G R A N G E 1 , P A T R I C E M E U N I E R

F R A N Ç O I S N A D A L

Precessional instability of a uid cylinder

In this paper, the instability of a uid inside a precessing cylinder is addressed theoretically and experimentally. The precessional motion forces Kelvin modes in the cylinder which can become resonant for given precessional frequencies and cylinder aspect ratios. When the Reynolds number is large enough, these forced resonant Kelvin modes eventually become unstable. A linear stability analysis based on a triadic resonance between a forced Kelvin mode and two additional free Kelvin modes is carried out. This analysis allows to predict the spatial structure of the instability and its threshold. These predictions are compared to the vorticity eld measured by Particle Image Velocimetry with an excellent agreement. When the Reynolds number is further increased, nonlinear eects appear. A weakly nonlinear theory is developed semi-empirically by introducing a geostrophic mode which is triggered by the nonlinear interaction of a free Kelvin mode with itself in the presence of viscosity. Amplitude equations are obtained coupling the forced Kelvin mode, the two free Kelvin modes and the geostrophic mode. They show that the instability saturates to a xed point just above threshold. Increasing the Reynolds number leads to a transition from a steady saturated regime to an intermittent ow in good agreement with experiments. Surprisingly, this weakly nonlinear model still gives a correct estimate of the mean ow inside the cylinder even far from the threshold when the ow is turbulent.

Introduction

The goal of this paper is to explain the onset of turbulence observed in a precessing cylinder completely lled by a uid.

The motion of precession is obtained when an object is in rotation around an axis which is itself rotating around a second axis. The knowledge of the ow forced by a precessional motion is of critical importance in several domains. In aeronautics (see [START_REF] Stewartson | On the stability of a spinning top containing liquid[END_REF][START_REF] Gans | Dynamics of a near-resonant uid-lled gyroscope[END_REF][START_REF] Lagrange | Spacecraft nutational instability prediction by energy dissipation measurments[END_REF][START_REF] Agrawal | Dynamics characteristics of liquid motion in partially lled tanks of a spinning spacecraft[END_REF][START_REF] Bao | Stability of a spinning liquid lled spacecraft[END_REF][START_REF] Lambelin | Non-resonant viscous theory for the stability of a uid-lled gyroscope[END_REF], the presence of a uid (such as a propellant liquid) inside a ying object (spacecraft, rocket, satellite...) can have dangerous consequences on the stability of this object. Indeed the presence of a weak precessing angle, due to a non-axisymmetry of the object, can lead to a large amplitude of the contained ow when it is resonant. This ow can in turn create a torque on the ying object which destabilizes the precessing angle and thus lead to the deviation of its trajectory. A good understanding of the behavior of such a uidstructure coupled system requires a precise knowledge of the hydrodynamics of the contained uid.

In external geophysics, atmospheric vortices such as hurricanes or tornadoes are also subject to precessional forcing. Indeed, their axes rotate with the earth rotation around the polar axis. The uid is thus in precession with an angle α equal to the co-latitude. This is exactly the kind of ow which is studied in this paper. Usually, the precession of atmospheric vortices is neglected and only Coriolis eects are taken into account. However, the question remains whether precession has a signicative eect on the cyclone dynamics and notably on its stability.

This problem is also relevant to internal geophysics. Indeed, the Earth precession induces a forcing on its liquid core and thus generates a strong turbulent ow. Because the Earth core is made of a conductive uid (melt iron) such a ow could be responsible for the geodynamo eect which generates the Earth magnetic eld, (see [START_REF] Malkus | Precession of the earth as the cause of geomagnetism[END_REF][START_REF] Kerswell | Upper bounds on the energy dissipation in turbulent precession[END_REF][START_REF] Tilgner | Precession driven dynamos[END_REF][START_REF] Tilgner | Kinematic dynamos with precession driven ow in a sphere[END_REF][START_REF] Wu | A precessionally-driven dynamo in a plane layer[END_REF], 2009). However, other eects such as convection, boundary layers, elliptic or tidal instability have also been proposed to explain the source of energy for the geodynamo.

The dynamics of a uid inside a precessing spheroidal cavity was rst studied by [START_REF] Poincaré | Sur la précession des corps déformables[END_REF]. He showed that the ow is a solid body rotation around an axis of rotation which is undetermined in the absence of viscosity. By introducing the viscosity in a thin boundary layer (Ekman layer), [START_REF] Busse | Steady uid ow in a precessing spheroidal shell[END_REF] determined this axis of rotation and the amplitude of the rotation. He also determined the critical latitudes where occurs a breakdown of the Ekman layer. This breakdown gives rise to strong shear zones in the bulk of the ow. These shear zones can be axisymmetric and are thus very well visualized in experiments (see [START_REF] Malkus | Precession of the earth as the cause of geomagnetism[END_REF][START_REF] Vanyo | Experiments on precessing ows in the earth's liquid core[END_REF][START_REF] Noir | Écoulement d'un uide dans une cavité en précession : approches numérique et expérimentale[END_REF]. [START_REF] Noir | Experimental evidence of nonlinear resonance eects between retrograde precession and the tilt-over mode within a spheroid[END_REF] numerically showed that they are due to nonlinear interactions in the Ekman layer. This breakdown of the ow also excites inertial waves which propagate along characteristic surfaces at a specic angle. They were predicted asymptotically by [START_REF] Kerswell | On the internal shear layers spawned by the critical regions in oscillatory ekman boudary layers[END_REF], studied numerically by [START_REF] Hollerbach | Oscillatory internal shear layers in rotating and precessing ows[END_REF]; Tilgner (1999a,b); Noir et al. (2001b) and observed experimentally by Noir et al. (2001a). Finally, when the precession rate is suciently large, [START_REF] Malkus | Precession of the earth as the cause of geomagnetism[END_REF] and [START_REF] Vanyo | Experiments on precessing ows in the earth's liquid core[END_REF] observed that the ow is unstable and rapidly degenerates into a turbulent state. However, it is still unclear whether this turbulent ow is due to a local instability in the Ekman layer or in the shear zones or if it results from a global instability such as a triadic resonance [START_REF] Lorenzani | Fluid instabilities in precessing spheroidal cavities[END_REF][START_REF] Kerswell | The instability of precessing ow[END_REF].

The presence of inertial waves in a precessing spheroid is simply due to the presence of the Coriolis force linked to the global rotation of the uid. In a cylindrical geometry, inertial waves are also present and interfere in order to construct global modes of the uid ow, known as Kelvin modes [START_REF] Kelvin | Vibrations of a columnar vortex[END_REF]. These modes are neutral in the absence of viscosity, with a frequency smaller than twice the angular velocity of the uid (Saman 1992). By taking into account viscous boundary layers on the walls of the cylinder [START_REF] Kudlick | On the transient motions in a contained rotating uid[END_REF] and [START_REF] Greenspan | The theory of rotating uids[END_REF] have extended the invscid theory to predict the viscous decay rates of a Kelvin mode. They were experimentally conrmed by [START_REF] Mcewan | Inertial oscillations in a rotating uid cylinder[END_REF] and [START_REF] Kobine | Inertial wave dynamics in a rotating and precessing cylinder[END_REF] and numerically by [START_REF] Kerswell | On the viscous decay rates of inertial waves in a rotating cylinder[END_REF].

The precession can excite the Kelvin modes if they have the same frequency as the precession. This was rst shown experimentally by [START_REF] Mcewan | Inertial oscillations in a rotating uid cylinder[END_REF] who used a tilted top end rotating at a dierent angular velocity than the cylinder, in order to model the forcing by precession. He observed through Kalliroscope visualisations that the ow becomes resonant at the frequencies of the Kelvin modes with the correct wavelength. This was further conrmed experimentally by [START_REF] Manasseh | Breakdown regimes of inertia waves in a precessing cylinder[END_REF] and [START_REF] Kobine | Inertial wave dynamics in a rotating and precessing cylinder[END_REF] for a real precessing cylinder. The amplitude of these modes can be predicted theoretically outside of their resonances by a simple linear and inviscid theory. However, at the resonance, viscous eects have to be added in order to predict the amplitude saturation. Gans (1970b) showed that this amplitude scales as the square root of the Reynolds number when viscous eects are taken into account because of Ekman pumping inside the boundary layers. [START_REF] Meunier | A rotating uid cylinder subject to weak precession[END_REF] have extended this result by adding the eect of nonlinear interactions of the Kelvin modes, which leads to the appearance of a geostrophic motion inside the cylinder. [START_REF] Mcewan | Inertial oscillations in a rotating uid cylinder[END_REF] and later [START_REF] Manasseh | Breakdown regimes of inertia waves in a precessing cylinder[END_REF] also showed that the ow is highly unstable in the vicinity of a resonant frequency. Indeed, after persisting in an apparently laminar form, they observed that the ow becomes unstable and degenerates abruptly into a ne-scale disordered state. They called this transition to turbulence the phenomenon of "resonant collapse" because it was associated with an amplitude decrease of the resonant Kelvin mode. This breakdown behavior was also observed by [START_REF] Thompson | Diurnal tides and shear instabiliies in a rotating cylinder[END_REF] who studied the case of a partly lled and tilted cylinder. By performing experiments in a precessing cylinder, [START_REF] Manasseh | Breakdown regimes of inertia waves in a precessing cylinder[END_REF] catalogued dierent breakdown regimes by a letter scheme (AG), where Type A breakdown results in the generation of turbulence with the smallest scales. [START_REF] Mcewan | Inertial oscillations in a rotating uid cylinder[END_REF], [START_REF] Thompson | Diurnal tides and shear instabiliies in a rotating cylinder[END_REF] and [START_REF] Manasseh | Breakdown regimes of inertia waves in a precessing cylinder[END_REF] also observed that for a precise range of parameters the unstable ow can relaminarize after breakdown leading to a cycle of breakdowns and relaminarizations. Because their experiments did not give any informations on global uid velocities, the physical mechanism leading to breakdown was unclear. [START_REF] Mahalov | The instability of rotating uid columns subjected to a weak external coriolis-force[END_REF] showed theoretically that the precessing ow inside an innite cylinder is unstable and that the instability mechanism is triggered by a triadic resonance between the vertical shear created by the precession force and two free Kelvin modes. However, this mechanism does not explain why the ow is more unstable at the resonance of a forced Kelvin mode. Recently, Lagrange et al. (2008) showed that the unstable ow indeed exhibits two free Kelvin modes which satisfy the conditions for a triadic resonance with the resonant forced Kelvin mode and not with the vertical shear. This mechanism will be used in this paper to carry out a linear stability analysis.

The mechanism of triadic resonance has been extensively studied to explain the elliptical instability (Moore & Saman 1975;[START_REF] Tsai | The stability of short waves on a straight vortex lament in a weak externally imposed strain eld[END_REF] as an interaction between the ellipticity of the vortex and two free Kelvin modes. The growth rate of this instability was later calculated analytically using local theories by [START_REF] Bayly | Three-dimensional instability of elliptical ow[END_REF] and Walee (1990). Theoretical predictions have been widely validated numerically [START_REF] Mason | Nonlinear evolution of the elliptical instability : an example of inertial breakdown[END_REF] and experimentally [START_REF] Eloy | Experimental study of the multipolar vortex instability[END_REF][START_REF] Eloy | Elliptic and triangular instabilities in rotating cylinders[END_REF]. Such a triadic resonance occurs when the dierence in the wavenumbers and frequencies between the two free Kelvin modes is equal to the wavenumber and frequency of the elliptical forcing (see the review by [START_REF] Kerswell | Elliptical instability[END_REF]. Such a mechanism can be generalized to the destabilization of a Kelvin mode by the interaction of three Kelvin modes, as proposed by [START_REF] Mason | Nonlinear evolution of the elliptical instability : an example of inertial breakdown[END_REF] and [START_REF] Kerswell | Secondary instabilities in rapidly rotating uids : inertial wave breakdown[END_REF] to explain the secondary instability of the elliptic instability. In this paper, we will use exactly the same procedure, except that the Kelvin mode is now forced by precession instead of being forced by an elliptic instability.

The paper is organized as follows. Section 2 presents the problem of a precessing cylinder and the experimental setup. Section 3 is dedicated to the base ow. The governing equations are derived and the classical linear and inviscid theory is recalled. Viscous effects are also added in order to calculate the base ow inside a resonant cylinder. A linear stability analysis based upon a mechanism of triadic resonance between Kelvin modes is developed in 4. The results (growth rate,stability diagram...) are discussed and compared with experiments in 5. Section 6 is dedicated to a weakly nonlinear theory taking into account the inuence of a geostrophic mode. This theory is also compared to experimental results. Conclusions and discussion in the more general context of rotating ows are nally given in 7.

Figure 1. Sketch of the experimental setup. A cylinder of radius Rc and height H rotates around its axis at the angular velocity Ω1. It is mounted on a platform which rotates at the angular velocity Ω2. The angle between the two axes of rotation is the precessing angle θ. A camera xed above the cylinder in the platform frame is used to perform PIV measurements in the luminous plane of a horizontal laser sheet. Polar coordinates (R, ϕ, Z) are dened in the cylinder rotating frame.

Presentation of the problem and experimental setup

An experimental setup has been built to study the ow inside a precessing cylinder. This experimental setup is sketched in gure 1 and described briey in the following. Readers should refer to [START_REF] Meunier | A rotating uid cylinder subject to weak precession[END_REF] for an extensive description.

A right cylinder of height H and radius R c lled with water rotates at the angular velocity Ω 1 around its axis and is mounted on a platform which rotates at the angular velocity Ω 2 . Each axis having its own driving motor, these angular velocities can be varied independently. The angular velocity Ω 1 can be increased up to 60 rad s -1 and is measured with an accuracy of 0.1%. The angular velocity of the platform Ω 2 can only be varied from 0.1 to 6 rad s -1 and is measured with an accuracy of 0.2% when the precession frequency is larger than 0.2 rad s -1 . Two dierent cylinders with the same aspect ratio h = H/R c = 1.62 ± 0.3% but dierent dimensions (H = 2.72 cm and H = 7.50 cm) are used. Because the Reynolds number depends on R c , these two cylinders allow to vary the Reynolds number over a large range [1000,50000].

The cylinder axis is tilted relative to the axis of the platform with an angle θ, which can be increased up to 15 • and which is determined with an absolute accuracy of ±0.1 • . A release device mounted on the platform and controlled externally allows to tilt the cylinder during the motion of the platform. The release device is composed of an electromagnet designed to keep the cylinder in a vertical position during the spinup phase. Once the electromagnet is turned o, a drawback spring pulls the cylinder into its tilted position.

PIV measurements have been performed by seeding the ow with small reecting polycrystalline particles (Optimage Ltd.) of mean diameter 50µm and density 1000 ± 20 kg m -3 . They are illuminated with a light sheet of thickness 2 mm created by a yttrium aluminum garner (YAG) pulsed laser. An external cylindrical lens is used to provide this laser sheet which is xed relative to the laboratory frame. However, because the precessing angle θ is small, the laser sheet can be considered normal to the cylinder axis at rst order. The height z of the laser sheet can be varied along the height of the cylinder. In our experiments we have chosen z = 0 (midheight of the cylinder) and z = h/4. These two positions allow to measure a maximum transverse velocity for the Kelvin modes observed in our experiments.

The images of particles are recorded by a PIV camera (Kodak Megaplus ES 1.0, 1008× 1018 pixels) mounted on the rotating platform and aligned with the axis of the cylinder. The time interval between two successive images is relatively large (from 2 ms to 50 ms) such that the cylinder rotates of approximately 20 degrees between the two images. This creates large displacements of the particles at the periphery of the cylinder (150 pixels), but the two images are rotated around the center of the cylinder in order to remove the solid body rotation of the particles. The PIV thus gives directly the velocity eld in the cylinder reference frame. This procedure allows to measure very small velocities down to 1% of the velocity of the cylinder wall. Such measurements would not have been possible without the image rotation. The pairs of images are then treated by a crosscorrelation algorithm detailed in [START_REF] Meunier | Analysis and minimization of errors due to high gradients in particule image velocimetry[END_REF] which gives velocity elds with 60 × 60 vectors. The power is brought up to the platform by a rotating collector through the vertical axis to supply the camera, the cylinder motor, the electro-magnet and the acquisition computer.

For the acquisition of a PIV eld we proceed as follows. The cylinder is rst kept vertical and rotates at Ω 1 . The platform rotates at Ω 2 . Once the spinup stage is completed (i.e the ow is in solid body rotation), the cylinder is tilted in precession and a rst acquisition of 85 PIV elds is launched. This rst acquisition allows the study of the transient stage of the instability, i.e. the growth of the Kelvin modes. A second PIV acquisition of 85 PIV elds is started when the transient stage is completed ; usually 10 minutes later. This second acquisition allows the study of the permanent stage. Some visualizations were also performed by using Kalliroscope particles which are known to reveal the structure of the ow in a rotating uid (see [START_REF] Mcewan | Inertial oscillations in a rotating uid cylinder[END_REF][START_REF] Manasseh | Breakdown regimes of inertia waves in a precessing cylinder[END_REF]). The tank is illuminated from the side with a vertical luminous sheet created by a 5 Watts Argon laser going through a cylindrical lens. A very slight shearing motion in any section of the cylinder is sucient to align the Kalliroscope particles, thus changing the light intensity seen by an observer. This provides an extremely sensitive indicator of relative uid motion. This method does not give any quantitative results but is a quick and ecient way to determine the wavelength of the base ow or to see if the ow is stable or unstable.

Base ow

In this section we give the governing equations for a precessing ow and we derive the classical solution which is valid before the appearance of the instability. This solution is linear and inviscid when the ow is non-resonant, but viscous eects have to be taken into account when one Kelvin mode is resonant. This solution will be used as a base ow for the linear stability analysis carried out in 4.

Formulation

A cylinder with radius R c and height H is completely lled of a uid with density ρ and kinematic viscosity ν. As shown in gure 1, the cylinder rotates around its own axis (O , k) at the constant angular frequency Ω 1 . The cylinder is mounted on a platform rotating at the constant angular frequency Ω 2 around an axis ẑ. There is an angle θ between the two axes of rotation creating a precessing motion of the cylinder. The rotation vector of the cylinder in the laboratory reference frame is given by Ω = Ω 1 k + Ω 2 ẑ and is time-dependent because k rotates around ẑ.

To have simple boundary conditions, it is easier to solve the problem in the cylinder frame of reference (O , , , k), (O being the center of mass of the cylinder) in which the radius vector R is dened by its cylindrical coordinates (R, ϕ, Z) (see Fig. 1). In this reference frame, the NavierStokes equations satised by the velocity eld U and the pressure eld P for an incompressible uid are

∂ U ∂ T + (U • ∇) U + 2 Ω × U + Ω × (Ω × R) + d Ω d T × R + Γ O = - 1 ρ ∇P + ν∆U, (3.1a) ∇ • U = 0, (3.1b)
with the viscous boundary condition U = 0 on the cylinder walls.

In equation (3.1a), the rst two terms are the usual inertial terms, the third and fourth term are the Coriolis and centrifugal accelerations, and the fth term is the acceleration of the rotation vector. The last left-hand side term is potential and refers to the acceleration of the centroid O of the cylinder. The two right-hand side terms of (3.1a) are the usual pressure and viscous terms.

By using R c and Ω -1 = (Ω 1 + Ω 2 cos θ) -1 as characteristic length and time, the NavierStokes equations for the dimensionless velocity eld u(r, t) become

∂ u ∂ t + 2 k × u + ∇p = -2Ro ωr cos(ωt + ϕ) k + u × (∇ × u) -2Ro δ × u + ∆u Re , (3.2a) ∇ • u = 0, (3.2b) with ω = Ω 1 Ω , Ro = Ω 2 sin θ Ω , Re = ΩR 2 c ν , δ = cos(ωt) -sin(ωt). (3.3ad )
In this dimensionless form, h = H/R c is the aspect ratio of the cylinder. The dimensionless pressure eld p(r, t) includes all the potential terms (see [START_REF] Meunier | A rotating uid cylinder subject to weak precession[END_REF], for the detail of p). The boundary condition of the velocity eld is u = 0 at the walls (r = 1 or z = ±h/2).

(3.4)

The NavierStokes equations (3.2 a, b) with the boundary condition (3.4) govern the ow inside a precessing cylinder. This set of equations has been obtained without any approximation and is thus valid for any value of the experimental parameters. As it appears from these equations, the problem is entirely governed by four dimensionless parameters. The amplitude of the forcing term is called a Rossby number Ro because it represents the ratio between the vorticity of the ow and the solid body rotation Ω.

The dimensionless frequency ω of the forcing term xes the frequency of the Kelvin modes which are excited. The cylinder aspect ratio h selects the structure of the Kelvin modes, and the Reynolds number Re quanties the viscous damping. In the following we will limit ourselves to the case of asymptotically small Rossby number Ro and large Reynolds number Re. This assumption is coherent with atmospherical and geophysical observations. It is also the relevant limit for a stable ow or at the onset of instability.

In the following the four-component and complex vector associated to the velocitypressure eld (u, p) will be noted v, such that (u, p) = v + v with v the complex conjugate of v. Using this formulation the NavierStokes equations (3.2 a, b) rewrite

∂ ∂t I + M v + c.c. = Ro F 0 e i(ωt+ϕ) + N(v, v) + Ro (D e i(ωt+ϕ) + c.c.)v + L v Re + c.c., (3.5)
where the operators D, I, L, M, the forcing vector F 0 and the bilinear function N are dened in Appendix A. The symbol c.c. stands for the complex conjugate.

3.2. Non-resonant cylinder In order to solve (3.5), an asymptotically small Rossby number Ro and a large Reynolds number Re are assumed. In this limit v is O(Ro) and the NavierStokes equation (3.5) at order O(Ro) becomes ωt+ϕ) .

∂ ∂t I + M v = Ro F 0 e i(
(3.6)

The no-slip boundary condition (3.4) becomes at this order a condition of no outward ow u • n = 0 at the walls (r = 1 or z = ±h/2), (3.7)

where n is an unitary vector normal to the wall. The set of equations (3.6) and (3.7) is a linear system whose second term represents the forcing due to precession. This system admits a particular solution v part. = 0, 0, Ro i r e i(ωt+ϕ) , 0 which is a shear along the cylinder axis. Because it does not satisfy the boundary condition (3.7) at z = ±h/2, it must be completed with a solution of the homogeneous equation (i.e. equation (3.6) without forcing term). Due to time and azimuthal dependence of the forcing and particular solution, the homogeneous solution can be written a sum of Kelvin modes of azimuthal wavenumber m = 1 and angular frequency ω (see [START_REF] Greenspan | The theory of rotating uids[END_REF]). Gathering the particular solution and the Kelvin modes yields the solution of (3.6) and (3.7)

v = v part. + ∞ i=1 ε i v 1,ω,k i , (3.8)
where v 1,ω,ki is a Kelvin mode of azimuthal wavenumber m = 1, frequency ω and axial wavenumber k i . Its structure is composed of two travelling waves of opposite wavenumber ±k i in order to form a stationary wave and match the parity of the forcing -kiz) .

v 1,ω,k i = u 1,ω,k i (r) e i(ωt+ϕ+kiz) -u 1,ω,-k i (r) e i(ωt+ϕ
(3.9)

The vector u m,ω,k i is found by solving the homogeneous equation and is given in Appendix A. The boundary condition in r = 1 imposes that the axial wavenumber k i is the positive root of the constitutive relation A PIV measurement of the rst Kelvin mode (i = 1) is shown in gure 2(a). It represents the axial and instantaneous vorticity eld ζ. This mode is characterized by two counter rotating vortices which induce a velocity along the y-axis (tilted axis). In the general case, the i-th Kelvin mode contains 2i vortices.

δ 2 i = 4 -ω 2 i ω 2 i k 2 i , ( 3
The amplitude ε i of each Kelvin mode is real and can be calculated by imposing a vanishing velocity at z = ±h/2. For this purpose, the z-velocity of the particular solution v part. is decomposed on the set of Bessel functions which characterize the z-component of the Kelvin modes. It leads to the inviscid and nonlinear amplitude ε i = Ro a i of the Kelvin modes, where a i is given in (A 8) of Appendix A. The linear forced response as a function of the forcing frequency ω is plotted in gure 2(b) for the rst Kelvin mode. It shows a series of divergences corresponding to the natural frequencies ω i,n of the cylinder (the notation ω i,n stands for the n-th resonance of the i-th Kelvin mode). These resonances occur when the z-velocity of the Kelvin mode vanishes, i.e. when the wavenumber k i of the Kelvin mode is equal to (2n -1)π/h with n the number of the resonance.

It can be demonstrated [START_REF] Kudlick | On the transient motions in a contained rotating uid[END_REF]) that there is always a Kelvin mode arbitrarily close to a resonance for any forcing frequency ω. It is thus necessary to add viscous eects in order to predict the saturation of the Kelvin mode amplitude ε i at a resonance. These results are recalled in the next section.

Resonant cylinder

To predict the saturation of the amplitude at a resonant frequency it is necessary to include the viscous eects (Gans 1970b) and/or the nonlinear eects [START_REF] Meunier | A rotating uid cylinder subject to weak precession[END_REF]). In the present paper, the amplitude is always saturated by viscous eects because |Re 1/2 Ro 2/3 | 1. In other words, the instability appears before nonlinearities become signicant in the base ow. Moreover, because volume viscous eects are O((m 2 + δ 2 i )Re -1 ) = O(Re -1 ) and surface viscous eects are O(Re -1/2 ) volume viscous eects will be neglected for the saturation of the resonant Kelvin mode amplitude. As shown by Gans (1970b), the resonant Kelvin mode of amplitude ε i generates a secondary ow at order O(ε i Re -1/2 ) in the core of the cylinder due to Ekman pumping in the viscous boundary layers. The correct scaling is obtained when this secondary ow is of the order of the forcing amplitude Ro. This gives a mode amplitude ε i = O(RoRe 1/2 ) which is Re 1/2 larger than the ow in the non-resonant case. Applying a solvability condition on the secondary ow allows to calculate analytically the amplitude of the resonant Kelvin

ε i = if Ro s Re 1/2 + i f ai , (3.12)
where the linear forcing parameter f is real and given in Appendix A. The surface viscous parameter s is a complex number with a positive real part and is given in Appendix B. The term 1/a i represents the detuning damping of the forced Kelvin mode if the frequency ratio ω is not exactly equal to a resonance frequency. This viscous amplitude is represented as a dashed line on gure 2(b) for Re = 3500. It is close to the inviscid amplitude ε i = Ro a i (solid line) far from the resonance and saturates at the resonance (i.e. when a i = ∞) at a nite value ε i = f Ro Re 1/2 /s represented by a point. It can be noted that the mode amplitude is maximum when the detuning of the forcing frequency compensates exactly the viscous detuning, i.e. f /a i = -Re -1/2 Im(s) and not at the exact resonance. At resonance, the base ow is an order Re 1/2 larger than far from the resonance and may thus be subject to stronger instabilities. In the following, the forcing frequency ω will be assumed to be a natural frequency of the cylinder such that the base ow is composed of a predominant Kelvin mode v 1,ω,k i with an amplitude ε i given by (3.12). For the sake of clarity the index i will be dropped such that the amplitude of the resonant Kelvin mode is noted ε instead of ε i . Its axial wavenumber is simply noted k and its radial wavenumber δ. We also assume that |RoRe 1/2 | 1 such that ε is a small parameter. This assumption allows to carry out an asymptotic and linear stability analysis of the base ow described here.

Linear stability analysis

As described in the literature (see [START_REF] Manasseh | Nonlinear behaviour of contained inertia waves[END_REF][START_REF] Kobine | Azimuthal ow associated with inertial wave resonance in a precessing cylinder[END_REF], when the Reynolds number or the Rossby number is increased above a certain threshold (which depends on h and ω), the ow inside the cylinder becomes unstable. PIV measurements in two dierent sections of the cylinder have revealed the three-dimensional structure of this instability (see [START_REF] Lagrange | Instability of a uid inside a precessing cylinder[END_REF]. It is composed of two free Kelvin modes whose wavenumbers and frequencies respect the conditions for a triadic resonance with the forced Kelvin mode, in a similar manner as for the elliptical instability (see [START_REF] Malkus | An experimental study of global instabilities due to tidal (elliptical) distortion of a rotating elastic cylinder[END_REF][START_REF] Eloy | Experimental study of the multipolar vortex instability[END_REF][START_REF] Kerswell | Elliptical instability[END_REF]. In this paper, a linear stability analysis based on such a mechanism is presented. It allows to predict the structure of the instability, its growth rate, and the stability threshold.

Governing equations for the perturbation

To perform a stability analysis, a perturbation dened by its four component and complex velocity-pressure eld v is added to the base ow. The total ow v in the precessing cylinder is thus the sum of the resonant base ow v 1,ω,k (given by (3.9)), whose amplitude ε is given by (3.12), the perturbation v and some lower order terms : the particular solution v part. and the non-resonant Kelvin modes

v 1,ω,k i v = εv 1,ω,k + v + v part. + i ε i v 1,ω,k i . (4.1)
We recall that the dominant term εv 1,ω,k is of order RoRe 1/2 . The other terms of the base ow (v part. and ε i v 1,ω,ki ) are of order Ro.

Substituting this decomposition (4.1) into (3.5) yields the equation satised by the perturbation v :

∂ ∂ t I + M v + c.c. = N ( v, εv 1,ω,k ) + N (εv 1,ω,k , v) + 1 Re L v + o.t. + c.c. (4.2)
We recall that I, M, N and L are operators given in Appendix A. The left-hand side term of equation (4.2) is the linear unsteady term of the homogeneous equation. The rst and second right-hand side terms represent the nonlinear interaction between the resonant Kelvin mode and the perturbation. They are of order O(RoRe 1/2 v). The third term is the classical viscous term for the perturbation. It is of order O(Re -1 v). The notation o.t. stands for `other terms' and includes the nonlinear interaction of the four-component vector v part. + ε i v 1,ω,ki + v with itself and the term Ro (D e i(ωt+ϕ) + c.c.)v. In these terms, the leading order is O(Ro v), which is negligible compared to the nonlinear terms of order O(RoRe 1/2 v) explicited in (4.2). Terms of order O(ReRo 2 ) and O( v 2 ) are also neglected in this linear analysis. Note that the forcing term Ro F 0 e i(ωt+ϕ) from (3.5) does not appear in equation ( 4.2) because it has been cancelled out by the unsteady term of the particular solution.

The ow is assumed to satisfy a no-slip boundary condition

v = 0 at the walls (r = 1 or z = ±h/2). (4.3)
The equation (4.2) and the boundary condition (4.3) describe the evolution of the linear perturbation v in the limit of small amplitudes ε.

In order to solve the system of equations (4.2-4.3) the four-component velocity-pressure eld v is expanded in powers of ε as follows

v = v (0) + εv (1) + O ε 2 . (4.4)
The equation for the growth rate of the instability is obtained by substituting the above expansion (4.4) into the NavierStokes equation (4.2) and examining its two rst orders.

Note that in this study we will consider the critical scaling Re -1/2 = O(ε) such that boundary viscous eects are of the same order as N ( v, εv 1,ω,k ) and N (εv 1,ω,k , v).

4.2. Description of the free Kelvin modes At order 0, equation (4.2) is the homogeneous equation

∂ ∂ t I + M v (0) = 0. (4.5)
Because viscous eects are of order Re -1/2 (boundary viscous eects) or Re -1 (volume viscous eects), an inviscid boundary condition is assumed at this order

v (0) • n = 0 at the walls (r = 1 or z = ±h/2). (4.6)
The resolution of the set of equations (4.5-4.6) is classical and its solution is a linear combination of free Kelvin modes. Each free Kelvin mode v j is composed of two travelling waves of opposite wavenumber k j in order to satisfy the boundary condition (4.6) at the top and bottom of the cylinder v j = u mj ,ωj ,kj (r) e i(ω j t+m j ϕ+k j z) ± u mj ,ωj ,-kj (r) e i(ω j t+m j ϕ-k j z) . (4.7)

We recall that the expression of the four-component vector u m j ,ω j ,k j is given in Appendix A. At this order the amplitude A j of a free Kelvin mode is undened but its azimuthal wavenumber m j , its frequency ω j and its axial wavenumber k j are connected through the dispersion relation D(m j , ω j , δ j ) = 0 given by equation (3.11) such that the radial velocity vanishes at r = 1. In gure 3 the rst branches of the dispersion relation are plotted for m j = 6 (solid line).

The condition of no normal ow at the top and the bottom discretizes the vertical wavenumber k j as a multiple of π/h. It also separates the free Kelvin modes into two categories with dierent parity, depending on the choice of sign between the two waves of opposite wavenumber k j in (4.7). The plus sign allows to consider free Kelvin modes whose axial velocity (resp. vorticity) is a sine (resp. cosine) function of z and the axial wavenumber satises k j = (2n -1)π/h with n an integer. The minus sign allows to consider free Kelvin modes whose axial velocity (resp. vorticity) is a cosine (resp. sine) function of z, with k j = 2nπ/h.

To examine the mechanism of triadic resonance, the perturbation v (0) will be assumed to be, at leading order, a combination of two free Kelvin modes v 1 and v 2 with unknown amplitudes

A 1 and A 2 v (0) = A 1 v 1 + A 2 v 2 , (4.8)
where the amplitudes are varying slowly with time such that time derivatives will appear at next order.

Inuence of the Ekman layers

It can be noted that the dispersion relation and the determination of the structure of the free Kelvin modes are obtained with an inviscid boundary condition on the cylinder walls (equation (4.6)). This condition does not match the viscous boundary condition given by equation (4.3). There will thus remain a wall parallel ow at the leading order for the free Kelvin modes. This problem can be solved by adding a viscous ow, in a viscous boundary layer of thickness O(Re -1/2 ) near the walls. However this resulting viscous ow has a component perpendicular to the walls, of order O(Re -1/2 ) called Ekman pumping. This pumping is an exponential decreasing function inside the Ekman layers. Note that in our problem we have an Ekman pumping due to the lateral wall and the top and bottom walls of the cylinder. The calculation of Ekman layers is classical. An exhaustive description is given in [START_REF] Greenspan | The theory of rotating uids[END_REF]. For the particular case of Kelvin modes, the reader could refer to [START_REF] Kudlick | On the transient motions in a contained rotating uid[END_REF].

Triadic resonance

Experiments have demonstrated (see [START_REF] Lagrange | Instability of a uid inside a precessing cylinder[END_REF]) that the instability of a uid inside a precessing cylinder is due to a triadic resonance between the resonant Kelvin mode εv 1,ω,k and two free Kelvin modes v 1 and v 2 . These three modes can be coupled by the nonlinear terms N ( v, εv 1,ω,k ) and N (εv 1,ω,k , v) in equation ( 4.2) when their Fourier components are chosen appropriately. The nonlinear interaction between the rst mode v 1 and the forced mode v 1,ω,k has a Fourier component e i(ω1t+m1ϕ±k1z) e i(ωt+ϕ±kz) . This term has a Fourier component e i(ω 2 t+m 2 ϕ±k 2 z) identical to that of the second mode v 2 only when the frequencies and wavenumbers satisfy the resonance condition

m 2 -m 1 = 1, (4.9a) ω 2 -ω 1 = ω, (4.9b) |k 2 -k 1 | = k. (4.9c)
If these conditions are satised, the nonlinear interaction between v 2 and the forced mode v 1,ω,k has also the same Fourier component as v 1 . It can be noted that the last equation of this system oers two possibilities which are

k 2 -k 1 = k and k 1 -k 2 = k.
However, in all our calculations the rst case has always been the most unstable one.

The rst condition imposes that the two free Kelvin modes have azimuthal wavenumbers separated by one. This is dierent from the elliptic instability where they should be separated by 2 [START_REF] Kerswell | Elliptical instability[END_REF]. This comes from the fact that, here, the forcing term has an azimuthal wavenumber m = 1, unlike the ellipticity which has an azimuthal wavenumber m = 2.

To nd Kelvin modes satisfying the resonance condition, we plot on the same graph the dispersion relation of the second mode v 2 and the dispersion relation of the rst mode v 1 translated by k along the abscissae and by ω along the ordinate. An example is shown in Fig. 3 for m 2 = 6 and m 1 = 5. At each crossing point between the dispersion relation of mode 1 (dashed lines) and mode 2 (solid line) the conditions of resonance are fullled. Since there is an innite number of branches for each mode and an innity of possible azimuthal wavenumber m 2 , there is a triple-innite number of resonances. We will label these resonant combinations (m 2 , l 1 , l 2 ), where l 1 (resp. l 2 ) is the branch number of the dispersion relation for the rst mode (resp. second mode). As an example, the lowest order resonant combination (m 2 = 6, l 1 = 1, l 2 = 1) is marked by a circle on Fig. 3. This combination will be found to be the most unstable and it will be extensively studied and compared to the experiments. It can be noted that the study can be restricted to free Kelvin modes with m j ≥ 0 because the dispersion relation (3.11) satises the following symmetry D(m j , ω j , δ j ) = -D(-m j , -ω j , δ j ).

An additional condition arises because the free Kelvin modes must have a zero normal velocity at the top and the bottom, which quanties the wavenumbers k j as multiples of π/h. It is the case in Fig. 3 where the resonant combination (6, 1, 1) is located on a dashdotted line (k 2 = 2π/h). This case of exact resonance only appears for specic aspect ratios (here h = 1.62) which allow an intersection between the three lines. The principal aspect ratios h allowing exact resonances are listed in Table 1 (resp. Table 2) for the combinations (m 2 , 1, 1) (resp. (m 2 , 2, 2)) when the rst Kelvin mode is excited at its rst resonance and where k 1 = π/h. Most of the resonances are obtained for small aspect ratios (for which the resonance frequency is large) and large azimuthal wavenumbers. Note that there does not exist any exactly resonant combinations (m 2 , 1, 1) (resp. (m 2 , 2, 2)) with m 2 ≤ 5 (resp. m 2 ≤ 3). This is why the instabilities observed in the experiments usually have high azimuthal wavenumbers. Finally, it can be noted that since the forced Kelvin mode is resonant, its axial wavenumber k is an odd multiple of π/h. Thus equation (4.9c) implies that the two free Kelvin modes have dierent parities with respect to z (i.e. free Kelvin modes with a dierent sign in equation (4.7)). By contrast, at the anti-resonance of the forced Kelvin mode, the axial wavenumber k is an even multiple of π/h. Thus the two free Kelvin modes have the same parities with respect to z. In this condition it can be shown that a mechanism of triadic resonance between Kelvin modes can not lead to the instability. Most of the time a combination (m 2 , l 1 , l 2 ) is not exactly resonant (see Manasseh 1996) because the crossing point is not on a dash-dotted line (i.e. k j = nπ/h, with n an integer). The free Kelvin modes thus do not satisfy the boundary condition (4.6) at the top and bottom of the cylinder and should not be considered. However, the problem can still be solved by introducing small detuning parameters ∆k 1 and ∆k 2 which measure the distance between the wavenumber k 1 (or k 2 ) and the closest nπ/h :

0 1 2 3 4 5 6 7 8 9 10 -1 -0.5 0 0.5 1 1.5 2 2 5 l = 2 1 l = 1 1 l = 1 5 l = (6,1,1) ω k                   2 6 m = 1 5 m = 1 m = • 2 k 2 ω h π 2 h π
∆k j = min n∈N * n π h -k j sgn n π h -k j . (4.10)
In gure 3 these detunings correspond to the relative distance between a crossing point and the closer vertical dash-dotted line. These detunings will only modify the problem at order ε if they are of order O(ε). Note that, as long as ω is a resonant frequency, the detuning parameters satisfy ∆k 1 = ∆k 2 . 1. Theoretical results considering the (m2, 1, 1) combinations with k2 = 2π/h and k1 = k = π/h. The combination is exactly resonant only for a specic aspect ratio h (given in the rst row) at its associated frequency ω1,1 (given in the second row) and for an azimuthal wavenumber of the second mode m2 (given in the third row). The fourth row gives the value of n1n2 which is the square of the inviscid growth rate. The last row gives the stability threshold without volume viscous eects and is calculated with equation ( 4

= k = π/h.
4.5. Amplitude Equations As mentioned previously, at leading order the perturbation v is a combination of two free Kelvin modes with unknown amplitudes. We now assume that A 1 and A 2 vary slowly with time such that ∂A j /∂t is of order ε. We will consider that this combination is not exactly resonant and that detuning eects ∆k j are also of order ε. This `critical scaling' allows to extend the theory in the presence of detuning eects (with this scaling detuning terms are of same order as destabilising terms). For larger detuning eects, the ow is no longer unstable.

At order ε equation (4.2) becomes

ε ∂ ∂ t I + M v (1) + c.c. = - ∂ ∂ t I + i∆k j M ∆ v (0) + N (v (0) , εv 1,ω,k ) + N (εv 1,ω,k , v (0) ) + Lv (0) Re + c.c. (4.11)
In equation (4.11) the left-hand side term is the linear operator applied to the solution at order ε. The rst right-hand side term is composed by the slow unsteady part of the perturbation of order 0 and by its detuning term. The operator M ∆ is dened in Appendix B and is obtained via a Taylor expansion of the Fourier component e i∆kjz of the free Kelvin modes. The second and third right hand side term represent the nonlinear interaction between the resonant Kelvin mode and the two free Kelvin modes. These terms represent the mechanism of triadic resonance. The last term of this equation is the volume viscous term. It should not appear at this order but at order ε 2 since ε = O Re -1/2 . However we have decided to take it into account in the analysis because its importance in saturating the growth rate of the instability has been shown in several papers [START_REF] Kerswell | On the viscous decay rates of inertial waves in a rotating cylinder[END_REF][START_REF] Eloy | Elliptic and triangular instabilities in rotating cylinders[END_REF][START_REF] Racz | Parametric instability in a rotating cylinder of gas subject to sinusoidal axial compression. part2. weakly nonlinear theory[END_REF].

The perturbation v (1) can be decomposed as

v (1) = 2 j=1 v (1) j = 2 j=1
u + j e i(ω j t+m j ϕ+k j z) ± u - j e i(ω j t+m j ϕ-k j z) . (4.12)

where the sign is chosen such that each component j has the same parity as the corresponding free Kelvin mode (a plus sign if there is a plus in (4.7)). In this case the left-hand side term in equation ( 4.11) has the same parity than the right-hand side terms.

Inserting the expression (4.12) for v (1) and the expression (4.8) for v (0) in (4.11) gives an equation for v j . A solvability condition is then obtained by forming the scalar product dened by (A 10) of v 1 (resp. v 2 ) with this equation. It yields two coupled amplitude equations for A 1 and A 2

∂A 1 ∂t = εn 1 A 2 - 1 Re 1/2 s 1 A 1 - 1 Re v 1 A 1 -iq 1 ∆k 1 A 1 , (4.13a) ∂A 2 ∂t = εn 2 A 1 - 1 Re 1/2 s 2 A 2 - 1 Re v 2 A 2 -iq 2 ∆k 2 A 2 . (4.13b)
The terms ∂A j /∂t come from the scalar product v j ∂ ∂t Iv j . The terms n 1 and n 2 are real and represent the interaction, through the nonlinear term of the NavierStokes equation, of a free Kelvin mode with the forced mode. These terms come from the scalar product v j [N (v (0) , εv 1,ω,k ) + N (εv 1,ω,k , v (0) )] + c.c. They are given in Appendix B.

The coecients s 1 and s 2 represent the surface viscous damping of the two free Kelvin modes due to Ekman boundary layers. They come from the rest of the scalar product 1) which almost vanishes because v j is in the kernel of the operator ∂ ∂ t I + M. This rest corresponds to the pressure of the free Kelvin mode p j times the normal velocity of v (1) j integrated over the cylinder walls. This normal velocity is given by the boundary condition (4.3) at order ε and is thus the opposite of the Ekman pumping associated with the free Kelvin modes, as explained in section 4.3. This is why these terms are proportional to Re -1/2 . The coecients s j are complex numbers with a positive real part and can be analytically calculated (see Appendix B and Kudlick 1966).

v j ∂ ∂ t I + M v (
The coecients v 1 and v 2 are real and represent the volume viscous damping of the two free Kelvin modes. They originate from the scalar product v j Lv j and can be decomposed into two parts : one proportional to k 2 j , another proportional to m 2 j as shown in Appendix B. The free Kelvin modes with a high azimuthal wavenumber are thus strongly damped by volume viscous eects. Although these terms should appear at higher order in the asymptotic expansion, they have be taken account here to allow for the selection of modes with the largest wavelengths.

The real coecients q 1 and q 2 represent the damping of the two free Kelvin modes by detuning eects. These terms originate from the scalar product v j M∆vj. They can be analytically calculated and are given in Appendix B. These terms vanish when the two free Kelvin modes are exactly resonant.

In the following, viscous and detuning eects are gathered in a single coecient (4.14) corresponding to the linear decay term.

α j = 1 Re 1/2 s j + 1 Re v j + iq j ∆k j ,
4.6. Growth rate of the instability Assuming that the amplitudes A 1 and A 2 are growing exponentially (A j ∼ e σt ), the equation for the complex growth rate σ is obtained by canceling the determinant of the linear system (4.13)

(σ + α 1 ) (σ + α 2 ) = |ε| 2 n 1 n 2 . (4.15)
This relation leads to an analytical expression for the complex growth rate. The temporal growth rate σ r of the instability is simply the real part of σ. The ow is thus unstable when σ r is positive for a given resonant Kelvin mode combination. The growth rate is a function of ε and Re only, or alternatively a function of Ro and Re.

For an exactly resonant combination (no detuning eect) and at innite Reynolds number, the linear saturation terms α j vanish such that the growth rate is simply given by σ r = |ε| (n 1 n 2 )

1/2 . As expected (based on similarities with the elliptic instability) it is proportional to the amplitude |ε| of the forced Kelvin mode. The ow is unstable if the coecient n 1 n 2 is positive, which is always true for Kelvin modes with dispersion relations of opposite slopes. This can be shown by looking at the signs of n 1 and n 2 (see Appendix B) and conrms the result by [START_REF] Fukumoto | The three dimensional instability of a strained vortex tube revisited[END_REF] proved with energetic methods. These coecients are given in Tables 1 and2 for the principal exact resonances. They increase drastically when the aspect ratio decreases. This comes from the fact that the velocity of the base ow increases and it does not mean that the precessing cylinder is more unstable as it will be seen later.

When linear saturation terms are introduced (α j = 0), the growth rate of the instability decreases. At innite Reynolds numbers only detuning eects play a role and the growth rate thus takes the simple form

σ r = |ε| 2 n 1 n 2 - 1 4 (q 1 ∆k 1 -q 2 ∆k 2 ) 2 1/2 . (4.16)
Detuning eects stabilize the ow and their inuence can be studied by varying the aspect ratio. Such a variation allows to observe several combinations (m 2 , l 1 , l 2 ) which become exactly resonant (i.e. ∆k j = 0) for particular values of h, listed in Table 1 and2.

For an exact resonance, detuning eects vanish and the instability is damped by viscous eects only. An asymptotic expression can be obtained for large Reynolds numbers (when

Im(α j ) ε(n 1 n 2 ) 1/2 ) σ r = |ε| (n 1 n 2 ) 1/2 - Re{s 1 + s 2 } 2Re 1/2 - v 1 + v 2 2Re , (4.17)
where the two damping terms are due to surface and volume viscous damping. These viscous eects stabilize the ow at low amplitude ε and thus allow to calculate the threshold of the instability. 4.7. Instability threshold Assuming that the real part of σ vanishes in (4.15) leads after some calculation to an expression for the amplitude ε crit at which the instability appears. Since this amplitude of the forced Kelvin mode is given by Eq. (3.12), we can thus determine the critical Rossby number at which the instability appears

Re Ro Re -3/2 Re -1 Re -1/2 Re -1 Re -2 Re 0 Stable ε f ∼Re -1 ε f ∼Re -1/2 ε f ∼Re 0 ε f ∼A 0f ∼Ro
|Ro crit | = 1 |f | α r 1 α r 2 n 1 n 2 1 + α i 1 -α i 2 α r 1 + α r 2 2 1/2 s Re 1/2 + if a i , (4.18)
where α r j and α i j are respectively the real and imaginary parts of α j dened in (4.14). We recall that f is the linear forcing parameter and s the surface viscous parameter of the forced Kelvin mode, given in Appendix A and B. The coecient a i corresponds to the inviscid amplitude of the forced Kelvin mode and is given in Appendix A. Note that equation (4.18) is not valid if α r 1 + α r 2 = 0, which means that the inviscid threshold is not the limit of the viscous threshold for large Re number. Dierent scalings for the critical Rossby number are obtained depending on the predominant damping eect in α j .

In the experiments, free Kelvin modes with fairly large azimuthal wavenumbers have been observed. In this case, the volume viscous damping of order (m 2 j + δ 2 j )Re -1 can be larger than surface viscous damping or order Re -1/2 if the Reynolds number is below a critical value dened by

Re 1 = v r 1 v r 2 s r 1 s r 2 1 + s i 1 -s i 2 2 (s r 1 + s r 2 ) 2 -1 . (4.19)
For Reynolds numbers between 1 and Re 1 , the critical Rossby number scales as Re -3/2 , as shown schematically in Fig. 4. A simple expression can be found in this regime

RoRe 3/2 crit = s f v 1 v 2 n 1 n 2 1/2 . (4.20)
At intermediate Reynolds numbers (Re Re 1 ), surface viscous eects become do-minant. The critical Rossby number then scales as Re -1 (see Fig. 4) and leads to an expression

|RoRe| crit = s f s r 1 s r 2 n 1 n 2 1 + s i 1 -s i 2 s r 1 + s r 2 2 1/2 . (4.21)
This regime is the most frequent in experiments and we thus give the values of the critical |RoRe| for the principal resonances in Tables 1 and2. The most unstable resonances (given by the smallest Rossby numbers) are found for the large aspect ratios h, although the inviscid growth rates (n 1 n 2 ) 1/2 are larger at small h. This is due to the strong increase of ε with h. When the combination is not exactly resonant, detuning eects can also damp the instability. They appear at high Reynolds numbers, when surface viscous terms become smaller than detuning terms, i.e. for Re Re 2 dened by

Re 2 = (s r 1 + s r 2 ) 2 + s i 1 -s i 2 2 (q 1 ∆k 1 -q 2 ∆k 2 ) 2 . (4.22)
In this regime, the critical Rossby number still depends on the Reynolds number since the forced Kelvin mode scales as Re 1/2 . The critical Rossby number thus scales as Re -1 and leads to a simple expression

RoRe 1/2 crit = s f (s r 1 s r 2 ) 1/2 s r 1 + s r 2 |q 1 ∆k 1 -q 2 ∆k 2 | (n 1 n 2 ) 1/2 . (4.23)
It can be noted that this expression is valid for Re asymptoting to innity and takes into account the values of the viscous coecients s j . It is dierent from the value that would be obtained at innite Reynolds number by assuming σ r = 0 in (4.16), but the two theories give identical threshold when s 1 = s 2 .

Discussion

In this section we discuss the general properties of the precessional instability as a function of the dimensionless parameters Re, h and ω and we compare these predictions to experimental results for a particular aspect ratio. We limit our study to the instabilities driven by the rst forced Kelvin mode.

Prediction of the unstable modes

In our previous paper [START_REF] Lagrange | Instability of a uid inside a precessing cylinder[END_REF], we have observed experimentally that the unstable ow exhibits two free Kelvin modes with azimuthal wavenumbers 5 and 6 at the rst resonance of the rst Kelvin mode (h = 1.62 and ω = 1.18). Their experimental axial and instantaneous vorticity elds are plotted in gures 5(a) and 5(b). They correspond exactly to the free Kelvin modes predicted theoretically for this aspect ratio. Indeed, the resonant combination (6, 1, 1) is more unstable than any other combinations (with dierent azimuthal wavenumbers m j and branch numbers l j ). The theoretical axial vorticity elds of the free Kelvin modes are shown in gures 5(c) and 5(d ) and look very similar to the experimental elds. They both have one ring of 10 (or 12) alternate vortices meaning that they correspond to the rst branch of the dispersion relations (l 1 = l 2 = 1) for m 1 = 5 and m 2 = 6. Furthermore, the radial position of these vortices seems to be well predicted : the maximum of the azimuthal average vorticity is located theoretically at r = 0.80 and experimentally at r ≈ 0.83 for the second mode with m 2 = 6. The agree- ment is not as good for the rst mode with m 1 = 5 (r = 0.70 theoretically compared to r ≈ 0.57 experimentally) because of the presence of the forced Kelvin mode (m = 1).

To better compare the structure of the unstable modes we have plotted in gure 6 the theoretical and the experimental averaged radial and azimuthal velocities of the free Kelvin modes as a function of r. They have been obtained by taking the azimuthal and the temporal average of the velocity elds and they are given in arbitrary units. The averaged radial velocity proles of the free Kelvin modes are very similar and look like bell-shaped proles. Because the two free Kelvin modes satisfy a condition of no outward ow at r = 1, their averaged radial velocities vanishes at r = 1. The averaged azimuthal velocity proles are more complicated. The averaged azimuthal velocity prole of the free Kelvin mode m 1 = 5 shows two extrema while for the mode m 2 = 6, it shows one extremum. Experiments roughly conrm these two dierent averaged azimuthal velocity proles.

Finally, theory also predicts that the free Kelvin mode m 1 = 5 (resp. m 2 = 6) has an axial vorticity which is a sine (resp. cosine) function of the altitude z. This parity is conrmed by experiments since the free Kelvin mode m 1 = 5 (resp. m 2 = 6) has been observed at z = 0 (resp. z = h/4).

These experimental observations conrm the mode predicted by the linear stability analysis and therefore validate the mechanism of triadic resonance proposed in this paper.

Prediction of the growth rate

In this section, we compare the theoretical growth rate σ r , with the experimental results. We restrict our analysis to the rst resonance of the rst Kelvin mode for h = 1.62, such that there is no detuning eects.

Figure 7 represents the evolution of the growth rate as a function of the forcing. In the inviscid case, the theory gives a constant value σ r /|ε| = (n 1 n 2 )

1/2 represented by a solid line. When surface viscous eects are added, it can be easily shown (dividing (4.15) by |ε| 2 ) that the rescaled growth rate σ r /|ε| is a function of |ε|Re 1/2 only. This is why we have plotted the growth rate in these coordinates. This viscous prediction is plotted as a dash-dotted line in Fig. 7. The growth rate increases with the forcing and tends toward the inviscid growth rate at innite Reynolds number. It vanishes at a specic value of |ε|Re 1/2 = 0.843 corresponding to the critical Rossby number (given in Table 1). This prediction is only valid for Reynolds numbers large compared to Re 1 . Below this Reynolds number, volume viscous eects must be taken into account. The theoretical prediction with all viscous terms is plotted as a dashed line for Re = 6000 and as a dotted line for Re = 1000. They show the same trend but remain weaker.

Experimental measurements of the growth rate are also plotted as symbols on the same graph. As expected, experimental measurements are below the growth rate with surface viscous terms only. They are represented as lled (resp. open) symbols when their Reynolds number is larger (resp. smaller) than the Reynolds number of the dashed line (Re = 6000). The symbols always fall within the correct band of growth rate, which is a good conrmation of the theoretical calculation. A quantitative comparison indicates that the measured growth rate is never farther than 50% from the theoretical prediction.

Experimentally, the growth rate was measured by an exponential t of the amplitude of the free Kelvin mode m 1 = 5 at the onset of the instability. Note that the instability can appear before the amplitude |ε| of the forced Kelvin mode has reached its stationary value given by equation (3.12). So here |ε| corresponds to the measured amplitude of the rst Kelvin mode at the onset of instability and not the value given by (3.12).

Critical Rossby number as a function of Re

Figure 8 represents the critical Rossby number Ro crit as a function of Re when the rst Kelvin mode is forced at its rst resonance. The prediction is issued from equation (4.18) and is represented by a solid line which divides the (Re, Ro)-plane into a stable domain and an unstable domain. Stable experiments are represented by circles and unstable experiments are represented by black symbols. These experiments were done on a very large range of precessing angle (0.25 • to 10 • ) and Reynolds number (10 3 to 5 × 10 4 ). It is striking to see such an agreement between experiments and theory with no tting parameter in the theory. However, there are some slight discrepancies at high Rossby numbers. They may come from non-linear eects in the base ow which arise because the amplitude of the forced Kelvin mode ε ∼ RoRe 1/2 is no longer a small parameter. Note also that the uncertainty is larger at small Rossby number because the precessing angle (equal to 0.25 • ) becomes comparable to the uncertainty of ±0.1 • .

Because the Reynolds number is varied over a large range (one decade), the experiments 5.4. Critical RoRe number as a function of h Now that the theory has been validated experimentally by the two previous sections, we use these predictions to study the general properties of the precessional instability as the parameters h and/or ω are varied. In this section, the aspect ratio is varied over a large range, from 0.2 to 10. However, the frequency ω always corresponds to the rst resonance of the rst Kelvin mode ω 1,1 .

Figure 9 shows the evolution of the threshold as a function of h. Several bands of instability corresponding to dierent resonant Kelvin mode combinations (m 2 , 1, 1) are observed. Each band reaches a minimum when the combination is exactly resonant, i.e. when detuning eects vanish. The most unstable combination (i.e. minimum threshold) is the (6, 1, 1) resonant combination for h = 1.62, studied experimentally in this paper.

We have seen in section 4.7 that, when volume viscous eects are neglected, the stability threshold only depends on the product RoRe. This is why we have plotted the threshold as |RoRe| in gure 9, such that the prediction without volume viscous eects is independent of the Reynolds number and plotted as a symbol for each exactly resonant combination. The gap between the symbol the corresponding solid line indicate the inuence of the volume viscous eects at Re = 6500.

The combinations (2, 1, 1) and (3, 1, 1) are special because they are not close to an exact resonance (which ensures a small detuning) although they are the most unstable for small aspect ratios. . Critical |RoRe| number of the most unstable modes as a function of h for the rst resonance of the rst Kevlin mode (ω = ω1,1). Each solid line corresponds to the full theory at Re = 6500 for a resonant combination (m2, 1, 1). Symbols correspond to the predictions without detuning and without volume viscous eects (thus valid at any Reynolds number, see Table 1) for the same combinations with m2 = 6 (•, ), m2 = 7 ( ), m2 = 8 (×), m2 = 9 (•), m2 = 10 (+), m2 = 11 ( * ), m2 = 12 ( ), m2 = 13 ( ). The dashed line (resp. dotted line) corresponds to a combination (3, 1, 1) (resp. (2, 1, 1)) which can not be exactly resonant.

Finally note that some resonant Kelvin mode combinations (m 2 , 2, 2) (not represented in gure 9) can be more unstable than the combinations (m 1 , 1, 1) represented in very small intervals of aspect ratios when h < 1.

Critical Rossby number as a function of ω

In this section the aspect ratio and the Reynolds number are xed and the frequency ω is varied around the rst resonance of the rst Kelvin mode. Figure 10 represents the critical Rossby number as a function of ω, for h = 1.62 and Re = 6500. Several bands of instability corresponding to dierent azimuthal wavenumbers m 2 = 7 (dotted line), m 2 = 6 (solid line) and m 2 = 5 (dashed line) are predicted. This is due to the change of ω which allow to couple dierent Kelvin modes (since ω 2 -ω 1 = ω). These predictions are well conrmed by PIV measurements which reveal the correct azimuthal wavenumber. Moreover, the theoretical threshold is in very good agreement with experiments.

The most unstable combination is the (6, 1, 1) mode and is obtained close to ω = 1.18. This can be easily understood because this is where the amplitude ε of the forced Kelvin mode is maximum (at its resonance). Around the resonance, the amplitude of the forced Kelvin mode decreases as 1/(ω -ω 1,1 ), which lead to weaker instabilities for the combinations (5, 1, 1) and (7, 1, 1).

This proves that the easiest way to trigger an instability is to force the base ow at a resonance frequency. For example, here, a precessing angle of θ ≈ 0.5 • is sucient to observe the (6, 1, 1) resonant Kelvin mode combination. Outside of the resonant frequency it is necessary to have a minimum precessing angle of θ ≈ 7 • (resp. θ ≈ 2 • ) to observe the (7, 1, 1) (resp. (5, 1, 1)) resonant Kelvin mode combination at the same Reynolds number.

Critical Rossby number as a function of h and ω

In this section we study the general problem of the precessional instability at a given Reynolds number when both the aspect ratio and the frequency are varied. We consider only instabilities triggered by the rst forced Kelvin mode. The critical Rossby number in the plane (h, ω) is represented in gure 11. It was obtained numerically by calculating the analytical coecients of the amplitude equations (4.13) for various azimuthal wavenumbers (m 2 = 1 to 15) and for various branches (l j = 1 to 5) on a very ne mesh of the plane (h, ω). This calculation took about a month on a standard computer.

On this gure, the most unstable regions correspond to the dark areas, i.e. where the critical Rossby number is small. There are two unstable regions, which are located around the resonances of the rst Kelvin mode, represented by white dashed lines. As explained in the previous section, this can be easily understood because the amplitude of the forced Kelvin mode ε is maximum there.

The black thin lines separate dierent resonant Kelvin mode combinations with azimuthal wavenumbers m 2 = 5 to 8. It is surprising to see that each combination creates a band which is almost parallel to the resonance curve. This explains why there are only a few combinations which are resonant (here only 4 combinations) when varying the aspect ratio. The combination (6, 1, 1) is the most unstable one because it is centered on the resonance curve almost over the whole range of aspect ratios considered here. It is curious to see that it is exactly resonant twice (for h = 1.62 and h = 3.6).

At higher Reynolds numbers, other combinations might become more unstable and thus create other bands of instabilities in between these combinations. However, one can have a rough idea of the stability diagram at any Reynolds number by simply assuming that the critical Rossby number is inversely proportional to the Reynolds number (as it is the case when surface viscous terms are considered).

Note that the previous gures correspond to the critical Rossby number along the white dashed line (for Fig. 9) and along the black dashed line (for Fig. 10). Note also that the combinations represented in gure 11 always satisfy k 2 -k 1 = k (combinations satisfying k 1 -k 2 = k are always more stable).

Weakly nonlinear theory

This section is devoted to the prediction of the instability saturation by nonlinear eects. The analysis will be restricted to the most unstable case, i.e. the exact resonance of the rst Kelvin mode (h = 1.62, ω = 1.18).

Geostrophic ow

Experiments have shown that the unstable Kelvin modes m 1 = 5 and m 2 = 6 mentioned previously give rise to a mode with a cylindrical symmetry which corresponds to a stationary azimuthal velocity eld. This geostrophic Kelvin mode appears at order O A j 2 Re -1/2 . It is due to the nonlinear interaction in the Ekman layers of the unstable Kelvin modes with their respective viscous ows (see [START_REF] Meunier | A rotating uid cylinder subject to weak precession[END_REF]. The role of the geostrophic mode is essential in the weakly nonlinear analysis because it saturates the amplitude of the resonant and the unstable Kelvin modes. Note that the nonlinear interaction of a Kelvin modes with itself has also a saturating eect. However, in our experiments, this saturating eect is less signicative than the saturation due to the geostrophic mode.

Finding an analytical expression of the geostrophic ow is a very complex problem. To avoid this diculty and to keep in the analysis the fewest number of mode possible, an empirical formulation has been used for the geostrophic mode based on experimental measurements. For h = 1.62 and ω = 1.18 (i.e. rst resonance of the rst Kelvin mode) experiments have shown that the free Kelvin modes m 1 = 5 and m 2 = 6 lead to a geostrophic Kelvin mode whose prole is close to the prole of the free Kelvin mode

v 0 = -J 5 (d 2 r) u ϕ , (6.1)
where u ϕ is the orthoradial unit vector and J 5 is the Bessel function of the rst kind.

The parameter d 2 is the second root of J 5 (i.e. d 2 = 12.339). We will assume that this geostrophic mode is added to the perturbation with an amplitude A 0 .

6.2. Weakly nonlinear amplitude equations Adding this geostrophic mode leads to a total ow

v = εv 1,ω,k + A 1 v 1 + A 2 v 2 + A 0 v 0 + o.t. (6.2)
We recall that ε is the amplitude of the forced Kelvin mode v 1,ω,k given by (3.9). Vectors v 1 and v 2 are the two free Kelvin modes of the triadic resonance, whose amplitudes are A 1 and A 2 . These vectors are given by equation (4.7). The notation o.t. stands for 'other terms' and includes the Ekman layer pumping ows and the non-resonant Kelvin modes. Inserting (6.2) into the NavierStokes equation (3.5) and forming the scalar product of this equation with v 1,ω,k , v 1 , v 2 and v 0 , leads to the following weakly nonlinear amplitude equations

∂ε ∂t = if Ro -αε -iξA 0 ε + λA 1 A 2 , ( 6.3a
)

∂A 1 ∂t = εn 1 A 2 -α 1 A 1 -iξ 1 A 0 A 1 -iσ 1 |A 1 | 2 A 1 , (6.3b) ∂A 2 ∂t = εn 2 A 1 -α 2 A 2 -iξ 2 A 0 A 2 -iσ 2 |A 2 | 2 A 2 , (6.3c) ∂A 0 ∂t = 1 Re 1/2 - 2 h A 0 + χ 1 |A 1 | 2 + χ 2 |A 2 | 2 . (6.3d )
The rst equation (6.3a) governs the temporal evolution of the forced Kelvin mode amplitude ε. All the coecients of this equation can be calculated analytically. The rst right hand-side term is the forcing due to the precessional motion, with a linear forcing parameter f given in Appendix A. The second term is due to the surface viscous damping of the amplitude of the forced Kelvin mode and is thus proportional to α = s/Re 1/2 (s being given in Appendix B). The third term comes from the nonlinear interaction of the forced Kelvin mode with the geostrophic mode. The parameter ξ can be calculated knowing the radial prole of the geostrophic mode given by (6.1) (see Appendix C). The last term is the saturating term due to the nonlinear interaction between the two free Kelvin modes with a parameter λ given in Appendix C. In the absence of instability, the amplitude ε grows and saturates due to viscous damping at a value if Ro/α. Equations (6.3b) and (6.3c) represent the temporal evolutions of the free Kelvin mode amplitudes A 1 and A 2 . All the parameters of these equations can also be calculated analytically. These equations are similar to the linear amplitude equations (4.13a) and (4.13b) with nonlinear eects added. The rst right hand-side term of these equations represents the nonlinear interaction of the forced Kelvin mode with a free Kelvin mode (n 1 and n 2 are given in Appendix B). The surface, volume viscous and detuning parameters are included in α 1 and α 2 given by equation (4.14). The third term of these equations was not considered before and comes from the nonlinear interactions of the free Kelvin modes with the geostrophic mode (parameters ξ j are given in Appendix C). The last f s ξ λ 0.226 1.86 -0.42i 0.165 6.503 3. Numerical values for the parameters appearing in the nonlinear amplitude equations (6.3). For these values, h = 1.62 and ω = 1.18 (rst resonance of the rst Kelvin mode). They correspond to the (6, 1, 1) resonant combination for the case

n 1 s 1 v 1 ξ 1 σ 1 χ 1 -1.
k 2 -k 1 = k.
term of equations (6.3b) and (6.3c) correspond to the interactions of a free Kevin mode with itself (parameters σ j are given in Appendix C).

Equation (6.3d) describes the evolution of the geostrophic Kelvin mode amplitude A 0 . The rst right hand-side term represents the surface viscous damping of the geostrophic Kelvin mode. Note that for this mode volume viscous eects have been neglected relative to surface viscous eects. The second and the third terms of this equation represent the nonlinear interaction of each free Kelvin mode with its Ekman pumping ow. The parameters χ 1 and χ 2 are dicult to calculate analytically. Because modes m 1 = 5 and m 2 = 6 are similar we shall assume that χ 1 and χ 2 are equal. Their value will be adjusted to t the experimental data.

We thus have a system of four nonlinear amplitude equations with only one tting parameter χ 1 = χ 2 . The numerical values of the parameters are given in Table 3. We will analyse in the following the properties of this dynamical system and compare them to experimental results.

Nonlinear evolution of the amplitudes

A numerical solution of the system (6.3) is shown in gure 12(a). This gure represents the temporal evolution of the amplitude of the free Kelvin mode m 1 = 5 for three Reynolds numbers. This corresponds to the most unstable case of the rst resonance of the rst Kelvin mode (h = 1.62, ω = 1.18), for which the numerical values of the parameters are given in Table 3.

In the stable regime (Re = 3500, •), the amplitude of the rst Kelvin mode A 1 remains equal to 0. The amplitudes of the second Kelvin mode A 2 and the geostrophic mode A 0 are equally null. However, the amplitude ε of the forced Kelvin mode grows and saturates at the viscous value given by equation (3.12) with a i = ∞.

Just above the instability threshold (Re = 4500, solid line) the amplitude A 1 grows and then saturates at a given value. This unstable and stationary ow thus exhibits a xed point that will be noted (ε f , A 1f , A 2f , A 0f ). Numerically, the instability saturation is found to be caused mainly by the nonlinear the term iξA 0 ε. In other words, the geostrophic mode is responsible of the saturation because it reduces the amplitude of the forced Kelvin mode.

At higher Reynolds number (Re = 6000, dashed line), the ow is still unstable but we also observe that the amplitudes are now oscillating, even in the permanent regime. This means that the xed point (ε f , A 1f , A 2f , A 0f ) has become unstable and the ow is said to be unstable and intermittent. Such a ow can be understood with the following reasoning. First, the amplitudes A 1 and A 2 grow exponentially due to the terms εn 1 A 2 and εn 2 A 1 in equations (6.3b) and (6.3c). Then, the geostrophic mode is forced through the coecients χ 1 and χ 2 appearing in equation (6.3d). This geostrophic mode grows slowly in time (since its characteristic time is of the order of hRe 1/2 ), which creates a delay to the saturation of the instability. This saturation is achieved by a decrease of ε through the coecient ξ in equation (6.3a). As soon as the amplitude ε gets smaller than the critical value for the onset of instability, the amplitudes of the free Kelvin modes vanish quickly. It leads to a slow decrease of A 0 and consequently to an increase of ε.

Then the instability can grow up again. This phenomenon being repetitive, amplitude oscillations are observed in time. Our experiments have clearly shown that the amplitude oscillations are due to the geostrophic term in equation (6.3a), which is delayed compared to the amplitudes of the free Kelvin modes. Figure 12(b) shows three experimental measurements of |A 1 | at the same Reynolds numbers as in gure 12(a). We nd again the three dierent regimes depending on the Reynolds number : stable, unstable and stationary, unstable and intermittent. The transition from stable to unstable ow is very well predicted theoretically, since it corresponds to the linear threshold of the instability : it was plotted on the stability diagram of Fig. 8 in the last section. The transition from an unstable stationary ow to an unstable intermittent ow is also in excellent agreement with the predictions. Indeed, this transition has been plotted in Fig. 8 as a dashed line. It clearly separates the stationary experiments ( ) from the intermittent experiments ( ). This agreement validates the nonlinear system (6.3).

However, the theory does not predict very well the quantitative value of the xed point A 1f in the stationary case : the experimental value is twice larger than the theoretical one. In a similar way, in the intermittent case, the amplitude of the oscillations are not very well predicted by the theory. Nevertheless, the theoretical period of these oscillations t th = 250 matches well the experimental one t exp = 290.

Fixed point

The system (6.3) admits two xed points. The rst one is trivial and corresponds to A 1f = A 2f = A 0f = 0 and ε f given by equation (3.12) with a i = ∞. The second one is obtained by looking for a solution of the system (6.3) which satises ∂ ε/∂ t = ∂ |A j | /∂ t = ∂ A 0 /∂ t = 0. This solution can be obtained analytically and its complete calculation is reported in Appendix C. In this section we only give asymptotical results by considering that viscous and detuning eects or geostrophic and nonlinear eects are dominant for the saturation of the instability.

Just above the threshold, the viscous and the detuning eects are stronger than geostrophic and nonlinear eects for the damping of the instability. Under this assumption, the terms ξ j and σ j can be neglected in equations (6.3b) and (6.3c). However, the amplitude of the forced Kelvin mode ε is still aected by the geostrophic and nonlinear term ξεA 0 . Then, a simple expression for the xed point ε f can be obtained

ε f = α r 1 α r 2 n 1 n 2 1 + α i 1 -α i 2 2 (α r 1 + α r 2 ) 2 1/2 . (6.4)
We recall that α r j and α i j are respectively the real and imaginary parts of the linear saturating term α j given in equation (4.14).

As for the instability threshold, the linear saturation term can be either volume viscous eects (Re Re 1 ), surface viscous eects (Re 1 Re Re 2 ) or detuning eects (Re Re 2 ). This leads to various scalings for the amplitude of the forced Kelvin mode ε f at the xed point. These scalings are indicated on Fig. 4.

Far from the threshold, the geostrophic and the nonlinear eects are stronger than the viscous and the detuning eects. Under this assumption, the terms α j can be neglected in equations (6.3b) and (6.3c), which leads to a simple expression for the xed point

ε f = (α r 1 α r 2 ) 1/2 |α r 1 + α r 2 | Rof (n 1 n 2 ) 1/2 ξ 1 -ξ 2 ξ 1/2 . (6.5)
The denition of the parameters ξ j is given in Appendix C. A simple expression for the xed point A 0f can also be obtained

A 0f =   |(α r 1 + α r 2 )| (α r 1 α r 2 ) 1/2 Rof (n 1 n 2 ) 1/2 ξ ξ 1 -ξ 2   1/2 . (6.6)
This xed point does not depend on the Reynolds number : it is completely determined by the nonlinear terms. From equations (6.5) and (6.6) we observe that ε f and A 0f scale as ∼ Ro 1/2 . The transition between the two regimes is shown schematically in gure 4 as a dashed line. It is simply found by equating (6.5) and (6.4) and leads again to three dierent scalings depending on the Reynolds number (see Fig. 4).

6.5. Fixed point and mean ow In this section we compare the theoretical xed point with the experimental results. The exact value of the xed point A 1,f can be calculated by solving numerically the polynomial equation (D 10). It is plotted in Fig. 13(a) as a function of Ro for Re = 3664 (thick line). We observe that the instability is weakly subcritical because the xed point can exist for Rossby numbers slightly smaller than the linear threshold represented by the rst vertical dash-dotted line. However, we were not able to assess the nature of the For high Reynolds numbers (i.e. Re Re 1 ) volume viscous eects can be neglected compared to surface viscous eects. Under this assumption (and in the absence of detuning eects), it can be shown that Re 1/4 A 1f only depends on ReRo by rescaling the nonlinear equations. This is why we have plotted this quantity in Fig. 13(b) in order to collapse all the experimental results on a single curve. The theoretical value of the xed point (shown as a line) increases and scales as Ro 1/4 far from the threshold.

The mean amplitude < |A 1 | > found experimentally is also plotted on the same graph and we observe that this mean amplitude is slightly overestimated by the xed point A 1f , especially at low ReRo. This may come from the eect of volume viscous terms which are neglected in this theory. However, the scaling for the mean ow is well predicted for high ReRo numbers. This means that the mean ow inside a precessing cylinder can be fairly well predicted even for very high Reynolds number (up to Re 40, 000). It is surprising to see that this weakly nonlinear model is still ecient one decade above the threshold of the instability, in a regime where the ow is turbulent. Figure 14(a) compares the xed point ε f with the experimental mean amplitude |ε| of the forced Kelvin mode. As above, neglecting the volume viscous terms implies that Re 1/2 ε f only depends on ReRo, which allow to collapse all the experimental results on a single curve. Below the threshold (rst dashdotted line, |ReRo| < 7.11) the ow is stable such that ε f is equal to the viscously saturated amplitude (3.12), which is plotted as a thick dashdotted line (extended by a dotted line above the threshold). Above the threshold the ow is unstable such that ε tends to the xed point ε f represented by a solid line. This xed point becomes unstable above the second dashdotted line (|ReRo| = 11.2), meaning that the ow becomes intermittent. The xed point is then plotted as a dashed line and scales as ε f ∼ Ro 1/2 (independently of the Reynolds number).

By comparing the xed point and the experiments one can observe that even if the experimental values are underestimated at low ReRo, the discrepancy decreases at high ReRo. Moreover, the scaling for the mean ow is fairly well predicted. As explained previously this good agreement is surprising because the ow is turbulent at these high Reynolds numbers.

Finally, Fig. 14(b) compares the xed point A 0f with the experimental mean amplitude A 0 of the geostrophic Kelvin mode. Once again, neglecting the volume viscous terms allow to collapse all the results on a single curve Re 1/2 A 0 as a function of ReRo. Below the threshold (rst dashdotted line, |ReRo| < 7.11) the ow is stable such that A 0f = 0.

Above the threshold the xed point (solid line) increases and becomes unstable above the second dashdotted line (dashed line). The xed point scales as A 0f ∼ Ro 1/2 at high Reynolds numbers. Even if the experimental values are strongly overestimated, the scaling for the mean ow is correctly predicted at high ReRo.

We have shown in this section that a weakly nonlinear model is able to predict the properties of the instability at the threshold but also the characteristics of the ow at high Reynolds number in a very turbulent regime.

Conclusion

In this paper the instability of a uid inside a precessing cylinder has been theoretically studied and compared with experiments.

Precession forces a Kelvin mode which is resonant for particular frequencies and thus is dominant compared to the rest of the ow. A linear stability analysis based on a mechanism of triadic resonance between Kelvin modes has been carried out and allows to obtain an analytical expression for the instability growth rate and threshold. We have shown that it is damped by volume viscous eects at low Reynolds numbers and by surface viscous eects due to Ekman layers at high Reynolds numbers. We have also shown that when a Kelvin mode combination is not exactly resonant the growth rate of the instability is also damped by detuning eects.

The structure of the unstable Kelvin modes, the growth rate of the instability and the stability diagram are in excellent agreement with experimental results. We deduce from this theory the general properties of the precessing instability for any aspect ratio and precession frequency. Several bands of instabilities with dierent azimuthal wavenumbers are predicted, which are conrmed experimentally. The instability always occur close to a resonance of a forced Kelvin mode since the amplitude of the base ow is larger than far from a resonance.

A weakly nonlinear analysis has also been carried out by taking into account the interactions between the two free Kelvin modes. The instability is saturated by the presence of a geostrophic mode which lowers the amplitude of the forced Kelvin mode. This weakly nonlinear model allows to predict the nature of the bifurcation and the properties of the unstable saturated ow just above the threshold. This model also predicts correctly the presence of an intermittent ow at higher Reynolds number. Finally, it is surprising to see that this low order model gives correct estimates for the mean velocity inside the cylinder at very high Reynolds numbers even when the ow is turbulent.

The nonlinear model is essentially based on the fact that the nonlinear interaction of a Kelvin mode with itself in the Ekman layers leads to the presence of a geostrophic mode. However, this interaction vanishes in the inviscid case and viscous eects are needed in order to predict the generation of the geostrophic mode. In this paper, this term was adjusted to the experiments by a tting parameter. A correct analysis of nonlinear and viscous interaction of Kelvin modes would thus be highly useful in order to get analytical expressions for the generation of geostrophic motion.

In the future, it would be interesting to see if this instability mechanism can be observed in an ellipsoid, as predicted theoretically by [START_REF] Kerswell | The instability of precessing ow[END_REF]. If it is the case, we expect the scalings observed in Fig. 4 and Fig. 8 to be still valid although the quantitative values will have to be calculated in a dierent manner. This could highly improve the comprehension of the ow inside the outer core of the earth and thus shed light on the geodynamo that takes place there.

It would also be interesting to see how these Kelvin modes interact with a magnetic eld if the liquid is conductor. Since the structure of the unstable ow is here predicted for any Reynolds number, a theoretical study of the growth of the magnetic eld in such a ow is now possible. This could explain the results by Gans (1970a) who observed an unstable magnetic eld in a precessing cylinder full of liquid sodium.

Finally, the destabilization of a uid-lled gyroscope by precession has been extensively studied under the assumption that the ow inside the gyroscope is stable and laminar. These results can now be extended to the case of an unstable uid cylinder owing to the model presented in this paper.

We would like to thank Wietze Herreman for very fruitful discussions, especially concerning the weakly nonlinear theory. This study was carried out under CEA-CNRS contract No. 012171.

Annexe A. Base ow calculation

The operators appearing in the complex NavierStokes equation (3.5) are dened by

I =     1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0     , (A 1) D =     0 0 -i 0 0 0 1 0 i -1 0 0 0 0 0 0     , (A 2) L =     ∆ -1 r 2 -2 r 2 ∂ ∂ϕ 0 0 2 r 2 ∂ ∂ϕ ∆ -1 r 2 0 0 0 0 ∆ 0 0 0 0 0     , ( A 3)
where (A 5)

∆ = 1 r ∂ ∂r + ∂ 2 ∂r 2 + 1 r 2 ∂ 2 ∂ϕ 2 + ∂ 2 ∂z 2 , ( A
The vectors F 0 and N(v 1 , v 2 ) are dened by

F 0 =     0 0 -rω 0     , ( A 6)
and

N (v 1 , v 2 ) = (v 1 + v 1 ) × (∇ × v 2 ) 0 . (A 7)
The linear and inviscid amplitude a i of the Kelvin modes v 1,ωi,ki (r) inside a precessing cylinder in the nonresonant case is

a i = ω 2 (ω -2) (k 2 i + 1) k i J 1 (δ i ) cos (k i h/2)
.

(A 8)

The vector u m i ,ω i ,k i (r) of a Kelvin mode appearing in equation (3.9) is dened by

u m i ,ω i ,k i (r) =     U m i ,ω i ,k i (r)
V mi,ωi,ki (r) W mi,ωi,ki (r) P mi,ωi,ki (r)

    =       -1 4-ω 2 i ω i δ i J m i (δ i r) + 2 mi r J m i (δ i r) -i 4-ω 2 i 2δ i J mi (δ i r) + ω i m i r J m i (δ i r) i k i ω i J m i (δ i r) -iJ mi (δ i r)       . (A 9)
We introduce the scalar product

X Y = V X r Y r + X ϕ Y ϕ + X z Y z + X p Y p d 3 V , ( A 10)
The forcing term f appearing in equation (3.12) is dened by

f = 1 i v 1,ω,k F 0 e i(ωt+ϕ) v 1,ω,k Iv 1,ω,k . (A 11)
For a resonant Kelvin mode, a simple expression of f is

f = (ω + 2) ω 4 -ω 2 2 δ 2 hJ 1 (δ) [ω 2 (ω + 2δ 2 -2) -4ω + 8] . ( A
n 1 = v 1 [N (v 1,ω,k , v 2 ) + N (v 2 , v 1,ω,k )] v 1 Iv 1 , ( B 1a
)

n 2 = v 2 [N (v 1,ω,k , v 1 ) + N (v 1 , v 1,ω,k )] v 2 Iv 2 , ( B 1b)
where the vectors v 1,ω,k and v j are respectively given by equations (3.9) and (4.7). Note that the denition of n j does not depend on the choice of the sign plus or minus in equation (4.7). By using the formula ∇×v K = 2 iω ∂ ∂z v K for a Kelvin mode v K , a development of (B 1a) and (B 1b) can be given as

n 1 = -8πh ± k 2 ω2 -k ω v 1 Iv 1 1 0 U m,ω,±k U m1,ω1,k1 U m2,ω2,k2 V m,ω,±k V m 1 ,ω 1 ,k 1 V m 2 ,ω 2 ,k 2 W m,ω,±k W m1,ω1,k1 W m2,ω2,k2
rdr, (B 2)

and

n 2 = 8πh ± k1 ω1 -k ω v 2 Iv 2 1 0 U m,ω,±k U m1,ω1,k1 U m 2 ,ω 2 ,k 2 V m,ω,±k V m 1 ,ω 1 ,k 1 V m 2 ,ω 2 ,k 2 W m,ω,±k W m1,ω1,k1 W m 2 ,ω 2 ,k 2 rdr, ( B 3)
where the operator |.| is the determinant. The plus sign corresponds to the case k 2 -k 1 = k.

The minus sign corresponds to the case k 1 -k 2 = k. We recall that equations (B 2) and (B 3) are only valid for free Kelvin modes with dierent parities with respect to z, i.e. at (or close to) a resonance of the forced Kelvin mode.

B.2. Viscous coecients The boundary viscous coecient s j appearing in equation (4.13) is dened by

s j Re 1/2 = v j ∂ ∂t I + M v (1) j v j Iv j , ( B 4)
where v

(1) j

is the vector appearing in equation (4.12). Integrating by part, we can show that

v j ∂ ∂t I + M v (1) j = -v (1) j ∂ ∂t I + M v j + surface v ⊥ j • v (1)p j + v p j • v (1)⊥ j dS, ( B 5)
where ⊥ stands for the perpendicular component to the wall and the subscript p stands for the pressure component. The rst right-hand side term of this last equation equals 0 because v j is in the kernel of the operator ∂ ∂t I + M . The rst term in the surface integral is also equal to 0 because v ⊥ j vanishes at the walls since it is a free Kelvin mode. The second term inside the integral can be calculated because v

(1)⊥ j is given by the no-slip boundary condition at order ε, which means that it is the opposite of the Ekman pumping (of order Re -1/2 ∼ ε) created by the Kelvin modes v j of order one. The determination of the velocity in the boundary layer is classical. The reader could refer to [START_REF] Greenspan | The theory of rotating uids[END_REF] or for the peculiar case of Kelvin modes to [START_REF] Kudlick | On the transient motions in a contained rotating uid[END_REF]. The viscous coecients can thus be calculated as

s j = 2 I r j + I z j v j Iv j , ( B 6)
where I r j = 2πhP mj ,ωj ,kj (1)

-i K j m j V mj ,ωj ,kj (1) + k j W mj ,ωj ,kj (1) , (B 7)

and

I z j = √ 2π (1 -i) ω 2 j J 2 mj (δ j ) (2 -ω j ) 3/2 + iJ 2 mj (δ j ) (2 + ω j ) 3/2 δ 2 j -m 2 j ω 2 j + 4m 2 j , ( B 8)
with

K j = |ω j | 2 1/2
(1 + i sgn(ω j )) . (B 9)

The same boundary viscous eects appear on the forced Kelvin mode v 1,ω,k and can be calculated in the same manner. If we note v 1,ω,k the Ekman pumping created by the Kelvin mode v 1,ω,k , the boundary viscous coecient s appearing in equation (3.12) is given by s

Re 1/2 = v 1,ω,k

∂ ∂t I + M v 1,ω,k v 1,ω,k Iv 1,ω,k = 2
Re 1/2

I r + I z v 1,ω,k Iv 1,ω,k , ( B 10)
where I r and I z are respectively given by I r j and I z j with m j = 1, ω j = ω and k j = k. The volume viscous coecient v j appearing in equation (4.13) is dened by

v j = - v j Lv j v j Iv j = k 2 j + δ 2 j . ( B 11)
B.3. Detuning coecients The detuning coecient q j appearing in equation (4.13) is dened by

q j = v j M∆vj v j Iv j , ( B 12)
with

M ∆ = 1 ik j     0 0 0 0 0 0 0 0 0 0 0 ∂ ∂z 0 0 ∂ ∂z 0     .
(B 13)

The calculation gives

q j = -8πh k j ω j 1 0 J 2 m j (δ j r) rdr v j Iv j . ( B 14)
Annexe C. Nonlinear calculation

The aim of this Appendix is to calculate the coecients σ j , ξ j and λ appearing in the weakly nonlinear amplitude equations (6.3).

C.1. Coecients σ j

The nonlinear interaction of a Kelvin mode v j with itself gives rise to a ow v 2j such that As shown by Walee (1989), v 2 j has two velocity components because v j is the sum of two waves with opposite axial wavenumbers +k j and -k j . For a Kelvin mode (m j , ω j , k j ) these two components are of the form (2m j , 2k j , 0) and (0, 0, 2k j ) and correspond to the two vectors 2 J 2 m j (δ j r) -J 2 m j (δ j ) r 2mj . (C 4)

v 2kj = ±     0 a 2k j cos (2k j z) 0 p 2k j cos (2k j z)     , v 2ωj = ±      im j r a 2ω j -1 2 da 2ω j dr 0 p 2ωj      , ( C
For a free Kelvin mode with a minus (resp. plus) sign in its denition (4.7) the plus (resp. minus) sign must be chosen in equation (C 2). The coecient σ j in (6.3) is separated into two parts. The rst one comes from the nonlinear interaction of v j and v 2kj σ 2k j = v j N v 2k j , v j + N v j , v 2k j + c.c. . (C 5)

The calculation gives σ 2k j = 8πh 1 0 2ik j a 2k j V m j ,ω j ,k j W m j ,ω j ,k j + a 2k j d U m j ,ω j ,k j V m j ,ω j ,k j dr rdr, (C 6)

with a 2k j given by equation (C 3).

The other term comes from the interaction of v j with v 2ω j σ 2ω j = v j N v 2ω j e 2i(ωj t+mj ϕ) , v j . (C 7)

The calculation gives σ 2ω j = 8πh ik j ω j 1 0 da 2ω j dr iU m j ,ω j ,k j W m j ,ω j ,k j + 2m j a 2ω j r V m j ,ω j ,k j W m j ,ω j ,k j rdr.

(C 8) The nal coecient σ j used in the weakly nonlinear amplitude equations (6.3) is

σ j = - σ 2k j + σ 2ω j i v j Iv j . ( C 9)
C.2. Coecients ξ j

The coecients ξ j of (6.3) describe the nonlinear interaction of the j-th free Kelvin mode with the geostrophic mode

ξ j = iv j N 1 2 v 0 , v j + N (v j , v 0 ) / v j Iv j . ( C 10)
We then obtain

ξ j = 8iπh 1 0 2kj ω j v 0 U m j ,ω j ,k j W m j ,ω j ,k j + 1 r d dr (rv 0 ) U m j ,ω j ,k j V m j ,ω j ,k j rdr v j Iv j , ( C 11)
where v 0 is the orthoradial component of the geostrophic ow given by (6.1). For the forced Kelvin mode, v j Iv j is replaced by v 1,ω,k Iv 1,ω,k in (C 11) in order to obtain the coecient ξ.

C.3. Coecient λ

The parameter λ describes the nonlinear interaction between the rst and the second free Kelvin mode. This parameter appears in equation (6.3a) and is given by

λ = v 1,ω,k [N (v 1 , v 2 ) + N (v 2 , v 1 )] v 1,ω,k Iv 1,ω,k . (C 12)
We obtain

  Figure 2. (a) Axial vorticity eld ζ of the rst Kelvin mode measured by PIV at its rst resonance (ω = 1.18), in the absence of instability. (b) Amplitude of the rst Kelvin mode as a function of ω. The solid line corresponds to the linear and inviscid theory given by equation (A 8) and the dashed line to the viscous theory given by equation (3.12). The point is the value of |ε1| for ω = 1.18 at the exact resonance. h = 1.62, Re ≈ 3500 and Ro = -0.0031.

Figure 3 .

 3 Figure3. Dispersion relations of the free Kelvin modes (h = 1.62, ω = 1.18). Solid lines (resp. dashed lines) correspond to the rst ve branches of the dispersion relation of the free Kelvin modes with an azimuthal wavenumber m2 = 6 (resp. m1 = 5). The dotted line represents the dispersion relation of the forced Kelvin mode (m = 1), which is used to translate the dispersion relations of the rst mode horizontally by k = π/h and vertically by ω = 1.18. Resonant free Kelvin modes correspond to a crossing point for the case k2 -k1 = k. Vertical dash-dotted lines indicate the discretisation of the axial wavenumber k2 = nπ/h for which the free Kelvin mode `ts' inside the cylinder.

Figure 4 .

 4 Figure 4. Schematic of the stability diagram in the presence of viscous and detuning eects. The solid line corresponds to the critical Rossby number at which the ow becomes linearly unstable ; it exhibits three dierent scaling laws depending on the value of the Reynolds number compared to the critical Reynolds numbers Re 1 and Re 2 given in (4.19) and (4.22). The non-linearly saturated value ε f of the amplitude of the base Kelvin mode is also shown. It corresponds to the stationary solution of the non-linear system (6.3) and is calculated analytically in Appendix D.

Figure 5 .

 5 Figure 5. Experimental (a, b) and theoretical (c, d) axial and instantaneous vorticity elds of the free Kelvin modes m 1 = 5 and m 2 = 6. h = 1.62, ω = 1.18 and Ro = -0.0031. (a) Re = 6000. (b) Re = 6500.

Figure 6 .

 6 Figure 6. Averaged radial (a) and azimuthal (b) velocity proles of the free Kelvin modes m1 = 5 (+, solid line), and m2 = 6 (•, dashed line), in arbitrary units. Symbols represent experimental data and lines are issued from the theory. h = 1.62, ω = 1.18, Re = 6000 for m1 = 5 ; Re = 6500 for m2 = 6.

Figure 7 .

 7 Figure7. Growth rate σ r /|ε| of the precessional instability for h = 1.62 and ω = 1.18. The inviscid prediction is represented by the horizontal solid line. The dashdotted line is the viscous prediction without volume viscous eects. The dashed line is the full viscous prediction for Re = 6000. The dotted line is the viscous prediction for Re = 1000. Filled symbols ( ) correspond to experiments with 6000 < Re < 24400. Open symbols ( ) correspond to experiments with 1000 < Re < 6000. The error bars correspond to the standard deviation on the tting parameter.

Figure 8 .

 8 Figure 8. Stability diagram of the ow inside a precessing cylinder for h = 1.62 and ω = 1.18.The stable and unstable domain are separated by the solid line which corresponds to the prediction (4.18). Open symbols (•) represent stable experiments and lled symbols represent unstable experiments. Unstable experiments can be either stationary ( ), intermittent ( ) or turbulent ( ). The error bars come from the uncertainty in the measurement of the precessing angle. The dashed line corresponds to the transition from a stationary to an intermittent ow in the weakly non-linear model.

  Figure9. Critical |RoRe| number of the most unstable modes as a function of h for the rst resonance of the rst Kevlin mode (ω = ω1,1). Each solid line corresponds to the full theory at Re = 6500 for a resonant combination (m2, 1, 1). Symbols correspond to the predictions without detuning and without volume viscous eects (thus valid at any Reynolds number, see Table1) for the same combinations with m2 = 6 (•, ), m2 = 7 ( ), m2 = 8 (×), m2 = 9 (•), m2 = 10 (+), m2 = 11 ( * ), m2 = 12 ( ), m2 = 13 ( ). The dashed line (resp. dotted line) corresponds to a combination (3, 1, 1) (resp. (2, 1, 1)) which can not be exactly resonant.

Figure 11 .

 11 Figure 11. Critical Rossby number as a function of h and ω, by considering that the rst Kelvin mode is forced by precession. The resonant Kelvin mode combinations (5, 1, 1), (6, 1, 1), (7, 1, 1), (8, 1, 1) are observed. The white dashed lines correspond to the rst (lower left) and second (upper right) resonance of the rst Kelvin mode. Fig. 10 corresponds to the vertical dashed line at h = 1.62. Re = 6500.

Figure 12 .

 12 Figure 12. Amplitude of the free Kelvin mode m1 = 5 as a function of the dimensionless time t showing the three dierent regimes of the nonlinear evolution of the instability. (a) Numerical simulation. (b) Experimental results. It is obtained at Re = 3500 (•), Re = 4500 (solid line) and Re = 6000 (dashed line). The frequency of acquisition f and the time delay ∆t between PIV pairs are f = 0.2 Hz, ∆t = 50 ms (•) ; f = 0.222 Hz, ∆t = 45 ms (solid line) ; f = 0.333 Hz, ∆t = 30 ms. h = 1.62, ω = 1.18, Ro = -0.0031.

Figure 13 .

 13 Figure 13. Amplitude of the free Kelvin mode m1 = 5 as a function of Ro for Re = 3664 (a) and as a function of RoRe without volume viscous eects (b). The xed point calculated from the weakly nonlinear theory is represented by a solid line in the unstable and stationary regime and by a dashed line in the unstable and intermittent regime. Dotted lines represent the maximum and the minimum of oscillations of A1. Dashdotted lines represent the two thresholds. Squares ( ) represent unstable and stationary experiments. Triangles ( ) represent the mean value of |A1| in the unstable and intermittent regime. Amplitude ranges indicate the maximum and minimum values of A1. h = 1.62, ω = 1.18.

Figure 14 .

 14 Figure 14. Amplitude of the forced Kelvin mode (a) and the geostrophic Kelvin mode (b) as a function of RoRe (volume viscous eects were neglected). The viscous saturating amplitude from equation (3.12) is represented by a dashdotted line below the threshold (rst vertical dashdotted line) and a dotted line above the threshold. The xed point (solid or dashed line) is compared to the experimental time averaged values |ε| (a) and A 0 (b). These values are represented by circles (•) (stable ow), squares ( ) (unstable and stationary ow) or triangles ( ) (unstable and intermittent ow). h = 1.62, ω = 1.18.

  12) Annexe B. Linear stability analysis coecients B.1. Nonlinear coecients The nonlinear coecients n 1 and n 2 appearing in equation (4.13) are dened by

  ∂ ∂t I + M v 2 j + c.c. = N (v j , v j ) + c.c. (C 1)

Table

	h	10.8	3.60 1.62 1.09 0.886 0.758
	ω 1,1	0.1639 0.5501 1.181 1.503 1.636 1.718
	m 2	7	6	6	7	8	9
	n 1 n 2	1.16	1.07 4.11 15.4 34.4 64.5
	|RoRe| crit 5.41	7.14 7.11 8.60 10.4 12.4
	h	0.669 0.601 0.547 0.504 0.467 0.436
	ω 1,1	1.772 1.811 1.841 1.863 1.881 1.895
	m 2	10	11	12	13	14	15
	n 1 n 2	108 170 253 358 496 666
	|RoRe| crit 14.5 16.8 19.2 21.7 24.3 27.1

Table 2 .

 2 .21). Theoretical results considering the (m 2 , 2, 2) combinations with k 2 = 2π/h and k 1

	h	9.57		1.13 0.880 0.744 0.651 0.581
	ω1,1	0.1867	1.477 1.640 1.727 1.783 1.822
	m2 n 1 n 2	4 3.15 × 10 -3 8.42 30.4 71.0 138 242 4 5 6 7 8
	|RoRe| crit	106		10.5 11.2 12.5 14.0 15.8
	h	0.527 0.484 0.448 0.417 0.391 0.368 0.348
	ω1,1	1.851 1.873 1.890 1.904 1.915 1.924 1.932
	m2	9	10	11	12	13	14	15
	n1n2	24.3 36.6 52.9 74.4 101 134 174
	|RoRe|crit 17.6 19.6 21.7 23.9 26.2 28.6 31.0

Table

	672 1.61 -0.06i 87.16 -0.066 -83.784 10000
	n2	s2	v2	ξ2	σ2	χ2
	-2.456 1.81 -0.13i 102.68 -0.365 48.987 10000

rdr.

(C 13)

The plus sign corresponds to the case k 2 -k 1 = k.

The minus sign corresponds to the case k 1 -k 2 = k.

Annexe D. Fixed point

To obtain a xed point (ε f , A 1f , A 2f , A 0f ) of the nonlinear amplitude equations ( 6.3), we search a solution (ε, A 1 , A 2 , A 0 ) such that ∂ε ∂t = ∂A 0 ∂t = 0 and A j = A 0 j e i ωt with ∂A 0 j ∂t = 0 and ω real. Then, the xed point satises the following system

)

)

Here, we have neglected the term λA 1 A 2 because it was very small in our case, but it can be easily added and the system can be solved in the same manner. The product of (D 1b) by (D 1c) is real such that the modulus of (D 1b) over (D 1c) equals the modulus of the real part of (D 1b) over the real part of (D 1c). Thus it comes

)

)

)

with

and

The product of equations (D 5b) and (D 5c) gives a complex equation, whose imaginary part gives the nonlinear frequency of the free Kelvin modes

and whose real part thus leads to

By equating (D 9) and the modulus of (D 5a) divided by (α + iξA 0 ) we obtain an equation of order 4 for

This equation has a unique real and positive solution which is A 0f , except in the subcritical regime, where it can have two positive roots. Note that if we take into account the term λ in equation (6.3a), the equation for A 0 becomes an equation of order 6 instead of 4. Once A 0f is known, ε f can be determined via equation (D 9) and A 1f and A 2f by equations (D 3) and (D 4).