Jordan Dezalay

Michel Broquier

Satchin Soorkia

Keisuke Hirata

Shun-Ichi Ishiuchi

Masaaki Fujii
email: mfujii@res.titech.ac.jp

Gilles Grégoire
email: gilles.gregoire@universite-paris-saclay.fr

Excited-state proton transfer in protonated adrenaline revealed by cryogenic UV photodissociation spectroscopy

We report a comprehensive study of the structures and deactivation porcesses of protonated adrenaline through cryogenic UV photodissociation spectroscopy. Single UV and double-resonance UV-UV hole burning spectroscopies have been performed and compared to coupled-cluster SCS-CC2 calculations done on the ground and first electronic states. Three conformers were assigned, the two lowest energy gauche conformers along with a higher energy conformer with an extended structure which is indeed the global minimum in solution. This demonstrates the kinetic trapping of this high energy gas phase conformer during the electrospray process. At the band origin of all conformers, the main fragmentation channel is the C-C bond cleavage, triggered by an excited state proton transfer to the catechol ring. Internal conversion leading to the water loss channel competes with the direct dissociation and tends to prevail with the increase of excess energy brought by the UV laser. Picosecond time-resolved pump-probe spectroscopy was achieved to measure the excited state lifetimes of the three conformers of AdH + , which decay with the increase of excess in the * state from 2 ns at the band origin down to few hundreds of picosecond 0.5 eV to the blue. Finally, about 0.8 eV above the band origin, the * state is directly reached leading to the opening of the H-loss channel.

Introduction

Adrenaline (Epinephrine), a human hormone belonging to the catecholamine family, is produced in response to acute stress and acts as a neurotransmitter involved in blood pressure regulation. Adrenaline binds to the  and  adrenergic receptors and activates G protein [START_REF] Fanelli | Computational Modeling Approaches to Structure-Function Analysis of G Protein-Coupled Receptors[END_REF][START_REF] Warne | The structural basis for agonist and partial agonist action on a β1-adrenergic receptor[END_REF] through multiple H-bonds and proton interaction. As many neurotransmitters, adrenaline is a flexible molecule which indeed assures the molecular recognition with its receptor. The search for pharmaceutical ligands binding to those receptors can benefit from the study of the neurotransmitter flexibility, which can be precisely addressed through gas phase laser spectroscopy and quantum chemistry calculations. For instance, -blockers bind to the adrenergic receptors and therefore inhibit the effect of adrenaline. [START_REF] Ramström | Chiral Recognition in Adrenergic Receptor Binding Mimics Prepared by Molecular Imprinting[END_REF] The spectroscopic study of gas phase conformers of neutral adrenaline was achieved by Carçabal et al. [START_REF] Çarçabal | A computational and spectroscopic study of the gas-phase conformers of adrenaline[END_REF] A dominant conformer was evidenced, with an extended sidechain structure stabilized by a COH to N intramolecular hydrogen bond. However, at physiological conditions, adrenaline is protonated at the methyl amino group [START_REF] Álvarez-Diduk | Adrenaline and noradrenaline: Protectors against oxidative stress or molecular targets?[END_REF] which should have a major impact on this intramolecular H-bond. Alagona et al. [START_REF] Alagona | Competitive H-bonds in vacuo and in aqueous solution for N-protonated adrenaline and its monohydrated complexes[END_REF] investigated theoretically the conformational landscape of protonated adrenaline (AdH + ) both in the gas phase and in aqueous solution. Three structural families were identified, differing by a 2/3 rotation of the alkyl amino chain along the C-C bond (see chart 1). In the gauche (folded) G1 conformer, both NH to COH and proton- interactions stabilize the gas phase structure while in aqueous solution, the trans (extended) T conformer is favored, due to the high solvation energy of the protonated methyl ammonium group by water. Both structures show a preference for NH to COH intramolecular hydrogen bond. The last family, in which the latter interaction is absent, the gauche G2 conformer, lies higher in energy both in the gas phase and in solution. Finally, for each family, four catechol ring orientations (STR1-4) are possible (see Chart 1) following the notation previously introduced. [START_REF] Alagona | Competitive H-bonds in vacuo and in aqueous solution for N-protonated adrenaline and its monohydrated complexes[END_REF] Structural investigation of flexible molecules in the gas phase is generally conducted through comparison of IR spectroscopy in the ground state with quantum chemistry calculations. For catecholamines, the distinct intramolecular H-bonds between protonated methyl ammonium and OH or catechol ring (gauche and trans forms) should normally provide sizeable spectral shifts that could be used for conformer assignment. However, for each of these rotamers, catechol ring orientation (STR1-4) has a little effect on the OH and NH stretching modes, precluding a definitive assignment of the structures through IR spectroscopy only. This has been recently documented for protonated noradrenaline, precursor of adrenaline in which the amine is not methylated, studied via cryogenic laser spectroscopy by Wako et al. [START_REF] Wako | A conformational study of protonated noradrenaline by UV-UV and IR dip double resonance laser spectroscopy combined with an electrospray and a cold ion trap method[END_REF] Interestingly, UV photodissociation spectroscopy of cold protonated molecules has emerged in the last decade and can nowadays be applied for structural assignment of medium size molecules through comparison with high-level excited state calculations. For instance, syn/anti conformation (position of the protonated amino group compared to the oxygen lone pair of the phenol ring) of protonated tyrosine 8 and synephrine [START_REF] Nieuwjaer | Photodissociation Spectroscopy of Cold Protonated Synephrine: Surprising Differences between IR-UV Hole-Burning and IR Photodissociation Spectroscopy of the O-H and N-H Modes[END_REF] have distinct vibronic spectra although their vibrational spectra in the 3 µm This journal is © The Royal Society of Chemistry 20xx

Please do not adjust margins Please do not adjust margins region (NH and OH stretches) cannot be discriminated, even via cryogenic IR spectroscopy. Beside the structural assignment of protonated molecules, UV photodissociation spectroscopy allows to decipher the deactivation processes at play following electronic excitation, where competition between internal conversion to the ground state and photoinduced reaction triggered in the excited state occurs. In particular, the specific C-C bond cleavage following UV excitation of protonated aromatic amino acids and peptides has recently been reviewed by Grégoire and coworkers. [START_REF] Soorkia | UV Photoinduced Dynamics of Conformer-Resolved Aromatic Peptides[END_REF] Such specific photofragmentation patterns could be useful for fast screening and identify biological molecules by mass spectrometry with complementary techniques as low energy collision (CID) or electron capture dissociation (ECD). [START_REF] Antoine | Comparison of the fragmentation pattern induced by collisions , laser excitation and electron capture . Influence of the initial excitation[END_REF] Furthermore, studying protonated systems cooled at cryogenic temperature reveals possible conformer selectivity which might be of prime importance for understanding the specific molecular recognition of such flexible neurotransmitters. 12 Chart 1 G1-STR1-4 structures of protonated adrenaline AdH + . The three rotamers G1, G2 and T correspond to 1 ≈ -60°, 60° and 180° respectively. For each rotamer, 2 ≈ ± 90° and 3,4 ≈ 0 or 180° define the different orientations of the catechol ring, noted STR1, STR2, STR3 and STR4.

In this work, the structures and conformer-resolved excited state properties of AdH + were investigated through cryogenic UV photodissociation spectroscopy in two laboratories. Conformer selected spectroscopies were recorded with high resolution nanosecond lasers in Tokyo, which includes UV photodissociation and UV-UV hole burning techniques. In Orsay, picosecond pump-probe photodissociation experiments were performed to record the excited state lifetime of the conformers assigned in Tokyo. We will discuss on the deactivation processes in the excited state and the evolution of the fragmentation branching ratio with the excess energy and the probed electronic state. The experimental data are analyzed by comparison with high level excited state calculations at the coupled-cluster SCS-CC2, including geometry optimization and frequency calculations.

Experimental and theoretical methods

The details of the experimental setup in Tokyo Institute of Technology are described elsewhere. [START_REF] Ishiuchi | High-coolingefficiency cryogenic quadrupole ion trap and UV-UV hole burning spectroscopy of protonated tyrosine[END_REF] Adrenaline (purchased from Sigma Aldrich) is dissolved at 10 µM into methanol containing 0.5% formic acid. The solution is sprayed from a nebulizer, and fine droplets were drawn into glass capillary heated at 60℃. The entrance of the glass capillary is coated by metal and high voltage (-4.5 kV) is applied to it. The desolvated ions are introduced to higher vacuum region via a skimmer and guided to a quadrupole mass spectrometer (Q-MS: Extrel) by a home-made hexapole ion guide whose pole diameter is 2 mm and operated at 4 MHz (RF generator CGC). Protonated adrenaline ions, mass-selected in the Q-MS, are introduced into a cryogenic 3-dimensional quadrupole ion trap (3D-QIT) made of copper via a home-made quadrupole ion deflector and a home-made octopole ion guide (pole diameter: 4 mm, RF: 1.6 MHz). The QIT is mounted on a second stage of a double stage closed cycle He cryostat (Sumitomo: RDK-408D2) cooled at 4 K. At the exit of the octopole ion guide, there is a pulsed lens electrode (20 Hz) to chop the ion beam to introduce the ions into the cryogenic QIT during 35 ms. He buffer gas at room temperature is introduced just before the entering of the ions and cooled down by collisions with the copper electrodes of the QIT. The trapped ions are thermalized at ~ 13 K by the cold He buffer gas. [START_REF] Ishiuchi | High-coolingefficiency cryogenic quadrupole ion trap and UV-UV hole burning spectroscopy of protonated tyrosine[END_REF] Then, a tunable UV laser (probe laser) is triggered at 37 ms and the photo fragments are ejected at 40 ms and detected by a home-made time of flight mass spectrometer (TOF-MS). The TOF-MS signal is amplified 10 times by a preamplifier (NF: BX-31A) and recorded by a fast digitizer board (NI: PXIe-5160) as a function of the wavelength of the UV laser. To measure UV-UV hole burning (UV-UV HB) spectra, another tunable UV laser (burn laser) is introduced to the cryogenic QIT 1 ms before the probe laser. During this 1 ms, a tickle RF pulse [START_REF] Wako | A conformational study of protonated noradrenaline by UV-UV and IR dip double resonance laser spectroscopy combined with an electrospray and a cold ion trap method[END_REF][START_REF] Ishiuchi | High-coolingefficiency cryogenic quadrupole ion trap and UV-UV hole burning spectroscopy of protonated tyrosine[END_REF][START_REF] Kang | New Method for Double-Resonance Spectroscopy in a Cold Quadrupole Ion Trap and Its Application to UV-UV Hole-Burning Spectroscopy of Protonated Adenine Dimer[END_REF] is applied to the entrance end-cup of the QIT to eject photo fragments produced by the burn laser. UV-UV HB spectra are recorded by monitoring the photo fragment signal due to the probe laser whose wavelength is fixed to a given band of the UVPD spectrum while scanning the wavelength of the burn laser. When the wavelength of the burn laser is resonant to any electronic transitions of the probed conformer, a decrease of ion signal is observed, which reflects depopulation of the zero vibrational level of this specific conformer. The probe laser is operated at 20 Hz, while the burn laser at 10 Hz. Therefore, probe-laser-only signal and double resonance signal were obtained alternatively. By dividing the latter by the former, long term fluctuations of the source condition, etc. could be cancelled. [START_REF] Okuzawa | Direct observation of second excited 1,3 (n,π*) states of pyrazine by UV-IR double resonance dip spectroscopy[END_REF] The setup installed in Orsay has already been described in details. [START_REF] Broquier | A comprehensive study of cold protonated tyramine: UV photodissociation experiments and ab initio calculations[END_REF] AdH + ions are produced in an electrospray source, first stored in an octopole ion trap for 100 ms and then extracted and accelerated at 200 V by a pulsed exit electrode and finally transferred in the cold 3D-QIT (Jordan Tof Inc.). The latter is biased at 200 V to ensure efficient trapping and avoid collision induced dissociation of the incoming ions with He buffer gas injected by a pulsed valve 1-2 ms before. The 3D-QIT is housed in a copper box directly mounted on a cold head of a compressed helium cryostat (CH-204 S, Sumitomo) that maintains the temperature around 15 K. A pulsed mass gate located before the entrance of the trap allows for massselecting the parent ion. The photodissociation laser is triggered at least 40 ms after the trapping of the ions when thermalization is achieved and helium buffer gas is mostly pumped. All ionic fragments and parent molecules are then extracted and accelerated for mass-analysis in a linear time-of-flight mass Please do not adjust margins Please do not adjust margins spectrometer and detected by MCP (Z-Gap, Jordan Tof Inc.). Fragmentation time can be recorded from 1 µs up to tens of ms by varying the delay between the photofragmentation laser and the extraction of the ions from the QIT. The photodissociation UV laser (pump) is the output of an optical parametric amplifier (EKSPLA-PG411) pumped by the third harmonic (355 nm) of a mode-locked picosecond Nd-YAG laser (EKSPLA-SL300) operated at 10 Hz which provides tunable visible/UV light in the 740-220 nm region. The spectral resolution is in the order of 8-10 cm -1 . For the 2 color experiments, the probe laser is the output of a second OPA pumped with the same laser. The probe laser has been scanned in the visible range without noticeable change and has been set at 550 nm. The picosecond time-resolved photodissociation spectroscopy [START_REF] Soorkia | Conformer-and Mode-Specific Excited State Lifetimes of Cold Protonated Tyrosine Ions[END_REF] is recorded by delaying the probe laser beam with a motorized optical delay line with 6.6 ps time step and a maximum delay time of 1.4 ns. Both beams are mildly focused by a 700 mm lens and cross each other in the center of the trap. Typical laser pulse energies are in the order of 40 µJ for the pump and 200 µJ for the probe beam. The cross correlation of the picosecond lasers is about 16 ps. Ab initio calculations have been performed with the TURBOMOLE program package 18 (v6.6) making use of the resolution-of-the-identity (RI) approximation for the evaluation of the electron-repulsion integrals. [START_REF] Christiansen | The secondorder approximate coupled cluster singles and doubles model CC2[END_REF][START_REF] Ahlrichs | Efficient evaluation of three-center twoelectron integrals over Gaussian functions[END_REF] The equilibrium geometries of AdH + in its electronic ground (S0) and excited (S1 and S2) states have been determined at the CC2 level [START_REF] Hättig | Geometry optimizations with the coupled-cluster model CC2 using the resolution-of-the-identity approximation[END_REF][START_REF] Köhn | Analytic gradients for excited states in the coupled-cluster model CC2 employing the resolution-of-the-identity approximation[END_REF] with the correlation-consistent polarized valence double- aug-cc-pVDZ basis set augmented with diffuse functions. [START_REF] Kendall | Electron affinities of the first-row atoms revisited. Systematic basis sets and wave functions[END_REF] In particular, the spin-component scaled approach (SCS) was applied since it improves the accuracy of CC2 calculations for the charge transfer states. [START_REF] Hellweg | Benchmarking the performance of spin-component scaled CC2 in ground and electronically excited states[END_REF] The vibrational modes of the ground and the first excited states have been calculated at the same level in order to obtain the adiabatic excitation energy corrected for the difference of zero-point energy between the two states. The calculated vibrationally-resolved electronic spectra were obtained with the PGOPHER spectra simulator package [START_REF] Western | PGOPHER: A program for simulating rotational, vibrational and electronic spectra[END_REF] using the calculated frequencies of the ground and excited states. For sake of comparison, the simulation was performed at 0 K and convoluted with a gaussian function of 3 cm -1 (full-width at half-maximum FWHM).

Results and discussion

1. Conformer assignment: UV photodissociation and UV-UV hole-burning spectra

The UV photodissociation mass spectrum of protonated adrenaline AdH + (m/z 184) shown in Fig. S1 corresponds to the difference mass spectrum (UV laser -no laser) with a delay between the laser excitation and the extraction of the ions from the QIT of 1 µs. In that conditions, only the fast fragmentation events are detected. The fragmentation channels can be sorted in two main groups. As for Noradrenaline, [START_REF] Rogalewicz | Decomposition of protonated noradrenaline and normetanephrine assisted by NH2 migration studied by electrospray tandem mass spectrometry and molecular orbital calculations[END_REF] the m/z 166 fragment corresponds to the water loss channel which is the main dissociation channel in the ground electronic state obtained through low energy collision and can eventually further fragment into its secondary fragment at m/z 135 (H2O + NH2CH3 loss) at longer fragmentation time. The other prompt fragmentation channels are issued from the C-C bond cleavage with the charge on the catechol ring (m/z 139) or on the amino alkyl chain moieties (m/z 46 and m/z 44) following H/H+ rearrangement. The last fragmentation channel at m/z 32 is assigned to the NH3CH3 + ionic fragment (C-N bond break). It should be stressed here that such fragmentation pattern, competition between CID like fragments and specific UV fragments issued from the C-C bond break, [START_REF] Soorkia | UV Photoinduced Dynamics of Conformer-Resolved Aromatic Peptides[END_REF] is commonly observed in the UV photodissociation of protonated aromatic amines, [START_REF] Féraud | Photofragmentation spectroscopy of cold protonated aromatic amines in the gas phase[END_REF] amino acids, 28-31 catecholamine 7 and short peptides. [START_REF] Dehon | Photofragmentation at 263 nm of small peptides containing tyrosine: the role of the charge transfer on CO[END_REF][START_REF] Zabuga | Fragmentation mechanism of UV-excited peptides in the gas phase[END_REF][START_REF] Soorkia | Ion-Induced Dipole Interactions and Fragmentation Times: Cα-Cβ Chromophore Bond Dissociation Channel[END_REF][START_REF] Burke | UV and IR spectroscopy of cold protonated leucine enkephalin[END_REF][START_REF] Kopysov | Nonstatistical UV Fragmentation of Gas-Phase Peptides Reveals Conformers and Their Structural Features[END_REF][START_REF] Deblase | Conformation-Specific Infrared and Ultraviolet Spectroscopy of Cold[END_REF] As it can be readily seen in Fig. S1, the fragmentation branching ratio is larger for the C-C bond cleavage events than for internal conversion. The electronic spectroscopy of AdH + reported in Fig. 1a has been recorded on the C-C bond cleavage fragment (m/z 139). The spectrum closely resembles the noradrenaline (NAdH + ) one. [START_REF] Wako | A conformational study of protonated noradrenaline by UV-UV and IR dip double resonance laser spectroscopy combined with an electrospray and a cold ion trap method[END_REF] The first weak vibronic transition is detected at 34 004 cm - 1 along with a long progression of low-frequency mode of 45 cm - 1 which extends up to hundreds of wavenumbers to the blue. Two sets of intense transitions are observed at 34 938 cm -1 and 35 046 cm -1 , which indeed correspond to the band origins of two other conformers as revealed by the UV-UV hole-burning spectroscopy reported in Fig. 1b. In this spectral region from 34 000 to 35 800 cm -1 , the excitation spectrum of AdH + is thus composed of three different conformers. Ab initio excited state calculations at the SCS-CC2/aug-cc-pVDZ level have been performed to assign the structure of the three conformers of AdH + . Following the same notation as for NAdH + , the conformers of AdH + can be sorted in three main families (G1, G2 and T) according to the 2/3 rotamers along the C-C bond. [START_REF] Alagona | Competitive H-bonds in vacuo and in aqueous solution for N-protonated adrenaline and its monohydrated complexes[END_REF] The G1-STR1 conformer, in which the protonated CH3NH2 + group is involved in two H-bonds with the catechol ring and the C-OH group, is the lowest energy minimum (see Chart 1). In this form, the protonated group is in syn position as compared to the oxygen lone pair of the hydroxyl groups of the This journal is © The Royal Society of Chemistry 20xx

Please do not adjust margins Please do not adjust margins catechol ring, above the OH group in meta position. The G1-STR3 conformer, iso energetic with G1-STR1, corresponds to an inversion of the CCCC 2 dihedral angle, so the NH2 group is no more on the top of the catechol OHs. For each conformer, the CCOH 3,4 dihedral angle of the catechol OHs can be inverted (G1-STR2 and G1-STR4, respectively) but their relative energies are significantly higher by 2.1 kcal/mol and 1.2 kcal/mol. In the T structures, the NH2 group forms a sole H-bond with the COH group in a trans orientation compared to the catechol ring (1≈180°). Such rotamers are among the lowest energy ones in aqueous solution [START_REF] Alagona | Competitive H-bonds in vacuo and in aqueous solution for N-protonated adrenaline and its monohydrated complexes[END_REF][START_REF] Wang | Hydrogen-bonding interactions in adrenaline-water complexes: DFT and QTAIM studies of structures, properties, and topologies[END_REF] but in the gas phase, due to the lack of proton- interaction, only the T-STR1 conformer has a relative low energy. Finally, all the conformers of the third rotamer family G2 (1 ≈ 60°), in which the NH2 group is not involved in Hbond with the COH group, have relative energies much higher and can be discarded from the analysis. In Table 1 are reported the ground state relative energies and -* adiabatic excitation energies of the most relevant conformers of AdH + . Conformer A has the reddest band origin, almost 1000 cm -1 below those of conformers B and C. The adiabatic excitation energy calculated for G1-STR1 is significantly lower than for all the other conformers. Its simulated Franck-Condon spectrum is reported in Fig. 2a and provides a fair agreement with the experimental spectrum of conformer A. Accordingly, conformer B of AdH+ can be confidently assign to the T-STR1 structure, giving the good agreement with the adiabatic excitation energy and the simulated vibronic spectrum (Fig. 2b). The extended structure T, in which the protonated methyl amino group does not interact with the catechol ring, lies relatively high in energy in the gas phase and should not be populated in the cold ion trap, unless kinetic trapping of the AdH + most stable solution phase structure occurs, as already observed in NAdH + . [START_REF] Wako | A conformational study of protonated noradrenaline by UV-UV and IR dip double resonance laser spectroscopy combined with an electrospray and a cold ion trap method[END_REF] Finally, conformer C is tentatively assigned to structure G1-STR3, although we cannot exclude the structure G1-STR4 which both provide satisfying adiabatic energy transitions and simulated FC spectra (Fig. 2c and Fig. S2). The active modes of the vibronic spectra of the 4 conformers are depicted in Fig. S3-6. One should stress here that for these two latter conformers, the intensity of the vibronic transitions 200 cm -1 above the band origin is significantly reduced compared to the predicted one while two intense bands show up at 523 cm -1 and 649 cm -1 . Such a cut-off was already observed in NAdH + , structure G1-STR3. [START_REF] Wako | A conformational study of protonated noradrenaline by UV-UV and IR dip double resonance laser spectroscopy combined with an electrospray and a cold ion trap method[END_REF] We do not have a simple explanation although this cut-off occurs for the same type of structure. Finally, it is worth mentioning that the absolute error in adiabatic excitation energies is quite low, less than 2%, which is totally consistent with the absolute error found at the CC2 level for protonated aromatic amino acids and related systems. In overall, the conformational landscape of AdH + and NAdH + are similar, although more conformers were observed for NAdH + Fig. 2 Conformer assignment of AdH + through comparison between the experimental and simulated vibronic spectra.

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Excited State lifetime

The excited state lifetimes of AdH + have been recorded through a picosecond pump-probe photodissociation scheme. In Fig. SI7, the vibronic spectrum recorded with the low resolution OPA picosecond laser is compared to the higher resolution spectrum (dye laser). The band origins of the three conformers A, B and C are readily observed along with the main vibronic transitions except for conformer B for which only the origin transition is clearly observed. So the pump-probe photodissociation dynamics have been monitored as a function of the pump excess energy for the two gauche conformers A and C and only at the band origin of the trans conformer B. In the two-color excitation scheme with the probe laser set at 550 nm, the photofragmentation pattern changes, with the opening of the H-loss channel at m/z 183 along with an net increase of the m/z 44, which is indeed the main fragment of the adrenaline radical cation. The difference mass spectrum (two-color excitation versus UV pump only) is reported in Fig. S2b. All the fragmentation channels observed through UV excitation are depleted following absorption of the probe beam, whatever the probe wavelength used in the visible from 420 nm to 650 nm or set at 355 nm. In Fig. 3a and Fig. 3b are reported the transients recorded on the H loss channel, average of the m/z 183 and m/z 44 signals, for the conformers A and C excited on the 00 0 transition and about 800 cm -1 further to the blue. In Fig. 3c, the dynamics recorded at the band origin of conformer B is plotted.

All the transients have been fitted by a mono exponential decay function with a time constant  convoluted by a gaussian function to take into account the temporal profile of the picosecond lasers (autocorrelation of 16 ps). The average uncertainty on the time constant issued from the fitting process is about 10-20 %. All conformers behave essentially the same with an excited state lifetime of about 2 ns, which smoothly decreases with the increase of excess energy imparted by the pump laser. It should be noted that at much higher pump energy, i.e. 37 000 cm -1 (270 nm), the time constant is significantly shorter, in the order of 370 ps (Fig. 3d), but consistent with an exponential decay of the excited state lifetime with the increase of excess energy.

The excited state lifetime of AdH + , in the order of 2 ns at the band origin, is significantly longer than in neutral catechol (about 10 ps) [START_REF] Livingstone | Time-resolved photoelectron imaging of excited state relaxation dynamics in phenol, catechol, resorcinol, and hydroquinone[END_REF] for which the lifetime is governed by strong coupling of the locally excited 1 * with the dissociative 1 OH* state along the O-H stretch. This coupling is strongly disturbed in water and crown ether complexes as revealed by the elongated lifetime of 10-12 ns. [START_REF] Morishima | Anomalous cage effect of the excited state dynamics of catechol in the 18-crown-6-catechol host-guest complex[END_REF] Interestingly, the lifetime of AdH + is similar to the lifetimes recorded for similar systems, such as protonated tyrosine [START_REF] Soorkia | Conformer-and Mode-Specific Excited State Lifetimes of Cold Protonated Tyrosine Ions[END_REF] or tyramine. [START_REF] Broquier | A comprehensive study of cold protonated tyramine: UV photodissociation experiments and ab initio calculations[END_REF] In these latter systems, excited state proton transfer process was clearly evidenced as the main deactivation process near the band origin while electronic coupling with charge transfer states (* or CO*) prevails higher in energy. As it will be discussed below, this general deactivation mechanism applied to AdH + . In Fig. 4 is reported the evolution of the fragmentation yields of the C-C bond cleavage, the water loss and the H loss on a large spectral range from the band origin of conformer A (34 000 cm - 1 , 4.2 eV) up to 45 000 cm -1 (5.6 eV). These spectra have been recorded with the OPA ps laser at a low spectral resolution. In Fig. SI 7, we compare the spectroscopy recorded on the m/z 139 fragment with the nanosecond (0.2 cm -1 resolution) and picosecond (10 cm -1 resolution). The main difference is the broad continuum obtained with the ps laser beneath the sharp vibronic transitions assigned to conformer A and C, which might be due to spectral congestion that cannot be accurately recorded with the OPA laser. Such broadening precludes the observation of all the transitions except the band origin assigned to the extended conformer B (around 34 900 cm -1 ) with the ps laser, which are nevertheless clearly detected with a 0.2 cm -1 dye laser. The C-C bond cleavage occurs at the band origin of the * state in all conformers. Such photoinduced reaction has previously been assigned in aromatic amines, amino acids and related systems such as synephrine to excited state proton transfer from the ammonium group to the aromatic ring. [START_REF] Soorkia | UV Photoinduced Dynamics of Conformer-Resolved Aromatic Peptides[END_REF] In the optimized structure of the * state of conformer A, a lengthening of the N-H bond (0.023 Å) pointing to the ring along with an out-of-plane bending of the CH (≈30°) facing the protonated group are observed (see Fig. S8). The Minimum Energy Path (MEP) for the excited state proton transfer reaction has been evaluated at the SCS-CC2/aug-cc-pVDZ level, i.e. optimization of the first excited state at fixed N-H bond lengths.

It should be stressed here that full optimization of the transferred form in the excited state leads to a curve crossing with the ground state, point where the optimization at the CC2 level fails to converge. So, in the course of the proton transfer reaction, a direct C-C bond cleavage is expected to occur, in competition with internal conversion leading to H2O loss, as it will be discussed below. A barrier for ESPT of 0.30 eV at NH=1.4 Å has been found, quite similar to the one calculated for

Please do not adjust margins Please do not adjust margins synephrine [START_REF] Nieuwjaer | Photodissociation Spectroscopy of Cold Protonated Synephrine: Surprising Differences between IR-UV Hole-Burning and IR Photodissociation Spectroscopy of the O-H and N-H Modes[END_REF] and twice larger than for tyrosine. [START_REF] Grégoire | Ab initio Study of the Excited-State Deactivation Pathways of Protonated Tryptophan and Tyrosine[END_REF][START_REF] Féraud | Excited State Dynamics of Protonated Phenylalanine and Tyrosine: Photo-Induced Reactions Following Electronic Excitation[END_REF] In these two latter systems, the ionic photo fragment associated to C-C bond cleavage has the aromatic ring with an extra proton and/or hydrogen attached to it, consistent with the proton transfer process. For AdH + , the fragment at m/z 139 corresponds to the ionic catechol ring following C-C bond rupture. This suggests that hydrogen rearrangement occurs after the initial proton transfer, as confirmed by the detection of two ionic fragments at m/z 44 and 46 (alkyl amino chain counterpart of the catechol ring). 3.3.2. Reaction path connecting the T and G1 conformers in the excited state. In the trans T-STR1 structure (conformer B), the methyl ammonium group does not interact with the catechol ring, so direct proton transfer can obviously not occur, although the C-C bond cleavage reaction is indeed observed (Fig. 1). We have calculated the reaction path connecting the two trans (T-STR1) and gauche (G1-STR1) structures in the excited state at the CC2-SCS/aug-cc-pVDZ. Fourteen intermediate structures were used for discretization of the path, which have been optimized throughout, leading to an energy barrier of 0.1 eV from the trans to the gauche conformer (see Fig. 5). Such a low energy transition state connecting the two rotamers could be overcome within the nanosecond excited state lifetime of the * state.

Competition between C-C bond cleavage and water loss.

At the band origin of conformer A (34 004 cm-1), the C-C bond break channels are more intense than the water loss channel, but the spectroscopy recorded on both is similar, with a bell shape structure of low frequency modes. Interestingly, the fragmentation branching ratio drastically evolves with the excess energy in the excited state, the spectroscopy recorded on the water loss fragment exhibits a broad, unstructured absorption band centred at 37 000 cm-1. From this excitation energy, the C-C bond cleavage channel closes.

The mechanism leading to water loss indeed remains unclear.

Water loss is the main dissociation channel in the ground state at low excitation energy, so internal conversion to the ground state is thought to be responsible for its detection following UV excitation. In the course of the excited state proton transfer, the S1/S0 energy gap vanishes so internal conversion could occur, in competition with the prompt C-C bond cleavage (see Fig. 4).

In that case, the photodissociation spectroscopy recorded on both fragments (m/z 166 and m/z 139) should be similar, reflecting the absorption of AdH + . This is indeed what happens at the band origin of all conformers. In the spectral region of the band origins of conformer B and C, conformer A still absorbs but the fragmentation branching ratio seems to evolve in favor of the water loss channel. The 00 0 transitions of conformer B and C are clearly detected on the m/z 139 fragment (background free) but buried in the water loss channel due to the contribution of the high energy absorption band of conformer A. Note that with the better spectral resolution provided by the dye laser, the band origin of these two latter conformers are observed at the m/z 166 fragment on the top of a broad fragmentation continuum. Such changes in the fragmentation branching ratio with the excess energy has already been observed and rationalized in protonated phenylethylamine and tyramine. [START_REF] Féraud | Photofragmentation spectroscopy of cold protonated aromatic amines in the gas phase[END_REF] Such charge transfer state in which the anti-bonding orbital is centred on the protonated amino group is dissociative through a barrier along the N-H stretch coordinate and will cross the ground electronic state (see Fig. 4). Since the H-loss ionic fragment is detected, it can be concluded that the */S0 curve crossing occurs at a lower energy than the excitation energy, otherwise hindered H-loss reaction leading to internal conversion would happen. [START_REF] Féraud | Photofragmentation spectroscopy of cold protonated aromatic amines in the gas phase[END_REF] The fragmentation yield on the H2O loss channel increases as well in the same spectral region, meaning that part of the population following the dissociative * surface will go back to the ground state at the */S0 crossing. It is noteworthy that the deactivation through the dissociative * state does not induce the C-C bond cleavage, since this latter fragmentation channel is not observed along with the H loss channel. So in overall, the nonradiative deactivation processes in AdH + involves proton transfer to the aromatic ring at the band origin while the coupling with the 1 NH2* charge transfer state prevails about 0.7 eV higher in energy. This mechanism is also found in protonated tyrosine and may be general for protonated catecholamines.

Conclusions

We have reported the UV photodissociation spectroscopy of cold AdH + . Three conformers have been identified through comparison of the UV-UV hole burning spectra with high-level calculations at the SCS-CC2 level in the ground and excited states. As for NAdH + , the relatively high energy trans conformer T, in which the alkyl side chain is in anti orientation compared to the catechol ring is observed, reflecting the kinetic trapping of this structure initially present in solution during the ESI process. The excited state lifetimes of the gauche and trans conformers have been determined around 2 ns at the band origin and smoothly decrease with the excess energy in the locally excited * state. Interestingly, the main deactivation process at the band origin of all AdH + conformers is the excited state proton transfer to the catechol ring, leading mostly to C-C bond cleavage fragments. In the ESPT structure, the S1/S0 energy gap vanishes so competition between direct C-C bond cleavage and internal conversion occurs. It turns out that internal conversion to the ground state leading to water loss prevails with the increase of excess energy in the * state. Less than 1 eV above the * state, direct excitation of the * state is evidenced by the opening of the H-loss channel.

Fig. 1

 1 Fig. 1 (a) UV photodissociation spectrum and (b) UV-UV hole burning spectra of AdH + recorded on the m/z 139 channel issued from the C-C bond cleavage.

Fig. 3 3 . 3 . Deactivation mechanism 3 . 3 . 1 .

 333331 Fig. 3 (a-c) Excited state lifetime of the three conformers of AdH+ (conformers A, C and B, respectively) as a function of the excitation wavelength recorded on the Hloss channel. The red curve is a monoexponential decay fitting function with a time constant , reflecting the excited state lifetime of the probed conformer at a given vibronic level. (d) pump wavelength set at 270 nm from which the C-C bond cleavage fragmentation channel closes.

Fig. 4

 4 Fig. 4 Fragmentation yields of AdH + recorded on (a) C-C bond break, (b) H2O loss and (c) H loss on a large spectral range from the band origin of the * transition up to 1.3 eV above. The fragmentation yields are plotted on the same vertical scale for sake of comparison. (d) Schematic representation of the PESs along the proton transfer coordinate and the H loss along the NH coordinate.

Fig. 5

 5 Fig. 5 Excited state reaction path connecting the trans (T-STR1) and gauche (G1-STR1) structures of AdH + .

Table 1 :

 1 Conformer assignment, relative energy (kcal/mol) in the ground state ES0, calculated and experimental * adiabatic excitation energy 00 0 with the absolute error  calc-exp (cm -1 ).

		Structure		00 0 (calc)	00 0 (exp)	 calc-exp
	Conf. A	G1-STR1	0	33 296	34 004	-707
	Conf. B	T-STR1	1.8	34 274	34 938	-663
	Conf. C	G1-STR3	0	34 470	35 046	-575
		G1-STR4	1.2	34 310

  C-C bond cleavage is the main or unique fragmentation channel at the band origin and closes few hundreds wavenumbers to the blue when ammonia loss, issued from internal conversion to the ground state, becomes the unique fragmentation channel. Conformer A (G1-STR1) of AdH + shares the same property, internal conversion to the ground state prevails as the excess energy in the * state increases, leading to the broad absorption band detected on the water loss channel. The onset of a second absorption band is clearly observed from 40 000 cm -1 , i.e. about 0.75 eV above the adiabatic excitation energy of the * state, with the opening of the H loss and H2O loss channels. The opening of the H loss channel 1 eV or less above the onset of the * transition has already been observed and assigned to the direct excitation of the * state.[START_REF] Kang | Photo-induced dissociation of protonated tryptophan TrpH + : A direct dissociation channel in the excited states controls the hydrogen atom loss[END_REF][START_REF] Kang | Control of Bond-Cleaving Reactions of Free Protonated Tryptophan Ion by Femtosecond Laser Pulses[END_REF][START_REF] Kang | Ultrafast deactivation mechanisms of protonated aromatic amino acids following UV excitation[END_REF] Vertical and adiabatic energies of the * state have been calculated for the three AdH + conformers and the results are reported in Table2. In all conformers, the S2 * state lies 0.6-0.8 eV above the S1 * state at the ground state geometry (vertical excitation energy). Optimization of the S2 * state leads to a true minimum (no imaginary frequency) of the potential energy surface for the gauche conformers A and C, with an adiabatic transition energy 0.73 eV and 0.61 eV above the band origin of the S1 * state, respectively. For the trans structure T-STR1 (conformer B), * geometry optimization leads to a curve crossing with the * state, 0.57 eV above its band origin. At this point, the optimization is restarted on the S1 electronic state but because of the large mixing of the * and * orbitals in the electronic configuration of the state, the optimization finally converges to the minimum of the * electronic state. The lower transition energy of the * state in the trans conformer can be qualitatively explained by the absence of proton- interaction in this structure, the NH2* orbital being destabilized when the protonated group is involved in two H-bonds like in the gauche conformers. Finally, it is worth noting that the calculated * This journal is © The Royal Society of Chemistry 20xxPlease do not adjust margins Please do not adjust margins adiabatic energy transition matches the onset of the second absorption band of AdH + leading to the H loss fragment about 0.75 eV above the onset of the * transition.

3.3.4. Opening of the H loss channel.

Table 2

 2 Vertical and adiabatic excitation energy of the * and * states of the three AdH + conformers. Adiabatic energy corrected by the difference in zero point energy in the ground and excited states. All values in eV. energy at the crossing point with * state, not a true minimum

	Structure		*	
		vert. energy	adiab. energy	adiab. energy
	G1-STR1 -conf A	4.52 / 5.35	4.13 / 4.87	0.73
	G1-STR3 -conf C	4.61 / 5.30	4.27 / 4.88	0.61
	T-STR1 -conf B	4.63 / 5.14	4.25 / 4.82 a	0.57 a

a

This journal is © The Royal Society of Chemistry 20xx J. Name., 2013, 00, 1-3 | 3

Acknowledgements

This work was supported in part by the ANR Research Grant (ANR2010BLANC040501-ESPEM), by KAKENHI (JP19H05527, JP18H01938, JP19K23624) of JSPS, World Research Hub Initiatives in Tokyo Institute of Technology, the Cooperative Research Program of the "Network Joint Research Center for Materials and Devices" from the Ministry of Education, Culture, Sports, Science and Technology (MEXT), Japan, and the RIKEN Pioneering Project, "Fundamental Principles Underlying the Hierarchy of Matter: A Comprehensive Experimental Study". GG thanks the Tokyo Institute of Technology for the invited professor grant. We also acknowledge the use of the computing facilities Méso-LUM of the LUMAT federation (LUMAT FR 2764) and MAGI of University Paris 13.

Conflicts of interest

There are no conflicts to declare.


Bruno Castelle
email: bruno.castelle@u-bordeaux.fr

Arthur Robinet

Déborah Idier

Maurizio D'anna

Modelling of embayed beach equilibrium planform and rotation signal

Keywords: embayed beach, hybrid shoreline model, headland length, rotation, equilibrium beach planform, headland sand bypassing

Embayed beaches are highly attractive sandy beaches bounded laterally by rigid boundaries, which deeply affect equilibrium beach planform and shoreline dynamics. We use LX-Shore, a state-of-the-art shoreline change model coupled with a spectral wave model to address embayed beach shoreline dynamics driven by longshore sediment transport processes. The model is applied to different idealized embayed beach configurations including variations in headland lengths. The model simulates a large range of equilibrium embayed beach planforms and associated spatial and temporal modes of shoreline variability. For short headlands enabling occasional headland sand bypassing, both embayed beach curvature and maximum erosion at the upwave side of the embayment increases with increasing headland length. Beach curvature also increases with increasing headland length for headlands long enough to prevent any headland sand bypassing. In contrast, at the same time, embayed beach becomes increasingly curved and symmetric, with maximum localised erosion within the embayment decreasing in intensity. When there is no headland sand bypassing, rotation signal decreases in amplitude and becomes increasingly symmetric with increasing headland length. The modal (timeinvariant) directional spreading of incident waves is critical to embayed beach behaviour, with the envelope and variance of cross-shore shoreline change and time-averaged shoreline curvature all increasing with decreasing modal directional spreading. Embayed beach rotation characteristic timescale increases with increasing embayed beach length, while the narrower the embayment the smaller the cross-shore amplitude of shoreline variability. Our simulations provide new insight into the influence of embayment characteristics and incident wave conditions on equilibrium planform and shoreline dynamics of embayed beaches. This work also implies that the degree of potential headland sand bypassing should be taken into account for modelling of beach rotational dynamics and embayed beach dynamic planform configuration.

Highlights

• Embayed beach shoreline response is simulated with a hybrid shoreline model

• Headland length and headland sediment bypassing control shoreline response

• Wave directional spreading is critical to both mean shoreline and rotation signal

• Embayment beach length controls rotation characteristic timescale

Introduction

Embayed beaches are sandy beaches bounded laterally and vertically by rigid boundaries (Loureiro et al., 2012a), which can deeply affect incident wave conditions and, in turn, breaking wave conditions (e.g., Daly et al., 2014a), beach morphodynamics (e.g., [START_REF] Castelle | The morphodynamics of rip channels on embayed beaches[END_REF][START_REF] Blossier | Nearshore sandbar rotation at single-barred embayed beaches[END_REF][START_REF] Mccarroll | Shoreface storm morphodynamics and mega-rip evolution at an embayed beach: Bondi Beach, NSW, Australia[END_REF] and shoreline variability (e.g. [START_REF] Ranasinghe | The Southern Oscillation Index, wave climate, and beach rotation[END_REF][START_REF] Ojeda | Shoreline dynamics and beach rotation of artificial embayed beaches[END_REF]Turki et al., 2013a;[START_REF] Harley | New insights into embayed beach rotation: the importance of wave exposure and cross-shore processes[END_REF]Robinet et al., 2020a). These physical boundaries include artificial coastal defences like groins, harbour walls and breakwater, and natural rocky headlands and submerged rocky platforms. Artificial coasts (e.g., approximately 30% of the French coastline, CEREMA, 2017) and hilly/mountainous coasts [START_REF] Short | Embayed and structurally controlled embayed beaches[END_REF]) constitute a large proportion of the world's coastlines, making embayed beaches ubiquitous globally. Due to the large natural variability of wave exposure and headland characteristics [START_REF] Fellowes | Morphometric classification of swell-dominated embayed beaches[END_REF], embayed beaches exhibit a striking planform variability (Figure 1). Therefore, understanding and modelling embayed beach dynamics is an important but complex research area.

Over the last decades, a lot of attention has been paid to describing the time-averaged, equilibrium planform of embayed beaches (e.g., [START_REF] Yasso | Plan geometry of headland bay beaches[END_REF]Sivester, 1970;[START_REF] Rea | Computer simulation models of a hooked beach shoreline configuration[END_REF]. The equilibrium beach concept was used to derive a number of static equilibrium models (Hsu et al., 1987;[START_REF] Hsu | Parabolic bay shapes and applications[END_REF], which are generally based on a single representative wave direction, a downcoast control point and a diffraction point. Parabolic bay shaped models [START_REF] Hsu | Parabolic bay shapes and applications[END_REF] were preferred to log-spiral models [START_REF] Yasso | Plan geometry of headland bay beaches[END_REF]Sivester, 1970), and became the most widely used approach to understanding the stability of embayed beaches [START_REF] Hsu | Geomorphic approach for mitigation beach erosion downdrift of littoral barrier[END_REF][START_REF] Jackson | Application of the equilibrium planform concept to natural beaches in Northern Ireland[END_REF]Cooper, 2010, Hsu et al., 2010). These equilibrium shoreline planform models, however, do not always show good skill [START_REF] Gonzalez | On the application of static equilibrium bay formations to natural and man-made beaches[END_REF]Klein et al., 2003a,b), for example near estuary mouths or where there is a strong geological control of beach morphology (e.g., intertidal to subtidal rocky platform, [START_REF] Jackson | Application of the equilibrium planform concept to natural beaches in Northern Ireland[END_REF]. Even when general settings are favourable to such model application, there are still limitations. For instance, a strong underlying assumption is that swell waves arrive persistently from a narrow range of directions, which is challenged on many coasts with multimodal wave climates. Such models also assume that alongshore wave height gradient is controlled by one diffraction point, while on natural coasts many diffraction points can co-exist and can even migrate across rocky platforms as the tide fluctuates [START_REF] Jackson | Application of the equilibrium planform concept to natural beaches in Northern Ireland[END_REF]. In addition, as noted by Daly et al. (2014a) diffraction is dominant only for strongly curved embayments when wave conditions are narrow-banded. Otherwise, refraction typically overwhelms diffraction effects. In addition, these models do not take into account headland sediment bypassing, which can have a profound impact on equilibrium shoreline planform as suggested by [START_REF] Goodwin | An insight into headland sand bypassing and wave climate variability from shoreface bathymetric change at Byron Bay, New South Wales, Australia[END_REF]. To overcome these issues and to address a wider spectrum of natural embayed beaches, equilibrium shoreline models are still under development (Elshinnawy et al., 2018a,b,c).

The processes governing the morphodynamics of embayed beaches on the timescales from storms to years/decades have been studied for decades (e.g., [START_REF] Klein | Short-term beach rotation processes in distinct headland bay beach systems[END_REF][START_REF] Ojeda | Shoreline dynamics and beach rotation of artificial embayed beaches[END_REF]. Day by day wave forcing is the primary driver of alongshore and cross-shore sediment transport, leading to complex patterns of erosion and accretion along the embayment. Clockwise/counterclockwise rotation of the embayed beach planform has been commonly observed as the dominant pattern of shoreline variability at embayed beaches [START_REF] Masselink | Seasonal changes in beach morphology along the sheltered coastline of Perth, Western Australia[END_REF][START_REF] Short | Decadal scale patterns in beach oscillation and rotation Narrabeen beach, Australia-time series, PCA and wavelet analysis[END_REF][START_REF] Ranasinghe | The Southern Oscillation Index, wave climate, and beach rotation[END_REF][START_REF] Martins | Small scale beach rotation process on a reflective beach[END_REF]Thomas et al., 2011;Loureiro et al., 2012a;Turki et al., 2013a;Van de Lageweg et al., 2013;[START_REF] Harley | Shoreline rotation and response to nourishment of a gravel embayed beach using a low-cost video monitoring technique: San Michele-Sassi Neri, central Italy[END_REF]Robinet et al., 2020a). [START_REF] Loureiro | Mechanisms and timescales of beach rotation[END_REF] reviewed the mechanisms and timescales of beach rotation at embayed beaches. Beach rotation has long been attributed to longshore transport processes, when enough sediment moves from one side of the embayment to the other to shift the mean orientation of the beach [START_REF] Ratliff | Modes and emergent time scales of embayed beach dynamics[END_REF]. However, storms and variations in incident wave energy cause crossshore sediment exchange and drive shoreline changes typically by tens of metres [START_REF] Yates | Equilibrium shoreline response: observations and modelling[END_REF][START_REF] Splinter | A relationship to describe the cumulative impact of storm beach[END_REF]. [START_REF] Harley | A reevaluation of coastal embayment rotation: the dominance of cross-shore versus alongshore sediment transport processes, Collaroy-Narrabeen Beach, southeast Australia[END_REF][START_REF] Harley | New insights into embayed beach rotation: the importance of wave exposure and cross-shore processes[END_REF] suggested a more complex conceptual model of beach rotation process at Narrabeen-Collaroy embayment (SE Australia) based on the analysis of an extensive dataset. They revealed that headland sheltering of oblique waves results in alongshore non-uniform wave exposure and cross-shore processes that, in turn, contribute significantly to observed beach rotation. [START_REF] Harley | New insights into embayed beach rotation: the importance of wave exposure and cross-shore processes[END_REF] therefore suggested that beach rotation occurs due to the complementary and interacting cross-shore and longshore processes at different timescales. Robinet et al. (2020a) used a state-of-the-art shoreline change model and demonstrated at the same coastal embayment that cross-shore processes drive a slightly alongshore variable cross-shore migration signal on the timescales from hours (storms) to months, while longshore processes primarily contribute to rotation on longer timescales of months to seasons at this coastal embayment.

In recent years, a new generation of shoreline change models have been developed, offering the opportunity to simulate shoreline evolution at wave-dominated sandy beaches on the timescales from hours (storm) to years and decades with reasonable skill [START_REF] Montaño | Blind testing of shoreline evolution models[END_REF]. Process-based models, which are computationally expensive, require accurate selection and sequencing of representative wave conditions to address embayed beach dynamics (Daly et al., 2014b). Instead, the so-called "hybrid" models are based on general principles (e.g., [START_REF] Yates | Equilibrium shoreline response: observations and modelling[END_REF][START_REF] Davidson | A simple equilibrium model for predicting shoreline change[END_REF][START_REF] Vitousek | A model integrating longshore and cross-shore processes for predicting long-term shoreline response to climate change[END_REF][START_REF] Robinet | A reduced-complexity shoreline change model combining longshore and cross-shore processes: The LX-Shore model[END_REF][START_REF] Antolínez | Predicting climate driven coastlines with a simple and efficient multi-scale model[END_REF]Tran and Barthelemy, 2020) rather than on the detailed description of nearshore hydrodynamics, sediment transport and morphological changes through conservation of mass and momentum. Hybrid models can therefore simulate shoreline change with low computational cost. Along coastlines that are longshore transport dominated with presence of rigid physical boundaries such as headlands, one-line models have emerged as a powerful tool (e.g., [START_REF] Hanson | Genesis -a generalized shoreline change numerical model[END_REF][START_REF] Larson | Simulation of regional longshore sediment transport and coastal evolution -the Cascade model[END_REF]Szmytkiewicz et al., 2000;[START_REF] Ratliff | Modes and emergent time scales of embayed beach dynamics[END_REF][START_REF] Hurst | Exploring the sensitivities of crenulate bay shorelines to wave climates using a new vector-based one-line model[END_REF]. All these models are essentially based on gradients in longshore sediment transport due to the alongshore variability in breaking wave conditions, but they vary in terms of numerical implementation and level of description on wave field transformation. [START_REF] Ratliff | Modes and emergent time scales of embayed beach dynamics[END_REF] identified two primary orthogonal modes of shoreline behaviour around the mean that are the well-known rotation mode and a newly identified breathing mode, which represents changes in shoreline curvature. [START_REF] Hurst | Exploring the sensitivities of crenulate bay shorelines to wave climates using a new vector-based one-line model[END_REF] examined the behaviour of embayed beaches forced by different directional wave climates.

They showed that embayed beach planform curvature decreases with the spread of wave directions.

However, in both studies headland sediment bypassing and the influence of headland length was not addressed. In addition, wave transformation was based on simple rules for the diffraction and refraction of waves where the coast was shadowed from incoming waves.

In this paper we use the state-of-the-art shoreline change model LX-Shore, which accounts for the detailed wave transformation to provide enhanced insight into embayed beach shoreline dynamics.

LX-Shore model [START_REF] Robinet | A reduced-complexity shoreline change model combining longshore and cross-shore processes: The LX-Shore model[END_REF] can simulate both the equilibrium embayed beach planform and a large range of spatial and temporal modes of shoreline variability even in complex natural embayed beaches, even with prominent rocky headland and submerged outcrops deeply affecting the incident wave field (Robinet et al., 2020a). The model is applied to different embayed beach configurations. We systematically address the influence of headland length on equilibrium embayed beach planform and shoreline dynamics on the timescales from hours to decades.

Method

LX-Shore model

LX-Shore is a two-dimensional planview cellular-based one-line shoreline change model for wavedominated sandy coasts developed by [START_REF] Robinet | A reduced-complexity shoreline change model combining longshore and cross-shore processes: The LX-Shore model[END_REF], with additional process implementation described in Robinet et al. (2020aRobinet et al. ( , 2020b). An overview of the model is provided in Figure 2 and the reader is referred to the aforementioned papers for more details. LX-Shore simulates shoreline change resulting from the combination of cross-shore transport driven by changes in incident wave energy and gradients in total longshore sediment transport (Figure 2b). Based on earlier work [START_REF] Ashton | Formation of coastline features by large-scale instabilities induced by high-angle waves[END_REF][START_REF] Ashton | High-angle wave instability and emergent shoreline shapes: 1. Modeling of sand waves, flying spits, and capes[END_REF]) LX-Shore computes changes in the relative amount of dry (i.e. land) surface in square cells discretizing horizontally the computation domain. The spatial resolution of grid cells ∆𝑥𝑥𝑥𝑥 is typically in the range of 10-100 m. Dry cell fraction ranges from 𝐹𝐹 = 0 (water cells) to 𝐹𝐹 = 1 for fully dry sandy and/or rocky cells. Indeed, LX-Shore also includes non-erodible areas such as coastal defences and headlands with time-invariant rocky cell fraction (𝐹𝐹 𝑅𝑅 ) contribution : 𝐹𝐹 = 𝐹𝐹 𝑆𝑆 + 𝐹𝐹 𝑅𝑅 , with 𝐹𝐹 𝑆𝑆 the sandy fraction. The shoreline is constructed using an interface reconstruction method, allowing the algorithm to handle complex shoreline geometries (e.g. sand spits, islands). Change in sediment fraction within the cells can result from longshore transport computed using the formula of [START_REF] Kamphuis | Alongshore sediment transport rate[END_REF] and from cross-shore transport using an adaptation of the equilibrium-based ShoreFor model [START_REF] Davidson | A simple equilibrium model for predicting shoreline change[END_REF][START_REF] Splinter | A relationship to describe the cumulative impact of storm beach[END_REF]. The sediment fraction within computation cells and resulting shoreline are updated at each time step (∆𝑡𝑡).

Critical to sediment transport and shoreline changes are the breaking wave conditions, which can be computed through the direct formula of [START_REF] Larson | Direct formula to compute wave height and angle at incipient breaking[END_REF], which is typically used in LX-Shore on academic cases or for open beaches with alongshore-uniform offshore bathymetry [START_REF] Robinet | A reduced-complexity shoreline change model combining longshore and cross-shore processes: The LX-Shore model[END_REF]. Otherwise, breaking wave conditions are computed using the spectral wave model SWAN [START_REF] Booij | A third-generation wave model for coastal regions: 1. Model description and validation[END_REF] to address configurations such as when complex wave shadowing or offshore wave transformation (e.g., energy focusing enforced by refraction, offshore wave energy dissipation through depth-induced breaking) affect breaking wave conditions and, in turn, sediment transport and shoreline change. At each time step LX-Shore provides an updated bathymetry that feeds back onto wave transformation. This updated bathymetry is reconstructed (Figure 2c) from the current shoreline position and the combination of two different idealized static equilibrium (Dean) beach profiles, for the sandy shores and for the rocky sectors.

Simulation set-up and analysis

LX-Shore was run for different embayed beach configurations schematised in Figure 3 with 𝑥𝑥 and 𝑥𝑥 the cross-shore and longshore coordinates, respectively. Embayed beaches were characterised by a length 𝑊𝑊 and headland length 𝐿𝐿, with the time-varying shoreline position denoted 𝑆𝑆(𝑥𝑥, 𝑡𝑡), with 𝑡𝑡 the time.

Periodic lateral boundary conditions were implemented so that headland sediment bypassing that occurs downdrift feeds the updrift part of the embayment. In other words, we assumed a circular system with constant sediment budget. This assumption will be discussed in Section 4. All simulations started from a uniform shoreline position 𝑆𝑆 = 0. Twenty years of wave hindcast were used, defined by time series of significant wave height 𝐻𝐻𝐻𝐻, peak wave period 𝑇𝑇𝑇𝑇, and peak angle of incidence 𝜃𝜃. A timeinvariant directional spreading 𝜎𝜎 𝜃𝜃 of 40° was used, although additional simulations were run for other values of 𝜎𝜎 𝜃𝜃 (see Section 3.3). We used a wave hindcast [START_REF] Durrant | A global wave hindcast focussed on the Central and South Pacific[END_REF] offshore of Sydney (January 1, 1994 -January 1, 2014). Such wave climate is known to drive important rotation signal [START_REF] Ranasinghe | The Southern Oscillation Index, wave climate, and beach rotation[END_REF][START_REF] Harley | New insights into embayed beach rotation: the importance of wave exposure and cross-shore processes[END_REF]Robinet et al., 2020a). Preliminary tests using 𝑊𝑊 = 1000 m (comprising a total headland width of 200 m) and different headland lengths 𝐿𝐿 indicated that model spin-up from the initial uniform shoreline typically lasted a few months. Therefore, the two first years of shoreline evolution were disregarded to insure that the simulated shoreline variability operates around a true dynamic equilibrium, and the subsequent 18 years (January 1, 1996 -January 1, 2014) were analysed.

Although cross-shore processes are found to affect embayment rotation, earlier work under the same wave climate as in Figure 3b (Robinet et al., 2020a) shows that cross-shore processes mainly drive a slightly alongshore variable cross-shore migration signal on the timescales from hours (storms) to months. In addition, the cross-shore transport module, which is based on ShoreFor [START_REF] Davidson | A simple equilibrium model for predicting shoreline change[END_REF][START_REF] Splinter | A relationship to describe the cumulative impact of storm beach[END_REF], uses free parameters that depend on breaking wave conditions. Using a different headland length, results in different sheltering effects and, in turn, different breaking wave conditions that require a new calibration effort even under the same offshore wave climate. However, these calibration data do not exist for such academic cases. For all these reasons, cross-shore processes were switched off in all our simulations, which will be further discussed in Section 4.

Wave transformation was computed by SWAN model with default settings. Diffraction was not activated given that it shows poor skill unless fine grid resolution is used [START_REF] Enet | Evaluation of diffraction behind a semi-infinite breakwater in the SWAN wave Model[END_REF], and that LX-Shore shows very good skill on natural beaches without diffraction (Robinet et al., 2020a). LX-Shore simulations were run with grid cell size ∆𝑥𝑥𝑥𝑥 = 50 m and a constant time step ∆𝑡𝑡 = 3 hours. SWAN cell size ("hydrodynamic grid") was 25 m. The water depth (ℎ) was retrieved at each time step using an equilibrium Dean profile [START_REF] Dean | Equilibrium beach profiles: characteristics and applications[END_REF] on sandy sectors, given by ℎ = 𝛼𝛼𝑑𝑑 𝛽𝛽 , with α and β two free model parameters. In the present study, we use values representative of Narrabeen beach (Robinet et al., 2020a), i.e. 𝛼𝛼 = 0.25 and 𝛽𝛽 = 0.67. The profile was projected offshore from the current shoreline position where 𝑑𝑑 is the distance offshore from the shoreline. The same applied for rocky shoreline with a steep profile also following a power law ℎ = 𝑎𝑎𝑑𝑑 𝑏𝑏 with 𝑎𝑎 = 1 and 𝑏𝑏 = 1 chosen arbitrary (linear profile, Figure 2c). When transects intersected, specific conditions including weighted averages were applied (for more detail see [START_REF] Robinet | A reduced-complexity shoreline change model combining longshore and cross-shore processes: The LX-Shore model[END_REF] to produce a consistent and smoothed bathymetry. In all simulations SWAN offshore boundary was located at 𝑥𝑥 = 800 m in approximately 16-18 m depth. All simulations were performed for an embayment beach length 𝑊𝑊 = 1000 m (comprising 200 m headland), although additional simulations were run for shorter and longer embayments (see Section 3.3). Headland length 𝐿𝐿 was varied from 0 to 700 m, every 50 m, with additional simulations for 𝐿𝐿 = 75 m, 125 m and 175 m to capture subtle changes in embayment shoreline response for this range of headland lengths.

Shoreline response was analysed using a set of variables (Figure 3), including the 18-year average (denoted (.̅ )), hereafter referred to as mean, shoreline position 𝑆𝑆 ̅ (𝑥𝑥), its most eroded position 𝑆𝑆 ̅ 𝑚𝑚𝑚𝑚𝑚𝑚 and corresponding longshore position 𝑥𝑥(𝑆𝑆 ̅ 𝑚𝑚𝑚𝑚𝑚𝑚 ), and the cross-shore standard deviation of shoreline position 𝜎𝜎 𝑆𝑆 (𝑥𝑥). We also computed the mean shoreline rotation 𝑅𝑅 � , which can also be referred to as orientation, as the slope of the trend-line fitted to 𝑆𝑆 ̅ . In order to address the spatial and temporal modes of shoreline variability, empirical orthogonal function (EOF) analysis of the shoreline deviation from the mean, i.e. 𝑑𝑑𝑆𝑆 = 𝑆𝑆 -𝑆𝑆 ̅ , was performed on the 18-year period. The analysis was performed at 150 m < 𝑥𝑥 < 850 m to ignore the influence of spurious shoreline interpolation near the headlands.

Finally, occurrence of headland sand bypassing was detected at each time step as 𝐹𝐹 𝑆𝑆 > 0 at the tip of the northern and/or southern headland(s). In order to address the wave control on embayed beach rotation we computed the power of the shore-parallel propagating wave component 𝑃𝑃𝑃𝑃, which is indicative of the portion of wave power available to drive alongshore currents in the surf zone [START_REF] Price | State dynamics of a double sandbar system[END_REF].

Results

Mean embayed beach planform

Figure 4 shows the mean shoreline 𝑆𝑆 ̅ obtained for different headland lengths, together with the alongshore-uniform shoreline without headland (𝐿𝐿 = 0). Results show that mean shoreline planform varies from a reasonably straight rotated shoreline (e.g., 𝐿𝐿 = 50 m), to a curved shoreline (e.g., 𝐿𝐿 = 700 m), through an asymmetric and slightly curved shoreline with large erosion in the southern sector (e.g., 𝐿𝐿 = 150 m). The mean embayed beach planforms also suggest headland sediment bypassing for certain headland lengths, e.g. for 𝐿𝐿 = 50 m, for which the mean shoreline position at the northern end of the embayment coincides with the headland extremity.

Figure 5 provides more insight into the influence of headland length on mean embayed beach planform characteristics and how they relate to headland sand bypassing. Two categories of mean shoreline planforms can be discriminated, with a separation at headland length of 𝐿𝐿 ≈ 150 m. For 𝐿𝐿 < 150 m, both shoreline orientation 𝑅𝑅 � and maximum erosion |𝑆𝑆 ̅ 𝑚𝑚𝑚𝑚𝑚𝑚 | rapidly increase with increasing headland length, with the opposite behaviour observed for 𝐿𝐿 > 150 m (Figure 5a,b). Such a threshold is, however, not clearly observed for the alongshore location of maximum erosion 𝑥𝑥(𝑆𝑆 ̅ 𝑚𝑚𝑚𝑚𝑚𝑚 ), which becomes increasingly close to the centre of the embayment as headland length increases (Figure 5c). The change in mean shoreline behaviour corresponds to the transition between occasional headland sand bypassing and no headland sand bypassing (Figure 5d). For 𝐿𝐿 = 150 m headland sediment bypassing only occurs 𝐵𝐵 = 0.2% of the time, which contrasts with shorter headlands (e.g. 𝐵𝐵 = 55.3% for 𝐿𝐿 = 50 m and 𝐵𝐵 = 23.7% for 𝐿𝐿 = 100 m), while headland sediment bypassing never occurs (𝐵𝐵 = 0) for 𝐿𝐿 > 150 m.

Overall, when headland sediment bypassing never occurs, the mean shoreline planform becomes increasingly symmetric and curved, with a less severe shoreline retreat in the embayment, with increasing headland length (Figure 4b). At the same time, embayed beach orientation and prevailing angle of offshore wave incidence are increasingly different. In contrast, for 𝐿𝐿 < 150 m as headland length is decreased allowing more frequent headland sediment bypassing, embayed beach planform becomes increasingly symmetric with its orientation increasingly deviating from the prevailing angle of wave incidence, while maximum erosion within the embayment at 𝑥𝑥 ≈200 m is increased (Figure 4b). 6e,f) and iii) long headlands with no headland sand bypassing (𝐿𝐿 = 600 m, Figure 6g,h). In the case of frequent sand bypassing, the mean shoreline is curved, asymmetric, with a more rectilinear form in the northern section of the embayment. The pivot zone, i.e. where 𝜎𝜎 𝑆𝑆 is minimized, is located at 𝑥𝑥 ≈ 600 m, that is, not in the centre of the embayment. The shoreline variability reaches its maximum against the northern headland, where both σS and cross-shore amplitude are locally maximised at y ≈ 800 m, and is more homogeneous along the southern section (y < 550 m) of the embayment (Figure 6c,d). In contrast, in the case of intermediate-length headlands with no headland sand bypassing, the mean shoreline is still strongly asymmetric, but with a pivot zone closer to the centre of the embayment and with most of shoreline variability approximately equally distributed between the northern and southern sectors (Figure 6e). The cross-shore amplitude of shoreline position for intermediate-length headlands is also larger than for the situation with headland sand bypassing, with an amplitude exceeding 65 m near the embayment extremities (Figure 6e,f). In the case of a longer headland, still with no headland sand bypassing, both the mean embayed beach planform and shoreline variance 𝜎𝜎 𝑆𝑆 are essentially symmetric (Figure 6g). Shoreline variability is still maximized at the extremities of the embayment, with a maximum amplitude decreasing to 18 m (Figure 6h). does not correspond to a pure rotation mode as it is associated with a change in curvature (Figure 7a).

Modes of shoreline variability

This mode of shoreline variability indicates that when the shoreline tends to rotate clockwise (dominant northerly longshore transport), the shoreline also tends to be stable at the southern end of the embayment, which is expected to be the signature of headland sand bypassing feeding the southern sector of the embayment. Compared to simulations with longer headland for which there is no headland sand bypassing, the temporal EOF mode shows increased amplitude, particularly in higher frequency variability, although interannual variability is still present.

The difference in shoreline behaviour observed with no headland sand bypassing between intermediate-length (Figure 6e) and long (Figure 6g) headlands can be explained by the difference in wave shadowing patterns by the upwave headland and resulting different alongshore variability in breaking wave height conditions. This is illustrated in Figure 8

Influence of directional spreading and embayed beach length

Figure 9 shows that for a headland length 𝐿𝐿 = 500 m (no headland sand bypassing) the mean embayed beach planform is much more curved and asymmetric for a time-invariant directional spreading 𝜎𝜎 𝜃𝜃 = 10° than for 𝜎𝜎 𝜃𝜃 = 40° (Figure 9a,d). Shoreline position also shows larger variance by a factor exceeding 3 (Figure 9c,f). Although the time-invariant directional spreading does not readily affect the temporal mode of rotation (Figure 9c,f), it affects the spatial mode as the rotation is associated with a slight change in overall shoreline curvature for 𝜎𝜎 𝜃𝜃 = 10° (Figure 9e). This can be explained by an inspection of the wave field for different directional spreadings. Figure 10 also that changes in directional wave spreading 𝜎𝜎 𝜃𝜃 have a profound impact on the wave shadowing zone and resulting alongshore variability in breaking wave height, with sharper gradients in breaking wave height for narrow-banded waves (Figure 10d). This is expected to favour changes in curvature.

Figure 11 shows the first temporal EOFs (rotation mode) for simulations with 𝑊𝑊 = 500 m and 𝑊𝑊 = 1000 m both for 𝐿𝐿 = 200 m (no headland sand bypassing around the headland). Results show that the narrowest embayment respond at much shorter timescales (Figure 11b). Rotation signal for 𝑊𝑊 = 1000 m correlates well with the 1-yr moving averaged 𝑃𝑃 � 𝑙𝑙 (Figure 11a), while the smallest embayment seems to respond on the timescales of days, in response to changes in 𝑃𝑃𝑃𝑃. Other simulations (not shown)

indicate that for longer embayments (𝑊𝑊 > 1000 m) the characteristic rotation timescale does not evolve much, instead it is the spatial mode that slowly evolves from a pure rotation pattern to rotation pattern associated with a change in shoreline curvature.

Discussion

Our simulations indicate that the onset of headland sediment bypassing is critical to both embayed beach equilibrium planform and the spatial and temporal modes of shoreline variability. These findings on the underrated role of headland sediment bypassing are in line with earlier observation on mean embayed beach planform [START_REF] Goodwin | An insight into headland sand bypassing and wave climate variability from shoreface bathymetric change at Byron Bay, New South Wales, Australia[END_REF]. Our simulations show that, in addition, headland sand bypassing changes shoreline variability from a pure rotation mode to a more complex rotation mode where curvature is also modified. Embayed beaches where headland sand bypassing occurs also show more variability on short timescales. These results will need to be compared with a systematic assessment of embayed beach response timescales for a wider range of embayed beaches and headland lengths. Shoreline and embayed beach rotation monitoring techniques from freely-available satellite imagery (e.g., [START_REF] Vos | Sub-annual to multi-decadal shoreline variability from publicly available satellite imagery[END_REF] will be critical to address such embayed beach behaviour globally. Overall, hybrid shoreline models like LX-Shore [START_REF] Robinet | A reduced-complexity shoreline change model combining longshore and cross-shore processes: The LX-Shore model[END_REF](Robinet et al., , 2020a)), and others (e.g. [START_REF] Hurst | Exploring the sensitivities of crenulate bay shorelines to wave climates using a new vector-based one-line model[END_REF], appear as relevant new tools to explore embayed beach shoreline dynamics, in parallel to more empirical approaches (e.g., Gonzales and Medina, 2001;Klein et al., 2003a;Elshinnawy et al., 2018a) looking at equilibrium beach planshape. [START_REF] Hurst | Exploring the sensitivities of crenulate bay shorelines to wave climates using a new vector-based one-line model[END_REF] tested different spreads of schematic and monochromatic wave conditions and showed that embayed beach planform curvature decreases with the spread of wave directions. Here, we only used a single time series of realistic wave conditions. Clearly, the considered wave climate offshore of Narrabeen shows a narrowly peaked angular distribution (Figure 3b), which limits embayed beach shoreline dynamics. A greater variety of wave climates will need to be considered in future modelling experiments, ranging from low to high angles waves. It is expected that the alternation between near shore-normally incident waves, exerting a shoreline straightening influence, and highlyobliquely incident waves will lead to larger stochastic variations in local effective diffusivity [START_REF] Ratliff | Modes and emergent time scales of embayed beach dynamics[END_REF]. We also anticipate that it will lead to larger changes in beach curvature and enhanced breathing mode. The role of multi-modality of wave climate on rotation and breathing modes should also be investigated. Recent work (e.g. Wiggins et al., 2019aWiggins et al., , 2019b) ) indicate that bimodal wave climate, which is common on many coasts [START_REF] Scott | Atmospheric controls and long range predictability of directional waves in the United Kingdom & Ireland[END_REF], can lead to dramatic beach rotation events from the timescale of storm to decades. In such bimodal wave climate, the absence of shorenormal condition resulting in vanishing shoreline diffusivity may also allow for greater-amplitude fluctuations in shoreline curvature. To the best of our knowledge, the role of such wave climate bimodality on the spatial and temporal modes of shoreline variability has never been studied. Instead of addressing the directional spread in wave climate, we investigated the influence of the timeinvariant directional spreading of each incident wave condition. Our results (Figure 9) show that the embayed beach planform is much more curved and asymmetric for narrow directional spreading, which is in line with Elshinnawy et al. (2018a) and consistent with the findings of [START_REF] Hurst | Exploring the sensitivities of crenulate bay shorelines to wave climates using a new vector-based one-line model[END_REF].

Shoreline position also shows larger variance for 𝜎𝜎 𝜃𝜃 = 10° than for 𝜎𝜎 𝜃𝜃 = 40°, and rotation signal is also associated with a change in overall curvature for a narrow directional spreading 𝜎𝜎 𝜃𝜃 = 10° (Figure 9e). This is consistent with the results of [START_REF] Ratliff | Modes and emergent time scales of embayed beach dynamics[END_REF] who found an important breathing signal, reflecting a change in shoreline curvature, at idealised embayed beaches where monochromatic obliquely incident waves resulted in sharp shadowed-exposed wave energy gradients. However, at this stage we cannot explain why rotation and curvature change signals are not orthogonal EOF modes of shoreline behaviour in our simulations, contrary to simulations in [START_REF] Ratliff | Modes and emergent time scales of embayed beach dynamics[END_REF] in which rotation and breathing modes were clearly discriminated. These results have strong implications, as they suggest that time series of parametric wave conditions (𝐻𝐻𝐻𝐻, 𝑇𝑇𝑇𝑇, 𝜃𝜃), which are typical products of global and regional wave hindcasts (e.g., [START_REF] Poli | Era-20C: An atmospheric reanalysis of the twentieth century[END_REF], are not enough to provide a fair description of both embayed beach equilibrium planform and rotation signal, and that directional spreading must also be accounted for.

Simulations were run to explore the role of 𝑊𝑊 on embayed beach response. Model spin-up was found to dramatically increase with increasing 𝑊𝑊, with for instance an approximately 10-yr model spin-up for 𝑊𝑊 = 2000 m (𝐿𝐿 = 300 m), compared to less than 2 years for 𝑊𝑊 = 1000 m. Computation cost can therefore become prohibitive for large coastal embayments. Empirical equilibrium embayed beach profile approaches (e.g., [START_REF] Hsu | Static bay beach concept for scientists and engineers: a review[END_REF], which are computationally cheap, should be preferred in such situations, bearing in mind they cannot address the time and spatial models of shoreline variability. Such long model spin-up is also in line with [START_REF] Hurst | Exploring the sensitivities of crenulate bay shorelines to wave climates using a new vector-based one-line model[END_REF] who showed that the time taken for modelled bays to reach equilibrium scales with the square of the distance between headlands. This also corroborates the results of [START_REF] Ratliff | Modes and emergent time scales of embayed beach dynamics[END_REF] who found a diffusing scaling, similar with the analytical findings of previous studies of e.g. Turki et al. (2013b), whereby increasing the alongshore length of embayed beach or decreasing the wave heights, will tend to result in nonlinear increases in the characteristic time scales. This emphasizes that, for a given headland length, small embayed beaches respond and recover from perturbations more rapidly than larger bays.

Prohibitive model spin-up time for long embayments using our model set-up prevented from running a comprehensive analysis on this topic.

In addition to some model limitations detailed in [START_REF] Robinet | A reduced-complexity shoreline change model combining longshore and cross-shore processes: The LX-Shore model[END_REF], and in Robinet et al. (2020a) specifically for embayed beaches, here cross-shore processes were switched off. Robinet et al. (2020a) indicated that cross-shore processes primarily control the cross-shore, slightly alongshore nonuniform, translation mode of variability at Narrabeen-Collaroy embayment. However, different embayment widths, headland lengths and wave climate may have a profound impact on the relative contributions of cross-shore and longshore processes to rotation signal. In addition, other morphodynamic processes (see review of [START_REF] Loureiro | Mechanisms and timescales of beach rotation[END_REF] can affect embayment rotation, such as for instance shoreline-sandbar-coupled rotation driven by strong longshore components during storms (van de Lageweg et al., 2013), or high-energy rips flowing against the headland (Loureiro et al., 2012b). More shoreline modelling is required on real embayed beaches with different hydrodynamic and geological settings, in order to better constrain academic modelling experiments and gain generic knowledge of the processes driving rotation.

We found that headland sand bypassing is important to embayed beach behaviour. However, it must be pointed out that sediment bypass only occurs in our model when the sandy shoreline reaches the tip of the rocky headland, while it is well established that headland sediment bypassing can occur in the surf zone or even farther offshore through boundary rips [START_REF] Castelle | Rip current types, circulation and hazard[END_REF][START_REF] Wiggins | Regionally-Coherent Embayment Rotation: Behavioural Response to Bi-Directional Waves and Atmospheric Forcing[END_REF][START_REF] Sous | Field Measurements of a High-Energy Headland Deflection Rip Current: Tidal Modulation, Very Low Frequency Pulsation and Vertical Structure[END_REF]. The degree of headland sand bypassing therefore strongly depends on the surf zone width relative to headland length [START_REF] Klein | Headland bypassing and overpassing: form, processes and applications[END_REF]. The critical headland length in the present study is likely underestimated and the 150 m value should be considered as low bound. Given that surf zone width depends on beach profile characteristics defined by a Dean profile based on Narrabeen characteristics (Robinet et al., 2020a), changing such profile characteristics should also affect headland sand bypassing and, in turn, mean embayment planshape and shoreline dynamics. To tackle these issues, a more realistic contribution of headland sand bypassing (e.g. [START_REF] Mccarroll | A general expression for wave-induced sediment bypassing of an isolated headland[END_REF] will need to be implemented in LX-Shore. We used synthetic headland shapes. However, headland geometry shows a striking natural variability [START_REF] Fellowes | Morphometric classification of swell-dominated embayed beaches[END_REF] ; Figure 1), which deeply affects wave shadowing regions and accommodation space for sand deposit and, in turn, embayed beach shoreline behaviour. We used periodic lateral conditions, meaning that the amount of sediment leaving the domain in the north enters the domain from the south and vice versa. In reality, this perfectly circular system behaviour does not exist. Different rates of sediment supply/sink due to different lengths and shapes of the headlands likely impacts the mean embayed beach planform and modes of shoreline variability. This is however difficult to implement numerically as the amount of sediment entering the system depends on many updrift beach variables, including headland length, beach width and orientation, which also depend on the further updrift embayment, and so on. Therefore our work is complementary to previous ones which disregarded headland sand bypassing and assumed embayed beaches as closed systems (e.g. [START_REF] Hurst | Exploring the sensitivities of crenulate bay shorelines to wave climates using a new vector-based one-line model[END_REF]. Our results provide general trends in embayed beach behaviour but encourage application of such modelling approach to natural embayed beaches for more quantitative local assessment, such as in Robinet et al. (2020a) at Narrabeen-Collaroy embayment.

Conclusions

Our simulations show that headland length controls both the embayed beach equilibrium planform and the spatial and temporal modes of shoreline variability. For short headlands promoting occasional headland sand bypassing, embayed beach curvature increases with increasing headland length while the opposite is observed for headlands long enough to prevent any headland sand bypassing occurrence. The onset of headland sand bypassing also controls embayed beach rotation signal. When there is no sand bypassing, rotation signal decreases in amplitude, resulting in less shoreline variance, and becomes increasingly symmetric with increasing headland length. The modal directional spreading of incident waves is also critical to embayed beach behaviour, with both shoreline embayment curvature and shoreline variance increasing with decreasing modal directional spreading. Embayed beach rotation characteristic timescale increases with increasing embayed beach length, while shoreline variance decreases across the entire embayment. Our simulations provide new insight into the influence of embayed beach characteristics and incident wave conditions on equilibrium planform and shoreline dynamics, implying that the potential occurrence of headland sand bypassing should be considered in future embayed beach classifications. State-of-the-art models coupling spectral wave and shoreline change modules such as LX-Shore [START_REF] Robinet | A reduced-complexity shoreline change model combining longshore and cross-shore processes: The LX-Shore model[END_REF] appear as a powerful tool to describe and further predict embayed beach behaviour, and to design coastal structures (e.g. groins, jetties), in parallel to the continuous development of empirical embayed beach planform models (e.g., [START_REF] Hsu | Static bay beach concept for scientists and engineers: a review[END_REF].

Figure 6

 6 Figure 6 provides insight into the spatial and temporal evolution of the embayment shoreline for three representative simulations: i) short headlands with headland sand bypassing (𝐿𝐿 = 100 m, 𝐵𝐵 = 23.7%, Figure 6c,d); ii) intermediate-length headlands with no headland sand bypassing (𝐿𝐿 = 250 m, Figure

Figure 7

 7 Figure 7 shows the first EOF mode for each of the same three simulations, providing additional insight into the spatial and temporal modes of shoreline variability for different headland lengths. It highlights a clear rotation response between the opposite ends of the embayment for both intermediate-length (𝐿𝐿 = 250 m) and long (𝐿𝐿 = 600 m) headlands, with a pivot zone at 𝑥𝑥 ≈ 600 m and 𝑥𝑥 ≈ 500 m, respectively. In each case, this rotation signal accounts for more than 95% of the total shoreline variability, with positive (negative) values of temporal mode 𝑐𝑐 1 (𝑡𝑡) signifying a clockwise (counterclockwise) rotation of the embayed beach planform. The simulation with intermediate-length (L = 250 m) exhibits a first temporal mode essentially similar in pattern with the one obtained with the long (L = 600 m) headland, with a large interannual variability. The first EOF mode of the short headland simulation (𝐿𝐿 = 100 m) explains less variability (74.9%) and, contrary to the two previous examples, it

  that shows the wave field for typical southeast swell conditions on May 16, 1995 (𝐻𝐻𝐻𝐻 = 1.2 m, 𝑇𝑇𝑇𝑇 = 7.72, 𝜃𝜃 = 132.9°) for different headland length 𝐿𝐿 = 250 m, 500 m and 600 m. Wave shadowing effects readily increase with increasing headland length (Figure8). For intermediate-length headland 𝐿𝐿 = 250 m (Figure8a) the embayment shoreline varies from fully exposed to wave conditions in the northern sector to well sheltered at the southern end of the embayment, maximizing alongshore gradients in longshore sediment transport and resulting shoreline change (Figure6e,f). For longer headlands, although wave shadowing and resulting alongshore variability in breaking wave height are large, the alongshore-averaged breaking wave height is largely reduced (Figure8b,c), therefore limiting longshore transport gradients and resulting shoreline change (Figure6g,h).

FiguresFigure 1 .

 1 Figures

  Figure 1. (a-d) Examples of typical embayed beach planforms from a relatively straight shoreline with (a) low and (c) high indentation at Porto Vecchio (Corsica) and Cassis (southeast France), respectively, to curved embayments at (b) Crozon (Brittany, northwest France) and (d) Guarapari (southeast Brazil). (e,f) Illustration of dramatic variability in shoreline orientation resulting from embayment rotation at Lloret de Mar (northeast Spain) (e) on June 2004 and (f) on March 2014 with in (f) the dotted the shoreline position in (e). Images adapted from Google Earth.

Figure 2 .

 2 Figure 2. Overview of features and processes included in LX-Shore (adapted from Robinet et al., 2018, 2020a) with (a) schematic planview coastal area with non-erodible rocky areas, sandy shores and incident waves; (b) zoom onto a sandy/rocky area showing where the longshore and cross-shore sediment transports are computed; (c) idealised rocky (grey) and sandy (sand) profiles used to design the updated nearshore bathymetry feeding back spectral wave model SWAN at each time step.

Figure 3 .

 3 Figure 3. (a) Schematic of embayed beach model set-up with primary input and output variables used for analysis and (b) significant wave height 𝐻𝐻𝐻𝐻 rose of wave conditions used as offshore boundary conditions. All variables in (a) are explained in the text.

Figure 4 .

 4 Figure 4. (a) Mean shoreline planform obtained for different headland lengths 𝐿𝐿 (coloured) with (b) zoom to emphasize variability.

Figure 5 .

 5 Figure 5. Mean embayed beach planform variables against headland length 𝐿𝐿: (a) rotation 𝑅𝑅 � ; (b) most eroded shoreline position 𝑆𝑆 ̅ 𝑚𝑚𝑚𝑚𝑚𝑚 ; (c) corresponding alongshore location 𝑥𝑥(𝑆𝑆 ̅ 𝑚𝑚𝑚𝑚𝑚𝑚 ) and (d) percentage of occurrence of sediment bypass around the headland 𝐵𝐵.

Figure 6 .

 6 Figure 6. Time series of (a) significant wave height 𝐻𝐻𝐻𝐻 and (b) alongshore component of offshore wave power 𝑃𝑃𝑃𝑃. Left-hand panels (c,e,g): mean (black line), cross-shore standard deviation 𝜎𝜎 𝑆𝑆 (grey line) and envelope (yellow area) of simulated shorelines; Right-hand panels (d,f,h): time evolution of shoreline deviation from the mean 𝑑𝑑𝑆𝑆 for different headland lengths (c,d) 𝐿𝐿 = 100 m; (e,f) 𝐿𝐿 = 250 m; (g,h) 𝐿𝐿 = 600 m.

Figure 7 .

 7 Figure 7. EOF analyses of 𝑑𝑑𝑆𝑆 showing the first (a) spatial and (b) temporal EOFs for simulations with 𝐿𝐿 = 100 m (green), 𝐿𝐿 = 250 m (blue) and 𝐿𝐿 = 600 m (red).

Figure 8 .

 8 Figure 8. Significant wave height 𝐻𝐻𝐻𝐻 field coloured on May 16, 1995 (𝐻𝐻𝐻𝐻 = 1.2 m, 𝑇𝑇𝑇𝑇 = 7.72, 𝜃𝜃 = 132.9°) for different headland length (a) 𝐿𝐿 = 250 m; (b) 𝐿𝐿 = 500 m; (c) 𝐿𝐿 = 600 m. In all panels, bathymetry is contoured in the background (black lines) at 2-m interval, with the thick grey line showing the embayment shoreline.

Figure 9 .

 9 Figure 9. Left-hand panels: shoreline envelope (yellow area), mean shoreline planform (black line) and cross-shore standard deviation 𝜎𝜎 𝑆𝑆 (blue line); Middle panels: spatial component of the first EOF of ; Right-hand panels: time evolution of shoreline deviation from the mean 𝑑𝑑𝑆𝑆 for 𝐿𝐿 = 500 m for different wave directional spreading (a-c) 𝜎𝜎 𝜃𝜃 = 40° and (d-f) 𝜎𝜎 𝜃𝜃 = 10°.

Figure 10 .

 10 Figure 10. Embayed beach morphology and wave field (significant wave height 𝐻𝐻𝐻𝐻 coloured) on December 26, 1995 for different time-invariant directional spreading values (a) 𝜎𝜎 𝜃𝜃 = 40°; (b) 𝜎𝜎 𝜃𝜃 = 30°; (c) 𝜎𝜎 𝜃𝜃 = 20°; (d) 𝜎𝜎 𝜃𝜃 = 10°.

Figure 11 .

 11 Figure 11. (a) Time series of alongshore component of offshore wave power 𝑃𝑃𝑃𝑃 (black) and its 1-yr moving average 𝑃𝑃 � 𝑃𝑃 (red); (b) first temporal EOFs of 𝑑𝑑𝑆𝑆 for simulations with 𝑊𝑊 = 500 m (red) and 𝑊𝑊 = 1000 m (blue) both for 𝐿𝐿 = 200 m (no headland sand bypassing).

Acknowledgments

BC funded by Agence Nationale de la Recherche (ANR) grant number ANR-17-CE01-0014; MD PhD scholarship funded by BRGM and Make Our Planet Great Again (MOPGA) national program. DI and AR funded by BRGM. We thank M.D. Harley and K.D. Splinter from Water Research Laboratory (UNSW) team for sharing the Sydney wave data kindly provided by the Manly Hydraulics Laboratory on behalf of the NSW OEH.

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