Sediment resuspension from shoaling internal solitary waves. Daniel Bourgault, Institut des sciences de la mer de Rimouski, Rimouski, QC, Canada

Transcription

1 Generated using V3.0 of the official AMS LATEX template journal page layout FOR AUTHOR USE ONLY, NOT FOR SUBMISSION! Sediment resuspension from shoaling internal solitary waves Daniel Bourgault, Institut des sciences de la mer de Rimouski, Rimouski, QC, Canada ABSTRACT A preliminary two-dimensional numerical experiment on shoaling internal solitary waves (ISWs) and sediment resuspension is presented. The results are compared qualitatively with a schematic diagram presented in the literature. Both representations are comparable and suggests that shoaling ISWs can resuspend muddy sediment at the breaking depth causing nepheloid layers, up-slope sediment transport and downslope turbid gravity currents Introduction This preliminary numerical experiment was motivated by the hypothesis put forward by Morsilli and Pomar (2011) that shoaling internal solitary waves (ISWs) may cause hummocky cross-stratification (HCS). A hydrodynamic-sediment model is used to simulate sediment resuspension caused by shoaling ISWs. 2. Methods a. Hydrodynamic model The two-dimensionnal hydrodynamic, fully non-linear and nonhydrostatic model used is described in Bourgault and Kelley (2004). Briefly, the model solves with secondorder finite differences on a z-coordinate (i.e. stepwise topography) the following equations: (Bu) + (Bu2 ) + (Bwu) = B p t x z ρ 0 x + ( ) u Bν h + ( ) u Bν v, (1) (Bw) + (Buw) + (Bw2 ) = B p t x z ρ 0 z + ρ Bg + ( ) w Bν h + ( ) w Bν v, (2) ρ 0 (Bu) x + (Bw) z = 0, (3) (Bρ) + (Buρ) + (Bwρ) = t ( ) x z ρ Bκ h + ( ) ρ Bκ v, (4) where t is time, x and z are, respectively, the along-channel and vertical axes, u the horizontal velocity, w the vertical velocity (positive downward), ρ the water density, ρ 0 a reference density, p the pressure, B the channel width, ν h and ν v, respectively, the horizontal and vertical eddy viscosity and κ h and κ v, respectively, the horizontal and vertical eddy diffusivity. An equation for the sea-surface η is obtained by integrating the continuity equation (see Bourgault and Kelley 2004). The equations above slightly differ from those presented in Bourgault and Kelley (2004). The main difference is that the advective terms are now written in the flux form (e.g. (wu)/ z instead of w u/ z ). Horizontal eddy viscosity is parametereized using the Smagorinsky scheme described in Bourgault and Kelley (2004), i.e. { (Cs x) ν h = 2 2S 2 N 2 if 2S 2 > N m 2 s 1 (5) otherwise, where x is the horizontal grid size, S 2 is the square of laterally-averaged strain rate tensor (see appendix in Bourgault and Kelley (2004) for details), N is the buoyancy frequency and C s is the Smagorinsky coefficient. Similarly, the vertical eddy viscosity is parameterized as { (Cs z) ν v = 2 2S 2 N 2 if 2S 2 > N m 2 s 1 (6) otherwise, where z is the vertical grid size. Note that when z = x, ν v = ν h. Similar expressions are set for the eddy diffusivities κ h and κ v except that the minimum values if 2S 2 < N 2 is 10 7 m 2 s 1. b. Sediment model The sediment model implemented is taken from Wang et al. (2005). It consists in an equation for the advection- 1

2 diffusion of sediment concentration C, (BC) t + (BuC) + (B(w + w s)c) = ( x ) z C Bκ h + ( C Bκ v ), (7) where w s is the settling velocity. The following bottom boundary condition is used for resuspension and deposition κ v C z w sc = E b, (8) Table 1. Parameters used in the hydrodynamic/sediment model. Parameter value Sediment density ρ s 1100 kg m 3 Settling velocity w s m s 1 Erosion coefficient E kg m 2 s 1 Critical shear stress τ c 0.02 Pa Bottom roughness length z m Smagorinsky coef. C s 0.2 where E b is the bottom sediment flux defined as, { Results E0 (τ E b = b /τ c 1) if τ b > τ c (resuspension), 61 C b w s (1 τ b /τ c ) if τ b τ c (deposition), The model results suggest that, under conditions sim- 62 (9) ulated here, a shoaling ISW can resuspend and transport 63 E 0 the erosion coefficient, C b the sediment concentration in muddy sediment (Figure 1). The resuspension starts prin- 64 the bottom layer, τ c the critical shear stress for resuspen- cipally at the breaking depth (H b 18 m) or, equivalently, 65 sion and τ b is the bottom shear stress parameterized in the the bolus formation depth (between 100 < x < 200 m in 66 hydrodynamic model with a bottom roughness length z 0 Figure 1). There is comparatively very little resuspension 67 (Bourgault and Kelley 2004). This boundary condition as- during the shoaling process prior to breaking. 68 sumes an unlimited sediment availability for resuspension. Sediment concentration reaches approximately max(c) The water density is adjusted following 1 kg m The results suggests that boluses can trans- 70 port sediment above the pycnocline-bottom intersection. It ρ = ρ w + (1 ρ w /ρ s ) C, (10) 71 has not been determined yet whether the sediment found 72 within boluses come from direct local resuspension, caused where ρ w is the density of clear water and ρ s is the density 73 by the boluses themselves, or whether it was resuspend of sediment. 74 earlier and then transported up-slope by the boluses. 75 While a portion of the sediment resuspended seems to c. Model settings 76 be transported up-slope by the boluses, another portion 77 A simulation based on conditions similar to those prespreads offshore at intermediate depths, causing what looks 78 vailing in the St. Lawrence Estuary is set (Bourgault and like a series of nepheloid layers. Finally, at t = 22.5 min 79 Kelley 2003; Bourgault et al. 2005, 2007, 2008). The bathymetry there seems to be indications of a turbid gravity current 80 consists of a 2 km long offshore flat bottom with depth flowing down the slope. H = 50 m and a shoaling zone of slope s = The 81 width is constant, i.e. B = constant in which cases all B s 4. Discussion and conclusion cancel out in the equations above. 82 Figure 2 presents a larger view of the sediment dis- The background density of clear water is 83 tribution simulated here after breaking and some run-up ρ w = ρ 1 + ρ 2 [1 + tanh ((z h 1)/δ)], (11) 84 (t = 22.5 min). The figure qualitatively supports the 85 schematic diagram presented in Morsilli and Pomar (2011) 86 (their Figure 4). Many elements are common in both repwith ρ 1 = kg m 3, ρ = 4.5 kg m 3 87, h 1 = 10 m resentations, with sediment being resuspended near the 88 and δ = 3.5 m. breaking depth, indications of nepheloid layers, and tur- 89 An ISW is generated in the flat-bottom offshore region bid downslope currents. One element that the model here 90 of the domain as in Bourgault and Kelley (2004) and Bour- suggests which is not represented in the Morsilli and Po- 91 gault et al. (2005). For the simulation presented below the mar (2011) diagram is the up-slope sediment transport by 92 wave, has an amplitude a 0 = 5 m. Relative to the surface the boluses. 93 layer thickness h 1 = 10 m the wave is characterized with The model being 2D cannot represent 3D turbulence 94 the nonlinearity parameter α = a 0 /h 1 = 0.5. so some of these results should be taken with a grain of 95 Parameters used for the sediment model are provided in salt, especially near the breaking depth. However, Bour- 96 Table 1. The sediment parameters used in this simulation gault et al. (2007) showed rather good qualitative as well 97 are, acccording to Wang et al. (2005), typical of muddy as quantitative agreement between field observations and 98 beds found in estuaries. simulations of a shoaling ISW in the St. Lawrence Estuary 2

3 99 (see their Figure 2). While some details about the exact 100 mixing taking place cannot be realistically captured with 101 this 2D model, some larger scale features, like boluses for- 102 mation and evolution, are well reproduced. 103 Many other simulations, playing with different waves 104 and sediment parameters and background conditions could 105 be considered. Whether these wave-induced sediment pro- 106 cesses could explain HCS is beyond my expertise at this 107 point! REFERENCES Bourgault, D., M. Blokhina, R. Mirshak, and D. E. Kelley, 2007: Evolution of a shoaling internal solitary wavetrain. Geophys. Res. Let., 34, L03 601, doi: /2006gl Bourgault, D. and D. E. Kelley, 2003: Wave-induced boundary mixing in a partially mixed estuary. J. Mar. Res., 61 (5), Bourgault, D. and D. E. Kelley, 2004: A laterally averaged nonhydrostatic ocean model. J. Atmos. Oceanic Technol., 21 (12), Bourgault, D., D. E. Kelley, and P. S. Galbraith, 2005: Interfacial solitary wave run-up in the St. Lawrence Estuary. J. Mar. Res., 63 (6), Bourgault, D., D. E. Kelley, and P. S. Galbraith, 2008: Turbulence and boluses on an internal beach. J. Mar. Res., 66 (5), Morsilli, M. and L. Pomar, 2011: Insight into the origin of hummocky cross-stratification (HCS): the role of internal waves (IWs). Terra Nova, submitted. Wang, X. H., D. S. Byun, X. L. Wang, and Y. K. Cho, 2005: Modelling tidal currents in a sediment stratified idealized estuary. Continental Shelf Research, 25, , doi: /j.csr

4 Fig. 1. Results of the simulation zoomed around the pycnocline-bottom intersection. (left) Sediment concentration C (kg m 3 ) and (right) horizontal velocity (m s 1 ). Isopyncals are also superimposed on each field as gray isolines 0.5 kg m 3 apart. 4

5 Fig. 2. Zoom on sediment distribution (C in kg m 3 ) at t = 22.5 min (see Figure 1). This figure is to be compared with Figure 4 in Morsilli and Pomar (2011). 5