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Modelling the formation and the long-term behavior of rip channel systems from the deformation of a longshore bar

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Modelling the formation and the long-term behavior of rip channel systems from the deformation of a longshore bar
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  Modelling the formation and the long-term behaviorof rip channel systems from the deformation of a longshore bar Roland Garnier, 1 Daniel Calvete, 2 Albert Falque´s, 2 and Nicholas Dodd 1 Received 9 November 2007; revised 21 February 2008; accepted 7 April 2008; published 31 July 2008. [ 1 ]  A nonlinear numerical model based on a wave- and depth-averaged shallow water equation solver with wave driver, sediment transport, and bed updating is used toinvestigate the long-term evolution of rip channel systems appearing from the deformationof a longshore bar. Linear and nonlinear regimes in the morphological evolution have beenstudied. In the linear regime, a crescentic bar system emerges as a free instability. Inthe nonlinear regime, merging/splitting in bars and saturation of the growth are obtained.In spite of excluding undertow and wave-asymmetry sediment transport, the initialcrescentic bar system reorganizes to form a large-scale and shore-attached transverse or oblique bar system, which is found to be a dynamical equilibrium state of the beachsystem. Thus the basic morphological transitions ‘‘Longshore Bar and Trough’’  ! ‘‘Rhythmic Bar and Beach’’ ! ‘‘Transverse Bar and Rip’’ described by earlier conceptualmodels are here reproduced. The study of the physical mechanisms allows us tounderstand the role of the different transport modes: The advective part induces theformation of crescentic bars and megacusps, and the bedslope transport damps theinstability. Both terms contribute to the attachment of the megacusps to the crescentic bars.Depending on the wave forcing, the bar wavelength ranges between 180 and 250 m (165and 320 m) in the linear (nonlinear) regime. Citation:  Garnier, R., D. Calvete, A. Falque´s, and N. Dodd (2008), Modelling the formation and the long-term behavior of ripchannel systems from the deformation of a longshore bar,  J. Geophys. Res. ,  113 , C07053, doi:10.1029/2007JC004632. 1. Introduction 1.1. Rip Channel Systems [ 2 ] Many text books on coastal sciences rely on thesimple concept of equilibrium beach profile as a cross-shore bathymetric profile which is invariant along the shore (at least at the length scale of a few times the surf zone width[see, e.g.,  Komar  , 1998]). However, present systematic andcareful observations reveal that rather than being the rulethis is an exception or maybe just a long-term average[ Short  , 1999;  van Enckevort et al. , 2004;  Castelle et al. ,2007;  Ribas and Kroon , 2007]. In fact, the nearshore infront of sandy beaches very often exhibits complex bathy-metric patterns with bars, shoals, troughs, channels andholes. As a result, the cross-shore profiles at distinct crosssections of the same beach may be very different.[ 3 ] The most simple of such bathymetric features are breaker bars which are shore parallel, straight in plan viewand with the bed level at their crest being approximatelyconstant alongshore [  Komar  , 1998;  Short  , 1999]. However,the bed level at the bar may also be alongshore oscillating insuch a way that wide shallow sections alternate with narrowdeep sections, so-called rip channels. The most important feature of such channels is that breaking waves force strong jet-like currents in them which are seaward directed andcalled rip currents. In many cases, when rip channels are present the bar is no longer straight in plan view, but meandering, so that the deepest sections are shifted offshoreand the shallowest sections are shifted onshore. Such barsare called crescentic [ van Enckevort et al. , 2004;  Lafon et al. , 2004, 2005;  Castelle et al. , 2007;  Ruessink et al. , 2007].The alongshore wavelength of the undulations in plan view(or the spacing between rip channels) is of the order of a fewtimes the distance of the bar to the shore and may be quiteregular (although not always). For this reason, such bathy-metric patterns are known as rhythmic topography.[ 4 ] Surf zone bars that are not parallel to the coast but form a certain angle with it have also been described and aregenerically known as transverse bars. More specifically, theterm ‘‘transverse’’ is used when the bars are shore-normaland the term ‘‘oblique’’ is used instead when the bars forman angle between 0 and 90   with the shore normal. Severalsuch bars may appear along the coast with a spacing of theorder of one to a few times the surf zone width. Again, thespacing may be quite regular so that the pattern is alsoknown as rhythmic topography. In many cases, the bars areconnected to a shore with a cuspate shoreline, the cusps(megacusps) being associated with the bar attachments[  Evans , 1938;  Komar  , 1998;  Short  , 1999].[ 5 ] While crescentic bars seem to be associated to a preexisting straight bar, the srcin of transverse bars is not  JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 113, C07053, doi:10.1029/2007JC004632, 2008 Click Here for Full Article 1 School of Civil Engineering, University of Nottingham, Nottingham,UK. 2 Departament de Fı´sica Aplicada, Universitat Polite´cnica de Catalunya,Barcelona, Spain.Copyright 2008 by the American Geophysical Union.0148-0227/08/2007JC004632$09.00 C07053  1 of 18  so clear and there are several types. Awell defined class arethe shoals that develop from the shallowest sections of acrescentic bar that progresses onshore until they attach tothe shoreline. This process and the corresponding morphol-ogy was described by  Wright and Short   [1984] in the caseof single-barred systems in macrotidal, wave-dominatedenvironments with relatively small bars and is essential totheir ‘‘TBR’’ (‘‘Transverse Bar and Rip’’) beach state. The barsmaybeexactlyshore-normalorslightlyoblique.Wewillrefer to them as ‘‘large-scale transverse bars’’ [  MacMahan et al. , 2005;  Castelle et al. , 2007]. Another type are theintertidal oblique bars, which are very common along theFrench Aquitanian coast  [  Lafon et al. , 2002, 2004, 2005;  De Melo Apoluceno et al. , 2002;  Castelle et al. , 2007]. Theywerepreviouslycalled‘‘ridgeandrunnelsystems’’sincetheyappear in the intertidal zone. However, they differ from thetypical ridge and runnel systems described for instance in Short   [1999];  Kroon and Masselink   [2002];  Masselink and  Kroon  [2006], because, in particular, (1) they exhibit a clear alongshore rhythmicity and (2) the crests are separated by adeep channel. They are actually similar to the large-scaletransverse bars except that they show an oblique down-current orientation, i.e., they appear skewed down current when viewed from the shore (Figure 1a). They will bereferred to as ‘‘large-scale oblique bars’’. In this paper weconsider these two types of transverse/oblique bars that wecall ‘‘large scale’’, to distinguish them from a third type that has been called ‘‘transverse/oblique finger bars’’ by  Ribasand Kroon  [2007], and was previously described by  Nieder-oda and Tanner   [1970];  Falque´s  [1989];  Komar   [1998];  Konicki and Holman  [2000]. These are thin and elongatedoffshore, and the spacing between them is smaller.[ 6 ] Although the existing literature set up a clear classi-fication of all these features [  Komar  , 1998] improvement inobservation techniques (e.g., Argus System) and the onset of mathematical morphodynamical models reveals that: i)the old classification is overwhelmed by the increasingcomplexity of observations that suggests more types andsubtypes than previously foreseen; and ii) different mor- phologies can be genetically linked, as they are different stages of the same morphodynamical process. Nevertheless,a common characteristic of all these surf zone bathymetric patterns that are alongshore rhythmic (or quasi-rhythmic) isthe alongshore succession of shoals and deeps, with break-ing waves forcing rip currents at the deeps. Thus we willrefer generically to such systems as ‘‘rip channel systems’’. 1.2. Modelling Rip Channel Systems [ 7 ] The srcin of rip channel systems was attributed in the past to the hydrodynamical forcing by infragravity edgewaves [  Bowen and Inman , 1971;  Holman and Bowen ,1982]. However, it has been more recently found that  positive feedbacks between waves, currents and morpholo-gy may render unstable an alongshore uniform surf zoneand may thus lead to the formation of rhythmic topography.Even if the hydrodynamic forcing could have some influ-ence as an initial triggering of those instabilities, the actualshape of the bathymetric patterns is eventually dominated by those feedbacks. This has been shown by numerousmodelling studies regarding the different types of barsduring the last decade. For example, the formation of ripchannel systems and transverse/oblique bars in planar  beaches (i.e., unbarred) has been examined by  Christensenet al.  [1994];  Falque´s et al.  [1996, 2000];  Caballeria et al. [2002];  Ribas et al.  [2003] and  van Leeuwen et al.  [2006].The tendency of a straight shore parallel bar to develop ripchannels and to become crescentic has been investigated by  Deigaard et al.  [1999];  Damgaard et al.  [2002];  Reniers et al.  [2004];  Calvete et al.  [2005, 2007];  Dronen and  Deigaard   [2007];  Klein and Schuttelaars  [2006], the latter study actually dealing with a double barred system.[ 8 ] Many of these studies use linear stability analysis toinvestigate the tendency to form rip channels from an initialfeatureless beach and to elucidate the nature of the feedback that is behind this process. To track the actual growth of therip channel systems up to an amplitude comparable withnatural systems a nonlinear stability analysis is necessaryand this is done in some of those studies. However, none of them is able to run for a long time, i.e., more than a fewtimes the typical growth time. Usually, the models break  Figure 1.  (a) Large-scale oblique bars (sometimes calledridge and runnel system). French Aquitanian coast. IGNParis 2007. (b) Coordinate system. The  x ,  y , and  z   axes (  x 1 ,  x 2 , and  x 3  axis, respectively) stand for the cross-shore, thelongshore, and the vertical directions, respectively. Thecoastline is at   x  = 0.  z  s  is the sea level,  D  is the water depth, h  is the bottom perturbation with respect to the long-itudinally uniform initial topography,  z   b0 is the initial bedlevel, and  z   b  is the bed level. C07053  GARNIER ET AL.: LONG-TERM BEHAVIOR OF RIP CHANNELS2 of 18 C07053  down before or just when the amplitude of the bars iscomparable to that observed in nature [ Caballeria et al. ,2002;  Dronen and Deigaard  , 2007]. In other models thewater depth above the bar crest keeps on decreasing untilzero and the model breaks down at this moment  [  Damgaard et al. , 2002]. As a consequence, none of these models candescribe the long-term behavior of the system and for thisreason they cannot describe the ‘‘finite amplitude dynamics’’and in particular, the transitions from one type of rhythmicsystem to another.[ 9 ] However,  Garnier et al.  [2006a] have recently shownthat an adequate treatment of the gravitational downslopesediment transport together with a simplified description of wave refraction allows for long-term runs (  100 day or more) of the morphodynamical nonlinear stability modelMORFO55. The model is able to describe the formation, thesaturation of the growth and the finite amplitude dynamicsof transverse/oblique bars on a planar beach and, despite thesimplifications, model results are fairly consistent withobservations. Preliminary computations for a barred beachhave shown that the model may also predict the formationof large-scale oblique bars from an initially straight bar,which becomes crescentic and further evolves into thetransverse/oblique bar system  Garnier et al.  [2006b]. Thisis one of the basic transitions described by  Wright and Short  [1984] (‘‘RBB’’  !  TBR, i.e., ‘‘Rhythmic Bar and Beach’’to Transverse Bar and Rip morphology), and conceptualmodels had been presented by  De Melo Apoluceno  [2002]; Castelle et al.  [2007]. The first modelling study has beenmade by  Ranasinghe et al.  [2004] who reproduce the RBB !  TBR transition of an event in Palm Beach, Australia. Inagreement with the observation, their numerical simulationsshow that this transition can occur for reduced incident wave conditions that follow a stronger event when the RBBstate formed. However, their initial topography is based on a preexisting RBB state given by Argus images and their model was unable to reach this state. 1.3. Objectives [ 10 ] The aim of the present contribution is to conduct asystematic nonlinear instability analysis of a single-barred beach by using the MORFO55 model. The objectives are:(1) properties of the saturation and nonlinear dynamics. In particular, quantification of the amplitude, existence of anew (dynamical?) equilibrium and systematic modelling of the RBB  !  TBR transition that was initiated in  Garnier et al.  [2006b], and (2) comparison of the initial formation of acrescenticbar with thelinear stabilitystudies of  Calvete et al. [2005]. In particular, possible formation of megacusps andtransverse bars at the shore coupled with the crescentic bar.[ 11 ] This paper is organized as follows: section 2 isdedicated to a description of the model and of the experi-mental setup, section 3 presents the main results; the physical mechanisms are explained in section 4, andfinally an overall discussion and a conclusion are given insections 5 and 6, respectively. 2. Numerical Model 2.1. Set of Equations [ 12 ] TheMORFO55modelisbasedonthephase-averagednonlinear shallowwaterequations [  Mei ,1989; Garnieretal. ,2006a]. It is applied to a rectilinear beach defined by thecoordinate system ( O ,  x ,  y ,  z  ), or ( O ,  x 1 ,  x 2 ,  x 3 ), where [ O ,  x )stands for the positive seaward cross-shore direction, [ O ,  y ),for the longshore direction and [ O ,  z  ), for the positiveupward vertical direction (see Figure 1b). The set of sixwave and depth-averaged equations comprises the water mass conservation equation (1), the momentum conserva-tion equation (2), the wave energy equation (3), the Snell’slaw (4) and the sediment mass conservation equation (5).They read (repeated indices indicate summation with  i ,  j   =1, 2;  t   is time): @   D @  t   þ  @ @   x  j   Dv   j    ¼ 0  ;  ð 1 Þ @  v  i @  t   þ v   j  @  v  i @   x  j  ¼  g  @   z  s @   x i   1 r  D @ @   x  j  S  0 ij   S  00 ij      t   b i r  D  ;  ð 2 Þ @   E  @  t   þ  @ @   x  j  v   j  þ c g  j     E    þ S  0 ij  @  v   j  @   x i ¼ e  ;  ð 3 Þ k   sin q ¼ k  0 sin q 0 ;  ð 4 Þ @   z   b @  t   þ @  q  j  @   x  j  ¼ 0  :  ð 5 Þ [ 13 ] The six time- and depth-averaged dynamicalunknowns are: the sea level  z  s (  x 1 ,  x 2 ,  t  ), the two components u  and  v   ( v  1  and  v  2 , respectively) of the horizontal velocity ~ v   (  x 1 ,  x 2 ,  t  ), the wave energy density  E  (  x 1 ,  x 2 ,  t  ), the waveangle  q (  x 1 ,  x 2 ,  t  ) and the bed level  z   b (  x 1 ,  x 2 ,  t  ).[ 14 ] The other variables are defined as follows.  D  is thetotal mean depth (  D  =  z  s   z   b ).  g   is the acceleration due togravity (  g   = 9.8 m s  2 ).  r  the water density ( r  = 1024 kgm  3 ).  S  0 is the wave radiation stress tensor from  Longuet- Higgins and Stewart   [1964].  S  00 is the turbulent Reynoldsstress tensor [  Battjes , 1975].  t   b  is the bed shear stress vector [  Mei , 1989;  Garnier  , 2007].[ 15 ] The relationship between the wave energy  E   and theroot mean square wave height   H  rms  is given by:  E   =  r  g  H  rms2 /8.  ~ c  g   is the group velocity vector.  e  is the dissipationrate because of wave breaking [ Thornton and Guza , 1983]and bottom friction [  Horikawa , 1988;  Garnier et al. ,2006a].[ 16 ] In the Snell’s law (4), the wave number   k   is themodulus of the wave vector   ~ k   and is computed using thedispersionrelation, q isdefinedastheanglebetweenthewaverays and the  x  axis,  k  0 and  q 0 are the wave number and thewave angle at the seaward boundary. Notice that by usingthis approximation, the wave topography interaction be-cause of wave refraction are not taken into account. Thisseems to be correct as the results by using the Eikonalequation do not essentially change. However, we decide tokeep the Snell’s law in order to simplify the equation systemand focus on the main mechanisms at the srcin of the beachinstability.[ 17 ] The horizontal sediment flux vector is based on thetotal load formula of Soulsby and Van Rijn [ Soulsby , 1997] C07053  GARNIER ET AL.: LONG-TERM BEHAVIOR OF RIP CHANNELS3 of 18 C07053  (see details in  Calvete et al.  [2005] and  Garnier et al. [2006a]). It reads: ~ q ¼ a  ~ v   g   u  b  ~ r h    ;  ð 6 Þ where  a  is the stirring factor,  g   is the bedslope coefficient, u  b  is the root mean square wave orbital velocity amplitudeat the bottom and  h  is the bed level deviation from initialequilibrium ( h  =  z   b   z   b0 , where  z   b0 is the initial bed level, seeFigure 1b). In order to simplify the notations, the bed porosity  p  = 0.4 has been included in  a : a ¼  11   p A S  u s  u c ð Þ 2 : 4 if   u s  >  u c ¼ 0 otherwise ; where  A s  and  u c  depend essentially on sediment character-istics and water depth [ Soulsby , 1997]. The stirring velocity u s  reads: u s  ¼ j ~ v  j 2 þ 0 : 018 c D u 2 b   1 = 2 ; c D  being the morphodynamical drag coefficient [ Soulsby ,1997].[ 18 ] There is an important degree of uncertainty in thevalue of the bedslope coefficient   g   [ Garnier et al. , 2006a].In the model it is chosen so as to have realistic results and to perform long-term evolutions. Here, it is fixed to  g   = 5which is higher than the default case of   Garnier et al. [2006a] ( g   = 1.5). With  g   = 1.5, the large-scale instabilitiesobtained here end up in an area that is too shallow close tothe shoreline leading to negative depth and model overflow.Since it depends on the mean current   ~ v   and on the bed perturbation  h , the onshore transport driven by wave non-linearity and undertow is excluded. In fact, this onshoretransport is assumed to be in balance with the gravitationaldownslope transport which takes into account the total beach slope (  ~ r  z   b ). The stirring factor   a  depends essentiallyon the water depth, on the current velocity magnitude, on  u  b and on the sediment characteristics. In particular, the sed-iment grain size has been chosen as  D 50  = 0.25 mm and the bed roughness length as  z  o  = 6 mm.[ 19 ] Periodic boundary conditions for each variable andfor its first   y  derivative are applied at the lateral boundaries.For more details on the offshore and shoreline boundary Figure 2.  Basic states. Cross-shore profiles at the longitudinally uniform equilibrium state obtained inthe case of nonperturbed initial topography (oblique wave incidence). Black thick line: default case.Dashed gray line: variation of wave period. Dashed black/gray line: variation of incident wave height.Gray line: variation of incident wave angle. (a) Hydrodynamical variables: root mean square wave height   H  rms , sea level  z  s , cross-shore velocity  v  , wave angle  q , and bed level  z   b . (b) Morphodynamical variables:stirring factor   a , potential stirring  C  , root mean square wave orbital velocity amplitude at the bottom  u  b ,and bed level  z   b . C07053  GARNIER ET AL.: LONG-TERM BEHAVIOR OF RIP CHANNELS4 of 18 C07053  conditions and on the numerical methods, please refer to Garnier et al.  [2006a] and  Garnier   [2007]. 2.2. Setup of the Numerical Simulations [ 20 ] The initial topography is assumed to be a longitudi-nally uniform longshore barred beach where a small per-turbation has been added. Thus the initial topography can bewritten as:  z   b  x ;  y ; t   ¼ 0 ð Þ¼  z  0 b  x ð Þþ h x ;  y ; t   ¼ 0 ð Þ  :  ð 7 Þ [ 21 ] The initial equilibrium barred beach profile  z   b0 (  x ) is based on the bar system at Duck, North Carolina [ Yu and Slinn , 2003]:  z  0 b  x ð Þ¼ a 0  a 1  1  b  2 b  1   tanh  b  1  xa 1    b  2  x þ a 2  exp   5  x   x c  x c   2 " #  ;  ð 8 Þ where  x c  is the bar location (  x c  = 80 m) and  a 2  is the bar amplitude (default case:  a 2  = 1.5 m). The height of the water depth at the swash/surf zone boundary is  a 0  = 25 cm and a 1  = 2.97 m. The shoreline and offshore slopes are  b  1  =0.075 and  b  2  = 0.0064, respectively. The bottom plot of Figures 2a and 2b show the equilibrium profile  z   b0 (  x ) whilethe top plot of Figure 3a shows the three-dimensional viewof a part of the initial bathymetry  z   b (  x ,  y ,  t   = 0). The same profile has been used in the linear stability analysis of  Calvete et al.  [2005], which provides a useful tool for validating our initial results.[ 22 ] The initial perturbation  h (  x ,  y ,  t   = 0) is a Dirac deltafunction in order to not excite preferentially a particular mode. It has been fixed at:  h (  x  = 50 m,  y  = 1000 m,  t   = 0) =3 cm and  h (  x  6¼  50 m,  y  6¼  1000 m,  t   = 0) = 0. Notice that the growth rate of the emerging instabilities does not dependeither on the location, or on the amplitude of the local peak.[ 23 ] Experiments have been done on the domain defined by: 0   x   L  x  = 250 m and 0   y   L  y   = 2000 m. The gridspacing is given by ( D  x ,  D  y ) = (5, 10) m. The hydrody-namical time step  D  t   = 0.05 s. The morphodynamical processes have been artificially accelerated by a factor 90[see  Caballeria et al. , 2002;  Garnier et al. , 2006a] so that the morphodynamical time step is  D  t  m  = 90  D  t   = 4.60 s. Notice that the use of a factor 1 does not change the initialgrowth rate of the instabilities, and the entire evolution isthe same by using a factor from 50 to 150. Results are givenup to 300 days of morphological evolution.[ 24 ] Two reference cases are described: (1) for normalwave incidence and (2) for oblique wave incidence with awave angle of   q (  L  x ,  y ,  t  ) =  q 0 = 4   at the seaward boundary(in 4.5 mdepth). The height of incident waves  H  rms (  L  x ,  y , t  ) =  H  rms0 = 1 m at the seaward boundary and the wave period T   = 6 s. Figure 3.  Normal wave incidence. Snapshots of a part of the topography during the formation,development, and growth saturation of crescentic bar system. (a) Tridimensional view of the bed level  z   b .(b) Top view of the bottom perturbation  h  (in meters). C07053  GARNIER ET AL.: LONG-TERM BEHAVIOR OF RIP CHANNELS5 of 18 C07053
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