G Karras

E Hertz

F Billard

B Lavorel

J.-M Hartmann

O Faucher

Erez Gershnabel

Yehiam Prior

Ilya Sh

Averbukh
email: ilya.averbukh@weizmann.ac.il

Orientation and Alignment Echoes

Keywords: numbers: 45.50.-j, 37.10.Vz, 42.50.Md

We present one of the simplest classical system featuring the echo phenomenon -a collection of randomly oriented free rotors with dispersed rotational velocities. Following excitation by a pair of time-delayed impulsive kicks, the mean orientation/alignment of the ensemble exhibits multiple echoes and fractional echoes. We elucidate the mechanism of the echo formation by kick-induced filamentation of phase space, and provide the first experimental demonstration of classical alignment echoes in a thermal gas of CO2 molecules excited by a pair of femtosecond laser pulses.

Echoes are common in many areas of physics. When an inhomogeneous ensemble of many nonlinear systems is impulsively kicked by an external force, the transient response to the kick vanishes fast due to dispersion in the properties of individual systems. The same happens when the ensemble is kicked again, however a new impulsive response shows up at twice the delay between the two pulses, -a celebrated echo phenomenon first observed by Hahn in precessing nuclear spins [1]. Echoes have been observed in a wide class of classical systems, including cyclotron echo [2,3], plasma wave echo [4], photon echo [5] and its mechanical analogs [6]. Echoes were predicted to occur in proton storage rings [START_REF] Stupakov | Echo Effect In Hadron Colliders[END_REF][START_REF] Stupakov | Echo Effect in Accelerators[END_REF], and were observed in high energy hadron beam experiments [START_REF] Spentzouris | [END_REF][START_REF] Bruning | Nonlinear and Collective Phenomena in Beam Physics[END_REF]. Echoenabled generation of short-wavelength radiation in freeelectron lasers [START_REF] Stupakov | [END_REF][12][13] was demonstrated (for a recent review, see [14]). Echoes are being also discussed in the context of cavity quantum electrodynamics [15,16] and cold atom systems [17][18][19][20].

In this Letter, we identify one of the simplest classical system featuring the echo phenomenon -a collection of free classical rotors stimulated by an external impulsive force. Using geometric arguments on the kick-induced transformations of the rotors phase-space, we predict multiple echoes in the mean orientation/alignment of the ensemble. Our qualitative analysis reveals the echo mechanism and predicts its dependence on parameters of the stimulating kicks. A full analytical theory of these echoes in 2D and 3D thermal ensembles of classical rotors, and its generalization to the quantum case will be published elsewhere [21]. Here, we discuss the physics behind the predicted echo effect and demonstrate it experimentally in a collection of CO 2 molecules stimulated by a pair of femtosecond laser pulses.

For a linear molecule having a permanent dipole mo-ment µ, and driven by a linearly polarized field, the interaction potential leading to orientation is V (θ, t) = -µE(t) cos(θ), where E(t) is the field amplitude, and θ is the polar angle between the molecular axis and the field direction. In the absence of a permanent dipole moment, the external field couples to the induced molecular polarization. For nonresonant laser fields, the angular-dependent interaction potential is V (θ, t) = -(∆α/4)E 2 (t) cos 2 (θ) [START_REF] Boyd | Nonlinear Optics[END_REF][START_REF] Friedrich | [END_REF], which leads to alignment along the field polarization (for reviews on molecular alignment see [24][25][26][27][28]). Here ∆α is polarizability anisotropy, and E(t) is the envelope of the laser pulse. The effects we present here are rather similar for the above two interaction types. Thus, in what follows, we use the dipole interaction form (V ∼ -cos(θ)) for discussion, and point out differences appearing in the case of alignment (V ∼ -cos 2 (θ)).

Consider an ensemble of 2D rotors kicked by a short orienting pulse at t = 0. The angular velocity, ω and angle θ of a rotor at time t are given by

ω = ω 0 -Ω sin(θ 0 ), θ = θ 0 + ωt (1)
Here ω 0 , θ 0 are the initial conditions, and Ω is proportional to the intensity of the kick. The orientation of the ensemble of rotors is quantified by the mean value of ⟨cos(θ)⟩, referred to as the orientation factor. Let us first assume that all the rotors are initially at rest and uniformly distributed in the angular interval [-π, π], as represented by the horizontal blue line in Fig. 1. A short kick in the θ = 0 direction does not move the rotors during its action, but induces rotation with angular speed -Ω sin(θ) (see Eq.( 1)). Immediately after the kick, the phase space distribution of the rotors takes a shape shown by the red curve in Fig. 1. In the course of the following free evolution, each point on the red curve moves horizontally towards θ = 0 with a velocity defined by its initial vertical position, as shown by the next two curves. After a certain delay, the curve experiences steepening at θ = 0 (light green curve), not unlike the accumulation of cars in congested traffic, which leads to a singularity in the angular distribution of the rotors [29]. Note, however, that the maximal value of the orientation factor ⟨cos(θ)⟩ is achieved not at this moment, but some time later, when the curve takes a typical folded shape (black curve in Fig. 1) leading to increased density of rotors in the region near θ = 0 [29]. This is a transient orientation, and the stronger the kick the shorter is the time needed to reach the maximally oriented state.

Next, we consider an ensemble that is initially uniformly dispersed in angle, and has a spread in angular velocity ω. For certainty, we assume a Gaussian distribution of angular velocities f (θ 0 , ω 0 ) ∼ exp[-ω 2 0 /(2σ 2 )]. The time-dependent orientation factor after the kick is found to be:

⟨cos(θ)(t)⟩ = exp [ -σ 2 t 2 /2 ] J 1 (Ωt)
, where J 1 (z) is the Bessel function of the first order. This pulsed response disappears fast as σt → ∞. By inverting the map (1) and using its area preserving property, one arrives at the following expression for the probability distribution function at time t:

f (ω, θ, t) = 1 2π 1 √ 2πσ exp [ - [ω -Ω sin(ωt -θ)] 2 2σ 2 ] (2)
Figure 2 shows the transformation of the initial distribution with time. Shortly after the kick (Fig. 2a) the density distribution takes a folded shape similar to the one shown in Fig. 1, resulting in the transient orientation. On the longer time scales, when the orientation signal ⟨cos(θ)⟩(t) vanishes, the probability density becomes rippled and develops multiple parallel filaments (see Fig. 2b). The number of these filaments grows with time, and their width is diminishing in order to keep the occupied phase space volume constant. Eventually, all the filaments tend to become almost horizontal and uniform in density. As follows from Eq.( 2), the neighboring filaments are separated in angular velocity by 2π/t, where t is the evolution time. At t = T , the ensemble is subject to another kick. Every filament in Fig. 2b is quite analogous to the initial uniform distribution of rotors shown in Fig. 1 by the blue line, but it has a non-zero mean velocity. With time, these filaments form the typical folded patterns leading to the transient orientation of the rotors in the filament. For every angle θ, the emerging patterns from neighboring filaments start moving with the velocity difference 2π/T with respect to each other. As a result, the kickinduced patterns are generally shifted with respect to each other most of the time after the kick, which results in a quasi-uniform total angular distribution considered as a function of θ only. However, at time τ ∼ T after the second kick, the folded filaments synchronously pile up near θ = 0 due to the above "quantization" of the angular velocities of the strips (see Fig. 3a). This results in an echo in the orientation factor ⟨cos(θ)⟩. Figure 3b demonstrates also orientation along θ = π that happens a little bit later in the same time region (anti-orientation echo).

These qualitative arguments have a considerable predictive power, in particular, the same mechanism should form the echo signals also at delays 2T, 3T, ... after the second pulse (higher order echoes). Moreover, it is expected that synchronization of the patterns from nonneighboring strips at τ = T /2, T /3, ... causes highly symmetric structures in the phase space, which may be associated with "fractional echoes". These echoes are not seen in a mere orientation signal ⟨cos(θ)⟩, but require measuring higher order observables ⟨cos(nθ)⟩ (n > 1). Moreover, just looking at the phase space pattern at the moment of the main echo, Fig. 3a, one may expect that the echo is best manifested if the delay T coincides with the time needed for achieving the maximal orientation in every strip. This means that for a given delay there exists a second-kick intensity that maximizes the echo signal. The simple 2D model considered here (see Eq.( 1)) allows for obtaining an analytical expression for the timedependent mean value of ⟨cos(nθ)⟩ where n is an integer:

⟨cos(nθ)⟩(τ ) = k=∞ ∑ k=0 (-1) k e -1 2 σ 2 (nτ -kT ) 2 J k+n [Ω 2 nτ ]J k [Ω 1 (nτ -kT )] (3)
For n = 1, 2 this quantity is the orientation factor and the alignment factor, respectively. Equation (3) presents a sequence of signals localized in time near τ = k n T where k is an integer. For n > 1, these are the above mentioned "fractional echoes", while n = 1 corresponds to the regular orientation echoes. Figure 4 presents calculated time-dependent orientation factor for two values of the intensity of the second kick. In Figure 4a, the intensity of the second kick, Ω 2 was optimized to achieve the maximal amplitude of the main echo at τ ∼ T . Although the second kick in Fig. 4a is three times weaker than that of Fig. 4b, it induces a considerably stronger echo, that even exceeds the initial response to the second kick. The peak and dip of the first echo in Fig. 4a correspond to the left and right panel of Fig. 3, respectively. As follows from Eq. ( 3), the amplitude of the echo is a decaying oscillatory function of the intensity of the second kick, Ω 2 after reaching the global maximum shown at Fig. 4a.

The same qualitative mechanism works in the case of the polarization-induced interaction (V ∼ -cos 2 (θ)), and it produces echoes in the alignment signal, ⟨cos 2 (θ)⟩(t). An analytical expression for the alignment echoes for 2D rotors is provided in the Supplementary Material [31], and it is quite similar to the Eq.( 3). The visualization of the phase space transformations in the case of alignment is more involved, but the main results remain the same, including the non-monotonous dependence of the echo signals on the intensity of the second kick.

We note the remarkable similarities between the physics of laser-induced alignment/orientation of molecules [24][25][26][27][28] and modern techniques for laser manipulations over electron beams in accelerator-based light sources [START_REF] Stupakov | [END_REF][12][13][14].

Both research fields aim at compressing the density of particles in certain areas (angular alignment/orientation vs electron bunching along the beam) by laser pulses, and in both cases phase-space folding is involved [START_REF] Stupakov | [END_REF]14,29,30]. The orientation/alignment echoes described here have much in common with the phenomena involved in echo-enabled harmonics generation (EEHG) in free-electron lasers. This connection is not merely formal, and may be beneficial for both fields. In particular, recent proposals to use several laser fields to significantly increase the electrons bunching factor [14] may be supported by a similar suggestion [29,30] in the context of molecular orientation/alignment by series of laser pulses. This concept was theoretically optimized [32], experimentally demonstrated [33][34][35], and even successfully applied to the enhanced focusing (bunching) of cold atoms interacting with multiple laser fields [36,37].

In what follows, we describe the first experimental observation of the rotational alignment echoes in laserkicked CO 2 molecules. The details of our experimental setup [38] are described in the Supplementary Material [31]. The two linearly polarized pump pulses P1 and P2 are derived from an amplified Ti:Sapphire laser (1 kHz, 800 nm, 100 fs FWHM) and properly delayed before being focused inside a CO 2 filled gas cell at a pressure of 0.2 bar. The alignment echoes are observed by time-resolved birefringence measurements [39], where a delayed, weak probe pulse, linearly polarized at +45 • relative to the pump pulses is linearly analyzed after passing through the cell by a polarizer set at -45 • relative to the pump pulses. The birefringence signal measured on the detector can be written as [START_REF] Faucher | it Progress in Ultrafast Intense Laser Science[END_REF]

S(τ ) ∝ ∫ ∞ -∞ I pr (t -τ ) [ ⟨cos 2 (θ)⟩ (t, I 1 , I 2 ) - 1 3 ] 2 dt, ( 4
)
where ⟨cos 2 (θ)⟩ (t, I 1 , I 2 ) is the alignment factor, and I 1 , I 2 , and I pr , are the intensity of P1, P2, and the probe pulse, respectively. Figure 5 presents the pump-probe signals obtained for three different sets of pump intensities with a delay T between P1 and P2 set to 1.6 ps. We measured the alignment signals till about one-half of the rotational revival time, T rev of the CO 2 molecule (T rev ∼ 42.7 ps). The classical echo effects discussed in this paper happen within the time interval of about T rev /4 where the alignment dynamics is well described by the classical mechanics. To guide the eye, the alignment signals caused by the pulses P1 and P2, and also their one-quarter and one-half revivals are colored in red and blue, respectively, whereas the echoes are highlighted with a yellow color. For each intensity set, an echo is produced after P2 at τ ∼ T . Its amplitude depends on the intensities of both P1 and P2. The maximum echo amplitude seen in Fig. 5c is achieved for a second pulse that is weaker than the first one, I 2 /I 1 ∼ 0.4, and this echo is stronger than the initial response induced by the second pulse P2. These observations resemble the theoretical results shown in Fig. 4, and they may be understood using the similar qualitative arguments. The second echo produced at τ ∼ 2T is also visible in Fig. 5c. On the longer time scale, replicas of these echoes are observed near the quarter and half-revival regions. They appear due to the interplay between the classical echo effect and the phenomenon of fractional quantum revivals [41]. These replicas will not be discussed in detail here, but they are nicely reproduced in the fully quantum theoretical description of our experiments (to be published). Related transient signals were also reported in Ref. [42] where the alignment of N 2 O molecules excited by two strong femtosecond laser pulses was studied. In these experiments the delay between the pulses was chosen close to the quarter-revival time, far beyond the region where the classical echo effects of the present paper exist. In order to reveal the non-monotonic dependence of the echo amplitude, we varied the intensity of the second pulse P2 while keeping the first one (P1) fixed. The results are summarized in Fig. 6 that demonstrates the existence of an optimal intensity of the second kick, and the oscillatory dependence of the echo amplitude on this intensity, in agreement with our theoretical considerations.

Summarizing, we showed that a textbook classical system -a collection of randomly oriented free rotors with dispersed rotational velocities -exhibits multiple echoes of different kinds in its orientation/alignment dynamics when kicked by a pair of delayed pulses. We provided qualitative and quantitative analysis of the phenomenon using a simple model system, and performed the first experimental demonstration of the classical alignment echoes in a thermal gas of CO 2 molecules excited by a pair of femtosecond laser pulses. The echoes were observed on a time scale much shorter than the quantum revival time, and their parameters were readily controllable by the intensity and delay of the excitation pulses. This makes the orientation/alignment echoes promising for exploring relaxation processes in high pressure gases and various liquids, including superfluid helium [43], where quantum rotational responses are considerably suppressed.

FIG. 1 .

 1 FIG. 1. (color online) Kick-induced orientation of an ensemble of rotors (see text for details).

FIG. 2 .

 2 FIG. 2. (color online) Filamentation of the phase space density distribution. Ω/σ = 1, (a) σt = 1, (b) σt = 30.

FIG. 4 .

 4 FIG. 4. (color online) Orientation factor versus time after the first kick. Ω1/σ = 1 , σT = 10. Panel (a): Ω2/Ω1 = 1/3. Panel (b): Ω2/Ω1 = 1. Here P1 and P2 denote transient responses to the first and the second kick, respectively. E1, E2 and E3 are echoes of the first, second, and third order, respectively. Insets show a magnified view of the second and third echoes.

2 FIG. 5 .

 25 FIG. 5. (color online) Birefringence signals as a function of the pump-probe delay τ for different intensities I1 and I2 of the pump pulses P1 and P2, respectively (see the inset). Trev: rotational revival time.

1 FIG. 6 .

 16 FIG. 6. (color online) Peak amplitude of the birefringence signal for the echo near τ = T versus I2/I1, with I1 set to 44 TW/cm 2 .

  This work was supported by the Conseil Régional de Bourgogne (PARI program), the CNRS, the Labex AC-TION program (contract ANR-11-LABX-01-01), and the French National Research Agency (ANR) through the CoConicS program (Contract No. ANR-13-BS08-0013). This research was also supported by DFG (Project No. LE 2138/2-1) and Minerva Foundation. I.A. acknowledges kind hospitality and support from the Université Paris Est Créteil during a one week stay at LISA.


Jantien Rutten
email: jrutten@iingen.unam.mx

B Dubarbier

T D Price

B G Ruessink

B Castelle

Alongshore variability in crescentic sandbar patterns at a strongly curved coast

Alongshore variability in crescentic sandbar patterns at a strongly curved coast

Introduction

Sandbars, submerged ridges of sand parallel to the shoreline, often possess a pronounced alongshore variability in cross-shore position and depth [START_REF] Sonu | Three-dimensional beach changes[END_REF][START_REF] Lippmann | Quantification of sand bar morphology: A video technique based on wave dissipation[END_REF][START_REF] Van Enckevort | Observations of nearshore crescentic sandbars[END_REF] that is related to the imposed wave energy, grain size and profile characteristics (Wright & Short, 1984;[START_REF] Calvete | Un)predictability in rip channel systems[END_REF]. These crescentic patterns are characterised by shallow landward protruding horns and deep seaward protruding bays with alongshore wavelengths of O(100 m) and cross-shore amplitudes of O(10 m) [START_REF] Van Enckevort | Observations of nearshore crescentic sandbars[END_REF]. Field observations show that crescentic patterns typically arise in a few days under low to moderately energetic conditions following a storm, and tend to disappear under high-energetic conditions (Wright & Short, 1984;[START_REF] Almar | Two-and three-dimensional double-sandbar system behaviour under intense wave forcing and a meso-macro tidal range[END_REF] or under oblique wave incidence [START_REF] Price | State dynamics of a double sandbar system[END_REF][START_REF] Contardo | Sandbar straightening under wind-sea and swell forcing[END_REF]. In some cases they may, however, persist for months or longer [START_REF] Van Enckevort | Video observations of nearshore bar behaviour. Part 2: alongshore non-uniform variability[END_REF]. Pattern formation and destruction often coincides with overall onshore and offshore sandbar migration, respectively.

The mechanism behind crescentic pattern formation includes horizontal circulation cells in the flow field, which preferably develop under shore-normal or near-normal waves (e.g. [START_REF] Falqués | A mechanism for the generation of wave-driven rythmic patterns in the surf zone[END_REF][START_REF] Calvete | Morphological development of rip channel systems: Normal and near-normal wave incidence[END_REF][START_REF] Thiébot | The influence of wave direction on the morphological response of a double sandbar system[END_REF]. Through a positive feedback between flow and morphology the characteristic crescentic patterning arises in the sandbar. Under increasing angles of incidence, circulation cells become skewed, growth rates decrease and wavelengths of the crescents increase [START_REF] Calvete | Morphological development of rip channel systems: Normal and near-normal wave incidence[END_REF]. In line with observations in the field, [START_REF] Thiébot | The influence of wave direction on the morphological response of a double sandbar system[END_REF] demonstrated with model simulations that crescentic patterns no longer arise when waves approach sufficiently oblique (θ > 11 • at their outer bar crest at ∼3.5 m depth, where θ is the wave incidence angle with the shore-normal) and the breaking-induced alongshore current starts to dominate over cell circulation. Besides, oblique waves have been observed to straighten sand-bars with pre-exisiting crescentic patterns [START_REF] Price | State dynamics of a double sandbar system[END_REF][START_REF] Price | Coupled sandbar patterns and obliquely incident waves[END_REF][START_REF] Contardo | Sandbar straightening under wind-sea and swell forcing[END_REF], whereof the underlying mechanism can be found in [START_REF] Garnier | A mechanism for sandbar straightening by oblique wave incidence[END_REF].

So far, the main focus has been on pattern formation under an alongshore uniform forcing. However, coasts that are concave, like embayed beaches, or convex, such as shoreline sandwaves and km-scale nourishments, impose an alongshore variation in forcing in the surf zone due to the refraction pattern over the curved depth contours (e.g. Castelle & Coco, 2012;[START_REF] Rutten | Observations on sandbar behaviour along a man-made curved coast[END_REF]. Similarly, offshore perturbations can create an alongshore variation in forcing (offshore bathymetric anomaly or offshore island; Castelle et al., 2012;[START_REF] Bryan | Beach rotation at two adjacent headland-enclosed beaches[END_REF] and accordingly, in bar behaviour. For example, the breaker height may vary alongshore, which was suggested by [START_REF] Short | Wave power and beach stages: a global model[END_REF] to generate an alongshore variation in sandbar characteristics. Also, the wave angle may vary alongshore, and enforce an alongshore difference in crescentic patterning. Such a relation between angle and patterning was found on a seasonal scale along the man-made curved coast of the Sand Engine, located at the roughly southwest-northeast oriented coastline of the Delfland coast in the Netherlands (Figure 1). Prolonged low-energetic north-northwestern waves in the spring-summer season [START_REF] Rutten | Observations on sandbar behaviour along a man-made curved coast[END_REF] initiated the formation of patterns only at the northern side of the Sand Engine. Under these conditions no patterns formed along the western side, where the waves were presumably much more oblique.

In the autumn-winter season, patterning at the northern side was erased, whereas patterns developed at the western side under storms passing from southwest to north-northwest.

Thus, patterns developed at the western side when actual shore-normal wave exposure was limited due to the varying angle. [START_REF] Castelle | Modeling formation and subsequent nonlinear evolution of rip channels : Time -varying versus time[END_REF] simulated the effect of a time-varying wave angle on crescentic patterns along a straight coast. Here, timevarying angles with low obliquity (θ<6 • , at 10.6 m water depth) resulted in crescents that were less pronounced than under time-invariant forcing, and moreover initiated an alongshore migration of the crescents that stimulated splitting and merging of the crescents. Time-varying angles including higher obliquity (θ> 6 • ), for at least 1 day, resulted in straightening of crescents by a strong alongshore current. Notwithstanding, how a spatiotemporal variation in wave angle, as occurring along a curved coast, contributes to pattern formation and destruction is yet unknown.

We hypothesise that a spatiotemporal variation in the local wave angle enforces an alongshore variation in the presence and growth rate of crescentic patterns, depending on the strength of the alongshore current. In this paper, we aim to systematically explore how curvature of a convex coast contributes to alongshore variability in the formation of crescentic bar patterns under time-varying forcing. We use the non-linear morphodynamic model of [START_REF] Dubarbier | Mechanisms controlling the complete accretionary beach state sequence[END_REF], wherein cross-shore and alongshore processes are included such that an initially alongshore-uniform bar can move onshore and develop alongshore variabilities simultaneously. Although the model setup is loosely based on observations of pattern formation at the Sand Engine, we do not aim to mimic the crescentic bar behaviour at this site. First, we outline the model formulation, its setup and the analysis method of the model results (Section 2). Then, we describe the effect of the offshore wave angle on pattern formation (Section 3). In Section 4, we discuss the effect of variations in wave characteristics and coastline curvature. Finally, we conclude our findings in Section 5.

Methodology

Model

The formation of crescentic patterns in sandbars and their evolution was simulated with a non-linear morphodynamic model consisting of four coupled modules [START_REF] Dubarbier | Mechanisms controlling the complete accretionary beach state sequence[END_REF]. In the first module, the statistical wave field was computed by the spectral wave model SWAN (version 41.10 Booij et al., 1999), wherein we chose the dissipation formulation of [START_REF] Ruessink | Calibration and verification of a parametric wave model on barred beaches[END_REF] and switched off local wave generation and the triplet and quadruplet wave-interaction source terms. In the second module, the 2D flow field was computed via the phase-averaged and depth-averaged non-linear shallow water equations, assuming balance of momentum and conservation of water mass, giving [START_REF] Phillips | The dynamics of the upper ocean[END_REF]:

∂Q i ∂t + ∂ ∂x j Q i Q j h = -gh ∂η ∂x i - 1 ρ ∂S ij ∂x j + 1 ρ ∂T ij ∂x j - τ b i ρ , (i, j = 1, 2) (1) ∂η ∂t = - ∂Q j ∂x j (2)
Using the Einstein convention, subscript i refers here to the two horizontal position coordinates (with X and Y the cross-shore an alongshore axis). This implies that terms containing an index twice include a summation over both indices. In these equations,

-5-manuscript submitted to JGR-Earth Surface Q i is the fluid volume transport, η is the mean free surface elevation, S ij is the radiation stress tensor, T ij is the lateral mixing term that describes the horizontal momentum exchange due to breaking-induced turbulence and the mean current, τ b i is bed shear stress, t is time, x is position, h is water depth, g is gravitational acceleration, and ρ =1000 kg/m 3 is water density. The wave return flow (undertow) was taken into account through the wave radiation stress formulation of [START_REF] Phillips | The dynamics of the upper ocean[END_REF]. In the third module, the total volumetric sediment transport q t was computed with an energetics-type transport model composed of three modes of transport, based on [START_REF] Hsu | Wave-induced sediment transport and -27-manuscript submitted to JGR-Earth Surface onshore sandbar migration[END_REF] and [START_REF] Dubarbier | Process-based modeling of cross-shore sandbar behavior[END_REF], as

q t = q w + q c + q g (3)
with a transport related to near-bed orbital velocity skewness q w , a transport related to the mean current q c , and a diffusion term q g representing the downslope gravitational transport that prevents unrealistic bar growth and/or unstable bar shapes. More specifically, q w accounts for wave non-linearity, but does not include infragravity or swash motions. Hereto, the intra-wave motion is reproduced using the robust parameterisation of [START_REF] Ruessink | On the parameterization of the free-stream non-linear wave orbital motion in nearshore morphodynamic models[END_REF] that relates values of wave-skewness and asymmetry to the local Ursell number, all derived from field measurements of the statistical wave field and mean water level. The sediment transports q w , q c , and q g contain both bedload and suspended load, with scaling coefficients of 0.135 and 0.015, respectively. The contribution of the three individual transport components to q t is scaled with coefficients C w , C c , and C g of 0.08, 0.08 and 0.24, respectively. For the specific definition of q w , q c , and q g , see [START_REF] Dubarbier | Mechanisms controlling the complete accretionary beach state sequence[END_REF]. In the fourth module, bed level change was computed, assuming conservation of sediment mass, as

∂z b ∂t = - 1 1 -p ∇ • q t (4)
with bed level z b and sediment porosity p =0.4.

By looping through the four modules, small perturbations in the bathymetry can grow and self-organise into rhythmic patterns through positive feedback between the bed level and the flow field. For further details on the model, see [START_REF] Dubarbier | Mechanisms controlling the complete accretionary beach state sequence[END_REF].

This model allows sandbars to develop crescentic patterns while moving onshore across the surfzone. The capability of the model to simulate cross-shore dynamics, besides alongshore dynamics, is important when investigating pattern formation along curved coasts, because alongshore variability in cross-shore migration may importantly affect pattern formation. For example, the separation distance of the bar from the shoreline is known to be critical to crescentic bar dynamics [START_REF] Calvete | Un)predictability in rip channel systems[END_REF]. Simulating accurately cross-shore bar migration is a challenge by itself and therefore often comes to compromises. In this state-of-the-art model, cross-shore migration speed and direction of the bar, depending on the velocity field and sediment characteristics (Dubarbier et al., 2017), can be tuned by changing the ratio and magnitude of the transport coefficients.

Furthermore, the model is able to properly simulate pattern formation under a wide range of local wave incidence, as observed along the curved coastline of the Sand Engine. Preliminary tests at a straight coast showed that crescentic patterns could develop under higher incidence angles using the nonlinear model of [START_REF] Dubarbier | Mechanisms controlling the complete accretionary beach state sequence[END_REF] than using a nonlinear model with the basic-state approach (e.g. Castelle et al., 2012;[START_REF] Garnier | Modelling the formation and the long-term behavior of rip channel systems from the deformation of a longshore bar[END_REF]. Although linear stability models, also based on a basic state, may produce crescentic patterns under higher incidence angles as well (e.g. [START_REF] Ribas | Modeling waves, currents and sandbars on natural beaches: The effect of surface rollers[END_REF], such models do not include cross-shore bar migration and thus are not suitable for this study.

Moreover, note that linear stability models are rather different from nonlinear models, and therefore results cannot be compared directly.

Model Setup

A synthetic bathymetry was created, based on in situ measurements of the Sand Engine in November 2014 (Figure 2). The synthetic bathymetry is a simplification of reality as the coastline does not show any asymmetry and the cross-shore profiles do not vary alongshore. First, a curved coastline was generated using a Gaussian shape function

X = p 1 + p 2 e - (Y +p 3 ) 2 2p 2 4
(5)

with alongshore distance Y , cross-shore distance X, and function coefficients p 1 = 253.36 m, p 2 = -698.37 m, p 3 = -494.33 m, and p 4 = -926.30 m following from the best nonlinear fit through the measured 0 m contour line (Mean Sea Level, MSL). The alongshore -7-manuscript submitted to JGR-Earth Surface axis was aligned with the regional coastline and the cross-shore axis pointed into the dunes (negative values offshore). Second, a barred cross-shore profile was generated using the double slope profile function of Yu and Slinn ( 2003) The influence of the significant wave height H s and a time-varying wave angle θ on nearshore pattern formation were tested by running the model with sets of scenarios, providing information for examining our hypothesis on the importance of a spatiotem-poral variation in local wave angle along curved coasts. Scenarios were inspired by wave conditions prevailing during the observed formation of crescentic patterns in the sandbar at the western side of the Sand Engine. As shown by [START_REF] Rutten | Crescentic bar patterns along curved coasts: observations and modelling[END_REF][START_REF] Rutten | Observations on sandbar behaviour along a man-made curved coast[END_REF], patterns developed at this side in autumn and winter with the passage of one or several storms.

z prof = x s - x s γ tanh tan β 1 X x s + tan β 1 X γ -b h e -bw( X-xc xc ) 2 (
During such storms, the wave angle typically changed from southwest to north-northwest, which can be attributed to the southwest-northeast oriented storm track in the North Sea. The forcing of every storm was slightly different, i.e. the dominant wave angles, the period and the function of the wave angle alternation as well as the wave height vary between the storms. To investigate how a series of storms with a time-varying angle may trigger pattern formation along a barred curved coast, various scenarios were generated whereof the wave conditions applied at the offshore boundary are summarised in Table 1.

The storms were schematised as an alteration of the wave angle θ between two directions (θ 1 and θ 2 ), and consequently a series of storms was schematised by repeating the angle alternation several times. The ranges in θ, significant wave height H s , and alternation period or function, covering the ranges in wave conditions observed at the Sand Engine [START_REF] Rutten | Crescentic bar patterns along curved coasts: observations and modelling[END_REF], allow to explore alongshore variability in crescentic pattern In all runs a directional spreading of 30 • was used. This value, assumed as typical for the North Sea, resulted in offshore leakage of wave energy at the offshore boundary un-der large wave angles (up to 9.3% for 55 • ). Bathymetric variations (Runs 40-48) were applied to test the aforementioned effect of coastline curvature on pattern formation. On a strongly curved coast, a gently curved coast and a straight coast, three scenarios were tested that differed in θ 1 , with θ 1 = 25 • (Runs 40-42), θ 1 = -25 • (Runs 43-45), and

θ 1 = 55 • (Runs 46-48).
In q w , q c , and q g a median grain diameter d 50 of 290 μm was used. Based on preliminary runs, scaling coefficients C w , C c and C f , corresponding to the individual sediment transports, were adjusted from their default 0.08, 0.08, and 0.24 to 0.02, 0.04, and 0.05, respectively. This setting allowed a formation and evolution of crescentic bar patterns that resemble the field observations at the Sand Engine in Rutten et al. ( 2018), but with an over predicted onshore bar migration. Realistic cross-shore migration rates were not pretended for this work. Cross-shore dynamics were speeded up to limit the computation time of the 50 runs on a 2.6 km by 7 km grid. Speeding up the cross-shore migration, consequently speeded up cross-shore profile change (e.g. depth of bar crest or trough). Since crescentic bar dynamics (i.e. rip spacing and growth rate) were found to relate to profile shape [START_REF] Calvete | Un)predictability in rip channel systems[END_REF], biased bar migration predictions could affect those dynamics. At the Sand Engine, rip spacing hardly changed while the bars moved onshore and offshore at weekly to monthly timescales [START_REF] Rutten | Observations on sandbar behaviour along a man-made curved coast[END_REF]. Also, [START_REF] Dubarbier | Mechanisms controlling the complete accretionary beach state sequence[END_REF] demonstrated that rip spacing and dynamics hardly changed during onshore bar migration under shore-normal wave incidence. Thus, the response time related to rip spacing seems to be larger than the response time related to cross-shore sandbar migration. Therefore we can speed up the onshore migration without harming the results on pattern formation.

Here, morphological change was simulated for a 20-day period, updating the bed level every 30 minutes. The 20-day period allows the growth rate of the patterns in the reference scenario to saturate. For each updated bathymetry, the corresponding wave field and flow field were computed in stationary mode. Periodic lateral boundaries conditions were used. In line with Castelle et al. (2012), the shoreline was allowed to evolve by computing the sediment fluxes at the cell centres and interpolate them at the cell interfaces. Accordingly, the sediment fluxes could transfer across the interface between dry and wet cells.

Analysis of Model Results

To reveal insight in the evolution of sandbar patterns and the underlying flow field, stacks of a time-varying transect were created along the alongshore-averaged bar crest position. The definition of the alongshore transect is not trivial. First, the model allows cross-shore sandbar migration, and thus the position of the alongshore transect needs to be updated every time step, following the cross-shore migration of the sandbar crest.

Second, alongshore differences in cross-shore bar migration exist at a curved coast, and thus the alongshore-averaged profiles cannot simply be a global alongshore average but needs to be determined at a local scale. Here, we introduce a six-step approach to create time stacks along a curved coast. First, the bathymetry, z b , was projected on local shore-normal transects, z b,n , following the curved coastline. Shore-normal transects were directed perpendicular to the tangent of the 10 m depth contour in the bathymetry at t = 0 without including perturbations. Second, alongshore-averaged profiles, z b,n , were computed by applying a moving average to the cross-shore profiles over an alongshore width of ∼500 m. Third, the second derivatives corresponding to the profiles, d 2 z b,n /dn 2 , were computed. Herein, smoothed profiles, using a 5-point Hanning window, were used, since the second derivative is rather sensitive to irregularities. Fourth, the bar crest positions were approximated by finding the minimum in d 2 z b,n /dn 2 that was nearest to the bar crest of the previous time step. Fifth, the bar crest location was determined at subgrid scale to allow for gradual onshore migration of the alongshore transect instead of discrete steps of 10 m corresponding to the grid resolution. Hereto a second order polynomial was fitted through d 2 z b,n /dn 2 at the bar crest approximation and its two neighbouring data points, giving a time-varying alongshore transect, (X, Y ) t . Sixth, bed elevation, the alongshore current and the total sediment transport, i.e. the variables of interest, were linearly interpolated along (X, Y ) t , and will be referred to as z b,c , U ls,c , and q t,c , respectively.

To analyse the growth rate of crescentic patterns, a measure was computed for the amplitude of the patterns on every time step. Previous studies [START_REF] Garnier | Generation and nonlinear evolution of shore oblique / transverse sand bars[END_REF][START_REF] Garnier | Mechanisms controlling crescentic bar amplitude[END_REF] computed the time-varying standard deviation of the bathymetry, a method known as global analysis. More specifically, they computed the root-mean-square deviation of the bathymetry from the time-and alongshore-averaged profile. Here, such a global approach is not sufficient, because the alongshore variability in pattern growth rate is of interest.

Therefore, we performed calculations at a local scale, in sections at the left straight coast, the left flank, the right flank, and the right straight coast (boxes I-IV in Figure 2b). Preliminary analysis showed that onshore bar migration at a curved coast introduces an alongshore variability related to migrational dynamics that contaminates the time-and intersectional average of the cross-shore profile and thereby the measure for pattern amplitude. Therefore, each section was divided into two subsections of about equal size. Then, the measure for pattern amplitude in each section followed from the weighted average of the standard deviation in the two corresponding subsections, also known as the pooled standard deviation. In equation-form, this measure for pattern amplitude ||h|| in a section reads as

||h|| = 1 N 1 + N 2 -2 N1 Z 2 p,1 + N2 Z 2 p,2 (7)
and increases with increasing potential energy density of the bedforms (0.5||h|| 2 , Vis- et al., 2008). N 1 and N 2 are the number of elements in subsections 1 and 2, respectively. The deviation from the alongshore-averaged profile in a subsection reads as

Star
Z p (N, t) = z b,n (N, t)-z b,n (t)
, with N as the shore-normal transect. Herein, bed elevations interpolated at the shore-normal transects, z b,n , were used.

The cross-shore and alongshore components of the flow vector field have often been found to be important variables to explain pattern formation (e.g. [START_REF] Garnier | A mechanism for sandbar straightening by oblique wave incidence[END_REF][START_REF] Price | Coupled sandbar patterns and obliquely incident waves[END_REF]. At a straight coast, the x-and y-component of a vector simply represent the cross-shore and alongshore component, respectively. Along a curved coast, however, the cross-shore and alongshore direction vary locally, and thus a local matrix rotation is needed to obtain a fair representation of both the cross-and alongshore component. The rotation angle φ in the rotation matrix was determined for every shore-normal transect. Then, the variable of interest, i.e. flow vector field, was linearly interpolated to shore-normal transects. Finally, the matrix rotation was applied to obtain the alongshore current U ls . Similar to a vector field, a curved coast complicates interpretation of -12-manuscript submitted to JGR-Earth Surface was defined as the local angle corresponding to the maximum wave height, hereafter referred to as the breaker wave height H b , at a shore-normal transect.

Results

Reference Scenarios

To interpret the results of the full set of scenarios, we first examine two reference scenarios, with a time-invariant angle and a time-varying angle, respectively.

Time-Invariant Angle Ref 1

Figure 3 shows that the onshore migrating sandbar develops crescentic patterns along a large part of the coast. On t = 19 days, several crescents have formed along the left flank and the straight coast, while rhythmic morphology lacks along the right flank.

Figure 4a-c shows the significant wave height H s , the total sediment transport q t , the bed level change rate Δz b and the wave and current vectors at t = 0 days when the sandbar is still alongshore uniform. For the same day, Figure 5 shows, for each cross-shore profile, the wave height at breaking H b , the local angle at breaking θ l,b , the alongshore current at the bar crest U ls,c and the total sediment transport at the bar crest q t,c . The abrupt cross-shore drop in H s in Figure 4a In Section II, the near-zero alongshore current converges (downward zero-crossing in Fig- ure 5c) and diverges (upward zero-crossing). At the convergence point, the opposing rightward and leftward directed alongshore current may have fed an offshore directed flow of small magnitude. The total sediment transport q t (Figure 4b, red arrows) is of small magnitude and slightly onshore directed along the straight coast, almost absent at the left flank, and relatively large ( q t,c ∼6.2 x10 -4 m 3 /m/s; Figure 5d) and directed along the sandbar crest at the right flank. Although the pattern in q t is largely in line with the flow vector field and thus dominated by sediment transport related to the mean-current ( q c ), also onshore directed sediment transport related to wave-skewness ( q w ) contributes to the total sediment transport and causes the cross-shore component of the total sediment transport to be zero (along right flank) or onshore directed (along straight coast and left flank).

Consequently, bed level change (Figure 4c), where erosion relates to positive gradients in the total sediment transport in the landward direction and deposition to negative gradients, results in a ∼30-40 m onshore bar migration along the entire coastline within the first four days. Thereafter, cross-shore migration starts to depend more strongly on the alongshore position. At the tip, the left flank and the straight coast onshore migration rates are about 6 m/day, while at the right flank only 2 m/day. Note that these migration rates are time-averaged and some temporal variation exists. For example, rates up to 9 m/day are found just left of the tip. While moving onshore, the bar steepens due to erosion at its seaward flank and deposition just shoreward of its crest (Figure 3 -15-manuscript submitted to JGR-Earth Surface

Time-Varying Angle Ref 2

By shifting θ at the offshore boundary, every day from 25 • to -25 • , the patterns in wave refraction, currents, sediment transport and bed level change are similar to the patterns described in Section 3.1.1 for t = 0, 2, 4, ...18 days but mirrored across the vertical in Y = 500 m for t = 1, 3, 5, ...19 days (e.g. black arrows in Figure 7). As a result, the sandbar migrates onshore and develops crescentic patterns along the straight coast but not along the flanks (Figure 7). Similar to Ref 1 the bar migrates by ∼35 m onshore in the first four days along the entire coastline, but only after t =12 days the migration rates start to show clear alongshore differences. At the straight coast and the tip rates increase up to 8 m/day compared to 2-3 m/day along the flanks. The lower rates at the flanks can be explained by a refraction-induced reduction of the wave height when the waves enter obliquely every other day at one of the flanks (wave divergence; Figure 5a). The strong alongshore current, leftward at the left flank (∼-0.5 m/s for t = 0, 2, 4, ...18 days; red line in Figure 5c) and rightward at the right flank (0.5 m/s; for t = 1, 3, 5, ...19 days; purple line in Figure 5c), may explain the absence of crescents here as they hinder the

Effect of Wave Angle

Variations on both the time-invariant and time-varying angle (Runs 1-21; Table 1)

show that alongshore variability in the presence of patterns change with the scenario.

Patterns arise where angles are low oblique throughout the simulation period and remain absent where angles are oblique for, at least, every other day. To summarise, the overall picture that arises from our simulations is that alongshore variability in the presence of crescentic patterns and their growth rate at a curved coast vary with the local wave angle. Low obliquity stimulates pattern formation through increased cell circulation because of a limited alongshore current and limited refractioninduced energy reduction (and thus high H b ).

Discussion

Comparison with Observations and Model Limitations

To show that the processes important to crescentic pattern formation are well included in the model, two comparisons are made with observations at the Sand Engine nourishment. Note that reaching exact quantitative agreement is beyond the scope of both our model and study aim, and that such agreement has not been reached with any other morphodynamic model applied to barred coasts under complex wave conditions.

Firstly, we analyse pattern formation under a time-invariant wave angle, comparing Run 8 with a 12-day period of north-northwestern waves with similar properties (H s =1.0 m, with the observations (on average ∼510 m; Figure 11). Alongshore migration rates are limited in both the predictions and the observations. Alike the time-invariant simulation, the largest differences are found in the cross-shore dynamics. The bar migrated on-shore with ∼7 m/day in both runs, whereas a 3 m offshore migration was observed at the Sand Engine. The shoreline at the Sand Engine retreated with 18 m, whereas the shoreline kept its position in the runs. After the patterns developed, the longshore-averaged bar crest depth in the runs was very similar to the observed one (2.2 m versus 2.3 m).

T p =6.
The differences in the simulated and observed onshore bar migration relate largely [START_REF] Van Enckevort | Effect of hydrodynamics and bathymetry on video estimates of nearshore sandbar position[END_REF][START_REF] Ribas | Assessing the suitability of video imaging for studying the dynamics of nearshore sandbars in tideless beaches[END_REF]. In addition, the observed shoreline position, used to compute the bar-shore distance in Figure 11, is only accurate up to O(10 m) [START_REF] Rutten | Observations on sandbar behaviour along a man-made curved coast[END_REF].

Furthermore, some of the differences between the simulated and observed bed evolution can be explained by the prescribed boundary conditions in the model. First, we use a rather high wave period and wave height. Assuming that the storms drive the most important morphologic change, we included only those in our scenarios without any calm period as observed in the field. Consequently, the wave height and wave period in the scenarios is higher than the time-averaged forcing during pattern formation at the Sand Engine. Although considering storm forcing only, the wave period is still relatively high for the North Sea. However, when using a smaller value no patterns developed. Earlier, [START_REF] Calvete | Morphological development of rip channel systems: Normal and near-normal wave incidence[END_REF] found with a morphodynamic stability model that pattern growth rate decreases substantially with decreasing wave period, especially for oblique waves (see their Figure 15). Second, we schematised the time-varying angle as an alternation between two angles; a simplification that allowed to systematically explore its effects on pattern formation. However, measurements at the Sand Engine show more complexity, in terms of the angle itself as well as the alternation function. At times, the angle alternates between three angles. The angle alternation function varies over time, sometimes better resembling a sawtooth and sometimes a cosine or abrupt function. In addition, the simplifications made when defining the synthetic bathymetry may have led to im-portant differences between the simulations and observations. We assumed an alongshore uniform profile, and thus without any variation in bar crest depth, bar height, bar width, bar position or shoreface slope. At the Sand Engine, some differences in profile existed between the left flank and the right flank [START_REF] Rutten | Observations on sandbar behaviour along a man-made curved coast[END_REF] and pattern formation events never started without any patterns somewhere along the coast. To illustrate this, the over prediction of the crescent wavelength at the right flank under a time-invariant angle may partly be related to such differences in the initial profile. Also, differences in bar behaviour at the straight coast adjacent to the right flank probably relate partly to the presence of a channel that connects the sea with the shallow lagoon at the Sand Engine (see Figure 2a) and wherein the flow reverses with every tide. Lastly, tides were neglected in our model setup. Morphologic change in the surf zone could be influenced by tide-induced water level variations [START_REF] Price | Coupled sandbar patterns and obliquely incident waves[END_REF], tide-driven currents, but also other tide-induced phenomena specific for curved coasts (e.g. tidal flow separation, [START_REF] Radermacher | Tidal flow separation at protruding beach nourishments[END_REF].

To summarise, the model can capture the formation of patterns with the right orders of magnitude (e.g. alongshore wavelength of crescents, pattern growth rate), despite some model limitations. The patterns produced under a time-varying angle do not always develop where they were observed at the Sand Engine, given that the used wave schematisation deviates from the observed wave conditions. For observed nearly-constant wave angles, the location is accurately reproduced by the model.

Effect of Time-Varying Wave Height

How alongshore differences in patterning are related to the local wave angle was investigated above for a time-invariant wave height of 2.0 m. At the Sand Engine, the offshore wave height varied during pattern formation [START_REF] Rutten | Crescentic bar patterns along curved coasts: observations and modelling[END_REF][START_REF] Rutten | Observations on sandbar behaviour along a man-made curved coast[END_REF]. Here, we describe how our results on the formation of crescentic patterns are affected by a timevarying wave height (Runs 36-39, Ref 2 ; Figure 12). Generally, an increase in the wave height results in increased growth rates, consistent with [START_REF] Calvete | Morphological development of rip channel systems: Normal and near-normal wave incidence[END_REF] and [START_REF] Castelle | Modeling formation and subsequent nonlinear evolution of rip channels : Time -varying versus time[END_REF]. Figure 12 also shows that a time-varying H s affects the pattern growth rate differently within the four sections. The alongshore variability in presence or absence of patterns, however, is not substantially affected, as patterns start to develop along the straight coast (sections I and IV, yellow and purple line in Figure 12) and slowly extend along the flanks (sections II and III, blue and red line in Figure 12) within all runs.

To summarise, a time-varying wave height influences the pattern growth rate and its alongshore variability, but to a relatively small extent in comparison to a time-varying wave angle (compare Figure 9 and 12).

Variations on Coastline Curvature

Runs 1-39 illustrate that a curved coast imposes alongshore differences in the local wave angle, the resulting flow field, and consequently pattern formation. Below, we discuss how variations on the curvature of the coastline (Runs 40-48) affect pattern formation, which is relevant in the design of km-scale nourishments and the anticipated alongshore diffusion of such coastline perturbations in perspective of swimmer safety (e.g. kmscale nourishments can modify the large-scale flow pattern and generate km-scale tidal eddies, [START_REF] Radermacher | Tidal flow separation at protruding beach nourishments[END_REF]. In fact, the coastline curvature is expected to affect crescentic pattern formation in a similar way as the offshore wave angle (i.e. influencing the alongshore variability in presence and growth rate), since for both sets of sce- ing decrease in growth rate cannot simply be explained by the difference between the offshore wave angle and the coastline orientation, since they both do not change within sections I and IV. However, H s reduces at the lee side of the curved coast (changing from left to right every day) due to divergence of the wave rays. This refraction-induced reduction in the wave height increases for coasts with stronger curvature, resulting in lower simulated an offshore migration at Sète beach, defining the ratio C w :C c an order of magnitude lower than in our work. To find a ratio that allows an offshore migration or a straightening of the bar at our site, the parameter space needs to be studied in more detail but this is beyond the scope of this article.

Conclusion

The formation of crescentic patterns was numerically simulated for an initially alongshoreuniform sandbar along a curved coast under a time-invariant and time-varying wave angle θ. We found that the presence and growth rate of patterns varied alongshore with -34-manuscript submitted to JGR-Earth Surface Here, ||h|| was computed for section I (yellow), section II (blue), section III (red), and section IV (purple), which positions are indicated in the panels with timestacks and in Figure 2b). Angle variations are schematised by the circles on the top, wherein the grey and black radius indicate θ1 and θ2, respectively.

  6) with bed elevation z prof , and function coefficients γ = tanβ 1 /tanβ 2 , β 1 = 0.0372, β 2 = 0.0112, x s = 3.63 m, x c = -222.6 m, b h = -2.39 m, and b w = 11.07 m, which followed from the best non-linear fit with the measured alongshore-averaged profile (over a 500 m box) at the western side of the Sand Engine in November 2014. In Equation (6), the first term and second term on the right-hand side create a profile with slope β 1 in the upper part, changing at cross-shore position x s into slope β 2 for the lower part of the profile. The last term creates a perturbation of amplitude b h and width b w at crossshore position x c , representing the sandbar. The profile extends up to +2 m MSL to anticipate for surged water levels. Third, the cross-shore profile was rotated along the Gaussianshaped 0 m contour, resulting in a bathymetry with an alongshore-uniform subtidal bar located at equal distance from the curved shoreline. Fourth, the generated bathymetry was linearly interpolated on a Cartesian grid, as required by the model. The computational grid extended 2600 m in the cross-shore and 7000 m in the alongshore direction, with equal cross-and alongshore grid sizes of 10 m. Fifth, depths beyond -15 m MSL were replaced by -15 m, and the bathymetry between the -12 m contour and the offshore boundary was recomputed to create a gradual fading of the Gaussian-shaped perturbation in these deep contour lines (see Figure2b). Finally, random perturbations of <0.01 m were added to the bathymetry to trigger pattern formation in the nearshore zone.Additional bathymetries were created with larger, smaller or zero coastline curvature to test the curvature effect on alongshore differences in crescentic bar patterns. The three additional bathymetries differed from the reference bathymetry in the cross-shore extension of the coastline perturbation, i.e. p 2 = -698.37 m in Equation (5) was adjusted to p 2 = -931.16 m (initial extension of the Sand Engine, in 2011), p 2 = -349.18 m (half of the extension of the reference bathymetry), and p 2 = 0 m (straight coast).

  formation along curved coasts under time-varying forcing in a general sense. The most simple scenario (Ref 1 ), not including any time-varying forcing, has an offshore wave angle that is shore-normal at the left flank with H s = 2.0 m and T p = 8.0 s and is expected to generate patterns at the left flank. Sensitivity to the wave angle itself was tested by variations on the time-invariant angle with θ 1 =θ 2 and -55 ≤ θ 1 ≤ 55 • (Runs 1-11). The simplest scenario with a time-varying angle had a symmetric forcing, i.e. the wave angle alternating abruptly every day from shore-normal at the right flank (θ 1 = 25 • ) to shore-normal at the left flank (θ 2 =-25 • ) of the perturbation and served as reference, named Ref 2 , for all time-varying scenarios throughout the article. Asymmetry in forcing and wave obliquity with respect to the flanks was tested in Runs 12-21, wherein θ 2 was fixed at -25 • and -55 ≤ θ 1 ≤ 55 • . Additional angle variations included the function of alternation (Runs 22-29) with gradual alternations from θ 1 to θ 2 following a cosine and sawtooth function, and the period of angle alternation with a duration Dur 1 and/or Dur 2 of 0.5, 1.5 and 2.0 days (Runs 30-35). Sensitivity to wave height was tested by varying 1.6 ≤ H s,1 ≤ 2.4 m while keeping H s,2 constant at 2.0 m (Runs 36-39).

  wave angles. The model gives the wave angle θ with respect to the global coastline orientation, while the wave angle with respect to the local coastline orientation may strongly differ along a curved coast. Here, the local wave angle θ l was computed from the difference between the global wave angle θ and φ, where 0 • denotes shore-normal incidence and positive values indicate waves coming from the right. The local breaker angle θ l,b

  Figure 5a). Such secondary variations in H b and H s are again related to refraction, but

Figure 4c (

 4c Figure 4c (blue arrows) and Figure 5c show that the alongshore current is of moderate strength (∼0.21 m/s) and rightward directed along the straight coast, nearly zero at the left flank, and strongest (∼0.5 m/s) at the right flank. In Section IV, the substantial differences in H b are not reflected in U ls,c because of the inverse effect of θ l,b on U ls,c .

  Figure3). More specifically, alongshore variabilities in the 2D horizontal flow field, i.e. onshore directed flow over shallower parts of the bar and seaward directed flow over the deeper parts, force the development of horizontal circulation cells, stimulating crescentic pattern formation through positive feedbacks. Note that the first rip channels (Y = -600 m and Y = 600 m) develop where the bar moved rapidly to shallower depths, letting patterns develop more easily through an increased cell circulation, and in absence of alongshore currents (U ls,c ∼0 m/s; Figure5). Subsequently, patterns start to develop in Section IV (t = 10 days, Figure3) and slightly later also in Section I. The small delay of Section I might be explained by a slightly higher U ls,c compared to Section IV.In Section II, the formation of a rip channel around Y = -1100 m may be partly stimulated by a small offshore directed flow, related to convergence of the alongshore current. However, this mechanism does not seem to have contributed substantially to pattern formation here, as patterns develop at nearly similar rate away from the convergence point in the flow field (e.g. sections I and II). The absence of crescents along the right flank of the perturbation can be explained by the strong alongshore-directed current (0.54 m/s) as the waves approach obliquely, hindering the development of cell circulation. The time evolution of the measure for pattern amplitude ||h|| is shown in Figure6for four sections of the coast, revealing the alongshore variability in growth rate of the crescentic patterns. Initially, ||h|| increases slowly. After t = 5 days, ||h|| rapidly increases at the left flank (section II, blue line). At t = 8 days, ||h|| starts also to increase for the sections with a straight coastline (sections I and IV; yellow and purple line). The growth rate slows down again at t = 16 days for the left flank (section II) and at t = 18 days at the right straight coast (section IV). At the right flank (section III), ||h|| does not show a strong increase that is typical for pattern formation within the 20-day simulation period. Thus, ||h||, estimated as pooled standard deviation, clearly describes the spatiotemporal variability in patterning as observed in Figure3.

  development of cell circulation. The first rip channels develop at the same location at the left and right straight coasts (Y = -2280 and Y = 3250 m). Figure 6b shows a slow increase in ||h|| until t = 8 days, then ||h|| increases rapidly for the sections with a straight coastline (sections I and IV; yellow and purple line). Compared to Ref 1 , this increase starts slightly later for Section IV (purple) while slightly earlier for Section I (yellow). After t = 18 days, the growth rate becomes negative for Section I. Both flanks (Sections II and III; red and blue line) lack the typical strong increase in ||h|| related to pattern formation within the 20-day simulation period. Thus Ref 2 shows that, in line with Ref 1 , patterns preferably develop under low-obliquity (limited alongshore current and refraction-induced reduction of the wave height). Such conditions stimulate the bar to move onshore to shallower depths where patterns develop more easily. Under a timevarying wave angle, certain stretches of the coast may be subjected alternately to low and high obliquity. Ref 2 shows that low obliquity throughout the simulation period is important to develop patterns under a time-varying angle.

  For the time-invariant angle scenarios, the position where patterns develop simply shifts with the imposed offshore wave angle (not shown). More specifically, hardly any pattern arises under -55 • or 55 • (Run 1, 11), but arise along the left flank (Section II) for -45 to -5 • (Runs 2-5, Ref 1 ), along the straight coast (Sections I, IV) for -25 to 25 • (Runs 4-8, Ref 1 ) and along the right flank (Section III) for 5-45 • (Runs 6-10). The interpretation of the time-varying angle scenarios (Runs 12-21) is less intuitive. Half of their simulation time the offshore angle equals -25 • ; in Ref 1 such an angle stimulated pattern formation along the left flank and the straight coast but prevented them to form along the right flank. During the other half of the time, the wave angle may be substantially different and thus stimulate or prevent pattern formation at entirely different positions. Figure 8 shows the bathymetries of the time-varying scenarios at t = 19 days. Within all scenarios, not any pattern develops along the right flank (Section III), where waves approach rather obliquely every other day (θ 2 = -25 • , θ l,b = -24 • ). As long as the forcing is from the left side only (Runs 12-16), crescentic patterns develop along the left flank (section II). In the two-sided scenarios (Runs 17-21), no patterns develop here, except for Run 17 with θ 1 = 5 • . Scenarios with one-as well as two-sided conditions create patterns along the straight coast (sections I and IV) if |θ 1 | ≤ 35 • (Runs 15-19, Ref 1 and Ref 2 ).Figure9shows the temporal evolution of the depth in the bar zone for the different time-varying scenarios (first column), and illustrates that the growth rates of crescentic patterns vary strongly both within and between the scenarios (second column). The time period at which the growth of alongshore variability in depth and position of the bar crest stabilises varies from several days to beyond the simulation period of 20 days. Generally, crescentic patterns start to clearly develop within the simulation period where maximum θ l,b <13 • , and they develop the fastest along stretches of the coast where wave obliquity is the smallest. For example, in Run 17 crescentic patterns develop first along the straight coast (sections I and IV), and subsequently start to form along the left flank (section II). This alongshore difference in growth rate can be clearly noted in the time evolution of ||h|| corresponding to the four sections (Figure9; Run 17). Here, waves approach rather obliquely half of the time (t = 0, 2, ..., 18 days) along the left flank (θ l,b ∼15 • ), whereas the straight coast is exposed to smaller obliquity (θ l,b ∼9 • on t = 1, 3, ..., 19 days). This confirms the preference of crescentic patterning for low obliquity. Furthermore, Figure9shows that crescents along the straight coast (sections I and IV) migrated alongshore in rightward direction (rates up to ∼20 m/day) in Runs 14-15 and Ref 1 wherein waves approached from the left side only. Zooming in shows that crescents in Runs 18-21 and Ref 2 migrated alternately leftward and rightward, correlating with the alternating wave angle between the left side and right side.In line with the reference scenarios, local obliquity may prevent or slow down the formation of crescentic patterns because of reduced local wave heights, and strong alongshore currents that inhibit cell circulation. The third column in Figure9shows the alongshore current U ls,c for the different time-varying scenarios on t = 0 days (blue line) and t = 1 days (red line), wherein positive and negative values indicate a rightward and leftward directed current, respectively. All one-sided scenarios (Runs 12-16) generate an alongshore current that is alternately small (condition 1: t = 0, 2, ...18 days) or near-zero (condition 2: t = 1, 3, ...19 days) along the left flank (section II). Presumably, horizontal cell-circulation prevails during the full 20-day simulation period (both condition 1 and 2), stimulating crescentic pattern formation within this section. At the same time, large continuously rightward directed U ls,c along the right flank (section III) prevents circulation cells and crescentic patterns to develop here. In Runs 19-21 both the flanks and the straight coast are subjected to large incidence angles (under either θ 1 or θ 2 ), and thus large U ls,c , inhibiting formation of distinct alongshore variability. Along the straight coast (sections I and IV), increased crescentic growth rates towards Runs 16-17 can be explained by the increase in shore-normal waves during condition 1 (θ 1 ), and thus decrease in the magnitude of U ls,c . Similarly, the fastest growth rate of crescentic patterns along the left flank (section II) can be observed in Ref 1 , when U ls,c ∼0 under both θ 1 and θ 2 . Near-normal waves stimulate the development of crescentic patterns also because wave energy barely redistributes due to refraction, resulting in a relatively high H b and consequently strong cell circulation. In addition, an offshore directed current related to convergence of the alongshore current may have stimulated rip channel formation at the convergence point (downward zero-crossing in right column of Figure9). This mechanism becomes more important for runs with a stronger alongshore current around the convergence point and thus a stronger offshore directed current. Run 16 crashed at t = 10.9 days, possibly due to the fast onshore bar migration near the tip of the curved coast, which resulted in a flow field too complex to be solved by the model. The results of Run 16 that are included above were obtained by performing the simulation with a smaller morphological time step of 15 minutes, which prevented crashing.The importance of low obliquity on the alongshore variability in presence and growth rates of crescentic patterns is corroborated by simulations with gradually varying θ, following either a sawtooth curve (Runs 22-25) or a cosine function. Figure10shows that ||h|| increases more rapidly at the straight coast (sections I and IV) for both sawtooth (dotted line) and cosine variations (dashed line). For example, ||h|| in section I increases beyond 0.05 m after 7.2 days in Run 22 (sawtooth) and after 7.9 days in Run 26 (cosine), while it takes 10.6 days in Run Ref 2 having the same θ 1 and θ 2 but with an abrupt alternation. In addition, the sawtooth variations give a relatively large ||h|| for the right flank (section III, red dotted lines in Figure10), which related to the cascading of patterns in section IV into section III. Higher ||h|| can be explained by the locally increased exposure time of near-normal waves, especially in the sawtooth variations, and thus shorter exposure to large U ls,c under a gradually varying θ. Although Run 23 crashed at t = 17.4 days and Run 27 at t = 18.7 days, the trends in ||h|| are clear. Therefore, no additional simulations were run to cover the full 20-day period. The influence of exposure to low obliquity on the pattern formation is additionally reflected in Runs 30-32, indicating that longer exposure (up to 2 days) of shore-normal waves at the right flank results in increased growth rates (not shown). The chronology in the time-varying forcing has no substantial effect on pattern formation for exposure times between 0.5 day (Run 33) and 2 days (Run 35; not shown).

  Figure11a-b). The largest differences between predictions and observations are found in the cross-shore dynamics. The simulated bar, initially located at ∼230 m from the 0 m contour at 2.6 m depth, migrated onshore with ∼7 m/day over 16 days. The observed bar was located only ∼120 m from the 0 m contour with its crest at 2.3 m depth and migrated 18 m onshore within the 12-day period of pattern formation.

  to the choice of the model coefficients. The relatively large coefficient for the wave-induced sediment transport (C w ) led, as intended, to an over predicted onshore migration speed of the bar and a speeding up of the computation time. Besides, some uncertainty exists in our observations of bar migration. The bar may have migrated further onshore than we observed, since the expected migration magnitude falls within the O(10 m) accuracy of the breaker line method used by[START_REF] Rutten | Observations on sandbar behaviour along a man-made curved coast[END_REF]. Especially migration under the passage of storms (November 2013) may not have been well captured by the method, as variations in wave height can affect the position of the breaker line(Van Enckevort &

  narios the local wave angles change as they are a function of the offshore wave angle and the coastline orientation. Here, we study the contribution of curvature on pattern formation under three wave climates only, whereof found in Section 3.2 to either create patterns along the straight coast (Run Ref 2 : θ 1 = 25 • and θ 2 = -25 • ) but not along the flanks, to create patterns along the left flank but not along the right flank (Run Ref 1 : θ 1 = θ 2 = -25 • ) or to create no patterns at all (Run 21: θ 1 = 55 • and θ 2 = -25 • ).

Figure 13

 13 Figure13shows that alongshore variability in the presence of crescentic patterns does not change substantially for the selected range of coastline curvatures, in contrast to the growth rate of ||h||. In Runs 40-48 the growth rate decreases for a smaller curvature, in particular for the straight coast scenarios (Runs 42 and 45). The decrease in growth rate in sections I and IV (Runs 40-42) relates to an increasing magnitude of U ls,c with decreasing coastline curvature, from a range of 0.15-0.22 m/s at the strongly curved coast (Run 40) to 0.24 m/s (Run 42) at the straight coast in these sections. Note that the range values are based on the condition with the largest U ls,c . The increase in U ls,c and result-

  the local breaker angle, θ l,b . Patterns arose within the 20-day simulation period where local obliquity was limited to θ l,b <13 • . Variations of θ, i.e. its value and the shape or period of its time-varying function, affected θ l,b and thereby the alongshore variability in presence of patterns and their growth rate. The preference of low obliquity for crescentic pattern formation can be attributed to the limited strength of alongshore currents and limited refraction-induced wave height reduction. Both positively affect the generation of horizontal circulation cells in the flow field that initiate crescentic pattern formation through positive feedbacks between the flow field and the bed. Simulations in which the coastline curvature was varied, from strongly curved to straight, confirm the important negative effect of the alongshore current on pattern formation. The presence and growth rate of crescentic bar patterns and associated rip channels increased with coastline curvature, if the percentage of locally near-normal incidence increased as well (e.g. wave climate with low obliquity

Figure 1 .

 1 Figure 1. Aerial picture of the Sand Engine in September 2014, looking in northeasterly direction. This km-scale nourishment, with 21.5 Mm 3 larger than regular nourishments (1-2 Mm 3 ), was constructed in July 2011 along the southwest-northeast oriented Delfland coast, The Netherlands, as a sustainable and nature-based protection measure against coastal erosion (Stive et al., 2013). Courtesy: Rijkswaterstaat, Joop van der Hout.

Figure 2 .

 2 Figure 2. (a) Measured bathymetry in November 2014 at the Sand Engine (Rutten et al., 2018), and (b) synthetic bathymetry with a curved coastline and alongshore-uniform sandbar, based on measurements in (a). The black contours correspond to the boxes I-IV for the analysis of pattern growth rates.

Figure 3 .

 3 Figure 3. Bathymetric evolution (panels 1, 3, 5 and 7) and bed level change rate Δzb (panels 2, 4, 6 and 8) of the reference scenario Ref1. Grey lines represent the depth contours, whereas the vectors indicate the total sediment transport. The arrows in the top left corner indicate the offshore wave angle.

Figure 4 .Figure 5 .

 45 Figure 4. Patterns in (a,d) significant wave height Hs, (b,e) total sediment transport qt, and (c,f) bed level change rate Δz b on t=0 days (a-c) and t=19 days (d-f) for the reference scenario Ref1. Vectors indicate the wave angle (black; a-f), current velocities (blue; a,d), and total sediment transport (red; b,e), whereas the grey lines represent the depth contours.

Figure 6 .

 6 Figure 6. Time evolution of the measure for pattern amplitude ||h|| showing the alongshore difference in growth rate of patterns in section I (yellow), section II (blue), section III (red), and section IV (purple; see Figure 2b) in reference scenario (a) Ref1 and (b) Ref2.

Figure 7 .

 7 Figure 7. Bathymetric evolution of reference scenario Ref2. Grey lines represent the depth contours, whereas the vectors indicate the total sediment transport. The arrows in the top left corner indicate the offshore wave angle.

Figure 8 .Figure 9 .Figure 10 .Figure 11 .Figure 12 .

 89101112 Figure 8. Bathymetric contours at t = 19 days for Runs 12-21, wherein θ1 varies from -55 to 55 • and θ2 is constant at -25 • . Angle variations are schematised by the circles on the left, wherein the grey and black radius indicate θ1 and θ2, respectively. Vertical solid lines indicate sections I-IV as shown in Figure 2b.-39-

Table 1 .

 1 ). Consequently, km-scale nourishments with a curved coast may enforce rip dynamics and associated flows that threaten swimmer safety. Wright, L. D.,& Short, A. D. (1984). Morphodynamic variability of surf zones and beaches: a synthesis. MarineGeology, 56 , 93-118. Yu, J., & Slinn, D. N. (2003). Effects of wave-current interaction on rip currents. Overview of the runs. In reference run Ref1 time-invariant wave conditions were simulated for a total duration of 20 days, while in Ref2 a bimodal wave field was simulated by alternating the angle θ abruptly every day from θ1 to θ2. Variations on Ref1 and Ref2 include the forcing (Runs 1-39) and bathymetry (Runs 40-48). Red colours indicate the differences with Ref2.

	Journal of Geophysical Research, 108 (C3). doi: 10.1029/2001JC001105.

A straightening and/or offshore migration of the bar, observed in the field (e.g. Lippmann & Holman, 1990;Gallagher et al., 1998;Holman et al., 2006;Price & Ruessink, 2011;Contardo & Symonds, 2015;Rutten et al., 2018), was roughly explored by running the model with a larger wave height, period or angle but without success. Running the model with another ratio of the transport coefficients probably allows such bar behaviour. Dubarbier et al. (2017) explored the parameter space of the transport coefficients for a bar-beach system based on the Gold Coast (Australia) and found that the migration direction depends on the ratio of C w and C c . Using the same model, Bouvier et al. (2019) -24-manuscript submitted to JGR-Earth Surface Acknowledgments J.R. and B.G.R. were supported by the Dutch Technology Foundation STW that is part of the Dutch Organisation for Scientific Research (NWO), and which is partly funded by the Ministry of Economic Affairs, under contract 12686 (Nature Coast: S1 Coastal Safety). T.D.P. was funded by the Dutch Organisation for Scientific Research (NWO), under contract 016.Veni.171.101. B.D. and B.C. were funded through projects CHIPO (ANR-14-ASTR-0004-01) and SONO (ANR-17-CE01-0014), respectively, from Agence National de la Recherche (ANR). J.R. acknowledges Alec Torres-Freyermuth for the financial support (CONACYT Project CB 2016/284430) during the review process of the article. Computer time for this study was provided by the computing facilities MCIA (Mésocentre de Calcul Intensif Aquitain) of the University of Bordeaux and the University of Pau and Pays de l'Adour. Field data used to create the synthetic bathymetry are from Rutten et al. (2018). Model output is available from Rutten et al. (2019).

θ l,b , H b , and thus lower U ls,c . In section II, the growth rate variation between Runs 43-45 also relates to U ls,c , which varies from 0.003-0.13 m/s (Ref 1 ) to 0.25 m/s (Run 45).

Here, the varying magnitude of U ls,c depends mainly on the coastline orientation in section II. Besides, the left flank in these runs is not as strongly subjected to refraction as in Runs 40-42 since waves do not approach as strongly obliquely here. In Figure 13, information of Run 43 is partly missing, because the run crashed at t = 10.7 days. Attempts to simulate the full 20-day period using a smaller morphological time step of 15 or 10 minutes were unsuccessful. No patterns arise within the 20-day simulation period in Runs 46-48, which can be explained by the relatively large U ls,c of 0.15-0.57 m/s along the entire coastline under either θ 1 or θ 2 , irrespective of a curved coastline.

Overall, our simulations demonstrate that rip channels, located between the lunateshaped shoals of the crescentic bar, may develop at faster rate and become deeper with increasing curvature of the coastline, if the latter produces an increase of the percentage of near-normal local incidence. Under a time-invariant and a time-varying wave climate with limited obliquity, we found that curved coasts impact rip channel dynamics along their flanks as well as their adjacent straight coastlines because of the alongshore varying coastline orientation and the global refraction pattern. Increased rip channel presence at the straight coasts adjacent to the curved coast can enforce localised beach and dune erosion [START_REF] Thornton | Rip currents, megacusps, and eroding dunes[END_REF]. Moreover, rip channels are associated with narrow and approximately offshore-directed flows (rip currents) which are the leading deadly hazard to recreational beach users worldwide [START_REF] Castelle | Rip current types, circulation and hazard[END_REF]. Accordingly, both the design and location of km-scale nourishments must be carefully examined in perspective of the prevailing wave climate and the primary beach entries at the foreseen site.

Offshore Bar Migration and Straightening


Tatiana Kondakova

Nadine Merlet-Machour

Josselin Bodilis

Frédéric Dionnet

Marc Feuilloley

Nicole Orange

Hermann J Heipieper

Cécile Duclairoir Poc

Effect of temperature on Pseudomonas phospholipid composition

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Pseudomonas are present in all natural environments, and able to quickly adapt to environmental modi cations. Physiologically, this adaptation depends essentially to the structure and organization of their envelope. The membrane functions are traditionally attributed to proteins that are immersed in the lipid bilayer, but lipids are not only a matrix to accommodate proteins, but play a major role in bacterial functioning. P. uorescens is a psychrophilic bacterium commonly found in soil and water. Some of P. uorescens members are able to growth at the human temperature, as members of the skin microbiota or behave as opportunistic pathogens.

Establish if the response to human temperature involves changes in the structure and the composition of P. uorescens phospholipids at di erent phases of growth Study of lipidomes of two P. uorescens strains from di erent ecological niches: one clinical isolate MFN1032 and one airborne isolate MFAF76a Characterization of the phospholipid composition of P. uorescens using an adapted HPTLC-MALDI-TOF MS Imaging tool

Conclusions

  Open questions Objectifs P. uorescens is able to adapt to human temperature using two lipidic pathways: 1. the increasing of degree of fatty acid unsaturation controls the membrane uidity 2. the increase of the synthesis of zwitteronic phospholipids results in the membrane charge changes. 1. The phosphatidylcholine role in Pseudomonas adaptation composition Temperature and degree of fatty acids' saturation At 37°C the both strains product more zwitteronic phospholipids, like phosphatidyléthanolamine and phosphatidylcholine To adapt to human temperature the degree of fatty acids' saturation increases in both P. uorescens strains P. uorescens adapts to temperature changes via membrane phospholipid modi cation. 1. The overproduction of zwitteronic phospholipids allows the changes of membrane charge 2. The phosphatidylcholine overproduction may be involved in P. uorescens -host interactions. 3. The increasing of degree of fatty acids' saturation allows P. uorescens to keep the optimal membrane