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Mechanisms controlling crescentic bar amplitude

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Mechanisms controlling crescentic bar amplitude
Roland Garnier,
1,2
Nicholas Dodd,
1
Albert Falqués,
3
and Daniel Calvete
3
Received 2 June 2009; revised 5 November 2009; accepted 17 December 2009; published 15 April 2010.
[
1
]
The formation of crescentic bars from self
‐
organization of an initially straight shore
‐
parallel bar for shore
‐
normal incident waves is simulated with a two
‐
dimensionalhorizontal morphodynamical model. The aim is to investigate the mechanisms behind thesaturation process defined as the transition between the linear regime (maximum andconstant growth of the crescentic pattern) and the saturated state (negligible growth). Theglobal properties of the morphodynamical patterns over the whole computationaldomain are studied (
“
global analysis
”
). In particular, consideration of the balance of the potential energy of the emerging bar gives its growth rate from the difference between a production term (related to the positive feedback leading to the instability) and a dampingterm (from the gravity
‐
driven downslope transport). The production is approximately proportional to the average over the domain of the cross
‐
shore flow velocity times the bed level perturbation. The damping is essential for the onset of the saturation, but it remains constant while the production decreases. Thus, it is notable that the saturationoccurs because of a weakening of the instability mechanism rather than an increase of thedamping. A reason for the saturation of the crescentic bar growth is the change in bar shapefrom its initial stage rather than the growth in amplitude itself. This change is mainlycharacterized by the narrowing of the rip channels, the onshore migration of the crests,and the change in the mean beach profile due to alongshore variability. These propertiesagree with observations of mature rip channel systems in nature.
Citation:
Garnier, R., N. Dodd, A. Falqués, and D. Calvete (2010), Mechanisms controlling crescentic bar amplitude,
J. Geophys. Res.
,
115
, F02007, doi:10.1029/2009JF001407.
1. Introduction
[
2
] The surf zone of sandy barred beaches is characterized by the presence of one or several shore
‐
parallel bars.They are not always straight in plan view but are oftenmeandering with deep and shallow sections alternatingalong the bars with a striking regularity. These bars arecalled crescentic bars (or lunate bars) and are probably themost documented and observed rhythmic features in thesurf zone [
Wright and Short
, 1984;
Short
, 1999;
van Enckevort et al.
, 2004;
Lafon et al.
, 2004;
Castelle et al.
, 2007;
Ruessink et al.
, 2007]. They are associated withthe typical current circulation of strong jet
‐
like offshore
‐
oriented currents (called rip currents) in the deep sections(called rip channels) and weaker wider onshore currents inthe shallow sections.[
3
] Understanding the formation and the evolution of crescentic bar systems is an active area of research.Although their formation had been attributed to the hydro-dynamical forcing of infragravity edge waves [
Bowenand Inman
, 1971;
Holman and Bowen
, 1982], it is wellaccepted nowadays that the feedback from the morphologyinto the flow is the primary cause of their formation[
Deigaardetal.
,1999;
Falquésetal.
,2000;
Damgaardetal.
,2002;
Reniers et al.
, 2004;
Klein and Schuttelaars
, 2006;
van Leeuwen et al.
, 2006;
Calvete et al.
, 2007;
Dronen and Deigaard
, 2007;
Garnier et al.
, 2008;
Smit et al.
, 2008].Apart from
Dronen and Deigaard
[2007], who used a quasi
‐
three
‐
dimensional area (Q3D) model, all these self
‐
organization studies are based on wave
‐
and depth
‐
averaged process
‐
based (two
‐
dimensional horizontal (2DH)) mor- phological modeling. They show that crescentic bar systemswould emerge from a free instability because of a positivefeedback between waves, currents, and morphology.[
4
] Under normal or near
‐
normal wave incidence, the positive feedback is explained by the bed surf mechanism[
Falqués et al.
, 2000;
Caballeria et al.
, 2002;
Ribas et al.
,2003;
Calvete et al.
, 2005;
Garnier et al.
, 2008]. Crescen-tic bars are sometimes viewed as two adjacent series of shoals and troughs. These series are antisymmetric withrespect to a line parallel to the coast. In particular, the bedsurf mechanism explains that these crescentic features canappear on an alongshore uniform beachif thedepth
‐
averagedsediment concentration profile admits a local maximum. The position of the maximum defines the antisymmetric axis,
1
Environmental Fluid Mechanics Research Centre, Process andEnvironmental Division, Faculty of Engineering, University of Nottingham, Nottingham, UK.
2
Now at Departament de Física Aplicada, Universitat Politécnica deCatalunya, Barcelona, Spain.
3
Departament de Física Aplicada, Universitat Politécnica de Catalunya,Barcelona, Spain.Copyright 2010 by the American Geophysical Union.0148
‐
0227/10/2009JF001407
JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 115, F02007, doi:10.1029/2009JF001407, 2010
F02007
1 of
14
which, for a barred beach, is close to the top of the bar.
Garnier et al.
[2008] showed that, for random waves over a barredbeach,theinnerseries(shoreward oftheaxis)emergesfrom a free instability but the outer is forced by the inner sothat its amplitude is weaker and crescentic bars are mainlydefined by the inner series.[
5
] Most of these modeling studies, based either on thelinear stability analysis or on nonlinear modeling, are lim-ited to the bar formation and to the earliest stage of the bar evolution because of the model formulation (linear models)or because the nonlinear models break down. The reason for this nonlinear model limitation is not clear and has not beeninvestigated. The reason is not necessarily numerical; it could be a lacuna in the physics, casting doubt on whether the saturation of the growth of crescentic bars can bedescribed by numerical models. Recently,
Garnier et al.
[2008] reproduced this saturation by using a simple 2DHmodel, but the physical mechanisms are still unexplored.They reached the nonlinear regime and were able to simulatethe finite amplitude behavior of crescentic bars, showing that the final bar shape differs from the initial one: the channels become narrower than the crests (this is associated with jet
‐
like rip currents), the channels tend to migrate offshore,and the shoals tend to migrate onshore.[
6
] Finite amplitude modeling of morphological featuresis fundamental for any comparison with observations. Thesaturation of the bed form growth has been obtained for other morphodynamical systems such as shoreface
‐
connectedsand ridges [
Calvete and de Swart
, 2003;
Roos et al.
, 2004;
Vis
‐
Star et al.
, 2008], sand ripples [
Marieu et al.
, 2008],and shore
‐
transverse sandbars [
Garnier et al.
, 2006]. Amethod to investigate the mechanisms of saturation wasintroduced by
Garnier et al.
[2006] and extended by
Vis
‐
Star et al.
[2008]; it is called the global analysis. It consists of studying the global properties of the bed forms over thewhole computational domain by deriving a potential energy balance of the bed forms. For both the shoreface
‐
connectedridges and the transverse bars, the saturation of the bedforms occurs because of the balance between a productionand a damping term. However, the reason for this balancehas not yet been explained.[
7
] The objective of this contribution is to investigate themechanisms behind the saturation of the growth of cres-centic bars from the numerical experiments of
Garnier et al.
[2008] by using the global analysis. It is organized asfollows. Section 2 presents the methodology by introducingthe equations and the hypothesis necessary to understandthe derivation of the global analysis. The general resultsand the main variables of the analysis are given in section 3.The physical mechanisms are explained in section 4.Section 5 gives the conclusions.
2. Methodology
2.1. Governing Equations
[
8
] The 2DH MORFO55 model solves the phase
‐
averagednonlinear shallow water equations with sediment transport and bed updating [
Mei
, 1989;
Caballeria et al.
, 2002;
Garnier et al.
, 2006, 2008]. The water mass (equation (1)),
the momentum (equation (2)), and the sediment mass(equation (3)) conservation equations read (repeated indicesindicate 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
D
@ @
x
j
S
0
ij
S
00
ij
bi
D
;
ð
2
Þ
@
z
b
@
t
þ
@
q
j
@
x
j
¼
0
;
ð
3
Þ
where
D
isthetotalmeandepth,
v
isthedepth
‐
averagedmeanvelocity vector (
v
= (
v
1
,
v
2
) = (
u
,
v
)),
g
is the acceleration dueto gravity (
g
= 9.8 m s
−
2
),
z
s
is the mean sea level,
r
is thewaterdensity(
r
=1024kgm
−
3
),
S
′
isthewaveradiationstresstensor,
S
″
istheturbulentReynoldsstresstensor,
t
b
isthebedshear stress vector,
z
b
is the bed level, and
q
is the horizontalsediment flux vector. Note that the bed porosity effect has been included in the sediment flux in order to simplify thenotations. The wave field is solved by using the wave energydissipation equation and Snell
’
s law; the wave current inter-action has been removed as it only has a small effect on the presented results. For more details, refer to
Garnier et al.
[2006, 2008].[
9
] An initial topography that is strictly alongshore uni-form forced by a stationary alongshore uniform wave field isconsidered. Since the dynamics of crescentic bars areassumed to be mainly governed by rip current circulation,the cross
‐
shore transport driven by undertow, wave non-linearities, and gravity is disregarded. This means that those contributions are considered to be in balance for this basic state, which is therefore an equilibrium state.When departures from this bathymetry develop, the sediment flux
q
does not vanish. It is based on the Soulsby
–
Van Rijntotal load formula [
Soulsby
, 1997] (see
Garnier et al.
[2008]for details) and reads
q
¼
v
u
b
r
h
ð Þ
;
ð
4
Þ
where
a
is the stirring factor, which includes the bed porosity
p
= 0.4 (
a
=
a
svr
/(1
−
p
));
g
is the bed slope coefficient;
u
b
isthe root
‐
mean
‐
square wave orbital velocity amplitude at the bottom; and
h
is the bed level deviation from initial equilib-rium (
h
=
z
b
−
z
b
0
,where
z
b
0
istheinitial bedlevel).The stirringfactor
a
svr
is computed as follows:
svr
¼
A
S
u
s
u
c
ð Þ
2
:
4
if
u
s
>
u
c
svr
¼
0 otherwise
;
according to
Soulsby
[1997], where
A
S
and
u
c
dependessentially on sediment characteristics 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]. Essentially,
a
v
describes the contribution fromthe circulation, and
−
ag
u
b
r
h
describes the contribution
GARNIER ET AL.: MECHANISMS CONTROLLING BAR AMPLITUDE
F02007F02007
2 of 14
that would bring the bathymetry back to the equilibrium profile if there were no circulation.
2.2. Bottom Evolution Equation Approximation
[
10
] The Bottom Evolution Equation (BEE) has beenintroduced by
Falqués et al.
[2000] as an approximateexpression of bed changes to facilitate understanding bedevolution in connection with hydrodynamics. For complete-ness,webrieflyrevisititinourcontext.Byusingthesediment transport formula (equation (4)), the sediment conservationequation (equation (3)) reads
@
h
@
t
¼rrrrrrr
v
ð Þþrrrrrrr
rrrrrrr
h
ð Þ
;
where
G
=
g a
u
b
. According to the water mass conservationequation (equation (1)),
rrrrrrr
v
ð Þ¼rrrrrrr
C D
v
ð Þ¼
D
v
rrrrrrr
C
C
@
D
@
t
;
where
C
is the equivalent depth
‐
averaged concentration(
C
=
a
/
D
), also called the potential stirring.[
11
] From the combination of the two previous equations,and by assuming
∣∂
D
/
∂
t
∣
’
∣∂
h
/
∂
t
∣
(the flow is assumed toadjust instantaneously to the bed changes), we obtain
1
C
ð Þ
@
h
@
t
’
D
v
rrrrrrr
C
þrrrrrrr
rrrrrrr
h
ð Þ
:
Finally, by using the approximation
C
1 (for instance,
C
’
0.001 from
Garnier et al.
[2008]), the BEE reads
@
h
@
t
’
D
v
rrrrrrr
C
þrrrrrrr
rrrrrrr
h
ð Þ
:
ð
5
Þ
2.3. Global Analysis
[
12
] The global analysis of beach evolution was intro-duced by
Garnier et al.
[2006] and consists of analyzingvariables that are integrated over the computational domain.It differs from the local analysis used by
Garnier et al.
[2008], which can only explain the formation of featuresand not the saturation of the growth. The limitation of thelocal analysis can be understood because the bars can still bein movement while their growth on average is already sat-urated, so that some sort of equilibrium is reached (we refer to this as a
“
dynamical equilibrium
”
). For instance, for oblique waves, an equilibrium state is reached, but the barsstill migrate; thus, the local analysis still predicts erosionand deposition at some locations. The global analysis for theevolution of transverse bar systems appearing on a planar beach [
Garnier et al.
, 2006] is extended here to the case of rip channels developing from the deformation of an initiallyalongshore uniform parallel bar obtained by
Garnier et al.
[2008].[
13
] We first introduce the overbar notation to define anaverage over the computational domain. It reads, for anyfunction
f
=
f
(
x
,
y
),
f
¼
1
L
x
L
y
Z
L
y
0
Z
L
x
0
f
d
x
d
y
;
where
L
x
(
L
y
) is the cross
‐
shore (alongshore) length of thecomputational domain. Following
Garnier et al.
[2006], the production
P
and the damping
D
can be defined by
P ¼
h
rrrrrrr
v
ð Þ
;
ð
6
Þ
¼
h
rrrrrrr
rrrrrrr
h
ð Þ
:
ð
7
Þ
The production term
P
comes from the first contribution of the sediment flux vector
q
(equation (4)) (advective part).According to BEE (equation (5)), it can be approximated by
P ’
hD
v
r
C
ð
8
Þ
and therefore measures the tendency for growth or decay of bars by the bed flow couplings and bed surf couplings[
Garnier et al.
, 2006]. The damping term
D
comes from thedownslope or diffusive contribution of
q
.[
14
] By using the same definition as
Vis
‐
Star et al.
[2008],the
“
global growth rate
”
s
of the instability reads
¼
1
k
h
k
2
d dt
12
k
h
k
2
;
ð
9
Þ
where
k
h
k
is the
L
2
norm of
h
and is defined as
k
h
k¼
h
2
1
=
2
;
so that
k
h
k
2
can be interpreted as the potential energydensity of the bed forms [
Vis
‐
Star et al.
, 2008].[
15
] To illustrate the physical meaning of
s
, let us con-sider a topographic perturbation like
h x
;
y
;
t
ð Þ¼
exp
0
t
ð Þ
H x
;
y
ð Þ
;
ð
10
Þ
which represents a topographic wave that exponentiallygrows with a growth rate
s
0
and which keeps a constant shape given by
H
(
x
,
y
), with
H
being an
L
y
periodic functionwith respect to
y
. This occurs, for instance, during the linear regime (initial stage) of the bar evolution. Then,
t
ð Þ¼
0
:
[
16
] More generally, the sign of
s
determines the different state of the bar evolution: the growth (decay) of the bars can be characterized by
s
0 (
s
< 0), while the saturation of bars occurs when
s
= 0. Moreover, the global growth ratecan be computed with the relationship
¼
1
k
h
k
2
P
ð Þ
;
ð
11
Þ
which can be obtained from the definition (equation (9)) bymultiplying the BEE (equation (5)) by
h
and integrating.
3. Model Results
[
17
] The morphodynamical evolution of an initiallyalongshore uniform parallel barred beach is studied for hundreds of days. A numerical experimental setup similar to that of
Garnier et al.
[2008] is used. Waves are assumedto arrive normal to the coast. At the offshore boundary
GARNIER ET AL.: MECHANISMS CONTROLLING BAR AMPLITUDE
F02007F02007
3 of 14
(
x
=
L
x
= 250 m) the height of the incident waves is
H
rms0
= 1 m, and the period is
T
= 6 s. By perturbingthe initial topography, instability develops, and an equilib-rium state is eventually reached. The final state is presentedin Figures 1a
–
1c for a random perturbation (perturbationamplitude
∼
1 mm). The evolution of the bed profile is dis- played in Figure 1h along a channel and along a crest (asindicated in Figure 1c). It shows that the shore
‐
parallel bar crest, initially at 2 m depth (black dots), subsequently risesup to 1.5 m depth on the crescentic horns (darkest solid line).[
18
] Figures 1d
–
1g show time series of variables takenfrom the longshore section defined at
x
= 50 m. Figure 1d is
Figure 1.
Final state (day 100): (a) top view of the bed level
z
b
and current vectors
v
and (b) top view of the bed level perturbation
h
and its contour. Solid (dashed) lines represent the crests (troughs). (c) Tridi-mensional view of the bed level
z
b
. Time series: (d)
h
(
x
= 50 m,
y
,
t
), bed level perturbation along thelongshore section
x
= 50 m (the darker colors represent the deeper areas); (e)
F
(
x
= 50 m,
l
,
t
), its Fourier transform (the darker colors correspond to the more predominant wavelengths); (f)
l
m
(
x
= 50 m,
t
),resulting predominant wavelength; and (g) growth rates computed with different formula. Black thick line,
s
, (global) growth rate computed with relationship (11). Gray thin lines,
s
m
, growth rates corresponding to
l
m
of Figure 1f. Different gray levels are used to distinguish the two wavelengths. (h) Bed level profileevolution of a crest (solid lines) and of a channel (dashed lines). The selected sections are shown inFigure 1c. The darker the lines are, the longer the evolution is. The dots indicate the initial bed level.
GARNIER ET AL.: MECHANISMS CONTROLLING BAR AMPLITUDE
F02007F02007
4 of 14
the bed level variation, Figure 1e is its Fourier analysis,Figure 1f is the predominant wavelength (
l
m
), and inFigure 1g the growth rates (predominant
s
m
and global
s
)are displayed. The predominant wavelength
l
m
(Figure 1f)corresponds to the maximum Fourier coefficient at eachtime step (plotted in Figure 1e). The growth rate (
s
m
)corresponding to
l
m
is displayed in Figure 1g. Notice that similar results are obtained if the Fourier analysis is made at another cross
‐
shore location where crescentic bars develop.Thus, these wavelengths correspond to the same unstablemode, that is, the crescentic bar system. At the final state,
l
m
= 180 m, and the corresponding initial growth rate(from Fourier analysis) is about
s
m
= 0.6 d
−
1
. This growthrate is similar to the initial growth rate corresponding to
l
m
= 200 m, which is dominant from day 3 to day 65. Notice that these initial growth rates are taken during the first period where they are constant in time: this period corre-sponds to the time when the mode amplitude grows expo-nentially, i.e., to the linear regime. Interestingly, thesegrowth rates obtained from Fourier analysis are similar tothe global growth rate
s
computed from relationship (11)(Figure 1g, thick black line), which takes into account the overall patterns, not only the features appearing in thesection
x
= 50 m.[
19
] The saturation of features begins when
s
decreases(Figure 2a). The saturated state is defined as the state for which for the first time
s
’
0. This state can be highlydynamical, in particular if there is merging of bars [
Garnier et al.
, 2006]. The equilibrium state is reached when
s
= 0 for all times. The time taken to reach the equilibrium state issometimes very long, as the growth rate during the saturatedstate is small; it can be an order of magnitude longer thanthe time corresponding to the saturation processes. Becauseequilibrium is sometimes not observed, its existence issometimes unknown. Here our interest is the saturation process, so we want to understand why
s
decreases. Thiscan be due to either the decrease of the production (moreexactly, of
P
/
k
h
k
2
) or the increase of the damping term
D
/
k
h
k
2
.[
20
] As Figure 3 shows, there are two ways to observe theevolution of
s
and all the variables: as a function of
t
(Figures2aand3a
–
3e)orof
k
h
k
(Figure2borFigures3f
–
3j).The condition of instability is given by
s
0; that is, the bars grow if
P
D
. It is found that
P
and
D
are verysimilar (Figures 3a and 3f), only their small differencesexplaining the instability (Figures 3b and 3g).[
21
] The difference in the normalized variables
P
/
k
h
k
2
and
D
/
k
h
k
2
, i.e.,
s
, should be constant in time for the initialgrowth of any linearly unstable mode. Because of non-linearities, it is not constant, and we see that the dynamics of these variables are better represented by analyzing their variations as a function of
k
h
k
(Figure 3h) rather than time.To be precise, we remark that
D
/
k
h
k
2
is constant during thesaturation process, while
P
/
k
h
k
2
is only constant at theinitial stage, decreasing thereafter until it balances
D
/
k
h
k
2
.[
22
] Thus, the saturation seems to be due to the reductionof the production term rather than an increase of thedamping. This will be analyzed in detail in section 4.
4. Physical Mechanisms
4.1. Analysis of the Damping Term
[
23
] Integrating by parts, and because of the boundaryconditions, i.e., because
a
(
x
= 0)
’
0,
h
(
x
=
L
x
)
’
0, andsince
h
is
L
y
periodic, we can write
=
k
h
k
2
¼
R
L
y
0
R
L
x
0
@
x
h
ð Þ
2
þ
@
y
h
2
n o
d
x
d
y
R
L
y
0
R
L
x
0
h
2
d
x
d
y
0
;
so that
D
will contribute to a loss of potential energy[
Vis
‐
Star et al.
, 2008].
4.2. Analysis of the Production Term4.2.1. Breaking Down the Production
[
24
] From equation (8),
P
can be broken down as
P
=
P
u
+
P
v
, with
P
u
¼
uhD
@
x
C
;
P
v
¼
vhD
@
y
C
;
which describe the role of the cross
‐
shore and the longshoreflow components on the production.
Figure 2.
Sketch of the saturation processes: (a)
s
as a function of
t
and (b)
s
as a function of
k
h
k
.
GARNIER ET AL.: MECHANISMS CONTROLLING BAR AMPLITUDE
F02007F02007
5 of 14

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