Soizic Terrien

Rémi Blandin

Christophe Vergez

Benoît Fabre

Regime change thresholds in flute-like instruments: influence of the mouth pressure dynamics

Since they correspond to a jump from a given note to another one, the mouth pressure thresholds leading to regime changes are particularly important quantities in flute-like instruments. In this paper, a comparison of such thresholds between an artificial mouth, an experienced flutist and a non player is provided. It highlights the ability of the experienced player to considerabily shift regime change thresholds, and thus to enlarge its control in terms of nuances and spectrum. Based on recent works on other wind instruments and on the theory of dynamic bifurcations, the hypothesis is tested experimentally and numerically that the dynamics of the blowing pressure influences regime change thresholds. The results highlight the strong influence of this parameter on thresholds, suggesting its wide use by experienced musicians. Starting from these observations and from an analysis of a physical model of flute-like instruments, involving numerical continuation methods and Floquet stability analysis, a phenomenological modelling of regime change is proposed and validated. It allows to predict the regime change thresholds in the dynamic case, in which time variations of the blowing pressure are taken into account.

Introduction and problem statement

In flute playing, the phenomenon of regime change is particularly important, both because it corresponds to a jump from a given note to another one (in most cases an octave higher or lower) and because it is related to the blowing pressure, directly controlled by the musician. As an example, a regime change from the first register to the second register (periodic oscillation regimes synchronized on the first and the second resonance mode of the instrument, respectively), occurs when the musician blows harder enough in the instrument, and is characterized by a frequency leap approximately an octave higher (see for example [START_REF] J W Coltman | Jet drive mechanisms in edge tones and organ pipes[END_REF]).

It is well known that register change is accompanied by hysteresis (see for example [START_REF] J W Coltman | Jet drive mechanisms in edge tones and organ pipes[END_REF][START_REF] Auvray | Regime change and oscillation thresholds in recorderlike instruments[END_REF][START_REF] Terrien | Flutelike musical instruments: A toy model investigated through numerical continuation[END_REF]): the mouth pressure at which the jump between two registers occurs (the so-called regime change threshold) is larger for rising pressures than for diminishing pressures. For musicians, the hysteresis allows a greater freedom in terms of musical performance. Indeed, it allows them both to play forte on the first register and piano on the second register, leading to a wider control in terms of nuance and timbre. Numerous studies have focused on both the prediction and the experimental detection of such thresholds [START_REF] J W Coltman | Jet drive mechanisms in edge tones and organ pipes[END_REF][START_REF] Auvray | Regime change and oscillation thresholds in recorderlike instruments[END_REF]. Other studies have focused on the influence of different parameters on regime change thresholds, such as the geometrical dimensions of channel, chamfers and excitation window of recorders or organ flue pipes [START_REF] Ségoufin | Experimental investigation of the flue channel geometry influence on edge-tone oscillations[END_REF][START_REF] Ségoufin | Experimental study of the influence of the mouth geometry on sound production in a recorder-like instrument: Windway length and chamfers[END_REF][START_REF] Fletcher | Sound production by organ flue pipes[END_REF], the importance of nonlinear losses [START_REF] Auvray | Regime change and oscillation thresholds in recorderlike instruments[END_REF], or the convection velocity of perturbations on the jet [START_REF] Auvray | Regime change and oscillation thresholds in recorderlike instruments[END_REF]. However, it seems that few studies have focused, in terms of regime change thresholds, on other control parameters (i.e. related to the musician) than the slowly varying blowing pressure.

Since it has important musical consequences, one can wonder if flute players develop strategies to change the values of regime change thresholds and to maximize the hysteresis. To test this hypothesis, increasing and decreasing profiles of blowing pressure (crescendo and decrescendo) were performed on the same alto recorder and for a given fingering (corresponding to the note F 4 ), by an experienced flutist, a non player, and an artifical mouth [START_REF] Ferrand | High-precision regulation of a pressure controlled artificial mouth: The case of recorderlike musical instruments[END_REF]. Both experienced musician and non musician have been instructed to stay as long as possible on the first register and on the second register for crescendo and decrescendo respectively. The different experimental setups will be described in section 2. The representation of the fundamental frequency of the sound with respect to the blowing pressure, displayed in figure 1, highlights that the musician obtained an increasing threshold 213 % higher and a decreasing threshold 214 % higher than the artificial mouth, whereas the differences between the non musician and the artificial mouth are of 9 % for the increasing threshold and 32 % for the decreasing threshold. As highlighted in figure 2, similar comparisons on other fingerings (G 4 , A 4 , B b 4 and B 4 ) show that thresholds reached by the musician are at least 95 % higher and up to 240 % higher than thresholds observed on the artificial mouth. On the other hand, thresholds obtained by the non musician are at most 13.3 % lower and 29 % higher than thresholds of the artificial mouth.

Figure 3 presents the comparison between the experienced flutist, the non musician and the artificial mouth in terms of hysteresis. For the three cases, the difference between the thresholds obtained performing an increasing and a decreasing blowing pressure ramp are represented for the five fingerings studied. One can observe that the musician reaches hysteresis between 169 % and 380 % wider than the artifical mouth for the F 4 , G 4 , A 4 and B b 4 fingerings, and up to 515 % wider than the artificial mouth for the B 4 fingering. The hysteresis observed for the non musician are between 27 % and 233 % wider than the hysteresis obtained with the artificial mouth. One can note that the maximum relative difference of 233 % is obtained for the B 4 fingering. For all the other fingerings, the relative differences with the artificial mouth remain between 27% and 65%. In all cases, the hysteresis obtained by the experienced flutist are at least 84 % wider than that observed for the non musician.

As a first conclusion, one can consider that the behaviour of a given instrument played by the artificial mouth and by a non musician is not significantly different in terms of increasing regime change thresholds. In terms of hysteresis, if the results are not significantly different for the F 4 , A 4 and B b 4 fingerings, more important differences are observed for both the G 4 and B 4 fingerings. However, the values measured for the experienced flutist remain significantly higher, both in terms of thresholds and hysteresis, than that obtained for the non player and the artificial mouth. An experienced flutist is able to significantly and systematically modify these thresholds, and thus to enlarge the hysteresis, which presents an obvious musical interest. Which parameters does the musician use to control the regime change thresholds?

If the influence of the blowing pressure has been widely studied under hypothesis of quasi-static variations [START_REF] J W Coltman | Jet drive mechanisms in edge tones and organ pipes[END_REF][START_REF] Auvray | Regime change and oscillation thresholds in recorderlike instruments[END_REF][START_REF] Terrien | Flutelike musical instruments: A toy model investigated through numerical continuation[END_REF][START_REF] Ségoufin | Experimental investigation of the flue channel geometry influence on edge-tone oscillations[END_REF][START_REF] Ségoufin | Experimental study of the influence of the mouth geometry on sound production in a recorder-like instrument: Windway length and chamfers[END_REF][START_REF] Fletcher | Sound production by organ flue pipes[END_REF][START_REF] J W Coltman | Time-domain simulation of the flute[END_REF][START_REF] Terrien | Calculation of the steady-state oscillations of a flute model using the orthogonal collocation method[END_REF] (called hereafter the static case), and if studies have focused on the measurement of various control parameters [START_REF] Fletcher | Acoustical correlates of flute performance technique[END_REF][START_REF] Garcia | Acquisition and study of blowing[END_REF]12] to the authors' knowledge, no study has ever focused on the influence of the blowing pressure dynamics on the behaviour of flute-like instruments. Moreover, recent works have shown the strong influence of this parameter on oscillation thresholds of reed instruments [13, 14], and thus suggest that it could be a control parameter for musicians. In the same way, as recent studies [START_REF] Terrien | Flutelike musical instruments: A toy model investigated through numerical continuation[END_REF][START_REF] Terrien | Calculation of the steady-state oscillations of a flute model using the orthogonal collocation method[END_REF] have highlighted that the phenomenon of register change in flute-like instruments is related to a bifurcation of the system, corresponding to a loss of stability of a periodic solution branch, it suggests to consider the results of the theory of dynamic bifurcations [15]. This theory takes into account time evolution of the bifurcation parameters. This paper focuses on the influence of the dynamics of linearly increasing and decreasing ramps of the blowing pressure on the regime change thresholds between the two first registers of flute-like instruments. In section 2, the stateof-the-art physical model for flutes is briefly presented, as well as the instrument used for experiments, and the numerical and experimental tools involved in this study. Experimental and numerical results are presented in section 3, highlighting the strong influence of the slope of a linear ramp of the blowing pressure on the thresholds. Finally, a phenomenological modelling of regime change is proposed and validated in section 4, which could lead to a prediction of regime change thresholds and associated hysteresis.

Experimental and numerical tools

In this section, experimental and numerical tools used throughout the article are introduced.

Measurements on musicians

For the present study, an alto Bressan Zen-On recorder adapted for different measurements and whose geometry is described in [START_REF] Lyons | Resonance frequencies of the recorder (english flute)[END_REF] has been played by the professional recorder player Marine Sablonnière. As illustrated in figure 4, two holes were drilled to allow a measurement of both the mouth pressure P m , through a capillary tube connected to a pressure sensor Honeywell ASCX01DN, and the acoustic pressure in the resonator (under the labium), through a differential pressure sensor endevco 8507C-2.

Pressure controlled artificial mouth

Such experiments with musicians do not allow a systematic and repeatable exploration of the instrument behaviour. To play the instrument without any instrumentalist, a pressure controlled artificial mouth is used [START_REF] Ferrand | High-precision regulation of a pressure controlled artificial mouth: The case of recorderlike musical instruments[END_REF]. This setup allows to control precisely the blowing pressure, and to freeze different parameters (such as the configuration of the vocal tract or the distance between the holes and the fingers) which continuously vary when a musician is playing. As described in figure 5, a servo valve connected to compressed air controls the flow injected in the instrument through a cavity representing the mouth of the musician. Every 40 µs, the desired pressure (the target) is compared to the pressure measured in the mouth through a differential pressure sensor endevco 8507C-1. The electric current sent to the servo valve, directly controlling its opening and thus the flow injected in the mouth, is then adjusted using a Proportional Integral Derivative controller scheme. It is implemented on a DSP card dSpace 1006 [START_REF] Ferrand | High-precision regulation of a pressure controlled artificial mouth: The case of recorderlike musical instruments[END_REF].

Physical model of the instrument

In parallel of experiments, the behaviour of the state-of-the-art model for flute-like instruments is studied through time-domain simulations and numerical continuation, and qualitatively compared below to experimental results, giving rise to a better understanding of the different phenomena involved.

As for other wind instruments, the mechanism of sound production in flute-like instruments can be described as a coupling between a nonlinear exciter and a linear, passive resonator, the later being constituted by the air column contained in the pipe [START_REF] H Von | On the sensation of tones[END_REF][START_REF] Mcintyre | On the oscillations of musical instruments[END_REF]. However, they differ from other wind instruments in the nature of their exciter: whereas it involves the vibration of a solid element for reed and brass instruments (a cane reed or the musician's lips), it is constituted here by the nonlinear interaction between an air jet and a sharp edge called labium (see for example [START_REF] Fabre | Physical modeling of flue instruments: A review of lumped models[END_REF]), as illustrated in figure 6.

More precisely, the auto-oscillation process is modeled as follows: when the musician blows into the instrument, a jet with velocity U j and semihalf width b is created at the channel exit. As the jet is naturally unstable, any perturbation is amplified while being convected along the jet, from the channel exit to the labium. The convection velocity c v of these perturbations on the jet is related to the jet velocity itself through: c v ≈ 0.4U j [START_REF] Rayleigh | The theory of sound second edition[END_REF][START_REF] De La Cuadra | Visualization and analysis of jet oscillation under transverse acoustic perturbation[END_REF][START_REF] Nolle | Sinuous instability of a planar jet: propagation parameters and acoustic excitation[END_REF]. The duration of convection introduces a delay τ in the system, related both to the distance W between the channel exit and the labium (see figure 6) and to the convection velocity c v through: τ = W cv . Due to its instability, the jet oscillates around the labium with a deflection amplitude η(t), leading to an alternate flow injection inside and outside the instrument. These two flow sources Q in and Q out in phase opposition (separated by a small distance δ d , whose value is evaluated by Verge in [START_REF] Verge | Jet oscillations and jet drive in recorder-like instruments[END_REF]) act as a dipolar pressure source difference ∆p src (t) on the resonator [START_REF] J W Coltman | Jet drive mechanisms in edge tones and organ pipes[END_REF][START_REF] Verge | Jet oscillations and jet drive in recorder-like instruments[END_REF][START_REF] Verge | Sound production in recorder-like instruments. ii. a simulation model[END_REF], represented through its admittance Y . The acoustic velocity v ac (t) of the waves created in the resonator disrupts back the air jet at the channel exit. As described above, this perturbation is convected and amplified along the jet, toward the labium. The instability is amplified through this feedback loop, leading to selfsustained oscillations. This mechanism of sound production can be represented by a feedback loop system, represented in figure 7. According to various studies describing the different physical phenomena involved ( [START_REF] Ségoufin | Experimental study of the influence of the mouth geometry on sound production in a recorder-like instrument: Windway length and chamfers[END_REF][START_REF] Rayleigh | The theory of sound second edition[END_REF][START_REF] De La Cuadra | Visualization and analysis of jet oscillation under transverse acoustic perturbation[END_REF][START_REF] Nolle | Sinuous instability of a planar jet: propagation parameters and acoustic excitation[END_REF][START_REF] De | The sound of oscillating air jets: Physics, modeling and simulation in flute-like instruments[END_REF] for the jet, [START_REF] J W Coltman | Jet drive mechanisms in edge tones and organ pipes[END_REF][START_REF] Verge | Jet oscillations and jet drive in recorder-like instruments[END_REF][START_REF] Verge | Sound production in recorder-like instruments. ii. a simulation model[END_REF] for the aeroacoustic source), the state-of-the-art model for flute-like instruments [START_REF] Fabre | Physical modeling of flue instruments: A review of lumped models[END_REF] is described through system 1, in which each equation is related to a given element of the feedback loop system of figure 7: Figure 7: Basic modeling of sound production mechanism in flute-like instruments, as a system with a feedback loop [START_REF] Chaigne | Acoustique des instruments de musique (Acoustics of musical instruments)[END_REF][START_REF] Fabre | Physical modeling of flue instruments: A review of lumped models[END_REF].

η(t) = h U j e α i W v ac (t -τ ) ∆p(t) =∆p src (t) + ∆p los (t) = ρδ d bU j W d dt tanh η(t) -y 0 b - ρ 2 v ac (t) α vc 2 sgn(v ac (t)) V ac (ω) = Y (ω) • P (ω) = a 0 b 0 jω + c 0 + p-1 k=1 a k jω ω 2 k -ω 2 + jω ω k Q k • P (ω) (1)
In these equations, α i is an empirical coefficient characterizing the amplification of the jet perturbations [START_REF] Rayleigh | The theory of sound second edition[END_REF][START_REF] De | The sound of oscillating air jets: Physics, modeling and simulation in flute-like instruments[END_REF], ρ is the air density, and y 0 the offset between the labium position and the jet centerline (see figure 6). V ac and P are respectively the frequency-domain expressions of the acoustic velocity at the pipe inlet v ac (t) and the pressure source δp(t).

In the second equation, the additional term

∆p los = -ρ 2 vac(t) αvc 2
sgn(v ac (t)) models nonlinear losses due to vortex shedding at the labium [START_REF] Fabre | Vortex shedding in steady oscillation of a flue organ pipe[END_REF]. α vc is a vena contracta factor (estimated at 0.6 in the case of a sharp edge), and sgn represents the sign function.

The admittance Y (ω) is represented in the frequency-domain as a sum of resonance modes, including a mode at zero frequency (the so-called uniform mode [START_REF] Chaigne | Acoustique des instruments de musique (Acoustics of musical instruments)[END_REF]). The coefficients a k , ω k and Q k are respectively the modal amplitude, the resonance pulsation and the quality factor of the k th resonance mode, ω is the pulsation, and a 0 , b 0 and c 0 are the coefficients of the uniform mode. For the different fingerings of the recorder used for experiments, these coefficients are estimated through a fit of the admittance. These admittances are estimated through the measure of the geometrical dimensions of the bore of the recorder and the use of the software WIAT [START_REF] Lefebvre | The wind instrument acoustic toolkit[END_REF]. The length corrections related to the excitation window of the recorder (see figure 6) are subsequently taken into account using the analytical formulas detailed in chapter 7 of [START_REF] Chaigne | Acoustique des instruments de musique (Acoustics of musical instruments)[END_REF].

Numerical resolution methods

Time-domain simulations of this model are carried out through a classical Runge-Kutta method of order 3, implemented in Simulink [START_REF] Bogacki | A 3(2) pair of Runge-Kutta formulas[END_REF]. A high sampling frequency f s = 23 × 44100 Hz is used. This value is chosen both because the solution is not significantly different for higher sampling frequencies, and because it allows an easy resampling at a frequency suitable for audio production systems.

In parallel, equilibrium and periodic steadystate solutions of the model are computed using orthogonal collocation (see for example [START_REF] Engelborghs | Collocation methods for the computation of periodic solutions of delay differential equations[END_REF]) and numerical continuation [START_REF] Krauskopf | Numerical continuation methods for dynamical systems[END_REF]. Starting from a given equilibrium or periodic solution, continuation methods, which rely on the implicit function theorem [START_REF] Doedel | Lecture notes on numerical analysis of nonlinear equations[END_REF], compute the neighbouring solution, i.e the solution for a slightly different value of the parameter of interest (the so-called continuation parameter), using a prediction-correction method. This iterative process is schematically represented in figure 8. It thus aims at following the corresponding branch (that is to say "family") of solutions when the continuation parameter varies. For more details on these methods and their adaptation to the state-of-the-art flute model, the reader is referred to [START_REF] Barton | Collocation schemes for periodic solutions of neutral delay differential equations[END_REF][START_REF] Barton | Bifurcation analysis tools for neutral delay equations: a case study[END_REF] and [START_REF] Terrien | Calculation of the steady-state oscillations of a flute model using the orthogonal collocation method[END_REF]. The stability properties of the different parts of the branches are subsequently determined using the Floquet theory (see for example [START_REF] Nayfeh | Applied Nonlinear Dynamics[END_REF]).

For a given dynamical system, the computation of both the different branches of equilibrium and periodic solutions and their stability, here achieved with the software DDE-Biftool [START_REF] Engelborghs | DDE Biftool: a Matlab package for bifurcation analysis of delay differential equations[END_REF][START_REF] Engelborghs | Numerical bifurcation analysis of delay differential equations using DDE-Biftool[END_REF][START_REF] Barton | Bifurcation analysis tools for neutral delay equations: a case study[END_REF], leads to bifurcation diagrams. Such diagrams ideally represent all the branches of equilibrium and periodic solutions as a function of the continuation parameter, and provide ac- Schematic representation of the principle of numerical continuation through a prediction-correction algorithm [START_REF] Krauskopf | Numerical continuation methods for dynamical systems[END_REF][START_REF] Engelborghs | DDE Biftool: a Matlab package for bifurcation analysis of delay differential equations[END_REF]. Starting from a known part of the branch, the neighbouring solution (for a slightly different value of the continuation parameter λ) is predicted and corrected. By succesive iterations, it leads to the computation of the complete solution branch of equilibrium or periodic solutions. x represents a characteristic of the solution, such as its frequency or its amplitude. cess to specific information that are not possible to access experimentally or in time-domain simulations: unstable parts of the branches, coexistence of different solutions, and bifurcations arising along the different branches. Thereby, a bifurcation diagram provides a more global knowledge of the system dynamics and an easier interpretation of different phenomena observed experimentally and in time-domain simulations, as illustrated for example in [START_REF] Karkar | Oscillation threshold of a clarinet model: a numerical continuation approach[END_REF][START_REF] Karkar | Toward the systematic investigation of periodic solutions in single reed woodwind instruments[END_REF][START_REF] Terrien | Calculation of the steady-state oscillations of a flute model using the orthogonal collocation method[END_REF]]. This will be illustrated by figure 11 provided in section 3, which represents such a diagram of the state-ofthe-art model of flute-like instruments, in terms of oscillation frequency of the periodic solutions with respect to the blowing pressure.

3 Linear ramps of the blowing pressure: experimental and numerical results

Influence of the slope of blowing pressure ramps on thresholds

As highlighted in section 1, important differences arise, in terms of regime change thresholds and hysteresis, between experienced flutist and artificial mouth or non musician, which remain unexplained. Recent works [13, 14] have demon-strated the strong influence of the dynamics of control parameters on the oscillation threshold of reed instruments. Particularly, it has highlighted, in such instruments, the phenomenon of bifurcation delay, corresponding to a shift of the oscillation threshold caused by the dynamics of the control parameter [15]. Although we focus here on transitions between the two first registers (i.e. between two different oscillation regimes), and although flute-like instruments are mathematically quite different dynamical systems from reed instruments, these former studies suggest that the temporal profile P m (t) of the pressure dynamics could considerabily influence the regime change thresholds. We focus in this section on the comparison of regime change thresholds between the static case and the dynamic case, the latter corresponding a time varying blowing pressure.

To test this hypothesis, linearly increasing and decreasing blowing pressure ramps P m (t) = P ini + a • t, with different slopes a, have been run both through time-domain simulations and experiments with the artificial mouth (the reader is referred to appendix A for a table of notations). Figure 9 represents, for the F 4 fingering, the dynamic pressure thresholds P dyn corresponding to the jump between the two first registers, with respect to the slope a. The positive and negative values of a correspond to increasing and decreasing ramps of P m (t) respectively. For each value of a, the experimental threshold is a mean value calculated for three realisations. In this paper, the value of P dyn is determined through a fundamental frequency detection using the software Yin [START_REF] De | Yin, a fundamental frequency estimator for speech and music[END_REF]: P dyn is defined as the value of P m at which a jump of the fundamental frequency is observed. The temporal resolution of the detection is 0.0016 s for experimental signals and 0.0007 s for simulation signals, which corresponds to a resolution of 0.8 and 0.36 P a (respectively), in the case of a slope a = 500 P a/s of the blowing pressure. Despite the dramatic simplifications of the model, these first results higlight that the real instrument and the model present similar qualitative behaviours. Surprisingly enough, the experimental and numerical behaviours are also quantitatively similar, with typical relative differences between 3 % and 28 % on the thresholds observed for rising pressure (called increasing pressure threshold P dyn 1→2 ). For the decreasing pressure threshold P dyn 2→1 , observed for diminishing pressure, the difference is more important, with a typical relative deviation of about 50 %. Moreover, the strong influence of a on both P dyn 1→2 and P dyn 2→1 is clearly pointed out: with the artificial mouth, a = 400 Pa/s leads to a value of P dyn 1→2 45% higher than a = 10 Pa/s, and to a value of P dyn 2→1 16% lower. Similarly, for time-domain simulations, a = 400 Pa/s leads to a value of P dyn 1→2 15.5% higher and to a value of P dyn 2→1 18% lower than a = 10 Pa/s. Increasing a thus enlarges the hysteresis; indeed P dyn 1→2 and P dyn 2→1 are respectively increased and decreased. This can be compared (at least qualitatively) with phenomena observed on an experienced flutist, presented in section 1.

Figure 10 represents, as previously, the mean value of the regime change thresholds P dyn 1→2 and P dyn 2→1 obtained for three experiments, with respect to the slope a, for the other fingerings already studied in section 1. It higlights that the behaviour observed in figure 9 for the F 4 fingering looks similar for other fingerings of the recorder. Indeed, depending on the fingering, the increase of a from 20 Pa/s to 400 Pa/s leads to an increase of P dyn 1→2 between 13 % and 43 % and to a decrease of P dyn 2→1 from 3 % to 15 %. Again, these results can be qualitatively compared with the results presented in section 1 for an experienced musician.

Influence of the slope of blowing pressure ramps on oscillation frequency and amplitude

As observed for the oscillation threshold in clarinet-like instruments [13], we show in this section that a modification of the regime change threshold does not imply a strong modification of the characteristic curves, observed in the static case, linking the oscillation amplitude and the oscillation frequency to the blowing pressure. For numerical results, this feature can be easily illustrated through a comparison between the results of time-domain simulations and the bifurcation diagrams obtained through numerical continuation. This is done in figure 11, in terms of frequency with respect to the blowing pressure P m , for modal coefficients corresponding to the G 4 fingering. In this figure, the two periodic solution branches correspond to the first and the second registers, and solid and dashed lines represent stable and unstable parts of the branches, respectively. As the computation of such a bifurcation diagram relies on the static bifurcation theory, the point where the first register becomes unstable, at P m = 311.5P a, corresponds to the static threshold P stat 1→2 from first to second register. It thus corresponds to the threshold that would be observed by choosing successive constant values of the blowing pressure, and letting the system reach a steady-state solution (here, the first or the second register). In the same way, the point at which the change of stability of the second register is observed corresponds to the static threshold from second to first register P stat 2→1 = 259P a. Figure 11 shows that for high values of a, the system follows the unstable part of the branch corresponding to the first register: the maximum relative difference between the frequency predicted by the bifurcation diagram and the results of time-domain simulations is 9 cents. In the dynamic case, the system thus remains on the periodic solution branch corresponding to the "starting" regime (the first register in figure 11), after it became unstable. Providing, for the A fingering, the oscillation amplitude as a function of P m for different values of a, figure 12 highlights that the same property is observed experimentally. In both cases, the value of a considerabily affects the register change thresholds. However, far enough from the jump between the two registers, the oscillation amplitude only depends on the value of P m , and does not appear significantly affected by the value of a.

In figure 12, the comparison between the two slowest ramps (20 Pa/s and 100 Pa/s) and the two others is particularly interesting. Indeed, for the two slowest ramps, an additional oscillation regime, corresponding to a quasiperiodic sound (called multiphonic sound in a musical context) [START_REF] Fletcher | Sound production by organ flue pipes[END_REF][START_REF] J W Coltman | Jet offset, harmonic content, and warble in the flute[END_REF][START_REF] Dalmont | Some aspects of tuning and clean intonation in reed instruments[END_REF][START_REF] Terrien | Flutelike musical instruments: A toy model investigated through numerical continuation[END_REF][START_REF] Terrien | Is the jet-drive flute model able to produce modulated sounds like flautas de chinos[END_REF], is observed for blowing pressure between 300 Pa and 400 Pa for a = 20 Pa/s, and between 340 and 400 Pa for a = 100 Pa/s. As this regime does not appear for higher slopes, it highlights that a modification of the blowing pressure dynamics can allow the musician to avoid (or conversely to favor) a given oscillation regime.

Influence of the pressure dynamics before the static threshold

To better understand the mechanisms involved in the case of a dynamic bifurcation between two registers, this section focuses on the influence, on the regime change thresholds, of the evolution of P m (t) before the static threshold P stat has been reached. In other words, the aim is to determine whether the way P m (t) evolves before the static threshold is reached impacts the dynamic regime change threshold.

To investigate this issue, different piecewise lin- ear ramps have been achieved both with the artificial mouth and in time-domain simulation. For rising pressures, these profiles are defined such as dPm dt = a 1 for P m < P knee and dPm dt = a 2 for P m > P knee (where a 1 and a 2 are constants) and P knee is a constant that may be adjusted. For diminishing pressure, they are such as dPm dt = a 1 for P m > P knee and dPm dt = a 2 for P m < P knee .

Experimental results

Experimentally, blowing pressure profiles constituted by two linear ramps with different slopes (a 1 = 350 Pa/s, a 2 = 40 Pa/s) have been achieved for the G 4 fingering. The pressure P knee at which the knee break occurs varies between the different realisations.

Figure 13 presents these experimental results in terms of P dyn 1→2 and P dyn 2→1 , with respect to P knee -P stat . Thereby, a zero abscissa corresponds to a change of slope from a 1 to a 2 at a pressure equal to P stat 1→2 for rising pressure, and equal to P stat 2→1 for diminishing pressure. It highlights that for rising pressure, P dyn 1→2 remains constant as long as P knee < P stat 1→2 (i.e. for negative values of the abscissa), and that this constant value (about 258 Pa) corresponds to the value of P dyn 1→2 previously observed for a linear ramp with constant slope a 2 = 40 Pa/s (see figure 10). Conversely, once P knee > P stat 1→2 , the value of P dyn 1→2 gradually increases to reach 295 Pa, which corresponds to the value observed for a linear ramp with a contant slope a 1 = 350 Pa/s. The same behaviour is observed for the decreasing threshold: as long as P knee > P stat 2→1 , the value of P dyn 2→1 is almost contant and close to that observed previously for a linear ramp of constant slope a 2 = 40 Pa/s (see figure 10). However, for P knee < P stat , the value of P dyn 2→1 progressively decreases to about 142 Pa, which corresponds to that observed for a linear ramp of constant slope a 1 = 350 Pa/s (see figure 10).

As a conclusion, as long as the slope break occurs before the static threshold has been reached, the dynamic threshold is driven by the slope of the second part of the blowing pressure profile. If it occurs just after the static threshold has been reached, the dynamic threshold lies between the dynamic thresholds corresponding to the two slopes of the blowing pressure profile. Finally, if the slope break occurs, for rising pressure, at a presure sufficiently higher (respectively lower for diminishing pressure) than the static threshold, the dynamic threshold is driven, as expected, by the slope of the first part of the blowing pressure Abscissa: difference between the pressure P knee at which the knee occurs and the static regime change threshold P stat . Dashed lines represent the dynamic regime change thresholds observed previously for linear ramps of constant slope a 1 and a 2 respectively. profile.

Results of time-domain simulations

These experimentally observed behaviours are also observed on numerical simulations of the model. For modal coefficients corresponding to the G 4 fingering, the comparison has been made between the dynamic thresholds obtained for three different cases:

• linear increasing ramps of P m (t), with slope a 2 .

• a first piecewise linear increasing ramp, with a slope change at P knee = 250P a, and a fixed value of a 1 = 500 Pa/s.

• a second piecewise linear increasing ramp, with a slope change at P knee = 250P a, and a fixed value of a 1 = 200 Pa/s.

It is worth noting that for the two kinds of piecewise linear ramps, P knee is lower than P stat 1→2 , predicted by a bifurcation diagram at 311.5 Pa (see figure 11). For each case, various simulations were achieved, for different values of a 2 . Figure 14 provides the comparison of value of P dyn 1→2 obtained for these three kinds of blowing pressure profiles as a function of a 2 . With a maximum relative difference of 3.5%, the thresholds obtained for the piecewise linear profiles are strongly similar to those obtained with linear ramps. As for the experimental results, if P knee < P stat 1→2 , the dynamic threshold P dyn 1→2 is thus driven by the second slope a 2 of the profile.

For the particular profile in which a 1 = 500 Pa/s and a 2 = 830 Pa/s, the influence of the value of P knee on P dyn 1→2 has been studied. The results are represented in figure 15 in the same way as the experimental results in figure 13. As experimentally, if P knee < P stat 1→2 , the value of P dyn 1→2 is driven by a 2 , and a constant threshold of about 385 Pa is observed, corresponding to the value obtained for a linear ramp with a slope equal to a 2 = 830 Pa (see figure 14). Conversely, when P knee > P stat1→2 , P dyn1→2 gradually shifts to finally achieve the value of 369 Pa, equal to that oberved for a linear ramp with a slope equal to a 1 = 500 Pa/s. The dynamic threshold is then driven by a 1 . Ordinate: dynamic threshold P dyn 1→2 . Abscissa: difference between the pressure P knee at which the knee occurs and the static regime change threshold P stat 1→2 . Dashed lines represent the values of P dyn 1→2 previously observed for linear ramps of slope a 1 and a 2 .

Comparison with the results of an experienced musician

The strong influence of the dynamics of P m (t) on thresholds and hysteresis suggests, by comparison with results presented in section 1, that musicians use this property to access to a wider control in terms of nuances and timbre. However, the comparison between the musician and the artificial mouth (see figures 2, 3 and 10) shows that the values of P dyn 1→2 obtained by the musician remains, for the different fingerings studied, between 61 % and 134 % higher than the maximal thresholds obtained with the artificial mouth for high values of the slope a. In the same way, the hysteresis obtained by the musician remains between 26% and 102% wider than the maximal hysteresis observed with the artifical mouth for the F 4 , G 4 , A 4 and B b 4 and up to 404 % wider for the B 4 fingering.

Discussion

These results bring out the strong influence of the dynamics of the blowing pressure on the oscillation regime thresholds in flute-like instruments. Comparisons between experimental and numerical results show that the substantial simplifi-cations involved in the state-of-the-art physical model of the instrument do not prevent it to faithfully reproduce the phenomena observed experimentally. Suprisingly enough, different results show good agreement not only qualitatively but also quantitatively. Moreover, both the experimental and numerical results show that the dynamic threshold does not depend on the dynamics of the blowing pressure before the static threshold has been reached.

Although the system studied here is mathematically very different from one that models reed instruments (see for example [START_REF] Auvray | Regime change and oscillation thresholds in recorderlike instruments[END_REF][START_REF] Terrien | Numerical resolution of a physical model of flute-like instruments: comparison between different approaches[END_REF][START_REF] Terrien | Calculation of the steady-state oscillations of a flute model using the orthogonal collocation method[END_REF]), and although focus is set here on bifurcations of periodic solutions, results can be compared with some phenomena highlighted by Bergeot et al. on the dynamic oscillation threshold of reed instruments [14,13]. As in the work of Bergeot, phenomena highlihted are not predicted by the static bifurcation theory, often involved in the study of musical instruments.

Moreover, the comparison between the results obtained with the artificial mouth and with an experienced flutist suggests that the musicians combine the dynamics of the blowing pressure with other control parameters in order to enlarge the hysteresis associated to regime change. Indeed, other works on flute-like instruments [START_REF] J W Coltman | Mouth resonance effects in the flute[END_REF][START_REF] Auvray | Influence of the fluctuations of the control pressure on the sound production in flute-like instruments[END_REF], together with different studies on other wind instruments [START_REF] Fréour | Vocal-tract influence in trombone performance[END_REF][START_REF] G P Scavone | Measurement of vocal-tract influence during saxophone performance[END_REF][START_REF] J M Chen | Pitch bending and glissandi on the clarinet: roles of the vocal tract and partial tone hole closure[END_REF] suggests that the vocal tract can also influence the regime change thresholds.

Toward a phenomenological model of register change

The different properties of the register change phenomenon, observed both experimentally and in simulations in the previous part, allow to propose a preliminary phenomenological modelling of this phenomenon.

Proposed model

Starting from the results presented in figures 13, 14, and 15, which lead to the conclusion that P dyn only depends on the dynamics of the blowing pressure after the static threshold has been reached, this modelling is based on the following hypothesis:

• The regime change starts when P m (t) = P stat .

• The regime change is not instantaneous, and has a duration t dyn during which the blowing pressure evolves from P stat to P dyn .

We thus write P dyn as the sum of the static threshold P stat and a correction term P corr related to the dynamics of the blowing pressure:

P dyn = P stat + P corr . (2)
Based on the two hypothesis cited above, we introduce a new dimensionless quantity, the fraction of regime change ζ(t). By definition, ζ = 0 when the regime change has not started (i.e. when P m (t) < P stat for rising pressure and when P m (t) > P stat for diminishing pressure), and ζ = 1 when the regime change is completed (i.e. when P m (t) = P dyn , which corresponds to the change of fundamental frequency, as defined in the previous section). ζ is consequently defined as:

ζ(t) = t tstat ∂ζ ∂t dt (3)
where t stat is the instant at which P m (t) = P stat . Defining the origin of time at t stat leads to t = -t stat , and thus gives:

ζ( t) = t 0 ∂ζ ∂ t d t (4)
As a simplifiying assumption, we consider that the rate of change ∂ζ ∂ t of the variable ζ( t) only depends on the gap ∆P ( t) = P m ( t) -P stat between the mouth pressure P m ( t) and the static regime change threshold:

∂ζ ∂ t = f (∆P ), (5)
where f is an unknown monotonous and continuous function.

According to the latest hypothesis, function f can be estimated at different points through the realisation of "steps" profiles of P m (t), from a value lower than P stat 1→2 , to a value larger than P stat 1→2 (see figure 16). Indeed, in such a case, for a step occuring at t = 0, ∆P ( t) corresponds to the difference between the pressure at the top of the step and P stat 1→2 , and is thus constant for t > 0. Consequently, f (∆P ) is constant with respect to time. From equations 4 and 5, one thus obtains for blowing pressure steps:

ζ( t) = t 0 f (∆P )d t =f (∆P ) t 0 d t =f (∆P ) • t (6)
Recalling that t dyn is the instant at which P m ( t) = P dyn , we have by definition ζ( t dyn ) = 1, and finally obtain for blowing pressure steps:

f (∆P ) = 1 t dyn (7)
For each value of the step amplitude, a different value of t dyn is obviously measured through a frequency detection: t dyn is defined as the time after which the oscillation frequency varies no more than two times the frequency resolution. Therefore, successive time-domain simulations of P m steps (see figure 16) with different amplitudes are carried out, to determine the function f (∆P ) through equation 7. Such simulations have been achieved for the two fingerings F 4 and G 4 , in both cases for transitions from the first to the second register. The results are represented in figure 17 with respect to ∆P . fingering and 0.97 for the G 4 fingering. Such results thus suggest to approximate the function f through:

f (∆P ) = α (∆P ); (8)
where the coefficient α depends on the considered fingering.

Assessment of the model

To check the validity of this modelling, the case of the linear pressure ramps studied in the previous section is now examined. In such a case, the difference between the blowing pressure and the static threshold is defined through ∆P ( t) = a • t, where a is the slope of the ramp in Pa/s. Recalling that ζ( t dyn ) = 1 and injecting equations 5 and 8 in equation 4 leads to:

t dyn 0 f (∆P ( t))d t = 1 t dyn 0 α ∆P ( t)d t = 1 t dyn 0 α a t • d t = 1 t dyn 0 td t = 1 α √ a t dyn = 3 2α √ a 2 3 (9)
Moreover, due to the expression of ∆P ( t) in the case of linear ramps, one can write from equations 2 and 9:

P corr =P dyn -P stat =∆P ( t dyn ) =a • t dyn = 3 2α a 2 3 (10)
According to this modelling, the value of P corr obtained with linear ramps should be proportional to the slope a to the power 2/3. Timedomain simulations for linear ramps of P m (t) with slope a are performed for two fingerings (F 4 and G 4 ). Figure 18 represents the threshold P dyn corresponding to the end of the transition from the first to the second register with respect to the slope a power 2/3. The results are correctly fit- ted by straight lines, with correlation coefficients higher than 0.99. This good agreement with the model prediction (equation 10) thus allows to validate the proposed modelling of the phenomenon of regime change. Moreover, on such a representation, the intercept of the fit with the y-axis provides a prediction of the static regime change threshold, which can not be exactly determined, strictly speaking, with linear ramps of the blowing pressure. The static thresholds thereby obtained are 294 Pa and 314 Pa for the F 4 and G 4 fingering respectively. These values present relative differences of 0.1% and 0.8% with the thresholds of 294.3 Pa and 311.5 Pa predicted by the bifurcation diagrams computed through numerical continuation (see figure 11 for the bifurcation diagram of the G 4 fingering), which supports the validity of the proposed modelisation.

Case of experimental data

The experimental thresholds displayed in figure 10 for the five fingerings studied are represented in figure 19 with respect to a 2/3 . Similarly to figure 18, the different curves are correctly fitted by straight lines, with linear correlation coefficients between 0.88 and 0.99. The fact that these coefficients are, in some cases, lower than those of simulations can be explained by the presence of noise and of small fluctuations of the mouth pressure during the experiment, sometimes prevents a threshold detection as accurate and systematic as in the case of numerical results. However, the good agreement of the experimental results with equation 10 also allows to validate the proposed phenomenological modelling of regime change.

Influence of the regime of arrival

In the case of time-domain simulations, for the G 4 fingering, starting from the second register and achieving linear decreasing ramps of P m (t) leads to a particular behaviour. As shown in figure 20, P dyn does not appear, at least in a first stage, to be proportional to the power 2/3 of the slope. However, this case is particular in the sense that different oscillation regimes are reached, depending on the slope a of the ramp. Thereby, as highlighted with circles in figure 20, low values of the slope (|a| < 20 Pa/s) lead to a transition from the second to the first register, whereas higher values of the slope lead to a transition from the second register to an aeolian regime, as represented with crosses in figure 20. In flute-like instruments, aeolian regimes corresponds to particular sounds, occuring at low values of the blowing pressure, and originating from the coupling between a mode of the resonator (here the 5 th ) and an hydrodynamic mode of the jet of order higher than 1 [START_REF] Fletcher | Sound production by organ flue pipes[END_REF][START_REF] Meissner | Aerodynamically excited acoustic oscillations in cavity resonator exposed to an air jet[END_REF][START_REF] Terrien | Flutelike musical instruments: A toy model investigated through numerical continuation[END_REF].

As highlighted in the same figure, considering the two different transitions separately allows to find, as previously, the linear dependance between P dyn and a 2/3 . Indeed, linear correlation coefficients of 0.98 for |a| < 20 Pa/s, and of 0.95 for |a| > 20 Pa/s are found. Since the corresponding slope is the inverse of 2 3 α to the power 2/3 (see equation 10), such results suggest that α does not only depend on the fingering, but also on the oscillation regimes involved in the transition.

The study of the Floquet exponents ρ m of the system supports this hypothesis. The Floquet exponents, computed for the system linearised around one of its periodic solutions, allow to estimate the (local) stability properties of the considered periodic solution [START_REF] Floquet | Sur les equations differentielles lineaires[END_REF][START_REF] Nayfeh | Applied Nonlinear Dynamics[END_REF]. More precisely, they provide information on whether a small perturbation superimposed on the solution will be amplified or attenuated with time. If all the Floquet exponents have negative real parts, any perturbation will be attenuated with time, and the considered solution is thus stable. Conversely, if at least one of the Floquet exponents has a positive real part, any perturbation will be amplified in the "direction" of the phase space corresponding to the eigenvector associated to this exponent, and the solution is thus unstable.

The real part of the Floquet exponents of the considered system, linearised around the periodic solution corresponding to the second register (i.e. 20 suggests that, in the case of a regime change, the "arrival" regime is driven by the Floquet exponent of the starting regime with the highest real part: indeed, as highlighted in figures 20 and 21, until the dynamics of P m (t) induces a regime change threshold higher than the pressure P cross for which the Floquet exponents intersect, one observes a transition to the first register. On the other hand, once the dynamics of P m (t) induces a threshold lower than P cross , the transition leads to the aeolian regime.

This interpretation seems furthermore to be consistent with the slope change observed in figure 20 and with the physical meaning of the real part of the Floquet exponents. Indeed, as the value of the real part of a Floquet exponent is related to the amplification of a perturbation with time, a high value of ℜ(ρ m ) should correspond to a small duration t dyn of the regime change, whereas a small value of ℜ(ρ m ) should correspond to a high value of t dyn . Therefore, by analogy with equations 7 and 8, coefficient α can be related to the evolution of ℜ(ρ m ) with P m . Thereby, a greater evolution of ℜ(ρ m ) with respect to ∆P should correspond to a higher value of α. Due to equation 10, valuable for linear ramps of P m (t), it finally corresponds to a smaller rate of change of the straight line linking P dyn and a 2/3 . This property is here verified by the comparison between figures 20 and 21: the real part of the "second" floquet exponent (in bold black dashed line in figure 21), related to a regime change to the aeolian regime, presents a greater evolution with ∆P = P m -P stat than the floquet exponent inducing a transition to the first register (in bold blue line in figure 21). In the same way, the rate of change of straight line related in figure 20 to the regime change toward the aeolian regime (dashed line) is smaller than that of the straight line related to the transition from the second to the first register (in solid line).

Surprisingly enough, these results thus highlight that bifurcation diagrams and associated Floquet stability analysis provide valuable information in the dynamic case, despite the fact that they involve the static bifurcation theory and a linearisation of the studied system around the "starting" periodic solution. In the dynamic case, they remain instructive on the following characteristics:

• the arrival regime resulting from the regime change.

• a qualitative indication on the duration of the regime change, through an estimation of the parameter α. It thus informs on both the dynamic threshold and its evolution with respect to the difference ∆P between the mouth pressure and the static regime change threshold.

• as highlighted in the previous section, the evolution of the oscillation amplitude and frequency with respect to the mouth pressure, even after the static threshold has been crossed.

Conclusion

Recent studies in the field of musical acoustics have demonstrated that musicians are able to modify strongly the behaviour of the instrument considered alone (see for example [START_REF] Fréour | Vocal-tract influence in trombone performance[END_REF][START_REF] J M Chen | Pitch bending and glissandi on the clarinet: roles of the vocal tract and partial tone hole closure[END_REF]), and thus argue for a wider consideration of the different control parameters.

A comparison between an experienced flutist, a non musician and an artificial mouth, in terms of regime change thresholds between the two first registers, and associated hysteresis, shows that the experienced musician seems to have developed strategies allowing him to significantly shift the regime change thresholds, and thus to enlarge the hysteresis, which presents an obvious musical interest. Conversely, for most fingerings studied, the behaviour observed when the recorder is played by a non musician and by an artifiical mouth do not present significant differences in terms of regime change thresholds.

The experimental and numerical results presented in this article highlight that the slope of linear increasing and decreasing ramps of the blowing pressure strongly affects the pressure regime change thresholds, and thus the hysteresis. Moreover, it appears that the important criterion lies only in the dynamics of the blowing pressure after the static regime change threshold has been reached. The modification of the dynamics of the blowing pressure can thus allow, in some cases, to avoid or conversely to favor a given oscillation regime, and thereby to select the "arrival" regime resulting from a regime change.

The phenomenological model proposed according to these observations allows to predict the dynamic regime change threshold from the knowledge of the temporal evolution of the blowing pressure. It highlights that the bifurcation diagrams and the associated Floquet stability analysis provide valuable information in the dynamic case, despite the fact that they involve a static hypothesis and a linearisation of the studied system.

However, taking into account the dynamics of the mouth pressure does not allow to shift the thresholds and to enlarge the hysteresis as much as the experienced flutist does. It thus suggests that flutists develop strategies to combine the effects of the dynamics with those of other control parameters, such as for example the vocal tract, whose influence on regime change thresholds has been recently studied [START_REF] Auvray | Influence of the fluctuations of the control pressure on the sound production in flute-like instruments[END_REF]. Moreover, the study presented here focuses on linear profiles of the mouth pressure. As such a temporal evolution does not seem realistic in a musical context (see for example [START_REF] Fabre | Structuring music in recorder playing: a hydrodynamical analysis of blowing control parameters[END_REF][START_REF] Garcia | Acquisition and study of blowing[END_REF]), it would be interesting to consider the effect of more complex temporal evolutions of the blowing pressure. Finally, it would be interesting to study more widely step profiles of the mouth pressure, whose importance is crucial in the playing of winf instruments.

Aknowledgment

The authors would like to thank Marine Sablonnière and Etienne Thoret for their participation in experiments.

A Table of notation

Figure 1 :

 1 Figure 1: Oscillation frequency with respect to the blowing pressure, for the F 4 fingering of an alto Zen recorder, played by an experienced flutist, a non musician and an artificial mouth. Oscillations around 350 Hz and 740 Hz correspond to the first and second register, respectively.

Figure 2 :

 2 Figure 2: Increasing pressure thresholds corresponding to the jump from the first to the second register of an alto recorder played by an experienced flutist, a non musician and an artificial mouth, for five fingerings.

Figure 3 :

 3 Figure 3: Hysteresis on the jump between the two first registers of an alto recorder played by an experienced flutist, a non musician and an artificial mouth, for five fingerings.

Figure 4 :

 4 Figure 4: Experimental setup with the adapted recorder, allowing to measure both the pressure in the mouth of the flutist and the acoustic pressure under the labium.

Figure 5 :

 5 Figure 5: Schematic representation of the principle of the artificial mouth. The opening of the servo valve, controlling the flow injected in the mouth, is adapted every 40 µs in order to minimize the difference between the measured and the desired values of the pressure in the mouth.

Figure 6 :

 6 Figure 6: Schematic representation of the jet behaviour, based on Fabre in [19]. (a) Perturbation of the jet at the channel exit by the acoustic field present in the resonator. (b) Convection and amplification of the perturbation, due to the unstable nature of the jet. (c) Jet-labium interaction: oscillation of the jet around the labium, which sustains the acoustic field.

Figure 8 :

 8 Figure 8:Schematic representation of the principle of numerical continuation through a prediction-correction algorithm[START_REF] Krauskopf | Numerical continuation methods for dynamical systems[END_REF][START_REF] Engelborghs | DDE Biftool: a Matlab package for bifurcation analysis of delay differential equations[END_REF]. Starting from a known part of the branch, the neighbouring solution (for a slightly different value of the continuation parameter λ) is predicted and corrected. By succesive iterations, it leads to the computation of the complete solution branch of equilibrium or periodic solutions. x represents a characteristic of the solution, such as its frequency or its amplitude.

Figure 9 :

 9 Figure 9: Dynamic regime change threshold between the two first registers of the F 4 fingering, with respect to the slope a of linear ramps: artificial mouth and time-domain simulation.

Figure 10 :

 10 Figure 10: Transition between the two first registers of an alto recorder played by an artificial mouth, for five different fingerings: representation of the dynamic regime change thresholds with respect to the slope a of linear ramps of the blowing pressure.

Figure 11 :Figure 12 :

 1112 Figure 11: Bifurcation diagram of the G fingering, superimposed with time-domain simulations of increasing linear ramps of the blowing pressure, for different values of the slope a: representation of the oscillation frequency with respect to the blowing pressure P m . For the bifurcation diagram, the two branches correspond to the first and the second register, solid and dashed lines represent stable and unstable parts of the branches, respectively. The vertical dotted lines highlight the static regime change thresholds P stat 1→2 and P stat 2→1

Figure 13 :

 13 Figure 13: Piecewise linear ramps of the blowing pressure (a 1 = 350 Pa/s and a 2 = 40 Pa/s), achieved on the G 4 fingering of an alto recorder played by an atificial mouth. Ordinate: dynamic threshold P dyn 1→2 (up) and P dyn 2→1 (down).Abscissa: difference between the pressure P knee at which the knee occurs and the static regime change threshold P stat . Dashed lines represent the dynamic regime change thresholds observed previously for linear ramps of constant slope a 1 and a 2 respectively.

Figure 14 :

 14 Figure14: Time-domain simulations of piecewise linear ramps of the blowing pressure with P knee = 270P a (a 1 = 500 Pa/s for squares and a 1 = 200 Pa/s for circles) and of linear ramps of the blowing pressure (crosses). Representation of the increasing dynamic regime change threshold P dyn 1→2 for the G 4 fingering, as a function of a 2 (slope of the second part of the blowing pressure profile for piecewise linear ramps, and slope of the linear ramps).

Figure 15 :

 15 Figure 15: Time-domain simulations of piecewise linear ramps of the blowing pressure (a 1 = 500 Pa/s and a 2 = 830 Pa/s), for the G 4 fingering.Ordinate: dynamic threshold P dyn 1→2 . Abscissa: difference between the pressure P knee at which the knee occurs and the static regime change threshold P stat 1→2 . Dashed lines represent the values of P dyn 1→2 previously observed for linear ramps of slope a 1 and a 2 .

Figure 16 :Figure 17 :

 1617 Figure 16: Illustration of the step profiles of the blowing pressure (up) achieved in time-domain simulations, and of the detection of the transient duration t dyn (down).

Figure 18 :

 18 Figure18: Time-domain simulations of linear increasing ramps of the blowing pressure, for both the F 4 fingering (+) and the G 4 fingering (x): representation of the dynamic regime change threshold P dyn 1→2 with respect to the power 2/3 of the slope a. Solid and dashed lines represent linear fit of the data, which both present linear correlation coefficients higher than 0.99.

Figure 19 :

 19 Figure19: Same data as in figure10: representation of the dynamic thresholds P dyn 1→2 and P dyn 2→1 , for five fingerings of an alto recorder played by an artificial mouth, with respect to the power 2/3 of the slope a of linear ramps of the blowing pressure. Solid lines represent linear fit of the data. Data present linear correlation coefficients between 0.88 and 0.99.

Figure 21 :

 21 Figure21: G 4 fingering: real parts of the Floquet exponents of the system linearised around the periodic solution corresponding to the second register, with respect to the blowing pressure P m . Floquet exponents provide information on the stability properties of the considered regime.

  pressure profiles in recorder playing. In Proceedings of the International Conference on New interfaces for Musical Expression, Oslo, Norway, 2011.[12] P de la Cuadra, B Fabre, N Montgermont, and L de Ryck. Analysis of flute control parameters: A comparison between a novice and an experienced flautist. In Forum Acusticum, Budapest, 2005.[13] B Bergeot, A Almeida, Ch Vergez, and B Gazengel. Prediction of the dynamic oscillation threshold in a clarinet model with a linearly increasing blowing pressure. Nonlinear Dynamics, 73(1-2):521-534, 2013. [14] B Bergeot, A Almeida, B Gazengel, Ch Vergez, and D Ferrand. Response of an artificially blown clarinet to different blowing pressure profiles. The Journal of the Acoustical Society of America, 135(1):479-490, 2014.


Denis Gapais
email: denis.gapais@univ-rennes1.fr

Justine Jaguin

Florence Cagnard

Philippe Boulvais

POP-DOWN TECTONICS, FLUID CHANNELLING AND ORE DEPOSITS WITHIN ANCIENT HOT OROGENS

Keywords: Tectonics, crustal shortening, weak lithosphere, fluid transfers, ore deposits, ancient cratons

Many Archaean and Paleoproterozoic deformation zones, often rich in ore resources, show particular structural patterns in particular marked by regional vertical stretch. These zones are not restricted to greenstone-bearing Archaean domains that may have suffered gravity-driven sagduction of heavy supra-crustals, as extensively discussed since the last twenties. Structures are actually best explained by pop-down tectonics of upper-crustal units within an underlying weak crust submitted to horizontal regional shortening. Here we present three complementary examples from two Archaean greenstone belts (Abitibi sub-Province, Quebec, and Murchison belt, South Africa) and one greenstone-lacking Paleoproterozoic belt (Thompson belt, Manitoba). In the three examples, ore is concentrated along steeply dipping deformation zones, rich in syntectonic deposits and marked by substantial sub-vertical crustal stretch. On the other hand, the three regions show differences in age, in metamorphic grade (from sub-greenschist facies to upper amphibolite facies), in metal contents (gold, antimony, nickel), in metal sources, transfers and concentration histories. Our analysis emphasizes that pop-down tectonics associated with horizontal shortening of weak lithospheres may account for observed structural patterns and provides a new and promising frame for the analysis of relationships between structural patterns and ore concentrations within old cratons.

Introduction

Tectonics that structured Precambrian cratons is a major ongoing debate in Earth Sciences. This relates to academic reasons linked to long-lasting and still debated controversies about the modes of lithospheric deformations through geological times [START_REF] Choukroune | Soft lithospheres during periods of Archaean crustal growth or crustal reworking[END_REF][START_REF] Burg | Orogeny through time: an overview[END_REF]Windley, 1997;[START_REF] Hamilton | An alternative Earth[END_REF][START_REF] Chardon | Flow of ultra-hot orogens: A view from the Precambrian, clues for the Phanerozoic[END_REF][START_REF] Gapais | Mountain building and exhumation processes through time: inferences from nature and models[END_REF][START_REF] Ganne | Modern-style plate subduction preserved in the Palaeoproterozoic West African craton[END_REF], and to economic ones, Precambrian cratons being particularly rich in ore concentrations.

Archaean and Paleoproterozoic cratons have remarkable structural specificities [START_REF] Choukroune | Soft lithospheres during periods of Archaean crustal growth or crustal reworking[END_REF][START_REF] Chardon | Flow of ultra-hot orogens: A view from the Precambrian, clues for the Phanerozoic[END_REF][START_REF] Gapais | Mountain building and exhumation processes through time: inferences from nature and models[END_REF]. In particular, strains marked by steeply dipping fabrics often bearing steeply plunging stretching lineations are widespread, from low grade upper crust to deeper granulitic and partial melting conditions [START_REF] Hudleston | Transpression in an Archean greenstone belt, northern Minesota[END_REF][START_REF] Fueten | Structural petrology along a transect across the Thompson Nickel Belt, Manitoba: dip-slip at the western Churchill-Superior boundary[END_REF][START_REF] Goscombe | Intense non-coaxial shear and the development of mega-scale sheath folds in the Arunta block, Central Australia[END_REF][START_REF] Chown | Tectonic evolution of the northern Volcanic Zone, Abitibi belt, Quebec[END_REF][START_REF] Bouhallier | Strain patterns in Archaean dome and basin structures. the Dharwar Craton (Karnakata, South India)[END_REF][START_REF] Choukroune | Soft lithospheres during periods of Archaean crustal growth or crustal reworking[END_REF][START_REF] Lonka | Ductile deformation of tonalite in the Suomusjarvi shear zone, southwestern Finland[END_REF][START_REF] Chardon | Archean granite-greenstone tectonics at Kolar (South India): Interplay of diapirism and bulk inhomogeneous contraction during juvenile magmatic accretion[END_REF][START_REF] Chardon | Precambrian continental strain and shear zone patterns: south Indian case[END_REF][START_REF] Chardon | Flow of ultra-hot orogens: A view from the Precambrian, clues for the Phanerozoic[END_REF]Cagnard et al., 2006a[START_REF] Cagnard | Collision tectonics involving juvenile crust: the example of the southern Svecofennides[END_REF][START_REF] Gapais | Paleoproterozoic tectonics in the Terre Adélie Craton (East Antartica)[END_REF][START_REF] Jaguin | The Murchison Greenstone Belt (South Africa): a general tectonic Framework[END_REF]. These regions contain deformation bands that show large strains despite lacks of major metamorphic jumps [START_REF] Chown | Tectonic evolution of the northern Volcanic Zone, Abitibi belt, Quebec[END_REF][START_REF] Bleeker | New structural-metamorphic constraints on early Proterozoic oblique collision along the Thompson Nickel Belt, Manitoba, Canada[END_REF][START_REF] Vearncombe | Geology, geophysics and mineralization of the Murchison Schist Belt, Rooiwater Complex and surrounding granitoids[END_REF][START_REF] Powell | Conditions and timing of metamorphism in the Southern Abitibi greenstone belt, Quebec[END_REF][START_REF] Gapais | Kinematics of long-lasting Palaeoproterozoic transpression within the Thompson Nickel Belt (Manitoba Canada)[END_REF][START_REF] Jaguin | The Murchison Greenstone Belt (South Africa): a general tectonic Framework[END_REF], and some are marked by important ore concentrations [START_REF] Bleeker | New structural-metamorphic constraints on early Proterozoic oblique collision along the Thompson Nickel Belt, Manitoba, Canada[END_REF][START_REF] Vearncombe | Geology, geophysics and mineralization of the Murchison Schist Belt, Rooiwater Complex and surrounding granitoids[END_REF][START_REF] Chardon | Archean granite-greenstone tectonics at Kolar (South India): Interplay of diapirism and bulk inhomogeneous contraction during juvenile magmatic accretion[END_REF][START_REF] Gapais | Kinematics of long-lasting Palaeoproterozoic transpression within the Thompson Nickel Belt (Manitoba Canada)[END_REF][START_REF] Böhm | Sm-Nd Isotope Technique as An Exploration Tool: Delineating the Northern Extension of the Thompson Nickel Belt, Manitoba, Canada[END_REF][START_REF] Dubé | Greenstone-hosted quartz-carbonate vein deposit[END_REF][START_REF] Jaguin | The Murchison Greenstone Belt (South Africa): a general tectonic Framework[END_REF][START_REF] Lin | Synchronous vertical and horizontal tectonism at late stages of Archean cratonization and genesis of Hemlo gold deposits, Superior craton, Ontario, Canada[END_REF]. Recent analogue experiments and field surveys have argued that vertical tectonics, marked by downward motion of upper-crustal rocks within a weak ductile underlying crust was a major mode of shortening of hot lithospheres (Cagnard et al., 2006a, b;Chardon et al., 2009 and refs. therein;[START_REF] Gapais | Mountain building and exhumation processes through time: inferences from nature and models[END_REF].

In this paper, we compare three orogenic sub-provinces marked by different metal contents: the Southernmost Abitibi Greenstone Belt (Abitibi Sub-Province, Quebec) (Au), the Antimony line within the Murchison Greenstone Belt (South Africa) (Sb), and the Thompson Nickel Belt (Canada) (Ni). Beside their different metal contents, the areas range from Archaean (Abitibi and Murchison belts) to Paleoproterozoic in age (Thompson belt). They further show different metamorphic grades, sub-greenschist facies to greenschist facies in southernmost Abitibi [START_REF] Powell | Conditions and timing of metamorphism in the Southern Abitibi greenstone belt, Quebec[END_REF], greenschist facies along the Antimony Line in the Murchison belt [START_REF] Vearncombe | Geology, geophysics and mineralization of the Murchison Schist Belt, Rooiwater Complex and surrounding granitoids[END_REF], upperamphibolite facies, up to partial melting and granulitic conditions, in the Thompson belt [START_REF] Bleeker | New structural-metamorphic constraints on early Proterozoic oblique collision along the Thompson Nickel Belt, Manitoba, Canada[END_REF]Zwanzig et al., 2007 and refs therein).

Our paper emphasizes that pop-down tectonics marked by downward motions of faultbounded upper crustal blocks was common during horizontal compression of weak continental lithospheres and may have been a first-order factor accounting for channelled fluid transfers and concentrations of ore deposits in ancient deformation belts.

Geology, structure, and ore deposits

At the map scale, the three examples show two first-order characteristics (Figs. 1 and 2). First, foliations are mainly steeply dipping, strike parallel to the belt (locally disturbed by syn-kinematic intrusions), and bear widespread steeply plunging lineations attesting to sub-vertical stretch; second, ore deposits are concentrated within or around belt-parallel alignments of supra-crustals (sediments or volcanics) and along narrow deformation zones parallel to the belts [START_REF] Chown | Tectonic evolution of the northern Volcanic Zone, Abitibi belt, Quebec[END_REF][START_REF] Vearncombe | Geology, geophysics and mineralization of the Murchison Schist Belt, Rooiwater Complex and surrounding granitoids[END_REF][START_REF] Gapais | Kinematics of long-lasting Palaeoproterozoic transpression within the Thompson Nickel Belt (Manitoba Canada)[END_REF][START_REF] Böhm | Sm-Nd Isotope Technique as An Exploration Tool: Delineating the Northern Extension of the Thompson Nickel Belt, Manitoba, Canada[END_REF][START_REF] Dubé | Greenstone-hosted quartz-carbonate vein deposit[END_REF][START_REF] Jaguin | The Murchison Greenstone Belt (South Africa): a general tectonic Framework[END_REF].

In the southernmost Quebec Abitibi sub-Province, major gold deposits mark the E-W striking steeply dipping Cadillac deformation zone that separates the Abitibi Greenstone Belt from the Pontiac domain to the South (Fig. 1a). Other E-W striking deformation zones also marked out by supra-crustals and gold deposits occur within the belt to the North (Fig. 1A) [START_REF] Dubé | Greenstone-hosted quartz-carbonate vein deposit[END_REF]. Evidence for vertical stretch components is widespread throughout the region [START_REF] Chown | Tectonic evolution of the northern Volcanic Zone, Abitibi belt, Quebec[END_REF][START_REF] Dubé | Greenstone-hosted quartz-carbonate vein deposit[END_REF] (Fig. 2a,b) where kinematics involved dominant top-to-the-South motions in an overall transpressive context [START_REF] Chown | Tectonic evolution of the northern Volcanic Zone, Abitibi belt, Quebec[END_REF]. However, metamorphic and structural data attest to limited syn-transpression offsets along the Cadillac zone [START_REF] Chown | Tectonic evolution of the northern Volcanic Zone, Abitibi belt, Quebec[END_REF][START_REF] Powell | Conditions and timing of metamorphism in the Southern Abitibi greenstone belt, Quebec[END_REF][START_REF] Dubé | Greenstone-hosted quartz-carbonate vein deposit[END_REF]. Gold concentrations are also observed along NW-SE striking zones recognized as late dextral strike-slip zones that offset lithologies to a maximum extend of the order of 10 th of km (Chown et al., 1992, Dubé and[START_REF] Dubé | Greenstone-hosted quartz-carbonate vein deposit[END_REF] (Fig. 1a). Gold is concentrated in quartz-carbonate ± tourmaline veins [START_REF] Dubé | Greenstone-hosted quartz-carbonate vein deposit[END_REF]Tremblay, 2011).

In the Murchison belt, sub-vertical fabrics bearing steeply plunging stretching lineations affect the entire belt, ore deposits being concentrated in quartz-carbonate veins along the so-called Antimony Line (Vearncombe et al., 1992) (Figs. 1b and 2b, c). At the regional scale, sub-vertical fabrics are associated with top-to-the-South motions with minor transpressive components and limited displacements along the Antimony line [START_REF] Vearncombe | Geology, geophysics and mineralization of the Murchison Schist Belt, Rooiwater Complex and surrounding granitoids[END_REF][START_REF] Jaguin | The Murchison Greenstone Belt (South Africa): a general tectonic Framework[END_REF]).

In the Thompson belt, Ni resources are mainly concentrated within sulphide facies iron Formations at the vicinity of shear zones that cut across at low angle or bound supracrustal metasediments along the western margin of the belt [START_REF] Böhm | Sm-Nd Isotope Technique as An Exploration Tool: Delineating the Northern Extension of the Thompson Nickel Belt, Manitoba, Canada[END_REF][START_REF] Zwanzig | Lithostratigraphy and Geochemistry of the High-Grade Metasedimentary Rocks in the Thompson Nickel Belt and Adjacent Kisseynew Domain, Manitoba: Implications for Nickel Exploration[END_REF] (Fig. 1c). These zones appear as latest active deformation zones within the belt that is marked by pervasive sub-vertical foliations bearing steeply plunging lineations [START_REF] Fueten | Structural petrology along a transect across the Thompson Nickel Belt, Manitoba: dip-slip at the western Churchill-Superior boundary[END_REF][START_REF] Bleeker | New structural-metamorphic constraints on early Proterozoic oblique collision along the Thompson Nickel Belt, Manitoba, Canada[END_REF][START_REF] Gapais | Kinematics of long-lasting Palaeoproterozoic transpression within the Thompson Nickel Belt (Manitoba Canada)[END_REF] (Figs. 1c and2e, f). The regional deformation was transpressive, with vertical components pointing to overall top-to-the West motions, combined with minor dextral wrenching [START_REF] Fueten | Structural petrology along a transect across the Thompson Nickel Belt, Manitoba: dip-slip at the western Churchill-Superior boundary[END_REF][START_REF] Gapais | Kinematics of long-lasting Palaeoproterozoic transpression within the Thompson Nickel Belt (Manitoba Canada)[END_REF].

A general model

None of the mineralized lineaments show evidence of major displacements along thrusts or normal detachments that would be marked by important metamorphic jumps.

They neither point to important wrench-type displacements that would be marked by pervasive horizontal stretching lineations and by significant horizontal offsets of tectonic units recorded at the map scale. Field evidence and analogue models have actually shown that shortening of weak lithospheres results in dominant vertical tectonics (Cagnard et al., 2006;[START_REF] Gapais | Kinematics of long-lasting Palaeoproterozoic transpression within the Thompson Nickel Belt (Manitoba Canada)[END_REF][START_REF] Gapais | Paleoproterozoic tectonics in the Terre Adélie Craton (East Antartica)[END_REF][START_REF] Gapais | Mountain building and exhumation processes through time: inferences from nature and models[END_REF]Cagnard et al., 2006aCagnard et al., , b, 2007;;[START_REF] Chardon | Flow of ultra-hot orogens: A view from the Precambrian, clues for the Phanerozoic[END_REF] (Fig. 3b), Analogue models have further shown that the onset of a local pop-down resulted in strain concentration and triggered the piling-up of other upper crustal popdowns along vertical deformation zones (Cagnard et al., 2006a).

Figure 3a shows our general model of tectonics, fluid flow and mineralization interactions at crustal scale, on the basis of analogue modelling of shortening of a weak lithosphere bearing a ductile lithospheric mantle (Fig. 3b). Syntectonic sediments concentrate within spaces provided at fault footwalls during first stages of pop-down motions of upper crust pieces. During their downward motion, upper-crustals are affected by ductile deformations. Fluids are produced during progressive burial through successive compaction, diagenesis, and metamorphic and melting reactions, steeply dipping foliations and shear planes favouring efficient upward circulation [START_REF] Sibson | High-angle reverse faults, fluid-pressure cycling, and mesothermal gold-quartz deposits[END_REF]) (Fig. 3a). As a positive feedback, fluid-induced strain weakening may contribute to further strain concentration along zones of pop-down piling-up. The final result of deformation-fluid interactions is the development of high-angle channels for localised fluid circulation [START_REF] Sibson | High-angle reverse faults, fluid-pressure cycling, and mesothermal gold-quartz deposits[END_REF][START_REF] Lin | Synchronous vertical and horizontal tectonism at late stages of Archean cratonization and genesis of Hemlo gold deposits, Superior craton, Ontario, Canada[END_REF] where large permeability is maintained at the grain scale by high strains [START_REF] Putnis | Mineral replacement reactions: from macroscopic observations to microscopic mechanisms[END_REF]. Mineral depositions may further concentrate in second order structures as small-scale tension gashes or pull-apart [START_REF] Sibson | High-angle reverse faults, fluid-pressure cycling, and mesothermal gold-quartz deposits[END_REF][START_REF] Bleeker | New structural-metamorphic constraints on early Proterozoic oblique collision along the Thompson Nickel Belt, Manitoba, Canada[END_REF][START_REF] Dubé | Greenstone-hosted quartz-carbonate vein deposit[END_REF]Tremblay, 2011).

Various fluid sources may be sampled during the process, from the crust surface to the mantle, provided deformation channels are long enough to connect potential reservoirs.

CO 2 degassing from the mantle has for example be documented within large shear zones of southern Madagascar [START_REF] Pili | Lithospheric shear zones and mantle-crust connections[END_REF], together with partial melting of the lower crust in the region [START_REF] Morteani | Geochemistry of phlogopite, diopside, calcite, anhydrite and apatite pegmatites and syenites of southern Madagascar. Evidence for crustal silicocarbonatitic (CSC) melt formation in a Panafrican collisional tectonic setting[END_REF]Martin et al., in press). As fluids of distinct origins may become connected within zones of high strains, chemical disequilibria occur at their meeting point, a classical mechanism for metal precipitation. For gold, mixing between oxidizing surface-derived fluids and Au-bearing deep fluids (e.g. [START_REF] Beaudoin | Stable isotope geochemistry of the Archean Val d'Or (Canada) orogenic gold vein field[END_REF] may for example induce the destabilisation of Au-bearing sulphide complexes by redox changes (e.g. [START_REF] Shenberg | Solubility of gold in aqueous sulfide solutions from 150 to 350°C[END_REF].

Geometric relationships between strain patterns associating vertical motions, alignments of supra-crustals rocks and ore deposits comparable to those described here have been reported for other Archaean domains [START_REF] Chardon | Archean granite-greenstone tectonics at Kolar (South India): Interplay of diapirism and bulk inhomogeneous contraction during juvenile magmatic accretion[END_REF][START_REF] Lin | Synchronous vertical and horizontal tectonism at late stages of Archean cratonization and genesis of Hemlo gold deposits, Superior craton, Ontario, Canada[END_REF]. However, attached interpretations were made in the light of models based on gravity-driven sinking (sagduction) of heavy greenstone belts within an underlying light and weak felsic crust (Raleigh-Taylor instabilities) [START_REF] Choukroune | Soft lithospheres during periods of Archaean crustal growth or crustal reworking[END_REF][START_REF] Chardon | Archean granite-greenstone tectonics at Kolar (South India): Interplay of diapirism and bulk inhomogeneous contraction during juvenile magmatic accretion[END_REF][START_REF] Lin | Synchronous vertical and horizontal tectonism at late stages of Archean cratonization and genesis of Hemlo gold deposits, Superior craton, Ontario, Canada[END_REF]. On the other hand, analogue models with normal density profiles increasing from top to bottom (Cagnard et al., 2006a, b;[START_REF] Gapais | Mountain building and exhumation processes through time: inferences from nature and models[END_REF] (Fig. 3b), and field evidence from Paleoproterozoic areas where heavy greenstone belts are lacking (Cagnard et al., 2006a;[START_REF] Gapais | Kinematics of long-lasting Palaeoproterozoic transpression within the Thompson Nickel Belt (Manitoba Canada)[END_REF]2008, 2009) demonstrated that pop-down tectonics as invoked here just required an underlying weak and hot lithosphere with a ductile sub-Moho mantle, and a regional compressive horizontal stress field. Gravity-driven sagduction of heavy greenstones deposited on a buoyant felsic crust may combine with pop-down tectonics and thus further favour downward motion of supra-crustals units [START_REF] Chardon | Flow of ultra-hot orogens: A view from the Precambrian, clues for the Phanerozoic[END_REF].

The three studied areas show first-order structural correlations between ore concentrations, regional-scale horizontal shortening involving important sub-vertical crustal stretch, and strain concentration along steeply dipping zones (Figs. 1 and2), their location having possibly been influenced by pre-existing discontinuities, like boundaries of accreting terrains [START_REF] Bleeker | New structural-metamorphic constraints on early Proterozoic oblique collision along the Thompson Nickel Belt, Manitoba, Canada[END_REF][START_REF] Chown | Tectonic evolution of the northern Volcanic Zone, Abitibi belt, Quebec[END_REF][START_REF] Vearncombe | Geology, geophysics and mineralization of the Murchison Schist Belt, Rooiwater Complex and surrounding granitoids[END_REF]. The three areas host distinct metal deposits (Au, Sb, Ni), notably because of distinct pressuretemperature conditions of mineralization. Another reason for this may lie in the fact that the local lithologies involved in the deformation history were enriched in a specific metal before the mineralization event. Ni is abundant in the ultramafics of the Thompson belt before its trapping in the iron formations (Zwantzig et al., 2007); metasediments of the Weigel Formation in the core of the Murchison belt are anomalously enriched in Sb [START_REF] Pearton | Antimony mineralisation in the Murchison Greenstone Belt -An overview[END_REF]; and various lithologies in the Abitibi greenstone belt may have provided Au for mineralization.

The three examples described here attest to transpressive motions during regional shortening [START_REF] Chown | Tectonic evolution of the northern Volcanic Zone, Abitibi belt, Quebec[END_REF][START_REF] Gapais | Kinematics of long-lasting Palaeoproterozoic transpression within the Thompson Nickel Belt (Manitoba Canada)[END_REF][START_REF] Jaguin | The Murchison Greenstone Belt (South Africa): a general tectonic Framework[END_REF]. Models (Cagnard et al., 2006a;[START_REF] Rey | Lithospheric scale gravitational flow: the impact of body forces on orogenic processes from Archaean to Phanaerozoic[END_REF] and field examples (Chardon et al., 2009 and refs. therein) have consistently shown that wrench components along such steep deformation zones may actually occur, basically because of transform motions due to horizontal longitudinal flow of the weak ductile crust that combines with regional horizontal shortening.

We show here that ore deposition does occur in quite comparable tectonic settings within greenstone-bearing Archaean belts and within younger greenstone-lacking Paleoproterozoic deformation zones. Consequently, we propose that vertical tectonics is a main driving process that concentrates supra-crustal materials, strains, fluid transfers, and metal transport along steeply dipping deformation zones. Reactivation of pre-existent structures may favour strain localisation and subsequent fluid and mineralization histories. It is worth noting that triple points within Archaean greenstones trapped between gneiss domes, as exemplified in the Pilbara area (Australia), show comparable relationships between vertical motions and ore concentrations [START_REF] Thébaud | Archean gravity-driven tectonics on hot and flooded continents: Controls on long-lived mineralised hydrothermal systems away from continental margins[END_REF].

Beyond further arguing for the main role of vertical tectonics during compression of hot lithospheres irrespective of their age, our analysis should provide a new and general tectonic frame for metallogenic exploration in ore-rich cratons. [START_REF] Chown | Tectonic evolution of the northern Volcanic Zone, Abitibi belt, Quebec[END_REF][START_REF] Gapais | Kinematics of long-lasting Palaeoproterozoic transpression within the Thompson Nickel Belt (Manitoba Canada)[END_REF][START_REF] Böhm | Sm-Nd Isotope Technique as An Exploration Tool: Delineating the Northern Extension of the Thompson Nickel Belt, Manitoba, Canada[END_REF][START_REF] Dubé | Greenstone-hosted quartz-carbonate vein deposit[END_REF][START_REF] Jaguin | The Murchison Greenstone Belt (South Africa): a general tectonic Framework[END_REF].

Figure captions

Fig. 1 .

 1 Fig. 1. Simplified geological and structural maps of studied areas. (a) Southern Abitibi

Fig. 2 .

 2 Fig. 2. Outcrop photographs and sterographic projections (equal area, lower hemisphere)

Fig. 3 .

 3 Fig. 3. Deformation modes of shortening of a weak continental lithosphere. (a) Sketch

  fig. a

Figure2

  Figure2

  Figure3

Acknowledgements D.G. thanks N. Machado, A. Potrel, M. Poujol and P. Roy for working opportunities in the three areas discussed in the paper. R. Daigneault and the Bureau de l'exploration géologique du Québec kindly allowed us to use their data of stretching lineations along the Cadillac fault zone. Stereograms were made using the software "Stereonet" (R. Allmendinger). Editors and anonymous reviewers are thanked for their constructive reviews of the initially submitted paper.