On the effect of wind waves on offshore sand wave characteristics

Transcription

1 Marine and River Dune Dynamics April 28 - Leeds, United Kingdom On the effect of ind aves on offshore sand ave characteristics F. Van der Meer & S.M.J.H. Hulscher Water Engineering and Management, University of Tente, the Netherlands N. Dodd University of Nottingham, UK ABSTRACT: Offshore sand aves are bed patterns that occur in shallo seas. They sho both spatial and temporal variation in their characteristics such as their height (1-1m), length (1-8m), asymmetry and migration rate (1-1m/y). The reason for this variation is still not fully understood. Due to the different environmental processes that are incorporated in our present sand ave model, parts of this variation can be modelled and understood in more detail (Németh et al. 27, Van der Meer et al. 27). At present, one of the largest drabacks of our model is the overprediction of the sand ave heights, especially in shallo ater (around 1 meter ater depth). A possible explanation for this overprediction is the absence of ind and eather influences. Though up till no ind aves are mostly assumed to be negligible, no studies are available investigating their effect on the evolution of sand aves in detail. Especially under storm conditions and in relatively shallo ater ind aves could influence the sand aves. Therefore e implemented ind aves in our non linear sand ave model to investigate the influence of ind aves on the various phases of sand ave evolution. Results sho that though ind ave energy increases over the sand ave crests, the effect of ind aves on the sand ave shape and evolution is negligible. 1 INTRODUCTION 1.1 Sand aves Offshore sand aves are bed patterns that occur in shallo seas. The avelengths of these bed forms vary beteen 1 and 8 metres, and heights can reach up to one third of the ater depth (i.e. a maximum of around 1 metres in 3 metres of ater). These characteristics, together ith the fact that sand aves can migrate several metres per year and that they cover the majority of e.g. the Southern North Sea (Van der Veen et al. 2), mean that they affect human activities in shallo seas. Therefore, e aim to model and so better understand the dynamics of these sand aves. 1.2 Wind aves Offshore in shallo seas, under normal conditions short surface aves rarely interact ith the sea bed. As the ater depth is in the order of tens of metres, sediment is mainly transported as bed load ith the current. Hoever, during storms, ind aves are considered to act as a stirring mechanism for sediment, such that the sediment transported by the currents increase if ave energy increases. In this case sand aves might be affected by the ind aves and, in return, ind aves might be affected by the sand aves. This can happen due to a changing ater depth, but also due to a changing current. If e take into account that sand aves can significantly change both the local ater depth and ith that the current velocity (Hennings et al. 2), ind ave characteristics are expected to change over sand aves and possibly influence the sand ave shape. Field observations confirm this idea. For example Passchier & Kleinhans (2) concluded that the sand ave morphology close to the coast of the Netherlands is a function of the general ind ave climate and measurement of Harris (1989) already sho that sand aves close to Australia change their asymmetry, most likely as a response to the ind driven currents during the trade and monsoon season. 1.3 Goal In this paper e are interested in the possible influence of ind aves on sand ave evolution. Therefore, e model ind ave effects in a non linear 227

2 sand ave model, to investigate the influence of ind aves on the various phases of sand ave evolution. We ill start by describing the general model and the extensions made to include the ind aves in Section 2. In Section 3 the results ill be presented, folloed by discussion of the results and conclusions in Section 4 and respectively. 2 METHOD In this section e ill briefly describe the background of the model (section 2.1), for further details e refer to Van den Berg & Van Damme (26) and Van der Meer et al. (27). In section 2.2 the inclusion of ind ave effects ill be described. 2.1 Sand ave model The Sand Wave Code (SWC) used in this project is based on an idealized model by Németh et al. (26) and further developed by Van den Berg (26). It is a to dimensional vertical model, hich is developed specifically to describe sand ave evolution from its generation, to its fully gron state. For sand ave fields in the Southern part of the North Sea the SWC has shon good results in describing the avelength, height, shape characteristics and migration (Van den Berg 26, Németh et al. 27, and Németh et al. 26). Sand ave formation is explained as self organisation due to interaction beteen a sandy seabed and a tidal flo. Sand aves occur as free instabilities in this system, i.e. there is no direct relation beteen the scales related to the forcing (tide) and those related to the morphological feature (sand ave) (Dodd et al. 23). Sand ave occurrence can be understood only if the feedback mechanism beteen the forcing and the seabed is taken into account. Hulscher (1996) described this mechanism of self organization for sand aves, here residual vertical vortices play a crucial role. In short the process is as follos. Starting from a flat bed ith an oscillating current, small perturbations of the sea floor cause small perturbations in the flo field and vice versa. The bed can be either stable, hich means that all bed perturbations ill be damped or unstable, hich means that certain bed perturbations ill gro and the sea bed is changed. If perturbations are unstable the flo field is changed such that, averaged over the tidal cycle, small vertical residual circulation cells occur (Fig. 1). These cells cause small net transport to the crests of the perturbation, thereby causing groth. Depending on the circumstances such as flo velocity and ater depth, perturbations ith different lengths ill sho different groth/decay rates. The fastest groing mode is the perturbation hich triggers the fastest initial groth. For small amplitude perturbations, groth can be described as linear, though as sand aves gro larger, non-linear effects become important. Hoever, there are several indicators that seem to imply that sand aves are only eakly non-linear: their amplitude is generally smaller than 2% of the ater depth and the predicted fastest groing ave length (groth in height) is close to the observed ave length. As their slopes are gentle there is no flo separation. Assuming eak non-linearity, the initially dominating avelength ill be close to the one dominating in the non-linear regime. Subsequently, the fastest groing mode provides the dominant sand avelength, hich is found to be close to the dominating one in reality for eakly non linear systems (Dodd et al. 23). The SWC simulates this process from the initial disturbance to the final sand ave shape that is in equilibrium ith the flo. The SWC consists of the hydrostatic flo equations for 2DV flo. The tidal flo is modelled as a sinusoidal current prescribed by means of a forcing. Boundary conditions at the bed disallo flo perpendicular to the bottom. Further, a partial slip condition compensates for the constant eddy viscosity, hich is knon to overestimates the eddy viscosity near the bed. At the ater surface there is no friction and no flo through the surface. Since the flo changes over a timescale of hours and the morphology over a timescale of years, the bathymetry is assumed to be invariant ithin a single tidal cycle. The flo and the sea bed are coupled through the continuity of sediment. Note that here the bathymetry depends on time, in contrast ith the calculations for the flo. Only bed load transport is taken into account, folloing Komarova & Hulscher (2). For equations e refer to Van den Berg & Van Damme (26) and Van der Meer et al. (27). ater depth (m) Figure 1. Residual circulation averages over one tidal cycle. In this unstable case the cells, formed due to the interaction beteen the flo and the bed perturbation, ill lead to groth of the bed perturbation 228

3 Marine and River Dune Dynamics April 28 - Leeds, United Kingdom 2.2 Wind aves As sand aves occur in relatively deep ater ith respect to ind aves, the ind aves are expected not to break. For the cases shon in Section 3 this holds though in some other cases this as not valid. Waves can break and even be blocked and sept back in the relatively deep ater over the sand ave due to the current that is in the opposite direction for half of the tidal cycle (see Section 4). To implement the effect of surface aves e use the linear ave theory, i.e. monochromatic aves for hich the linear approximation holds (ak<<1, a/h<<1 and a/k 2 h 3 <<1, here a is amplitude, k avenumber and h local depth). We assume that the aves and 2DV current are collinear. The absolute frequency of the aves, ω, is assumed to be constant and the ave action, E/σ, is a conserved quantity (because there is no ave breaking considered in the model). Furthermore e assume that the currents ill influence the ave characteristics, hile the aves don t influence the currents (Mei 1999). With a given incoming ave period the ave number is calculated using Equation 1. Knoing k, e can find the ave energy per location over the sand ave ith Equation 2. Uk represents the Doppler effect. This holds especially for small ave periods and high current velocities. ω = Uk + d dx ( U + C ) gk tanh( kh) g E = σ ω 2kh C g = 1 + 2k sinh(2kh) 2π ω = T (1) (2) (3) (4) Here ω is the absolute ave frequency, U is the depth averaged current velocity, k the ave number of the surface aves, g the gravitational force, and h the ater depth. The ave energy is denoted by E, σ is the intrinsic ave frequency and C g and T the group velocity and the ave period of the surface aves respectively. With equation e can find the surface ave height H at different locations in time. With this ave height the shear stress at the bed due to the ind aves is defined using the ave orbital velocity (Equations 6 and 7). 1 2 E = ρgh 8 ωh u = 2sinh( kh) () (6) 229 τ f = 1 2 f u (7) (8) Here u is the flo velocity due to the surface aves just above the bed, τ is the bed shear stress due to u, f is the bed friction factor according to Soulsby (1997), a is the surface ave amplitude (equal to half of H ) and D is the medium grain size. Note that e use the volumetric bed shear stress. To combine both the current and the ave shear stress, e follo Soulsby (1997): τ τ t = τ c τ + τ c (9) Where τ t and τ c represent the total bed shear stress and the bed shear stress due to the current respectively. As can be seen in the equations, the linear approximations don't hold for aves that are blocked or sept back by the current, leading to ave breaking and other non-linear effects. This occurs especially for higher opposing currents and small ave periods (Peregrine 1976, Chala & Kirby 22). Cases in hich this happens are excluded from the results in Section 3 as they can not be described ith the used ind ave model. 3 RESULTS 2 a = D.2 The effect of ind aves is investigated and a sensitivity analysis is carried out for the influence of the current speed, the ater depth, the ind ave period and initial height, and storm periods. Table 1 shos the initial parameter settings for the simulations. Table 1. Parameter settings for the model runs unless stated different in the text. Parameter value dimension U 1. m/s A v.3 m 2 /s S.1 m/s λ h 1 m H ind 1. m T 7. s 3.2

4 position of sand ave crest and through (m) E(kg s 2 ) Figure 2. Sand ave groth (top) and shape (bottom) ith (left) and ithout (right) aves. Water depth is 1 m, averaged tidal current 1. m/s H ind (m) τ ind (m 2 s 2 ) x sand ave shape after years 3.1 Including ind aves Figure 2 shos the effect of ind aves on a sand ave in 1 m ater depth. When ind aves are included the ave length stays the same and also the shape is mostly identical beteen the case ith and ithout ind aves. Hoever, the sand ave height is loer if ind aves are included (22 m instead of 24 m). Note that this height is still far from realistic values for 1 m ater depth (in the order of m, Wilkens 1997). Figure 3 shos the ave energy over the sand ave during one tidal ave. The thick line indicates the average magnitude. The figure clearly shos the increase of ave energy over the top of the sand ave. Because of that the ind ave height increases ith.3 m at maximum tide, leading to an increase of the bed shear stress. For the maximum current in the direction of the aves, the ave height decreases by.1 m. This increase/decrease is due to the shalloer ater and the increasing opposite current. Figure 4 shos the individual effect of these to factors. Taking C g as a constant (second box) shos that the variation during the tide is smaller. The migration increases leading to a more asymmetrical shape of both the sand ave shape (not shon) and ave energy distribution. The maximum ave energy is still at the sand ave crest. Taking U as zero in Equation 2 results in a constant value over the tide, but different depending on the location (third box). No migration occurs in this situation. Though there is a clear increase in ave energy the effect on the sand ave form and evolution is small. Wave length and shape are unaffected and the ave height decreases ith approximately 1% Figure 3. Wave energy, height and bed shear stress over a sand ave. The envelope in values indicates the different values over one tidal cycle. E(kg s 2 ) E(kg s 2 ), C g const E(kg s 2 ), U const Figure 4. Wave energy over a sand ave, including both the influence of ater depth and current (top), and excluding ater depth (second) or current (third) effects. The envelope in values indicates the different values over one tidal cycle. Note that the sand ave shapes for the three situations are different, leading to a different location for the maximum ave energy (see text for more explanation). 23

5 Marine and River Dune Dynamics April 28 - Leeds, United Kingdom position of sand ave crest and through (m) Figure 6 shos the effect of different ind ave conditions, ranging from very mild eather (ind ave period of 4 seconds and ave height of. m) to extreme ind aves (period 12 s and height 2 m). Figure 6 shos that the influence on sand aves position of sand ave crest and through (m) Figure. Sand ave groth (top) and shape (bottom) ith (left) and ithout (right) aves. Water depth is 3 m, averaged tidal current 1. m/s Figure 6. Sand ave groth (top) and shape (bottom) ith a ave height and period of.m and 4 s (left), 1.m and 7 s (middle) and 2 m and 12 s (right) Figure 7. Sand ave groth (top) and shape (bottom) ith a ave height of m (left),.m (middle) and 7. m (right), all for a ater depth of 1 m and a depth averaged current of. m/s at maximal tide. 3.2 Sensitivity Figure shos the sand ave groth for a ater depth of 3 m, ith and ithout aves. Though different from the 1 m ater depth case, the ind aves don t play an important role in this difference. Again sand ave length stays the same hen ind aves are included, and the sand ave height is identical. The shape is slightly smoother ithout the ind aves, but not significantly different. When the tidal current speed is changed from 1. to. m/s (maximal current, depth averaged), e see an initial sloer groth of the sand ave, but the same final shape is reached after about 3 years (results are not shon). The influence of the ind aves is not increasing for a decreasing current Figure 8. Sand ave groth (top) and shape (bottom) ith a ave height of m (left),.m (middle) and 7. m (right), all for a ater depth of 3 m and a depth averaged current of. m/s at maximal tide. stays negligible even for the largest ind aves. No constant loering of the sand ave is observed, and only the shape seems to be changed and sloly deforms to a flatter and longer crest. Hoever, the final sand ave forms sho a clear migration in the direction of the ind aves, hich increases for increasing ind ave energy. 3.3 Storm periods In reality ind aves of long period and large height only occur in short periods during the year. Therefore simulations ere done, in hich the aves ere only included 1 eeks per year. This is still an overestimation of the storm season, but gives a good 231

6 indication of the effect and also keeps the fast computation time. The results are shon in Figures 7 and 8, for 1 and 3 m ater depth respectively, both ith a current of. m/s. The Figures sho different effects. First, for a ater depth of 1 m, the ind aves slightly increase the groth rate but don t change the final sand ave height. Second, in the 3 m ater depth case, the main difference is the difference in sand ave height, ere the cases ith ind aves sho loer sand aves and more instabilities. It seems that the periods ith ind aves destabilise the sand aves and loer the crest ith approximately %. Though difficult to see in the Figures 7 and 8, the ind aves cause migration of the sand aves in the direction of the ind aves. On average the migration is.2 and.3 m/y for the 3 m ater depth case, respectively for a ind ave height of and 7 m, and.2 m/y for both ind ave cases in 1 m ater depth. Note that both a ave height of and 7 m is unrealistic for a time period of 1 eeks, but are used here as an extreme research case. 4 DISCUSSION To test the influence of ave breaking Equation 2 as extended to Equation 1 ith D as defined in Equation 11, folloing Chala (22) as the ave breaking ill occur due to the opposing current instead of due to the ater depth. E + t D = 8 d dx β π ( U + C ) ρ g H 3 g E D = σ σ gk tanh kh (1) (11) In this, D is the energy dissipation rate per unit area, and β is a non dimensional parameter (.1). No changes ere observed in the simulations, larger than 1-2%. Still in some cases the included ind ave model as insufficient. This occurred hen ind aves ere blocked or even sept back by an opposing current. This as for example the case for a 3 m ater depth and a ave height and period of. m and 4 s (results not shon) and for the inclusion of a residual current (not shon). In both cases the currents ere strong enough to block the ind aves. In this cases the strong current is most likely overruling the effect of the relatively small ind aves on the sand aves. Hoever the used ind ave model is unable to simulate this non linear ind ave behaviour. Even for the largest ind aves tested, the effect on the sand ave evolution is small. The total bed shear stress is determined by the bed shear stress due to the current and the bed shear stress due to the ind aves (Eq. 9). Even for severe ind aves the ind ave influence is smaller than the current influence. When the ind ave bed shear stress as artificially increased by a factor 1, the result sho a tendency to decrease the sand ave height ith approximately 2% compared ith a case ithout ind aves. Here again the model as unable to find a final state. Though the effect on the sand ave shape is small, the ind aves are able to cause migration of the sand aves ith several decimetres per year in the case of storm and calm eather periods. This coincides ith observed sand ave migration during storms, though horizontal precision of the measurements is often less accurate than this (Németh 22, Dorst et al. 26 and references herein). The overestimation of the sand ave height seems not to be caused by the exclusion of ind aves. Another option is the effect of suspended load that is currently not taking into account. Including suspended sediment might also increase the effect of ind aves as sediment is transported over larger distances higher in the ater column, hich may add to the diffusive effects. CONCLUSIONS Modelling sand aves in shallo ater shos an overestimation of the sand ave height. In this study the possible influence of ind aves on the sand ave characteristics is investigated. Though the ind ave energy clearly increases over the sand ave crest, the sand ave height, length and shape is hardly affected by this increase in ave energy. The overestimation of the sand aves therefore must be caused by another factor, e.g. a combination of ind aves and suspended sediment transport. 6 ACKNOWLEDGEMENTS This research is supported by EncoraNL and the Technology Foundation STW, applied science division of NWO and the technology programme of the Ministry of Economic Affairs. Their support is gratefully acknoledged. 7 REFERENCES Chala, A. and Kirby, J. T. (22). Monochromatic and random ave breaking at blocking points. Journal of Geophysical Research, 17(C7, doi:1.129/21jc142). Dodd, N., Blondeaux, P., Calvete, D., de Sart, H., Falques, A., Hulscher, S. J. M. H., Rozynski, G., and Vittori, G. 232

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