Keywords: tidal sand transport; tidal sediment concentrations; tidal sediment import; effect of channel deepening

Transcription

1 EFFECT OF CHANNEL DEEPENING ON TIDAL FLOW AND SEDIMENT TRANSPORT; PART I: SANDY CHANNELS (Pblished in Jornal Ocean Dynamics, Agst 18; Doi.org/1.17/s ) L.C. van Rijn; LeoVanRijn-sediment Consltancy, Blokzijl, The Netherlands; B. Grasmeijer (Deltares, Delft, The Netherlands); and L.M. Perk, (WaterProof, Lelystad, The Netherlands); abstract Natral tidal channels often need deepening for navigation prposes (to facilitate larger vessels). Deepening often leads to tidal amplification, salinity intrsion and increasing sand and md import. These effects can be modelled and stdied by sing detailed 3D-models. Reliable simplified models for a first qick evalation are however lacking. This paper presents a simplified model for sand transport in prismatic and converging tidal channels. The simplified model is a local model neglecting horizontal sand transport gradients. The latter can be inclded by copling (as postprocessing) the simplified model to a DH or 3D flow model. Basic sand transport processes in stratified tidal flow are stdied based on the typical example of the tidal Rotterdam Waterway in The Netherlands. The objective is to gain qantitative nderstanding of the effects of channel deepening on tidal penetration, salinity intrsion, tidal asymmetry, residal density-driven flow and the net tideintegrated sand transport. We firstly stdy the most relevant tidal parameters at the moth and along the channel with simple linear tidal models and nmerical DH and 3D tidal models. We then present a simplified model describing the transport of sand (TSAND) in tidal channels. The TSAND-model can be sed to compte the variation of the depth-integrated sspended sand transport and total sand transport (incl. bed-load transport) over the tidal cycle. The model can either be sed in standalone mode or with compted near-bed velocities from a 3D hydrodynamic model as inpt data. Keywords: tidal sand transport; tidal sediment concentrations; tidal sediment import; effect of channel deepening 1. Introdction Most allvial estaries have a converging (fnnel shaped) planform with a decreasing width in the pstream (landward) direction. The bottom of the tide-dominated section is often fairly horizontal in the entrance region de to dredging for navigation. The passage of large vessels reqire water depths of abot 1 to m. Some tidal channels have an almost prismatic planform de to the presence of dikes and/or groins. Generally, these strctres keep the channel velocities relatively high to prevent deposition. Tidal systems are affected by varios processes: inertia, amplification (fnnelling), friction-related damping, reflections and varying river discharges (Van Rijn 11a, page 8.19 and 11b). If the fresh water river discharge is sfficiently large and mixing rates are relatively low, the tidal system is stratified with density-driven residal flow near the bed in landward direction. The combined tide-driven and density-driven velocities are often sfficient to mobilize sand and md from the bed, the side slopes and the banks to be transported p and down the tidal channel. The transport of fine sand particles generally is in dynamic eqilibrim with the prevailing flow velocity as the pward transport by trblence and the downward transport by settling proceeds relatively fast (within 15 mintes) to create satrated transport conditions (as described by sand transport formlae). An example of a prismatic tidal channel with large water depths is the tidal Rotterdam Waterway in The Netherlands, which is the artificial moth of the river Rhine and has a length of abot 3 km, see Figre 1. This tidal channel is briefly described here, as field data from this case has been sed for verification of the varios models applied in this Part I. More information related to md is given in Part II. The water depth (below mean sea level) of the Waterway decreases stepwise from abot 18 m at the moth to abot 11 m near the city of Rotterdam at abot 3 km from the moth and to 8 m more pstream. The tidal range at the moth is abot 1.85 m dring mean tidal conditions at present (17). The annal-mean fresh water discharge throgh the Waterway is abot Q waterway,5%=131 m 3 /s (pstream river discharge at German border of Q pstream,5%= m 3 /s). The bed of the Waterway channel sections is relatively flat (reglar dredging) and consists of fine to medim sand (.-.4 mm) with a gradally increasing percentage of md from 5% at the moth to 35% at abot km from the moth. Dring the past decades, the Rotterdam Waterway has ndergone a process of constant deepening to accommodate larger vessels calling at the ports of Rotterdam. Plans have been made by the Port Athorities for frther deepening (from 15 m to 16.3 m between km 11 and 13, see Figre 1) of some sections of the main navigation channel. 1

2 Figre 1 Plan view and bathymetry (December 14) of Rotterdam Waterway, The Netherlands (NAP Mean Sea level), (Arcadis 15) Engineering stdies have been done to explore the conseqences of the channel deepening of the Rotterdam Waterway (Arcadis 15). More general research qestions related to channel deepening are addressed in these papers (Part I and II): what is the effect of channel deepening on the tide, salinity and sediment transport characteristics in prismatic and converging tidal channels? Is there a risk of regime change from import to export of sediment (or vice versa), nder what conditions and what are the net annal transport rates involved? Most attention is focssed on prismatic channels based on the example of the Rotterdam Waterway, bt converging channels are also addressed. Varios papers of the effects of channel deepening on hydrodynamics and sediment dynamics are available in the international literatre, see overview of Van Maren et al. (15a). Amin (1983) has pointed to changes in tidal constitents de to local channel modifications, channel deepening, and altered hydralic roghness. DiLorenzo et al. (1993) have fond that the mean tidal range near the head of the Delaware Estary has increased sbstantially (two-fold) de to dredging activities in the middle/pper estary over the past centry. They stated: The tidal phase-lag between the head of the tide and the entrance decreased from abot 9 hors to 6.4 hors. An additional increase of 1.5 m of the navigational channel depth (below Philadelphia) is fond (based on nmerical simlations) to have a comparatively small or negligible impact on the tidal regime. Familkhalili and Talke (16) have sed an idealized nmerical model to stdy the effects of channel depth changes on tides and storm srges in the Cape Fear River Estary. Their model reslts show that the tidal range in Wilmington is dobled over the last 13 years de to the increase of the local channel depth from 7 m tot 15.5 m. Channel deepening not only leads to an increase of tidal penetration, larger tidal ranges (amplification) and peak tidal velocities, bt may also lead to an increase of tidal velocity asymmetry, salt intrsion, density-driven residal flow near the bed in landward direction and larger sspended sediment concentrations, transport and deposition rates, as discssed for the Ems river by Talke et al. (9), Chernetsky et al. (1), Winterwerp (11), Winterwerp and Wang (13), De Jonge et al. (14) and Van Maren et al. (15a). A prononced tidal asymmetry and density-driven near-bed flow in landward direction were observed in the Ems river (Herrling and Niemeyer 8; Chernetsky et al. 1; Talke et al. 9). The increasing flooddominated asymmetry strengthens varios sediment importing mechanisms in the Ems river related to resspension, vertical mixing and flocclation (Winterwerp 11) and lag effects (Chernetsky et al. 1). Recent semi-analytical model stdies show an p-estary shift of the Estarine Trbidity Maximm (ETM) de to channel deepening, changes in tidal asymmetry (Chernetsky et al. 1) and bed roghness (de Jonge et al. 14) for the Ems river. Most of these papers are focssed on mddy tidal channels; less attention has been paid to sandy tidal channels, which is the focs of the present paper. Frthermore, most of these papers are site-specific, whereas the present paper has a more general focs sing schematic cases of prismatic and converging channels. This paper stdies the effect of channel deepening on the tidal hydrodynamics (Section 3) and sand transport processes (particles > 63 m) in tidal channels sing both simplified and detailed modelling approaches (Sections, 3 and 4). This is a follow-p of the papers by Van Rijn (11b) and Kijper and Van Rijn (11). The simplified sand transport model (TSAND) has been applied with inpt from a hydrodynamic DH or 3D-model for the prismatic Rotterdam Waterway case, bt the model has also been sed in standalone mode. Available data on annal sand

3 dredging volmes (Arcadis 15) of the prismatic Rotterdam Waterway has been sed for overall verification of the compted reslts (Section 4.1). The proposed simplified model for sand transport in tidal flow (standalone mode or copled to hydrodynamic models) is a new development and can be sed for qick evalation of the most relevant parameters affecting channel deepening. Based on this, the parameter range can be narrowed down sbstantially to redce the nmber of detailed nmerical model rns. The standalone model has been sed to prodce a general graph of net tide-integrated sand transport rates as fnction of water depth and river discharge for sand of.5 mm (Section 4.). This general plot clearly shows the effect of channel deepening on the import or export of sand at the moth of a tidal channel with a tidal range of abot m. This provides both qalitative and qantitative insights of channel deepening effects on sand transport processes. The plot can be sed to make a first assessment of the effects of channel deepening on the tide-integrated sand transport rates and hence on dredging volmes. In Part II, a similar approach for md transport (particles < 63 m) is presented. Typical examples of mddy rivers with trbidity maxima (increased md concentrations) in Erope are discssed in Part II: tidal Elbe river in Germany; tidal Ems river in Germany; Loire river in France; Western Scheldt tidal estary and Rotterdam Waterway (Rhine moth) in The Netherlands. A simplied md transport model (TMUD) is described and applied in Part II.. Model description.1 Detailed models Varios detailed hydrodynamic and sediment transport models have been sed by the athors. Herein, the leading detailed model is the non-linear Delft3D-model for shallow waters, which has been applied in 3D and in DH-mode. The 3D-Flow modle solves the nsteady shallow-water eqations in one (1DH), two (DV or DH) or three dimensions (3D). The system of eqations consists of the horizontal momentm eqations, the continity eqation, the salinity transport eqations, and varios trblence closre models. The vertical momentm eqation is redced to the hydrostatic pressre relation as vertical accelerations are assmed to be small compared to the gravitational acceleration and are not taken into accont. This makes the 3D model sitable for predicting the flow in shallow seas, coastal areas, estaries, lagoons, rivers, and lakes. The ser may choose whether to solve the hydrodynamic eqations on a Cartesian rectanglar, orthogonal crvilinear (bondary fitted), spherical grid or flexible mesh grid. In three-dimensional simlations a bondary fitted ( -coordinate) approach as well as a fixed grid are available for the vertical grid direction. The 3D model has been sed in combination with sbmodels for salinity and sediment transport. Varios simple sediment transport formlae and the detailed transport model for sand and md (Van Rijn 1984, 7) have been implemented. The non-linear DH and 3D-models of the Delft-model system have been sed and verified extensively in nmeros stdies (Lesser et al. 4, Van Rijn 7, Van Rijn 11a,b, Van Maren et al. 15a,b,). The 3D-hydrodynamic model of the Port Athority of Rotterdam has also been sed. This hydrodynamic model is based on the same eqations as the Delft3D-model, bt ses a different schematization with a mch finer horizontal grid to better inclde all geometric details of the Port of Rotterdam with all its docks at the expense of very large rn times. This latter model was careflly calibrated to represent the observed water levels (errors within.1 m) and the observed salinity distribtion (errors within 1% for salt intrsion length) along the Rotterdam Waterway (Arcadis 15)... Simplified models Simple approximations for tidal penetration, peak tidal velocities, tidal asymmetry, salinity-intrsion, density-driven circlation velocities, and tidal sand transport in prismatic and converging models are sed (Van Rijn 11b). In the case of converging channels, the width is represented by b=b o exp(-x/ ), with: b= channel width, b o=channel width at moth, = converging length scale. A tidal channel is almost prismatic for > 3 km (width redction of abot 65% over 3 km). Strongly converging channels have < 3 km (width redction of 65% over 3 km). Weakly converging channels for between 3 and 3 km...1 Linear tidal model The simplest way to stdy the effect of water depth and channel deepening on the tidal parameters along prismatic and converging tidal channels is by sing a linear tidal model. The classical soltion of the linearized mass and momentm balance eqations for a prismatic channel of constant depth and width is well-known (Lorentz 19, 196; Dronkers 1964; Van Rijn 11a,b). This soltion for a prismatic channel represents an exponentially damped sinsoidal wave which dies ot gradally in a channel with an open end or is reflected in a channel with a closed landward end. In a frictionless system with depth h o both the incoming and reflected wave have a phase speed of c o= (gh o).5 and have eqal amplitdes reslting in a standing wave with a virtal wave speed eqal to infinity de to sperposition of the incoming and reflected wave. Inclding (linear) friction, the wave speed of each wave is smaller than the classical vale c o (damped co-oscillation). The linear approach can also be sed for exponentially converging channels, which has been explained and verified in detail in earlier work (Van Rijn 11a,b). 3

4 It is noted that the linearized soltion cannot deal with the varios sorces of non-linearity sch as: qadratic friction, large ratio of tidal amplitde and water depth, convective acceleration, variation of the water depth nder the crest and trogh, effects of river discharges and effects of tidal flats casing differences in wave speed and hence wave deformation. Therefore, the non-linear Delft3D-model (in DH-mode) has also been sed for some cases. Neglecting the convective acceleration term ( / x; very small in tidal channels; see Van Rijn 11a,b), the eqations of continity and motion for depth-averaged flow in a prismatic channel with a rectanglar cross-section are: b η Q + = (1) t x 1 Q g η Q Q + + = () A t x C A R in which: A = b h o = area of cross-section, b = srface width, h o = depth to MSL (mean sea level), R = hydralic radis ( h o if b>>h o), η = water level to mean sea level, Q= discharge= A, =cross-section-averaged velocity of the main channel, see Figre, A= area of cross-section of main channel and C = Chézy-coefficient =5.75 g.5 log(1r/k s), k s= bed roghness height of Nikradse (1933). The eqations of continity and momentm for a wide, prismatic channel with rectanglar cross-section and constant mean depth (h = h o+ ) can be linearized (Lorentz-method) by replacing the qadratic friction term by a linear term, as follows: η h o + = (3) t x g η + + m û = (4) t x in which: = cross-section-averaged velocity, û = amplitde of the tidal crrent velocity, m = linear friction coefficient = (8g û )/(3πC R), R= hydralic radis (R h o for a very wide channel b>>h o). Very similar eqations can be specified for a compond channel (Figre ) with tidal flats (rectanglar cross-sections of flats and main channel; Van Rijn 11a). Amplitde HW= High Water MSL = Mean Sea Level LW=Low Water h o b channel b flats Figre Schematized cross-section of compond channel with main channel and tidal flats In the case of a compond cross-section consisting of a main channel and tidal flats it may be assmed that the flow over the tidal flats is of minor importance and only contribtes to the tidal storage. The discharge is conveyed throgh the main channel. This can to some extent be represented by sing: c o= (g h eff).5 with: h eff = A c/b s = h h c with h = b c/b s; A c= area of main channel (=b ch c), h c = depth of main channel (redces to h o withot flats), b c= width of main channel, b f= width of flats and b s= b c+b f=srface width. The transfer of momentm from the main flow to the flow over the tidal flats can be seen as additional drag exerted on the main flow (by shear stresses in the side planes between the main channel and the tidal flats). This effect can be inclded to 4

5 some extent by increasing the friction in the main channel slightly. If the hydralic radis (R) is sed to compte the friction parameters (m and C) and the wave propagation depth (h eff= R), the tidal wave propagation in a compond channel will be similar to that in a rectanglar channel with the same cross-section A. The soltion for a prismatic channel is an exponentially damped wave de to bottom friction, as follows: x,t = ˆη o [e x ] [cos( t kx)] x,t = û o [e x ] cos( t kx+ ) û o= -( ˆη o/h o) ( /k) [cos ] = -( ˆη o/h o) c [cos ] (5a) (5b) (5c) with: = friction parameter, k = wave nmber, c= wave speed, = phase lead of velocity, H= ˆη o = tidal range, tan = /k, x = horizontal coordinate; positive in landward direction, û o= peak tidal velocity of linear model. The detailed soltions for prismatic and converging channels are similar (Van Rijn 11a,b). It is noted that the linear model is not very accrate for very small water depths (ratio ˆη o/h o shold be <<1)... Simplified tidal sand transport model (TSAND-model) Definitions We have developed a simplified sand transport model for tidal flow to describe the transport of sand (TSAND). The simplified approach is based on the detailed sediment transport formlations by Van Rijn (7), which have been verified extensively. The detailed sediment transport formlations have been implemented in the Delft3D-model. The TSAND-model can be sed in standalone mode or in post-processing mode (copled to hydrodynamic model) to compte the variation of the depth-integrated sspended sand transport and total transport (incl. bed-load transport) over the tidal cycle. It is assmed that the fine sand concentrations are qickly adjsting to local flow conditions (no significant nder or overloading). The velocities and sand concentrations are compted as a fnction of z and t; z=height above bed and t=time (time step of 5 min). The grid points over the depth (5 points) are distribted exponentially, as follows: z=a[h/a] (k-1)/(n-1) with: a= reference height above bed (inpt vale), h=h o+ = water depth, h o= depth between bed and mean sea level, = tidal water level, k= index nmber of point k, N= total nmber of grid points. Used as a standalone model, the basic hydrodynamic parameters shold be specified by the ser. Used as a post-processing model, the hydrodynamic inpt may come from a 1D, DH or 3D-model. TSAND can be sed for prismatic and converging channels. Tidal water levels, flow and asymmetry The tidal water level at each time t is represented, as: flood,t = flood,max sin{ (t- T)/T flood)} for t < T flood (6a) flood,max = a (H/) and T flood= ( ebb,max/h)t ebb,t = ebb,max sin{ (t-t flood- T)/T ebb} for t > T flood (6b) ebb,max = H - flood,max and T ebb= ( flood,max/h)t with: H = flood,max + ebb,max = tidal range between the levels of HW and LW; a= asymmetry factor (inpt vale a=1 tot 1. see Figre 8, a=1 yields a symmetrical tide), T= tidal period (inpt vale), T= phase shift (velocity is ahead of water level, inpt vale), T flood= flood period, T ebb= ebb period. In standalone mode, only one representative tide of the neap-spring cycle is considered (H, a, T specified as inpt by ser); the tidal range H comes from the linear tidal model (Section..1). The depth-averaged velocity at each time t is represented, as: t = r + max,flood sin( t/t flood) for t < T flood (7a) max,flood = a ( max) t = r + max,ebb sin{ (t-t flood)/t ebb} for t > T flood (7b) max,ebb = -1/ a ( max) with: t=depth-averaged velocity at time t (m/s), r= net tide-averaged and depth-averaged velocity = Q/(b h), b= flow width of channel (inpt vale), h= water depth (inpt vale), max== peak tidal velocity of sinsoidal tide (inpt). 5

6 Eqation (7) is sed to describe the tidal motion of one single tide in the case that the model is sed in standalone mode. An asymmetrical tide can be generated, sing the asymmetry factor a. Tidal flow in varios type of channels has been stdied to determine the asymmetry factor a, (Section 3.3). If the sediment transport model is sed as a post-processing model, the nearbed velocity of the hydrodynamic model (1D, DH or 3D) is sed as inpt. The vertical distribtion of the velocity at each time t is represented, as: z,t= rs,z + [ t/(-1+ln(h/z o))] ln(z/z o) (8) with: rs,z= residal flow velocity de to horizontal salinity-indced density gradient based on Eqation (9), z= level above bed (m), h= = h o+ t = water depth at time t (m), h o= tide-averaged water depth (m), t = tidal water level (m), t = depth-averaged flow velocity at time t de to tide + steady river crrent (m/s), r= Q/(bh)= river velocity, z o=.33k s,c = zero-velocity level, k s,c = crrent-related bed roghness (wave-crrent interaction is neglected). The first part of Eqation (8) represents the residal flow de to the density gradient. The depth-integration yields zero flow velocity. The second part represents the tidal velocity profile and the river flow velocity profile de to fresh water discharge. The vertical velocity distribtion of tidal and river flow is represented by a logarithmic fnction. De to density-driven gravitational circlation, a tide-averaged residal flow is generated in a prismatic channel with tidal flow from the seaside and fresh river water flow from the landside. The residal flow is landward near the bottom and seaward near the water srface. Residal density-indced flow in a prismatic tidal channel can be determined from the tide-averaged momentm eqation. There are two main contribtions: the free convection contribtion arising from the density difference between salt water and fresh water and the fresh water discharge contribtion. The sips-mechanism related to tidal straining (Section 1 of Part II, see also Brchard and Bamert 1998) which enhances the landward-directed near-bed flow is neglected. Based on the tide-averaged eqation of momentm (inclding eddy viscosity concept) and no net flow over the depth, the residal (secondary) velocity profile related to density-driven flow can be derived (Hansen and Rattray 1965, Chatwin, 1976 and Prandle, 1985, 4, 9). Herein, a similar approach based on a constant vertical eddy viscosity concept is sed, reslting in (Van Rijn 11a): rs,z = Mh o [ (1/6) (z/h o) 3 + (1/) (z/h o) (1/4) (z/h o)] (9) M = (g h o/e) (1/ m) ( m/ x) = [g.5 C/{ ( max + r ) h o}][(h o/ fresh) ( m/ x)] with: z= level above bed (m), h o= tide-averaged water depth (m), max= peak tidal velocity, r= Q/(bh)= river velocity, z o=.33k s,c = zero-velocity level, k s,c = crrent-related bed roghness of Nikradse, M=salinity factor, E = * h o= mixing coefficient in near bed zone (constant), *= (g.5 /C) ( U max + r ), C= Chézy-coefficient=5.75g.5 log(1h/k s,c), sea= density of sea water (kg/m 3 ), m/ x= salinity-indced horizontal density gradient, m= depth-averaged and tide-averaged flid density de to salinity= fresh+.77 S (kg/m 3 ), fresh = fresh water density, S= salinity vale ( to 3 kg/m 3 ), =coefficient (.3 based on calibration, see Part II). The maximm residal velocity of Eqation (9) is approximately: s,max =.35Mh o at abot z/h o=.3. Eqation (9) is valid in the moth region of a tidal channel where the vertical density profile is partially mixed over the depth. The residal velocities are maximm in the near-bed region and show increasing residal velocities for increasing water depth and increasing river discharge. The vertical mixing coefficient E (or ) is sed as calibration parameter, which is discssed in Part II based on measred and compted (3D-model) data of the Rotterdam Waterway. Eqation (9) is very similar to that of Hansen and Rattray (1965), Chatwin (1976) and Prandle (1985, 4, 9). The latter derived for a partially-mixed estary: s,max = Mh o [ (1/6) (z/h o) (z/h o).37 (z/h o)-.9], which yields s,max.3mh o at abot z =.1h. It is noted that the expression of Prandle yields a finite residal velocity vale at z= (at the bed), whereas Eqation (9 ) yields a zero residal velocity at the bed. The residal velocities generated by density differences de to strong horizontal sediment concentration gradients, which may occr in the high trbidity zone (Talke et al., 9), have been neglected. Hence, the model is less accrate when it is sed in standalone mode for the estarine trbidity maximm (ETM) with relatively high horizontal sediment density variations. 6

7 Sediment concentrations and transport The sand concentrations are compted sing a mlti-fraction method (Van Rijn 7). The mlti-fraction method is based on N=6 fractions ( mm,.15-. mm,.-.3 mm,.3-.5 mm,.5-1 mm, 1- mm). The single fraction method is obtained for N=1. The sand concentration profile at each time t is represented by an analytical expression, as: c i=c a,i[((h-z)/z)(a/(h-a))] ZSi (1) with: c i= concentration of fraction i (kg/m 3 ), ZS i= sspension nmber of sediment fraction i (-). The reference concentration c a of fraction i at reference level z=a is represented, as (Van Rijn 7, Part II): c a,i=.15 ca (1-p md) (d i/a) (T i) 1.5 (D *,i) -.3 (11) with: ca= correction/scaling coefficient (defalt=1), d 5= median particle diameter, a= inpt vale, D *,i= d i[(s-1)g/ ].333 = dimensionless particle parameter, T i=[ i( b / - I b,cr,d5)]/[ (d i/d 5) b,cr,d5]= dimensionless bed-shear stress parameter of fraction i, b,cr,d5= critical bed-shear stress of sand based on d 5, b/ = c b,c+ w b,w= effective bed-shear stress de to crrent and waves, s= s/ = relative density, =(1+p gravel)(1+p md) 3 = factor representing effect of md and gravel on the critical shear stress of sand, i=(d i/d 5).5 = roghness correction factor of fraction i, i= (d 5/d i).5 = hiding-exposre factor of fraction i (maximm and minimm.5), = kinematic viscosity coefficient, s= sediment density (inpt vale), = flid density, p md= fraction of md (<.63 mm) of top layer of bed ( to.3), p gravel= fraction of gravel (> mm) of top layer of bed ( to.1). Eqation (11) can be sed for both sand (Part I), silt and md (Part II). The critical bed-shear stress of sand is represented by (Shields based on d 5): b,cr,sand= ( s- ) g d 5 [.3/D * +.55(1-e -.D* )] (1) The critical bed-shear stress for erosion of md (Part II) is an inpt parameter to deal with site-specific conditions (no general relationship is available). The bed-shear stresses at time t de to crrents and waves are represented as: b,c= ( *,c) with *,c= z /ln(3z/k s,c) (13) b,w=.5 f w (U w) (14) with: =.4 and z= flow velocity at z.1 h, U w= ( /T p)a w=peak orbital velocity (linear wave theory), A w= peak orbital excrsion, T p= peak wave period, f w= exp(-6+5.(a w/k s,w) -.19 )= wave-related friction coefficient, = flid density inclding salinity effect; k s,c= crrent-related roghness, k s,w= wave-related roghness. The effect of srface waves are inclded to deal with waves in the moth of a tidal channel. The effective bed-shear stresses at time t for sediment transport are represented as: / b,c = c b,c (15) b,w/ = w b,w (16) with: c=f c / /f c= crrent-related efficiency factor, f c=.4/(log(1h/k s,c)) = crrent-related friction coefficient, f c / =.4/(log(1h/d 9)) =grain-related friction coefficient, d 9= grain size, w=.7/d *= wave-related efficiency factor ( w,min=.14, w,max=.35). 7

8 The sspension nmber ZS of sand is represented as: ZS sand,i= w sand,i,o/( *,cw) +.5(w sand,o/ *,cw).8 (c a,sand/c o).4 + ( / fresh - 1).4 (17) with: w sand,i,o= fall velocity of fraction i (based on formla Van Rijn 1993), =.4, = 1+(w sand,i/ *,cw) = coefficient ( max=1.5), *,cw=( *,c + *,w ).5 = bed-shear velocity de to crrent and waves, *,c= ( b,c/ ).5 = crrent-related bed-shear stress, *,w= ( b,w/ ).5 = wave-related bed-shear stress, c a= reference concentration, c o=.65=maximm bed concentration, = flid density inclding salinity effects, fresh = flid density of fresh water (= 1 kg/m 3 ). Eqation (17) represents the effects of downward gravity settling, pward trblence mixing (first term), empirical damping de to vertical sediment concentration gradients (second term) and empirical damping de to vertical salinity gradients (third term), (Winterwerp 1; Van Rijn 1993). The damping de to vertical salinity/density gradients mainly occrs dring the flood period when the salt wedge penetrates landward. A larger ZS-vale yields smaller concentrations. The time lag effect of the sspended concentrations and sspended transport rate can be represented to some extent by applying an exponential adjstment of the reference concentration at time t based on: dc a/dt =-A(c a,t - c a,t,eq), reslting in (Van Rijn 15): c a,t=c a,t- t + dc a (18) c a,t=[1/(1 + A t)][c a,t- t + A t c a,eq,t] with: t= time step (5 min), c a,t- t= sspended reference concentration at previos time ( c a,i for mltifraction method), c a,eq,t= eqilibrim reference concentration at time t, A= c.5(1/h)(w sand,o/ *,cw)(1+w sand,o/ *,cw)(1+h s/h), A minimm=.5, c= calibration coefficient (range.5 to ; defalt =1). The sspended transport q s can be compted by integration of the prodct of velocity and concentration over the water depth: q s = o N a h ( c) dz with N= 6= nmber of fractions (smmation over fractions and depth) (19) The bed load transport (in kg/m/s) is compted by sing a simplified expression (Van Rijn 7). The total load transport at each time t is compted as: q tot=q s+q b The tide-integrated sediment transport rates are compted by integration over the tidal cycle. Verification of sand transport model The simplified approach presented here is based on the detailed sediment transport formlations by Van Rijn (1984, 1993, 7, 15), which have been verified extensively for river and tidal flow. The detailed formlations have been implemented in the Delft3D modelling site and the approach presented here is basically a simplified version of the detailed nmerical model. As the detailed sand transport model has been verified many times before (Van Rijn 1984, 1987, 7; Lesser et al. 4). Herein, only one high-qality data set for tidal flow is considered. This case refers to measred sand concentrations and velocities in a tidal channel of the Eastern Scheldt estary in the sothwest part of The Netherlands (Krammer, Station, 8 April 1987; Voogt et al., 1991). The data set based on classical mechanical sampling instrments represents a wide range of velocities (1.4 to 1.9 m/s) and is a high-qality data set. The measred sspended transport has a relative error of abot 3% (Voogt et al., 1991). The water depths are in the range of 7 to 8 m. Water depth variations are de to tidal water level variations and the presence of mega-ripples (with height of abot.5 m). The bed material is sand with d 5=.3 mm and d 9=.6 mm (six fractions: p 1=.4; p =.15, p 3=.35, p 4=.45, p 5=.1, p 6=). Model validation reslts are given in Figre 3 presenting the measred water depths, measred depth-mean velocities, measred sspended transport rates and the compted sspended transport rates based on the single fraction and mlti fraction methods (TSAND-model) withot any calibration. The bed roghness is taken to be k s=.4 m based on measred velocity data. The depth-mean velocity increases from abot 1.1 m/s to abot 1.9 m/s at peak flow conditions reslting in a strong increase of the sspended transport. The compted sspended transport rates of the mlti fraction method are in good agreement with measred vales (all reslts within factor ), althogh the compted model reslts are, on average, somewhat too small. The compted sspended transport rates based on the single fraction method with d ss=d 5=.3 mm are too small (factor ). The single fraction method prodces good reslts if the sspended sediment size is calibrated to be d ss=.8d 5 =.4 mm, which is in agreement with earlier findings (Van Rijn 1984). 8

9 Water depth (m), velocity (m/s), sspended transport (kg/m/s) Figre measred water depth measred depth-mean velocity measred sspended transport (3% inaccracy) compted sspended transport mlti fraction compted sspended transport single fraction dss=.3 mm compted sspended transport single fraction dss=.4 mm Time (hors) Measred and compted sspended sand transport in tidal channel Krammer, Eastern Scheldt, The Netherlands (Voogt et al. 1991) 3. Effect of channel deepening on tidal parameters Location: Krammer Eastern Scheldt Estary (NL) Date: 8 April 1987 To operate the TSAND-model in standalone mode for schematic and practical cases (Section 4.), it is essential to have knowledge of the peak tidal velocities in the moth region (length<.1 tidal wave length) and the velocity asymmetries involved. We will therefore now stdy these parameters and the effects of channel deepening on these parameters by sing linear and non-linear models focssing on varios schematic cases with varying water depth. 3.1 Tidal range and tidal penetration In the case of a prismatic channel, the linear model soltion is a damped tidal wave with a damping rate depending on the water depth and the bed roghness. In the case of a converging tidal channel, the soltion can be a damped tidal wave or an amplified tidal wave depending on the converging length scale, the water depth and the bed roghness (Friedrichs 1, Van Rijn 11a,b). It has been shown that the linear model works very well for a wide range of conditions (Western Scheldt estary (Netherlands), Hooghly estary (India), Delware estary (USA) and the Yangtze estary in China (Van Rijn 11a,b). We have sed the linear model to compte the redction of the tidal range (H/H o) in a prismatic channel with water depths in the range of 11 to 17 m and channel width of 5 m, see Figre 4. Two cases with different depth conditions are distingished: Rotterdam Waterway in the year 191 with a depth in the range of 8 to 1 m and in the year 17 with a depth range of 8 to 18 m. The tidal range at the moth is H o 1.65 m in 191 (Hollebrandse 5) and 1.85 m in 17 (Rijkswaterstaat 17). The tidal period is 447 s. The applied bed roghness is set to.5 m. Measred tidal range vales in 17 are available for 4 stations (Figres 1 and 4) and in 191 for 1 station Rotterdam (Rijkwaterstaat 17; Van de Kreeke and Haring 1979). The longitdinal bottom profile of the Rotterdam Waterway in 17 consists of varios sections with different water depths (h i) between 18 and 8 m to MSL (mean sea level) and different length (L i) between 4 and 19 km (L t= 45 km = total length). As tidal penetration is fond to be proportional to h 1.8 (see below), the effective water depth has been determined as h e= (h i 1.8 L i/l t) reslting in the weighted vale h e= 13.5 m for the present sitation in 17. The effective depth in 191 is abot 1 m. The depth-mean river velocity (river discharge divided by cross-sectional area) is abot.15 m/s, which is small compared to the peak tidal velocity of abot 1 m/s. The river velocity effect on the tide cannot be taken into accont by the linear model. Using the effective depth vales, the linear model yields a tidal range redction of abot 18% over 3 km (Rotterdam) in 191 and 13% in 17, which is somewhat higher than the measred vales of 14% in 191 and 11% in 17 (Figre 4). This can easily be improved by sing a smaller bed roghness vale (calibration). Echo sonding reslts of the bed srface shows a rather flat bed withot the presence of bed forms (Arcadis 15). Hence, a small roghness vale in the range of.1 to.5 m is realistic.the linear model and the measred data show that the damping is less for a larger depth, bt the effects are relatively small. 9

10 Tidal penetration length (km) Relative tidal range H/H o (-) Figre Hook of Holland Maasslis Vlaardingen Rotterdam Compted h=13 m Compted h=15 m Compted h=17 m Compted h= 11 m Measred 17 Measred Distance from moth (m) Effect of water depth on the redction of tidal range along the prismatic tidal channel of the Rotterdam Waterway in 191 and 17 based on linear tidal model (k s=.5 m) Hereafter, the linear model is sed to stdy the tidal penetration distance (L p) as fnction of water depth, bed roghness and the presence of tidal flats (compond prismatic channel) for a range of conditions, see Figre 5. The tidal penetration distance is defined as the distance after which the tidal amplitde is smaller than 1% of that at the moth. Water depth, bed roghness and tidal flats are the most inflencial parameters. It is clear that tidal penetration increases strongly for increasing water depth (less friction); roghly L p h o 1.8 for k s=.1 m (see Figre 5) and L p h o 1.5 for k s=.1 m (see Figre 5) for sitations with no tidal flats. Tidal penetration increases for decreasing bed roghness k s (from.1 to.1 m). Tidal penetration is strongly affected by the presence of tidal flats. The bed of the tidal flats is assmed to be between LW (low water level) and MSL (mean sea level), see Figre. We have sed two vales of the width of the tidal flats: b flats=.5 b channel and b flats= b channel. The channel width is set to 5 m. As can be seen, the presence of tidal flats yields a significant redction (more than 5%) of the tidal penetration length. This nderlines the importance of the tidal flats (see also Friedrichs and Abrey 1988). Removal of tidal flats leads to larger tidal penetration and associated salinity intrsion. In practice, the effect of tidal flats is mch less as tidal flats are sally only present in the oter part of the estary. Tidal flats in the middle part of the estary are often reclaimed for commercial se ks=.1 m, no tidal flats ks=.1 m, no tidal flats ks=.1 m, with tidal flats (b-flats=.5 b-channel) ks=.1 m, with tidal flats (b-flats= b-channel) Tidal range at moth= m Figre Water depth (m) Effect of water depth, bed roghness and tidal flats on tidal penetration for a prismatic tidal channel based on linear tidal model 1

11 Peak tidal velocity at moth (m/s) 3. Peak tidal velocity at moth The magnitde and direction of tidal sand transport in the moth region strongly depends on the peak velocities of the tidal system in that region and are discssed in this section. The horizontal distribtion of the peak tidal velocities along the tidal channel beyond the moth region is discssed in Section 3.4. We have sed the linear model and non-linear models to compte the peak tidal velocity at the moth of varios tidal channels and the phase shift between the vertical and horizontal tides. The peak tidal velocity at the moth based on the linear model primarily depends on the type of channel (prismatic or converging), the tidal range, the water depth and the bed roghness, see Figre 6 for the linear model reslts and Figre 7 for both linear and non-linear models. The converging case is a strongly converging tidal channel with a length of 6 km (<< tidal wave length) and a moth width of 5 km redcing (exponentially) to abot km at 6 km (resembling the Western Scheldt, The Netherlands). In the case of prismatic channels, the peak velocity at the moth increases from abot.8 m/s at a tidal range of m to abot m/s at a tidal range of 8 m based on Eqation (5c). The effect of the water depth (between 1 and m) is marginal for prismatic channels; the peak velocity is slightly smaller for a larger depth. A smaller bed roghness of k s=.1 m (factor 5 smaller) leads to somewhat larger velocities (abot 5%). A larger bed roghness of k s=.5 m (factor 5) leads to somewhat smaller velocities (abot 7%). The phase lead between the horizontal and vertical tide in prismatic channels decreases from abot 1. hors to.7 hors for increasing water depths between 5 and 15 m (not shown). The phase lead is almost zero for a depth of 5 m. In a prismatic channel, the phase lead of the linear model only depends on the bottom friction and is fairly accrate. In the case of a strongly converging channel with a moth width of 5 km redcing (exponentially) to abot km at 6 km from the moth, the peak velocities at the moth are mch smaller compared to those in prismatic channels. The effect of the water depth is mch stronger. Increasing the water depth from 1 m to m leads to a velocity decrease of abot 4%. In a converging channel, the predicted phase lead of the linear model is less accrate as it depends on bottom friction and on the converging length scale, which introdces additional schematization errors (Van Rijn 11a,b). A comparison of linear and non-linear model reslts is given in Figre 7, which shows the peak tidal velocity as fnction of the water depth for a constant tidal range of 4. m (abot the middle of the tidal range interval of Figre 6). The peak velocity of the non-linear DH-model represents the average vale of the peak flood and ebb vales. The compted reslts of the linear model and the non-linear DH-Delft model for a tidal range of 4. m are fairly similar. The linear model, althogh strictly valid for ˆη o<<ho) does srprisingly well for shallow depths ( ˆη o/h o.3-.4). Figre Prismatic; water depth= 1 m (linear) Prismatic; water depth= 15 m (linear) Prismatic; water depth= m (linear) Converging; water depth= 1 m (linear) Converging; water depth= 15 m (linear) Converging; water depth= m (linear) Prismatic channels Converging channels Tidal range (m) Effect of water depth and tidal range on the peak tidal velocity at the moth for prismatic and converging channels based on linear model (bed roghness=.5 m) The relationship between the peak tidal velocity at the moth and the water depth in prismatic channels can be determined analytically sing Eqation (5c) of the linear model. Based on sensitivity comptations sing the linear model, the wave speed c in water depth between 5 and m is roghly proportional to c h o.7 and the phase lead (cos ) is proportional to cos( ) h o.1. The phase lead is abot 1.5 hors for a very shallow prismatic channel (relatively large friction) and abot zero for a very deep channel (almost no friction). The vale of cos( ) varies very weakly between.8 and 1. Given these reslts, the peak tidal 11

12 Peak tidal velocity at moth (m/s) -. velocity (Eqation 5c) at the moth of a prismatic channel withot flats varies roghly as: û o h o yielding a weakly decreasing crve for increasing channel depths, as shown in (Figre 7). Ths, channel deepening has a very marginal effect on the peak tidal velocity in a relatively wide prismatic channel. In a strongly converging tidal channel, the peak tidal velocity redces by abot 3% if the water depth increases (factor 3) from 5 to 15 m. Other configrations can be easily explored by the analytical linear model eqations. Figre Prismatic Ho=4. m: Peak velocity; non-linear model Prismatic Ho=4. m: Peak velocity; linear model Converging Ho=4. m: Peak velocity; linear model Converging Ho=4. m: Peak velocity; non-linear model Prismatic Converging Linear Non-linear Non-linear Linear Water depth (m) Effect of water depth on the peak tidal velocity at the moth for prismatic and converging channels based on linear and non-linear DH Delft-model (tidal range= 4. m; bed roghness=.5 m) 3.3 Tidal asymmetry at the moth The analytical soltions of the linearized eqations of continity and momentm presented above do not express the non-linear effects. All non-linear terms ( / x; ( )/ x and ) have been neglected or linearized. Ths, the wave speed is constant and the peak flood and ebb velocities are eqal (no asymmetry). Friedrichs and Abrey (1988) have shown for a short estary withot pstream river flow that the velocity asymmetry depends on the relative tidal amplitde ( ˆη o/h o) and the ratio V s/v c with ˆη o= tidal amplitde, h o = water depth to mean sea level, V s= intertidal water volme between HW and LW above the flats and V c=water volme in main channel below mean sea level. A ratio V s/v c means the presence of a channel withot tidal flats reslting in flood dominance ( a= û flood/ û ebb > 1). Ebb dominance can occr if the propagation of the tide dring flood is slowed down sfficiently by the water mass above the tidal flats. Dring ebb, the flats are dry and the tide propagates throgh the main channel at a smaller water depth with larger flow velocities reslting in ebb dominance. It is shown that ebb-dominant sand transport prevails at the moth of shallow estaries for ˆη o/h o <.3 with V s/v c >.3, which cover many small-scale estaries in the USA. Flood dominance prevails for all conditions withot flats. Ths, flood-dominance occrs in shallow, narrow channels and ebb-dominance in deeper channels with flats. Hence, shallow channels tend become shallower de to sediment import (flood dominance) and deeper channels tend to become deeper de to sediment export (ebb dominance) for tidal channels withot river flow (no river flshing). In practice, the morphodynamic behavior also depends on the location of the channel within the estary. Some flood or ebbdominated channels may show (qasi) morphodynamic eqilibrim de to sediment spply from adjacent channels. Frthermore, relatively short tidal inlet channels may behave in a different way than longer estary channels. We stdied the velocity asymmetry effects in the moth region of tidal channels based on compted data available in the literatre and new additional Delft3D model simlations. The focs is to obtain a simple relationship between tidal velocity asymmetry ( a= û flood/ û ebb) and the relative tidal amplitde ( ˆη o/h o), which can be sed if the TSAND-model is sed in standalone mode (feasibility stdies/qick-scan stdies), see Figre 8 and Table 1 with all data sorces inclded. The data refer to compted peak tidal velocities in the moth of varios tidal channel sections of major Eropean rivers based on nonlinear DH and 3D-model reslts. For almost all natral tidal rivers/channels the velocity asymmetry is > 1 (flood dominance), bt the asymmetry factor remains small between 1 and 1.. The model simlations for the Ems river (Van Maren et al. 15b) are based on two Manning-coefficients (relatively smooth: n=.1 or C= 13 m.5 /s; relatively rogh: n=. or C= 65 m.5 /s). 1

13 Tidal velocity asymmetry factor (-) The pper error bar of the Ems river data (Figre 8) refers to smooth bed conditions and the lower error bar refers to rogh bed conditions. The velocity asymmetry is abot 1% larger for smooth bed conditions. The model reslts of Herrling et al. (14) for the Ems-river show a somewhat smaller asymmetry vale. The Elbe tidal channel at Cxhaven (moth region) is an exception with a velocity asymmetry factor of.9 (Figre 8), which is most likely related to the presence of wide tidal flats (5 to 1 km wide) in the moth region. The water volme above the flats drains throgh the main channel dring ebb with relatively small water depths reslting in a strong increase of the peak tidal velocity dring ebb. The ratio V s/v c is abot 1 and ˆη o/h o is abot.1 which yields ebb-dominance based on the reslts of Friedrichs and Abrey (1988). Some reslts of the linear model are also shown in Figre 8. A first order estimation of the tidal asymmetry can be obtained by sing the linear model with the same settings at high water and at low water. The tidal asymmetry can be expressed as A=c max/c min with c max= wave speed at high water and c min= wave speed at low water. As the peak tidal velocity is proportional to the wave speed, see Eqation (5c), it follows that A= c max/c min û flood/ û ebb. Very reasonable reslts can be obtained by defining an effective high water depth and low water depth as h max h o+.5 ˆη o and h min h o-.5 ˆη o. The linear model has also been applied for a compond channel with a main channel (b c=5 m; k s=.1 m) and tidal flats (see Figre ) with V s/v c in the range of. to.5, yielding an asymmetry factor between.85 and 1 (ebb dominance) for b flats/b channel= 1. The asymmetry is smallest (.85) for a deep main channel. The asymmetry is abot.75 for b flats/b channel= 4. The asymmetry factor approaches 1 for a shallow main channel with tidal flats. The flood-related asymmetry for prismatic and converging channels withot large-scale tidal flats (trend crve of Figre 8) can be represented by the following fit: a=1+( ˆη o/h o) 1.5. This latter expression (goodness of fit r.4) is sed in the standalone TSAND-model for the comptation of net tide-integrated sand transport rates (Section 4.). Available field data sets shold be stdied to confirm the proposed velocity asymmetry relationship, which is mainly based on model data. Figre Linear model for prismatic channels Linear model prismatic channels with tidal flats Trendcrve Delft DH model for prismatic channel (Van Rijn 11) Delft DH model for strongly converging channel (Van Rijn 11) Eastern Scheldt moth converging channel (Deltares 198) Elbe moth converging channel with flats (BAW 6) Western Scheldt moth converging channel (Deltares 11) Rotterdam Waterway moth prismatic channel (Arcadis 15) Ems moth converging channel (Herrling et al 14) Seine moth converging channel (Breno-LeHir 1999) Delft3D-model Ems river (Pogm, Terborg), (Van Maren 15) Ratio of tidal amplitde and water depth (-) Effect of relative water depth (ratio of tidal amplitde and water depth) on tidal velocity asymmetry in the moth region of prismatic and converging tidal channels based on model reslts 13