R Émi Manceau
email: remi.manceau@univ-pau.fr

Rodolphe Perrin
email: rodolphe.perrin@univ-poitiers.fr

Muhamed Hadžiabdi
email: mhadziabdic@ius.edu.ba

Sofiane Benhamadouche
email: sofiane.benhamadouche@edf.fr

ANALYSIS OF THE EFFECTS OF ROTATION ON AN AXISYMMETRIC WALL JET

The case of a turbulent round jet impinging perpendicularly onto a rotating, heated disc is investigated, aiming at understanding the mechanisms at the origin of the influence of rotation on the radial wall jet and associated heat transfer. The present study is based on the complementary use of an analysis of the order of magnitudes of the terms of the mean momentum and Reynolds stress transport equations, available experiments and dedicated Reynolds-Averaged Navier-Stokes (RANS) computations with refined turbulence models. The analysis identifies a scenario for the appearance of rotational effects when the rotation rate is gradually increased.

INTRODUCTION

The case of a jet impinging on a rotating disk, relevant to cooling processes in many industrial applications, implies a complex interaction between the radial wall jet and the azimuthal boundary layer, as shown in Fig. 1. The present collaborative work was initiated in the frame of the workshop organized at TU Graz, Austria, under the auspices of the Special Interest Group 15 turbulence modelling of ERCOFTAC [START_REF] Steiner | 13th ERCOFTAC workshop on refined turbulence modelling[END_REF], and provides an example of cross-fertilization of analytical, experimental and numerical studies. The configuration, based on flow field and heat transfer measurements performed by [START_REF] Minagawa | Development of turbulent impinging jet on a rotating disk[END_REF] and [START_REF] Popiel | Local heat transfer from a rotating disk in an impinging round jet[END_REF], respectively, is issuing with the bulk velocity U j from a pipe of diameter D (Re j = U j D/ν = 14,500), at H = 5D from a rotating disk of diameter R = 6D. Several rotation rates are investigated, ωD/U j = 0, 0.12, 0.24 and 0.48.

The computations were performed using two in-house codes, T-FlowS [START_REF] Ničeno | Modeling and Simulation of Turbulent Heat Transfer, chap. Unstructured large-eddyand conjugate heat transfer simulations of wall-bounded flows[END_REF] developed at Delft University of Technology and the open-source (www.code-saturne.org) Code Saturne developed at EDF [START_REF] Archambeau | Code Saturne: A Finite Volume Code for the Computation of Turbulent Incompressible flows -Industrial Applications[END_REF]. As usual in the ERCOFTAC SIG-15 workshop series, the groups computed the cases independently and performed grid refinement studies. The ζ -f model [START_REF] Hanjalić | A robust near-wall elliptic relaxation eddy-viscosity turbulence model for CFD[END_REF] was used in T-FlowS (International university of Sarajevo); the k-ω-SST [START_REF] Menter | Two-equation eddy-viscosity turbulence models for engineering applications[END_REF] and ϕ -f [START_REF] Hanjalić | v 2 /k)f turbulence model and its application to forced and natural convection[END_REF] models were used in Code Saturne (EDF); the Elliptic-Blending Reynolds stress model (EB-RSM, [START_REF] Manceau | Elliptic Blending Model: A New Near-Wall Reynolds-Stress Turbulence Closure[END_REF] was also used in Code Saturne (Institute Pprime/LMA). As concerns heat transfer modeling, the eddy-viscosity models were associated with the SGDH, while the Reynolds stress models were associated with the GGDH.

The rms radial velocities for cases without and with rotation are shown in Fig. 2. The models have difficulties to reproduce the impingement region, a behaviour analyzed in detail in a previous paper [START_REF] Manceau | Turbulent jet impinging onto a rotating disc: A collaborative evaluation of RANS models[END_REF], mainly linked to the overestimation of the spreading of the free jet before impingement. The present paper focuses on the radial wall jet, i.e., the region r/D > 2.5.

Ouside of the region r/D < 2.5, dominated by pressure and convection terms, the influence of rotation increases with the rotation rate and the distance to the symmetry axis (Fig. 2). The modification of ∂V r /∂ z due to both the acceleration and the thinning of the wall jet implies an increase of turbulent production, which generates strong radial fluctuations v ′ r . Such an effect is captured by all the models, but eddy-viscosity models give a much too early (r/D ≈ 2 rather than r/D ≈ 4) increase of v ′ r , which shows that this intermediate region is more difficult to reproduce.

The present paper aims at extending the limited knowledge and understanding of this case, in particular as concerns the mechanisms of the migration from a self-similar axisymmetric solution to a rotation-dominated flow.

STRUCTURE OF THE WALL JET

In the absence of rotation, the wall jet can be characterized by the maximum velocity V rmax and the thickness δ

δ = 1 V rmax ∞ 0 V r (z)dz.
(1)

Due to entrainment, the flow rate Q(r) across the surface at a fixed r is larger than the flow rate issuing from the pipe Q j = π D 2 4 U j , and the entrainment ratio is defined as e = Q(r)/Q j . A reduced thickness δ ′ = 8δ /e is introduced, such that

V rmax = U j D 2 rδ ′ . (2)
The wall jet can be considered a two-layer flow: the region located between the wall and δ rmax (location of the velocity maximum), called the inner region, exhibits characteristics close to those of a boundary layer; the region beyond δ rmax , called the outer region, is similar to a free shear flow, and is governed by inviscid equations.

The standard definition of a scale of change in the rdirection, L, used for plane flows, ∂U/∂ x ∼ U max /L (where ∼ stands for of the order of magnitude of ), is replaced by

1 r ∂ rV r ∂ r ∼ V rmax L where 1 L = 1 δ ′ ∂ δ ′ ∂ r . (3)
The standard analysis of this nearly parallel flow shows that the radial wall jet satisfies standard boundary layer equations, which admit a self-similar solution with power-law evolutions of δ , δ ′ and e in the radial direction, and a lin- ear growth of L, and, consequently, power law evolutions of both V rmax and ∆T = T w -T ∞ . Finally, the reduced thickness δ ′ /D exhibits a very slow variation in the radial direction, and can be, for order of magnitude evaluations below, considered a constant and close to unity. Consequently, Eq. (2) leads to

V rmax U j ≈ r D -1 . ( 4
)
As suggested by [START_REF] Glauert | The wall jet[END_REF], the inner layer can be described using Kármán's method, such that the friction coefficient and the Nusselt number can be shown to behave as

C f = τ w 1 2 ρU 2 j = V 2 rmax U 2 j τ w 1 2 ρV 2 rmax ∝ 1 Re 1/4 j r D -1.95 (5)
and

Nu = qw D (λ ∆T ) ∝ Re 3/4 j r D -1 . ( 6
)

INFLUENCE OF ROTATION

In the case of the boundary layer developing on a rotating disk in a quiescent environment, it can be shown (e.g., [START_REF] Dorfman | Hydrodynamic resistance and the heat loss of rotating solids[END_REF] that the thickness ℓ of the boundary layer scales with the local Reynolds number Re r = r 2 ω/ν

ℓ r ∝ r 2 ω ν -1 5 = α -1 5 Re -1 5 j where α = r 2 ω U j D (7)
For weak rotation rates, rotation only moderately affects the thickness δ of the wall jet (see Fig. 3) and Fig. 4 ROTC shows that the thickness of the azimuthal boundary layer satisfies Eq. ( 7). Thus, their ratio

ℓ δ ∝ r 2 ω ν -1/5 = α -1/5 Re -1/5 j r D 0.2 (8)
is small, owing to the high Reynolds number, as can be seen in Fig. 3, where ℓ = 1 rω ∞ 0 V θ dz. Since ℓ decreases with the rotation rate, the increasing influence of rotation on the outer layer of the wall jet cannot be attributed to a growing of ℓ, but rather to an interaction between the rotationinduced boundary layer and the inner layer of the wall jet, which modifies the equilibrium between the inner and outer layers.

Limit of strong rotations

In the inner layer, centrifugal acceleration becomes dominant in the radial momentum equation in the limit ω → ∞, leading to

V r ∂V r ∂ r ∼ V 2 r r - V 2 θ r ∼ rω 2 = 0, (9)
which yields V r ∼ rω. The primary effect of rotation is thus an acceleration of the inner layer. Although the centrifugal acceleration does not directly affect the outer layer, the acceleration of the inner layer necessarily leads to a suction of fluid at the bottom of the outer layer, by mass conservation, such that V r in the outer layer also scales as V r ∼ V rmax ∼ rω.

Since rV rmax is constant in the absence of rotation, the influence of rotation is characterized by

rV rmax (rV rmax ) 0 ∼ rV rmax U j D ∼ α , (10)
where the index 0 denotes the non-rotational case. Since Eq. ( 2) is exact, this result is equivalent to

δ ′ δ ′ 0 ∼ 1 α , (11)
showing a thinning of the wall jet. Fig. 5 confirms the scaling laws ( 10) and ( 11), even though discrepancies are observed in the slopes predicted by the different models.

For large α's, the dynamics is dominated by centrifugal acceleration, and the influence of the turbulence model is weak. Therefore, as can be seen in Fig. 2, reproducing the effects of rotation is more challenging for turbulence models in the region located between the impingement region and the rotation-dominated region, and for intermediate rotation rates.

It is confirmed by Fig. 7, for both experiments and EB-RSM results, that the rotation of the disk does not directly affect the turbulence in the outer layer (down to z/D ≈ 0.3). Consequently, as can be seen in Fig. 9, the entrainment ratio is independent of the rotation rate. The situation is different for eddy-viscosity models, because the turbulent shear

ζ -f ϕ -f k-ω-SST EB-RSM α δ ′ /δ ′ 0 (b)
Figure 5. Scaling with α of (a) rV rmax /(rV rmax ) 0 and (b) δ ′ /δ ′ 0 . Lines: theoretical relations fitted to the experimental results.

stress is assumed proportional to the mean shear, which is modified by the acceleration and thinning of the wall jet.

Since the thickness δ of the jet is related to the entrainment ratio e and the reduced thickness δ ′ by δ = eδ ′ /8, and e is independent of the rotation rate, we have also δ /δ 0 ∼ α -1 . Fig. 6a confirms this scaling for strong rota- tion rates (α > 5). [START_REF] Minagawa | Development of turbulent impinging jet on a rotating disk[END_REF] conjectured this scaling, but, due to limitations inherent to the experimental approach, they monitored the variation of the location of the velocity maximum, δ rmax /δ rmax0 . As shown in Fig. 6b, this quantity does not scale very well with α -1 , contrary to δ . Fig. 8 shows the evolution of the Nusselt number based on the radial location Nu r = hr/λ as a function of the rotational Reynolds number Re r = ωr 2 /ν = α Re j . It can be observed that the results of the ζ -f and k-ω-SST models well satisfy the scaling proposed by [START_REF] Popiel | Local heat transfer from a rotating disk in an impinging round jet[END_REF], Nu r = f (Re r ), contrary to the ϕ -f model that shows some scatter, due to the lack of self-similarity of the solution. For high values of Re r , it is expected that the Nusselt number approaches the asymptote Nu r ∝ Re 0.8 r applicable to a rotating disk in still air [START_REF] Dorfman | Hydrodynamic resistance and the heat loss of rotating solids[END_REF]. It can be seen that the present computations, for all the models, seems to approach the correct asymptotic behavior.

Appearance of rotational effects

Momentum equation As long as the rotation rate is sufficiently weak, the self-similar solution for the outer layer holds, and the description of the inner layer given above is valid. In the inner layer, the scales are u τ and ν/u τ , where u τ can be obtained from Eq. ( 5), δ rmax /δ rmax0 . Lines: theoretical relations fitted to the experimental and numerical results.

u τ ∼ 1 Re 1/8 j U j D r r D 0.025 ∼ 1 Re 1/8 j U j D r . (12)
The factor (r/D) 0.025 is ignored in this relation since its order of magnitude is unity.

Since the azimuthal velocity is V θ ∼ rω, the radial momentum equation reads, at leading order,

- V 2 θ r ∼ rω 2 = ν ∂ 2 V r ∂ z 2 ∼ Re 5/8 j U 2 j D 2 r 3 - ∂ v r v z ∂ z ∼ Re 5/8 j U 2 j D 2 r 3 (13)
Consequently, the relative weight of the centrifugal acceleration in this equation is

r 4 ω 2 Re 5/8 j U 2 j D 2 = α 2 Re 5/8 j , (14)
such that the centrifugal acceleration remains negligible for α ≪ Re Production terms For the non-rotating case, the components of the production tensor reduce to

P rr = P (1) rr + P (2) rr = -2v 2 r ∂V r ∂ r -2v r v z ∂V r ∂ z P θ θ = P (1) θ θ = -2v 2 θ V r r P zz = P (1) zz + P (2) zz = -2v r v z ∂V z ∂ r -2v 2 z ∂V z ∂ z P rz = P (1) rz + P (2) rz + P (3) rz = -v 2 r ∂V z ∂ r + v r v z V r r -v 2 z ∂V r ∂ z P rθ = P θ z = 0 ( 15
)
(the underlined terms are those related to ∂V r /∂ z and are dominant). When the disk is rotated, additional terms appear,

P (r) rr = 2v r v θ V θ r P (r) θ θ = P (r1) θ θ + P (r2) θ θ = -2v r v θ ∂V θ ∂ r -2v θ v z ∂V θ ∂ z P (r) rθ = P (r1) rθ + P (r2) rθ + P (r3) rθ + P (r4) rθ + P (r5) rθ = -v 2 r ∂V θ ∂ r + v r v θ ∂V z ∂ z -v r v z ∂V θ ∂ z + v 2 θ V θ r -v θ v z ∂V r ∂ z P (r) θ z = P (r1) θ z + P (r2) θ z + P (r3) θ z + P (r4) θ z = -v r v θ ∂V z ∂ r + v θ v z ∂V r ∂ r -v r v z ∂V θ ∂ r -v 2 z ∂V θ ∂ z , P (r) rz = v θ v z V θ r (16)
Some of the production terms, evaluated from the EB-RSM computations, are plotted in Fig. 11, at the edge of the disk. The orders of magnitude of the different terms can be readily deduced from previous sections, and depend on the region of the flow.

Outer layer Since we have shown that the outer layer is not directly affected by the azimuthal boundary layer, the production tensor in the outer layer must keep the same structure as in the case without rotation, and the following orders of magnitude are obtained P rr ∼ P ∼ u 2 V rmax δ ; P θ θ ≪ P; P zz ≪ P; P rz ∼ P (17

)
where the P ∼ 1 2 P rr is the kinetic energy production. In the outer region, approximately located between z/D = 0.1 and z/D = 1 (Fig. 11), production is dominated by P zz , related to the radial expansion of the jet, are lower at least by a factor of δ /r. Fig. 11 shows that in the outer layer, this description of the production tensor remains valid for all the rotation rates, which confirms that the outer layer is not directly affected by the rotation-induced boundary layer. The production terms are however increased as a consequence of the modification of the velocity gradient ∂V r /∂ z.

Consequently, the budgets of the Reynolds stresses (Fig. 12) are very similar to the case of a standard boundary layer, even when the disk rotates: the velocity gradient ∂V r /∂ z induces turbulent production on the longitudinal component (here, v r v r ), and energy is then redistributed to other components via the pressure term. However, due to the radial expansion of the jet, convection is active and P θ θ and P zz are not zero.

Inner layer

In this region, located between the wall and z/D ≈ 0.1 (Fig. 11a0), scales are u τ and ν/u τ , with u τ given by Eq. ( 12), such that we have

P rr ∼ Re 1/2 j U 3 j D 3
r 4 ; P θ θ ≪ P; P zz ≪ P; P rz ∼ P (18)

Contrary to the outer layer, additional terms appear in the production tensor (Fig. 11), and, aiming at identifying a ROTC threshold, the analysis below considers the introduction in the equations of a gradually increasing rotation rate. A straightforward, albeit long, analysis provides the following evaluations for the production terms due to rotation:

P (r) rr ∼ P (r) rz ∼ α 2 Re 3/4 j U j D rℓ u 2 τ P (r) θ θ ∼ α 2 Re 3/4 j U j D ℓ 2 u 2 τ P (r) rθ ∼ P (r) θ z ∼ α U j D rℓ u 2 τ . (19)
As long as α is sufficiently weak, the production terms due to rotation are negligible compared to those already present in the non-rotating case except for P (r)

rθ and P (r) θ z , since P rθ = P θ z = 0.
When α is increased, the main effect is the rapid increase of P (r2) θ θ . Since, in the absence of rotation, the production term P θ θ is small compared to P (and negative), v 2 θ essentially receives energy from other terms (redistribution, convection, diffusion), and these terms are of the same order of magnitude as production P. The correct criterion to evaluate the influence of rotation on v 2 θ is then the ratio

P (r) θ θ /P ∼ α 12/5 Re -11/10 j . (20)
It can be seen in Fig. 11a1 and b1 that, when rotation is increased, the term

P (r2)
θ θ rapidly becomes much stronger than P rr , such that the turbulence anisotropy is completely modified and, as can be seen in Fig. 12, energy transfer mechanisms are reversed (production of v 2 θ and redistribution towards other components). The order of magnitude analysis leading to Eq. ( 20) thus shows that the value of α for which P (r2) θ θ becomes significant scales with Re 11/24 j , but a closer inspection of the level of P (r2) θ θ as a function of α in our numerical results suggests that the threshold is about α ≈ 0.01Re 11/24 j . To say the obvious, such effects of rotation on turbulence cannot be reproduced by linear eddyviscosity models, which are completely insensitive to rotation: they only reproduce the increase of v 2 θ with rotation that results from the increase of V r due to the centrifugal acceleration.

CONCLUSIONS

The order of magnitude analysis supports a scenario for the gradual increase of the influence of rotation. For low rotation rates, such that α < 0.01 Re 11/24 j , the flow is in a superposition regime (see Fig. 10), in which the rotation-induced boundary layer does not significantly affect the self-similar wall jet. For intermediate rotation rates, 0.01 Re 11/24 j < α < 0.25 Re 5/16 j , an interaction regime is observed, in which the inner layer resemble an Ekman layer, while the outer layer remains weakly perturbed. Finally, for strong rotation rates, α > 0.25 Re 5/16 j , in the rotationdominated regime, centrifugal acceleration becomes dominant and causes an intensification and thinning of the jet.

Therefore, depending on the value of the parameters α and Re j , the modification of the wall jet can be linked to the influence of the centrifugal acceleration, or to more complex effects due to the anisotropic transfer of energy from the mean flow to the turbulent field. The linear eddyviscosity models are able to reproduce the former effects, in the rotation-dominated regime: acceleration and thinning of the jet (independent of turbulence modeling); intensification of the shear, and consequently of turbulence production. The more complex effects appearing in the interaction regime, when the skewing of the inner layer of the wall jet completely modifies the structure of the different terms in the Reynolds stress budgets, require the use of a more elaborate model. ωD/U j = 0 ωD/U j = 0.12 ωD/U j = 0.24 ωD/U j = 0.48

Figure 1 .

 1 Figure 1. Artist view of the different regions of the flow.

Figure 2 .

 2 Figure 2. Profiles of the rms radial velocity at z/D = 0.032. Black: no rotation; Red: ωD/U j = 0.48.

Figure 3 .

 3 Figure 3. Thicknesses of the wall jet and the azimuthal boundary layer (EB-RSM).

Figure 4 .

 4 Figure 4. Influence of rotation on ℓ/r (EB-RSM).

Figure 6 .

 6 Figure 6. Scaling with α of (a) δ /δ 0 and (b) δ rmax /δ rmax0 . Lines: theoretical relations fitted to the

  6 show a transition in the region α ≈ 5, which suggests the threshold α ≈ 0.25 Re 5/16 j .

Figure 7 .

 7 Figure 7. Influence of rotation on the radial rms velocity at the edge of the disk.

Figure 8 .Figure 9 .

 89 Figure 8. Nusselt number Nu r = hr/λ as a function of Re r .

Figure 10 .Figure 11 .

 1011 Figure 10. Different regimes in the plane α-Re j .ωD/U j = 0 ωD/U j = 0.12 ωD/U j = 0.24 ωD/U j = 0.48

Figure 12 .

 12 Figure12. Budgets of the Reynolds stresses at the edge of the disk for the four rotation rates (EB-RSM computations), made non-dimensional by U j and D. The legend is in the plot (b3).


T Tran

R Perrin

R Manceau

J Borée

P Jordan

FLOW SOURCES OF WALL PRESSURE FLUCTUATIONS RESOLVED BY DDES IN A REATTACHING FLOW REGION

This paper is concerned with the analysis of the flow structures responsible for the wall pressure fluctuations predicted by DDES near a reattachment. To this aim, a simulation of the flow over a thick plate is considered. It observed that the linear part of sources is dominant upstream of the reattachment, and that quadratic sources dominate downstream. This is associated with the rapid loss of coherence of the vortices shed from the separated shear layer in the downstream region. Furthermore, it is shown that linear stochastic estimation of the velocity field from the wall pressure well reproduce the linear sources, with negligible quadratic sources.

INTRODUCTION

The work reported in this paper is concerned, on a general level, with the problem posed by our increasing ability to compute and/or measure turbulent flows with high space-and time-resolution: perspicacious analysis and data-reduction methodologies are paramount. From the full complexity of the turbulence, measured or computed, it is necessary to distill the essential mechanisms with regard to the problem at hand, or an observable of particular interest. It is by means of such distillation that reduced complexity models can be proposed, eventually with a view to addressing the problem of control. We consider flow over a blunt flat plate (fig. 1) at high Reynolds number, which we compute using DDES, and our goal is to probe the flow with regard to the flow structures that underpin the wall pressure fluctuations; the latter phenomenon is treated as an observable that we use to educe, from the broad range of turbulent motions, only those by which it is driven. This two-dimensional separation-reattachment flow has been the subject of much research in the past [START_REF] Kiya | Structure of a turbulent separation bubble[END_REF], Cherry, Hiller and Latour 1984[START_REF] Hudy | Stochastic estimation of a separated-flow field using wall-pressure-array measurements[END_REF], and the question regarding the turbulence flow sources of hydrodynamic wall pressure fluctuations continues to be a subject of academic interest. The mean reattachment is responsible for a rapid longitudinal evolution of the spatiotemporal organisation of these sources; the flow thus presents a considerable challenge in terms of both its computation and its analysis.

Possibly one of the most widely-used feature-eduction techniques found in the literature is conditional analysis, perfectly suited for the task described above: an observable is chosen and used to conditionally sample some related field. Linear stochastic estimation is an interesting tool in this regard: it allows the computation of an approximation of a conditional field using an observable that can be spatiotemporally distributed. Two recent examples can be found in [START_REF] Sicot | Wall pressure and conditional flow structures in the reattachment region of the flow over a thick plate[END_REF] and [START_REF] Kerhervé | Educing the source mechanism associated with dowstream radiation in subsonic jets[END_REF]. In the former paper an experimental data base, acquired in the same research project that supported the present work (ANR DIB), was processed using an adapted multi-time, multipoint stochastic estimation of the velocity field associated with the fluctuating wall pressure; the conditional field obtained comprised large-scale vortices shed from the separated shear layer.

In this paper we consider the sources of hydrodynamic wall pressure fluctuations predicted by DDES. Using the velocity field computed by the DDES we compute both the linear and quadratic contributions to the wall pressure fluctuations; we do so using a technique taken from the aeroacoustics literature [START_REF] Lee | Direct correlation of noise and flow of a jet[END_REF], where the contributions are computed by means of pressure-velocity correlations: the observable is thus used to compute its source. The nature of the sources, and in particular the relative contributions of the linear and quadratic components, are shown to evolve strongly from upstream to downstream of the reattachment point. The same pressure-velocity correlations are then used to compute a conditional velocity field, by means of linear stochastic estimation, with respect to the pressure fluctuations at the reattachment point; the pressure source terms associated with the conditional field are computed and compared with those of the complete flow. The linear source terms of the full and conditional velocity fields are identical (this can be demonstrated formally), suggesting that the conditional field computed by stochastic estimation may provide access to the space-time flow motions associated with the linear pressure term of the complete flow (these motions are not directly accessible from the raw data ; i.e. while it is straightforward to compute the linear source terms from a given flow realisation, it is not straightforward to educe the space-time field that contributes to the source term.) The conditional field does not, on the other hand, comprise a quadratic source term.

FLOW CONFIGURATION AND NUMERICAL ASPECTS

The flow configuration is presented in figure 1. The Reynolds number, based on the upstream velocity U ∞ and the plate thickness e, is Re = U ∞ e/ν= 80, 000. The extent of the computational domain is Lx = 70e, Ly = 31e and Lz = 3.5e, with an inlet located 30e upstream of the stagnation point. Preliminary 2D URANS tests have shown that such an extent is necessary to ensure that the results are independent of the size of the domain. The boundary conditions are inflow/outflow in the x-direction and free-slip in the y-direction. Since the reference experiment was performed using a relatively large aspect ratio (≃ 30), homogeneity in the z-direction in the central part of the plate is assumed, such that periodicity is imposed in the z-direction.

Figure 1 : Flow configuration

The simulations have been performed using Code_Saturne, developed at EDF [START_REF] Archambeau | Code Saturne: A Finite Volume Code for the Computation of Turbulent Incompressible Flows -Industrial Applications[END_REF], an unstructured, collocated finite-volume code, which is second order accurate in time and space, and uses a SIMPLEC type algorithm, with a Rhie and Chow procedure. The flow has been investigated using URANS and DDES approaches both based on the SST model [START_REF] Menter | Two-equation eddy-viscosity turbulence models for engineering aplications[END_REF]. The DDES model that has been implemented uses the shield function proposed by [START_REF] Spalart | A new version of detached-eddy simulation, resistant to ambiguous grid densities[END_REF], rather than the function F 2 of the SST model, mainly because the protected RANS region has been found less extended using the Spalart function, hence accelerating the transition from a RANS to a LES behavior. For the convection scheme, the blending proposed by [START_REF] Travin | Physical and numerical upgrades in the detachededdy simulation of complex turbulent flows[END_REF] which switches from a TVD scheme in the RANS region to a centered scheme in LES regions has been used. The grid contains approximately 2.5 million cells, refined near the wall to achieve a y + value of order 1. The time step was set to 10 -3 e/U ∞ and independence of the statistical quantities with regards to the time step have been checked. After a transient period of approximately 100e/U ∞ , statistical averaging has been performed over a period of 400 e/U ∞ , which represents approximately 50 shedding periods downstream of the mean reattachment. The convergence of the statistics is also enhanced by averaging along the z direction.

The simulation is found in good agreement with the experimental study [START_REF] Sicot | Wall pressure and conditional flow structures in the reattachment region of the flow over a thick plate[END_REF] carried out during the same project (ANR DIB). The mean reattachment length is found at x≈5.7 e, in good agreement with the experiment (≈5.5e) and with the literature. Figure 2 shows longitudinal mean velocity profiles at different x positions. As expected, the agreement is found much better than that achieved with a RANS model, even if the boundary layer recovery seems to be slightly anticipated with DDES. Spectra of velocity components in the outer layer at the mean reattachment have been plotted in Figure 3, and a good agreement is again achieved with experiment, even if the broadband contribution corresponding to vortex shedding downstream of the reattachment, is found in a slightly higher Strouhal number range (St=f e/U ∞ ≈0.15-0.2) that in the experiment of [START_REF] Sicot | Wall pressure and conditional flow structures in the reattachment region of the flow over a thick plate[END_REF] (St=f e/U ∞ ≈0.1-0.15). A more detailed analysis of the simulation compared to experiment can be found in [START_REF] Tran | Modélisation Hybride RANS/LES d'écoulements massivement décollés en régime turbulent. Etude des corrélations pression/vitesse et confrontation à l'expérimentation[END_REF], where it was concluded that the large scales of the fluctuating motion is well predicted by DDES. Figure 4 shows the longitudinal evolution of the fluctuating wall pressure coefficient Cp'. In agreement with the literature, the maximum is found just upstream of the mean reattachment, and a strong decrease is observed downstream. This has been attributed to a rapid loss of coherence, after the reattachment, of the shed structures emanating from the shear layer [START_REF] Sicot | Wall pressure and conditional flow structures in the reattachment region of the flow over a thick plate[END_REF]. The maximum value of Cp' is slightly overestimated by the simulation. A possible explanation will be given in the following section.

( ) ( ) ∫ - - = V d t q t p x' x x' x' x , 2 , π ρ where ( ) i j j i i j j i i j j i u u u u U u q , , , , , , 2 + - + - = ρ ρ Q L q q + =
(1) L q and Q q are respectively the linear and quadratic contributions to p. Strictly speaking, using DDES velocity fields, this equation should be completed by terms coming from the modeled turbulent stresses. For a first approach, we have chosen to neglect these terms, which seems to be justified by the fact that the integration of the source terms described in the following restitutes correct values of ' p C . The focus of this paper is thus the flow sources of wall pressure fluctuations resolved by the present DDES approach in a reattaching flow region. We insist on the fact that such methodology is valid to test any hybrid RANS-LES or LES results in such a challenging situation.

Following eq. ( 1), the variance ( )

x 2 p reads: ( ) ( ) ( ) ∫ ∫ - - = V V d d t q t q p ' ' , ' , 4 2 2 2 x' x' x x' x x' x' x' x π ρ ( ) ∫ ∫ = V V d d S ' 4 2 2 x' x' ' x' , x' x π ρ (2) In relation (2), Q Q L Q Q L L L
q q q q q q q q qq + + + = 6a -and( )

0 , 02 . 0 , ' R R L L = x - fig.
0 , 1 . 0 , ' R R L L = x -fig. 6b -respectively.
Equation ( 2) can therefore be written in a simpler form as : [START_REF] Adrian | Conditional eddies in isotropic turbulence[END_REF]. Figure 10 shows that this conditional structure has the shape of an hairpin. This observation is consistent with the experimental findings of [START_REF] Kiya | Structure of large scale vortices and unsteady reverse flow in the reattaching zone of a turbulent separation bubble[END_REF]. A refined analysis shows that the spanwise distance between the "legs" of this conditional structure is 0.08L R . In fig. 9d, the spanwise distance between the dominant positive contribution to ( )

( ) ( ) ( ) ∫ - - = V d t p t q p x' x x' x x' x , , 2 2 π ρ . ( ) ∫ = V d D x' x' x ( ) ( ) [ ] ∫ + = V Q L d D D x' x' x' x x (3) (a) (b)
x' x D ˆ
and the localized lateral negative contributions is 0.04L R . Such a distance could therefore correspond to the average spanwise location of significant straining regions when a vortex tube is located above the considered wall location and contributes significantly to 2 p . This shows that the correct prediction of flow sources of wall pressure fluctuations in the reattachment region (and downstream it) critically depends on the ability of the DDES model to predict such longitudinal near wall structuration. In particular, many authors (e.g. [START_REF] Shur | A hybrid RANS LES approach with delayed-DES and wall-modelled LES capabilities[END_REF][START_REF] Piomelli | The inner outer layer interface in large-eddy simulations with wall-layer models[END_REF] have shown that in channel flow simulations, a DES model predicts unphysical very elongated streaks. Even if the situation is different in this case (reattachment), it is likely that the non-linear terms are strongly affected by these artificial structures. This can also explain the overestimation of Cp' observed in fig. 4. Therefore, the sources educed here have to be interpreted as sources predicted by a DDES model and an extrapolation to physical mechanisms would require further tests, using models that does not predict such structures, for instance IDDES [START_REF] Shur | A hybrid RANS LES approach with delayed-DES and wall-modelled LES capabilities[END_REF] or PITM (Fadai-Gotbi & al, 2010). This shows that the linear contribution of estimated fields is equal to the exact linear contribution. This statement has been confirmed by computing flow sources from a multi-time and multi-channel stochastic estimation of the DDES velocity field correlated with the wall pressure in Tran, 2012. Corresponding graphs are not shown here for brevity. Moreover, as any linearly estimated velocity field appears to be filtered at large scales, we have observed that the quadratic contribution of i u ~ is negligible. The rapid loss of coherence of estimated flow structures downstream the reattachment region [START_REF] Sicot | Wall pressure and conditional flow structures in the reattachment region of the flow over a thick plate[END_REF]) is therefore associated to the strong decay of linear flow sources after reattachment.

CONCLUSION

Flow sources responsible for the wall pressure fluctuations were evaluated on the basis of a DDES simulation. It has been shown that the linear sources dominate upstream of the mean reattachment, while the quadratic terms dominate downstream, due to the rapid loss of coherence of the shed vortices in this region.

This paper is restricted to the analysis of the flow predicted by a DDES model. Further studies should consider simulations using different hybrid approaches, especially in order to evaluate the influence of the model on the quadratic part. More generally, the same methodology can also be applied to LES or DNS at lower Reynolds number.

Figure 2 :

 2 Figure 2 : Mean longitudinal velocity profiles.

Figure 3 :

 3 Figure 3 : Spectra of velocity in the shear layer above the mean reattachment. Blue line: experiment, red line: simulation. (top: u component, bottom: v component)

Figure 4 :

 4 Figure 4 : Predicted Cp' versus experimental data DETERMINATION AND ANALYSIS OF THE SOURCES OF WALL FLUCTUATING PRESSURE The wall region analysed here surrounds the mean reattachment point located at e L x R 5 . 5 = = . Neglecting the contribution of the leading edge and of the Stokes terms at the wall (Naguib and Koochesfahani 2004), the wall fluctuating pressure at a location x reads :

.

  fig. 5b and 6b. R L y 02 . 0 = approximately corresponds to the average boundary of the "LES region" of the DDES computation while R L y 1 . 0 = approximately corresponds to the location of the maximum mean shear y U ∂ ∂ at meanreattachment. We clearly see that the cross correlations between linear and quadratic terms are weak when compared to linear-linear and quadratic-quadratic correlations. They will therefore be neglected in what follows. Moreover, we see that the two-point contributions rapidly decay in space when the separation between x'' and x' increases. "Integral length scales" can be computed from the corresponding correlation coefficients and are found to be of order 0.02L R .

Fig

  Fig. 7 : Iso-surface of

  and explain the lateral coherence observed in fig.7a and b.

D

  Figure 8 : Distributions of

D

  Figure 9: Distributions of

Fig. 10 :

 10 Fig. 10 : Conditional Q and wall fluctuating pressure isocontours. The condition for an event of second quadrant is rms rms v u v u -< .at

Fig

  Fig. 11 : Longitudinal evolution of

ACKNOWLEDGMENT

This study was conducted during the french ANR contract DIB (Dynamic Unsteadiness Noise) under the contract ANR-07-BLAN-0177.

This work was granted access to the HPC resources of IDRIS under the allocation 2010-020912 made by GENCI (Grand Equipement National de Calcul Intensif).


Alexandre Boritchev
email: boritchev@math.polytechnique.fr

Konstantin Khanin
email: khanin@math.toronto.edu

On hyperbolicity of minimizers for 1D random Lagrangian systems

Keywords: AMS classification scheme numbers: Primary 35Q53, Secondary 35R60, 35Q35, 37H10, 76M35

.

Introduction

A large body of work on the random forced Burgers equation and Burgers turbulence in the last 10 years (see [START_REF] Bec | Burgers Turbulence[END_REF] and further references therein) has motivated closely related studies of random Lagrangian systems [START_REF] Weinan | Invariant measures for Burgers equation with stochastic forcing[END_REF][START_REF] Iturriaga | Burgers Turbulence and Random Lagrangian Systems[END_REF]. The main object of analysis is a Lagrangian system which depends smoothly on position x and velocity v, but quite irregularly on time t:

L ω (x, v, t) = L 0 (x, v) + F ω (x, t), (1)
where F ω (x, t) is a stationary random process in t. The Lagrangian is defined on the tangent bundle T M to a connected d-dimensional Riemannian manifold M . Most rigorous results available at the moment require that M be compact, which will also be the standing assumption in this paper. Since the potential F ω (x, t) is smooth in x the most natural continuous time model is given by

F ω (x, t) = K k=1 Ẇ ω k (t)F k (x), (2)
where F k (x) are smooth non-random potentials on M , and Ẇ ω k (t) are independent white noises. One can also consider "kicked" models:

F ω (x, t) = +∞ j=-∞ F ω (j)(x)δ(t -j), (3)
where {F ω (j)(x), j ∈ Z} is a stationary sequence of random potentials. We shall assume that potentials F ω (j) are picked independently for different j ∈ Z according to a given probability distribution µ on C n (M ), where n is big enough. The Lagrangian dynamics corresponding to (3) can be described as follows. For non-integer times t the system evolves according to a non-random Lagrangian L 0 , and at integer times t = j ∈ Z the velocity changes discontinuously:

v(j + 0) = v(j -0) + ∇F ω (j)(x).
Although the two models ( 2) and (3) look rather different, the theory and results for both cases are parallel.

Lagrangian systems [START_REF] Bec | Burgers Turbulence[END_REF] are related to random forced Hamilton-Jacobi equations. One has to first define the Hamiltonian

H ω (x, p, t) = max v [p • v -L ω (x, v, t)] = H 0 (x, p) -F ω (x, t),
and then to consider the corresponding Hamilton-Jacobi equation

φ t + H ω (x, ∇φ, t) = 0. (4)
One of the most studied cases corresponds to L 0 = v 2 /2. In this case H 0 = p 2 /2 and the Hamilton-Jacobi equation (4) takes the form

φ t (x, t) + 1 2 |∇φ| 2 -F ω (x, t) = 0.
Then for the velocity field v(x, t) = ∇φ(x, t) one gets the inviscid Burgers equation:

v t (x, t) + (v • ∇)v(x, t) -∇F ω (x, t) = 0.
Although all the results of this paper hold for any Lagrangian L 0 which is convex in v and grows super-linearly as |v| → ∞, below we only consider the case

L 0 = v 2 /2.
It is well-known that minimizers for the Lagrangian L ω generate the viscosity solution of the Hamilton-Jacobi equation (4). This connection is especially useful and important for the study of global solutions, that is solutions for t ∈ (-∞, ∞). In order to discuss a global solution one has to fix the value of the first integral b = M ∇φ(x, t)dx.

(5)

The theory developed in [START_REF] Iturriaga | Burgers Turbulence and Random Lagrangian Systems[END_REF] states that under extremely mild conditions, with probability 1, for every value of the first integral b ∈ R d , there exists a unique (up to an additive constant) global solution to the Hamilton-Jacobi equation. This unique global solution can be viewed as a stationary solution. It plays the role of a global attractor for the dynamics corresponding to the Cauchy problem for the Hamilton-Jacobi equation.

Under additional assumptions of non-degeneracy one can also prove that for every value of b ∈ R d , with probability 1, there exists a unique global minimizer for the Lagrangian L ω (see [START_REF] Iturriaga | Burgers Turbulence and Random Lagrangian Systems[END_REF]). A global minimizer can be defined as a smooth curve γ : (-∞, ∞) → M such that for any compact perturbation γ the difference between Lagrangian actions corresponding to γ and to γ is non-negative. Namely, if γ -γ is supported on

[T 1 , T 2 ], then A ω,b (γ) -A ω,b (γ) = T 2 T 1 L ω (γ, γ -b, t)dt - T 2 T 1 L ω (γ, γ -b, t)dt ≥ 0.
It is expected that the global minimizer is a hyperbolic trajectory of the Lagrangian flow.

Unfortunately such a result is not available at present in the multi-dimensional case d > 1. In our view hyperbolicity of the global minimizer is one of the most important open problems in the theory of random Lagrangian systems on compact manifolds. In the one-dimensional case hyperbolicity was established in [START_REF] Weinan | Invariant measures for Burgers equation with stochastic forcing[END_REF]. However the proof in [START_REF] Weinan | Invariant measures for Burgers equation with stochastic forcing[END_REF] is unnecessarily complicated and conditions are too restrictive. In this paper we present a new proof which is both elementary and conceptual. Here, conditions for hyperbolicity are almost the same as the conditions for uniqueness of a global minimizer (see [START_REF] Iturriaga | Burgers Turbulence and Random Lagrangian Systems[END_REF]). This is another important advantage of the approach used in this article.

The following property is crucial for establishing hyperbolicity of the global minimizer. Define first backward minimizers as minimizers on semi-infinite time intervals (-∞, t] with one end point at t fixed. They can be defined in the same way as global minimizers. Now consider all backward minimizers which originate at time t, and denote by Ω s,t the set of all points x which are reached by some backward minimizer at time s ≤ t. We prove that the diameter of Ω s,t tends to zero exponentially as t → ∞. This property implies hyperbolicity by the standard argument, which also allows to construct corresponding stable and unstable manifolds. We shall not discuss these issues in the present paper and refer the readers to [START_REF] Weinan | Invariant measures for Burgers equation with stochastic forcing[END_REF]. Instead, here we shall only deal with the key shrinking property formulated above.

We finish this section with several general remarks. First, we want to emphasize the importance of hyperbolicity of the global minimizer. It immediately implies many fundamental properties of the global solution to the Hamilton-Jacobi equation, such as piecewise smoothness, exponential rate of convergence to the global solution, and many others. It also allows to study the structure of singularities (shocks) (see [START_REF] Bec | Burgers Turbulence[END_REF]).

Our second remark is related to a general problem of hyperbolicity of minimizers for generic non-random Lagrangian systems. This is one of the central problems of the Aubry-Mather theory. Randomness is another way to introduce the notion of genericity. In this setting generic stands for properties which hold for almost all systems (with probability 1). Note however that in many respects, random and nonrandom (autonomous, or depending on time periodically) Lagrangian systems are very different.

In particular, all number-theoretical aspects of the Aubry-Mather theory disappear in the random case.

Finally, we want to say a few words about the non-compact case. At present there are almost no rigorous results in that setting. It is believed that if the system exhibits any form of translation invariance, global minimizers do not exist. However, it is likely that backward minimizers do exist, and the study of their asymptotic scaling properties is an extremely interesting and important problem.

Hyperbolicity assumptions and main results

We begin by formulating the assumptions on potentials: Assumption 2.1. In the "kicked" case, we assume the following. (i) The kicks at integer times j are of the form

F ω (j) = K k=1 c ω k (j)F k ,
where F k are smooth potentials on S 1 = R/Z. The random vectors (c ω k (j)) 1≤k≤K are independent identically distributed R K -valued random variables. Their distribution on R K , denoted by µ, is assumed to be absolutely continuous with respect to the Lebesgue measure. (ii) 0 belongs to Supp µ. (iii) The mapping from S 1 to R K defined by x → (F 1 (x), ..., F K (x)) is an embedding. Remark 2.1. Let g be the function defined by

g(c 1 , ..., c K ) = K k=1 c k F k .
We denote by ν the corresponding push-forward measure ν = g * (µ) on a smooth Sobolev space. The assumption 0 ∈ Supp µ can then be replaced by the slightly weaker assumption 0 ∈ Supp ν. Assumption 2.2. In the case of the white force potential, we assume the following. (i) The forcing has the form

F ω (x, t) = K k=1 Ẇ ω k (t)F k (x),
where F k are smooth potentials on S 1 , and Ẇ ω k are independent white noises, i.e. weak time derivatives of independent Wiener processes W ω k (t). (ii) The mapping from S 1 to R K defined by

x → (F 1 (x), ..., F K (x))
is an embedding.

We denote by G an antiderivative in time of the forcing:

G ω (x, t) = K k=1 W ω k (t)F k (x),
where W ω k (t) are independent standard Wiener processes with W ω k (0) = 0. Since we will only consider time differences of G, the particular choice of antiderivative has no importance.

In both cases, F ω will be abbreviated as F , and in the white force case F (•, t) will be abbreviated as F (t), and similarly for G.

Remark 2.2. The embedding conditions are consistent with the condition for uniqueness of the global minimizer (see [START_REF] Iturriaga | Burgers Turbulence and Random Lagrangian Systems[END_REF]). In the "kicked" case, the condition for uniqueness in [START_REF] Iturriaga | Burgers Turbulence and Random Lagrangian Systems[END_REF] is slightly weaker: the map x → (F 1 (x), ..., F K (x)) is only required to be one-to-one. However, we need to assume the embedding to prove hyperbolicity.

The following property, called the separation property, plays a crucial role in our construction.

Property 2.1. There exist α 0 > 0, three pairwise disjoint open intervals J i , i = 1, 2, 3, and three potentials Fi , i = 1, 2, 3 with the following properties. 1) In the "kicked" case, we have Fi ∈ Supp ν for every i. In the white force case, each Fi is a linear combination of the F k .

2) Each of the functions -Fi reaches its minimum, denoted by m i , at a single point x i .

3) For every α, 0 < α ≤ α 0 , there exist three open intervals

I i (α), I i ⊂ J i , i = 1, 2, 3 such that Fi (S 1 -I i ) ⊂ (-∞, -m i -α].
Note that for every i and α, the point x i where min(-Fi ) is reached belongs to I i . Lemma 2.1. Assumptions 2.1 or 2.2 imply the separation property.

Proof of Lemma 2.1: "Kicked" case: We start by showing that, for Lebesgue-a.e. vector (c j ) 1≤j≤K , the maximum of

K j=1 c j F j (x)
is reached at a single point x ∈ S 1 . This follows from a rather standard argument (see [START_REF] Iturriaga | Burgers Turbulence and Random Lagrangian Systems[END_REF]Corollary 5]). Indeed, the function

Φ : (c 1 , . . . , c K ) → max x∈S 1 K j=1 c j F j (x)
is Lipschitz and therefore differentiable a.e., with respect to the Lebesgue measure µ Leb . On the other hand, at a point of differentiability of Φ,

∇Φ(x max ) = (F 1 (x max ), . . . , F K (x max ))
for every point of maximum x max . Hence the embedding assumption 2.1 (iii) implies that the point of maximum is unique. Since µ is absolutely continuous with respect to µ Leb , the maximum uniqueness set O 1 ⊂ R K has full µ-measure.

Furthermore, by the Lebesgue points theorem [4, Theorem 7.7], c = (c j ) j is a Lebesgue point for the density q = dµ dµ Leb on a set O of full µ Leb -measure, and thus of full µ-measure.

Denote by O 2 ⊂ O the set of Lebesgue points c for q such that q(c) > 0. By definition, they belong to Supp µ, and O 2 has full µ-measure.

Now consider c 1 = (c 1 j ) j ∈ O 1 ∩ O 2 .
Denote by x 1 the point where the maximum of

F1 = K j=1 c 1 j F j (x) is reached: x 1 = argmax F1 . Denote by V the set of vectors (c j ) j such that K j=1 c j dF j dx (x 1 ) = 0.
Denote by B n the open ball with radius 1/n centered at c 1 . We will also need

B n = B n ∩ (c 1 + V ) ∩ O 1 ∩ O 2 . By the embedding assumption 2.1 (iii), B n ∩ (c 1 + V ) is just B n itself with a removed hyperplane. Thus, since µ is continuous with respect to µ Leb , we have µ(B n ) = µ(B n ).
Using [4, Theorem 7.7] one more time, we obtain that there exists a constant N 0 such that for n ≥ N 0 ,

µ(B n ) = µ(B n ) ≥ q(c 1 ) 2 µ Leb (B n ) > 0.
On the other hand, for small enough > 0 there exists N 1 ( ) such that for n ≥ N 1 , if (c j ) j ∈ B n , then K j=1 c j F j reaches its (unique) maximum in a point of theneighbourhood of x different from x itself. Considering a smaller neighbourhood at each step, this argument can be repeated any finite number of times. It enables us to construct any number of potentials contained in Supp ν and attaining their respective maxima at different points: three suffice for our purposes. Denote them by F1 , F2 , F3 . Let J 1 , J 2 , J 3 be three non-intersecting open intervals around their respective points of maximum. Take as α 0 the minimum of max( Fi ) -max( Fi | S 1 -J i ). It is obvious that for any α ∈ (0, α 0 ] we can construct the required intervals I i (α).

White force case: The proof follows the same lines, but is much simpler since measure-theoretic arguments are trivialised. Definition 2.1. Consider a closed subset Z of S 1 . Let m(Z) denote the maximal length of a connected component of S 1 -Z. We define the diameter of Z as

d(Z) = 1 -m(Z).
The diameter of Z can be thought of as the minimal length of an interval on S 1 containing Z.

In what follows we use the function ψ ω , either deterministic or random, as an initial condition at time s. Everywhere below, the value of the first integral b (see ( 5)) is fixed. For simplicity, we do not indicate dependence on b in our notation. Definition 2.2. For a given value of b ∈ R, a curve γ y,x s,t (τ ) is a minimizer if it minimizes the action

A(γ) = 1 2 t s ( γ(τ ) -b) 2 dτ + n∈[s,t) -F (n)(γ(n))
in the "kicked" case and the action

A(γ) = 1 2 t s ( γ(τ ) -b) 2 dτ + t s γ(τ ) ∂G ∂x (γ(τ ), τ ) - ∂G ∂x (γ(τ ), t) dτ + G(γ(s), s) -G(γ(s), t)
in the white force case, respectively, over all absolutely continuous curves with endpoints x at time t and y at time s.

Definition 2.3. For any time interval [s, t] and any continuous function ψ :

S 1 → R, a curve γ x s,t,ψ (τ ) : [s, t] → S 1 is a ψ-minimizer if it minimizes A(γ) + ψ(γ(s)
) over all absolutely continuous curves with endpoint x at time t. Definition 2.4. For -∞ < r < s ≤ t < +∞ and for a fixed function ψ(•, r) : S 1 → R, let Ω r,s,t,ψ be the set of points reached, at the time s, by ψ-minimizers on [r, t]:

Ω r,s,t,ψ = {γ x r,t,ψ (s), x ∈ S 1 }. Remark 2.3.
In what follows, the initial condition ψ will always be fixed, while t will increase to +∞. It is important that we shall consider both deterministic and random initial conditions ψ. In the latter case, ψ should be measurable with respect to the past σ-algebra B r = B (-∞,r] , which is defined in a standard way. It is important to take r smaller then s. Everywhere below, we set r = s -1. To simplify notation, Ω s-1,s,t,ψ will be denoted by Ω s,t .

It is well-known that Ω s,t is a closed set. Obviously, Ω s,t 1 ⊇ Ω s,t 2 for all s ≤ t 1 ≤ t 2 . It follows that t → d(Ω s,t ) is a non-increasing function.

We are now able to formulate the main results of this paper which are the following theorem and its corollary. Both results hold for a given value of b ∈ R. However, all constants are uniformly bounded if b stays bounded. It is easy to see that in the "kicked" case, b is effectively defined modulo 1, since the action is invariant under the transformation (b, γ) → (b + 1, γ + t). Thus in this case all constants are uniformly bounded for all b.

Theorem 2.1. Assume that the separation property holds. Then there exist constants λ,

C > 0 such that if -∞ < s ≤ t < +∞, then E(d(Ω s,t )) ≤ C exp(-λ(t -s)),
where E(•) stands for the expectation with respect to the distribution of potentials.

Corollary 2.1. Assume that the separation property holds. Fix s ∈ R. Then, for a.e. ω, there exists a random constant C(s, ω) > 0 such that

d(Ω s,t ) ≤ C(s, ω) exp(-λ(t -s)/2), t ≥ s.
Here, λ is the same as in Theorem 2.1.

As we have already pointed out in the introduction, Corollary 2.1 implies hyperbolicity (see [START_REF] Weinan | Invariant measures for Burgers equation with stochastic forcing[END_REF] for details). The following lemma, called the main lemma, is proved in Section 3: the proof is quite involved, with additional technical difficulties in the white force case.

Main Lemma. Assume that the separation property holds. Fix b ∈ R. Then there exist constants c, T > 0 such that if -∞ < s ≤ t < +∞, then the following inequality holds a.s.:

P d(Ω s,t+T ) ≤ d(Ω s,t ) 2 | B t ≥ c.
We finish this section by deriving Theorem 2.1 and Corollary 2.1 from the main lemma.

Proof of Theorem 2.1 : Consider the function

d(t) = E(d(Ω s,t )) exp(λ(t -s)),
where λ is a fixed positive number, chosen later.

Since t → d(Ω s,t ) is non-increasing, the main lemma implies that

E(d(Ω s,t+T )) ≤ c E(d(Ω s,t )) 2 + (1 -c)E(d(Ω s,t )).
Thus

d(t + T ) ≤ exp(λT ) 1 - c 2 d(t). Now put λ = - 1 T ln 1 - c 2 .
It follows that d(t + T ) ≤ d(t). But d(s) = 1. Therefore, for t ∈ s + T N, we have d(t) ≤ 1. Consequently, since t → d(Ω s,t ) is non-increasing, we have

E(d(Ω s,t )) ≤ C exp(-λ(t -s)), t ≥ s, with C = exp(λT ) = (1 -c 2 ) -1
. This proves the theorem's assertion. Proof of Corollary 2.1 assuming Theorem 2.1: In the same way as in the previous proof, it is enough to prove the statement for t ∈ s + T N. By Theorem 2.1 and Chebyshev's inequality, for every X > 0,

P(d(Ω s,s+nT ) ≥ X exp(-λnT /2)) ≤ C X exp(-nλT /2), n ≥ 0.
An application of the Borel-Cantelli lemma ends the proof.

Proof of the main lemma

For all s < t, let us define a map S t s from S 1 to S 1 , which can be viewed as a coordinate projection at time t of the generalized Lagrangian flow corresponding to the Burgers equation. It certainly depends on the initial condition ψ at time s -1.

If, at time s, a point y belonging to S 1 is reached by a ψ-minimizer on [s -1, t] starting in x at time t, then S t s (y) is equal to the point x. Note that such an x is unique, since minimizers on the time interval [s -1, t] cannot intersect outside of endpoints s -1 and t.

If a point y is not reached by such a ψ-minimizer, then it belongs to a closed interval corresponding to a shock at time t. In this case S t s (y) is equal to the corresponding shock position. To define an interval at time s corresponding to a shock at time t, one has to consider rightmost and leftmost minimizers originating at (x, t). Intersections of those minimizers with S 1 × {s} generate a space interval of points absorbed by the shock (x, t). It is easy to see that every point (y, s) is reached by a minimizer or belongs to a shock interval generated by a uniquely defined shock.

Note that some points may correspond to both cases considered above. Namely, points corresponding to minimizers which originate from the shock positions. However, even in this case the map S t s is still uniquely defined.

Proof in the "kicked" case

Put C = 3 max i∈1,2,3 Fi C 1 + 1 . (6)
Then put

α = min α 0 , 1 10C (7)
(see the separation property for the definition of α 0 .) We keep in mind that α < 1/30. Consider integers

N ∈ 2 + 1 α 3 , 2 α 3 ; N ∈ 2 α 10 + 1, 4 α 10 . ( 8
)
Denote by E 1 the event

F (t + k) ∞ ≤ α 20 , 0 ≤ k ≤ N -1. (9)
By Assumption 2.1 (ii) the zero potential belongs to Supp ν. It follows that E 1 has positive probability. Put l = 1-d(Ω s,t ). If Ω s,t = S 1 , consider a connected component (y 1 , y 2 ) of S 1 -Ω s,t which has maximal length l. Let y 3 be the center of (y 1 , y 2 ), and let y 4 be the point diametrically opposite to y 3 . If Ω s,t = S 1 , let y 3 and y 4 be any pair of diametrically opposite points in S 1 . Then consider z 1 = S

γ(τ ) = x + b(τ -t -N ), τ ∈ [t + N, t + 2N -1]
for some x ∈ S 1 . We claim that there exist (at least)

N different integers 0 = n 0 < . . . < n N -1 ≤ N -1 such that we have max j,j ∈[0,N -1] |γ(t + N + n j ) -γ(t + N + n j )| ≤ α 7 .
(10) Indeed, by the pigeonhole principle, since N ≤ α 7 N , there exist integers 0 ≤ ñ0 < . .

. < ñN -1 ≤ N -1 such that max j,j ∈[0,N -1] |γ(t + N + ñj ) -γ(t + N + ñj )| ≤ α 7 .
Then it suffices to take, for every j, n j = ñj -ñ0 . By definition of C and α, (10) yields that

max j,j ∈[0,N -1] | F1 (γ(t + N + n j )) -F1 (γ(t + N + n j ))| ≤ α 7 F1 C 1 ≤ α 6 /10. ( 11
)
Now consider the event E 2 defined by the system of inequalities:

     F (t + N + n j ) -F1 ∞ ≤ α 20 , 0 ≤ j ≤ N -1. F (t + N + k) ∞ ≤ α 20 , k ∈ [0, N -1] -{n 0 , . . . , n N -1 }. ( 12
)
Since F1 and 0 belong to Supp ν, this event (independent from E 1 ) also has positive probability.

It remains to prove that for ω ∈ E 1 ∩ E 2 all minimizers on [t, t + 2N ] pass through I 1 (α) at time t + N , which follows from Lemma 3.1 and Lemma 3.2. Indeed, if this statement holds, no such minimizers can pass through [z 1 , z 2 ] at t + N , since I 1 (α) ⊂ J 1 . Consequently all the points that are in [y 3 , y 4 ] at time s will not be reached by minimizers originating at time t+2N . In particular, it follows that [y 3 , y 4 ] is contained in an interval generated by some shock at time t+2N . Therefore (y

1 , y 2 )∪[y 3 , y 4 ] = (y 1 , y 4 ] is contained in a connected component of S 1 -Ω s,t+2N . Thus d(Ω s,t+2N ) ≤ 1 - 1 + l 2 = 1 2 d(Ω s,t )
with a positive conditional probability which equals at least P(E 1 )P(E 2 ). This proves the lemma's assertion.

Lemma 3.1. Assume that ω ∈ E 2 . Then for every minimizer γ 1 on [t + N, t + 2N ] there exists j,

1 ≤ j ≤ N -1, such that -F1 (γ 1 (t + N + n j )) ≤ m 1 + α 2 .
Proof: We argue by contradiction. Suppose that there exists a minimizer γ 1 on

[t + N, t + 2N ] such that -F1 (γ 1 (t + N + n j )) > m 1 + α 2 , 1 ≤ j ≤ N -1. (13)
Consider a curve γ 2 with the same endpoints as γ 1 , linear on intervals

[t + N + k, t + N + k + 1]. Moreover we suppose that γ 2 = x 1 + b(τ -t -N ) on [t + N + n 1 , t + N + n N -1 ]
(x 1 being the point where F1 reaches its maximum), and that

| γ2 -b| ≤ 1/2n 1 and | γ2 -b| ≤ 1/2(N -n N -1 ) on the extremal intervals [t + N, t + N + n 1 ] and [t + N + n N -1 , t + 2N ],
respectively. From now on, for a curve γ we denote γ -b by γb . We recall that the "kicked" case action A for γ :

[t 1 , t 2 ] → S 1 equals A(γ) = 1 2 t 2 t 1 ( γb (τ )) 2 dτ - n∈[t 1 ,t 2 ) F [γ(n)].
The first part of the right-hand side, corresponding to the kinetic energy, will be denoted by A k . The remaining part, corresponding to the potential energy, will be denoted by A p . We observe that

A k (γ| [t 1 ,t 3 ] ) = A k (γ| [t 1 ,t 2 ] ) + A k (γ| [t 2 ,t 3 ] ), (14)
and similarly for A p . We have

A k (γ 1 ) ≥ 0; A k (γ 2 ) ≤ 1 4 .
On the other hand, using the inequalities (11-13), we get

A p (γ 1 ) ≥ (N -1)(m 1 + α 2 -α 20 ) -(N -N )α 20 -F (γ(t + N )), A p (γ 2 ) ≤ (N -1)(m 1 + α 6 /10 + α 20 ) + (N -N )α 20 -F (γ(t + N )).
The constants C, α, N , N are the same as in the proof of the "kicked" case, with the exception that now

α = min α 0 , 1 10C , 1 10(b + 1) 2 , (15)
and that the definitions of N and N change accordingly. Denote by E 1 the event

sup t 1 ,t 2 ∈[t,t+N ] G(t 1 ) -G(t 2 ) C 1 ≤ α 40 . (16)
By classical properties of the Wiener process, E 1 has positive probability, uniformly in t. Now we proceed exactly in the same way as in the "kicked" case, supposing with the same notation and without loss of generality that

[z 1 , z 2 ] ∩ J 1 = ∅.
We assume that for every j, j ∈ [0, N -1] (we take n N = N ), G satisfies:

                   G(t + N + n j+1 ) -G(t + N + n j ) -F1 C 1 ≤ α 40 . G(t + N + n j+1 ) -G(t + N + n j + τ ) C 1 ≤ α 40 , τ ∈ [α 40 , n j+1 -n j ]. G(t + N + n j + τ ) -G(t + N + n j + τ ) C 1 ≤ 3 2 F1 C 1 ≤ C 2 , τ, τ ∈ [0, n j+1 -n j ]. (17)
This event, denoted by E 2 , has positive probability and is independent from E 1 . Finally, in the same way as in the "kicked" case, the lemma's assertion follows from Lemma 3.3 and Lemma 3.4.

Lemma 3.3. Consider a minimizer γ 1 on [t + N, t + 2N ]. Then, if ω ∈ E 2 , we have -F1 (γ 1 (t + N + n j )) ≤ m 1 + α 2 (18)
for some j, 1 ≤ j ≤ N -1.

Proof: As previously, we argue by contradiction, considering a minimizer γ 1 on [t + N, t + 2N ] such that (18) does not hold for any j, 1 ≤ j ≤ N -1. We recall that the action is given by:

A(γ) = 1 2 t 2 t 1 γb (τ ) 2 dτ + t 2 t 1 γ(τ ) ∂G ∂x (γ(τ ), τ ) - ∂G ∂x (γ(τ ), t 2 ) dτ + G(γ(t 1 ), t 1 ) -G(γ(t 1 ), t 2 ) .
The first term of the right-hand side, i.e. the kinetic energy, will be denoted by A 1 .

The second and the third terms, whose sum is the potential energy, will be denoted by A 2 and A 3 , respectively. We observe that A as well as the quantities A 1 and A

∂G ∂x (γ 1 (τ ), τ ) - ∂G ∂x (γ 1 (τ ), t + N + n j+1 ) 2 dτ .
The first term of the right-hand side can be estimated by observing that the restriction of γ 1 to [t + N + n j , t + N + n j+1 ] is still a minimizer, and that A 3 t+N +n j ,t+N +n j+1 , which only depends on the endpoint of the curve at t + N + n j , is the same for γ 1 and γ 3 .

On the other hand, the second and the third terms of the right-hand side can be estimated by using (17). Thus we obtain that

R ≥ - 2α 20 C -R + A 1 t+N +n j ,t+N +n j+1 (γ 3 ) + A 2 t+N +n j ,t+N +n j+1 (γ 3 ) + b 2 N 2 - C 2α 20 α 40 C 2 4 + (n j+1 -n j -α 40 )α 80 ≥ - 2α 20 C -R + 1 8(n j+1 -n j ) + t+N +n j+1 t+N +n j γ3 (τ ) ∂G ∂x (γ 3 (τ ), τ ) - ∂G ∂x (γ 3 (τ ), t + N + n j+1 ) dτ + b 2 N 2 -α 20 C 3 . Consequently, R ≥ 2α 20 C R - α 20 4C - 2α 20 C b + 1 2 t+N +n j+1 t+N +n j ∂G ∂x (γ 3 (τ ), τ ) - ∂G ∂x (γ 3 (τ ), t + N + n j+1 ) dτ - b 2 N α 20 C -α 20 C 3 .
Using (6), ( 15), (8), and (17), we get

R ≥ 2α 20 C R - α 20 4C -b + 1 2 N α 20 - b 2 N α 20 C -α 20 C 3 ≥ 2α 20 C R -N α 20 (b + 1) 2 -C 3 + 1 4C α 20 ≥ 2α 20 C R - 4α 9 10 - 2α 17 10 3 ≥ 2α 20 C R - α 9 2 . Consequently, R ≥ -1 - 2α 20 C -1 α 9 2 ≥ -α 9 .
By (11), it follows that for j ∈ [1, N -2] we have

A t+N +n j ,t+N +n j+1 (γ 2 ) -A t+N +n j ,t+N +n j+1 (γ 1 ) = (A 1 t+N +n j ,t+N +n j+1 (γ 2 ) -A 1 t+N +n j ,t+N +n j+1 (γ 1 )) + (A 2 t+N +n j ,t+N +n j+1 (γ 2 ) -A 2 t+N +n j ,t+N +n j+1 (γ 1 )) + (A 3 t+N +n j ,t+N +n j+1 (γ 2 ) -A 3 t+N +n j ,t+N +n j+1 (γ 1 )) ≤ (0 -0) + b(Cα 40 /2 + N α 40 ) -(-α 9 ) + (m 1 + α 6 /10 + α 40 ) -(m 1 + α 2 -α 40 ) ≤ - α 2 2 . ( 19
)
Here, the estimate of A 2 t+N +n j ,t+N +n j+1 (γ 2 ) follows from (17). Similarly, since A 3 t+N,t+N +n 1 (γ 2 ) = A 3 t+N,t+N +n 1 (γ 1 ), we have

A t+N,t+N +n 1 (γ 2 ) -A t+N,t+N +n 1 (γ 1 ) ≤ 1 8 + A 2 t+N,t+N +n 1 (γ 2 ) + α 9 ≤ 1, (20)
and

A t+N +n N -1 ,t+2N (γ 2 ) -A t+N +n N -1 ,t+2N (γ 1 ) ≤ 1 8 + A 2 t+N +n N -1 ,t+2N (γ 2 ) + α 9 + C ≤ 2C. (21)
Here, we get

A 2 t+N,t+N +n 1 (γ 2 ), A 2 t+N +n N -1 ,t+2N (γ 2 ) ≤ (b + 1/2)(Cα 40 /2 + N α 40
) in the same way as for the estimate of A 2 t+N +n j ,t+N +n j+1 (γ 2 ) above. It remains to add together the inequalities (19-21). Using (15) and (8) we get

A t+N,t+2N (γ 2 ) -A t+N,t+2N (γ 1 ) ≤ 2C + 1 -(N -2) α 2 2 ≤ 2C + 1 - 1 2α < 0.
This inequality is in contradiction with the fact that γ 1 is a minimizer. This proves the lemma's assertion.

Lemma 3.4. For ω ∈ E 1 ∩E 2 , if for some minimizer γ 1 on [t, t+N +n j ], 1 ≤ j ≤ N -1, y = γ 1 (t + N + n j ) satisfies: -F1 (y) ≤ m 1 + α 2 ,
then we have

γ 1 (t + N ) ∈ I 1 (α).
Proof: In the same way as in the proof of Lemma 3.2, we consider a "bad" minimizer γ 1 . Without loss of generality, we assume that

-F1 [γ 1 (t + N + n j )] > m 1 + α 2 , 1 ≤ j < j. (22)
We define γ 2 with the same endpoints as γ 1 in the same way as in the proof of Lemma 3.2, i.e. such that γ2 b = 0 between t + N and t + N + n j , linear between t and t + N , and satisfying

| γ2 b | ≤ 1 2N . We get A 3 t,t+N (γ 2 ) = A 3 t,t+N (γ 1 ). A 1 t,t+N (γ 2 ) -A 1 t,t+N (γ 1 ) ≤ N 8N 2 -0 ≤ α 10 16 . A 2 t,t+N (γ 2 ) ≤ b + 1 2N t+N t ∂G ∂x (γ 2 (τ ), τ ) - ∂G ∂x (γ 2 (τ ), t + N ) dτ ≤ α 29 .
The last inequality follows from (15), (8), and (16).

To estimate the quantity R = A 2 t,t+N (γ 1 ), we proceed in the same way as for A 2 t+N +n j ,t+N +n j+1 (γ 1 ) in Lemma 3. The estimates of A 2 t+N +n j ,t+N +n j +1 (γ 1 ), 0 ≤ j < j in Lemma 3.3 still hold in our case. Therefore A 2 t+N,t+N +n j (γ 1 ) ≥ -N α 9 ≥ -2α 6 . By (11) and ( 22), for 1 ≤ j ≤ j -1 we get A 3 t+N +n j ,t+N +n j +1 (γ 2 ) -A 3 t+N +n j ,t+N +n j +1 (γ 1 )

≤ (m 1 + α 2 + α 6 10 + α 40 ) -(m 1 + α 2 -α 40 ) ≤ α 6 .
Finally, since we have supposed that γ 1 (t + N ) / ∈ I 1 (α), we have Combining all these inequalities with (23) we get U ≤ 2α 9 + α 26 + 2α 6 + (N -1)α 6 -α/2 < 0.

We have a contradiction with the fact that γ 1 is a minimizer. This proves the lemma's assertion.

y 4 .

 4 Since the J i (see the separation property for their definition) are pairwise disjoint, one of the J i has an empty intersection with one of [z 1 , z 2 ] and [z 2 , z 1 ]. Without loss of generality, we may suppose that [z 1 , z 2 ] ∩ J 1 = ∅. Now consider the straight line defined by

A 3

 3 t+N,t+N +n 1 (γ 2 ) -A 3 t+N,t+N +n 1 (γ 1 ) ≤ (m 1 + α 2 + α 6 10 + α 40 ) -(m 1 + α -α 40 ) ≤ -α/2.

  2 + A 3 satisfy a relation of the same type as (14). To see it for A 2 + A 3 , it suffices to write down this sum as a stochastic integral. A(γ| [s,t] ) is denoted by A s,t (γ), and similarly forA i , i = 1, 2, 3.Consider a curve γ 2 with the same endpoints as γ 1 , defined exactly in the same way as in the proof of Lemma 3.1. Namely,γ 2 = x 1 +b(τ -t-N ) on [t+N +n 1 , t+N +n N -1 ],and γ 2 is linear on [t + N, t + N + n 1 ] and on[t + N + n N -1 , t + 2N ] with | γb 2 | ≤ 1/2n 1 and | γb 2 | ≤ 1/2(N -n N -1), respectively. Now, for every j ∈ [0, N -1], consider a straight line γ 3 connecting γ 1 (t + N + n j ) and γ 1 (t+N +n j+1 ) with constant velocity γ3 , | γ3 b | ≤ 1/2(n j+1 -n j ) (we take n N = N ). Denote by R the quantity R = A 2 t+N +n j ,t+N +n j+1 (γ 1 ).

	Since
	a(τ )b(τ )dτ ≥ -	2α 20 C		a(τ ) 2	2	dτ -	C 2α 20 b(τ ) 2 dτ ,
	then we have
	R ≥ -	2α 20 C	t+N +n j+1	γb 1 (τ ) + b 2	2	dτ
				t+N +n j
			t+N +n j+1
	-	C 2α 20		∂G ∂x	(γ 1 (τ ), τ ) -	∂G ∂x	(γ 1 (τ ), t + N + n j+1 )	2	dτ
			t+N +n j
	≥ -	2α 20 C	A 1 t+N +n j ,t+N +n j+1 (γ 1 ) +	b 2 (n j+1 -n j ) 2
				t+N +n j +α 40
	-	C 2α 20				∂G ∂x	(γ 1 (τ ), τ ) -	∂G ∂x	(γ 1 (τ ), t + N + n j+1 )	2	dτ
				t+N +n j
		t+N +n j+1
	+
	t+N +n j +α 40

  [START_REF] Iturriaga | Burgers Turbulence and Random Lagrangian Systems[END_REF]. Namely, we consider a straight line γ 3 with the same endpoints asγ 1 | [t,t+N ] satisfying | γ3 b | ≤ 1/2N .Since a restriction of γ 1 is still a minimizer, we getR ≥ -2α 20 A 1 t,t+N (γ 3 ) + A 2 t,t+N (γ 3 ) -R + ) ≥ -α 9 .On the other hand, we haveA 1 t+N,t+N +n j (γ 2 ) -A 1 t+N,t+N +n j (γ 1 ) ≤ 0.By definition, the action differenceU = A t,t+N +n j (γ 2 ) -A t,t+N +n j (γ 1 ) satisfies U = (A 1 t,t+N (γ 2 ) -A 1 t,t+N (γ 1 )) + (A 1 t+N,t+N +n j (γ 2 ) -A 1 t+N,t+N +n j (γ 1 )) + (A 2 t,t+N (γ 2 ) -A 2 t,t+N (γ 1 )) + (A 3 t,t+N (γ 2 ) -A 3 t,t+N (γ 1 )) + (A 2 t+N,t+N +n j (γ 2 ) + A 3 t+N,t+N +n j (γ 2 ) -A 2t+N,t+N +n j (γ 1 ) -A 3 t+N,t+N +n j (γ 1 )).In the same way as previously, we get A 2 t+N,t+N +n j (γ 2 ) ≤ bj Cα 40 2 + N α 40 ≤ 5bN α 30 ≤ α 26 .

	≥ -2α 20	t+N	( γb 1 (τ )) 2 + b 2 2	dτ
				t
			t+N
	-	1 2α 20				∂G ∂x	(γ 1 (τ ), τ ) -	∂G ∂x	(γ 1 (τ ), t + N )	2	dτ .
			t
									b 2 N
									2
			t+N
	-	1 2α 20				∂G ∂x	(γ 1 (τ ), τ ) -	∂G ∂x	(γ 1 (τ ), t + N )	2	dτ
			t
	≥ -2α 20 N 8N 2 + b +	1 2N
		t+N
	×		∂G ∂x	(γ 3 (τ ), τ ) -	∂G ∂x	(γ 3 (τ ), t + N ) dτ
		t
	-R +	b 2 N 2	-	N 2α 20 α 80
	≥ 2α 20 R -2α 20 α 10 16	+ (b + 1)N α 40 +	2b 2 α 10 -	N α 60 2
	≥ 2α 20 R -(5b 2 + 1)α 10 ≥ 2α 20 R -	α 9 2	.
	Therefore
	We have t,t+N (γ 1 Consequently, R ≥ -2α 20 A 2 U ≤ α 10 16 + 0 + α t+N γ1 (τ ) 2	2	dτ
				t
			t+N
	-	1 2α 20							2	dτ

t ∂G ∂x (γ 1 (τ ), τ ) -∂G ∂x (γ 1 (τ ), t + N ) 29 + α 9 + 0 + (A 2 t+N,t+N +n j (γ 2 ) + A 3 t+N,t+N +n j (γ 2 ) -A 2 t+N,t+N +n j (γ 1 ) -A 3 t+N,t+N +n j (γ 1 )) ≤ 2α 9 + (A 2 t+N,t+N +n j (γ 2 ) -A 2 t+N,t+N +n j (γ 1 ) + A 3 t+N,t+N +n j (γ 2 ) -A 3 t+N,t+N +n j (γ 1 )].

(23)

Acknowledgments

We would like to thank R. Iturriaga and S. Kuksin for numerous discussions. A big part of the work was carried out while both authors were visiting Observatoire de la Côte d'Azur in Nice. We are very grateful to U. Frisch, J. Bec and other colleagues from the Observatoire for their hospitality.

Therefore, by (7-8), we get

Thus we have a contradiction with the fact that γ 1 is a minimizer. This proves the lemma's assertion.

Then we have

Proof: We argue by contradiction, supposing that γ 1 (t + N ) / ∈ I 1 (α). We may also assume that

Indeed, otherwise we could consider a smaller value of j. In the same way as previously, we want to prove that γ 1 cannot be a minimizer, and we consider a curve γ 2 with the same endpoints as γ 1 . Namely, we suppose that γ 2 satisfies γb 2 = 0 between t + N and t + N + n j , γ 2 is linear between t and t + N , and moreover | γb 2 | ≤ 1/2N . We have the inequalities

On the other hand, using the separation property, (9), (11), and (12), we get

Therefore, by (7-8), we obtain that

Again, we have a contradiction. This proves the lemma's assertion.

Proof in the white force case

The scheme of the proof is very similar to the one in the "kicked" case. The major differences are auxiliary lemmas which are technically more involved and the conditions on the forcing, in some way much more restrictive.