= 1}(i.e., $\underset{{a}_{ref}}{\overset{\delta {z}_{k=1}}{\int}}}c(z)dz={C}_{k=1}\delta {z}_{k=1$). This leads to:

#### 2.1.3. Suspended Sediment Horizontal Flux

_{k}is calculated by the model in each layer following:

_{s}, U

_{k}, C

_{k}and δz

_{k}are respectively the sediment density, the velocity, the sediment concentration and the thickness of the layer k (Figure 1). Note that since the eddy diffusivity is based on the Prandtl mixing length formulation, velocity profile is logarithmic.

_{1}in Equation (15). A similar correction procedure has been previously proposed by Waeles et al. [34]. The transport rate thus needs to be corrected to compensate for the low vertical resolution.

_{k}

_{= 1}in the bottom layer (i.e., at z = δz

_{1}/2), ensuring the adequacy of the coupling between the numerical model and the analytical model.

#### 2.1.4. Erosion Fluxes

_{a}represents the so-called reference concentration.

_{d}, and the erosion flux can then be expressed as $E={W}_{s}{c}_{a}$. More generally, some authors propose to express the erosion flux following this latter equation even when equilibrium is not reached (e.g., [24,25,31]). When the erosion boundary condition is expressed with a reference concentration, attention has to be paid to which reference height it refers: naturally, it has to be the same as the one selected for the deposition term.

_{ref}= 2.5D) was changed to match the reference height corresponding to the erosion boundary condition (if prescribed).

#### 2.2. Transport Rate Validation Strategy

**Table 1.**Abbreviations used in this paper for transport models and data. Siam, simulation of multivariate advection; 1DV, one-dimension vertical.

Abbreviations | References |
---|---|

Erosion flux formulations used with Siam 1DV | |

Siam-ERODI | [23] |

Siam-VR84 | [21] |

Siam-EF76 | [25] |

Siam-ZF94 | [31] |

Current only transport formulation | |

EH67 | [37] |

Y73 | [38] |

VR84 | [39] |

Sediment transport formulations used by Davies et al. [19] | |

STP | [40] |

TKE | [41,42] |

BIJKER A | [43,44] |

SEDFLUX | [1,45,46] |

Dibajnia Watanabe | [47] |

TRANSPORT | [48] |

Bagnold-Bailard | [49,50] |

Sediment transport formulations used by Camenen and Larroudé [51] | |

Bijker | [52] |

Bailard | [50] |

van Rijn | [53] |

Dibajnia Watanabe | [47] |

Ribberink | [54] |

Sediment transport data | |

krammer | [55] |

scheldt | [55] |

alsalem | [56,57] |

janssen | [58] |

dibajnia | [47,59] |

#### 2.2.1. Current Only

#### 2.2.2. Wave and Current

_{s}> 2 m and wave period T > 7 s) or when the flow velocity reaches 2 m/s, the conditions were considered to correspond to the plane-bed (sheet-flow) regime (i.e., ${k}_{s}=2.5D=0.\text{625mm}$). In our study, we use the results of Davies et al. [19], which include two 1DV models (with high vertical resolution), the STP model [40] and a TKE model [41,42], as well as engineering formulations called BIJKER A [43,44], SEDFLUX [1,45,46], Dibajnia and Watanabe [47], TRANSPORT [48] and Bagnold-Bailard [49,50]. Like in Davies et al. (2002) [19], our model is tested and compared to other models for a set of five hydrodynamic configurations listed in Table 2.

**Table 2.**Configurations used for the comparison of sediment transport rates computed from various models [19]. Current velocities range between 0 and 2 m/s for each test. Water depth h is fixed (h = 5 m).

Forcing | H_{s} (m) | T (s) |
---|---|---|

Current only | 0 | |

Current + wave 1 | 0.5 | 5 |

Current + wave 2 | 1 | 6 |

Current + wave 3 | 2 | 7 |

Current + wave 4 | 3 | 8 |

_{s,num}) is considered to be acceptable when it approaches the experimental flux (q

_{s,data}) within a factor 2 (here, “a factor n” means between 1/n times and n times the measured value). For each model, the percentage of values predicted within a factor 2 (i.e., error lower than 50%) and within a factor 1.25 (i.e., error lower than 20%) is computed for “current only” data and “wave-current” data (indicated as Cc50, Cc80, Cw50 and Cw80, respectively). When the experimental flux is negative (current and wave are in opposite directions), the absolute value is shown on the graph if the predicted flux direction is correct. If not, the error is considered to be infinite.

## 3. Results

#### 3.1. Evaluation of the 1DV Model Transport Rates in the “Current Only” Case

#### 3.2. New Empirical Formulation of Erosion Flux

_{ref}= k

_{sf}= 2.5D and z

_{0}= k

_{sf}/30. The current velocity, the sediment concentration and the transport rate are computed for three grain sizes (100, 200 and 500 μm) and reported in Figure 3.

**Figure 2.**Sand transport rate as a function of sand diameter for a current velocity of 1.8 m/s. Water depth h is fixed (h = 7.2 m). Engineering models VR84, EH67 and Y73 are tested (black lines). Siam 1DV with different erosion flux formulations (Siam-VR84, Siam-EF76, Siam-ZF94, Siam-ERODI) is also tested (grey lines). Scheldt and krammer data with velocity ranging from 1.8 to 1.84 m/s and h ranging from 6.7 to 7.3 m are added (black circles).

**Figure 3.**Profiles obtained for three different sand sizes, when using a simple logarithmic velocity profile and a Rouse profile for sand concentration. Mean current velocity is 1.8 m/s and water depth h = 7.2 m. (

**a**) Current velocity; (

**b**) relative concentration profile (i.e., concentration normalized by the reference concentration c

_{bot}). The Rouse number R

_{sf}is indicated for each relative concentration profile. (

**c**) Relative transport rate (i.e., transport rate normalized by the reference concentration c

_{bot}). The depth integrated transport rate is indicated next to each profile.

_{bot}, the total transport rates for 100 μm and 200 μm sands are two orders of magnitude different (Figure 3). Further, there is only one order of magnitude between the transport rate for 200 μm and 500 μm. According to formulations EH67, VR84 and Y73, transport rates from 100 μm to 500 μm should be within the same order of magnitude (Figure 2). With this simple model, in order to balance the transport rate for the three different grain sizes, a large range of reference concentrations is required, especially between 100 and 200 μm. Since at equilibrium, the erosion flux is proportional to the settling speed, which increases with diameter, and to the reference concentration (E = W

_{s}c

_{bot}), a large dependency of the erosion flux with the diameter is needed. In the erosion flux formulation that we used previously (ERODI and VR84), there is less than one order of magnitude difference from 100 μm to 500 μm (Figure 4). This is likely to explain the discrepancies observed in the Siam 1DV model while using erosion flux formulation from the literature.

**Figure 4.**Erosion parameter E

_{0}as a function of sand diameter for several erosion flux formulations. The line labelled “this study” corresponds to Equations (19) and (20). (N.B.: the various orders of magnitude between different formulations are likely to be partly compensated by different powers of the non-dimensional excess shear stress).

_{1}= 18,586 kg·m

^{−3}·s

^{−1}and b

_{1}= −3.38 kg·m

^{−2}·s

^{−1},

_{2}= 59,658 kg·m

^{−3}·s

^{−1}and b

_{2}= −9.86 kg·m

^{−2}·s

^{−1}.

_{0}that depends on the sand diameter (Figure 4). The erosion factor E

_{0}increases from 0.03 to 21 kg·m

^{−2}·s

^{−1}for fine sands between 100 and 500 μm. The critical shear stress τ

_{ce}is computed from the classical critical Shields’ parameter for sand movement.

#### 3.3. Validation of the New Erosion Flux Formulation

**Figure 5.**Transport rate as a function of sand diameter for three current velocity configurations (grey areas). Engineering models VR84, EH67 and Y73 and the 1DV model (Siam) using the new erosion law (Equations (18)–(20)) are tested.

#### 3.3.1. Overall Performance

Model | Cc50: Current only | Cw50: Wave + Current |
---|---|---|

Bijker [52] | 66 | 18 |

Bailard [50] | 82 | 35 |

van Rijn [53] | 70 | 45 |

Dibajnia Watanabe [47] | 84 | 48 |

Ribberink [54] | 60 | 45 |

Siam 1DV + erosion law (this paper) | 67 | 37 |

**Figure 6.**(

**a**) Comparison of total sand transport rates between the Van Rijn [53] formula (q

_{s,num}) and experimental data (q

_{s,data}). From Camenen and Larroudé [51]. (

**b**) Comparison of total sand transport rates between Siam 1DV (q

_{s,num}) and experimental data (q

_{s,data}). See the explanations of the legend and symbols Cc and Cw in the Methods Section. Values predicted within a factor 2 sit in between the two dashed lines (q

_{s,num}= 0.5q

_{s,data}and q

_{s,num}= 2q

_{s,data}).

**Figure 7.**Comparison of sand transport rates as a function of sand diameter in the case of prevailing waves (

**a**); the case of wave and current (

**b**) and the case of current only (

**c**). Bold black curves, corresponding to Siam 1DV transport rates using the new erosion law (Equations (18)–(20)), are added to other models’ results (copied from Camenen and Larroudé [51]). U

_{c}is the mean current velocity; h the water depth; T

_{w}is the wave period; and U

_{w}is the wave orbital velocity at the bottom.

#### 3.3.2. Validation for Varying Grain Size

**Figure 8.**Comparison of sand transport rates for different wave and current conditions (cf., Table 2) (sand diameter 250 μm). Bold black curves, corresponding to Siam 1DV transport rates using the new erosion law (Equations (18)–(20)), are added to other models’ results (copied from Davies et al. [19]).

#### 3.3.3. Validation for Varying Hydrodynamic Conditions

#### 3.4. Relevance of Analytical Refinements in the Bottom Layer

_{d}= W

_{s}C

_{k}

_{= 1}). In this configuration, the sand fluxes in the case of current only are highly dependent on the vertical resolution (Figure 9). For two different resolutions of the bottom layer (20 cm and 5 m), the transport rates (grey lines) vary by about one order of magnitude. On the contrary, transport rates calculated with the model presented in this study are nearly independent of the resolution of the model (Figure 9). We can however note that some discontinuities, related to the appearance of the sheet-flow regime, are encountered. The change of roughness under sheet-flow and its impact on the suspended flux depend on the model resolution in our model.

**Figure 9.**Siam 1DV sand transport rates for current only with varying velocity (sand diameter 200 μm, water depth h = 10 m). Several configurations of the model are tested. The resolution of the bottom layer k = 1 is 20 cm for the solid line and 5 m for the dotted line. The results of the model presented in this study are in black. The results of the numerical model without analytical refinement in the bottom layer (i.e., f = 1 and F

_{d}= W

_{s}C

_{k}

_{= 1}) are in grey.

## 4. Discussion

- Bed load has not been accounted for in our model. For the experiment leading to Figure 2, adding the bed load in the model leads to the same conclusion. For instance, using VR84, which gives both suspended load and bed load, it appears that, whatever the velocity, the bed load only accounts for 6%–22% of the total transport from the finer to the coarser sand (following VR84, the ratio of the suspended load to the total load is independent of the velocity). Moreover, for the “current only” experiment (Figure 2), the ratio ${u}_{\ast}/{W}_{s}$ is always greater than one, which means that the suspended load is the predominant mode of transport [61]. The bed load is therefore not expected to explain the discrepancies observed when using classical erosion flux formulations, at least for this experiment.
- The 1DV model could suffer from weaknesses, mostly in the way the boundary layer is formulated. It is for instance not fully clear if the Rouse profile holds very close to the bottom. Different tests were conducted with varying parameterization (reference height, hindered settling velocity or the parameterization of vertical mixing were modified) and led to the same conclusion. A simple Rouse concentration profile (assuming a single bottom boundary layer) has also been tested, but did not enable us to match the total flux range given by engineering models. Further, we cannot take into account the impact of sediment concentration on settling velocity near the bottom in the model. This is due to the fact that we are using an analytical solution for the bottom layer of the model (a Rouse profile), which is only valid for constant settling velocity (with depth). We acknowledge that this could induce biases in the transport rates computed by the model.
- The erosion fluxes proposed in the literature could be inconsistent with our modelling strategy. We suggest here that this hypothesis is the most relevant one. Our approach was thus to search for an empirical erosion law able to produce results that match the horizontal flux range given by empirical engineering models.

_{0}with D is much more amplified than in other formulations (Figure 4). We acknowledge that this erosion flux formulation may be dependent on the conceptual model of the boundary layer that we adopted in this study. Transposing this erosion flux formulation to other studies would therefore require using the same boundary layer conceptual model. However, we suggest that the strong variation of E

_{0}with D is not specific to our model and that the same behaviour would be observed for any other comparable model. This aspect would require further investigations. We also suggest that the same validation approach should be carried out when using multi-class sediment transport models using advection-diffusion equations for sand. Further, due to the strong variation of E

_{0}with D and the discontinuity in the formulation around 180 μm, we acknowledge that some discontinuities in the transport rates can be produced (e.g., Figure 7c). This, however, should not be an issue for implementation in 3D models using a discrete form of E

_{0}(i.e., using only a few sand classes).

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix

#### A. Bottom Shear Stress Calculation

_{s}= k

_{sf}is the bedform roughness given in Appendix B.

_{w}is evaluated according to the Swart formulation [63]:

_{b}and A are respectively the orbital velocity and half excursion above the wave boundary layer, ($A=\frac{{U}_{b}T}{2\pi}$, with T the wave period).

_{m}represents the mean (wave-averaged) shear stress in the direction of the current and φ is the angle between current and wave directions. τ is the maximum shear stress generated during a wave period and is either used for the expression of the skin ${\tau}_{sf}$ or the hydraulic shear stress ${\tau}_{ff}$, depending on the roughness used (k

_{s}= k

_{ss}or k

_{sf}).

#### B. Bedform Predictor

_{sf}is predicted following Yalin [64]:

#### C. Erosion Flux Formulations

_{ce}the classical critical Shields’ parameter for sand movement.

#### D. Sand Transport Formulations

_{50}is the median grain size and D

_{90}is the 90th percentile of the grain size distribution.

#### E. Determination of the erosion flux formulation

_{0}and α parameters. We first started by determining α. To do so before E

_{0}is even determined, we worked on the ratio between the transport for various velocities and diameters q(U,D) and the transport q(U = 0.5,D) for U = 0.5 m/s. This ratio reads:

_{0}in the Siam 1DV model (cf. section 2.1), R does not depend on E

_{0}and is a function of α

_{ref}obtained from VR84, EH67 and Y73 (Figure A1a). We computed the Root Mean Square Error (RMSE) between R(U,D) and R

_{ref},(U,D) for $U\in [0.5,2]$m/s and $D\in [100,500]$μm for a range of α values (Figure A1c). We finally looked for α minimizing the RMSE, and retained α = 0.50 as the best value for the erosion flux formulation.

**Figure A1.**Ratio between the sand transport q(U,D) and the transport q(U = 0.5 m/s,D): (

**a**) is the average ratio for VR84, EH67 and Y73 and (

**b**) is the ratio for Siam 1DV with α = 0.5. (

**c**) Root Mean Square Error between R and R

_{ref}for different values of α.

_{0}. We found out that E

_{0}depend strongly on D. Therefore for each different diameter (ranging from 100 to 500 μm) we computed the E

_{0}value minimizing the RMSE between the Siam 1DV transport and the average transport obtained from VR84, EH67 and Y73 for current velocities ranging from 0 to 2 m/s. A curve of best fit has then been computed to present E

_{0}as a continuous function of D.

## References

- Soulsby, R.L. Dynamics of Marine Sands. A Manual for Practical Applications; Thomas Telford: London, UK, 1997; p. 249. [Google Scholar]
- Whitehouse, R.; Soulsby, R.L.; Roberts, W.; Mitchener, H. Dynamics of Estuarine Muds; Thomas Telford: London, UK, 2000; p. 210. [Google Scholar]
- Van Rijn, L.C. Sediment transport: Part I: Bed load transport. J. Hydraul. Ing. Proc. Am. Soc. Civ. Eng.
**1984**, 110, 1431–1456. [Google Scholar] [CrossRef] - Einstein, H. The Bed-Load Function for Sediment Transportation in Open Channel Flows; United States Department of Agriculture, Soil Conservation Services: Washington, DC, USA, 1950.
- Meyer-Peter, E.; Müller, R. Formulas for Bed-Load Transport. In Proceedings of the 2nd Meeting of the International Association for Hydraulic Structure Research, Stockholm, Sweden, 7–9 June 1948; pp. 39–64.
- Mitchener, H.; Torfs, H. Erosion of mud/sand mixtures. Coast. Eng.
**1996**, 29, 1–25. [Google Scholar] [CrossRef] - Panagiotopoulos, I.; Voulgaris, G.; Collins, M.B. The influence of clay on the threshold of movement of fine sandy beds. Coast. Eng.
**1997**, 32, 19–43. [Google Scholar] [CrossRef] - Migniot, C. Tassement et rhéologie des vases—Première partie. Houille Blanche
**1989**, 1, 11–29. [Google Scholar] [CrossRef] - Van Ledden, M.; Wang, Z.B. Sand-Mud Morphodynamics in an Estuary. In Proceedings of the 2nd Symposium on River, Coastal and Estuarine Morphodynamics, Obihiro, Japan, 10–14 September 2001; pp. 505–514.
- Chesher, T.J.; Ockenden, M. Numerical modelling of mud and sand mixture. In Cohesive Sediments; John Wiley & Sons: New York, NY, USA, 1997; pp. 395–406. [Google Scholar]
- Le Hir, P.; Cayocca, F.; Waeles, B. Dynamics of sand and mud mixtures: A multiprocess-based modelling strategy. Cont. Shelf Res.
**2011**, 31, S135–S149. [Google Scholar] [CrossRef] - Dufois, F.; Verney, R.; Le Hir, P.; Dumas, F.; Charmasson, S. Impact of winter storms on sediment erosion in the rhone river prodelta and fate of sediment in the gulf of lions (north western mediterranean sea). Cont. Shelf Res.
**2014**, 72, 57–72. [Google Scholar] [CrossRef] - Waeles, B.; Le Hir, P.; Lesueur, P. A 3D morphodynamic process-based modelling of a mixed sand/mud coastal environment: The seine estuary, France. Proc. Mar. Sci.
**2008**, 9, 477–498. [Google Scholar] - Sanford, L.P. Modeling a dynamically varying mixed sediment bed with erosion, deposition, bioturbation, consolidation, and armoring. Comput. Geosci.
**2008**, 34, 1263–1283. [Google Scholar] [CrossRef] - Van Ledden, M. A process-based sand-mud model. Proc. Mar. Sci.
**2002**, 5, 577–594. [Google Scholar] - Harris, C.K.; Wiberg, P.L. A two-dimensional, time-dependent model of suspended sediment transport and bed reworking for continental shelves. Comput. Geosci.
**2001**, 27, 675–690. [Google Scholar] [CrossRef] - Ulses, C.; Estournel, C.; Durrieu de Madron, X.; Palanques, A. Suspended sediment transport in the gulf of lion (NW mediterranean): Impact of extreme storms and floods. Cont. Shelf Res.
**2008**, 28, 2048–2070. [Google Scholar] [CrossRef] - Sherwood, C.R.; Book, J.W.; Carniel, S.; Cavaleri, L.; Chiggiato, J.; Das, H.; Doyle, J.D.; Harris, C.K.; Niedoroda, A.W.; Perkins, H.; et al. Sediment dynamics in the Adriatic sea investigated with coupled models. Oceanography
**2004**, 17, 58–69. [Google Scholar] [CrossRef] - Davies, A.G.; van Rijn, L.C.; Damgaard, J.S.; van de Graaff, J.; Ribberink, J.S. Intercomparison of research and practical sand transport models. Coast. Eng.
**2002**, 46, 1–23. [Google Scholar] [CrossRef] - Waeles, B. Modélisation Morphodynamique de L’embouchure de la Seine. Ph.D. Thesis, Université de Caen, Caen, France, 2005. [Google Scholar]
- Van Rijn, L.C. Sediment pick-up functions. J. Hydraul. Eng.
**1984**, 110, 1494–1502. [Google Scholar] [CrossRef] - Smith, J.D.; McLean, S.R. Spatially averaged flow over a wavy surface. J. Geophys. Res.
**1977**, 82, 1735–1746. [Google Scholar] [CrossRef] - Le Hir, P.; Cann, P.; Waeles, B.; Jestin, H.; Bassoulet, P. Erodibility of natural sediments: Experiments on sand/mud mixtures from laboratory and field erosion tests. Proc. Mar. Sci.
**2008**, 9, 137–153. [Google Scholar] - Van Rijn, L.C. Sediment transport: Part II: Suspended load transport. J. Hydraul. Ing. Proc. Am. Soc. Civ. Eng.
**1984**, 110, 1613–1641. [Google Scholar] [CrossRef] - Engelund, F.; Fredsøe, J. A sediment transport model for straight alluvial channels. Nord. Hydrol.
**1976**, 7, 293–306. [Google Scholar] - Nielsen, P. Coastal Bottom Boundary Layers and Sediment Transport; World Scientific: Singapore, Singapore, 1992; p. 324. [Google Scholar]
- Lesser, G.; Roelvink, J.; Van Kester, J.; Stelling, G. Development and validation of a three-dimensional morphological model. Coast. Eng.
**2004**, 51, 883–915. [Google Scholar] [CrossRef] - Papanicolaou, A.N.; Elhakeem, M.; Krallis, G.; Prakash, S.; Edinger, J. Sediment transport modeling review—Current and future developments. J. Hydraul. Eng.
**2008**, 134, 1–14. [Google Scholar] [CrossRef] - Villaret, C.; Hervouet, J.-M.; Kopmann, R.; Merkel, U.; Davies, A.G. Morphodynamic modeling using the telemac finite-element system. Comput. Geosci.
**2013**, 53, 105–113. [Google Scholar] [CrossRef] - Warner, J.C.; Sherwood, C.R.; Signell, R.P.; Harris, C.K.; Arango, H.G. Development of a three-dimensional, regional, coupled wave, current, and sediment-transport model. Comput. Geosci.
**2008**, 34, 1284–1306. [Google Scholar] [CrossRef] - Zyserman, J.A.; Fredsøe, J. Data analysis of bed concentration of suspended sediment. J. Hydraul. Eng.
**1994**, 120, 1021–1042. [Google Scholar] [CrossRef] - Le Hir, P.; Ficht, A.; Jacinto, R.S.; Lesueur, P.; Dupont, J.-P.; Lafite, R.; Brenon, I.; Thouvenin, B.; Cugier, P. Fine sediment transport and accumulations at the mouth of the seine estuary (France). Estuaries
**2001**, 24, 950–963. [Google Scholar] [CrossRef] - Brenon, I.; Le Hir, P. Modelling the turbidity maximum in the seine estuary (France): Identification of formation processes. Estuar. Coast. Shelf Sci.
**1999**, 49, 525–544. [Google Scholar] [CrossRef] - Waeles, B.; Le Hir, P.; Lesueur, P.; Delsinne, N. Modelling sand/mud transport and morphodynamics in the seine river mouth (France): An attempt using a process-based approach. Hydrobiologia
**2007**, 588, 69–82. [Google Scholar] [CrossRef] - Reed, C.W.; Niedoroda, A.W.; Swift, D.J. Modeling sediment entrainment and transport processes limited by bed armoring. Mar. Geol.
**1999**, 154, 143–154. [Google Scholar] [CrossRef] - Li, Z. Direct skin friction measurements and stress partitioning over movable sand ripples. J. Geophys. Res. Ocean. (1978–2012)
**1994**, 99, 791–799. [Google Scholar] - Engelund, F.; Hansen, A. A Monograph on Sediment Transport in Alluvial Streams; Verlag Technik: Copenhagen, Denmark, 1967; p. 62. [Google Scholar]
- Yang, C. Incipient motion and sediment transport. J. Hydraul. Div. Proc. Am. Soc. Civ. Eng.
**1973**, 99, 1679–1703. [Google Scholar] - Van Rijn, L.C. Sediment transport: Part III: Bed forms and alluvial roughness. J. Hydraul. Ing. Proc. Am. Soc. Civ. Eng.
**1984**, 110, 1733–1754. [Google Scholar] [CrossRef] - Fredsøe, J.; Andersen, O.H.; Silberg, S. Distribution of suspended sediment in large waves. J. Waterw. Port Coast. Ocean Eng.
**1985**, 111, 104–1059. [Google Scholar] [CrossRef] - Davies, A.G.; Villaret, C. Sand Transport by Waves and Currents: Predictions of Research and Engineering Models. In Proceedings of the 27th International Conference on Coastal Engineering, Sydney, Australia, ASCE, 16–21 July 2000; pp. 2481–2494.
- Davies, A.G.; Li, Z. Modelling sediment transport beneath regular symmetrical and asymmetrical waves above a plane bed. Cont. Shelf Res.
**1997**, 17, 555–582. [Google Scholar] [CrossRef] - Bijker, E.W. Longshore transport computations. J. Waterw. Harb. Coast. Eng. Div.
**1971**, 97, 687–701. [Google Scholar] - Bijker, E.W. Mechanics of Sediment Transport by the Combination of Waves and Current. In Proceedings of the 23rd International Conference on Coastal Engineering—Design and Reliability of Coastal Structures, Venice, Italy, ASCE, 1–3 October 1992; pp. 147–173.
- Damgaard, J.S.; Stripling, S.; Soulsby, R.L. Numerical Modelling of Coastal Shingle Transport; HR Wallingford: Wallingford, UK, 1996. [Google Scholar]
- Damgaard, J.S.; Hall, L.J.; Soulsby, R.L. General engineering sand transport model: Sedflux. In Sediment Transport Modelling in Marine Coastal Environments; van Rijn, L.C., Davies, A.G., van de Graaff, J., Ribberink, J.S., Eds.; Aqua Publications: Amsterdam, The Nederlands, 2001. [Google Scholar]
- Dibajnia, M.; Watanabe, A. Sheet Flow under Nonlinear Waves and Currents. In Proceedings of the 23rd International Conference on Coastal Engineering, Venice, Italy, ASCE, 1–3 October 1992; pp. 2015–2028.
- Van Rijn, L.C. General View on Sand Transport by Currents and Waves; Delft Hydraulics: Delft, The Netherlands, 2000. [Google Scholar]
- Bagnold, R.A. An approach to the sediment transport problem from general physics. U.S. Geol. Surv.
**1966**, 442, 37. [Google Scholar] - Bailard, J.A. An energetics total load sediment transport model for plane sloping beach. J. Geophys. Res.
**1981**, 86, 10938–10954. [Google Scholar] [CrossRef] - Camenen, B.; Larroudé, P. Comparison of sediment transport formulae for the coastal environment. Coast. Eng.
**2003**, 48, 111–132. [Google Scholar] [CrossRef] - Bijker, E.W. Littoral Drift as Function of Waves and Current. In Proceedings of the 11th International Conference on Coastal Engineering, London, UK, ASCE, September 1968; pp. 415–435.
- Van Rijn, L.C. Handbook Sediment Transport by Currents and Waves; Delft Hydraulics: Delft, The Netherlands, 1989. [Google Scholar]
- Ribberink, J. Bed-load transport for steady flows and unsteady oscillatory flows. Coast. Eng.
**1998**, 34, 59–82. [Google Scholar] [CrossRef] - Voogt, L.; van Rijn, L.C.; van den Berg, J.H. Sediment transport of fine sands at high velocities. J. Hydraul. Eng.
**1991**, 117, 869–890. [Google Scholar] [CrossRef] - Al Salem, A. Sediment Transport in Oscillatory Boundary Layers under Sheet Flow Conditions. Ph.D. Thesis, Delft Hydraulics, The Netherlands, 1993. [Google Scholar]
- Ribberink, J.; Al Salem, A. Sediment transport in oscillatory boundary layers in cases of rippled beds and sheet flow. J. Geophys. Res.
**1994**, 99, 707–727. [Google Scholar] [CrossRef] - Dohmen-Janssen, M. Grain Size Influence on Sediment Transport in Oscillatory Sheet Flow, Phase-Lags and Mobile-Bed Effects. PhD Thesis, Delft University of Technology, Delft, The Netherlands, 1999. [Google Scholar]
- Dibajnia, M. Sheet flow transport formula extended and applied to horizontal plane problems. Coast. Eng. Jpn.
**1995**, 38, 178–194. [Google Scholar] - Van Rijn, L.C. Principles of Sediment Transport in Rivers, Estuaries and Coastal Seas; Aqua Publications: Amsterdam, The Netherlands, 1993. [Google Scholar]
- Julien, P.Y. Erosion and Sedimentation, 2nd ed.; Cambridge University Press: Cambridge, UK, 1995. [Google Scholar]
- Villaret, C. Intercomparaison des Formules de Transport Solide: Etude de Fonctionnalités Supplémentaires du Logiciel Sisyphe; EDF: Chatou, France, 2003. [Google Scholar]
- Swart, D.H. Offshore Sediment Transport and Equilibrium Beach Profiles; Delft Hydraulics: Delft, The Netherlands, 1974. [Google Scholar]
- Yalin, M.S. Geometrical properties of sand waves. J. Hydraul. Div. Proc. Am. Soc. Civ. Eng.
**1964**, 90, 105–119. [Google Scholar]

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**MDPI and ACS Style**

Dufois, F.; Hir, P.L.
Formulating Fine to Medium Sand Erosion for Suspended Sediment Transport Models. *J. Mar. Sci. Eng.* **2015**, *3*, 906-934.
https://doi.org/10.3390/jmse3030906

**AMA Style**

Dufois F, Hir PL.
Formulating Fine to Medium Sand Erosion for Suspended Sediment Transport Models. *Journal of Marine Science and Engineering*. 2015; 3(3):906-934.
https://doi.org/10.3390/jmse3030906

**Chicago/Turabian Style**

Dufois, François, and Pierre Le Hir.
2015. "Formulating Fine to Medium Sand Erosion for Suspended Sediment Transport Models" *Journal of Marine Science and Engineering* 3, no. 3: 906-934.
https://doi.org/10.3390/jmse3030906