Sébastien Rouzeau

Françoise Méchin
email: fmechin@insa-lyon.fr

Jean-Pierre Pascault

Benoît Magny

Cross-Linked Polymer Microparticles : Characterization by Size Exclusion Chromatography Using Universal Calibration

Keywords: cross-linked polymer microparticle, microgels, size exclusion chromatography, universal calibration

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Introduction

Cross-linked Polymer Microparticles (CPMs), are defined as intramolecularly cross-linked macromolecules. They have been commonly known as microgels for many years, and can be viewed as a new, fourth class of polymers (besides linear, branched macromolecules and macroscopic polymer networks), on the border between molecules and particles [START_REF] Graham | Nanogels and microgels: the new polymeric materials playground[END_REF][START_REF] Funke | Microgels. Intramolecularly crosslinked macromolecules with a globular structure[END_REF]. Such macromolecular objects are based on at least one monomer with a functionality greater than 2, as a cross-linking agent. Microgels can be found for example in natural rubber, but numerous synthetic methods are also available, among which emulsion or microemulsion free radical polymerization, dispersion polyaddition, highly dilute, semi-dilute solution or bulk polymerization [START_REF] Graham | Nanogels and microgels: the new polymeric materials playground[END_REF][START_REF] Funke | Microgels. Intramolecularly crosslinked macromolecules with a globular structure[END_REF][START_REF] De Groot | Water-soluble microgels made by radical polymerization in solution[END_REF][START_REF] Radhakrishnan | Synthesis of uniform polyurethane particles by step-growth polymerization in a dispersed medium[END_REF][START_REF] Ramanathan | Preparation of polyurethane microspheres via dispersion polycondensation using poly(1,4-isoprene)-block-poly(ethylene oxide) as steric stabilizer[END_REF]. They now find applications in several practical and/or industrial fields: i) Coatings: because of their small size and globular shape, microgels can form colloidal solutions with high solid contents and low viscosities; therefore they are often incorporated as components of the binder for organic coatings, with the aim of adjusting the rheological behavior and preventing sagging, controlling the surface properties and reducing shrinkage, or filling pores and irregularities of some substrates (wood) [START_REF] Funke | Microgels. Intramolecularly crosslinked macromolecules with a globular structure[END_REF]. ii)

When sufficiently small (diameter below 100 nm), microgels can also be used as nanometric carriers for numerous dyes, pharmaceutical or biochemical compounds, and therefore play an essential role for therapeutic and diagnostic applications. Proteins and enzymes can be covalently linked to the microgel surface either directly or through a spacer in order to develop immuno assays and controlled drug delivery systems [START_REF] De Groot | Water-soluble microgels made by radical polymerization in solution[END_REF], especially as some microgels (based on poly(acrylic acid) or poly(N-alkylacrylamide)) can display pH-and temperature-sensitive swelling/deswelling behavior in aqueous media [START_REF] Alvarez-Lorenzo | Reversible adsorption by a pH-and temperaturesensitive acrylic hydrogel[END_REF][START_REF] Soppimath | Chemically modified polyacrylamide-g-guar gum-based crosslinked anionic microgels as pH-sensitive drug delivery systems: preparation and characterization[END_REF][START_REF] Kazakov | UV-induced gelation on nanometer scale using liposome reactor[END_REF]. Moreover, other microgels can also be molecularly imprinted towards various chemical probes [START_REF] Ye | Molecular imprinting on microgel spheres[END_REF]. Bigger microgels can also exhibit high ion-exchange capacity for cationic hydrophilic drugs [START_REF] Bromberg | Smart microgel studies. Polyelectrolyte and drug-absorbing properties of microgels from polyether-modified poly(acrylic acid)[END_REF]. Finally when doped with specific nanoparticles (semiconducting, superparamagnetic…), hybrid submicronic microgels have promising potential applications in photonics, catalysis or separation technologies [START_REF] Zhang | Polymer microgels: reactors for semiconductor, metal, and magnetic nanoparticles[END_REF]. iii) Microgels can finally be incorporated as high performance fillers in plastics, thermosetting polymers or coatings [START_REF] Valette | Use of acrylic functionalized (meth)acrylic crosslinked polymer microparticles in photopolymerized acrylic films[END_REF][START_REF] Valette | Use of functional (meth)acrylic crosslinked polymer microparticles as toughening agents for epoxy/diamine thermosets[END_REF], in order to improve shock or abrasion resistance for instance. The "microgels" described above can have real nanometric sizes, but can also be rather micrometric particles, somewhat different from the definition given by Funke et al. [START_REF] Funke | Microgels. Intramolecularly crosslinked macromolecules with a globular structure[END_REF]. In contrast, this paper is especially devoted to the characterization of CPMs or microgels with very small diameters (below 50 nm). It describes a new, reliable method to measure molar masses of CPMs with Size Exclusion Chromatography (SEC) using universal calibration (UC). In this study, CPMs and their homologous linear polymers (LP), prepared by copolymerizing lauryl acrylate (LA), butyl acrylate (BA) and cardura acrylate (CA), with hexanediol diacrylate (HDDA) as a cross-linking agent in the case of CPMs, were characterized by SEC-UC method. The synthesis of the CPMs we have used is described in other papers [14].

Background

Size Exclusion Chromatography

As individual polymer particles, CPMs have the particularity to keep their globular shape in solution, although they remain soluble in convenient solvents, just like linear macromolecules [START_REF] Funke | Microgels. Intramolecularly crosslinked macromolecules with a globular structure[END_REF]. This solubility allows the analysis of CPMs by Size Exclusion Chromatography, SEC. Indeed, determination of CPMs' molar mass is a critical issue, due to the fact that they are non linear polymers and also mostly copolymers. SEC is based on the separation of macromolecules according to their hydrodynamic volumes, partially or totally swelled, in the eluting solvent. In most cases, it is processed using just a concentration detector (e.g. refractometer). For accurate measurements, columns must be calibrated with a set of well-defined polymer standards having a chemical nature and a structure identical with the polymer to be analyzed. In practice, such polymer standards are usually not available, except for a few polymers like poly(methyl methacrylate) (PMMA) and polystyrene (PS). Conventional calibration with PS standards is widely used, but in the case of CPMs, molar masses obtained in this way can differ from the true ones by a factor 5 [START_REF] Graham | Nanogels and microgels: the new polymeric materials playground[END_REF][START_REF] Pille | Studies on microgels. I. Microgel formation studied by gelpermeation chromatography with an on-line light scattering detector[END_REF][START_REF] Pille | Studies on microgels. II. Analysis of the reaction between "living" poly(4-t-butylstyrene) and dimethacrylates by size exclusion chromatography coupled with DRI, UV and MALLS detectors[END_REF]. This is due to the different chemical nature and also to the compact structure of CPMs as compared to similar linear macromolecules.

SEC-MALLS

In order to determine an accurate molar mass for any polymer, a so-called absolute technique can be used: SEC coupled with an on-line Multi Angle Laser Light Scattering (MALLS) detector [17]. The light scattering signal is directly proportional to the molar mass of the polymer. Consequently, there is no need to calibrate columns. Molar masses are calculated from Eq. ( 1):

C A M R KC w 2 0 2 1 + = (1)
where R 0 is the Rayleigh ratio at zero angle scattering (obtained by extrapolating data from other angles to zero angle), C is the concentration of the solution, K is an optical constant proportional to the square of the refractive index increment (dn/dc) 2 , and A 2 is the second virial coefficient. In SEC-MALLS, the LS signal intensity is proportional to (dn/dc) 2 and concentration, and M w is also inversely proportional to (dn/dc) 2 . This parameter must thus be determined very precisely if accurate molar masses are requested. For polymers with a very low dn/dc, like in our case PBA in THF at 25°C (dn/dc=0.06) [START_REF] Penzel | Solution properties of polyacrylic esters. I. Light scattering and viscosity measurements in tetrahydrofuran[END_REF], the concentration has to be about 10 times as high as for PS in THF at 25°C (dn/dc≈0.185) [19] to obtain a similar LS signal. But as in contrast, eq. ( 1) is only valid at very low concentrations, the adjustment of the concentration of the injected samples is quite delicate. In addition to molar mass, SEC-MALLS can also provide another very important parameter: the zaverage radius of gyration R z of the particles, provided they are not smaller than 15nm. SEC-MALLS has been widely used in the determination of molar mass and size of hyperbranched polymers [START_REF] Simon | Characterization of highly branched poly(methyl methacrylate) by solution viscosity and viscoelastic spectroscopy[END_REF][START_REF] Baskaran | Hyperbranched polymers from divinylbenzene and 1,3-diisopropenylbenzene through anionic self-condensing vinyl polymerization[END_REF] and in CPM characterization [3,14-16,22-24].

SEC with Universal Calibration

Another method to get accurate values of molar masses is SEC with universal calibration (SEC-UC). Unlike conventional calibration, UC is supposed to be valid for all polymers [START_REF] Kostanski | Size exclusion chromatography -a review of calibration methodologies[END_REF][START_REF] Grubisic | A universal calibration for gel permeation chromatography[END_REF]. It is based on Fox-Flory's law [26], which states that the hydrodynamic volume of a macromolecule in solution is proportional to the product of its intrinsic viscosity, [η] and its molar mass, M (eq. 2):

[η]M = Φ 〈r g 2 〉 3/2 ∝ R h 3 (2)
When applying this relationship to a given set of SEC columns, there is a direct relationship with the elution volume V (eq. 3):

M V ] [η ∝ (3)
This relation is valid for any polymer. Column calibration can thus be obtained from any set of welldefined polymer standards, e.g. polystyrene, and the calibration curve is given by ( ) ( )

V f M = ] [ log η
. With an on-line viscosity detector, the elution volume and intrinsic viscosity of any unknown polymer sample are measured directly, and molar masses can be deduced from the calibration curve. This method also allows the determination of molar masses of any type of copolymer, which cannot be achieved with conventional SEC. Therefore it has been widely used to characterize linear copolymers, graft copolymers, hyper-branched polymers [START_REF] Simon | Characterization of highly branched poly(methyl methacrylate) by solution viscosity and viscoelastic spectroscopy[END_REF][START_REF] Ambler | Randomly branched styrene/divinylbenzene copolymers. I. Preparation, molecular weight characterization, and GPC analysis[END_REF][START_REF] Kaivez | Molecular structure characterization of linear and branched polystyrene blends by size exclusion chromatography coupled with viscosimetry[END_REF][START_REF] Geladé | Molecular structure characterization of hyperbranched polyesteramides[END_REF][START_REF] Ye | Newtonian flow behavior of hyperbranched high-molecular-weight polyethylenes produced with a Pd-diimine catalyst and its dependence on chain topology[END_REF], dendrimers and proteins, all of which are complex systems that cannot be studied with conventional SEC. Even a few studies on CPMs by Ishizu et al. resorted to SEC-UC to determine molar masses [START_REF] Chen | Preparation of soluble microgel by the copolymerization of methyl methacrylate with p-divinylbenzene in the presence of tetrabromomethane[END_REF][START_REF] Ishizu | Kinetics of crosslinking reactions. I. Reaction rate of intramolecular crosslinkings by using a soluble microgel[END_REF][START_REF] Ishizu | Architecture of nanostructured polymers[END_REF], but in this case viscosity measurements were not made on-line. With on-line measured intrinsic viscosities, it is possible to study the relationship between [η] and M for the sample, referring to the well-known Mark-Houwink-Sakurada (MHS) equation (eq. 4):

a M K ⋅ = ] [η (4)
The exponent a can assume values varying between 0 and 2, corresponding to spherical and rod-like macromolecules, respectively [START_REF] Netopilik | Polystyrene-equivalent molecular weight versus true molecular weight in size-exclusion chromatography[END_REF]. For non-ionic macromolecules with linear flexible chains and random coil conformation, a can vary between 0.5 in theta-solvents and 0.8 in thermodynamically very good solvents. For a given polymer, a depends on the "quality" of the solvent and on the architecture of the macromolecules: the more branching on the polymer chains, the lower the value of a. Several recent studies on hyperbranched polymers showed their exponent a was lower than 0.5 in good solvents [START_REF] Simon | Characterization of highly branched poly(methyl methacrylate) by solution viscosity and viscoelastic spectroscopy[END_REF][START_REF] Baskaran | Hyperbranched polymers from divinylbenzene and 1,3-diisopropenylbenzene through anionic self-condensing vinyl polymerization[END_REF][START_REF] Geladé | Molecular structure characterization of hyperbranched polyesteramides[END_REF][START_REF] Kharchenko | Role of architecture on the conformation, rheology, and orientation behavior of linear, star, and hyperbranched polymer melts. I. Synthesis and molecular characterization[END_REF]. Finally CPMs also keep a rather globular shape in solution, and as a result of this compact structure, their viscosity in solution is much lower than that of similar linear macromolecules with the same molar mass. Thus, MHS exponent a is also smaller than 0.5 for CPMs [START_REF] Funke | Microgels. Intramolecularly crosslinked macromolecules with a globular structure[END_REF].

Finally, SEC-UC also provides radii of gyration of polymers, using the Fox-Flory relation for polymer coils (eq. 5) [26,36,37]:

0 2 3 3 6 ] [ Φ = M R g η (5)
Contrarily to SEC-MALLS, this method allows the determination of the radius of gyration even for small macromolecules, due to their better detection through viscosity measurement than using MALLS detection.

Experimental

Materials

Monomers are presented in Table 1 and were used as supplied. Polystyrene (PS) standards were obtained from Sigma-Aldrich. The solvent used for SEC was HPLC grade THF from SDS.

Syntheses of LP and CPMs

LP and CPMs were synthesized by free-radical polymerization in an organic solvent (heptane, methyl ethyl ketone or heptane/isopropanol mixture) (initial monomer concentration: 25wt.-%). The initiator was 2,2'-AzobisMethylButyroNitrile (AMBN) at 10 mmol/L. The completion of the reaction was achieved within 6 hours at 70°C. Polymerization conditions are detailed in other papers [14].

Characterization SEC-UC method was processed with three Styragel HR 5E columns (mixed bed: extended range of porosity) from Waters in series, heated at 35°C. The detectors used for refractive index (RI) and intrinsic viscosity (IV) were respectively VE 3580 and T60A from Viscotek.. Polymer solutions were prepared with concentrations from 1 to 4 mg/mL in THF. 100 µL of solution was injected onto the columns (1 mL/min) for each measurement. The refractometer and viscosimeter were calibrated with different PS standards of known concentrations and viscosities. SEC-MALLS was processed using a Dawn detector from Wyatt Technology, with laser light tuned on 632.8 nm. The eluent was THF at 1.0 mL/min.

Results and discussion

Molar mass determination with SEC-UC

A linear copolymer and its corresponding CPM were characterized by SEC-UC and SEC-MALLS in order to validate the universal calibration method. The molar composition of the copolymer was LA/BA/CA/HDDA with 10/70-x/20/x, x=0 and 5 mol-% for linear copolymer and CPM, respectively. Table 2 shows values obtained for molar masses with both SEC methods. Results slightly differ, by about 15 to 20%. This probably comes from the fact that, in SEC-MALLS, the refractive index increment dn/dc of the sample has to be known very precisely. The value that was used here was that of pure poly(butyl acrylate) in THF at 25°C (dn/dc=0.06), available from the literature [START_REF] Penzel | Solution properties of polyacrylic esters. I. Light scattering and viscosity measurements in tetrahydrofuran[END_REF]. This is a good approximation, due to the high content in BA of the copolymers, but it is certainly not precise enough to allow the exact calculation of the molar masses.

Under our experimental conditions, SEC-UC can thus be considered as a convenient method to measure molar masses of CPMs, which is not possible with conventional SEC. Though it does not seem to be a better method than SEC-MALLS, we will show how useful it can be to characterize CPMs' compactness and to determine their radius of gyration.

Determination of Mark-Houwink-Sakurada exponent, a

For each slice of the chromatogram obtained by SEC-UC, the intrinsic viscosity, [η], and the molar mass, M, of the corresponding polymer are known. It is thus possible to plot log[η] versus logM. Mark-Houwink-Sakurada (MHS) equation can be re-written as follows (eq. 4bis):

K M a log log ] log[ + = η (4bis)
As a result, the value of exponent a is given by the slope of the linear plot log[η] versus logM. Curves were plotted for the linear copolymer and the CPM previously described (Figure 1). The value of the MHS exponent a is 0.62 for the linear copolymer. First of all, a is greater than 0.5, which proves that THF at 35°C is a relatively good solvent for the copolymers we intend to characterize. For further discussion, Table 3 shows values of exponent a for homo-PBA and copolymers of BA with CA and LA. The value obtained for exponent a of homo-PBA is practically equal to 0.8 and is consistent with the value reported in the literature for a good solvent [START_REF] Pille | Studies on microgels. II. Analysis of the reaction between "living" poly(4-t-butylstyrene) and dimethacrylates by size exclusion chromatography coupled with DRI, UV and MALLS detectors[END_REF]. One can note a strong decrease in a for copolymers containing CA, or both CA and LA. This is probably due to the presence of long dangling aliphatic chains coming from the co-monomers LA and CA (C 12 and C 9 dangling chains), which make THF at 35°C a poorer solvent [START_REF] Bohdanecký | The Mark-Houwink-Kuhn-Sakurada exponent of polymers with long side groups: is a 0 = 1/2 a reliable criterion of the theta state?[END_REF]. No value of a for homo-poly(cardura acrylate) seems to be available from the literature, but it has been reported to be 0.58 for homo-poly(lauryl acrylate) in THF at 25°C [START_REF] Pille | Studies on microgels. II. Analysis of the reaction between "living" poly(4-t-butylstyrene) and dimethacrylates by size exclusion chromatography coupled with DRI, UV and MALLS detectors[END_REF]. This is consistent with values observed for copolymers in this study. For the CPM, the value of a is 0.37, which is definitely lower than 0.5, and characteristic from dense particles in solution in a good solvent. Unlike the corresponding linear copolymer, the CPM keeps its globular shape in THF at 35°C. The only difference between the two macromolecules is the presence of 5 wt.-% HDDA as a cross-linking agent in the case of CPM. It can thus be concluded that the CPM is really an intramolecularly cross-linked, or at least highly branched macromolecule. Furthermore, the difference in slope between CPMs and linear copolymers (0.37 vs 0.62) is much more pronounced than what was observed for example for a hyperbranched polyurethane (based on an in situ-generated 3,5-bis(hydroxyethoxy)phenyl isocyanate) and its linear analog (prepared from diethylene glycol and toluene 2,4-diisocyanate), i.e. 0.41 vs 0.47 in THF [START_REF] Hong | Studies on hyperbranched polyurethane by multidetector size exclusion chromatography[END_REF]. This could confirm the real densely crosslinked character of our particles. Another point which is important in the case of CPMs raises from the fact that the curve ( )

M f log ] log[ = η
is a straight line over the entire range of molar mass. It implies that the value of a is the same for molecules with low or high molar masses. This shows that all the macromolecules have the same degree of crosslinking for their structure, whatever their size. The experimental conditions used for the synthesis lead only to a definite type of CPM molecules, and not to a mixture of linear, branched copolymers and cross-linked macromolecules. Even though the molar mass distribution is broad (PDI=7.5), the density of the particles is homogeneous.

Particles size -Radius of gyration

After these first results, the relation between the radius of gyration and the molar mass was determined, both for LP and CPMs. Therefore LP and CPMs with a wide range of molar masses were synthesized by changing the experimental conditions, and their radii (R z ) were measured by SEC-UC. Plots of R z against molar mass are shown in Figure 2. In both cases, the log-log plot of radius of gyration versus molar mass is a straight line. The slopes are ν=0.59 and ν=0.53, respectively for LP and CPM, where ν is defined as the excluded volume index.

Thus, the radius of gyration increases faster with respect to molar mass in the case of linear copolymers. This is consistent with differences in compacity between linear and cross-linked macromolecules. The density of the latter is higher than that of a linear macromolecule with a similar molar mass. Attempts to measure radii of gyration were also made using SEC-MALLS measurements. However, SEC-MALLS is unable to measure reliable radii smaller than 15nm, which is the case for most of our LP and CPMs. It is all the more critical, as dn/dc is very low for our polymers; this increases incertitude concerning small macromolecules. Results could only be obtained for CPMs, and are compared to those from SEC-UC in Table 4. It can be seen that radii of gyration measured by SEC-MALLS are larger than those obtained from SEC-UC. However, the difference is bigger for smaller molar masses. For CPMs with M w =10 3 kg/mol, identical values of R z are obtained from both methods. SEC-MALLS obviously overestimates radii of gyration of small-sized particles. SEC-UC is thus more convenient to obtain accurate values, and exponent a is directly given by [η]-M w relationship.

Conclusion

Size exclusion chromatography with universal calibration (SEC-UC) has proved to be a very useful method to characterize cross-linked polymer particles (CPMs). Not only does it give access to molar masses, but also to the radius of gyration and to the value of Mark-Houwink-Sakurada exponent a. Radii of gyration obtained in this way are much more accurate than those measured by SEC-MALLS for small CPMs (with R z <15nm and M w <150 kg/mol). The MHS exponent a is directly characteristic of the crosslinking density of macromolecules in the solvent. In this study, it was shown that a polymer containing 5mol-% of cross-linking agent has an exponent a lower than 0.5, whereas it is higher than 0.5 for a linear polymer of the otherwise same composition. SEC-UC is thus an efficient method to prove CPMs are intramolecularly cross-linked, or at least very highly branched macromolecules.

Table 4 .Figure 1 .

 41 Figure captions

Figure 2 .

 2 Figure 1. Plot of log([η]) versus log(M) for a linear polymer, LP, and its corresponding cross-linked polymer particle CPM ([η] in dL/g and M in g/mol). Figure2. Plot of radius of gyration versus molar mass for linear polymers, LP, and their corresponding cross-linked polymer particles, CPMs (R z in nm and M w in g/mol).

Figure 1 .

 1 Figure 1. Plot of log([η]) versus log(M) for a linear polymer, LP, and its corresponding cross-linked polymer particle CPM ([η] in dL/g and M in g/mol).

Figure 2 .

 2 Figure 2. Plot of radius of gyration versus molar mass for linear polymers, LP, and their corresponding cross-linked polymer particles, CPMs (R z in nm and M w in g/mol).

Table 1 . Monomers used in this study

 1

	Name	Designation	Structure	M (g/mol)	Supplier
			O
	butyl acrylate	BA		O	128	Aldrich
		O
	lauryl acrylate	LA	O		240	Aldrich
			OH
		O
	cardura acrylate	CA	O	O	300	Cray Valley
				O
			O
	hexanediol diacrylate	HDDA	O	O	226	Aldrich
				O

Table 2 . Determination of molar masses for a linear polymer, LP, and its cross-linked polymer particle counterpart, CPM, with SEC-UC and SEC-MALLS. PDI=polydispersity index.

 2

			M w (kg/mol)
	Sample	dn/dc			PDI
			SEC-MALLS	SEC-UC
	LP	0.06	14	16.5	1.7
	CPM	0.06	106	124	6.5

Table 3 . Influence of comonomers on MHS exponent a in copolymers based on BA. PDI=polydispersity index.

 3

		Composition	M w (kg/mol)	PDI	a
		BA	156	3	0.79
	LP	CA/BA (20/80)	16.5	1.7	0.67
		LA/BA/CA (10/70/20)	16.5	1.7	0.62
	CPM	LA/BA/CA/HDDA (10/65/20/5)	124	7.5	0.37


E Pelinovsky

• K Belibassakis

• J Touboul

J Touboul
email: julien.touboul@mio.osupytheas.fr

Foreword to special issue on Rogue Waves

and "three sisters"), which can produce serious damages to offshore industry and navigation. Rogue waves have been part of the marine folklore for centuries, but, since the seventies of the last century, oceanographers have started to believe them. Observations gathered by the oil and shipping industries suggest that there really is something like a true monster of the deep that devours ships and sailors without mercy or warning. Several catalogues of rogue-wave observations in the World Ocean are published now demonstrating their reality and wide distribution [START_REF] Liu | A chronology of freaque wave encounters[END_REF][START_REF] Didenkulova | Freak waves in 2005[END_REF]Nikolkina andDidenkulova 2011, 2012;[START_REF] Baschek | Rogue wave observations off the US west coast[END_REF][START_REF] O'brien | Extreme wave events in Ireland: 2012-2016[END_REF]. During the last years, various physical models of the rogue-wave phenomenon have been intensively developed and many laboratory experiments conducted. The main goal of these investigations is to understand the physics of the huge wave appearance and its relation to environmental conditions and to provide the "design" of freak wave needed for engineering purposes. Several books and review papers have been published in the literature [START_REF] Olagnon | Rogue waves 2000[END_REF][START_REF] Olagnon | Rogue waves 2004[END_REF], 2009; 123 [START_REF] Dysthe | Oceanic rogue waves[END_REF][START_REF] Ch | Rogue waves in the ocean[END_REF][START_REF] Akhmediev | Discussion & debate: Rogue waves-towards a unifying concept?[END_REF]Slunyaev et al. 2011;[START_REF] Onorato | Rogue waves and their generating mechanisms in different physical contexts[END_REF][START_REF] Onorato | Rogue and Shock Waves in Nonlinear Dispersive Media, Springer, Berlin Slunyaev A, Didenkulova I, Pelinovsky E[END_REF]. Now, the study of the freak wave phenomenon is shifted to ocean engineering and the present issue is aimed at showing the progress that was made in the field of rogue wave in the last years. The selection of papers in this issue contains methods based on fully nonlinear potential models, or popular studies based on specific models as Zakharov, NLS, or Witham equations, and results concerning the probability of occurrence of extreme events. In particular, Athanassoulis et al. (2017) studied the effects of sloping bottom and spectral bandwidth on rogue-wave dynamics and kinematics, in finite depth, by means of an exact Hamiltonian coupled-mode system modeling nonlinear refocusing of transient wave groups. The latter are obtained by linear backpropagation of high-amplitude wave trains constructed by the theory of quasi-determinism with application to extreme design waves over variable bathymetry. In the same context, Ducrozet and Gouin ( 2017), using a nonlinear potential flow solver based on the High-Order Spectral method, examined the influence of varying bathymetry in the occurrence of rogue waves within unidirectional and directional sea-states, focusing on the effect of the directional spreading on the wave statistics. van Groesen and Wijaya (2017) using results from study presented by van Groesen et al. ( 2017), demonstrated a detection system, based on Xband radars, that could be implemented onboard ships and provide warning up to 60 s before the event.

Considering unidirectional propagation, Shemer and Chernyshova (2017) studied the nonlinear evolution of narrow-banded wave trains along wave flume both experimentally and numerically using Zakharov equation. The effect of the spectral width on the evolution along the tank is considered for bi-modal initial spectra, and for spectra consisting of a carrier wave and two finite-amplitude sidebands. Good agreement between the experimental results and numerical simulations is obtained. On the other hand, Chabchoub et al. ( 2017) present results from experimental study describing higher order stage of modulation instability, starting from five-wave systems (main wave frequency and two pairs of unstable sidebands). Laboratory data confirm the NLS wave dynamics forecast by Akhmediev breather solutions. Moreover, Athanassoulis et al. (2017) model rogue waves as localized instabilities emerging from homogeneous and stationary background wavefields under NLS dynamics. The Wigner transform and Penrose's method are used to recover spatially periodic unstable modes, considered as generalized Benjamin-Feir modes, and a criterion for the emergence of rogue waves is formulated. Based on Hamiltonian formalism, Dyachenko et al. ( 2017) derived a "super compact" equation, amenable to describe nonlinear dynamics of wave group, up to the initial stages of wave breaking. This equation allows definition of exact wave envelop, independently of the spectrum width. Another nonlinear equation was derived by [START_REF] Kharif | Rogue waves in shallow water in the presence of a vertically sheared current[END_REF], who generalized the Whitham equation to take vorticity into account. They considered the behavior of focusing wave groups in the presence of vorticity, and compared their results with fully nonlinear simulations, demonstrating the strong influence of underneath vorticity on extreme wave dynamics.

Furthermore, comparing linear and nonlinear simulations, [START_REF] Chalikov | Estimation of wave height probability based on the statistics of significant wave height[END_REF] have confirmed that probability distributions of water elevation are sensible to nonlinearity, but they observed no effect when considering cumulative probability of wave height, concluding that it could be computed based on results by phase-averaged models, such as WaveWatch III. Finally, Slunyaev and Kokorina (2017) demonstrated that the presence of self-coherent, long-living nonlinear structures that are envelope solitons, whenever present in the wave field, has a strong influence on water waves statistics, making them extreme.

We hope this special issue, presenting some of the latest improvements in understanding rogue-wave formation, from both statistical and deterministic point of view, together with latest developments in rogue-wave prediction will be interesting for specialists in ocean engineering, and help in the development of reliable warning systems.


Julien Touboul
email: julien.touboul@mio.osupytheas.fr

Christian Kharif
email: kharif@irphe.univ-mrs.fr

Focusing Wave Group Propagating in Finite Depth in the Presence of Surface Current and Vorticity

The kinematics of two-dimensional focusing wave trains on a shearing flow in water of finite depth are investigated analytically. In the absence of waves, the vorticity due to the vertical gradient of the horizontal current velocity is assumed constant. A linear kinematic model based on the spatio-temporal evolution of the frequency is derived predicting the focusing distance and time of a chirped wave packet in the presence of constant vorticity, and surface current. Based on this model, the kinematic behavior of the transient wave packet is analyzed, and described in terms of spreading of the focusing point into a wider area. The effects of bathymetry, vorticity, and surface current are analyzed and discussed. Two main results are obtained: (i) the combined effects of surface current and vorticity, in deep water, are nontrivial, highly depending on the presence of surface current (ii) the effects of bathymetry, in the presence of shear, are also counterintuitive in the presence of vorticity, since significant effects can be observed when considering high values of the depth parameter.

Introduction

Freak, rogue, or giant waves are extreme events, localized in time and space. Most of the time, they are defined by a wave height briefly exceeding some statistical properties of the wave field, such as twice the significant wave height. They are characterized by their unpredictability, which explains that they are known as "waves from nowhere". They are responsible for an important number of large damages, caused to ships or offshore rigs, which explains they have focused the attention of the scientific community for the last 30 years.

Formation of these waves might be explained through various mechanisms, such as spatio-temporal focusing Kharif et al. [START_REF] Kharif | Focusing of nonlinear wave groups in deep water[END_REF], Johannessen and Swan [START_REF] Johannessen | On the nonlinear dynamics of wave groups produced by the focusing of surface water waves[END_REF], Brown [START_REF] Brown | Space-time surface gravity wave caustics: structurally stable extreme events[END_REF], Brown and Jensen [START_REF] Brown | Experiments on focusing unidirectional water waves[END_REF], nonlinear or modulational instability Benjamin and Feir [START_REF] Benjamin | The disintegration of wave train on deep water[END_REF], Dyachenko and Zakharov [START_REF] Dyachenko | Modulation instability of Stokes waves -freak wave[END_REF], envelope soliton and breather interactions Clamond and Grue [START_REF] Clamond | Interaction between envelope solitons as a model for freak wave formations[END_REF]. These mechanisms have been reviewed by Kharif and Pelinovsky [START_REF] Kharif | Physical mechanisms of the rogue wave phenomenon[END_REF] and by Dysthe [START_REF] Dysthe | Modelling a Rogue Wave -Speculations or a realistic possibility? Rogue Waves, Brest[END_REF].

Wave-current interaction contributes also in the freak wave formation and historically; this mechanism was the first to explain the origin of freak waves Lavrenov [START_REF] Lavrenov | The wave energy concentration at the Agulhas current of South Africa[END_REF], White and Fornberg [START_REF] White | On the chance of freak waves at sea[END_REF]. This is due to an important modification of water waves kinematics in the presence of currents reviewed extensively in Peregrine [START_REF] Peregrine | Interactions of water waves and currents[END_REF] and Jonsson [START_REF] Jonsson | Wave-current interactions[END_REF].

Since all these mechanisms can exist simultaneously at sea, it is interesting to analyze their robustness when considered together. This, for instance, was achieved for studying the influence of sheared currents on modulational instability in Johnson [START_REF] Johnson | On the modulation of water waves on shear flows[END_REF]. The influence of wind on dispersive focusing was studied in Touboul et al. [START_REF] Touboul | Freak waves under the action of wind: experiments and simulations[END_REF], and in Kharif et al. [START_REF] Kharif | Influence of wind on extreme wave events: experimental and numerical approaches[END_REF], while the evolution of modulation instability under wind action was studied in [START_REF] Kharif | Under which conditions the Benjamin-Feir instability may spawn an extreme wave event: a fully nonlinear approach[END_REF][START_REF] Touboul | On the interaction of wind and extreme gravity waves due to modulational instability[END_REF][START_REF] Touboul | Nonlinear evolution of the modulational instability under weak forcing and damping[END_REF].

Such a question can be asked about the dispersive focusing under the action of currents. This was initially addressed by Touboul et al. [START_REF] Touboul | Nonlinear focusing wave groups on current[END_REF], who investigated the modification of the dispersive focusing mechanism for water waves propagating in deep water in the presence of uniform currents. But generally, the characteristic length scales of wind waves and swells are much smaller than the spatial scales of horizontal variations in oceanic currents. On the opposite, there are many circumstances in which this claim cannot be applied to current velocity variations in vertical direction. Consequently, the vorticity due to the vertical gradient of the horizontal current velocity which may have an important effect on the dynamics of ocean surface waves cannot be ignored. Various studies of water waves propagating in such conditions can be found in the literature. Among them, one can cite Tsao [START_REF] Tsao | Behavior of surface waves on a linearly varying current[END_REF], Dalrymple [START_REF] Dalrymple | A finite amplitude wave on linear shear current[END_REF], Brevik [START_REF] Brevik | Higher-order waves propagating on constant vorticity currents in deep water[END_REF], Simmen [START_REF] Simmen | Steady deep-water waves on a linear shear current[END_REF], Simmen and Saffman [START_REF] Simmen | Steady deep-water waves on a linear shear current[END_REF] Teles da Silva and Peregrine [START_REF] Da Silva | Steep, steady surface waves on water of finite depth with constant vorticity[END_REF], Kishida and Sobey [START_REF] Kishida | Stokes theory for waves on linear shear current[END_REF], Pak and Chow [START_REF] Pak | Free surface waves on shear currents with non-uniform vorticity : third order solutions[END_REF], Constantin [START_REF] Constantin | Two-dimensionality of gravity water flows of constant non zero vorticity beneath a surface wave train[END_REF], etc. Thus, Touboul and Kharif [START_REF] Touboul | Effect of vorticity on the generation of rogue waves due to dispersive focusing[END_REF] extended their previous study [START_REF] Touboul | Nonlinear focusing wave groups on current[END_REF] to this more realistic case of water waves propagating in the presence of vorticity.

Besides, such vortical flows are observed in coastal areas [START_REF] Belibassakis | A coupled-mode model for water wave scattering by vertically sheared currents in variable bathymetry regions[END_REF][START_REF] Touboul | Extended mild-slope equation for surface waves interacting with a vertically sheared current[END_REF]. As pointed out by Professor Ezersky, the effect of bathymetry on dispersive properties should be taken into account. This study was realized experimentally and numerically for water waves propagating in waters of finite depth, in the presence of uniform currents in Merkoune et al. [START_REF] Merkoune | Focusing wave group on a current of finite depth[END_REF].

The purpose of this work is to analyze both the effects of surface current and vorticity for rogue waves propagating in waters of finite depth, extending results presented in [START_REF] Merkoune | Focusing wave group on a current of finite depth[END_REF] to take vorticity into account. To achieve this goal, attention is focused on the kinematic behavior of a focusing wave group. The components should merge at a given point in both time and space, producing a rogue wave. The transformation of this point into a wider area, under the combined effects of surface current, vorticity, and depth is investigated here. The chapter is organized as follows: in Sect. 2, the kinematic model is formulated, focusing on the computation of the coordinates of the focusing point without and with current and vorticity, in finite depth. In Sect. 3, the results are discussed in detail, and briefly summarized in Sect. 4, describing the main conclusions of the work.

The Kinematic Model

Since water waves are known to be dispersive, short waves propagating in front of longer ones will be overtaken, and a large amplitude wave can occur at a fixed point. A linear approach of the problem would lead to consider sea surface as a superposition of linear waves of frequencies ω (x, t). The following nonlinear hyperbolic equation governs the spatiotemporal evolution of these components, as pointed out by Brown [START_REF] Benjamin | The disintegration of wave train on deep water[END_REF] and Whitham [START_REF] Whitham | Linear and nonlinear waves[END_REF],

∂ω ∂t + C g (ω) ∂ω ∂x 0, (1)
where C g is the group velocity, defined as C g ∂ω/∂k. The boundary value problem for this equation can be solved using the method of characteristics. Its solution is

ω (x, t) ω 0 (τ ) , on t τ + x/C g (ω) , (2)
where ω 0 corresponds to the temporal frequency distribution of the wave train at x 0. By differentiating the frequency, it comes

∂ω ∂t ∂ω 0 ∂τ 1 -x C 2 g ∂C g ∂ω d ω 0 d τ (3)
The latter equation can be solved once the dispersion equation is known, and various examples of solutions can be found in the literature. In the classical formulation of water propagating in deep water, for instance, this dispersion equation reads

ω 2 gk, ( 4
)
where k stands for the wavenumber, and g is the acceleration due to gravity. We can now obtain an expression of the group velocity given by C g g/ (2ω). Equation (3) might thus be rewritten

∂ω ∂t ∂ω 0 ∂τ 1 -2x g d ω 0 d τ , ( 5
)
and one can notice that the case d ω 0 /d τ < 0, which corresponds to the case of short waves emitted before longer waves, leads to a singularity. This singularity nizar.abcha@unicaen.fr corresponds to the focusing of several waves at t T f and x X f . Given the expression of the group velocity in the case at hand, we can find out the frequency to impose to a wave maker located at x 0, and for 0 < t < T , given by ω (0, t) g 2

T f -t X f (6)
This frequency modulation, varying linearly from ω min to ω max , provides the optimal focusing of the linear wave packets in still water of infinite depth, and is very often applied in the laboratory conditions. Such a wave train will involve components which will propagate and all merge at the same place X f , and at the same time T f . Coordinates of the focusing point in the (xt) plane would thus be given by

X f gT 2 1 ω max -ω min , and T f 2ω max g X f . ( 7
)
If we now consider water waves propagating in deep water, in the presence of uniform currents, the dispersion Eq. ( 4) can be modified to take a Doppler shift into account, and now reads

(ω -kU ) 2 gk. ( 8
)
This specific case was investigated in Touboul et al. [START_REF] Touboul | Nonlinear focusing wave groups on current[END_REF], where Eq. ( 5) was demonstrated not to be solution anymore, but to be replaced with ∂ω ∂t

d ω 0 d τ 1 + 2gx [g+2U 0 (ω-kU 0)] 2 d ω 0 d τ (9)
This is true, since the group velocity admits a new expression, C g U 0 + g/ (2ω -kU 0 ). Equation ( 9) shows a difference in the kinematics of the wave group, which becomes more complicated. Indeed, the denominator is now a function of time, and is equal to zero for several values of space and time. The focusing point is not a unique singularity anymore, and the waves present in the group do not merge at a single location of time and space. The singularity is now spread over a focusing area, ranging from L min and L max , where

L min X f 1 + 2U 0 σ min g 2 , and L max X f 1 + 2U 0 σ max g 2 , ( 10
)
where σ min and σ max are the intrinsic, Doppler shifted, frequencies, respectively given by σ min (ω mink min U 0 ) √ gk min , and σ max (ω maxk max U 0 ) √ gk max . In a recent work, Touboul and Kharif [START_REF] Touboul | Effect of vorticity on the generation of rogue waves due to dispersive focusing[END_REF] investigated the evolution of a chirped wave packet in the presence of a horizontally constant current presenting linear variations with respect to depth, so that

U (z) U 0 + Sz, ( 11
)
where U 0 is the current velocity at the undisturbed free surface, and S is the value of the shear of the current. As an immediate consequence, the vorticity vector associated to this flow field is given by ∇ ∧ U (0, S, 0). In such a configuration, the dispersion equation is also known analytically (see, e.g., [START_REF] Jonsson | Wave-current interactions[END_REF][START_REF] Kirby | Surface waves on vertically sheared flows: approximate dispersion relations[END_REF][START_REF] Touboul | Extended mild-slope equation for surface waves interacting with a vertically sheared current[END_REF])

(ω -kU 0 ) (ω -kU 0 + S) g |k| , ( 12
)
and the related group velocity now reads

C g U 0 + g/ [σ 0 + σ 2 ]
, where σ 0 and σ 2 are respectively the Doppler shifted intrinsic frequencies (ω -kU 0 ) and

(ω -kU 2 ) (ω -k (U 0 -S/k)).
In such configuration, neither Eq. ( 5) nor Eq. ( 9) is the solution of Eq. ( 1) anymore. Instead, the solution (3) reads ∂ω ∂t

d ω 0 dt 1 + 2gx [g+U 0 (σ0+σ2)] 2 d ω 0 dt . ( 13
)
Here again, the focusing point turns out to be a focusing area, where the values of L min and L max are now given by

L min X f 1 + U 0 σ 0,min + σ 2,min g 2 , andL max X f 1 + U 0 σ 0,max + σ 2,max g 2 (14)
where σ 0,min , σ 2,min , σ 0,max and σ 2,max are respectively the minimum and maximum of intrinsic frequencies σ 0 and σ 2 .

The latter formulation might reduce to both previous cases. Indeed, for S = 0, we find the result σ 0 σ 2 , and Eq. ( 14) reduces to Eq. [START_REF] Dysthe | Modelling a Rogue Wave -Speculations or a realistic possibility? Rogue Waves, Brest[END_REF]. Furthermore, with U 0 0, this equation further reduces to L min L max X f , correspondingly to the solution (7) of Eq. [START_REF] Brown | Experiments on focusing unidirectional water waves[END_REF].

Still, as pointed out in Touboul and Kharif [START_REF] Touboul | Effect of vorticity on the generation of rogue waves due to dispersive focusing[END_REF], solution ( 14) also has an interesting behavior, when investigating the only effect of vorticity S, independently of any surface current U 0 . Indeed, when considering the absence of surface current, U 0 0, a nonzero vorticity S 0 turns out to change the focusing area in a single point. All frequencies have group velocities affected in such a way that the focusing point is not affected by the value of the vorticity. On the other hand, the focusing time is significantly affected, and all components of the chirped wave packet will reach the focusing point X f at a time T given by

T T f + S g X f ( 15
)
This result is very different from what was observed in the presence of uniform current. Indeed, when uniform current was considered, each component was affected nizar.abcha@unicaen.fr by Doppler shift, so that the focusing was not optimal. Here, the components are also affected, but the focusing point remains unchanged. This means that the focusing of energy remains optimal, even if delayed or leaded.

These three configurations are interesting, since they admit analytical solutions, allowing to emphasize the effect of a uniform current, or a sheared current, on the focusing behavior of a chirped wave packet. But another effect, of possible great significance, was not considered here. This effect is the influence of the water depth, which has, as it is classically known, a significant impact on dispersive properties of water waves. This idea was raised by Pr. Ezersky, and investigated both numerically and experimentally in Merkoune et al. [START_REF] Merkoune | Focusing wave group on a current of finite depth[END_REF]. In this work, we extend this analysis to the presence of a linearly sheared current.

In the latter case, the above derivation of C g and ∂C g /∂ω can still be performed. The dispersion equation now reads σ 0 σ 2 gktanh (kh), where σ 0 and σ 2 are still the previous Doppler shifted frequencies, but now read respectively (ω -kU 0 ) and (ω -kU 0 + Sth (kh)). This new dispersion equation provides the following expression:

C g U 0 + gtanh (kh) σ 0 + σ 2 + gkh + σ 0 Sh σ 0 + σ 2 1 -tanh (kh) 2 (16)
Starting from this group velocity, and after some algebra, one may derive the expression of ∂C g /∂ω. We obtain

∂C g ∂ω -2 C g -U 0 2 (σ 0 + σ 2 )C g + 2 1 -tanh (kh) 2 (σ 0 + σ 2 ) C g gh + gkh 2 + σ 0 Sh 2 tanh (kh) (17)
Finally, introducing these values in the denominator of Eq. ( 3) leads to the following relationship:

x C 2 g ∂C g ∂ω d ω 0 d τ - gx X f C g + U 0 2 (σ 0 + σ 2 ) C 3 g + gx X f 1 -tanh (kh) 2 (σ 0 + σ 2 ) C 3 g gh + gkh 2 + σ 0 Sh 2 tanh (kh) (18)
Here again, we notice the complexity of Eq. ( 18), which is time dependent. Indeed, the denominator of Eq. ( 3) now admits an infinity of singularities, ranging from

L min X f σ 0,min + σ 2,min C 3 g (ω min ) g × F (ω min , k min ) ,
and

L max X f σ 0,max + σ 2,max C 3 g (ω max ) g × F (ω max , k max ) , ( 19
)
where the function F (ω, k) is given by

F (ω, k) 1 C g (ω) + U 0 2 + gh + gkh 2 + σ 0 (ω, k) Sh 2 tanh (kh) 1 -tanh (kh) 2 . ( 20
)
nizar.abcha@unicaen.fr

Results and Discussion

Results presented here aim to describe the influence of bathymetry, surface current, and vorticity on the focusing behavior of a wave packet. To achieve this goal, the chirped wave packet considered here corresponds to the packet studied in a previous work by Touboul and Kharif [START_REF] Touboul | Effect of vorticity on the generation of rogue waves due to dispersive focusing[END_REF]. The focusing wave group considered had a frequency varying from f max 1.3 Hz to f min 0.7 Hz. The peak frequency is thus given by ω mean 2π (f max + f min ) /2 2πrad/s. This chirped wave packet, when propagating in deep water, in the absence of current and vorticity, presented a focusing location of X f 16.26 m and a focusing time of T f 27 s. To illustrate the kinematic effects of bathymetry, surface current, and vorticity, we focus in the following subsections on the spreading of the focusing point. Based on the expression of the minimum and maximum distance for caustic formation, L min and L max , provided by Eq. ( 19), the spreading of the focusing area can be described as

A spread L max -L min X f ( 20
)
This quantity will provide good insights on how vorticity, surface current, and bathymetry affect dispersion.

a. Effect of vorticity and surface current

In this section, we focus on the combined effects of vorticity and a surface current, homogeneous with depth, on the spreading area A spread . In every case considered, the bathymetry effects are neglected, and water waves propagating in deep water are considered.

Figure 1 depicts the spreading area A spread of the chirped wave packet as a function of the normalized vorticity S/ω mean . In the figure, various lines are represented. These lines are colored as a function of the reference current velocity U 0 /c mean . The word "reference" here means the wavenumber considered that corresponds to the wavenumber computed in the absence of vorticity (S/ω mean 0) for the mean pulsation of the wave packet ω mean .

This figure shows the strong dependence of the focusing area to both the vorticity and the surface current, in a nontrivial coupling. Indeed, it appears that for strong currents, either positive or negative, the width of the focusing area is sensitive to the vorticity. For positive values of the surface current, the width of the area increases with the vorticity, while it decreases when considering negative values of the surface current. Furthermore, when considering counter flowing currents (negative values of U 0 /c mean ), the spreading of the focusing area tends to be limited. This is connected to the limiting value of the blocking current velocity. When considering positive values of the current, this limiting behavior is not observed, and the focusing area seems to be unbounded. One may observe it is greater than 1, meaning the width of the focusing is as wide as the focusing distance, for relatively weak values of the surface current. In the meantime, when considering zero surface current (this case is emphasized by the thick black line in Fig. 1), the focusing area turns out to remain a singularity, corresponding to a focusing point, and to be insensitive to the vorticity. This is a confirmation of the result initially obtained in Touboul and Kharif [START_REF] Touboul | Effect of vorticity on the generation of rogue waves due to dispersive focusing[END_REF]. This result appears not to remain valid for constant, but nonzero values of the surface current.

The dependence of the width of this focusing area can also be analyzed as a function of the normalized effective surface current velocity. The word "effective" here means the phase velocity considered for normalization is still the phase velocity of the average pulsation of the group, ω mean , but now accounts for the actual value of vorticity. This behavior is presented in Fig. 2, where the spreading area is plotted versus the normalized effective current velocity, U 0 /c mean . In this figure, various lines appear, colored as a function of the normalized vorticity, S/ω mean .

Confirming the previous result, every curve intersects in A spread 0 for the value of surface current U 0 /c mean 0. But beyond this point, it is interesting to notice that the area of focusing always increases with the velocity of the surface current (in modulus). Though, from these curves, it also appears the vorticity has an influence on this behavior, since for strong positive vorticities, the increase rate is smaller than it is when considering strong negative vorticities.

These results are summarized in Fig. 3, where the evolution of the spreading area (in modulus) of the focusing wave packet is presented in the (U 0 /c mean , S/ω mean ) plane. From this figure, the dependence of the focusing area to both surface current velocity and vorticity might become more intuitive. It appears that for zero values of the surface current, the focusing area is always zero, corresponding to optimal focusing. But this focusing area is found to depend on both surface velocity and vorticity. However, this coupling is nontrivial, since the enlargement of the focusing area is more sensitive to the surface current when considering strong negative

b. Effect of Vorticity and Bathymetry

In this section, we focus on the combined effects of bathymetry and vorticity on the spreading area A spread . In every case considered, the surface current velocity is taken equal to zero. These lines are colored as a function of the reference depth parameter kh. The word "reference" means the wavenumber considered that corresponds to the wavenumber computed in the absence of vorticity (S/ω mean 0) for the mean pulsation of the wave packet ω mean . From this figure, various behaviors appear for large or small values of kh. Indeed, for large values of the depth parameter, the spreading area A spread is zero for almost every value of the vorticity. This is a confirmation of the result initially obtained in Touboul and Kharif [START_REF] Touboul | Effect of vorticity on the generation of rogue waves due to dispersive focusing[END_REF] for wave packets propagating in deep water. In this study, the focusing location was found to be unaffected, whatever the value of the vorticity. The vorticity, in deep water, has no effect on the dispersive behavior of the chirped wave packet.

On the other hand, for strong negative values of the vorticity S/ω mean , Fig. 4 shows a spreading of the focusing area, even for the strongest values of the kh parameter (up to values of kh 48). These values of the vorticity correspond to a strong coflowing current, which will intuitively result in large values of the wavelength. Thus, the effective wavenumber, taking the influence of the vorticity into account, will be smaller, and the influence of the bathymetry will have a significant effect on the dispersive behavior of the wave packet, resulting in a spreading of the focusing area.

When considering the smallest values of the reference depth parameter kh, it turns out that the spreading area is not equal to zero. This means no optimal focusing can be reached in such conditions, and the influence of bathymetry is predominant, regardless to the value of vorticity. Nevertheless, it seems the value of zero will eventually be reached asymptotically, for values of the vorticity large enough. Given the previous results, it is also interesting to consider the influence of effective depth parameter kh on the focusing area A spread . This is the purpose of Fig. 5, which presents the evolution of the spreading area as a function of the effective depth parameter. The word "effective" here means kh is evaluated using the real wavenumber k, computed using the mean pulsation of the wave packet ω mean , and taking the real vorticity value into account. In this figure, several lines are observed, colored as a function of the normalized vorticity magnitude, S/ω mean .

In this figure, we observe various behaviors given the magnitude of the vorticity considered. Indeed, when considering strong vorticity, the spreading area appears to tend to a zero value, meaning the parameter kh has less influence when vorticity is important. On the opposite, for strong, but negative values of the vorticity, the parameter kh is found to have a strong influence on the focusing area of the chirped wave packet.

The values of the parameter kh are striking. Indeed, in the absence of vorticity, depicted by the green lines in Fig. 5, the bathymetry seems to have an influence up to kh 7. This is explained through the frequency width involved in the chirped wave packet. But in the meantime, these values are larger than 10 when vorticity becomes important, but negative.

In Fig. 6, the evolution of the spreading area (in modulus) of the focusing wave packet is presented in the (S/ω mean , kh) plane. From this figure, it appears that the focusing area is zero, corresponding to optimal focusing, for deep water conditions, and for positive values of the vorticity. On the other hand, the important influence of the depth parameter on the focusing area appears clearly on that figure, when considering important opposing vorticities. This observation is probably explained through a broadening of the spectral width of the group in the presence of vorticity. Thus, the mean frequency is not accurate enough to provide a good indicator of the depth effects on the spreading area. Indeed, components involved in the wave packet are now sensitive to finite depth effects, resulting in a spreading of the focusing area. Same remark can be done when considering the vorticity effects for waves propagating in shallow water (small values of the kh parameter). When components of the group tend to become nondispersive, due to shallowness, vorticity has no influence anymore, and the focusing cannot be reached anymore.

Conclusion

The kinematic behavior of a focusing wave group propagating in finite depth, in the presence of constant vorticity and surface currents is studied analytically. Within the framework of the linear theory we use an approximate kinematic model allowing to describe the focusing point, and its transformation to a focusing area.

Confirming previous results by Touboul and Kharif [START_REF] Whitham | Linear and nonlinear waves[END_REF], the effect of vorticity in deep water is found to be surprising, and very different from what is expected in the presence of surface current, varying uniformly with depth. Indeed, when the only vorticity is present, the spreading of the focusing area is not expected anymore. On the other hand, when both current and vorticity are present the focusing area of the wave packets exhibits a dependence on the vorticity. Namely, the modulus of the focusing area always increases with the modulus of the vorticity, whatever its sign. On the other hand, this area is always found to depend on the surface current intensity, with or without vorticity. However, the rate of this dependence is found to be sensitive to the value of vorticity considered.

The effect of bathymetry on the focusing area is also investigated. Here again, results are surprising, since bathymetry turns out to have a strong impact on the focusing area, even for very large values of kh. Indeed, this can be explained by a broadening of the wavelength within the wave group. In the presence of strong negative vorticities, the spatial spectral width is increased in such a way the reference wave number might be misleading. Thus, some parts of the components suffer the influence of the bathymetry where we should not expect it. Influence of bathymetry is thus found, indirectly, to have even more significant impact on the dispersive behavior of the chirped wave packet than surface currents do.

Fig. 1

 1 Fig. 1 Spreading area A spread of the focusing wave packet plotted versus normalized vorticity S/ω mean . Various lines are colored as a function of the normalized reference current velocity U 0 /c mean

Fig. 2 Fig. 3

 23 Fig. 2 Spreading area A spread of the focusing wave packet plotted versus normalized effective current velocity U 0 /c mean . Various lines are colored as a function of the normalized vorticity S/ω mean

Fig. 4

 4 Fig. 4 Spreading area A spread of the focusing wave packet plotted versus normalized vorticity. Various lines are colored as a function of the reference depth parameter kh

Fig. 5

 5 Fig. 5 Spreading area A spread of the focusing wave packet plotted versus depth parameter kh. Various lines are colored as a function of the normalized vorticity S/ω mean

Fig. 6

 6 Fig. 6 Evolution of the modulus of the focusing area in the (S/ω mean , kh) plane

nizar.abcha@unicaen.fr

Acknowledgements This work is the continuation of a collaboration initiated jointly with Prof. Alexander Ezersky. The authors are also thankful to the editors for organizing such a manuscript. The authors would like to thank the French DGA, who supported this work through the ANR grant ANR-13-ASTR-0007.