Dynamics of shoreface-connected and inactive sand ridges on a shelf, Part 2: The role of sea level rise and associated changes in shelf geometry

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 Dynamics of shoreface-connected and inactive sand ridges on a shelf, Part 2: The role of sea level rise and associated changes in shelf geometry A. Nnafiea, H.E. de Swarta, D. Calveteb, R. Garnierc aInstitute for Marine and Atmospheric research, Utrecht University, Princetonplein 5, 3584 CC Utrecht, The Netherlands bIDepartament de F´ısica Aplicada, Universitat Polit`ecnica de Catalunya, Campus Nord 08034 Barcelona, Spain cEnvironmental Hydraulics Institute (IH Cantabria), Universidad de Cantabria, PCTCAN, C/ Isabel Torres 15, 39011 Santander, Spain Abstract Many inner continental shelves are characterized by the presence of large 1 rhythmic bedforms, such as shoreface-connected ridges and the more offshore 2 located sand ridges, which have heights of several meters and are spaced sev3 eral kilometers apart. This study focuses on explaining the observed orienta4 tion difference between shoreface-connected sand ridges and the more offshore 5 located ridges. For this, an existing idealized morphodynamic model is used, 6 but modified such that sea level rise simultaneously induces a steepening of 7 the inner shelf and a retreating shoreface. Different settings (rate of sea level 8 rise; landward depth of the inner shelf) are systematically explored. For 9 each setting, the gross characteristics of ridges (growth rate, height, migra10 tion, orientation) during their initial formation and long-term evolution are 11 quantified. Model results show that a rising sea level and associated shoreface 12 retreat and shelf steepening lead to new ridges in the shallow area of the inner 13 shelf, which remain active in time (i.e. ongoing growth and downstream mi14 Email addresses: [email protected] (A. Nnafie ), [email protected] (H.E. de Swart), [email protected] (D. Calvete), [email protected] (R. Garnier) *Revised Manuscript with Changes Marked Click here to download Revised Manuscript with No changes marked: ManuScript_SLRPart2_Revision2_Final.pdfClick here to view linked References 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 gration in time). Old ridges that were already formed in the antecedent area15 of the shelf and which in the course of time experience deeper water, become16 less active with the rising sea level. In the case that migration of the offshore17 parts of the ridges vanishes, these parts change orientation to become more18 shore-parallel compared with the active onshore parts of these ridges. In the19 case of small landward depths of the inner shelf and a decreasing rate of20 sea level rise, the active onshore parts migrate too fast, thereby causing the21 drowned offshore parts to detach and to become inactive. The characteris-22 tics of modeled shore-oblique shoreface-connected and more parallel offshore23 located ridges agree with those of observed sand ridges.24 Keywords: Holocene, inner shelf, sand ridges, connected, inactive, detached, sea level rise, shoreface retreat, steepening 1. Introduction25 The rise of mean sea level (MSL) during the Holocene has had a profound26 impact on the evolution of continental shelves and shores of coastal seas. This27 is evident from studies (cf. Rampino and Sanders, 1980; Swift and Field, 1981;28 Stubblefield et al., 1984; Hapke et al., 2010; Schwab et al., 2013) in which29 coastal evolution was reconstructed from field data. As argued by e.g. Duane30 et al. (1972) and McBride and Moslow (1991), sea level rise has also had a31 major impact on the evolution of shoreface-connected sand ridges (hereafter32 abbreviated as sfcr). These large-scale rhythmic bedforms have heights of33 up to 12 m, they are spaced apart by 2-10 km, they have a shore-oblique34 orientation, they evolve on a timescale of centuries and they migrate several35 meters per year along the coast. Sfcr occur on continental shelves where36 2 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 storms generate waveand wind-induced currents (Duane et al., 1972; Swift 37 et al., 1978). The continental shelves of the the Mid Atlantic Bight are 38 examples where sfcr occur. Fig. 1 shows sfcr on one of these shelves, viz. the 39 shelf of Long Island off the coast of Fire Island. Field data (Swift et al., 1978; 40 Niedoroda et al., 1984) indicate that sfcr undergo an intermittent process of 41 development, which is associated with storm wave activity and storm-driven 42 currents. 43 Understanding the dynamics of sfcr is of high interest, because they mod44 ify wind-generated surface waves, thereby causing a complex wave pattern 45 that influences coastal sediment transport and related morphological changes 46 (Hayes and Nairn, 2004). Any morphodynamic change in these bedforms may 47 have large impact on beach erosion patterns. Also, recent studies hypothe48 size that these ridges may be an important source of sediment to maintain 49 beach stability (Hapke et al., 2010; Schwab et al., 2013). Moreover, because 50 of their proximity to the coast, the ridges are also considered as potential 51 locations for future offshore wind mill park (Barrie and Conway, 2013). 52 Many studies focused on gaining fundamental insight into the formation 53 and long-term evolution of sfcr (Dyer and Huntley, 1999; Hayes and Nairn, 54 2004). McClennen and McMaster (1971) proposed that they are relict fea55 tures from before the Holocene transgression and became submerged during 56 a period of sea level rise. Duane et al. (1972) and Swift et al. (1973, 1978) 57 concluded that the sand ridges evolve from a initial sand source as the latter 58 became submerged by the rising sea level and reworked by wave and currents. 59 As the coast retreated in response to sea level rise, the ridges experienced 60 larger water depth to become a field of isolated bedforms. McBride and 61 3 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 Moslow (1991) postulated that one of the initial sand sources is a segment62 of an ebb-tidal delta abandoned by inlet migration. However, these models63 did not explain the shore-oblique orientation and the migration rates of sfcr.64 The latter two aspects were explained by Trowbridge (1995), who anal-65 ysed a simple process-based model and showed that bedforms resembling sfcr66 can form as a result of positive feedbacks between a storm-driven longshore67 flow and the sandy bed. The underlying mechanism is that a storm-driven68 flow moving over an upcurrent-rotated ridge (seaward end of the crest is69 shifted upstream with respect to its landward end) is deflected seaward, as70 a result of mass conservation. The offshore flow component and the related71 sand transport decrease with increasing distance to the coast, because of the72 larger depths, thereby resulting in deposition of sand. Thus, a crucial factor73 in this model is the occurrence of a transverse bottom slope. The offshore74 veering of the current over the ridges is supported by field data (Swift et al.,75 1978; Warner et al., 2014). A drawback of this model is that it was not76 able to simulate the correct time scales related to growth and migration of77 these bedforms. Calvete et al. (2001) resolved this problem by including78 both bedload and suspended load sediment transport and by adding depth-79 dependent stirring of sediment by waves. More recent studies by Calvete and80 de Swart (2003), Vis-Star et al. (2008) and Nnafie et al. (2011) describe also81 the long-term evolution of sfcr towards finite heights.82 Although these models successfully describe many features of sfcr, they83 are not able to explain the fact that shoreface-connected sand ridges are84 in general more obliquely oriented with respect to the present shoreline,85 while the more offshore located sand ridges (sometimes also called shoreface-86 4 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 detached ridges, or ’drowned’ ridges (Snedden et al., 2011) or moribund ridges 87 (Goff and Duncan, 2012)) are more parallel to the shoreline (Fig. 1). 88 Nnafie et al. (2014) used an idealized model to study the impact of sea 89 level rise on the characteristics of sand ridges during their initial and long90 term evolution. Different scenarios (rates of sea level rise, geometry of inner 91 shelf) were examined. Their results showed that with increasing sea level the 92 height of sand ridges increases and their migration decreases until they even93 tually drown. Furthermore, their model indicates that if shoreface retreat due 94 to sea level rise is included, new ridges appear in the landward part of the 95 inner shelf that remain active in time. Old ridges that were already formed 96 in the antecedent part of the inner shelf, which gets located further offshore, 97 become less active and drown in the course of time. However, the latter 98 result was based on a rather simple scenario, in which geometrical parame99 ters of the inner shelf (slope, width and water depth) have their present-day 100 values, and the rate of sea level rise is 1 mm/yr. However, geological records 101 (as e.g. presented in Cowell et al., 2003; van Heteren et al., 2011) reveal 102 strong variations in width and steepness of the shelf and shoreface at time 103 scales of millennia. These variations result from processes like flooding by 104 sea level rise, sediment reworking by waves and tides and sand supply by 105 rivers. Hutton et al. (2013) demonstrated that sea level rise, in combination 106 with landward migration of the coastline, leads to shelf steepening due to a 107 seaward increasing water loading on the shelf in the newly created accom108 modation space. Variations in width and slope of the shelf will have a strong 109 impact on the evolution of sfcr, as Vis-Star et al. (2008) already demon110 strated that steeper bottom slopes result in larger growth rates and smaller 111 5 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 migration rates of ridges. Thus, when considering sea level rise, retreat and112 steepening of the inner shelf, new ridges that form on the landward side of113 the shelf will grow and migrate differently than ridges that formed on the114 antecedent part of the shelf.115 Of primary interest in the present work is the fundamental understanding116 of the observed orientation difference between the shoreface-connected ridges117 and the more offshore located ridges. The key hypothesis in this study is118 that observed orientation difference between the shoreface-connected ridges119 and the more offshore located ridges is the consequence of their differential120 migration rates caused by the rising sea level and the retreating shoreface. To121 test this hypothesis, runs are conducted with the numerical morphodynamic122 model used by Nnafie et al. (2014) (called MORFO56), but modified by123 implementing an equilibrium beach profile that allows for a combined effect124 of shoreface retreat and shelf steepening due to sea level rise. The Long-125 Island inner shelf is taken as a study area where both shoreface-connected as126 well as the more offshore located sand ridges are observed (Fig. 1). With this127 model, first, the impact of a retreating shoreface and a changing inner shelf128 geometry (increasing width and slope) on the characteristics (growth rate,129 height, migration, orientation) of the sand ridges is investigated. Next, the130 sensitivity of model results to different rates of sea level rise and to different131 values of the landward water depth of the shelf is examined.132 An overview of the model formulation, the setup of the model experiments133 and tools to analyze model output are given in Section 2. Results are pre-134 sented in Section 3, followed by a discussion (Section 4) and the conclusions135 (Section 5).136 6 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 2. Material and Methods 137 2.1. Model formulation 138 The model governs feedbacks between waves, currents and bottom evo139 lution on the inner and outer shelf. The inner shelf is the transition area 140 between the relatively steeply sloping nearshore zone and the more gently 141 sloping outer shelf (Fig. 2). A Cartesian coordinate system is used, with xa142 cross-shore, yan alongshore, and za vertical coordinate. The bed level z=zb143 and the sea level z=zsare defined with respect to a reference level z= 0, 144 which corresponds to the mean bottom level (i.e. averaged in the longshore 145 direction and over a hydrodynamic time scale in the order of days) of the 146 outer shelf. The reference bed level hzbiis defined as the mean value of zb.147 Perturbations of the bottom with respect to the reference bed level zb=hzbi148 are denoted as h(x, y, t), i.e. zb=hzbi+h. The sea level z=zs=hzsi+ξ,149 where hzsiindicates the mean sea level (i.e. averaged over a hydrodynamic 150 time scale in the order of days) and ξis the free surface elevation with respect 151 to z=hzsi. Furthermore, His a reference water depth: H=hzsi−hzbi, and 152 Dis the total water depth: D(x, y, t) = zs−zb.153 The nearshore zone (xc≤x≤xi, with xcthe position of the coastline and 154 xithe transition between nearshore zone and inner shelf, Fig. 2) is assumed 155 to have the equilibrium bottom profile hzbi=Hs−a(x−xc)2/3, with aa156 sediment scale parameter, which is of the order 0.1 m1 3for medium to fine 157 sand (Dean, 1987). The mean bed level hzbiof the entire coastal zone (i.e., 158 7 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 nearshore zone, inner shelf and outer shelf) is given by159 hzbi=              Hs−a(x−xc)2/3if xc≤x≤xi, β(xs−x) if xi≤x≤xs, 0 if x>xs. (1) Here, βis a constant slope given by β= (Hs−Hi)/(xs−xi), with Hiand160 Hsthe reference water depths at the transitions between nearshore zone and161 inner shelf (xi), and of the outer shelf (xs), respectively (Fig. 2). In Sec-162 tion 2.2, equations for variables Hs,Hi,xsand xiare given. The alongshore163 length of the domain is indicated by Ly. The cross-shore length of the do-164 main, which increases in time, is equal to Lx−xi(t), with Lxits length at165 t= 0. At location of x=Lxit is assumed that bed level zbequals its166 longshore averaged value hzbi.167 The wave model is based on linear wave theory (Holthuijsen, 2007). In168 this study only swell waves are considered. The state variables are the wave169 frequency ω(= 2π/T, with Tthe wave period), the wavenumber κ, the angle170 of wave incidence θ(see Fig. 2), and the root-mean-square wave height Hrms.171 From the solutions of the system of equations describing these variables, the172 root-mean-square amplitude of the near-bed wave orbital velocity urms is173 computed. Explicit formulations are given in the Electronic Supplement174 (Section A).175 Currents (called storm-driven currents) are described by depthand wave-176 averaged shallow water equations, which are forced by a combination of a177 longshore surface wind stress τw, an alongshore pressure gradient and diver-178 gence of the radiation stresses produced by waves. Note that the contribution179 8 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 to storm-driven flow due to divergence of the radiation stresses is in general 180 small. Only when wave breaking occurs, as in the case of a small landward 181 part of the inner shelf, this additional forcing becomes more important. The 182 unknown quantities are the mass transport velocity ~v (components u, v) and 183 the free surface elevation ξ.184 The evolution of the bottom is a result of convergences and divergences in 185 the sediment transport, which is assumed to exist only during storm events 186 (Calvete et al., 2001). During these events, sediment transport results from 187 the combined action of high waves, which stir sand at the bottom, and storm188 driven flow, which subsequently transport this sand as bedload (~qb) and sus189 pended load (~qs). In addition, a threshold near-bed wave orbital velocity uc190 for erosion is included to account for the fact that sediment transport oc191 curs only if the shear stress exerted on the bed exceeds a critical value. If 192 the near-bed wave orbital velocity urms is smaller than uc, sediment trans193 port vanishes. Explicit formulations for the above variables and the used 194 numerical scheme are given in the Electronic Supplement (Section A). 195 2.2. Shelf evolution in response to sea level rise 196 Studies of the Holocene evolution of the continental shelf of Long-Island 197 (Rampino and Sanders, 1980; Panageotou and Leatherman, 1986) suggest 198 that the shoreline migrated several kilometers landward as a result of shoreface 199 retreat induced by sea level rise. A paleo-reconstruction of the shelf evolution 200 on geological time scales is rather complex, viz. shelf and coastal morpholog201 ical changes result from the interaction of a complex array of processes and 202 mechanisms acting over a variety of temporal and spatial scales (Niedoroda 203 et al., 1995). In this study, the reconstruction method applied is highly ide204 9 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 appear, which have heights that increase with time (panel a). Panel e reveals339 that if rate R≥1.5 mm/yr, drowned ridges also occur in the course time.340 Note that the increasing height of the latter ridges is due to the fact that with341 increasing sea level the drowning area (i.e. area in which urms < uc) increas-342 ingly covers the upper part of the inner shelf. Furthermore, it is seen that343 increasing Rleads to a larger growth of the active ridges, it speeds up their344 formation (panel b) as well as their drowning (panels f and g) and it causes345 the magnitude of their migration rates to decline more rapidly with time346 (panel c). Interestingly, for all rates R, the times at which bottom perturba-347 tions start to grow (indicated by the solid grey line in panel b) correspond348 with a slope of ∼0.43 ×10−3. Furthermore, from Fig. 7d,h, it appears that349 the angle of orientation θDrowned of the drowned ridges is approximately 8o 350 smaller than that of the active ones. This difference implies that the drowned351 ridges are oriented more parallel to the coast compared with active ones. As352 an example, Fig. 8 presents snapshots of how the spatial distribution of bot-353 tom perturbations h(x, y, t) evolves in time in the case that sea level rises at a354 rate Rof 2.5 mm/yr. The orientation difference between active and drowned355 ridges can clearly be seen from panels c and d of this figure (indicated by356 dashed dotted lines).357 Results of experiment ’SensRate2’ (Table 1), in which Rvaries during358 the simulation period (R= 2.5 mm/yr for t < 5000 years, followed by359 R= 1 mm/yr for t≥5000 years ), are depicted in Fig. 9 (red lines). By360 way of comparison, results for two fixed rates, R= 1 mm/yr (black lines)361 and R= 2.5 mm/yr (blue lines) are included as well. Note that for t <362 5000 years, the blue and red lines are the same. Fig. 9a,b demonstrate that,363 16 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 after decreasing Rto 1 mm/yr at t= 5000 years, the increase in the height of 364 active and drowned ridges (red solid and dashed lines, respectively) reduces 365 compared with that in the case that Rwould remain constant after this 366 time period (i.e. R= 2.5 mm/yr for t≥5000 years, blue lines). Moreover, 367 panel c shows that in the former case, the reduction of migration rates of the 368 active ridges is weaker after t= 5000 years (red solid line) than that in the 369 latter case (blue solid line). Another noticeable difference between a fixed 370 an a mixed rate of sea level rise is that the angle of orientation of drowned 371 ridges decreases after decreasing rate Rat t= 5000 years (red dashed line in 372 panel d), thereby further increasing the orientation difference between these 373 ridges and the active ones. 374 3.2.2. Landward depth H0375 Sensitivity of model results with respect to smaller and larger landward 376 depths (indicated by depth H0at the transition x=xibetween nearshore377 zone inner shelf; experiment ’SensDepth’, Table 1) is investigated in this 378 section. Results in the case of R= 1 mm/yr are shown in Fig 10. Here, 379 variables hrms,σ,|Vm|and angle of orientation θare constructed in the 380 H0−tspace for both active (panels a-d) and drowned (panels e-h) ridges. 381 This figure reveals that an increasing depth H0leads to ridges with larger 382 heights (panel a), and it initiates their formation more rapidly (panels b). 383 Besides the more rapid formation of the ridges with increasing H0, the time 384 period of this formation lasts also longer. In the case of large H0(= 16 m, 385 18 m) the ridges drown in the course of time (Fig 10e). If the landward 386 depth is too small (H0<11 m, grey area), active ridges do not form. No 387 drowned ridges are observed if depth H0<16 m and R= 1 mm/yr. With 388 17 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 increasing depth H0, the magnitude of the migration rate Vmof active ridges389 declines more rapidly in the course of time (Fig 10c). Also, in the case390 of the formation of drowned ridges (H0≥16 m), an orientation difference391 ∆θbetween the latter (panel h) and the active (panel d) ridges is induced,392 which implies that the drowned ridges are orientated more parallel to the393 coast compared with the active ones.394 A new feature revealed by experiment ’SensDepth’ (Fig. 11) is that in395 the case of a landward depth H0of 11 m and using a mixed rate of sea level396 rise ([2.5→1] mm/yr), the drowned offshore parts detach from the active397 onshore ones at t∼8000 years (panel b) to form a series of inactive ridges398 on the shelf floor in the subsequent time period (panels c, d). From Fig. 12a,399 it is found that the active onshore parts experience larger migration rates in400 the case of a smaller landward depth (red solid line) compared with those in401 the case of a larger depth (blue solid line). In the end, the inactive ridges and402 the attached ones (sfcr) have an orientation difference of ∼200with respect403 to the coast Fig. 12b.404 3.2.3. Initial shelf width L0and longshore domain length Ly 405 Model results for a smaller initial width of the inner shelf (L0= 3 km, ex-406 periment ’SensWidth’, Table 1) are presented in the Electronic Supplement407 (Fig. E1). This figure shows a qualitatively similar behavior in the former408 case compared with that in the default case, i.e. active ridges appear on the409 inner shelf (panels a and b of Fig. E1) with heights that increase and migra-410 tion rates that decrease in time (panel c). The main quantitative differences411 between experiments ’Default’ and ’SensWidth’, are that in the latter case,412 the formation of the ridges is faster, their growth is stronger (red lines in413 18 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 panels a and b), the decline in the magnitude of their migration rates is 414 weaker (red lines in panel c), and the angle of their orientation with respect 415 to the coast is slightly higher (red lines in panel d) than those in the former 416 case. 417 With regard to the sensitivity of model results to different longshore 418 lengths of the domain (experiment SensLength’, Table 1), the results do not 419 fundamentally differ from those of the default case (Fig. E2 in the Electronic 420 Supplement). With increasing shelf length Ly, ridges have larger heights 421 (panel a), they emerge faster (panel b), their migration is smaller (panel c) 422 and the decrease in the angle of their orientation with time is smaller. 423 4. Discussion 424 An important result revealed by the model is that a drowned (i.e. van425 ishing growth and migration) offshore part of a ridge is aligned more parallel 426 to the coast compared with its active onshore part. Furthermore, in the case 427 of a small landward depth of the inner shelf and a decreasing rate of sea 428 level rise with time, an orientation difference between the active onshore and 429 drowned offshore parts of the ridges was found. Besides, the latter parts 430 also detach from the former ones to form a field of inactive bedforms. This 431 section addresses the physical mechanisms responsible for these properties. 432 4.1. Orientation difference and detachment 433 A conceptual model for the change in the orientation and detachment of 434 an offshore ridge is presented in Fig. 13. As the sea level rises, the near435 bed orbital velocity urms decreases. In the course of time, orbital velocity 436 urms will be below the critical velocity for erosion ucin an offshore part of 437 19 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 the inner shelf (grey area in Fig. 13) that extends towards the coast with438 increasing sea level. As the migration of the offshore part of the ridge in439 this area vanishes, this part of the ridge (dashed red lines) lags behind the440 more onshore part (solid red lines) of the ridge, which meanwhile keeps on441 migrating in the downstream direction. This process continues in the course442 of time at the location where urms =uc(Fig. 13, panels c,d).443 As was demonstrated by Nnafie et al. (2014), the migration of bedforms444 scales as Vm∼e U2 rms −u2 c/e Urms e H, with e Hand e Urms typical values of445 the width and depth of the inner shelf and of the near-bed wave orbital446 velocity, respectively. This velocity scale demonstrates that bedforms mi-447 grate faster in the case of smaller landward depths of the inner shelf (and448 consequently larger e Urms) than in the case of larger depths. Thus, in the449 former case, a larger differential migration rate exists between the drowned450 and active parts of the ridge than those in the latter case. If the differential451 migration rate is too large, as in the case of a landward depth H0of 11 m,452 the drowned offshore parts of the ridges can not maintain their attachment453 to the active onshore parts. Consequently, the former parts detach from the454 latter ones to become inactive sand ridges that are oriented more parallel455 to the coast compared with ridges that are still attached to the shoreface456 (sfcr). Such an event appears particularly in the case that the rate of sea457 level rise decreases in the course of time. This is because a smaller rate (and458 consequently a weaker steepening and a smaller widening of the inner shelf)459 reduces the decrease in the migration of the active onshore part of the ridge,460 thereby creating a larger differential migration rate between this part and461 the drowned offshore part.462 20 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 4.2. Shelf steepening 463 With respect to the preceding paper of Nnafie et al. (2014), where no 464 bending and detachment of ridges were found in the case that rate of sea 465 level rise is 1 mm/yr, the question rises whether the simultaneous sim466 ulation of the steepening of the shelf and the shoreface retreat is a ne467 cessity to reach the bending and detachment of the drowned ridges. For 468 this, additional experiments were conducted with the model of Nnafie et al. 469 (2014), which does not account for shelf steepening, for three different slopes 470 (β= [0.40; 0.75; 1.0] ×10−3) where the landward depth H0= 11 m and 471 R= [2.5→1] mm/yr were chosen. Bending of the offshore part of the ridge 472 occurred in all these experiments, but detachment was observed only in the 473 case of β= 0.75 ×10−3. These results indicate that steepening of the shelf, 474 albeit causing differential migration speeds of sfcr because of reasons pre475 sented in section 1, is not crucial for bending and detachment of the offshore 476 part of the ridges. 477 4.3. Comparison with field observations 478 Observed sfcr have an oblique orientation with respect to the coast (typ479 ical orientation ∼30 −50o, Schwab et al., 2013), whereas further seaward, 480 sand ridges are in general more coast-parallel (Swift and Freeland, 1978; Stub481 blefield et al., 1984). Furthermore, sand ridges are asymmetric, with steeper 482 (milder) seaward (landward) flanks. In general, with increasing depth, ridges 483 are higher and they become increasingly asymmetric (Swift and Field, 1981; 484 Stubblefield et al., 1984; Goff and Duncan, 2012). 485 In the case of a decreasing rate of sea level rise with time and a small 486 depth of the landward part of the inner shelf (∼11 m), the model simulates 487 21 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 a field of shore-oblique (θ∼40o) attached and more shore-parallel (θ∼20o)488 inactive sand ridges in shallow and deep waters, respectively. Fig. 14, which489 shows snapshots of the alongshore profiles of bottom perturbations hat two490 different cross-shore locations (x=−2 km, blue line; x= 5 km, red line)491 demonstrates that ridge heights are larger in the deep part of the inner shelf492 compared with those in the shallow part. Moreover, from this figure it is493 seen that ridges are steepest on their seaward flanks, and that their offshore494 parts have steeper seaward flanks than onshore ones. These model findings495 are in qualitative agreement with field observations. It should be stressed496 that this model is a gross simplification of reality (Section 4.4) and, thus,497 such a comparison is not straightforward. In reality, the geometry of the498 continental shelf such as that of Long Island and the observed patterns of499 the ridges are quite complex500 citepschwab2013,schwab2014. Many ridges are characterized by the presence501 of small-scale bottom features, with wavelengths in the order of a few hun-502 dreds meters. Furthermore, these ridges may be connected to a nearshore bar503 system (Schwab et al., 2000). This suggests that other mechanisms, besides504 those included in the model, may play a role. Additionally, the applied sce-505 narios of shelf evolution in response to rising sea level and retreating shoreface506 are highly idealized. In reality, shelf and coastal morphological changes re-507 sult from the interaction of a complex array of processes and mechanisms508 acting over a variety of time and space scales. Nevertheless, it is encouraging509 to see that this model, which yields comparable rates of shoreface retreat510 as those observed (order of meters per year, McBride and Moslow, 1991,511 Hapke et al., 2011, captures well some mean characteristics (such as height512 22 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 and asymmetry) of the observed shelf ridges. Moreover, the model is capable 513 of simulating shore-oblique shoreface-connected and more parallel offshore 514 located sand ridges. 515 4.4. Model simplifications 516 The present model is based on several assumptions. The major simpli517 fications are described below. Other simplifications are discussed in Nnafie 518 et al. (2014). 519 First, the model that describes the landward migration and steepening 520 of the inner shelf in response to rising sea level (Eq. 3) is highly idealized. 521 In reality, shelf and coastal morphological changes result from the interac522 tion of a complex array of processes and mechanisms acting over a variety of 523 temporal and spatial scales. The shelf slope is primarily a function of the an524 tecedent stratigraphy being eroded by the processes associated with marine 525 transgression. Sediment eroded from the shoreface during transgression does 526 not form a sheet of sand offshore, as is the case of the scenarios considered 527 in this study (Fig. 3). Furthermore, the definition of the offshore ridge orien528 tations is relative to the modern shoreline position, which may be different 529 than the shoreline position when those ridges were formed. Another aspect 530 to mention is that there are observations of cross-shore sediment transport 531 processes (e.g. offshore directed storm-driven undertow during stormy con532 ditions (Niedoroda et al., 1984; Hayes and Nairn, 2004)) that can transport 533 sediment from the nearshore zone to less active regions located far offshore. 534 This sediment transport will increase the morphodynamic activity of drowned 535 ridges that are located in the latter areas. 536 Secondly, effects of the changing seafloor on wave characteristics (wave537 23 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 topography feedbacks) are not accounted for in the present study. Vis-Star538 et al. (2007) and Nnafie et al. (2011), who used a wave transformation model539 that is based on linear wave theory (as in the present model), showed that540 the inclusion of these feedbacks causes enhanced wave stirring in the area541 upstream of the ridges due to focusing of wave rays in this area. As a con-542 sequence, the ridges grow and migrate too fast, which might be due to ne-543 glecting directional spreading of waves in these models. To account properly544 for these feedbacks, a spectral wave model, (e.g., SWAN,Holthuijsen, 2007)545 would be required.546 Thirdly, in observational studies it is stressed that storms are highly547 episodic events (Lentz et al., 2013). In this study the formation and evolu-548 tion of the ridges is investigated for continuous stormy conditions, where the549 underlying assumption is that morphodynamic activity of the ridges mainly550 takes place during storms, whereas during fair weather conditions it is as-551 sumed that there is hardly any activity. In reality wave asymmetry might552 also cause onshore sediment transport during fair weather conditions (Hayes553 and Nairn, 2004). Furthermore, storms will involve different wind intensities554 and directions. Within the context of the present study, a changing storm555 and wave climate would imply that the drowned offshore parts during mild556 stormy conditions will become active during severe storms. This would cause557 irregularities in the growth, migration and orientation of the ridges.558 5. Conclusions559 The main aim of this study was to understand the observed orientation560 difference between shoreface-connected sand ridges and the more offshore lo-561 24 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 cated ridges. For this, the model of Nnafie et al. (2014) was modified by 562 implementing an equilibrium beach profile equation that allows a simultane563 ous simulation of steepening of the inner shelf and shoreface retreat under a 564 rising sea level. Various scenarios were considered, in which different rates 565 of sea level rises and different landward depths of the inner shelf were used 566 to systematically explore their effects on the characteristics (growth rate, 567 height, migration, orientation) of sand ridges on an inner shelf. 568 For a model setting that resembles the Long Island inner shelf, results 569 show that coast-oblique ridges appear in the shallow part of the inner shelf, 570 which remain active in time (i.e. ongoing growth and migration). Ridges, 571 initially formed in shallow waters, become located progressively further sea572 ward due the retreating shoreface and the rising sea level. In the course of 573 time these ridges become decreasingly active until they eventually drown, 574 i.e. their growth and migration vanish. In addition, higher rates and larger 575 landward depths induce an orientation difference between the active onshore 576 and drowned offshore parts of the ridges, such that the latter parts are ori577 ented more parallel to the coast compared with the onshore ones. This is 578 due to the fact that the drowned offshore parts of the ridges lag behind the 579 more onshore ones, which meanwhile keep on migrating in the downstream 580 direction. If this differential migration rate is too large, as in the case of a 581 decreasing rate of sea level rise with time and a shallow landward part of 582 the inner shelf, the drowned offshore parts cannot maintain their attachment 583 to the active onshore parts (sfcr). As a result, the former parts detach and 584 strand on the shelf floor to become inactive. The simulated field of shore585 oblique shoreface-connected sand ridges and more parallel offshore located 586 25 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 Swift, D.J.P., Parker, G., Lanfredi, N.W., Perillo, G., Figge, K., 1978. 721 Shoreface-connected sand ridges on American and European shelves: a 722 comparison. Estuarine and Coastal Marine Science 7, 257–273.723 Trowbridge, J.H., 1995. A mechanism for the formation and maintenance724 of shore-oblique sand ridges on storm-dominated shelves. Journal of Geo-725 physical Research 100, 16071–16086.726 Vis-Star, N.C., de Swart, H.E., Calvete, D., 2007. Effect of wave-topography727 interactions on the formation of sand ridges on the shelf. Journal of Geo-728 physical Research 112, 1978–2012. doi:10.1029/2006JC003844.729 Vis-Star, N.C., de Swart, H.E., Calvete, D., 2008. Patch behaviour and730 predictability properties of modelled finite-amplitude sand ridges on the731 inner shelf. Nonlinear Processes in Geophysics 15, 943–955.732 Warner, J.C., List, J.H., Schwab, W.C., Voulgaris, G., Armstrong, B., Mar-733 shall, N., 2014. Inner-shelf circulation and sediment dynamics on a series734 of shoreface-connected ridges offshore of Fire Island, NY. Ocean Dynamics735 64, 1767–1781. doi:10.1007/s10236-014-0781-y.736 32 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 List of Figures 737 1 Bathymetric map of the Long Island continental shelf. Insert 738 on top left: large scale map. Active shore-oblique sand ridges 739 (sfcr) are located in shallow waters offshore of Fire Island. 740 Further seaward (indicated by the white 20m-isobath), more 741 shore-parallel sand ridges are observed. According to Goff 742 et al. (1999), beyond a depth of ∼20 m sand ridges become 743 less active. Map based on data from NOAA (2013). . . . . . 36 744 2 Schematic view of the study area. For an explanation of the 745 symbols see the text. . . . . . . . . . . . . . . . . . . . . . . . 37 746 3 Schematic view of the change of the coastal zone geometry 747 under sea level rise within a small time increment ∆t, under 748 the condition that sand volume in area A equals that in area B 749 (Vis-Star et al., 2008). The inner shelf migrates landward over 750 a distance ∆xi(= xi(t+∆t)−xi(t)) while its slope βincreases. 751 The location of the transition inner-outer shelf (xs), the depth 752 Hi(= H0) at the transition nearshore zone-inner shelf (x=xi)753 and the width w(= xi−xc) of the nearshore zone are kept fixed. 38 754 4 Rate of shoreface retreat dxi/dt (panel a) and bottom slope β755 (panel b) in the R−tspace in the case of present-day values 756 of the shelf (Section 2.3.1). . . . . . . . . . . . . . . . . . . . . 39 757 5 (a-d) Snapshots of the spatial distribution of bottom pertur758 bations h(x, y, t) (m) in the x-ydomain for the default case 759 (H0= 14 m and R= 1 mm/yr) at times t= 0 years (panel a), 760 t= 3000 years (panel b), t= 6000 years (panel c), and 761 t= 9000 years (panel d). Crests and troughs are indicated 762 by red and blue colors, respectively. Dashed black lines in763 dicate location of the initial transition nearshore zone-inner 764 shelf (xi= 0) and solid black lines denote that of the new 765 transition (xi). The part of the domain that is not yet part of 766 the inner shelf is indicated by grey. The black arrows in panel 767 (a) indicate directions of storm-driven current and waves. (e) 768 Cross-shore profile of mean bed level hzbiat t= 9000 years. . . 40769 33 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 6 (a) The root-mean-square height hrms of the sand ridges aver-770 aged over the entire inner shelf (i.e. xi≤x≤xs), which re-771 mains active during the full simulation time period of 10000 years.772 (b) As in (a), but for the global growth rate σ. (c) As in (a),773 but for the global migration rate Vm. (d) As in (a), but for the774 angle of orientation θof the ridges with respect to the coast.775 Variables Vm(panel c) and θ(panel d) are computed from the776 time (t∼3000 years, solid grey vertical lines) when bottom777 perturbations hstart to grow (σ > 0). In panel a, the dashed778 grey line indicates the time (∼9500 years) at which the slope779 βhas its present-day value (∼1.1×10−3)............ 41780 7 Left (a-d): Contour plots of (a) height hrms,Active, (b) growth781 rate σActive, (c) absolute value of migration rate Vm,Active and782 (d) angle of orientation θActive of the active ridges in the R−t783 space (experiment ’SensRate1’, Table 1). Right (e-h): As in784 the left panels, but for drowned ridges. The dashed grey line in785 panel a denotes the times at which the slope βhas its present-786 day value (∼1.1×10−3). In area ’A’, active ridges have not787 yet appeared (σ≤0). In area ’C’, drowned ridges are not788 observed yet (urms > uc, with uc= 0.35 m/yr), and in the789 grey area, they do not appear during the entire simulation790 period. In area ’B’, no solutions were obtained with the model. 42791 8 As in Fig. 5, but for a higher rate Rof 2.5 mm/yr. In panels792 c and d, the dashed dotted lines indicate the orientation of793 active and drowned ridges with respect to the coast. The794 dashed white lines mark the transition between the drowned795 and active part of the inner shelf (urms =uc). ......... 43796 9 As in Fig. 6 (black lines), but including results for a mixed rate797 (red lines; R= [2.5→1] mm/yr, i.e. R= 2.5 mm/yr for t <798 5000 years and R= 1 mm/yr for t≥5000 years, experiment799 ’SensRate2’, Table 1) and a fixed rate R= 2.5 mm/yr (blue800 lines). Solid and dashed lines represent active and drowned801 ridges, respectively. Note that for t < 5000 years, the blue802 and red lines are the same. . . . . . . . . . . . . . . . . . . . 44803 10 (a-h): As in Fig. 7, but in the H0−tspace (experiment ’Sens-804 Depth’, Table 1). In the grey areas in the left panels, active805 ridges do not form, and in those in the right panels ridges do806 not drown. Rate of sea level rise R= 1 mm/yr. . . . . . . . . 45 807 34 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 11 Snapshots of the spatial distribution of bottom perturbations 808 hin the case of a smaller depth and a mixed rate of sea level 809 rise (H0= 11 m and [2.5→1] mm/yr; part of experiment 810 ’SensDepth’) at times t= 7500 years (panel a), t= 8000 years 811 (panel b), t= 9000 years (panel c) and t= 10000 years (panel d). 46 812 12 Migration Vm(panel a, red lines) and angle θ(panel b, red 813 lines) in the case that landward depth H0= 11 m and using 814 a mixed rate of sea level rise (R= [2.5→1] mm/yr), versus 815 time. Solid and dashed lines represent active and drowned 816 ridges, respectively. For the sake of comparison, results in the 817 case that H0= 14 m and R= [2.5→1] mm/yr are plotted as 818 well(bluelines). ......................... 47 819 13 Schematic view illustrating the change in the orientation of the 820 offshore part (dashed red lines) of ridges under a rising sea level 821 and a retreating shoreface (the latter is not plotted for reasons 822 of clarity) at four successive times t1(panel a), t2(panel b), t3823 (panel c) and t4(panel d). Red and grey lines denote positions 824 of the ridge crest at present and at earlier times, respectively. 825 At t=t2, in the grey area, wave orbital velocity urms is below 826 critical velocity for erosion uc, and thus the migration of the 827 offshore part of the ridge in this area vanishes. Meanwhile, the 828 onshore part of the ridge outside this area keeps on migrating. 829 The result is that the offshore part rotates counter-clockwise. 830 At times t=t3and t=t4, the rising sea level increases the 831 grey area in the onshore direction, thereby causing the process 832 of differential migration rate between the parts of the ridge 833 inside and outside this area to repeat further up the inner shelf. 48 834 14 (a) Snapshots of alongshore profiles of hat x=−2 km (blue 835 line) and at x= 5 km (red line) at time t= 9000 years in the 836 defaultcase............................. 49837 35 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 Atlantic Ocean New York Long Island Fire Island Shore-oblique sand ridges More shoreparallel sand ridges (m) 4004′ 7 N 4004′ 2 N 4003′ 6 N 4003′ 0 N 4002′ 3 N 7304′ 7 W 7303′ 5 W 7302′ 4 W 7301′ 2 W 7300′ 0 W Figure 1: Bathymetric map of the Long Island continental shelf. Insert on top left: large scale map. Active shore-oblique sand ridges (sfcr) are located in shallow waters offshore of Fire Island. Further seaward (indicated by the white 20m-isobath), more shore-parallel sand ridges are observed. According to Goff et al. (1999), beyond a depth of ∼20 m sand ridges become less active. Map based on data from NOAA (2013). 36 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 outer shelf inner shelf Hs Hi nearshore zone storm-driven flow  v θ wave ray x=xs z y x x=Lx wind stress τ w H(x,t) D h zb ξ z=zs y=Ly z=zb z=zs x=xc x=xi y=0, z=0 Figure 2: Schematic view of the study area. For an explanation of the symbols see the text. 37 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 xc(t+Δt) xc(t) xi(t+Δt) xi(t) Hi(t)=H0 Hi(t+Δt)=H0 zbt zbt+Δt H(x,t) H(x,t+Δt) Hs(t) Hs(t+Δt) xs L(t) L(t+Δt) z=0 x z zst zst+Δt R d ′ t t t+Δt ∫ A" B inner shelf nearshore zone outer shelf Figure 3: Schematic view of the change of the coastal zone geometry under sea level rise within a small time increment ∆t, under the condition that sand volume in area A equals that in area B (Vis-Star et al., 2008). The inner shelf migrates landward over a distance ∆xi(= xi(t+ ∆t)−xi(t)) while its slope βincreases. The location of the transition innerouter shelf (xs), the depth Hi(= H0) at the transition nearshore zone-inner shelf (x=xi) and the width w(= xi−xc) of the nearshore zone are kept fixed. 38 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 Figure 4: Rate of shoreface retreat dxi/dt (panel a) and bottom slope β(panel b) in the R−tspace in the case of present-day values of the shelf (Section 2.3.1). 39 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 retreat Hi=14m xi xs retreat xi retreat (m) (m) (m) (m) v waves H0=14 m; R = 1 mm/yr Active Active Active Active Figure 5: (a-d) Snapshots of the spatial distribution of bottom perturbations h(x, y, t) (m) in the x-ydomain for the default case (H0= 14 m and R= 1 mm/yr) at times t= 0 years (panel a), t= 3000 years (panel b), t= 6000 years (panel c), and t= 9000 years (panel d). Crests and troughs are indicated by red and blue colors, respectively. Dashed black lines indicate location of the initial transition nearshore zone-inner shelf (xi= 0) and solid black lines denote that of the new transition (xi). The part of the domain that is not yet part of the inner shelf is indicated by grey. The black arrows in panel (a) indicate directions of storm-driven current and waves. (e) Cross-shore profile of mean bed level hzbiat t= 9000 years. 40 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 No growth (σ = 0) No growth (σ = 0) No growth (σ = 0) Figure 6: (a) The root-mean-square height hrms of the sand ridges averaged over the entire inner shelf (i.e. xi≤x≤xs), which remains active during the full simulation time period of 10000 years. (b) As in (a), but for the global growth rate σ. (c) As in (a), but for the global migration rate Vm. (d) As in (a), but for the angle of orientation θof the ridges with respect to the coast. Variables Vm(panel c) and θ(panel d) are computed from the time (t∼3000 years, solid grey vertical lines) when bottom perturbations hstart to grow (σ > 0). In panel a, the dashed grey line indicates the time (∼9500 years) at which the slope βhas its present-day value (∼1.1×10−3). 41 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 Active ridge urms ≤ucVm=0  Vm,Active storm-driven flow  V t1 t2>t1 t3>t2 t4>t3  Vm,Active Active Drowned Active Active  Vm,Active  Vm,Active y x urms ≤ucVm=0 Drowned urms ≤ucVm=0 Drowned Figure 13: Schematic view illustrating the change in the orientation of the offshore part (dashed red lines) of ridges under a rising sea level and a retreating shoreface (the latter is not plotted for reasons of clarity) at four successive times t1(panel a), t2(panel b), t3(panel c) and t4(panel d). Red and grey lines denote positions of the ridge crest at present and at earlier times, respectively. At t=t2, in the grey area, wave orbital velocity urms is below critical velocity for erosion uc, and thus the migration of the offshore part of the ridge in this area vanishes. Meanwhile, the onshore part of the ridge outside this area keeps on migrating. The result is that the offshore part rotates counter-clockwise. At times t=t3and t=t4, the rising sea level increases the grey area in the onshore direction, thereby causing the process of differential migration rate between the parts of the ridge inside and outside this area to repeat further up the inner shelf. 48 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 Figure 14: (a) Snapshots of alongshore profiles of hat x=−2 km (blue line) and at x= 5 km (red line) at time t= 9000 years in the default case. 49 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 List of Tables838 1 Experiments of sea level rise. Changes in parameter values are839 indicatedbyred. ......................... 51 840 50 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 Table 1: Experiments of sea level rise. Changes in parameter values are indicated by red. Experiment R(mm/yr) H0(m) L0(km) Ly(km) default 1 14 5.5 4.1 SensRate1 [0.5; 1.5; 2; 2.5; 3; 3.5; 4; 4.5; 5] 14 5.5 4.1 SensRate2 [2.5→1]: 14 5.5 4.1 2.5, for 0 <t<5000 years 1, for t > 5000 years SensDepth [1; 2.5; 5][2.5→1] [6,8,10,11,5.5 4.1 12,16,18] SensWidth 1 14 34.1 SensLength 1 14 5.5 [3,6,8] 51