Chapter : Modelling the derease in wave height over the shorefae due to slope-indued hanges in bottom frition. Abstrat Wave height-redution on the shorefae is partly indued by frition at the bottom. The bottom frition depends on the slope of the shorefae and on the seabed grain size and morphology. The slope of the shorefae profile from prograded shorefae deposits in a oastperpendiular ross setion is determined from isohrons. The slope of the shorefae inreased from.5º during initial progradation to.59º during final progradation. The wave-height redution from to 5 m below palaeo-mean sea level on the reonstruted shorefae profiles has been alulated. The alulations show that high, long-period waves are more dampened than low short period waves. On the gentle initial-prograded shorefae slope about 35% of the high waves remain, on the steep slope of the modern shorefae % of the high waves remain. The differene in wave-height redution indued by differenes in travel lenght over the shorefae slope an well explain observed variation in the shorefae deposits. Shorefae deposits from the initial prograded oast onsist predominantly of mud with thin wavelaminated storm sand and silt layers. The shorefae deposits indiate relatively quiet depositional onditions during initial progradation. Shorefae deposits of the final prograded oast onsist of dm parallel-laminated thik storm beds of sand. The shorefae deposits indiate high-energy depositional onditions during final progradation. 33
Chapter. Introdution Holoene oastal deposits in the Western Netherlands onsist of transgressive oastal deposits overlain by prograded wave-dominated deposits (Chapter ). The deposits of the prograded wave-dominated oastal system onsist of shorefae, breaker bar and beah deposits (Chapter ). Deposition prograded over tidal deposits of the transgressive oastal system. The Haarlem ross setion runs perpendiular to the oast from the initial prograded deposits to the modern shoreline (Figure 3.). In the ross setion a marked hange in the distribution and harater of the shorefae deposits from initial to final progradation is visible. The initial prograded shorefae deposits onsist of mud with thin storm sand layers of fine sand and silt. The storm layers usually ontain wave-ripple lamination. The shorefae deposits from the end of progradation onsist of thik storm beds of middle to fine sand, with sare thin mud layers. The storm sand beds usually ontain horizontal parallel lamination. The transition from predominantly mud with thin storm sands to predominately sand is gradual. The depth up to whih mud and thin storm sands are found inreases in the diretion of progradation and is assoiated with an inrease in the thikness and abundane of storm sand beds. Van Straaten (95) observed that the depth at whih mud layers were found in prograded shorefae deposits inrease seaward, similar to the trend in the Haarlem ross setion (figure 3.). This trend was also observed in the Wassenaar ross setion (Van Someren, 988, and Chapter 3, figure 3.). The hange in the harater of the deposits during oastal progradation an be regarded as the result of an inrease in wave energy during progradation. Initially mud was deposited under relatively quiet onditions, while thin storm sand layers were deposited during higher energeti events. During progradation the deposition of mud ontinued to our under relatively quiet onditions, but most of the mud was during storm events. During the storm events more and more sand was reworked into thik storm sand layers. Changes in the overall wave-limate of the North Sea during the Holoene have been modelled by Stive (987). The model outome showed that the wave limate hanged little during the last years. Changes in the overall wave limate an thus be ruled out as a possible ause of hange in the harater of shorefae deposits. Van Straaten (95) envisaged a bar on the lower shorefae during initial oastal progradation, that sheltered the middle and upper shorefae from intense wave ation. There is indeed evidene for the presene of suh bars during the earlier stages of oastal evolution (Van Someren, 988, and Chapter 3). Other possible mehanisms for the lower wave-energy during initial progradation are hanges indued by the shorefae profile itself, i.e., redution of wave height due to bottom frition. Two fators play a role in the redution of wave height on the shorefae profile: the slope of the shorefae and the frition fator, whih is a ombination of seabed morphology (rippled versus plane bed) and grain-size (Nielsen, 983, Van Rijn and Houwman, 999). In this hapter we test the assumption that the low-energy harater of shorefae deposits from the initial stage of progradation has resulted from wave-height redution due to the relatively large travel lenght imposed by a gentle shorefae slope. We do this by alulating the loss in wave height over shorefaes with different slopes using Nielsen s expliit wave formulae (Nielsen, 98, 983). We use geologial data to estimate the slopes of the shorefae from the start to the end of progradation. In addition we disuss the role of the frition fator, the influene of wave refration, the shorefae-slope development, the onsequenes for sediment transport and deposition. Previous explanations for the low-energy shorefae deposits are briefly disussed. 3
Wave-height redution Depth (m NAP) W -5 - -5.5º Non-deposition/Erosion surfae Modern ( BP).59º 3 BP 3 BP 35 BP 9 BP 5 BP BP.3º Unit Transgressive deposits 35 BP Unit 3 Prograded deposits Haarlem ross setion 37 BP BP BP Unit Transgressive deposits Unit Pleistoene - -. 3...... 3.. 5.. 7. Distane to shoreline (km) Shoreline.5º BP E -5 - -5 Figure.: The Haarlem ross setion with shemati stratigraphy; shaded grey marks the preprogradation deposits that form the substrate for progradation. For information on the stratigraphi units to 3 see Chapter 3 and. In the prograded shorefae deposits the isohrons, based on AMS C dates on single shells are indiated in grey (see Chapter 3, figure 3). The blak lines indiate the reonstruted shorefae profiles. The slopes of the profiles are indiated as well, as is the sea level from that period. The isohron of 35 oinides with the reonstruted shorefae profile... Hypothesis The transition from low-energy shorefae deposition of thin sandy storm layers in mud to high-energy shorefae deposition of thik sandy storm beds without mud results from an inrease in wave-energy during oastal progradation. We assume that the inrease in wave ation during progradation of the oastal system is related to the observed inrease in slope of the lower shorefae during progradation. A gentle slope of the shorefae means that the distane over whih waves feel the influene of the bottom is larger and thus that the redution of wave height due to frition at the seafloor is large. On a steep slope the distane over whih the influene of the bottom is felt is muh smaller and hene, the redution of the wave height is more limited. The hange in the shorefae slope is indued by the nearly level substrate for progradation, by the rise of sea-level, and by the progradation itself.. Methods.. Profile reonstrution The Haarlem ross setion (Chapter 3 and, figure 3. and.) is used to reonstrut the shorefae profiles during progradation. We reonstrut the shorefae profile during initial progradation over the top of transgressive oastal deposits (Figure.). The transition of transgression to progradation ourred around 7 to BP. The age of the prograded deposits has been determined with AMS C dating of single shells and was used to onstrut isohrons (Chapter 3). We have used the isohrons from the prograded shorefae, breaker bar and beah deposits to reonstrut the slope of the shorefae. The upper boundary of the shorefae profile is delimited by the sea level of that time (from Jelgersma, 979). The lower boundary is given by the top of the underlying transgressive deposits. In the westernmost half of the ross setion this an nondepositional/erosional surfae. The shorefae profiles with their slopes, whih were used for the alulations are indiated in figure.. We have used the isohrons of and 35 BP that streth from the lower to the upper boundary. The small nik in the BP isohron was smoothed. The 3 BP timeline was onstruted slightly different, beause the atual isohron only overs the lowest part of the ross setion. For the upper boundary of the 3 35
Chapter BP ross setion a position between the isohrons of 9 and 5 BP was interpolated. The modern shorefae profile was averaged over its steepest part, from to m NAP. The shorefae profiles were used to alulate the slope of the shorefae. The wave-height redution over the interval of to 5 m below palaeo-mean sea level was alulated.. Wave limate We use modern wave statistis at m waterdepth in the North Sea as starting onditions for the alulations of the loss of wave height. The wave statistis are an average over the year period 975-98 from Meetpost Noordwijk (Hokke and Roskam, 987) as presented by Stive and De Vriend (995). Kohsiek (988) gives an average (5%) wave for the Duth oast of. m with a wave period of 3.7 s. Model alulations of Stive (987) indiate that the wave limate of the North Sea around 5 BP was muh similar to the modern wave limate. Despite the inrease in waterdepth due to the rise in sea level the average wave height and wave period were only slightly less (below 5%) at 5 BP. Only the wave skewness hanged onsiderably, but this parameter is not onsidered in the alulations. Table.: Average wave height, wave period and ourrene frequeny of waves at m below mean sea level over a year period, from 975 to 98 from the platform Meetpost Noordwijk, whih is representative for the oast of the Western Netherlands (Hokke and Roskam, 987; ited by Stive and De Vriend, 995). H (m) T (s) Frequeny (%).5..778.75.5 3..5 5..85.75 5.3.7.5 5.5 5..75..7 3.5.5. 3.75 7...5 7.5.3.75 8.. 5.5 8.5. 5.75 9...3 Wave height redution The loss of wave height over a gently sloping shorefae profile was alulated with Nielsen s (983) expliit wave formulae. X axis is positive in the diretion of wave propagation (landward) and depth is positive (so beah slope is negative) g os α g os α = + H H βh I g os α k f e β = dh 3π dx (.) (.) 3
Wave-height redution π k h = gt h (.3) I = 5 with α= the analytial solution of I beomes h h.5.5 ( k h ) + ( k h ) ( k h ) o h h.5 o.5 3 o.75 h h.75 (.) following Nielsen (98) we approximate g / with the intermediate water depth (h/l o <.) equation H g π h L e h π L = (.5) If we onsider g and g onstant, equation beomes for α= : h π k f e h L = H + H π e I dh L 3 π dx (.).3 Results.3. Wave height The loss in wave height on the shorefae slopes of.5,.5 and.59 is alulated for all types of waves in table.. For all simulations an idential value of.5 is taken for the frition fator (f e ). The results are presented in figure.. The original wave height at m, the remaining wave height at 5 m and the remaining perentage of the waves is given for all waves. The redution of the wave height is larger on the gentle slopes. The wave-height redution is muh more effetive for the long-period high waves than for the short-period low waves. On the most gentle slope only 5% of the highest long-period waves remains, against 57% on the steepest slopes. In other words, the gentle shorefae from the period of initial progradation resulted in altogether lower waves and dampened out all the large waves. The results of the simulations are also expressed in figure.3, where the wave-height distribution after their redution over the lower shorefae is plotted. The pattern in the modern wave distribution at 5 m is omparable to that of the original wave-height distribution at m. The distribution after wave-height redution on the gentle slope shorefae from BP shows no waves over m at 5 m. The distribution after wave-height redution on the intermediate slopes holds an intermediate position.. Disussion The outome of the alulations shows that the slope of the shorefae has a marked influene on the wave-height redution due to the hanges in travel lenght over the seafloor. The gentle lower-shorefae slope of the initial prograding oast redued the impat of wave on the shorefae and beah muh more than the steep slope of today. 37
Chapter A: Gentle slope (.5 ) ( BP) 8 B: Intermediate slope (.5 ) (3 BP) 8 8 8 C: Steep slope (.59 ) (modern, BP) 8 8 Figure.: Graph of the derease in wave height from m to 5 m palaeo-mean sea level on shorefae profiles with different slopes. The frition fator is onstant at.5 for all alulations. The light bars indiate the wave height at m that is used as input for all three graphs. The dark bars represent the wave-height at 5 m, after wave-height redution due to bottom frition. The dotted line indiates the remaining wave-heights as a perentage of the original wave height, i.e., it represents the differenes between the light bars and the dark bars. Figure.A represents the situation at BP, with a gentle shorefae slope of 5º. In figure. B the situation at 3 BP is represented, with a shorefae of slope of.5º. This situation is roughly similar to the situation of 35 BP. In. C the modern situation is shown, with a relatively steep shorefae slope of.59º... Shorefae slope We have used a simplifiation of the C -based isohrons to estimate the slope of the shorefae. The atual slopes of the lower and middle shorefae may deviate in detail from the slopes we reonstruted, but the trend, from gentle to steep during progradation, is learly present. We alulated the wave-height derease over the deeper water depths (- to 5 m 38
Wave-height redution palaeo-mean sea level), beause in the upper part of the shorefae profile other proesses (wave-breaking, breaker-bar formation, swash, et.) start to dominate the dynamis. The lak of isohrons in the deeper part of the western half of the ross setion makes it diffiult to reonstrut the evolution of the shorefae slope between 35 and BP. The reonstruted 3 BP shorefae profile is relatively gentle, ompared to the 5 and 9 BP isohrons. The atual 3 BP profile may well have been steeper than the reonstruted profile, giving a more gradual shift from the gentle 35 profile to the steep modern profile. It is also worth to notie that onsiderable alongshore variation ours in the modern shorefae slope (Postma and Kroon, 98). The Haarlem ross setion has a relatively steep ross setion (ompare the modern shorefae of the Haarlem and the Wassenaar ross setion in figure 3.3 and 3.7, and notie that the modern Wassenaar shorefae is relatively gentle (slope down to m is.º).. Frition fator (f e ) The frition fator or energy-dissipation fator is an important omponent in the alulation of the wave-height redution (Nielsen, 983, Van Rijn and Houwman, 999). In the frition fator the bottom frition due to grain and form roughness are ombined. In the alulations above we have used a onstant frition fator, beause we onentrate on the influene of shorefae slope alone. However, from the lithology and the sedimentary harateristis we an infer that the grain size and the morphology of the seabed have hanged onsiderably in during progradation. The initial prograded deposits ontain mainly mud with some thin finesand layers with wave-ripple lamination. The final prograded shorefae deposits onsist of dm thik parallel-laminated storm-sand beds, likely deposited under sheetflow onditions (Chapter 5). In other words, the grain-size has inreased during progradation and the seabed morphology has shifted from wave ripple bedding to plane bedding. It is likely that the frition fator has hanged along with these hanges. Depth (m NAP) W -5 Frequeny (%) 3 Input wave frequeny at - m (below palaeo sea level) Wave height (m) Frequeny (%) 3 Wave frequeny at -5 m (below palaeo sea level) modern: BP Wave height (m) Modern ( BP).59º Frequeny (%) 3 Wave frequeny at -5 m (below palaeo sea level) 3 BP Wave height (m) Frequeny (%) 3 BP 35 BP 3 Wave frequeny at -5 m (below palaeo sea level) 35 BP Wave height (m) Frequeny (%) 3 Wave height (m) E BP Wave frequeny at -5 m (below palaeo sea level) BP -5 -.5º.3º.5º - -5-5 - -. 3...... 3.. 5.. 7. Distane to shoreline (km) Shoreline Figure.3: The Haarlem ross setion with the reonstruted lower shorefae profiles and with the frequeny of the wave heights of different lasses at the 5 m palaeo-mean sea level. The input distribution at m palaeo-mean sea level is shown as well. On the modern shorefae the large waves are redued, but the distribution of waves still resembles the pattern at the m. On the BP shorefae profile all large waves have been redued and only waves smaller than m remain at 5 m. 39
Chapter There is no shared opinion on the best method for the determination and alulation of the frition fator and several approahes may be applied to alulate grain and form roughness (Van Rijn, 993, Van Rijn and Houwman, 999). The form roughness depends on the wave-ripple height and length. Usually the relation onsists of the square of the ripple height, divided by the ripple length and multiplied by a fator. The fator an vary from 8 (Nielsen, 983) to 8 (Grant and Madsen, 98). The grain roughness is always expressed as a funtion of the grain size (expressed as d 5 or d 9 ), in some ases ombined with the skin frition Shield s parameter θ (Van Rijn, 993). In the sheetflow regime the form roughness is not of importane, the bottom roughness and frition fator under sheetflow onditions is related to sediment-onentration gradients and the sheet-flow layer. Aording to Van Rijn (989, 993), the grain roughness of the sheetflow bed is in the order of the sheetflow layer thikness or the boundary layer thikness. The frition fator does not exeed a ertain threshold value, that is usually onsidered to be.3 (Nielsen, 983, Van Rijn, 993). Researh of Van Rijn and Houwman (999) demonstrated that a onstant bed-roughness fator lead to improved preditions of urrent veloities in a -DV flow model over frition fators alulated with the methods outlined above. The value of.5 used in the alulations is high ompared to the value of. obtained by Van Rijn and Houwman (999), and ompared to values normally used in the shorefae zone (also around., Stive, personal ommuniation). Apart from knowledge of the grain size and the seabed morphology, the alulation of the frition fator requires knowledge of the water semi-exursion and the horizontal orbital veloity at the bottom. To keep out of irular reasoning, either a rigid relation between frition fator, the hydrauli roughness and the grainsize is introdued that is based on laboratory observations (Nielsen, 983), or the grain roughness is solved by iteration (Van Rijn, 993). Beause the alulation of the frition fator is ompliated and beause the multitude of methods gives several possible outomes for similar situations we refrain from alulating the frition fator. To demonstrate the influene of the frition fator on the wave height redution four different senarios are presented in the Appendix. We have used equation., but now with arbitrary values of.3 and of.5 for the frition fator for the steepest and gentlest shorefae slope from the reonstrutions. The extreme differenes in the frition fator lead to differenes of up to % in the wave-height redution. Changes in the energy-dissipation may thus have influenes on the wave-height redution in the same order of magnitude as the hanges in shorefae slope...3 Quantitative? The outome of the alulations should be onsidered as an indiation of the influene of the shorefae slope on the shallow-water wave limate, but may not be regarded as the absolute palaeo-wave limate at 5 m below palaeo-sea level. The influene of the frition fator f e has been disussed above. The diretion at whih the waves reah the oast and refration on irregular substrates affets the wave-limate too. All alulations have been done for waves that approah the oast with wave rests aligned to the shoreline. For waves with different angles to the oast the redution of wave height will be larger, beause the path over whih the waves travel is longer and the bottom frition larger (Nielsen, 983). This is the ase for the modern wave limate, where moderate storm waves and fair-weather waves predominantly arrive from the west, and high storm waves predominantly arrive from the northwest (Van Straaten, 9). We have no estimates of the diretions of the palaeo-wave limate.
Wave-height redution Loally-generated wind waves are not inorporated in the alulations. Suh waves may have a onsiderable effet on the shallow-water wave limate. The effet of loallygenerated waves was larger for the gentle shorefae slope, beause the length over whih the wind affets the waves inreases. The transgressive oastal evolution prior to oastal progradation has led to the development of humps and bumps on the lower shorefae surfae (see for instane the 85 BP isohron in the Haarlem ross setion, figure.). Wave refration on the irregularities result in differenes in the wave-height at the shoreline. The wave refration leads to loal hanges in the wave height. The importane of wave refration dereases with progradation, beause the humps silt up and the bumps are eroded, leading to an overall smoother lower shorefae. For a quantitative estimate of the wave-height redution on the different shorefae onfigurations during progradation, more sophistiated spatial alulations of the wave-height redution and refration are required. The diretion of the wave limate, loally generated wind waves and the omplex morphology of the shorefae should be inluded in these alulations. This is not a straightforward exerise, beause it requires knowledge of the palaeo-wind limate and of the subsurfae morphology in 3-dimensions... Other explanations Van Straaten (95) suggested that the relatively quiet onditions during initial oastal progradation resulted from sheltering of the shorefae and beah by an offshore submarine bar (his figure ). The bars ated as offshore breakwaters that redued wave ation on the oast. Evidene for submarine bars were oarse-grained deposits that originated from an offshore environment seaward of the shorefae. This bar was present during initial progradation. During later stages of progradation Van Straaten envisaged a bar more seaward. All diret evidene for the seond bar would have been erased during later oastal erosion. Despite improved offshore oring tehniques and an inreased knowledge of the deposits below the modern shorefae remains of suh a bar have never been found (Beets et al., 995). The absene of remains of these bars makes this explanation less likely. An overall quieter wave limate ould explain the low energeti onditions during initial progradation. However, the model alulations of Stive (987) do not show a large influene of a lower sea level on the wave limate. A different wind intensity and diretion may have tampered the wave limate, but indiations for suh limati hanges are absent. The main argument against hanges in wave limate is found in the prograded deposits themselves. The shift from low energeti to high energeti deposits is found in all prograded deposits. The timing of progradation differs along the oastal streth (Beets et al., 99) and hene the timing of the shift from high- to low-energeti deposition has differed along the oast. This is a strong argument against a hanging wave limate beause this would affet the whole oastal streth simultaneously. Changes in the shorefae slope are favoured over other explanations for the hange in the energy harater of the shorefae deposits, beause the hange in slope is inherent to the progradation and the rise of sea level...5 Sediment transport and sedimentation rates The indiations of a hange in wave limate indue speulations about the influene of the lower shorefae slope on sediment transport and deposition. The low wave heights on the gentle shorefae profile result in low near bottom veloities. If near-bottom veloities are
Chapter suffiiently low mud an be deposited and preserved on the shorefae. The deposition of mud inreases the sedimentation rate on the shorefae. Aretion or erosion of beahes is related to fair-weather versus storm onditions (Van den Berg, 977). The derease of the wave height due to the gentle lower shorefae slope during the start of progradation may have led to a greater proportion of fair-weather onditions on the beah and thus have inreased aretion. The modern wave limate shows a relation between the wave height and the angle of inidene on deep water. High storm waves approah the oast on average from the northwest, while moderate storm waves and fairweather waves on average approah the oast from the west (Van Straaten, 9). Beause the gentle lower shorefae slope redued the higher waves more effetively then the lower waves, the average diretion of the longshore transport may have been different. Assuming a similar relation between wave height and wave inidene for 5 BP this means redued wave influenes from the northwest and therefore less longshore transport to the south. The most important effet of the shorefae slope on sediment transport and deposition is that the plae where most wave energy is dissipated hanges. On a steep shorefae slope most wave energy is dissipated on the upper shorefae and beah, while on a gentle shorefae slope muh energy is dissipated further offshore. In other words, as sediment transport is diretly related to the wave-energy dissipation, the sediment transport on a steep shorefae slope will be restrited to the upper shorefae and beah. On a gentle shorefae slope the sediment transport will be extended over a muh larger part of the shorefae..5 Conlusions The gentle-sloped lower shorefae of the initial progradational oast redued the wave height of the upper shorefae and beah more than today s shorefae, due to the inreased bottom frition. On steep shorefae slopes, like the modern shorefae, the derease of the wave height due to bottom frition is limited. Observed differenes in shorefae deposition during progradation, relatively fine-grained deposition with wave ripple lamination during initial progradation and relatively oarse-grained plane-bed storm sands during final progradation, are explained by the hange in slope of the shorefae profile during progradation. List of symbols Wave-phase veloity g Wave-group veloity f e Frition fator or Energy dissipation fator g Aeleration of gravity H Wave height h Water depth k Wave number (π/l) L Wave length T Wave period α Angle between wave rests and bottom ontours β see equation. subsript means deep water value subsript at starting depth h subsript at final depth h
Wave-height redution Appendix Bottom frition fator We have alulated the ontribution of f e to the loss in wave height. We have alulated the wave height redution of the waves in table. for two values of f e, on steep slopes and gentle slopes, using equation.. The results are presented in fig.a. The redution of the wave-height on the gentle slope with the low frition fator ranges up to 35% for the highest long-period waves. On the steep slope the redution of the high long-period waves is up to %. For the high frition fator the redution is up to 9% on the gentle slope and up to 7% on the steep slope. In other words, on the gentle slope and on the steep slope an inrease in the frition fator from.5 to.3 leads to an extra redution of the wave height of about 55 to 3 %. A: Low Fe (.5) Gentle slope (.5 ) 8 9 8 7 5 C: Low Fe (.5) Steep slope (.59 ) 8 8 B: High Fe (.3) Gentle slope (.5 ) 8 8 D: High Fe (.3) Steep slope (.59 ) 8 8 Figure.A: Graph of the derease in wave height from m to 5 m palaeo-mean sea level on shorefae profiles with different slopes and with different frition fators. The graphs have a similar onstrution as in figure.; the light bars indiate the wave height at m that is used as input for all three graphs, while the dark bars represent the wave height at 5 m, after wave-height redution due to bottom frition. The dotted line indiates the remaining wave heights as a perentage of the original wave height, i.e., it represents the differenes between the light bars and the dark bars. In figure.a. A and B the wave-height redution on a gentle slope (.5º) is depited, with a low frition fator (.5 ina) and a large frition fator (.3 in B). The differene in wave-height redution of the largest waves is about 5 %. Similar differenes in frition fator are depited in figure.a. B and C on a steep slope (.59º). The high frition fator in D (.3) leads to about 3% more redution of the wave height than the low frition fator in C (.5). 3