Ruslan Sadykov

François Clautiaux

Matthieu Gérard

Artur Pessoa

Issam Tahiri

François Vanderbeck

Eduardo Uchoa

Recent results for column generation based diving heuristics

Assume a bounded integer program with decomposable structure:

[P] ≡ min c x :

A y ≥ a y = k ∈K x k x k ∈ X k = { B k x k ≥ b k x k ∈ N n(k ) }, ∀k ∈ K Assume that subproblems [SP] k ≡ min{c x k : x k ∈ X k } (1)
are "relatively easy" to solve compared to problem [P]. Then,

X k = {z q } q∈Q(k ) conv(X k ) = x k ∈ R n(k ) + : q∈Q(k ) z q λ q , q∈Q(k ) λ q = 1, λ q ≥ 0 q ∈ Q(k )
The Branch-and-Price approach (2)

Reformulation as the master program (Dantzig-Wolfe reformulation):

[M] ≡ min

k ∈K q∈Q(k ) (cz q ) λ k q : k ∈K q∈Q(k ) (Az q )λ k q ≥ a q∈Q(k ) λ k q = 1, ∀k ∈ K λ k q ∈ {0, 1}, k ∈ K, q ∈ Q(k ).
Aggregation of identical blocks in K:

[AM] ≡ min q∈Q (cz q ) λ q : q∈Q (Az q )λ q ≥ a q∈Q λ q = K , λ q ∈ N, q ∈ Q.

Contents Introduction

Various heuristics based on diving in λ variables space

Computational comparison of heuristics

Computational results for Generalized Assignment

Computational results for Multi-Activity Tour Scheduling

Conclusions

Rounding heuristics in λ var space Rounding a variable λ q → new dual variable Adding upper bound λ q ≤ u q : If dual variable is ignored, λ q might be wrongly regenerated as best.

If enforced, significant modifications to pricing.

Adding lower bound λ q ≥ l q : If ignored, λ q 's reduced cost is overestimated, hence not regenerated Adapted to Column Generation: if one only uses λ q ≥ l q

Remark

Fixing λ q ← λq as a partial solution is equivalent to setting a lower bound on λ q

Diving heuristics in λ var space

The residual master problem may become infeasible after rounding, as the partial solution may not satisfy the master constraints;

the partial solution may not be completed with columns generated so far.

Solution 1

One should work with proper columns, i.e. columns that could take a non-zero value in a master integer solution (may be harder to price such columns).

Solution 2

Diving, i.e. further column generation after rounding is a generic way to restore feasibility, i.e. to generate "missing" complementary columns.

Pure Diving [START_REF] Harvey | Limited discrepancy search[END_REF])

MaxDiscrepancy = 2, MaxDepth = 3
At each node, we have a tabu list of columns forbidden to be added to the partial solution. Tour Scheduling: Description A tour scheduling problem 1. Needs: to perform at best a limited list of activities (workload) during a planning horizon (a week).

2. Human Ressources: list of employees with skills, individualised contract and personal preferences/obligations.

Main objective [START_REF] Chan | La planification du personnel: acteurs, actions et termes multiples pour une planification opérationnelle des personnes[END_REF] To design a JuSTE planning: Juridical, Social, Technical, Economical.

→ feasibility and optimisation problem.

Daily workload for a production activity (time period = 15 min) Piecewise linear cost function for each period -production activity

/ 30

[START_REF] Danna | Exploring relaxation induced neighborhoods to improve MIP solutions[END_REF].

Optimum

Diving for feasibility

We are doing backtracking in diving until a feasible solution is found, corresponds to Diving with LDS with parameters MaxDiscrepancy = 1, MaxDepth = ∞

Strong Diving

The candidate columns for selection are evaluated (as in strong branching). We choose a candidate which deteriorates the least the column generation bound.

Diving with sub-MIPing

Run Diving

Run Restricted Master Heuristic with all columns generated during diving A variant with "local branching" [START_REF] Fischetti | Local branching[END_REF] The following constraint is added to the restricted master:


C Brouzet

E Ermanyuk

S Joubaud

G Pillet

T Dauxois

Internal wave attractors: different scenarios of instability

This paper presents an experimental study of different instability scenarios in a parallelogram-shaped internal wave attractor in a trapezoidal domain filled with a uniformly stratified fluid. Energy is injected into the system via the oscillatory motion of a vertical wall of the trapezoidal domain. Whole-field velocity measurements are performed with the conventional PIV technique.

 for the limiting case of rectangular domain. In the present paper, we describe also a new intermediate scenario for the case of weak focusing.

We explore the long-term behaviour of cascades of triadic instabilities in wave attractors and show a persistent trend toward formation of standing-wave patterns corresponding to some discrete peaks of the frequency spectrum. At sufficiently high level of energy injection the system exhibits a "mixing box" regime which has certain qualitatively universal properties regardless to the choice of the operating point in the Arnold tongue. In particular, for this regime, we observe a statistics of events with high horizontal vorticity, which serve as kinematic indicators of mixing.

Introduction.

Internal waves are ubiquitous in large geophysical systems such as oceans, seas and lakes. All these systems are geometrically confined. The importance of confinement depends on the ratio between the scale of the internal wave motion and the size of the domain, and on the rate of energy dissipation. Typical limiting cases include purely standing waves in a fully confined reservoir (e.g. internal seiches in lakes [START_REF] Hutter | Lakes as oscillators[END_REF], and purely propagating waves in laterally unbounded domains (e.g. baroclinic tides emitted by an isolated bottom topography [START_REF] Garrett | Internal tide generation in deep ocean[END_REF]). Experimental and geophysical reality often lies inconveniently between these two extremities.

Dynamics of internal waves in confined domains filled with continuously stratified fluid is significantly enriched by the specific form of the dispersion relation. Uniformly stratified fluid (with constant buoyancy frequency N ), a conventional simplification for a continuous stratification, supports internal waves propagating in form of oblique beams with the phase (group) velocity vector tilted at an angle θ to the vertical (horizontal) defined by the dispersion relation sin θ = ± ω/N = ± Ω, where ω is the forcing frequency, and Ω is its nondimensional counterpart normalized by N = [(-g/ρ)(dρ/dz)] 1/2 , with g the gravity acceleration, ρ(z) the density distribution along the vertical coordinate z and ρ a reference value [START_REF] Mowbray | A theoretical and experimental investigation of the phase configuration of internal waves of small amplitude in a density-stratified fluid[END_REF][START_REF] Turner | Buoyancy effects in fluids[END_REF]. This anisotropic dispersion relation represents a strong geometric restriction since it requires preservation of the angle θ to the horizontal upon reflection of the wave beam at a rigid wall. For vertical or horizontal walls, reflection is similar to the classic case of specular reflection considered in geometrical optics. Accordingly, a typical internal wave regime in rectangular domains is represented by standing waves (normal modes) as described in [START_REF] Turner | Buoyancy effects in fluids[END_REF]. These waves can be obtained as a sum of four identical monochromatic (in time and space) propagating waves corresponding to four possible choices of θ in the domain with depth (resp. length) equal to integer number of halfwavelength in vertical (resp. horizontal) direction. Considering the limit of infinitely thin wave beams as geometric rays, one obtains the web of rays mapped exactly on themselves upon reflection (global resonance).

If the confined system has a sloping wall, focusing/defocusing occurs [START_REF] Dauxois | Near-critical refection of internal waves[END_REF], the width of the wave beam decreases/increases upon reflection. Assuming a certain particular geometry in a two-dimensional problem, one can obtain a full classification of the observed wave regimes as it is done for parabolic and trapezoidal domains [START_REF] Maas | Geometric focusing of internal waves[END_REF][START_REF] Maas | Observations of an internal wave attractor in a confined stably stratified fluid[END_REF]. Careful investigation shows that focusing prevails: internal wave rays converge to closed trajectories called internal wave attractors. A convenient classification of the observed regimes can be done [START_REF] Maas | Observations of an internal wave attractor in a confined stably stratified fluid[END_REF]) in terms of Lyapunov exponents that characterize quantitatively the rate of convergence of the ray trajectories toward limiting cycles. The Lyapunov exponents can be plotted as a function of two parameters, where one parameter d controls the shape of the domain while another parameter τ controls the geometric ray pattern for a given d (see figure 1). By using the idea of affine similitude, the control over the ray pattern can be achieved by the change of Ω or by rescaling the depth of fluid. The map of the Lyapunov exponents as a function of these two parameters shows clearly defined regions of strong convergence, the so called Arnold tongues [START_REF] Manders | Observations of inertial waves in a rectangular basin with one sloping boundary[END_REF][START_REF] Maas | Wave attractors: linear yet nonlinear[END_REF] (appearing e.g. in the study of the Mathieu equation) corresponding to wave attractors of different complexity. The Arnold tongues are separated by regions of low convergence where the Lyapunov exponents exhibit a fan-shaped pattern of alternating dark-and light-grey regions, where dark regions correspond to almost zero Lyapunov exponents. It is important that the map contains embedded lines corresponding to global resonances (seiche modes). At these lines, the Lyapunov exponents are identically zero so that the ray pattern is mapped exactly on itself. It should be stressed that the pattern of dark-grey regions is much more dense that the pattern of lines corresponding to global resonance.

The concept of wave attractor emerging as a result of ray tracing assumes that the attractor represents a propagating wave. In a viscous fluid, the width of the attractor beams is set by the competition between geometric focusing and viscous broadening [START_REF] Ogilvie | Wave attractors and the asymptotic dissipation rate of tidal disturbances[END_REF][START_REF] Hazewinkel | Observations on the wavenumber spectrum and evolution of an internal wave attractor[END_REF][START_REF] Grisouard | Numerical simulation of a two-dimensional internal wave attractor[END_REF] what once again assumes a propagating wave: one can even put forth an analogy with a wave beam emitted by an oscillating object as discussed in [START_REF] Grisouard | Numerical simulation of a two-dimensional internal wave attractor[END_REF] and [START_REF] Jouve | Direct numerical simulations of an inertial wave attractor in linear and nonlinear regimes[END_REF]. This concept works well if the width-to-length ratio for all beams of the attractor is small. This "narrow-beam" approximation is violated for geometrically 2 with a = 5 mm). The primary wave frequency is plotted with a filled and larger symbol. The subharmonics (respectively superharmonics) are represented by white (respectively black) symbols. degenerated systems: for example, when the shape of the fluid domain does not support focusing (a rectangular domain) or when the attractor becomes very elongated and ultimately collapses onto a line (see figure 1).

In the present paper, we consider experimentally the effect of geometric degeneracy on the structure of the Arnold tongue for a generic case of (1, 1) attractor in trapezoidal geometry. (1, 1) attractor represents a parallelogram with one reflecting point at each side of the trapezoid [START_REF] Maas | Observations of an internal wave attractor in a confined stably stratified fluid[END_REF]. This attractor emerges when the forcing frequency falls into the range whose upper and lower limits are defined by the slopes of the diagonals of the trapezoid. The diagonals represent the degenerated forms of (1, 1) attractor. In addition, the trapezoidal geometry does not support focusing when it degenerates into a rectangle. To investigate the strength of attractors as a function of their location in the Arnold tongue, we introduce the total kinetic energy of the confined system as a variable, which allows to quantify the "susceptibility" of the system to forcing and to discriminate between nearly standing and nearly propagating waves.

Further, we explore the scenario of the wave instability via triadic resonance as a function of the operational point of the experimental system at the Arnold tongue: as introduced in Brouzet et al. (2016b), we will use the acronym TRI for the abbreviation of Triadic Resonance Instability.

Two scenarios of TRI in a confined fluid domain have been described in literature so far.

i) TRI in a rectangular basin has been studied theoretically and experimentally in [START_REF] Mcewan | Degeneration of resonantly excited standing internal gravity waves[END_REF]. The effect of confinement in this case is fully present since all waves in the resonant triad are assumed to be standing.

ii) TRI in a trapezoidal domain has been studied experimentally for (1, 1) attractors in [START_REF] Scolan | Nonlinear fate of internal waves attractors[END_REF] and numerically in Brouzet et al. (2016b). They describe the onset of the resonance as a local event in the most energetic branch of the attractor similar to [START_REF] Koudella | Instability mechanisms of a two-dimensional progressive internal gravity wave[END_REF] and [START_REF] Bourget | Experimental study of parametric subharmonic instability for internal plane waves[END_REF], with the effect of finite width of the wave beam involved [START_REF] Karimi | Parametric subharmonic instability of internal waves: locally confined beams versus monochromatic wave trains[END_REF][START_REF] Bourget | Finite-size effects in parametric subharmonic instability[END_REF]. The primary and secondary wave vectors have been measured and no clear effect of the confinement has been detected. Note that all waves involved into the measured triad are propagating. Thus the confinement in this scenario appears only indirectly as a factor defining the overall shape of the attractor and the width of the primary wave beam in accordance with the mechanism revealed by [START_REF] Ogilvie | Wave attractors and the asymptotic dissipation rate of tidal disturbances[END_REF], [START_REF] Hazewinkel | Observations on the wavenumber spectrum and evolution of an internal wave attractor[END_REF] and [START_REF] Grisouard | Numerical simulation of a two-dimensional internal wave attractor[END_REF].

In this paper, we show that the choice of the operational point in the Arnold tongue allows to observe the scenario of instability that is intermediate between the two abovementioned cases. Further, we analyse the properties of the wave regimes which develop as result of triadic instability by considering their time-frequency diagrams, Hilbert transforms and probability density functions (PDFs) of horizontal vorticity.

The present paper is organized as follows. In Section 2, we describe the experimental set-up. In Section 3, we describe the structure of the Arnold tongue of a stable (1, 1) attractor. Section 4 presents three scenarios of the onset of TRI as a function of the choice of the operating point of the experimental system in the Arnold tongue. We describe in detail a new scenario which is intermediate between the cases considered in [START_REF] Scolan | Nonlinear fate of internal waves attractors[END_REF] and [START_REF] Mcewan | Degeneration of resonantly excited standing internal gravity waves[END_REF]. This new scenario shows the emergence of standing secondary waves due to weak TRI in a weakly-focused attractor. We analyse then in Section 5 the long-term behaviour of a cascade of TRI in a well-focused attractor. We demonstrate that some secondary waves in the rich wave pattern emerging as a result of the cascade correspond to quasi-global resonances (standing waves). The results presented in Sections 4 and 5 show a strong general trend toward emergence of standing-wave components as a result of a simple one-stage triadic resonance or more sophisticated multi-stage TRI cascades. Finally, in Section 6, we consider the statistical properties of the internal wave regimes obtained as a result of well-developed cascades of triadic interactions. We show that at sufficiently large forcing the system reaches the regime of an "internal wave mixing box". The statistics of extreme events in the system and the resulting mixing efficiency can be efficiently controlled by the amount of the input energy and by the choice of the operating point in the Arnold tongue. The main findings of this study are summarized in Section 7.

Experimental set-up

The experimental set-up used in the present work and sketched in figure 2 is similar to the one described in [START_REF] Scolan | Nonlinear fate of internal waves attractors[END_REF] and Brouzet et al. (2016b). Experiments are conducted in the rectangular test tank of size 800 × 170 × 425 mm 3 filled with uniformly stratified fluid using the conventional double-bucket technique. Salt is used as a stratifying agent. The density profile is measured prior and after experiments by a conductivity probe attached to a vertical traverse mechanism. The value of the buoyancy frequency N is evaluated from the measured density profile. The trapezoidal fluid domain of length L (measured along the bottom) and depth H is delimited by a sliding sloping wall, inclined at the angle α. The wall is slowly inserted into the fluid after the end of the filling procedure. The global Archimedes number representing the ratio of the Reynolds and Froude numbers based on the fluid depth is kept fixed throughout all the experiments so that Ar = H 2 N/ν = 85000, where ν is kinematic viscosity. Thus, at the global scale of the experimental setup the effect of viscosity is reasonably weak, allowing an energy cascade toward smaller dissipative scales.

The input forcing is introduced into the system by an internal wave generator described in [START_REF] Gostiaux | A novel internal waves generator[END_REF], [START_REF] Mercier | New wave generation[END_REF] and [START_REF] Joubaud | Experimental parametric subharmonic instability in stratifed fuids[END_REF]. The time- dependent vertical profile of the generator is prescribed in the form ζ(z, t) = a sin(ω 0 t) cos(πz/H), where a and ω 0 are the amplitude and frequency of oscillations, respectively. In a horizontally semi-infinite domain, the motion of the generator would generate the first vertical mode of internal waves. The profile is reproduced in discrete form by the horizontal motion of a stack of 47 plates.

The whole-field velocity measurements are performed via the standard PIV technique. The fluid is seeded with light-reflecting hollow glass spheres of size 8 µm and density 1100 kg/m 3 . The sedimentation velocity of particles is found to be very low, with negligible effect on results of velocity measurements. The longitudinal (x, y = 0, z) mid-plane of the test section is illuminated by a vertical laser sheet coming through the side of the tank. The velocity field is calculated with the help of the cross-correlation technique (Fincham and Delerce 2000) with a typical resolution of 1 velocity vector per area of about 3 × 3 mm 2 . The mesh of measurements is found to be sufficient to resolve the small-scale details of the wave field.

A computer-controlled video AVT (Allied Vision Technologies) Stingray or Pike camera with CCD matrix of 2452×2054 pixels is used for video recording of the fluid motion. The camera is located at a distance of 2850 mm from the test tank. For the PIV measurements in short-term experiments, the camera operates at the constant frame rate of 4 Hz. After the PIV treatment, 4 velocity fields per second are obtained, yielding typically around 40 fields per period of forcing. In long-term experiments, the camera operates in burst mode taking two images separated by 0.125 s and waiting 0.375 s before the subsequent burst. The PIV treatment in this case yields 2 velocity fields per second. This sampling rate is found to provide a sufficient resolution of the significant frequency components of the signal.

Arnold tongue of a stable (1, 1) attractor

To quantify the strength of (1, 1) attractor, we consider the total dimensionless kinetic energy of the fluid confined in the trapezoidal domain. Experimentally, this quantity is measured in the vertical longitudinal mid-plane and defined as follows

K = S dxdz 1 2 (v 2 x + v 2 z ) 1 2 (aω 0 ) 2 S , (3.1)
where v x and v z are the horizontal and vertical velocity components and S is the area of the trapezoidal domain. Brouzet et al. (2016b) show that the internal wave pattern in the experimental set-up is reasonably two-dimensional. Thus, the measurements performed in the vertical mid-plane are representative for the whole volume except thin near-wall boundary layers. Figure 3 presents a typical experimental time-history of K. It can be seen that after a transient, which has a typical duration of about 25 periods, the process reaches saturation and kinetic energy oscillates between certain well-defined minimum and maximum values. We denote its corresponding time-averaged value as K , performed in the saturated regime. The quantity K can be interpreted as a susceptibility, i.e. a dynamic response of the experimental system to the prescribed forcing of unit amplitude a. In addition, one can introduce R = K min /K max , the ratio of minimum to maximum kinetic energy as a measure characterizing a particular wave regime as standing or propagating waves. Similar to K , this quantity is defined for the saturated regime. For purely standing waves, R = 0, while R → 1 for purely propagating waves with vanishingly thin wave beams (in the linear regime with a vanishingly small viscosity). In realistic systems with wave beams of finite thickness, we observe 0 < R < 1.

Using K and R as variables, we performed a series of 50 short-term experiments (parameters are given in Table 1) with stable (1, 1) attractors to study the structure of their domain of existence, the so-called Arnold tongue, as a function of the two parame- 1 11.3 0.552 0.73 1.93 26 31.2 0.719 0.20 1.24 2 11.3 0.554 0.73 1.90 27 31.2 0.731 0.20 1.22 3 11.3 0.574 0.73 1.86 38 34.6 0.530 0.09 1.93 4 11.3 0.588 0.73 1.83 39 34.6 0.545 0.09 1.85 5 11.3 0.602 0.73 1 4 and studied in this paper. For all these experiments, the working depth is 300 mm, the working bottom length is 455±10 mm, and the amplitude of the generator is a = 15 mm. # is the number of the experiment (from 1 to 50), α the angle (in •) of the slope, Ω0 the nondimensional forcing frequency, d and τ the corresponding nondimensional parameters of the diagram presented in figure 1.

# α Ω0 d τ # α Ω0 d τ
ters (d, τ ) controlling the convergence of wave rays. For the particular case of trapezoidal geometry, [START_REF] Maas | Observations of an internal wave attractor in a confined stably stratified fluid[END_REF] introduce d = 1 -(2H/L) tan α as the control parameter of the slope and τ = (2H/L) 1/Ω 2 0 -1 as the control parameter of the wave-ray pattern. The limiting case of triangle geometry (d = -1) is typically characterized by the presence of a point attractor at a vertex of the triangle, while the case of rectangular geometry (d = 1) corresponds to classic normal modes [START_REF] Mcewan | Degeneration of resonantly excited standing internal gravity waves[END_REF] for a discrete set of τ . Originally, (d, τ ) diagrams have been suggested in [START_REF] Maas | Observations of an internal wave attractor in a confined stably stratified fluid[END_REF] as a highly convenient tool for the classification of the inviscid wave regimes in a trapezoidal domain filled with a linearly stratified fluid. Lyapunov exponents have been used as a variable characterizing the convergence of wave rays toward limit cycles. The plot of the Lyapunov exponents as a function of (d, τ ) is shown in figures 1 and 4 in greyscale. The white (respectively grey) domains correspond to strong (respectively weak) convergence onto wave attractors. Wave attractors with high rate of convergence toward a simple well-defined structure are found in white regions (as anticipated, the so-called Arnold tongues). The dark-and light-grey regions have a highly complicated structure where the Lyapunov exponent varies in a wide range, occasionally reaching nearly-zero values (quasi-resonances). Ray tracing reveals for these regions a slow convergence toward attractors of high complexity with a high number of boundary reflections. Identically zero Lyapunov exponents are found at a discrete set of lines corresponding to global resonances (seiche modes). for the weakly-focused (1, 1) attractor described here (see § 4.3). The solid black line shows the contour at the value R = 0.5, enclosing the region with nearly propagating waves (NP), while nearly standing waves (NS) are outside this region.

It can be seen that the domain of existence of (1, 1) attractor in figure 4 has a triangular shape. In a linear viscous regime at fixed values of Ar and H/L, the quantities K and R are the functions of a coordinate in (d, τ ) plane. Combining the information presented in terms of K and R, we can conclude that there are two distinct regions with different regimes for R: i) the central part of the Arnold tongue with high values of R: it is indicating a propagating wave system (denoted NP in the remainder of the text) corresponding to a classic case of wave attractor with thin well-defined branches. It is clearly seen in the left panel of figure 4 that there exists an optimum range in (d, τ )-space, where the mean kinetic energy of a well-focused attractor is maximized. As the focusing increases (negative d), the excitation of high-energy attractors is hindered by increased dissipation in narrow wave beams.

ii) the border regions of the Arnold tongue are typically characterised by low values of R due to geometric degeneration of the attractor. In that domain referred as NS, waves are nearly standing and the mean kinetic energy of attractor is low except, the right tip of the Arnold tongue, where we observed high values of K and low values of R, typical features for standing waves. Indeed, the right tip of the Arnold tongue corresponds to d = 1 and τ = 2, i.e. to an exact standing mode. Thus, the secondary maximum of K in the vicinity of the right corner of the Arnold tongue is associated with optimum condition of excitation of a nearly standing wave in a nearly rectangular domain.

Three scenarios for the onset of triadic resonance in a trapezoidal domain

In this section, we discuss how the choice of the operating point in the Arnold tongue of (1, 1) attractors affects the observed scenario for the onset of triadic resonance instability (TRI). In Table 2 * and m * are respectively the horizontal and vertical forcing wavenumbers. The horizontal and vertical wave vector components are denoted ( i, mi) in which the subscript i refers to the primary (i = 0) or to the secondary waves (i = 1 and 2). Ωi is the corresponding nondimensionalized frequency. In [START_REF] Mcewan | Degeneration of resonantly excited standing internal gravity waves[END_REF], we have chosen the data with the frequency triplet that is close to those considered in Brouzet et al. (2016a) and in the present paper. The values of wave vectors are evaluated from the parameters of standing waves presented in [START_REF] Mcewan | Degeneration of resonantly excited standing internal gravity waves[END_REF].

TRI with purely standing waves: the degenerate case of the rectangular domain

The case of TRI in standing waves in a rectangular domain has been studied in [START_REF] Mcewan | Degeneration of resonantly excited standing internal gravity waves[END_REF], represented by the lozenges and ♦ in figure 4. These cases fall on the vertical line d = 1 where R = 0. The theoretical analysis presented in [START_REF] Mcewan | Degeneration of resonantly excited standing internal gravity waves[END_REF] assumes that standing primary wave oscillating at frequency Ω 0 can feed two standing secondary waves oscillating at frequencies Ω 1 and Ω 2 provided spatial and temporal conditions of the triadic resonance are satisfied and provided the damping in the system is below a given threshold. The damping is supposed to be primarily associated with the loss of energy in boundary layers at the side walls of the domain. The temporal resonance condition of the TRI amounts to Ω 0 = Ω 1 ± Ω 2 . Since all the waves involved into triadic resonance are standing ones, their half-lengths in horizontal and vertical directions should be equal to L/M and H/N where M and N are integer numbers. Then, the spatial resonance condition can be written as

M 0 = M 1 ± M 2 and N 0 = N 1 ± N 2 .
In practice, high energetic contents are encountered for the triads with "plus" and "minus" signs in temporal and spatial resonance conditions, respectively, i.e. with the cases where secondary waves oscillate at a smaller frequency than the primary wave, and the energy is transferred to shorter spatial scales, privileging the direct cascade. All three waves (j = 0, 1, 2) should satisfy the dispersion relation:

(N j /M j ) 2 (L/H) 2 = (1 -Ω 2 j )/Ω 2 j .
In the experiments described in [START_REF] Mcewan | Degeneration of resonantly excited standing internal gravity waves[END_REF], the forcing typically corresponds to low mode with N 0 = 1 and M 0 varying from 1 to 5. It has been shown experimentally that, at the laboratory scale, it is possible to transfer energy to the waves that are a few in figure 4 for a = 5 mm. For this regime, K is large and R 0.5. Accordingly, the (1, 1) attractor has thin well-localised branches. [START_REF] Scolan | Nonlinear fate of internal waves attractors[END_REF] and Brouzet et al. (2016b) have shown that the onset of instability in such a wave attractor occurs locally first in the most energetic wave beam which has the largest amplitude but the smallest width. This scenario of instability is consistent with the one discussed in [START_REF] Bourget | Experimental study of parametric subharmonic instability for internal plane waves[END_REF][START_REF] Bourget | Finite-size effects in parametric subharmonic instability[END_REF] and [START_REF] Karimi | Parametric subharmonic instability of internal waves: locally confined beams versus monochromatic wave trains[END_REF]. This is also what is observed in Brouzet et al. (2016a), at the very beginning of the triadic cascade, as shown in figure 5.

Owing to local character of instability [START_REF] Scolan | Nonlinear fate of internal waves attractors[END_REF]Brouzet et al. 2016a,b), the confinement of fluid domain does not affect the instability directly. The focusing provides the energy transfer from the input perturbation which has the scale of vertical size of the fluid domain to the scale associated with the width of the attractor beams, which serves as a primary wave. Subsequent triadic resonance transfers energy to secondary waves. The overall efficiency of the energy transfer from large-to small-scale motions is remarkably high, even at the laboratory scale. [START_REF] Scolan | Nonlinear fate of internal waves attractors[END_REF] describe the case where the typical wavelength of the primary wave represented by the beam of the attractor is roughly 9 times shorter than the wavelength of the input forcing, and one of the secondary waves has even a wavelength roughly 3 times shorter than the primary wave, leading to a reduction factor of 25. This is also visible in figure 5 where the primary and the two secondary waves have been disentangled using the Hilbert transform [START_REF] Mercier | Refection and diffraction of internal waves analysed with the Hilbert transform[END_REF]. 4.3. TRI with nearly standing waves: the case of weakly focused attractor.

In this subsection, we describe a new scenario for the onset of instability which is intermediate between those described in §4.1. and §4.2. This new scenario is typical for weak focusing. The observation of triadic resonance has been performed for (d, τ ) = (0.65, 1.80) and a = 5 mm. This experiment is represented by the symbol in figure 4.

The onset of instability

The snapshot of wave pattern after 200 periods of oscillations (approximately 30 minutes of observations) is shown in figure 6(a). The time history of oscillations measured at the point A, defined in figure 6(a), is presented in figure 6(b). After a rather short transient of approximately 25 periods, the signal is stable: the attractor is set. Then, one can clearly see the apparent instability developing after 400 forcing periods, i.e. after more than 1 hour of observations. The development of the spectrum of wave motion over time is presented in figure 7 2 for parameters).

in [START_REF] Bourget | Experimental study of parametric subharmonic instability for internal plane waves[END_REF], with the formula

S r (Ω, t) = +∞ -∞ v r (x, z, τ )e iΩN τ h(t -τ ) dτ 2 xz , (4.1)
where h is an Hamming window and v r the component of the velocity field, r = x or z.

The calculations are performed with the Matlab toolbox described in [START_REF] Flandrin | Time-Frequency/Time-Scale Analysis, Time-Frequency Toolbox for Matlab[END_REF].

The appropriate choice of the length of the Hamming window allows to tune the resolution in frequency and time. Typically, the data are averaged over a finite analysing area which, as a limit, can extend over the whole fluid domain. Figure 7(a) presents the time-frequency diagram for the case illustrated in figure 6. The diagram is calculated for the data averaged over the rectangle shown in figure 6(a) around point A. It can be seen that, initially, the signal is entirely dominated by the forcing frequency Ω 0 = 0.60 corresponding to the primary (carrier) wave. Oscillations with frequencies Ω 1 = 0.33 and Ω 2 = 0.27 slowly develop with time. These frequencies correspond to two secondary waves generated by TRI. They satisfy the frequency conditions for the triadic resonance:

Ω 1 + Ω 2 = Ω 0 . (4.2)
In addition, one can also see two peaks satisfying differential conditions:

Ω 3 -Ω 1 = Ω 0 and Ω 4 -Ω 2 = Ω 0 . (4.3)

The components of the wave pattern

Figure 8 shows the decomposition of the wave field into the components corresponding to the secondary waves while the primary wave has already been exhibited in figure 6(a). 2 for the parameters). Note that the components oscillating at Ω1 and Ω2 correspond to nearly standing waves as is clearly seen on the sequences of wave profiles shown on the right of each picture.

The decomposition, performed with the help of Hilbert transform [START_REF] Mercier | Refection and diffraction of internal waves analysed with the Hilbert transform[END_REF], reveals node-antinode patterns of amplitudes of secondary waves, similar to Chladni figures. These patterns are present soon in the experiment, but their amplitude increases gradually. Thus, this is very different from TRI presented in [START_REF] Scolan | Nonlinear fate of internal waves attractors[END_REF] where the waves are created on a very localized region of the tank. Here, the secondary waves appear directly everywhere in the tank, in a shape presented in figure 8 The standing wave patterns can be further decomposed into the sums of propagating waves whose wave vectors can be measured (see data in Table 2). All vectors satisfy the dispersion relation individually. The length of the primary wave can be estimated [START_REF] Mercier | Refection and diffraction of internal waves analysed with the Hilbert transform[END_REF] Therefore, in the present case, the primary wave has a length scale associated with the global geometry of attractor. The length scale associated with the beam width does not clearly manifest itself. Indeed, weakly focused attractors with elongated shape have two parallel beams which can interact with each other since the width of beams and the inter-beam distance are comparable quantities. As result the notion of the beam width is difficult to define for such attractors, in contrast with spatially localised beams of wellfocused attractors. In the latter case described in [START_REF] Scolan | Nonlinear fate of internal waves attractors[END_REF] (see § 4.2 and symbol in figure 4), we have seen that the length of the primary wave is associated with the width of the attractor beam.

The growth rate

Horizontal cuts in figure 7 at Ω 1 and Ω 2 are plotted in figure 9. At low amplitude of secondary waves, the linear trend is obscured by the instrumental noise, while toward the end of the experiment there is a trend to saturation of amplitudes of secondary waves. The vertical logarithmic representation emphasizes a remarkably linear growth for about three decades (!) in the magnitude of the spectral peaks corresponding to secondary waves. The growth rate of the instability can be estimated to be around σ = 6.5 × 10 -4 s -1 , which leads to a characteristic growth time of the instability around 1500 s (i.e. half an hour or 150T 0 ). The theoretical value for monochromatic plane wave, taking account viscosity [START_REF] Koudella | Instability mechanisms of a two-dimensional progressive internal gravity wave[END_REF][START_REF] Bourget | Experimental study of parametric subharmonic instability for internal plane waves[END_REF]) is equal to 2.1 × 10 -2 s -1 , that is two order of magnitudes larger than the experimental value. The difference is due to the spatial confinement of the attractor, which differs clearly from monochromatic plane wave, as underlined by [START_REF] Karimi | Parametric subharmonic instability of internal waves: locally confined beams versus monochromatic wave trains[END_REF] and [START_REF] Bourget | Finite-size effects in parametric subharmonic instability[END_REF].

The instability growth time has to be compared with the time scale τ w set by the limited size of the test tank. It can be defined as the time for secondary waves to perform a horizontal return-trip in the tank: τ w = 2L/c gx , where c gx is the group velocity in the horizontal direction. This group velocity can be measured using the wave numbers of the secondary waves estimated via Hilbert transform and wave number filtering. It gives τ w ≈ 60 s, which is very small in comparison with the instability growth time of 1500 s. It is worth to note that the horizontal group velocity is a good measure of the return-trip time of secondary waves because these waves are only slightly inclined. Indeed, their propagation angle, with respect to the horizontal direction, is close to 20 • since Ω 1 and Ω 2 are around Ω 0 /2 = 0.3.

The data presented in [START_REF] Scolan | Nonlinear fate of internal waves attractors[END_REF] or in figure 5 clearly shows that the triadic resonance develops first in the most energetic branch of the wave attractor, the one connecting the inclined slope to the surface. The instability growth time is of 30 s. The data on the wave-vector components presented in [START_REF] Scolan | Nonlinear fate of internal waves attractors[END_REF] yield τ w around 180 and 320 s for the two secondary waves involved into the triplet which is measured at time around 300 s after emergence of detectable secondary waves. Therefore, the secondary waves could make only between 1 and 2 return trips, and owing to the presence of the vertical component of the group velocity they could not return to their generation site. So, the onset of the instability described in [START_REF] Scolan | Nonlinear fate of internal waves attractors[END_REF], and also seen in the experiment represented by the on the (d, τ ) diagram is to a good approximation a spatially isolated local event, while the onset and subsequent development of instability described in the present Section is a global event, which is inherently related to the confinement. This event has many common features with the instability in a rectangular domain described in [START_REF] Mcewan | Degeneration of resonantly excited standing internal gravity waves[END_REF].

Well-developed TRI cascade in well-focused attractor: presence of secondary standing waves

In the previous Section, we have demonstrated the effect of the ratio between the typical growth time of the primary instability τ * and the duration of the horizontal return-trip τ w of secondary waves in the tank. If τ * τ w , the onset of instability is local, while for τ * τ w the secondary waves are more likely to appear in form of standing waves. However, τ * and τ w do not form an exhaustive set of time scales. The evolution of the system to saturated state where the energy injection is in balance with dissipation may have a very long time-scale τ sat . Over the time span 0 < t < τ sat , the system may experience a cascade of triadic resonance instabilities generating new time-scales of the return trips τ w . For the full study of dynamics, the system should be observed during the time τ obs > τ sat τ w . Thus, long-term experiments with τ obs τ w can reveal the effects of confinement for the system with τ * τ w similar to the one described in [START_REF] Scolan | Nonlinear fate of internal waves attractors[END_REF]. In this section, we analyse the long-term evolution of the spectrum in the case with strong focusing ( on (d, τ ) diagram) and present for the first time the evidence of the effects due to confinement.

Time-frequency diagram

The time-frequency diagram of the long-term experiment with well-focused attractor ( ) at a = 5 mm is presented in figure 10. One can identify at least eight couples of secondary waves in panel (a). The four most energetic couples in the spectrum are labelled in panel (b). The secondary frequencies are named using two indices as follows. The first index i indicates the position in the couple, i = 1 for frequencies higher than Ω 0 /2 or i = 2 for frequencies smaller than Ω 0 /2. The second index n corresponds to the number of the couple. Thus, n varies from 1 to 8, as one can observe at least eight different couples. For each couple n, one has Ω 1,n + Ω 2,n = Ω 0 .

(5.1)

The index n classifies the different couples from the most intense (n = 1) to the less intense (n = 8). The first couple is generated by the attractor itself, as a standard TRI described in [START_REF] Scolan | Nonlinear fate of internal waves attractors[END_REF] or in § 4.2. Then, other couples appear and all the frequencies in the tank are linked by 3-wave interactions. The cascade is created as follows. Once the first couple has been created by the attractor, the two frequencies of this couple (Ω 1 = Ω 1,1 and Ω 2 = Ω 2,1 ) interact together to create a third frequency (Ω 2,2 ), which belongs to the second couple. The second couple is completed by the interaction between the third frequency and the frequency of the attractor, Ω 0 . These reactions are given by

Ω 1,1 -Ω 2,1 = Ω 2,2 , (5.2) Ω 0 -Ω 2,2 = Ω 1,2 .
(5.3) Thus, the second couple is complete. Now, several interactions are possible between the different frequencies. An interaction between the first and the second couples leads to a third one:

Ω 1,2 -Ω 1,1 = Ω 2,3 , (5.4
) 2 for the parameters).

Ω 0 -Ω 2,3 = Ω 1,3 , (5.5)
and an interaction between the frequencies of the second couple creates a forth couple:

Ω 1,2 -Ω 2,2 = Ω 1,4 , (5.6)
Ω 0 -Ω 1,4 = Ω 2,4 .

(5.7)

Here, one has four different couples, so eight frequencies. One can easily continue to create other frequencies by combining the different frequencies of the different couples. This mechanism leads to an infinite set of discrete frequencies and the first eight couples are at least visible in figure 10. Note that each coupling satisfies also the spatial resonant condition and that all the waves fulfill the dispersion relation. At the end, each frequency is linked with a large number of other frequencies by a three-wave interaction (Brouzet et al. (2016a)). Experimentally, we observe the fullfilment of the above equations with the accuracy in non-dimensional frequency of ±0.001.

Observational evidence of standing secondary waves

Although the instability starts and evolves differently in well-and weakly-focused attractors, they can exhibit some common features. Figure 8 shows that the secondary waves generated by the weakly-focused attractor are standing waves. Some frequencies created by a well-focused attractor in long-term experiments can also be standing waves as shown in figure 11. For example, the fluid motions at frequencies Ω 2,1 and Ω 2,2 are standing waves as it is clear from the wave patterns filtered at these frequencies. Note that not all the frequencies present in figure 10 correspond to standing waves.

With the frequencies corresponding to the peaks of the spectrum shown in figure 10(b), one can compute the values of parameters τ associated to these waves. These frequencies are shown by different symbols in the diagram (d, τ ) presented in figure 1. It can be seen that the τ parameters for the standing secondary waves are close to black zones in the (d, τ ) diagram. Thus, at least some frequencies created by the weakly-and wellfocused attractors could correspond to quasi-resonant modes. In absence of commonly accepted terminology, we can tentatively call the dark regions (see figure 1) as geometric quasi-resonances. Previously quasi-resonances have been discussed assuming non-zero resonance width (see [START_REF] Nazarenko | Wave Turbulence[END_REF]). Here we refer to possibility of global quasiresonances assuming that the Lyapunov exponent can take very small negative non-zero value. In a realistic system with weak viscous dissipation, we can expect that some of the quasi-resonances can exhibit a seiche-like behavior similar to exact seiche modes discussed in [START_REF] Maas | Observations of an internal wave attractor in a confined stably stratified fluid[END_REF] and [START_REF] Manders | Observations of inertial waves in a rectangular basin with one sloping boundary[END_REF]. Therefore, at the conceptual level, the key driving processes in long-term experiments with unstable wave attractors can be considered as a combination of "local" [START_REF] Scolan | Nonlinear fate of internal waves attractors[END_REF] and "global-scale" (present paper) TRI events, with long-term transients between the two.

Interesting parallels can be found in the dynamics of rotating fluids. For instance, [START_REF] Beardsley | An experimental study of inertial waves in a closed cone Stud[END_REF], [START_REF] Thompson | Diurnal tides and shear instabilities in a rotating cylinder[END_REF] and [START_REF] Duguet | Oscillatory jets and instabilities in a rotating cylinder[END_REF] show rich multi-peak spectra for inertial oscillations in a compressible fluid confined in a rotating cylindrical vessel; they relate the frequencies associated with the observed peaks to global modes of the fluid motion. Similarly, [START_REF] Favier | Generation and maintenence of bulk turbulence by libration-driven elliptical instability[END_REF] analyse the frequency spectrum of fluid motion generated by libration-driven elliptic instability in a rotating ellipsoid and show the presence of the eigenfrequencies of linear and quadratic inertial modes.

The "wave mixing box" regime in NP and NS attractors

In the previous Sections, we have shown that the scenario of instability in wave attractors and the parameters of wave triplets involved into resonant interactions strongly depend on the choice of the operating points in (d, τ ) plane. However, one can expect a certain level of universality for fully developed cascades of triadic interactions transferring energy to small-scale internal waves and ultimately to mixing events. A high mixing efficiency of triadic cascades operating in confined domains has been demonstrated so far in [START_REF] Mcewan | Internal mixing in stratified fluids[END_REF] for normal modes and in Brouzet et al. (2016a) for well-focused attractors, in rectangular and trapezoidal domains, respectively.

Below, we compare the case of well-focused (1, 1) attractor described in Brouzet et al. (2016a) ( ) with the case of a weakly focused (1, 1) attractor ( ). The time-frequency diagrams of two unstable attractors obtained at high amplitude of forcing (a = 10 mm) in regimes with measurable mixing are shown in figure 12. Both diagrams have qualitatively similar features: several discrete frequency peaks of large magnitude are embedded into a continuous frequency spectrum, which has significantly smaller magnitude. The magnitudes of discrete peaks strongly fluctuate in time. Brouzet et al. (2016a) show that the normalized horizontal y-component of the vorticity field,

Ξ y = ξ y (x, z, t) N = 1 N ∂v x ∂z - ∂v z ∂x , (6.1)
represents a useful quantity indicating the occurence of extreme events in the experimental system. This quantity has a clear physical meaning as a ratio of the destabilizing effect of vorticity to the stabilizing effect of stratification. Obviously, we can expect some mixing in the system when a significant statistics of events with high values of |Ξ y | is observed.

Figure 13 represents the probability density functions (PDFs) of Ξ y for (1, 1) attrac- 2).

tors with weak and strong focusing for two different values of the forcing amplitude a.

Comparing the data obtained at fixed a, we see that quantitatively the system with weak focusing shows consistently lower statistics of extreme events than the system with high focusing. Qualitatively, the shape of vorticity PDFs in attractors with high and low focusing is similar. Moreover, the long-term erosion of stratification is also similar as discussed below.

The significant statistics of extreme events with high horizontal vorticity arises from the spontaneous summation of the frequency components of the wave field. It is important to note that the primary (monochromatic) wave alone cannot produce extreme events. This is attested by figures 13(c) and (d) that represent the PDFs of Ξ y for the primary wave field filtered at Ω 0 . It can be seen that these PDFs have no tails extending to the domains with |Ξ y | > 2. The threshold value |Ξ y | > 2 has been proposed in Brouzet et al. (2016a) as extension of the classic Miles-Howard criterion for stability of horizontal shear flows of continuously stratified fluid. In fact, a possibility of extension of the Miles-Howard criterion to a wider context has been discussed in some classic works, e.g. [START_REF] Phillips | The Dynamics of the Upper Ocean[END_REF]. Using |Ξ y | = 2 as a threshold value one can make some qualitative conclusions.

In figures 13(e) and (f), we show the ratios of the final to initial stratifications observed in different experiments described in the present paper. In both cases at large amplitude of forcing we observe a well-measurable mixing when comparing the initial and final density distributions. The probability of having an extreme event with intensity |Ξ y | > 2 is given by the integral over the corresponding tails of the PDF. The cases with measurable mixing depicted in figures 13(e) and (f) correspond to drastically increased probability of extreme events. It increases from 2‰ for a = 5 mm to 19‰ for a = 10 mm in the "propagating waves" case, and from 0.7‰ for a = 5 mm to 5.6‰ for a = 10 mm in the "standing waves" case. We observe a comparable increase of probability, by factor 10 and 8, in both cases. Although the statistics of high-vorticity events seems to represent a kinematic indicator for the occurence of mixing, at the present stage it is difficult to relate this statistics directly to the overall mixing efficiency. Additional parameters The PDF are calculated from all images between 480 and 500 T0, for the two propagating (panel a, in figure 4) or standing (panel b, in figure 4) wave experiments presented in figure 12. Panel (c) and (d) present the same quantity for the vorticity filtered at Ω0. In all panels, dashed black and solid red lines correspond to a = 5 mm and a = 10 mm. should be measured, most importantly, the typical scale of overturning structures. As an alternative, one can make an attempt to estimate the turbulent diffusion coefficient in the experimental system similar to [START_REF] Dossmann | Mixing by internal waves quantified using combined PIV/PLIF technique[END_REF].

Conclusions

Geometric focusing of internal wave motions in a confined fluid domain leads to energy concentration at a closed loop, an internal wave attractor [START_REF] Maas | Geometric focusing of internal waves[END_REF][START_REF] Maas | Observations of an internal wave attractor in a confined stably stratified fluid[END_REF]. The rate of convergence of internal wave rays toward limiting cycle can be quantified in terms of Lyapunov exponents. Using this variable, one can define the domains of existence (Arnold tongues) of wave attractors.

In the present paper, we explore the structure of the Arnold tongue corresponding to a simple parallelogram-shaped (1, 1) attractor in a trapezoidal domain of a uniformly stratified fluid. We show that the kinetic energy of the confined fluid system represents an appropriate global variable that allows to classify the observed wave regimes. A typical Arnold tongue is shown to have a central region corresponding to "classic" well focused wave attractors with thin branches where the wave energy is concentrated. These attractors typically have a large total kinetic energy, and the wave motion in the attractor branches is represented by a propagating wave. The behaviour of "classic" wave attractors has been studied in great details, both in linear [START_REF] Maas | Geometric focusing of internal waves[END_REF][START_REF] Maas | Observations of an internal wave attractor in a confined stably stratified fluid[END_REF][START_REF] Rieutord | Inertial waves in a rotating spherical shell: attractors and asymptotic spectrum[END_REF][START_REF] Ogilvie | Wave attractors and the asymptotic dissipation rate of tidal disturbances[END_REF]Hazewinkel et Importantly, a typical Arnold tongue has border regions corresponding to zones of geometric degeneracy which can be of two types: i) attractor collapses onto a line representing a diagonal of the trapezoid (see figure 1) or ii) the system itself does not support effective focusing. In this latter case, each internal wave orbit is periodic contrary to attracting cases, for which only a few periodic orbits exist and are named attractors. Moreover, this case corresponds, as a limit, to well-studied configuration of standing internal waves in a rectangular domain filled with uniformly stratified fluid [START_REF] Mcewan | Degeneration of resonantly excited standing internal gravity waves[END_REF]. The intermediate case of an elongated attractor in the domain with weak focusing is studied in detail in the present paper. For this case, the wave motion in the attractor has both the properties of standing and propagating waves. Importantly, increased energy concentration at the loop predicted by the ray tracing indicates that the propagating component of the wave field does matter. We show that in this configuration under appropriate forcing one can observe a scenario of triadic resonance with very low growth rate of secondary waves: the exponential growth for nearly three decades in amplitude at the time-scale of one hour has been detected experimentally. The secondary waves are found to be represented by nearly standing waves. Thus, at the border regions of the Arnold tongue the onset of triadic instability represents an intermediate case between the cases of purely propagating [START_REF] Scolan | Nonlinear fate of internal waves attractors[END_REF]) and purely standing [START_REF] Mcewan | Degeneration of resonantly excited standing internal gravity waves[END_REF] wave triplets.

Further, we explore the long-term behaviour of instability detected in Scolan, Ermanyuk & Dauxois (2013) and Brouzet et al. (2016a). We show that some frequencies in the spectrum of wave motions correspond to standing and nearly-standing waves with high vertical modes. These observations may have an important consequence to the dynamics of large stratified geophysical systems. Such geophysical systems (lakes for instance) may be subject to very complex forcing. The cascade of triadic instabilities that develops as a response to forcing may effectively transfer energy to standing and nearly-standing waves of high vertical modes. When the forcing stops, the attractor-like components of the internal wave motion quickly disappear as described in [START_REF] Hazewinkel | Observations on the wavenumber spectrum and evolution of an internal wave attractor[END_REF] and [START_REF] Grisouard | Numerical simulation of a two-dimensional internal wave attractor[END_REF]. In absence of energy injection, at low wave numbers, the spectrum of wave motions in attractors quickly shifts toward high wave numbers due to focusing, and the high wave numbers quickly decay due to viscosity. In contrast, the normal vertical (global resonance) modes conserve their length scale and decay at much longer time-scale owing to purely viscous mechanism. Similar dynamics is expected for quasi-resonant modes characterized by vanishingly weak focusing. Such a scenario can explain the presence of high vertical modes in limnological observations.

As the energy input into a confined domain increases, one can expect a cascade of energy from global to small scales. As it is shown for well-focused "classic" wave attractors (Brouzet et al. 2016a), such a cascade operates via a hierarchy of triadic wave-wave interactions transferring energy to small scale dissipation and mixing events. We show that under sufficiently large forcing this is a common fate for wave attractors with strong and weak focusing, with qualitatively similar statistics of extreme events leading to mixing. Interestingly, even in the limit of rectangular box the cascade of triadic interactions can transfer a significant portion of energy to mixing as shown in [START_REF] Mcewan | Internal mixing in stratified fluids[END_REF]. The choice of the operating point inside the Arnold tongue and the amplitude of the input perturbation provide extreme flexibility to control the parameters of the energy cascade in the system. We believe that this "mixing box" configuration has a strong potential to mimic many aspects of the energy cascade in confined geophysical systems both experimentally and numerically.

Interestingly, in the end, both well and weakly focused attractors display similar mixing properties. This generic result is of importance to argue that any continuously-stratified and continuously-forced fluid repeatedly goes through a sequence of focusing, mixing and restratification. The presence of non-uniform stratification and mean flows has sometimes been interpreted as presenting counter-evidence of the occurrence of wave attractors [START_REF] Gerkema | Absence of internal tidal beams due to non-uniform stratification[END_REF]. In the light of the present result, it may be worth to reconsider this interpretation.

In the present paper, we have measured the kinetic energy of stable attractors as function of their location at the Arnold tongue at the laboratory scale. At large scale we expect the increase of the central region of the Arnold tongue corresponding to classic "narrow-beam" attractors with propagating wave motion, and decrease of the border regions corresponding to attractors with considerable standing-wave component. However, such a qualitative extrapolation to large scales applies only to stable attractors in linear viscous regime [START_REF] Hazewinkel | Observations on the wavenumber spectrum and evolution of an internal wave attractor[END_REF][START_REF] Grisouard | Numerical simulation of a two-dimensional internal wave attractor[END_REF]. Recent results presented in [START_REF] Guo | Internal wave attractors over random, small-amplitude topography[END_REF] suggest that for typical bathymetry of the ocean bottom in a two-dimensional model, one can expect about 10 attractors per 1000 km. Interestingly, this study shows that in the case of small-scale bathymetry the attractors with very elongated loops represent the most probable configuration, justifying our interest to attractors with a "degenerated" geometry. The realistic wave regime in large-scale stratified reservoirs in presence of wave attractors is likely to be nonlinear, with signatures of internal-wave turbulence and mixing as described in Brouzet et al. (2016a) and the present paper. The relevance of such a scenario to a three-dimensional topography [START_REF] Manders | On the three-dimensional structure of the inertial wave field in a rectangular basin with one sloping boundary[END_REF]; [START_REF] Drijfhout | Impact of channel geometry and rotation on the trapping of internal tides[END_REF]; [START_REF] Wang | Three dimensional simulation of internal wave attractors in the Luzon strait[END_REF] and to inertial-wave attractors in spherical liquid shells [START_REF] Favier | Non-linear evolution of tidally forced inertial waves in rotating fluid bodies[END_REF] remains to be explored.

Figure 1 .

 1 Figure 1. Right panel presents the (d,τ ) diagram obtained by Maas et al. (1997). Left panels present two examples of the corresponding (1,1) attractor. The different symbols in the right panel correspond to the different peak frequencies for the 2 experiments ( and in Table2with a = 5 mm). The primary wave frequency is plotted with a filled and larger symbol. The subharmonics (respectively superharmonics) are represented by white (respectively black) symbols.

Figure 2 .

 2 Figure 2. Sketch of the experimental set-up showing the wave generator and the sloping wall (inclined at an angle α with respect to the vertical) inside the immobile tank of size 800 × 170 × 425 mm 3 . The working bottom length of the section, the depth and the sloping angle are given within the text.

KFigure 3 .

 3 Figure 3. Typical experimental time-history of the nondimensional kinetic energy. The vertical dotted dashed line indicates the instant where the kinetic energy saturates. The average of the signal, computed from this instant and to the end of the experiment, is represented by the dashed white line. The two horizontal dashed black lines represent Kmin and Kmax, the minimal and maximal values bounding the oscillations of the kinetic energy during the saturated state. The parameters for this experiment are: Ω0=0.59, H=300 mm, L=450 mm, α = 27.3 • , τ = 1.84, d = 0.31 and a = 1.5 mm. For this experiment, K = 5 and R = 0.65.

Figure 4 .

 4 Figure 4. Structure of Arnold tongue of a stable (1, 1) attractor in terms of the mean normalized kinetic energy K (panel a) and the ratio of minimum to maximum kinetic energy R (panel b). The results are plotted from the 50 experiments described in Table 1. Superimposed on this picture are three different symbols corresponding to the location of the three main long-term experiments discussed throughout this paper: and ♦ for the case of the rectangular domain discussed by McEwan (1971) (see § 4.1), for the well-focused (1, 1) attractor discussed in Brouzet et al. (2016a) with propagating waves (see § 4.2),for the weakly-focused (1, 1) attractor described here (see § 4.3). The solid black line shows the contour at the value R = 0.5, enclosing the region with nearly propagating waves (NP), while nearly standing waves (NS) are outside this region.

Figure 5 .

 5 Figure 5. Horizontal velocity fields for the experiment indicated by the symbol in figure 4 and Table 2 for a = 5 mm, filtered at Ω0 (panel a), Ω1 (panel b) and Ω2 (panel c) in frequency and in space to keep only the relevant directions. The velocity field is displayed only where the wave amplitude is larger than 15% of the maximum. The wave vectors are represented by the arrows. Black dashed lines show the billiard geometric prediction of the attractor.

Figure 6 .

 6 Figure 6. Velocity field for the experiment indicated by the symbol in figure 4 with a=5 mm (see Table 2 for the parameters). (a) Picture of internal wave field, filtered around Ω0, 200 T0 after the start of the wave maker and black lines show the billiard geometric prediction of the attractor. The picture presents the absolute value of the norm of the velocity vectors. The large value around the upper corner of the attractor is due to the presence of a small homogenous layer above the stratified domain. (b) the time-history of oscillations at the point A shown in (a). The vertical dashed line in (b) shows the instant of the picture (a) while the dashed square in (a) shows the region where the time-frequency diagram in figure 7 has been calculated. The initial transient is due to the setting of the attractor, while the instability is noticeable after roughly 400 periods of oscillations.

Figure 7 .

 7 Figure 7. Weak TRI. (a) Time-frequency diagram of internal-wave field, obtained from a 50 mm side square region, surrounding the point A shown in figure 6(a); (b) vertical cut of the diagram along the frequency axis at t = 400T0, indicated by the white dashed line. S0 corresponds to the time average of the frequency component associated with the primary wave Ω0. Results are presented for the experiment indicated by the symbol in figure 4 (see Table2for parameters).

Figure 8 .

 8 Figure 8. Components of the wave field corresponding to frequencies Ω1 (panel a) and Ω2 (panel b), obtained with the Hilbert transform around 620 T0 using a time window 85 T0 long. Results are presented for the experiment indicated by the symbol in figure 4 with a=5 mm (see Table2for the parameters). Note that the components oscillating at Ω1 and Ω2 correspond to nearly standing waves as is clearly seen on the sequences of wave profiles shown on the right of each picture.

  . The sequences of wave profiles measured along the vertical line indicated in the images are also shown in the right panels. It can be seen that the secondary-wave motion is represented by standing waves of high vertical modes. They do not correspond to global eigenmodes of the trapezoidal basin (represented by the black lines of the (d, ) diagram) but to quasi-global resonances (dark regions of the (d, τ ) diagram, as shown in figure 1(c). Diffusion induces a certain thickness to the branches and thus allows the secondary waves to be quasi-standing and to correspond to quasi-global modes of the trapezoidal basin.

  as λ 0 = 2π/|k 0 | = 39.8 cm with horizontal and vertical components λ x0 = 66.1 cm and λ z0 = 49.7 cm. The quantities λ x0 /2 and λ z0 /2 are reasonably close to the distances from the left top corner of the trapezoidal fluid domain to the reflection points of the attractor. The ray tracing yields the distances to the reflection points 28 and 21.6 cm in horizontal and vertical directions, respectively.

Figure 9 .

 9 Figure 9. Cuts along the time-axis of the time-frequency diagram presented in figure 7 at frequencies Ω1 (solid red line) and Ω2 (dashed blue line).

Figure 10 .

 10 Figure 10. TRI cascade. (a) Time-frequency diagram of internal-wave field, obtained from a 5 × 5 cm 2 side square region in the most energetic wave beam; (b) Vertical cut of the diagram along the frequency axis at t = 400T0, indicated by the black dashed line. S0 corresponds to the time average of the frequency component associated with the primary wave Ω0. The different frequency peaks corresponding to the four first couples are labelled. Results are presented for the experiment indicated by the symbol in figure 4 with a=5 mm (see Table2for the parameters).

Figure 11 .

 11 Figure 11. Components of the wave field corresponding to frequencies Ω2,1 (a) and Ω2,2 (b), obtained with the Hilbert transform around 620 T0 using a time window 85 T0 long. Note that the components oscillating at Ω2,1 and Ω2,2 correspond to standing waves as is clearly seen on the sequences of wave profiles shown on the right of each picture. Results are presented for the experiment indicated by the symbol in figure 4 with a = 5 mm (see Table2for the parameters).

Figure 12 .

 12 Figure 12. Time-frequency diagrams of two signals recorded in unstable wave attractors corresponding to different operating points. Both attractors are unstable and the wave regime is such that it induces significant mixing. The left panel corresponds to the experiment indicated by the symbol in figure 4, while the right one by the (with a = 10 mm and other parameters listed in Table2).

Figure 13 .

 13 Figure13. Panels (a) to (d) present the probability density functions of the vorticity in the tank, while panels (e) and (f) exhibit the ratio of the final (after 700 periods) to initial stratifications. The PDF are calculated from all images between 480 and 500 T0, for the two propagating (panel a, in figure4) or standing (panel b, in figure4) wave experiments presented in figure12. Panel (c) and (d) present the same quantity for the vorticity filtered at Ω0. In all panels, dashed black and solid red lines correspond to a = 5 mm and a = 10 mm.

  al. 2008; Grisouard, Staquet & Pairaud 2008; Hazewinkel, Tsimitri, Maas & Dalziel 2010; Echeverri, Yokossi, Balmforth & Peacock 2011; Guo & Holmes-Cerfon 2016) and non-linear (Ogilvie 2005; Scolan, Ermanyuk & Dauxois 2013; Jouve & Ogilvie 2014) regimes including the onset of triadic resonance and formation of energy cascades of triadic wave-wave interactions.

Table 1 .

 1 Parameters for the series of experiments represented in figure

	.81	30 34.6 0.588 0.09 1.61
	6 11.3 0.609 0.73 1.79	31 34.6 0.643 0.09 1.43
	7 24.2 0.551 0.41 1.94	32 34.6 0.643 0.09 1.41
	8 24.2 0.559 0.41 1.89	33 34.6 0.669 0.09 1.34
	9 24.2 0.587 0.41 1.85	34 34.6 0.707 0.09 1.15
	10 24.2 0.615 0.41 1.74	35 37.5 0.673 0.00 1.85
	11 24.2 0.642 0.41 1.42	36 37.5 0.707 0.00 1.78
	12 24.2 0.668 0.41 1.37	37 37.5 0.743 0.00 1.42
	13 28.5 0.554 0.28 1.90	38 37.5 0.755 0.00 1.30
	14 28.5 0.574 0.28 1.83	39 37.5 0.766 0.00 1.04
	15 28.5 0.602 0.28 1.71	40 40.3 0.552 -0.10 1.88
	16 28.5 0.622 0.28 1.62	41 40.3 0.559 -0.10 1.85
	17 28.5 0.656 0.28 1.49	42 40.3 0.602 -0.10 1.55
	18 28.5 0.695 0.28 1.32	43 40.3 0.616 -0.10 1.50
	19 28.5 0.707 0.28 1.30	44 40.3 0.643 -0.10 1.40
	20 31.2 0.554 0.20 1.90	45 40.3 0.731 -0.10 1.22
	21 31.2 0.574 0.20 1.83	46 45.0 0.530 -0.30 1.86
	22 31.2 0.602 0.20 1.68	47 45.0 0.559 -0.30 1.85
	23 31.2 0.636 0.20 1.54	48 45.0 0.602 -0.30 1.50
	24 31.2 0.669 0.20 1.42	49 45.0 0.629 -0.30 1.40
	25 31.2 0.707 0.20 1.24	50 45.0 0.669 -0.30 1.36

Table 2 .

 2 are specified the experimental conditions of the three main long-term experiments represented by the symbols , and in figure 4. Parameters for the three main and long term experiments discussed in this paper.

	Reference	Unit	McEwan (1971) Brouzet et al. (2016a) Present paper (2016)
	Section	§	4.1	4.2	4.3
	Symbol in Fig. 4	-
	H	mm	326	303	303
	L	mm	377	444	444
	a	mm	4.8	5 or 10	5 or 10
	α	•	90	25.4	14.3
	Ω0	-	0.655	0.61	0.60
	d	-	1	0.35	0.65
	τ	-	0.66	1.80	1.80
	Ω1	-	0.397	0.36	0.33
	Ω2	-	0.277	0.25	0.27
	| * |	rad•m -1	25.0	7.8	7.8
	|m * |	rad•m -1	9.6	10.4	10.4
	| 0|	rad•m -1	25.0	33	9.5
	|m0|	rad•m -1	9.6	37	12.6
	| 1|	rad•m -1	50.0	80	25.3
	|m1|	rad•m -1	38.5	200	66.4
	| 2|	rad•m -1	25.0	45	15.8
	|m2|	rad•m -1	28.9	170	53.8

Acknowledgments

EVE gratefully acknowledges his appointment as a Marie Curie incoming fellow at Laboratoire de physique ENS de Lyon. This work was supported by the LABEX iMUST (ANR-10-LABX-0064) of Université de Lyon, within the program "Investissements d'Avenir" (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR). This work has achieved thanks to the resources of PSMN from ENS de Lyon. We thank L. Maas and H. Scolan for helpful discussions.


Lazhar Fekih-Ahmed

ON THE POWER SERIES EXPANSION OF THE RECIPROCAL GAMMA FUNCTION

Keywords: 2010 Mathematics Subject Classification. Primary 41A60, 30E15, 33B15, 11Y35, 11Y60 Special functions, Reciprocal Gamma function, Taylor expansion, asymptotic expansion. 1

Using the reflection formula of the Gamma function, we derive a new formula for the Taylor coefficients of the reciprocal Gamma function. The new formula provides effective asymptotic values for the coefficients even for very small values of the indices. Both the sign oscillations and the leading order of growth are given.

Introduction

The reciprocal Gamma function is an entire function with a Taylor series given by

(1.1) 1 Γ(z) = ∞ n=1 a n z n .
It has been a challenge since the time of Weierstrass to compute or at least estimate the coefficients of the reciprocal Gamma function. The main reason is the ubiquitous presence of the reciprocal gamma function in analytic number theory and its various connections to other transcendental functions (for example the Riemann zeta function). Since Bourguet [START_REF] Bourguet | Sur les integrales Euleriennes et quelques autres fonctions uniformes[END_REF] who was the first to calculate the first 23 coefficients, there has been very few publications, to the author's knowledge, on accurate calculations of the coefficients beyond a 50 .

Knowing that an effective asymptotic formula is always useful as an independent check for the sign and value of the coefficients for very large values of n, it is important to have such a formula in order to enhance the calculations. The only asymptotic formula that is known to date is that of Hayman [START_REF] Hayman | A Generalisation of Stirling's Formula[END_REF].

With regard to the computation of the coefficients of the reciprocal Gamma function, there are basically three known methods [START_REF] Wrench | Concerning Two Series for the Gamma Function[END_REF][START_REF] Bourguet | Sur les integrales Euleriennes et quelques autres fonctions uniformes[END_REF][START_REF] Bornemann | Accuracy and stability of computing high-order derivatives of analytic functions by Cauchy integrals[END_REF]. The first method is due to Bourguet [START_REF] Bourguet | Sur les integrales Euleriennes et quelques autres fonctions uniformes[END_REF]. It consists in exploiting the recursive formula

(1.2) na n = γa n-1 -ζ(2)a n-2 + ζ(3)a n-3 -• • • + (-1) n+1 ζ(k); n > 2,
with a 1 = 1, a 2 = γ, the Euler constant, and ζ(k) is the zeta function of Riemann. It has been noticed in [START_REF] Bornemann | Accuracy and stability of computing high-order derivatives of analytic functions by Cauchy integrals[END_REF] that this method suffers from severe numerical instability; all digits are lost from n ≥ 27.

The second method is based on Cauchy's formula for the coefficients of Taylor series using circular contours:

a n = 1 2πi |z|=r 1 z n+1 Γ(z) dz = 1 2πr n 2π 0 e -inθ Γ(re iθ )
dθ, (1.3) where r can be between 0 and ∞ since the reciprocal Gamma function is entire. The integral (1.3) can be evaluated with the many existing quadrature rules such as the trapezoidal rule or the Gauss-Legendre quadrature. Particular attention is given to the method discovered by Lyness [START_REF] Lyness | Numerical Differentiation of Analytic Functions[END_REF] which uses the trapezoidal rule in conjunction with the discrete Fourier transform. It is very fast and provides good results [START_REF] Trefethen | The Exponentially Convergent Trapezoidal Rule[END_REF] as long as the radius of the contour is properly selected.

Although the radius r of the contour can theoretically be arbitrarily chosen, the effects of the value of r on approximation and round-off errors are numerically very different. A comprehensive investigation for choosing a good radius r has been carried out in [START_REF] Bornemann | Accuracy and stability of computing high-order derivatives of analytic functions by Cauchy integrals[END_REF], where it has been shown that as n increases so does the good r.

In [START_REF] Schmelzer | Computing the Gamma Function Using Contour Integrals and Rational Approximations[END_REF], different quadrature formulas, using also the method of contour integration, have been investigated for the calculation of a n . The contour chosen is no longer circular but chosen as the Hankel contour. The reciprocal Gamma function is represented using Heine's formula [START_REF] Heine | Einige Anwendungen der Residuenrechnung von Cauchy[END_REF]:

(1.4) 1 Γ(z) = 1 2πi C e t t -z dt,
where C consists of the three parts C = C + ∪ C ǫ ∪ C -: a path which extends from (∞, ǫ), around the origin counter clockwise on a circle of center the origin and of radius ǫ and back to (ǫ, ∞), where ǫ is an arbitrarily small positive number.

Lastly, the third method for calculating the coefficients of the reciprocal Gamma function for large values of n is not a numerical one. It consists in approximating the coefficients using an asymptotic formula. The first attempt was initiated by Bourguet [START_REF] Bourguet | Sur les integrales Euleriennes et quelques autres fonctions uniformes[END_REF] who found the following upper bound a n ≤ (-1) n πΓ(n + 1)

eπ n+1 n + 1 + 4 π 2 Γ(n + 1) 4 π 2 Γ(n + 1) . (1.5)
But the first systematic study to obtain an asymptotic formula for the coefficients was carried out by Hayman [START_REF] Hayman | A Generalisation of Stirling's Formula[END_REF] theoretically, and by Bornemann [START_REF] Bornemann | Accuracy and stability of computing high-order derivatives of analytic functions by Cauchy integrals[END_REF] numerically (see also [START_REF] Berry | Universal Oscillations of High Derivatives[END_REF] for the related phenomenon of oscillations of the derivatives).

In this paper, we will give a new effective asymptotic formula for the coefficients a n . With the formula, we obtain the sign oscillations and the leading order of growth of the coefficients. We will show that our results can be considered very accurate even for very small values of n.

2. An Integral Formula For The Coefficients a n Let's replace z by z -1 into the series (1.1), we have

(2.1) 1 Γ(z -1) = a 1 (z -1) + a 2 (z -1) 2 + a 3 (z -1) 3 • • • ,
and dividing both sides by z -1, we get

(2.2) 1 Γ(z) = 1 z -1 a 1 (z -1) + a 2 (z -1) 2 + a 3 (z -1) 3 • • •
To obtain an integral formula for the reciprocal Gamma function, we start from Euler's reflection formula

(2.3) Γ(z)Γ(1 -z) = π sin(πz) to get (2.4) 1 Γ(z) = sin(πz) π Γ(1 -z). Now, for Re(z) < 2, we can write 1 Γ(z) = sin(πz) π Γ(1 -z) = sin(π(z -1)) π(z -1) Γ(2 -z) = sin(π(z -1)) π(z -1) ∞ 0 e -t t 1-z dt. (2.5)
By observing that sin(π(z -1)) = e iπ(z-1) -e -iπ(z-1)

2i

, we can rewrite (2.5) as

(2.6) 1 Γ(z) = 1 z -1 1 2πi
∞ 0 e -t e (z-1)(-log(t)-iπ)e (z-1)(-log(t)+iπ) dt.

And if we compare the two equations (2.6) and (2.2), we deduce that the coefficients a n for n ≥ 1 are given by

a n = 1 2πin! ∞ 0 e -t lim z→1 d n dz n e (z-1)(-log(t)+iπ) -e (z-1)(-log(t)-iπ) dt = 1 2πin! ∞ 0 e -t {(-log(t) + iπ) n -(-log(t) -iπ) n } dt = 1 πn! ∞ 0 e -t Im {(-log(t) + iπ) n } dt, (2.7)
where Im stands for the imaginary part. This is our expression of the coefficients a n , described in the following Theorem 2.1. The coefficients a n are given by

(2.8) a n = (-1) n πn! ∞ 0 e -t Im (log t -iπ) n dt.
Theorem 2.1 permits an effective asymptotic evaluation of the constants a n

1

. It is the subject of the next section.

Asymptotic Estimates of the Coefficients

This section is dedicated to approximating the complex-valued integral (3.1)

I(n) = ∞ 0 e -t (log t -iπ) n dt
using the saddle-point method [START_REF] Copson | Asymptotic Expansions[END_REF][START_REF] De Bruijn | Asymptotic Methods in Analysis[END_REF]. By the change of variables t = nz, our integral becomes

I(n) = n ∞ 0 e -nz {log (nz) -iπ} n dz = n ∞ 0 e n{-z+log[log(nz)-iπ]} dz. (3.2) If we define (3.3) f (z) = -z + log (log (nz) -iπ) ,
then the saddle-point method consists in deforming the path of integration into a path which goes through a saddle-point at which the derivative f ′ (z), vanishes. If z 0 is the saddle-point at which the real part of f (z) takes the greatest value, the neighborhood of z 0 the dominant part of the integral as n → ∞ [4, p. 91-93]. This dominant part provides an approximation of the integral and it is given by the formula

I(n) ≈ ne nf (z0) -2π nf ′′ (z 0 ) 1 2 . (3.4)
In our case, we have

f ′ (z) = -1 + 1 z (log (nz) -iπ)
, and (3.5)

f ′′ (z) = -1 z 2 (log (nz) -iπ) - 1 z 2 (log (nz) -iπ) 2 . (3.6)
The saddle-point z 0 should verify the equation

z 0 (log (nz 0 ) -iπ) = 1 ⇔ nz 0 e -iπ log nz 0 e -iπ = ne -iπ . (3.7)
1 The theorem is almost evident and easy to derive. It is hard to believe that it has not been discovered before. To the author's knowledge, the integral formula (2.8) is new and seems to be nonexistent in the literature.

The last equation is of the form v log v = b whose solution can be explicitly written using the branch k = -1 of the Lambert W -function 2 [START_REF] Corles | On the Lambert W Function[END_REF]:

v = e W-1(b) . (3.8)
The saddle-point solution to our equation (3.7) is given by

z 0 = e -iπ n e W-1(ne -iπ ) = e W-1 (-n) -n , (3.9)
and at the saddle-point, we have the values

f (z 0 ) = -z 0 -log z 0 (3.10) f ′′ (z 0 ) = -1 - 1 z 0 . (3.11)
Therefore, the saddle-point approximation of our integral (3.1) is given by (3.12)

I(n) ≈ √ 2πne -nz0 z 1 2 -n 0 √ 1 + z 0 .
Now since a n = (-1) n πn! Im {I(n)}, we arrive at our main result:

Theorem 3.1. Let z 0 = e W -1 (-n) -n
, where W -1 is the branch k = -1 of the Lambert W -function. For n large enough, the Taylor coefficients of the reciprocal Gamma function can be approximated by

(3.13) a n ≈ (-1) n 2 π √ n n! Im e -nz0 z 1 2 -n 0 √ 1 + z 0 .
Bornemann's derivation [START_REF] Bornemann | Accuracy and stability of computing high-order derivatives of analytic functions by Cauchy integrals[END_REF] of Hayman's asymptotic formula for the coefficients a n is given by 3

a n ∼ √ 2 πn 1 |Γ (r n e iθn ) |r n n cos φ n , (3.14)
where 2 The principal branch of the Lambert W -function is denoted by W 0 (z) = W (z). The principal branch W 0 (z) and the branch W -1 (z) are the only branches of W that take on real values. The other branches of W have the negative real axis as the only branch cut closed on the top for counter clockwise continuity. In our equation (3.7), the argument is -π and not π and so the solution belongs to the branch of W -1 . See [START_REF] Corles | On the Lambert W Function[END_REF] for a excellent discussion and explanation of all the branches of W . 3 The formula of Bornemann differs from that of Hayman in the phase approximation. The orignal approximation given by Hayman is φn = n -1 2 sin 2 θn θn -θn . For the calculations, both phase approximations give essentially the same results.

z n = r n e iθn = e W ( 1 2 -n) (3.15) φ n = n - 1 2 sin 2 θ n θ n -θ n - 1 2 cot θ n -θ n csc 2 θ n . (3.16)
Note that both formulas use the lambert W -function. Our formula will be compared to Hayman's formula in the next section.

We can also find an asymptotic formula of our a n as a function of n only by resorting to the following asymptotic development of the branch of W -1 (z) [START_REF] Corles | On the Lambert W Function[END_REF]:

(3.17) W -1 (z) = log(z -2πi) -log (log z -2πi) + • • •
For n ≫ 1 we can write 3.21) is a rough approximation. It will not be used for calculations. It only provides the leading order of growth and the sign oscillations of the coefficients. However, one can use the approximation (3.21) to easily prove that the order of the reciprocal gamma function is 1 and that its type is maximal.

z 0 ∼ -n -2πi -n log(-n -2πi) ∼ 1 log n -πi ∼ e -i arctan( π log n ) (log n) 2 + π 2 ∼ e i π log n log n , (3.18) z 1 2 -n 0 √ 1 + z 0 ∼ 1 z n 0 ∼ (log n) n e i nπ log n , (3.19)

Numerical Results and Conclusion

We implemented the formula of Theorem 3.1 and Hayman's formula (3.14) The approximations (3.13) and (3.14) were examined and compared to the exact values for n from 1 to 20 given in [START_REF] Wrench | Concerning Two Series for the Gamma Function[END_REF]. They are displayed in Table 1. Table 2, displays the approximate value of a n and exact values for higher values of n.

For n = 4 Hayman's formula gives the wrong sign. It provides a value of the coefficient with an error of 18.5% for n = 15, and for n = 13, 16, the error is almost 96%. Moreover, we can see that for at least the values of 2 ≤ n ≤ 20 Hayman's formula is not as good an approximation to the exact value as the formula of Theorem 3.1. Table 2. The coefficients a n for different higher values of n and their approximations using the asymptotic formula of Hayman (3.14) and the asymptotic formula of Theorem 3.1.

From the trend of the values in Table 1 and Table 2, we conclude that for small values of n our formula outperforms Hayman's formula and that for larger values of n both formulas give the same sign but differ slightly in magnitude. The asymptotic formula of this paper has the advantage that it does not depend on the radius r n of the circular contour, a real advancement in estimating the coefficients of the Taylor series of the reciprocal Gamma function.

As a final remark, the asymptotic formulas of this paper can of course be used to find an asymptotic formula for the related constants b n defined by the power series

  and e -nz0 ∼ e -n log n e -i π log n ∼ e -n log n , (3.20) and using Stirling formula n! ∼ √ 2πn n e n , this yields the second approximation a n ∼ (-1) n+1 π e -n log n+n log log n+n-n log n sin nπ log n . (3.21) Equation (

  in Maple TM . For a given value of n, the following code computes the value of a n : w0 := LambertW(-1, n*exp(-I*Pi)): z0 := exp(-I*Pi)*exp(w0)/n: f := -z0-ln(z0): fpp := -1-1/z0: evalf((-1)**n*Im(n*sqrt((-2*Pi)*(1/(n*fpp))) *exp(n*f))/(Pi*factorial(n)));

  1 + z) b 0 + b 1 z + b 2 z 2 • • • ,where the coefficients a n and b n are connected by the relation(4.2) a n = b n-1 + b n-2 ; n ≥ 2.

Table 1 .

 1 TM Maple is a trademark of Waterloo Maple Inc. First 20 coefficients and their approximate values given by Theorem 3.1 and formula (3.14). .46251758839.10 -1431 -2.460396773.10 -1431 -2.5852781. -1431 1400 -6.07622638292.10 -2792 -6.074000773.10 -2792 -6.5759375.10 -2792

	n	a n	Formula of Theorem 3.1	Hayman's formula
	2	0.577215664		0.471315586	0.318527853
	3	-0.655878071		-0.634156618	-0.745580393
	4	-0.042002635		-0.024878383	0.035835755
	5	0.166538611		0.1586548367	0.170422513
	6	-0.042197734		-0.0422409922	-0.055165293
	7	-0.009621971		-0.0088055266	-0.006842089
	8	0.007218943		0.0070070400	0.007791124
	9	-0.001165167		-0.0011689459	-0.001538105
	10	-0.000215241		-0.0002013214	-0.000162310
	11	0.000128050		0.0001248855	0.000137477
	12	-0.000020134		-0.0000200451	-0.000025104
	13	-0.00000125		-0.000001139	-0.000000054
	14	0.000001133		0.0000011053	0.000001178
	15 -2.0563384.10 -7 16 6.11609510.10 -9		-2.034656492.10 -7 6.506886194.10 -9	-2.410634519.10 -7 1.201994777.10 -8
	17	5.00200764.10 -9		4.864046460.10 -9	4.859838872.10 -9
	18 -1.18127457.10 -9 19 1.043426711.10 -10		-1.164373917.10 -9 1.043634325.10 -10	-1.3136121.10 -9 1.3322234.10 -10
	20 7.782263439.10 -12		7.415156531.10 -12	5.436583518.10 -12
	n	a n		Formula of Theorem 3.1 Hayman's formula
	30	1.7144063219.10 -20		1.708720889.10 -20	2.072558647.10 -20
	40 50 100	-1.1245843492.10 -30 -1.0562331785.10 -41 6.6158100911.10 -106	-1.110270738.10 -30 -1.051407032.10 -41 6.599969140.10 -106	-1.143814145.10 -30 -1.211991030.10 -41 7.56758012.10 -106
	150	1.1936904502.10 -179	1.193587226.10 -179	1.4412588.10 -179
	250 300	-2.4488582032.10 -343 2.90203183445.10 -431	-2.446740476.10 -343 2.900143434.10 -431	-2.8028909.10 -343 3.3306712.10 -431
	800 -2

Acknowledgement

I am grateful to Dr. Fredrik Johansson for providing the exact values in Table 2 to 20D for a n , n ≥ 30. The algorithm he used for the computations is described in his thesis [11].