Ian Frigaard

Cherif Nouar

Onset of flow in a vibrated thin viscoplastic layer

Keywords:

The onset of flow in vibrated layer of viscoplastic fluid is investigated theoretically, using a lubrication approximation. The rheological behavior of the fluid is described by the Herschel-Bulkley model. The equation describing the evolution of the free surface is derived. Four different regimes are found depending on the ratio χ of the gravitational acceleration to the acceleration of vibration and the ratio of the yield stress to the shear-stress at the vibrated wall. Furthermore, a necessary condition for instability is derived. The different regimes are illustrated in the case of a 1D problem.

Introduction

When a horizontal fluid layer is subjected to vertical oscillations, its free surface becomes unstable to standing waves beyond a certain threshold. This phenomenon was first observed by Faraday [START_REF] Faraday | On the forms and states assumed by fluids in contact with vibrating elastic surfaces[END_REF]. He, noticed that the vibrating frequency of the waves is half of that of the forcing. The onset of these waves was first determined by Benjamin and Ursell [START_REF] Benjamin | The stability of the plane free surface of a liquid in a vertical periodic motion[END_REF] for an incompressible inviscid fluid. Using linear stability analysis, they showed that for an inviscid infinite layer of fluid, the onset problem reduces to analysis of Mathieu's equation, which allows harmonic and subharmonic responses. Kumar and Tuckerman [START_REF] Kumar | Parametric instability of the interface between two fluids[END_REF] extended the linear stability analysis of Benjamin and Ursell [START_REF] Benjamin | The stability of the plane free surface of a liquid in a vertical periodic motion[END_REF] to viscous fluids. In the last few decades, several studies have been devoted to the Faraday instability in Newtonian fluids and focused on pattern formation. According to the driving frequency and amplitude and the fluid viscosity, the free surface may exhibit stripes, squares, hexagons and quasi-crystalline patterns; see e.g. [START_REF] Douady | Pattern selection in Faraday instability[END_REF][START_REF] Douady | Experimental study of the Faraday instability[END_REF][START_REF] Christiansen | Ordered capillary-waves states: quasicrystals, hexagons, and radial waves[END_REF][START_REF] Edwards | Patterns and quasipatterns in the Faraday experiment[END_REF][START_REF] Kumar | Competing patterns in the faraday experiment[END_REF][START_REF] Kudrolli | Patterns and spatiotemporal chaos in parametrically forced surface waves: a systematic survey at large aspect ratio[END_REF].

Recently, interest has grown in the effect of vertical vibrations on a layer of a complex fluid both theoretically and experimentally. Raynal et al. [START_REF] Raynal | Faraday instability with a polymer solution[END_REF] provided the first experimental data for polymer solutions. Their work concentrates on dilute polymer solutions where the influence of elasticity is small. They found a slight shift of the critical accelerations; the critical wave numbers were not affected. Muller & Zimmermann [START_REF] Hw Müller | Faraday instability in a linear viscoelastic fluid[END_REF] presented a linear stability analysis for a Maxwell fluid. They found that when the inverse of the relaxation time compares to the driving frequency, a harmonic response of the fluid surface might become unstable rather than the classical subharmonic response. The signature of elasticity in the Faraday instability was demonstrated by Ballesta & Manneville [START_REF] Ballesta | Signature of elasticity in the faraday instability[END_REF] using a wormlike micelle solution. They observed that the critical accelerations and wavenumbers present oscillations as a function of driving frequency and fluid height. For shear-thickening fluids, an interesting phenomenon was discovered by Merkt et al. [START_REF] Merkt | Persistent holes in a fluid[END_REF]. Using a cornstarch and glass micro-sphere suspension, the authors found that holes which were created by applying a finite amplitude perturbation to the fluid layer were persistent in a certain range of frequency and acceleration. The stabilizing mechanism of persistent holes is currently unknown. Merkt et al. [START_REF] Merkt | Persistent holes in a fluid[END_REF] argued that the increase of the viscosity with the applied shear stress is the key factor. Deegan [START_REF] Robert D Deegan | Stress hysteresis as the cause of persistent holes in particulate suspensions[END_REF] suggested a different mechanism, independent of whether the fluid is shear-thinning or shear-thickening, based on stress hysteresis. The class of complex fluids that support persistent holes was widened to emulsions by Falcon et al. [START_REF] Falcón | Localized structures in vibrated emulsions[END_REF]. Thus, it seems that shear-thickening is unnecessary for these structures to form, contrary to initial claims [START_REF] Merkt | Persistent holes in a fluid[END_REF].

Here we consider the case of viscoplastic fluids, i.e. fluids that exhibit yield stress behaviour. A comprehensive review of the physics of yield stress materials and how it is related to the microstructure was recently presented by Bonn et al. [START_REF] Daniel Bonn | Yield stress materials in soft condensed matter[END_REF] for a broad range of materials (emulsion, gels, colloidal suspensions, ...). Using a polymeric gel (ultrasound gel Dane-gel R1) with a significant yield stress (146 ± 6 Pa), Shiba et al. [START_REF] Shiba | Elastic convection in vibrated viscoplastic fluids[END_REF] observed a new pattern which consists in a pair of counter rotating vortices. Qualitatively, similar results were also obtained using shaving gel or toothpaste. To characterize the onset of convection, Shiba et al. [START_REF] Shiba | Elastic convection in vibrated viscoplastic fluids[END_REF] introduced a non-dimensional parameter Σ, representing the ratio of the stress exerted by the plate due to inertia of the material, divided by the yield stress. The onset of convection occurred for Σ = O(1). The convective rolls appear if the blob of gel is either placed directly on a horizontal plate without sidebands, or within a vertical oscillating cylinder. The list of complex fluids for which persistent holes can be observed, when they are submitted to vertical mechanical vibration, increased further after the experiments of Wolf et al. [START_REF] Wolf | Vibrationinduced geometric patterns of persistent holes in carbopol gels[END_REF]. The authors used Carbopol gel (a high molecular weight, hydrophilic, and crosslinked polyacrylic acid polymer) as a test fluid, reporting yield stresses in the range 34 -184 Pa (depending on the measurement and fitting method).

There are two main contributions in [START_REF] Wolf | Vibrationinduced geometric patterns of persistent holes in carbopol gels[END_REF]. Experimentally, the authors observed the formation of persistent holes in a regular geometric pattern in the surface of initially circular discs of Carbopol solutions, subjected to vertical oscillations of varying frequency and amplitude. Here the hole patterns are formed within an intermediate linearly increasing band of acceleration versus frequency. Above this band, they report behaviour analogous to [START_REF] Shiba | Elastic convection in vibrated viscoplastic fluids[END_REF], although not explored here, and below the band no flow occurred. Within the band, increasing oscillation frequency produced an increasing number of modes in the patterns which could be stably observed and which grew eventually into persistent holes within the sample. In sense therefore, the study of [START_REF] Wolf | Vibrationinduced geometric patterns of persistent holes in carbopol gels[END_REF] can be thought of as a more in depth study of the transition of [START_REF] Shiba | Elastic convection in vibrated viscoplastic fluids[END_REF], which appears not to be a sharp transition to the roll-structure, but instead has a rich intermediate patterning structure.

It is hard to interpret the effects of the vertical vibration directly on the rheology, in this type of experiment. Yield stress behaviours arise in many different materials [START_REF] Daniel Bonn | Yield stress materials in soft condensed matter[END_REF] and depend on the structure of the material. The application of a vertical vibration to certain types of yield stress materials may modify significantly the internal structure leading to a strong reduction of the yield stress. This has been observed for concrete by Tattersall & Baker [START_REF] Tattersall | The effect of vibration on the rheological properties of fresh concrete[END_REF] and for highly concentrated suspension of milk chocolate by Ouriev & Uriev [START_REF] Boris | Influence of vibration on structurerheological properties of a highly concentrated suspension[END_REF]. Carbopol is not however a suspension and this type of multiphase mechanism is probably not present. On the other hand, rheological properties of Carbopol solutions are strongly dependent on the type of Carbopol used, its concentration and the preparation method. Elastic effects are certainly present at low shear, although there is no suggestion that the onset in [START_REF] Wolf | Vibrationinduced geometric patterns of persistent holes in carbopol gels[END_REF] is due to elasticity in the same sense as [START_REF] Ballesta | Signature of elasticity in the faraday instability[END_REF]. Thixotropic hysteresis has also been observed for some Carbopol solutions. If this provides a rheology reduction mechanism for these experiments is less clear: many thixotropic models require a strain rate or viscous dissipation to initiate destruction, i.e. flow is needed for destruction. If however there is a reduction in yield stress, it is still noticeable that there appears to be a yield-stress dependent onset threshold in the experiments of [START_REF] Shiba | Elastic convection in vibrated viscoplastic fluids[END_REF][START_REF] Wolf | Vibrationinduced geometric patterns of persistent holes in carbopol gels[END_REF], below which there is no motion. Thus, onset of motion for a fluid with a yield stress is of interest.

The second part of [START_REF] Wolf | Vibrationinduced geometric patterns of persistent holes in carbopol gels[END_REF] presents a form of lubrication flow analysis. The final form of evolution equation for the surface is a parabolic equation in which an acceleration dependent viscosity is made separable from the horizontal Laplacian operator. This separable structure evidently leads to a modal solutions and the authors are able to produce qualitatively similar patterns to their experiments. Two aspects of [START_REF] Wolf | Vibrationinduced geometric patterns of persistent holes in carbopol gels[END_REF] are however unsatisfactory in our opinion. First, having studied a material that is clearly non-Newtonian the authors commence by assuming the fluid is Newtonian, but then later introduce an unspecified dependency of the viscosity on the acceleration. Both points are obscure. Second, the linearized equations appear to assume that the fluids are always flowing, which means that the issue of onset in terms of how the yield stress is overcome is not addressed, i.e. this is pattern formation in a yielded fluid. Indeed the role of the yield stress is unclear in the model although the experimental study reports significant regimes of no flow. While the pattern formation aspects are of interest, this leaves the question of flow/no-flow untreated.

In this paper we develop a thin-film/lubrication approach and model only flow onset in a Faraday setup. The thin-film approach is classical and results in a model (different to that in [START_REF] Wolf | Vibrationinduced geometric patterns of persistent holes in carbopol gels[END_REF]) in which the acceleration acts via the surface gradients to induce horizontal motion in the flow, i.e. film spreading. We discuss the scaling under which this mode takes place, the different qualitative behaviours that result and the potential onset of instability. The aim of this short paper is simply to show what results from this style of analysis and the assumptions of an ideal visco-plastic fluid model. The behaviours predicted are somewhat different to the experiments of [START_REF] Shiba | Elastic convection in vibrated viscoplastic fluids[END_REF][START_REF] Wolf | Vibrationinduced geometric patterns of persistent holes in carbopol gels[END_REF], although of course the model situation is also different. We end the paper by discussing some of the challenges to improving this.

Governing equations

We consider a layer of an incompressible yield stress fluid on a horizontal plate which is subjected to a vertical sinusoidal oscillation of amplitude â and a frequency f . The mean depth of the fluid layer is denoted ĥ0 . In the frame of reference which moves with the oscillating plate, the gravitational acceleration Ĝ is temporarily modulated,

Ĝ = - ( ĝ -â cos ωt ) e z .
(

Here, e z is the unit-vector in the vertical z-direction, and ω = 2π f . The fluid satisfies the continuity and momentum equations:

∇ • û = 0, (2)
ρ D D t û = -∇p + ∇ • τ + ρ Ĝ, (3)
where û denotes the velocity, p is the pressure and τ is the deviatoric stress tensor, Throughout the paper we shall adopt the convention of denoting dimensional quantities and variables with the "hat" symbol, i.e. •.

We assume that the rheological behavior of the yield stress fluid is described by the Herschel-Bulkley model:

       τ = ( κγ (û) n-1 + τy γ(û) ) γ(û), if τ > τy γ(û) = 0 if τ ≤ τy , ( 4
)
where τy , κ and n are the yield stress, consistency and power-law index of the fluid, respectively. The tensor γ(û) is the rate of strain tensor associated with the velocity field û, defined component wise as

γij (û) := ∂ ûi ∂ xj + ∂ ûj ∂ xi , (û, v, ŵ) = (û 1 , û2 , û3 ), (x, ŷ, ẑ) = (x 1 , x2 , x3 ). ( 5
)
γ and τ are norms of γ and τ , defined as

γ = 1 2 ∑ ij γ2 ij and τ = 1 2 ∑ ij τ 2 ij . ( 6
)
For boundary conditions, on the plate surface the no-slip and no-penetration boundary conditions take the form:

û = 0 at ẑ = -ĥ0 . ( 7
)
The free surface of the fluid layer is at ẑ = ξ(x, ŷ, t), where ξ has zero mean, i.e. ĥ0 is the mean depth of the fluid layer. The motion of the free surface is governed by the kinematic condition:

∂ ξ ∂ t + ûH • ∇H ξ = ŵ, (8)
where ûH and ∇H denote the horizontal components of the velocity and divergence operator, respectively. At the interface between air and fluid, the diference between stresses is balanced by surface tension:

[ σa -σ] • n = γs [ ∇ • n]n, at ẑ = ξ, (9)
where σa = -p a δ, pa is the atmospheric pressure, σ = -pδ + τ , and

n = ( - ∂ ξ ∂ x , - ∂ ξ ∂ ŷ , 1 )   1 + ( ∂ ξ ∂ x ) 2 + ( ∂ ξ ∂ ŷ ) 2   1/2 .
Denoting by t any tengential vector perpendicular to n, the dot product of (9) with t leads to

t • (τ • n) = 0. ( 10
)
In the following, we assume that the amplitude of the free-surface deformation is much smaller than the characteristic length of the horizontal layer:

∇ H ξ << 1. (11)
Combining [START_REF] Raynal | Faraday instability with a polymer solution[END_REF] with [START_REF] Hw Müller | Faraday instability in a linear viscoelastic fluid[END_REF] shows that the tangential stress components vanish, τxz = 0 and τyz = 0 at ẑ = ξ. ( 12)

Stationary layers

Before proceeding, let us examine the stationary state that we expect to exist for sufficiently small oscillations. We assume that û = 0 and that ξ is independent of t. The momentum equations reduce to:

0 = -∇p + ∇ • τ + ρ Ĝ, (13)
and for the moment lets assume that τ = 0. Thus, the pressure is given by:

p = pa + γs [ ∇ • n], at ẑ = ξ(x, ŷ), (14) p
(x, ŷ, ẑ) = pa + γs [ ∇ • n] + ρ[ĝ -â cos ωt ][ ξ(x, ŷ) -ẑ] (15)
We observe that horizontal pressure gradients now exist within the fluid layer:

∇H p(x, ŷ, ẑ) = ∇H ( γs [ ∇ • n] + ρ[ĝ -â cos ωt ] ξ) , ( 16
)
induced by horizontal gradients in elevation and surface curvature.

Evidently, (13) cannot be satisfied in the absence of deviatoric stresses within the layer, i.e. τ ̸ = 0. This does not invalidate the potential for a stationary layer, but it is necessary that the deviatoric stresses lie below the yield stress of the fluid layer, i.e. and consequently that there is a non-zero yield stress.

In general the stresses are indeterminate in this situation. The above analysis serves to indicate the magnitude of the deviatoric stresses induced by horizontal gradients. As in all Faraday type problems the deviatoric stresses come from gradients in ξ, both in the surface tension and spreading terms within [START_REF] Daniel Bonn | Yield stress materials in soft condensed matter[END_REF]. With ∇H ξ = 0, we should expect no motion at all in any ideal viscoplastic fluid, i.e. regardless of | Ĝ|. Furthermore, for a given size of horizontal surface gradient we can make a preliminary estimate of the required yield stress to prevent motion.

Dimensionless setting

Let us suppose that the initial surface profile ξ has maximal amplitude ξ0 and varies over a horizontal length-scale L0 . We shall make the assumption that δ = ĥ0 / L0 ≪ 1 and seek to develop a thin-film style of model. We assume that lengths and velocities scale as:

(û, v, ŵ) = Û0 (u, v, δw), (x, ŷ, ẑ) = L0 (x, y, δz), t = L0 Û0 t,
and write ξ = ĥ0 ξ(x, y, t) and ξ0 = ĥ0 ξ 0 .

The static layer analysis suggests that the pressure consists of 2 components: a time varying (static) part and a component due to the surface profile. Accordingly define the dimensionless pressure p via:

p = pa -ρ[ĝ -â cos ωt ]ẑ + ρ[ĝ + â] ξ0 p. ( 17
)
Here we have implicitly taken the buoyant component of pressure rather than the surface tension as the dominant driving force. We observe that the maximal horizontal pressure gradients should scale approximately like:

| ∇H p| ∼ O ( ξ0 ρ(ĝ + â) L0 ) [ 1 + O ( γs ρ(ĝ + â) L2 0 )]
.

Assuming a balance of the horizontal pressure gradients primarily through shear stress gradients across the depth of the fluid layer, this suggests that shear stresses imposed on the fluid layer should scale like:

|τ | ∼ O ( ĥ0 ξ0 ρ(ĝ + â) L0 ) [ 1 + O ( γs ρ(ĝ + â) L2 0 )]
. These imposed stresses are opposed by the yield stress of the fluid and by viscous stresses. We define the velocity scale Û0 by: Û0 = ĥ0

( ĥ0 ξ0 ρ(ĝ + â) L0 κ

) 1/n , ( 18
)
and use this to define a typical viscous scale τ0 :

τ0 ≡ κ ( Û0 ĥ0
) n = ĥ0 ξ0 ρ(ĝ + â) L0 . ( 19
)
The Bingham number B is defined as:

B = L0 τy ĥ0 ξ0 ρ(ĝ + â) = τy τ0 , ( 20
)
which is observed to be the ratio of yield stress to buoyancy stress, or alternatively yield to viscous stress. Although strictly speaking the stress scaling is derived in order to balance the principal shear stresses, many studies have shown that long-thin geometries of yield stress fluids are characterized by pseudo-plug regions in which the extensional components of τ have similar magnitude to the shear components. Thus, we assume that

τzx = τxz ∼ τ0 , τzy = τyz ∼ τ0 , τyx = τxy ∼ δτ 0 , τxx ∼ τ0 , τyy ∼ τ0 , τzz ∼ τ0 ,
noting that in fully yielded fluid regions the extensional stresses would drop to O(δ).

The scaled momentum equations are

δRe D Dt (u, v) = -∇ H p + ∂ ∂z [τ xz , τ yz ] + O(δ), ( 21
)
δ 3 Re D Dt w = - ∂p ∂z + O(δ), ( 22
)
where the Reynolds number Re is defined by:

Re = L2 0 ξ0 (ĝ + â) ( ĥ0 ξ0 ρ(ĝ + â) L0 κ ) 2/n , f = L0 ω0 2π Û0 .
The neglected terms in (21) & (22) correspond to the next largest deviatoric stress terms. Thus, even with the assumed scaling of the extensional stresses they still remain absent in the leading order problem, as δ → 0 at fixed Re:

∇ H p = ∂ ∂z [τ xz , τ yz ], ( 23
) ∂p ∂z = 0. ( 24
)
From (24) we see that the left-hand side of ( 23) is independent of z and hence integrate (24) to give:

[τ xz , τ yz ] = [z -ξ(x, y)]∇ H p (25)
In regions where the fluid is yielded the velocity scaling implies that the largest strain rates are

γxz ∼ ∂u ∂z + O(δ 2 ), γyz ∼ ∂v ∂z + O(δ 2 ).
Therefore, the leading order constitutive laws are:

(τ xz , τ yz ) = [ γn-1 + B γ ] ( γxz , γyz ), ⇔ |(τ xz , τ yz )| > B, ( 26
) ( γxz , γyz ) = 0, ⇔ |(τ xz , τ yz )| ≤ B. ( 27
)
Here note that:

γ =   ( ∂u ∂z ) 2 + ( ∂v ∂z ) 2   1/2 .
We observe that [τ xz , τ yz ] is a vector that locally points in the direction of ∇ H p, which is independent of z. By orienting a local coordinate system in the direction of -∇ H p, say s, we see that the shear stress normal to s is zero and hence also the velocity component. If the velocity component in the s-direction is denoted υ, then we have the following 1D problem:

τ sz = [ξ(x, y) -z]|∇ H p|, ( 28
)
τ sz =   ∂υ ∂z n-1 + B ∂υ ∂z   ∂υ ∂z , ⇔ |τ sz | > B, ( 29
)
∂υ ∂z = 0, ⇔ |τ sz | ≤ B. ( 30
)
We solve for υ using the no-slip condition at z = 0, and integrate across the fluid layer to find the areal flux:

∫ ξ -1 υ dz = |∇ H p| 1/n (1 -θ Y ) 1+1/n + (1 + ξ) 2+1/n 2 + 1/n ( 1 + θ Y 1 1 + 1/n ) , ( 31
)
θ Y = B (1 + ξ)|∇ H p| . ( 32
)
Note that (1 -θ Y ) + = max{0, 1 -θ Y }, i.e. the positive part. Physically, θ Y gives the ratio of yield stress to basal shear stress (at the bottom of the fluid layer): if θ Y ≥ 1 then υ = 0.

Returning now to the two-dimensional problem with [τ xz , τ yz ] in the direction of ∇ H p, we find that:

∫ ξ -1 (u, v) dz = -∇ H p |∇ H p| 1/n-1 (1 -θ Y ) 1+1/n + (1 + ξ) 2+1/n 2 + 1/n ( 1 + θ Y 1 + 1/n ) . ( 33
)
Combining the kinematic equation with the mass conservation equation gives:

∂ξ ∂t = w -(u, v) • ∇ H ξ = -∇ H • ∫ ξ -1 (u, v) dz + ∫ ξ -1 ∇ • u dz, ⇒ 0 = ∂ξ ∂t + ∇ H • ∫ ξ -1 (u, v) dz. (34)
Finally, we integrate the z-momentum equation to find the pressure from the normal stress condition. In dimensionless form this is:

p = α(t)ξ - ∇ 2 H ξ Ca , ( 35
)
where

α(t) = [ĝ -â cos ωt] [ĝ + â] , ω = ω L0 Û0 , ( 36
)
Ca = ρ[ĝ + â] L2 0 γs = 1 ξ 0 δ 2 τ0 γs L0 (37)
with Ca denoting the capillary number. The plate oscillation is captured in the term α(t). We observe that

∇ H p = α(t)∇ H ξ - ∇ 3 H ξ Ca , ( 38
)
which depends on ∇ H ξ , ∇ 3 H ξ, α(t) and Ca. Therefore, we may write:

∫ ξ -1 (u, v) dz ≡ q(ξ, ∇ H ξ, ∇ 3 H ξ, B, n, α(t), Ca), (39)
defined by (33). Evolution of the surface is governed by solving:

∂ξ ∂t + ∇ H • q(ξ, ∇ H ξ, ∇ 3 H ξ, B, n, α(t), Ca) = 0. ( 40
)
In comparison to the lubrication model in [START_REF] Wolf | Vibrationinduced geometric patterns of persistent holes in carbopol gels[END_REF], we note that even neglecting the surface tension term (Ca = ∞) these evolution equations are quite different. Equation (40) can not be linearised about a fixed film thickness and result in the separable non-conservative form given in [START_REF] Wolf | Vibrationinduced geometric patterns of persistent holes in carbopol gels[END_REF]. In particular, whether the fluid is flowing or not is governed by the term (1 -θ Y ) + in (33) in which the gradient of the layer thickness appears and not the layer thickness. Indeed this feature is missing from [START_REF] Wolf | Vibrationinduced geometric patterns of persistent holes in carbopol gels[END_REF] as the authors assume that their fluid is described rheologically as through a time-dependent Newtonian viscosity.

1D and no surface tension

For simplicity, for the remainder of the paper we consider the 1D model with no surface tension Ca = ∞:

0 = ∂ξ ∂t + ∂ ∂x q(ξ, ∇ H ξ, B, n, α(t)), ( 41
) q = -α(t)ξ x |α(t)ξ x | 1/n-1 (1 -θ Y ) 1+1/n + (1 + ξ) 2+1/n 2 + 1/n ( 1 + θ Y 1 + 1/n ) , (42) θ Y = B |α(t)ξ x |(1 + ξ) . ( 43
)
A similar analysis could be carried out for the 1D axisymmetric version of (40) with broadly similar conclusions.

Discounting the α(t) in front, the divergence of -q is a quasilinear elliptic operator. Thus, when α(t) > 0, (41) is a quasilinear parabolic equation, describing the (modified) gravitational spreading of the surface. This diffusive spreading is enhanced by both larger α(t) > 0 and stronger gradients in the liquid layer. We can see however that diffusive spreading approaches zero if θ Y → 1 -, which happens as either the gradients or the acceleration α(t) weaken sufficiently, i.e. at a constant α(t) we expect that the interface evens out such that θ Y → 1 -and the spreading stops. On the other hand, for any θ Y < 1, if α(t) < 0, the spreading is no longer diffusive but anti-diffusive, i.e. small wavelengths are amplified and the surface motion is expected to become unstable. We now explore these regimes and where they occur.

Solutions and regimes

It helps to illustrate the motions by considering 3 example initial surface profiles, as follows:

(1) Linear profile :

ξ = 2ξ 0 ( 1 4 -|x| ) (2) Cosine profile ξ = 1 2 ξ 0 cos(2πx) (3) Parabolic profile ξ = 4ξ 0 [ x 2 -1 12 ]
Note that each profile has zero mean and the parameter ξ 0 has been chosen to represent the amplitude of the surface variation, i.e. ξ 0 = ξ max -ξ min . The 3 initial conditions are displayed in Fig. 1a.

Static layers

We first consider whether or not the initial surface flows at all. The dimensionless acceleration α(t) is given by:

α(t) = χ -cos(2πf t) χ + 1 ∈ [ χ -1 χ + 1 , 1 ] ,
where χ = ĝ/â. Thus, the minimal value of θ Y is found when α(t) = 1 and at the position where |ξ x |(1 + ξ) is maximal. We denote:

A(ξ 0 ) = max x∈[-1/2,1/2] {|ξ x |(1 + ξ)},
which we find to be:

A(ξ 0 ) = 2ξ 0 (1 + ξ 0 /2)
, for the linear profile, (44)

A(ξ 0 ) = πξ 0 ( 1 + ξ 0 2 cos(2πx * ) )
sin(2πx * ), for the cosine profile, ( 45)

A(ξ 0 ) = 4ξ 0 ( 1 + 2 3 ξ 0 )
, for the parabolic profile. The functions A(ξ 0 ) are plotted versus ξ 0 in Figure 1b, from which we conclude that the parabolic profile is most likely to move.

We observe from (43) that the minimal θ Y is given by θ Y,min = B/A(ξ 0 ) and consequently the layer is static for

1 ≤ B A(ξ 0 ) = L0 τy ĥ2 0 ρ(ĝ + â) 1 ξ 0 A(ξ 0 ) (48)
This criterion can be understood physically as follows. The maximal inertial stress per unit area is [ ĥ0 ρ(ĝ + â)]. Due to the assumed mechanism of a dominant shear motion, this is scaled with a representative surface slope ξ 0 ĥ0 / L0 , to give a scale for the maximal generated shear stresses (multiplying again by A(ξ 0 )). This is compared now with the yield stress, which it needs to exceed in order to flow.

Spreading regimes

Lets now assume that B/A(ξ 0 ) < 1 so that there is some part of the oscillation during which the minimal θ Y satisfies θ Y,min < 1 and there is at least some part of the oscillation for which the layer yields. Given the range of α(t) we see that as θ Y → 1 -, we will have α(t) > 0, i.e. due to gravitational acceleration, α(t) > 0 during more of each oscillation than it is negative. We see that if χ ≥ 1 then α(t) ≥ 0 at all times and (41) remains parabolic. We call this regime, strong spreading, given that the spreading motion is always diffusive.

In contrast, if B/A(ξ 0 ) < 1 and χ < 1 then for part of the oscillation we have α(t) < 0. We expect negative diffusion during this period if also θ Y < 1, and label such flows unstable. To avoid unstable regime we need to ensure that θ Y ≥ 1 while α(t) < 0. The marginal condition is found to be when:

B A(ξ 0 ) ≥ 1 -χ χ + 1 . ( 49
)
Surfaces and parameters satisfying (49), for which also χ < 1 and B < A(ξ 0 ), we classify as weak spreading. This has the physical meaning that although the net acceleration is upwards for a portion of the oscillation, the stresses generated during that part of the oscillation are insufficient to yield the fluid.

Example numerical solutions

To illustrate the spreading behaviours, we solve (41) numerically. For this we use a simple explicit finite difference method:

ξ n+1 j = ξ n j + ∆t ∆x ( q n j-1/2 -q n j+1/2 ) , ( 50
)
where ξ n j is the height of the free surface at time t n = n×∆t and axial position x j = j × ∆ x. The timestep ∆t ∝ ∆x 2 is selected to maintain stability, by estimating the maximum diffusivity in (41).

Our examples below all take the parabolic initial profile (2), which we saw was the most mobile, and fix ξ 0 = 0.5. First we show in Fig. 2 the results for frequency f = 1, for both a strong (χ = ĝ/â = 2) and weak (χ = 0.8) spreading regime. We mark the initial surface and that at t = 1/f , as well as selected intermediate profiles. The weak spreading profile has a slower evolution at the start and finish of the cycle (due to α), but similar evolution in the minimum of the cycle as α(t) ≈ 1 in both cases. Although the initial states are identical and the states at t = 1/f are different (on close inspection), although they may be evolving to the same final state. The nonlinear nature of (41) is also evident, i.e. not all the surface spreads initially and the diffusivity is very local. For example, near the lowest parts of the surface the surface remains stationary until the (near linear) part of the profile propagates in from the outside, yielding the flow.

Figure 3 shows the same examples, except now for a higher frequency (f = 10). Again we see the slower evolution of the weak spreading case, particularly at start and finish of the oscillation, and the stationary surface at the lowest part. The main effect of increasing the frequency is a shortening of the time for evolution of the surface.

Summary of flow regimes

To summarise the flow regimes, we have identified the following:

(1) Static layer: if θ Y ≥ 1, ∀t .

(2) Strong spreading: if α(t) ≥ 0 ∀t (found when χ ≥ 1), and θ Y < 1 for part of the oscillation. (3) Weak spreading: if α(t) < 0 for part of the oscillation, but only when θ Y ≥ 1. (4) Unstable: otherwise. These regimes are illustrated in Fig. 4, plotted in the (χ, B/A(ξ 0 ))-plane.

Working from the most stable regime to least stable, the regime boundaries are a little conservative in that the temporal aspect is not included explicitly. To clarify, α(t) = [χ -cos(2πf t)]/[χ + 1] is initially minimal so might not move, whereas with a shift of time by half a period the spreading regimes will be driven by the maximal acceleration initially. 2), which are the surface profiles at t = 0 and t = 1/f , respectively.

Final states

A different temporal aspect is to note that all our stable spreading regimes appear to evolve towards a final regime that is static. Depending on the initial condition, it could be that some parts of the surface simply do not evolve at all, e.g. as the lower parts of the surface in Fig. 3 (although these may evolve at longer t than that shown). Discounting this possibility and assuming that all parts of the surface move, we might return to (41) and examine when the spreading stops. It would appear to be as θ Y → 1 -. As the spreading occurs whenever θ Y < 1, the final interval of spreading occurs when α(t) = 1, from which we see that in this limit. Integrating this expression we find the final shape of the surface, say ξ ∞ (x):

|ξ x |(1 + ξ) = B,
ξ ∞ (x) = √ 2B|x| + (ξ ∞ (0) + 1) 2 -1. (51)
The constant ξ ∞ (0) > -1 is to be found from the condition that the mean interface position is zero. As B → 0 we find that ξ ∞ (x) = 0. For small B we find the approximation:

ξ ∞ (x) ∼ B(|x| -1/4) + O(B 2 ). ( 52
)
We note the linear profile of the last computed shape in Fig. 2. The above analysis suggests that these may evolve to the same (approximately linear) final profile. Regardless of the small B approximation, it is notable that ξ ∞ (0) and hence ξ ∞ (x) should depend only on B.

The above analysis is reminiscent of slump flow approximations other predictions of final shapes: they are all conservative in the sense that they rely on surface motion continuing until the final shape is filled out, i.e. requiring that the fluid keeps moving until the final shape is achieved. At the same time in such problems, it is not hard to construct initial shapes that have segments where the slope is too shallow and that will remain stationary.

Unstable flows

Lastly we have some comments to make regarding the unstable flows. Principally, we wish to emphasize that the transition to unstable flows in Fig. 4 is not independent of the temporal features of the oscillation and represents only necessary conditions for instability. As we cross marginally into the unstable regime, the short period for which α(t) < 0 is balanced by a longer period for which α(t) > 0. It is possible that the diffusive spreading occurring during the period of positive α is sufficient to increase θ Y ≥ 1. We have explored this feature numerically using different frequencies f and different phase shifts of the oscillation, all close under the curve B/A(ξ 0 ) = (1 -χ)/(ξ + 1). We are able to find the above phenomenon marginally below this line, but the surface instability when it occurs is extreme and our simple explicit scheme is itself susceptible to numerical instabilities in these regimes. This makes numerical simulation the wrong tool for investigating further. An interesting avenue here could be to reintroduce the surface tension terms to stabilize the short wavelengths.

Discussion

In this paper we have outlined a lubrication analysis of the typical Faraday instability scenario for a yield stress fluid layer. Our focus has been to work form the perspective of the stable static layer, understanding how yielding works and how stability is potentially restored during the spreading regimes.

It is notable that the weak and strong spreading regimes that we have studied numerically evolve in one direction only: towards more stable and eventually static configurations. We have also derived the necessary conditions for instability at the limit of these spreading regimes.

Considering the experimental studies of [START_REF] Shiba | Elastic convection in vibrated viscoplastic fluids[END_REF][START_REF] Wolf | Vibrationinduced geometric patterns of persistent holes in carbopol gels[END_REF] we feel that our analysis and model may partly relate to the static layer states observed. We note however that in neither study is there any detail on control of the initial layer uniformity (critical here in our analysis), the aspect ratio of height to diameter is not typically asymptotically small and the discs/blobs placed on the oscillating plates have edges, where the surface gradients are necessarily large. However, new experiments could be performed in which the initial surface shape was explicitly molded.

Another difference here is with the lubrication model in [START_REF] Wolf | Vibrationinduced geometric patterns of persistent holes in carbopol gels[END_REF], which does produce qualitatively reasonable pattern shapes. The model presented in [START_REF] Wolf | Vibrationinduced geometric patterns of persistent holes in carbopol gels[END_REF] assumes a Newtonian viscosity for the gel, with a time dependency related to the acceleration. We feel this simplification misses some key features of the rheology, but so of course does an ideal visco-plastic fluid (as here). The type of separable model derived in [START_REF] Wolf | Vibrationinduced geometric patterns of persistent holes in carbopol gels[END_REF] might conceivably result from a constitutive model of the sub-yield (low shear) behaviour as either a very viscous fluid or perhaps as a linear viscoelastic gel, although this is also not obvious to us. The model does not represent any generalised Newtonian fluid. In recent years there have been many constitutive models advanced to better explain the behaviour of hydrogels such as those in [START_REF] Shiba | Elastic convection in vibrated viscoplastic fluids[END_REF][START_REF] Wolf | Vibrationinduced geometric patterns of persistent holes in carbopol gels[END_REF], within these regimes. Evidently, the Faraday instability is a setup in which sub-yield behaviour could be critical in determining onset.

Our analysis simply sets the scene for what happens in the ideal case of a Hershel-Bulkley fluid. An interesting challenge is to better model the sub-yield rheological behaviour and/or to consider a different scaling for the dominant strain rates in a thin film model, i.e. compression/extension. We are considering this type of model as the next step. However, we feel that the analysis presented should be valid for suitably thin layers.

Fig. 1 .

 1 Fig. 1. (a) The 3 initial surface profiles: (1) Linear; (2) parabolic and (3) sinusoidal. (b) The function A(ξ 0 ) vs ξ 0 , that defines the minimal θ Y .

Fig. 2 .

 2 Fig.2. Time evolution of the free surface in strong and weak spreading regimes at f = 1: a) strong spreading at χ = 2; b) weak spreading at χ = 2. For both we have B = 0.2A(ξ 0 ) and have marked curves (1) and (2), which are the surface profiles at t = 0 and t = 1/f , respectively.

Fig. 3 .

 3 Fig.3. Time evolution of the free surface in strong and weak spreading regimes at f = 10: a) strong spreading at χ = 2; b) weak spreading at χ = 2. For both we have B = 0.2A(ξ 0 ) and have marked curves (1) and (2), which are the surface profiles at t = 0 and t = 1/f , respectively.

Fig. The 4

 4 Fig. The 4 different regimes identified.

Preprint submitted to Elsevier Preprint


I Rème

Dandouras

Suping Duan
email: spduan@nssc.ac.cn

Lei Dai

Chi Wang

Chunlin Cai

Zhaohai He

Yongcun Zhang

H Reme

I Dandouras

Conjunction Observations of Energetic Oxygen Ions O + Accumulated in the Sequential

whether they are published or not. The documents may come

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Introduction

The cusp is one of important source regions of single charge oxygen ions, O + , in the magnetosphere. The outflow of oxygen ions can be observed in the whole cusp region [e.g., [START_REF] Yau | Energetic auroral and polar ion outflow at DE 1 altitudes: Magnitude, composition, magnetic activity dependence, and long-term variations[END_REF]Yau and Andre,1997;Yau etal.,2007]. The energy of the outflow oxygen ions is around 300 eV at the middle altitude cusp [e.g., [START_REF] Yau | The polar wind: Recent observations[END_REF]. But oxygen ions with energy above 1 keV have been observed in the high altitude cusp regions [e.g., [START_REF] Slapak | O+ heating associated with strong wave activity in the high altitude cusp and mantle[END_REF]Kristler, et al., 2010]. There are many debates on the source of energetic oxygen ions in the high altitude cusp region [e.g., [START_REF] Chen | Energetic oxygen ions of ionospheric origin observed in the cusp[END_REF][START_REF] Fritz | Energetic ions, large diamagnetic cavities, and Chapman-Ferraro cusp[END_REF]Lindstedt et al., 2011;[START_REF] Trattner | Cluster observations of bow shockenergetic ion transport through the magnetosheath into the cusp[END_REF]. [START_REF] Lindstedt | Oxygen energization by localized perpendicular electric fields at the cusp boundary[END_REF] proposed that the oxygen ions can be locally accelerated by the perpendicular electric field in the high altitude cusp. [START_REF] Chen | Energetic oxygen ions of ionospheric origin observed in the cusp[END_REF] reported that energetic oxygen ions were from the ionosphere and energized at the high altitude cusp. On the other hand, [START_REF] Trattner | Cluster observations of bow shockenergetic ion transport through the magnetosheath into the cusp[END_REF] pointed out that the energetic oxygen ions were originated from the forth shock ahead magnetosheath.

The Flux Transfer Events (FTEs) originated from the low latitude dayside magnetopause can move along the magnetopause to the high latitude cusp region [e.g., [START_REF] Akhavan-Tafti | MMS examination of FTEs at the Earth's subsolar magnetopause[END_REF][START_REF] Cowley | The causes of convection in the Earth's magnetosphere: A review of developments during IMS[END_REF][START_REF] Lee | A theory of magnetic flux transfer at the Earth's magnetopause[END_REF][START_REF] Roux | What is the nature of magnetosheath FTEs?[END_REF]. Flux ropes or open flux tubes are usually adopted as the description of FTEs [e.g., [START_REF] Duan | Cluster observations of unusually high concentration of energetic O+ carried by flux ropes in the nightside high-latitude magnetosheath during a storm initial phase[END_REF][START_REF] Eastwood | Ion-scale secondary flux ropes generated by magnetopause reconnection as resolved by MMS[END_REF][START_REF] Lee | A theory of magnetic flux transfer at the Earth's magnetopause[END_REF][START_REF] Roux | What is the nature of magnetosheath FTEs?[END_REF][START_REF] Sun | Large-scale characteristics of flux transfer events on the dayside magnetopause[END_REF][START_REF] Xiao | Inferring of flux rope orientation with the minimum variance analysis technique[END_REF][START_REF] Zhang | Evidence that crater flux transfer events are initial stages of typical flux transfer events[END_REF]. [START_REF] Lee | A theory of magnetic flux transfer at the Earth's magnetopause[END_REF] reported that multiple X-line reconnection gave rise to multiple flux ropes at the dayside magnetopause. [START_REF] Omidi | Flux transfer events in the cusp[END_REF] proposed that multiple FTEs with a various sizes are formed at LLBL and traveled along the magnetopause surface into high latitude polar region during southward IMF. The periodicity of sequential FTEs is in the range from 1.5 min to 18.5 min with average value of 8 min and common value of 3 min [e.g., [START_REF] Trattner | Evidence of multiple reconnection lines at the magnetopause from cusp observations[END_REF][START_REF] Rijnbeek | A survey of dayside flux transfer events, observed by the ISEE-1 and -2 magnetometers[END_REF]Lockwood and Wild, 1994]. [START_REF] Cowley | The causes of convection in the Earth's magnetosphere: A review of developments during IMS[END_REF] reported that the travel time of FTEs from the subsolar magnetic reconnection region to the vicinity of the cusp was around 5 min. The low latitude flux ropes propagate towards southward and northward high latitude region with different bipolar signatures [e.g., [START_REF] Berchem | Flux transfer events on the magnetopause: Spatial distribution and controlling factors[END_REF][START_REF] Le | Flux transfer events simultaneously observed by Polar and Cluster: Flux rope in the subsolar region and flux tube addition to the polar cusp[END_REF]. [START_REF] Le | Flux transfer events simultaneously observed by Polar and Cluster: Flux rope in the subsolar region and flux tube addition to the polar cusp[END_REF] pointed out that the southward moving flux tube produced a bipolar signature with inward/outward (-/+) polarity.

The scale size of flux ropes at the dayside magnetopause is in the range from a few hundred kilometers to about an Earth radius [e.g., [START_REF] Akhavan-Tafti | MMS examination of FTEs at the Earth's subsolar magnetopause[END_REF][START_REF] Lee | A theory of magnetic flux transfer at the Earth's magnetopause[END_REF][START_REF] Eastwood | Ion-scale secondary flux ropes generated by magnetopause reconnection as resolved by MMS[END_REF][START_REF] Sun | Large-scale characteristics of flux transfer events on the dayside magnetopause[END_REF]. [START_REF] Akhavan-Tafti | MMS examination of FTEs at the Earth's subsolar magnetopause[END_REF] reported that the mean diameter of FTEs was ~1700km at the Earth's subsolar magnetopause. [START_REF] Eastwood | Ion-scale secondary flux ropes generated by magnetopause reconnection as resolved by MMS[END_REF] reported that ion-scale flux ropes with scale size ~1100 km were observed by MMS at the dayside magnetopause. Ten keV energetic oxygen ions near the dayside magnetopause have gyroradius around ~1000 km.

These energetic oxygen ions may be captured into the flux ropes and transported form low latitude to high latitude region. [START_REF] Phan | Cluster observations of continuous reconnection at the magnetopause under steady interplanetary magnetic field conditions[END_REF] reported that energetic O+ with energy larger than 3keV in the reconnection jets are observed by Cluster at the duskside mid-latitude magnetopause under steady southward IMF condition.

In this paper, the conjunction observations of TC-1 and Cluster present a clear chain for sequential FTEs transport from the LLBL to the high altitude cusp. We focus on the sequential FTEs accumulated with energetic oxygen ions in the high altitude cusp observed by Cluster. They both observed sequential FTEs with interval time period of ~ 3 min. In the second section the observations of sequential FTEs near the dayside magnetopause by TC-1 and FTEs accumulated with energetic oxygen ions in the high altitude cusp by Cluster are presented, respectively.

Discussions and summary are shown in the last section.

Conjunction observations of FTEs

Sequential Flux ropes observed by TC-1 at the LLBL

Figure 1 shows plasma and magnetic field data from TC-1 located near the dayside magnetopause.

From top to bottom, the panels show that (a) ions number density, (b) ions temperature, (c) plasma beta value , ( is the ratio of plasma thermal pressure to the magnetic pressure, =2μ 0 nT/B 2 ), (d) to (f) the x, y, and z component of ions velocity, (g) to (i) the z, x, and y component (grey line) and the total magnitude (black line) of magnetic field. The position of TC-1 located near dayside low latitude magnetopause is presented at the bottom in Figure 1. The two vertical red lines mark the time intervals of the continuous negative B z component from 10:53:00 UT to 11:14:00 UT. The minimum value of B z component is about -40 nT. During this interval the ions number density is high (larger than 10 cm -3 ) as shown in Figure 1a. The maximum value of ion number density is about 25 cm -3 . Ion temperature is low (less than 1 keV) as shown in Figure 1b. Ions velocity in the X direction is negative and slow, Vx ~ -100 km/s. Those parameters indicate that TC-1 is located at the dayside magnetosheath-side of the low latitude boundary layer. Figure 1f presents that the B x component has bipolar signatures. This may be associated with the magnetic reconnection at the dayside magnetopause. So we display the magnetic field and ions velocity in the local boundary normal coordinate system (LMN) in Figure 2.

Figure 2 shows three components of the magnetic field and proton velocity in LMN coordinate system. From top to bottom, the panels display the B N , B M , B L component , and the total magnitude Bt of the magnetic field, the v N ,v M , v L component and the total magnitude vt of the ions velocity, respectively. The normal vector N = (0.93, 0.26, -0.25) (GSM) is determined as the minimal variation direction of the magnetic field in the interval from 10:56:50 UT to 10:59:56 UT.

The vectors M and L are defined as M = N×Z GSM and L = M×N. In Figure 2 three green shade regions demonstrate that the B N firstly decreases then increases, the B M sharply increases and the significant increase in the magnetic total value, Bt. These three regions mark three flux ropes observed by TC-1 near the dayside low-latitude magnetopause. They are signed as FR A1, FR B1 and FR C1 shown at the topside of Figure 6, respectively. The first flux rope marked by FR A1 has large negative B N then decrease suddenly at 10:52:20 UT. The second and third flux ropes marked by FR B1 and FR C1 have distinct B N bipolar signature from negative to positive and larger total value of magnetic field at10:56:20 UT and 10:58:50 UT respectively. The location of TC-1, ~ (10.51,-0.28,-1.57) R E , at the bottom of Figure 2 shows that the spacecraft located in the southward hemisphere side of subsolar point. The negative to positive bipolar signatures imply that flux ropes are from the subsolar point magnetic reconnection and propagate toward the southward hemisphere [START_REF] Le | Flux transfer events simultaneously observed by Polar and Cluster: Flux rope in the subsolar region and flux tube addition to the polar cusp[END_REF]. These three flux ropes are all accompanied with down-tail ions bulk flow, as shown in Figure 2e and2g.

Energetic oxygen ions within Sequential Flux ropes in the high-altitude cusp

Figure 3 presents the plasma and magnetic field observed by Cluster C4 at southern hemisphere high latitude region around (3.7,-4.4,-9.8) Re. From top to bottom, the panels show the number density of (a) proton, (b) helium ions He ++ , and (c) oxygen ions O + , (d) proton temperature, (e) plasma beta value, , (f) the x component of proton velocity, (g) to (i) the x and z component of the magnetic field, the magnetic field total magnitude (black line) and the By component (blue line), respectively. During the intervals of 10:57 UT to 11:09 UT, Figure 3b presents that the He ++ number density is larger than 0.5 cm -3 . At the same time period, Figure 3f displays the proton vx component value decrease to around ~100 km/s and in the tailward direction. Figure 3i shows the total magnetic field magnitude decreases and with significant fluctuations. Those parameters demonstrate that Cluster C4 is crossing the high altitude cusp. At 10:57:30 UT, the Bx and Bz components change from positive value to negative value which as marked by the first pink vertical dashed line. After this time Cluster C4 enters into the southern high altitude cusp region. 4a clearly demonstrates that energetic oxygen ions with energy larger than 10 keV are found in the high altitude cusp region. They are especially marked by the three dotted vertical lines in Figure 4.

The pitch angle distribution of high energy oxygen ions, >10 keV, presented in Figure 4d, shows that these O + ions are dominated in the pitch angle less than 90 degree. It means that those O + ions almost outflow from the low altitude cusp in the southern hemisphere. On the other hand, oxygen ions with energy less than 10 keV are distributed in a wide pitch angle range from 0 to 180 degree, as shows in Figure 4b and4c. According to the energetic O + ions with sharp increase in the energy flux and number density, we present the magnetic field and proton velocity in the LMN coordinate system in the cusp, as shown in Figure 5. The wide range of the pitch angle distribution of energetic oxygen ions in the high altitude cusp shown in Figure 4b and 4c means that these oxygen ions are from two different regions.

The 2D cut of oxygen ions velocity distributions can clear reveal that the energetic O + ion are characterized with the counter-streaming flows. Figure 6 shows the velocity distributions of oxygen ions with energy in the range of 40 eV to 40 keV at 10:59:24 UT, 11:02:28 UT and 11:05:53 UT, respectively. Within these three flux ropes oxygen ions have counter-streaming features, which are both presented in Figure 6b and6c. The right three panels of Figure 6 show that oxygen ions have large counter -direction velocity in the Z direction at 10:59:24 UT, 11:02:28 UT and 11:05:53 UT respectively. The total magnitude of O + ions velocity is around 600 km/s. These three times are in the time intervals of the magnetic flux ropes in the high altitude cusp region which are shown in Figure 5 as three green shade regions, respectively. They have one-to-one corresponding relationship. These counter-streaming oxygen ions in three flux ropes demonstrate that they are from two difference regions. They consist of the ionosphere outflow and magnetopause magnetic reconnection precipitation.

Summary and discussions

The above conjunction observations of TC-1 at the LLBL and Cluster at the high altitude cusp regions are adopted to investigate the sequential FTEs transporting from the subsolar dayside magnetopause to the high altitude cusp. The results demonstrate that the energetic oxygen ions have counter-streaming in the high altitude cusp regions. These oxygen ions have two different source origins. One is from the ionosphere outflow. The other is the oxygen ions precipitation originating from the magnetic reconnection region at the low latitude dayside magnetopause. As displayed in Figure 5, there are three flux ropes observed by Cluster C4 at high-altitude cusps about ~5-6 minutes delay from those by TC-1 at 10:58:20 UT, 11:01:20 UT and 11:04:50 UT, respectively, which are presented by three green shaded regions marked by FR A , FR B and FR C , respectively. Thus, these conjunction observations of sequential FTEs suggest that the high-latitude Flux ropes in the cusp are from the dayside low-latitude magnetopause reconnection.

On the other hand, the velocity distributions in these flux ropes as shown in Figure 6 present ions precipitation in the high-altitude cusps with the large v z value. Ions precipitation along the magnetic field has been used to induce the magnetic reconnection process. The ions precipitation also implies that these three cusp flux ropes are origin from the dayside low-latitude magnetopause reconnections. [START_REF] Phan | Cluster observations of continuous reconnection at the magnetopause under steady interplanetary magnetic field conditions[END_REF] reported that energetic oxygen ions with high density 0.2-0.3 cm -3 in the reconnection jets are observed by Cluster at the dayside magnetopause under the steady southward IMF. Our results that the sequential FTEs accumulated with high number density 0.25 cm -3 of energetic oxygen ions in the high altitude cusp maybe associated with the similar reconnection at the LLBL reported by [START_REF] Phan | Cluster observations of continuous reconnection at the magnetopause under steady interplanetary magnetic field conditions[END_REF].

The delay time ~ 5-6 min from the LLBL flux ropes (FR A1 , FR B1 and FR C1 ) observed by TC-1 to the high altitude cusp flux ropes (FR A , FR B and FR C ) observed by Cluster is consisted with the propagating time of FTEs from LLBL to cusp in previous reports [e.g., [START_REF] Cowley | The causes of convection in the Earth's magnetosphere: A review of developments during IMS[END_REF]. According toTC-1 observations, the background magnetic field magnitude 60 nT and the ions density 16 cm -3 as presented in Figure 1i and 1a, the Alfvén speed can be calculated is ~ 327 km/s. Based on the locations of TC-1 at the bottom of Figure 1, TC-1 observed FTEs at the LLBL with X~10.4 R E , we can calculate the time interval of FTEs transporting from the LLBL to the high altitude cusp being ~ 5.3 min. This calculating time interval of FTEs transporting from the LLBL to the high altitude cusp is consistent with that of conjunction observations between TC-1 at the LLBL and Cluster in the cusp time intervals, from 10:52:20 UT to10:58:40 UT, 10:56:20 UT to 11:01:20 UT and 10:58:50 UT to 11:04:50 UT.

Energetic oxygen ions with tens keV energy have large Larmor radius, ~ several thousand km, in the high altitude cusp. This spatial scale is closed to FTEs radius size in a few thousands km.

Those energetic oxygen ions can be trapped in these flux ropes and move along the magnetopause to the high altitude cusp with the same velocity of FTEs. [START_REF] Eastwood | Ion-scale secondary flux ropes generated by magnetopause reconnection as resolved by MMS[END_REF] reported that ion-scale flux ropes with ~1100 km generated by the magnetic reconnection are observed by MMS at dayside low latitude magnetopause. Using 2.5-D global hybrid simulations, [START_REF] Omidi | Flux transfer events in the cusp[END_REF] proposed that multiple FTEs with a various sizes are formed at LLBL and traveled along the magnetopause surface into high latitude polar region during southward IMF. The energetic oxygen ions in our event maybe origin from ring current during the recovery phase of the median storm on 9 March, 2004 (Dst ~-80 nT). They escape into the low latitude magnetopause by guiding center magnetic field. These energetic oxygen ions located in flux ropes as the reconnection jets. The energetic oxygen ions with high number density observed in the reconnection jet are also reported by [START_REF] Phan | Cluster observations of continuous reconnection at the magnetopause under steady interplanetary magnetic field conditions[END_REF]. Tens keV energetic oxygen ions with large gyroradius of ~ 1000km, which is close the radius of flux ropes, are trapped in FTEs propagating to high latitude cusp and then to high latitude magnetotail region. [START_REF] Duan | Cluster observations of unusually high concentration of energetic O+ carried by flux ropes in the nightside high-latitude magnetosheath during a storm initial phase[END_REF] reported that flux ropes accumulated with energetic oxygen ions observed by Cluster at high latitude nightside magnetotheath were from the high altitude cusp.

As addressed above, the conjunction observations of the sequential flux ropes at the low latitude boundary layer by TC-1 and at the high altitude cusp by Cluster are investigated in our paper. The main conclusions of our observations can be drawn as follows. The delay time of these sequential flux ropes from the low latitude magnetopause to the high altitude cusp is around 5-6 min. These sequential flux ropes are accumulated with energetic oxygen ions observed by Cluster in the high altitude cusp. The number density of energetic O + ions, detected by CIS/CODF instrument on Cluster, is very high in these sequential flux ropes, ~ 0.25 cm -3 . These O + ions with different energy have different pitch angle distribution in the high altitude cusp. The oxygen ions with energy lager than 10 keV are dominantly distributed in the quasi-parallel direction along the background magnetic field in the southern high altitude cusp. While O + ions with lower energy less than 10 keV have a wide range of pitch angle distribution from 0 degree to 180 degree. This reveals that oxygen ions with energy less than 10 keV in the high altitude cusp are from two different regions, the dayside low latitude magnetopause and low altitude cusp region. Our investigations present evidence that the sequential flux ropes at dayside low latitude magnetopause can carry oxygen ions into the high altitude cusp region.

Figures and Captions

Figure

  Figure 3c presents three sharp increases in the number density of O + ions in the high altitude cusp, as marked by three vertical red lines in Figure 3. The maximum value of O + ions number density is about 0.27 cm -3 in the cusp. Corresponding these oxygen ions number density increase, the magnetic field total value also increases and the B z component has bipolar signature as shown in the last two panels, Figure 3i and 3h, respectively.

Figure 4

 4 Figure 4 presents the oxygen ions energy flux spectrum and pitch angle distribution in the intervals of 10:30 UT to 11:30 UT. The first panel shows the oxygen ions energy flux from 40 eV to 40 keV. The next three panels show oxygen ions pitch angle distributions in three energy channel 40 eV to1 keV, 1 keV to 10 keV, 10 keV to 40 keV, respectively. The interval between the two pink vertical dashed lines is the time of Cluster crossing the high altitude cusp. Figure 4a

  Figure

Figure 5

 5 Figure 5 presents three flux ropes observed by Cluster C4 in the high altitude cusp region, which are marked by three green shade regions, FR A , FR B and FR C (shown at the topside of this figure) , respectively. From top to bottom, the panels show oxygen ions density, helium ions density, three components of the magnetic field and proton velocity in a LMN coordinate system, B N , B M and B L , v N ,v M and v L , respectively. The normal vector N = (-0.43, 0.02, 0.90) (GSM) is determined as the minimal variation direction of the magnetic field in the intervals from 11:00:45 UT to 11:01:45 UT. Figure 5a shows that these three flux ropes are accumulated within the high density of oxygen ions, n o+ ~0.25 cm -3 , in the high altitude cusp. The B N component has bipolar signals at around 10:58:20UT, 11:02:20UT and 11:04:50UT, as shown in Figure 5c. The total magnetic magnitude increases significantly at these three times, as presented in Figure5f. These flux ropes all have negative V L value as shown in the bottom panel. These features of the magnetic field show that three flux ropes are observed in the high altitude cusp regions with a period of ~ 3 min.

Figure

  Figure1The plasma and magnetic field parameters are from TC-1 HIA and FGM during the

  Figure 1 The plasma and magnetic field parameters are from TC-1 HIA and FGM during the period of 10:30 UT to 11:30 UT 10 March 2004. From top to bottom, panels are (a) ions number density, (b) ions temperature, (c) plasma beta value, , (d) to (f) the vx ,vy and vz component of ions, respectively,(e) plasma beta value, , (f) the x component of proton velocity, (g) the B x component of magnetic field, (h) the B z component, and (i) the B y (blue line)and total magnitude of magnetic field.

Figure 2

 2 Figure 2 Sequential Flux ropes observed byTC-1 at the dayside low latitude boundary layer under the southward Bz component. From top to bottom, the panels are (a) the magnetic field B N component, (b) the B M component, (c) the B L component, (d) the total magnitude of the magnetic field, B t , (e) to (h) proton velocity v N ,v M ,v L component and the total magnitude v t , respectively. Three green shade regions mark the intervals of sequential flux ropes observed by TC-1 at LLBL as FR A1 , FR B1 , FR C1 at the top side of Figure 2, respectively.

Figure 3

 3 Figure 3 The plasma and magnetic field parameters are from Cluster C4 CIS/CODIF and FGM during the period of 10:30 UT to 11:30 UT 10 March 2004. From top to bottom, panels are (a) proton number density, (b) He ++ number density, (c) O + number density, (d) proton temperature, (e) plasma beta value, , (f) the x component of proton velocity, (g) the B x component of magnetic field, (h) the B z component, and (i) the B y (blue line) and total magnitude of magnetic field. Three red dashed vertical lines mark the time of high number density of O+ ions in the high altitude cusp.

Figure 4

 4 Figure 4 The O + ion energy spectrogram flux and pitch angle distribution obtained from CODIF (energy range is from 40 eV to 40 keV) on 10 March 2004. (a) O+ energy spectrogram of omnidirectional flux, (b) to (d) O + pitch angle distribution with energy in the range of 40 eV to 1 keV, 1 keV to 10 keV, and 10 keV to 40 keV, respectively.

Figure 5

 5 Figure 5 Sequential Flux ropes accumulated by energetic oxygen ions observed by Cluster C4 in southward high altitude cusp region, which are marked by FR A , FR B , FR C at the top, respectively. From top to bottom, panels are (a) O + number density, (b) He ++ number density, (c) the magnetic field B N component, (d) the B M component, (e) the B L component, (f) the total magnitude B t , (g) to (i) proton velocity v N ,v M ,v L component, respectively. Three green shade regions mark the intervals of flux ropes and high number density of O + ions as FR A , FR B , FR C at the topside of Figure 5, respectively.

Figure 6

 6 Figure 6 Two-dimensional cuts of the three-dimensional O + ion velocity distributions obtained by CODIF/C4 on 10 March 2004. Panels (a) to (c) show the O + ion velocity distributions in three flux ropes FR A , FR B , and FR C at 10:59:24 UT, 11:02:28 UT, and 11:05:53 UT, respectively.

Acknowledgments

We acknowledge the use of data from the ESA Cluster Science Archive (http://www.cosmos. esa. int /web/csa).We thank the FGM, CIS, EFW instrument teams. This work is supported by the National Natural Science Foundation of China grants 41874196, 41731070, 41574161and 41574159; the Strategic Pioneer Program on Space Science, Chinese Academy of Sciences, grants XDA15052500, XDA15350201 and XDA15011401; the NSSC Research Fund for Key Development Directions and in part by the Specialized Research Fund for State Key Laboratories.


Laurent Moutet
email: laurent.moutet@ac-amiens.fr

The extended theoretical framework of Mathematical Working Space (extended MWS): Potentialities in physics

Keywords: Mathematics, physics, extended MWS, modelling

The aim is to show how the extended Mathematical Working Space (extended MWS) theoretical framework makes it possible to analyse the tasks implemented during a few stages of a modelling cycle in physics. The study begins with a special relativity teaching sequence using a diagrammatic approach in "Terminale S" in France (Grade 12). The analysis using the extended MWS theoretical framework allows to highlight the learning advantages of this diagrammatic approach during a complete didactic engineering. This work is proposed for TWG6 to CERME11.

Presentation of the extended MWS theoretical framework

The Mathematical Working space (MWS) was developed to better understand the didactic issues around mathematical work in a school environment by [START_REF] Kuzniak | Mathematical working spaces in schooling: An introduction[END_REF]. The MWS has two levels: one of a cognitive nature in relation to the student and another of an epistemological nature in relation to the mathematical content studied. The epistemological plane contains a set of representations (signs used), a set of artefacts (drawing instruments or software) and a theoretical reference set (definitions and properties). The cognitive plane contains a visualization process (representation of space in the case of geometry), a construction process (function of the tools used) and a discursive process (arguments and evidence). Mathematical work results from an articulation between the cognitive and epistemological planes through instrumental genesis (operationalization of artefacts), semiotic genesis (based on the register of semiotic representations) and discursive genesis (presentation of mathematical reasoning). The different phases of mathematical work associated with a task can be highlighted by representing three vertical planes on the ETM diagram. Semiotic-instrumental interactions lead to a process of discovery and exploration of a given academic problem. Those of an instrumental-discursive type favour mathematical reasoning in relation to experimental evidence. Finally, those of the semioticdiscursive type are characteristic of the communication of mathematical results as well as of more elaborate reasoning. The MWS diagram was transformed by [START_REF] Moutet | Analyse d'une séquence d'enseignement de la relativité restreinte: L'apport du modèle de l'ETM étendu[END_REF] by adding an epistemological plane corresponding to the rationality framework of physics (Figure 1). It was chosen to keep only one cognitive plane. We hypothesized that students' conceptions in mathematics and physics can be highlighted using a students' personal extended MWS. The functioning of the schemes [START_REF] Vergnaud | Pourquoi la théorie des champs conceptuels ? Infancia y aprendizaje[END_REF] specific to these conceptions could appear during the activation of the various geneses. The Concepts in mathematics would be highlighted by the activation of geneses, between the cognitive plane and the epistemological plane of mathematics, those in physics would be highlighted by the activation of geneses, between the same cognitive plane and the epistemological plane of physics. A single cognitive plane would therefore be sufficient to describe conceptions in mathematics, physics and to analyse student tasks from both a mathematics and a physics perspective. The term "proving" is retained, but in the case of the extended MWS, it will take a narrower meaning of justification or reasoning. The extended MWS framework makes it possible to analyse in detail the interactions between the various rationality frameworks, and the student's cognitive plane and then to qualify the nature of the work accomplished by the student.

The epistemological plane of mathematics studied here concerns Euclidean geometry. The tasks to be carried out are associated with the construction of lines, segments, symmetries parallel to a line, projections passing through a point and parallel to an axis. The epistemological plane of physics is associated with the reference frame of special relativity studied in class. These are the notion of the Galilean reference frame (two reference frames moving in a straight line and at a constant speed with respect to each other), of an event (a point located on a space-time diagram and associated with a position x and a time t), of Einstein's two postulates, of the notion of proper duration and improper duration as well as the relationship of dilation of durations. Lorentz transformations are not studied in Grade 12. From the "model situation" to the "real results" during a special relativity modelling A teaching sequence developed by [START_REF] Moutet | Analyse d'une séquence d'enseignement de la relativité restreinte: L'apport du modèle de l'ETM étendu[END_REF] is destined for students in "Terminale S" (Grade 12) on the topic of special relativity following the work of [START_REF] De Hosson | Exploring students' understanding of reference frames and time in Galilean and special relativity[END_REF]. It follows the learning of the course, and the correction of exercises in the manual, and participates in the conceptualisation of notions of special relativity, by allowing them to be reinvested in a different and unknown context [START_REF] Vergnaud | La théorie des champs conceptuels[END_REF].

Two research questions guided this work: 1) How does the extended MWS framework allow the analysis of the sets of rationality frameworks between mathematics and physics during a sequence dealing with special relativity with "Terminale S" students via a geometric approach? 2) To what extent does the analysis of the use of dynamic geometry software by the extended MWS framework show that it promotes conceptualisation in students?

The teaching sequence can be described in the following context: Armineh drives a car moving at a speed v near the speed of light relative to Daniel. The latter is on the roadside next to three flash lights S 1 , S 2 and S 3 associated with three events E 1 , E 2 , E 3 and initially known in Daniel's reference frame (Figure 2). An event E i corresponds to a coordinate point (x i , c.t i ) in the Minkowski spacetime diagram. S 1 and S 2 have the same position in Daniel's reference frame. We can use the modelling cycle proposed by [START_REF] Blum | Filling Up "the problem of independence-preserving teacher interventions in lessons with demanding modelling tasks[END_REF] to position the teaching sequence on the chronological order of events as a function of the reference frame in the context of special relativity. We carried out a preliminary study of a sequence by studying the passage real model → real results with the objective of developing a longer-term sequence covering a complete modelling cycle real situation → real results. In this case, it will not be appropriate to use a "2CV" as a vehicle to favour the passages real situation → situation model and situation model → mathematical results.

We can also take the point of view of [START_REF] Vince | Simuler pour modéliser. Le cas du son[END_REF] who studied the articulation between the world of objects and events and that of theories and models in the physical sciences. The objects resulting from the simulation are observable and can be manipulated. They are not real objects or elements of the world of theories and models. They act as intermediaries, facilitating the transition, through the problem-solving activity, between the world of objects/events and the world of models.

The Minkowski diagram is a space-time diagram allowing to know the space-time coordinates of an event in a reference frame of Armineh or Daniel. In the Minkowski diagram, the line x = 0 is described by the axis (Oc.t) in the Daniel reference frame. Similarly, the line x' = 0 is described by the axis (Oc.t') in the Armineh reference frame. c is the speed of light in vacuum or air. Projections on this type of diagram are made parallel to the axes. The axis (Ox') is the symmetric of the axis (Oc.t') with respect to the line x = c.t. It is the same for the axes (Ox) and (Oc.t).

The dynamic geometry software GeoGebra and the Minkowski space-time diagram were used by the students. It represents the reference frame (xOc.t) relative to Daniel's reference frame and the reference frame (x'Oc.t') relative to Armineh's reference frame. The latter moves at the speed v of 0.6 times the speed of light in vacuum with respect to Daniel along an axis (Ox). The straight lines (Ox) or (Ox') correspond to the route in Daniel's or Armineh's reference frames (Figure 3).

With respect to Daniel, the chronological order of production of the three events is E 1 , E 2 and E 3 while with respect to Armineh the order is E 1 , E 3 and E 2 . The GeoGebra cursor allows to modify the experimental conditions by changing the speed v and thus observe a change of chronological order of events production in the Armineh frame of reference. The epistemological plane of mathematics and the cognitive plane are mobilised during the construction of the cursor because it is necessary to know how the Ox' and Oc.t' axes are modified according to the speed v. The two axes are symmetric with respect to the line x' = c.t' which is also confounded with the line x = c.t. The epistemological plane of physics is also mobilised when the students conclude on the chronological order of events according to the two reference frames (Figure 4).

A class of 34 students from a secondary school in Picardy participated in the experiment. A first paper-pencil session was performed to construct and use the Minkowski diagram with strong teacher guidance. The work focused on the phenomena of time dilation for a given speed from one reference frame to another. The activity, corresponding to the second session, was then given to groups as a homework. The speed conditions were different from one group to another. The students had to model in autonomy the situation where the speed of one reference frame compared to another is modified and close to the speed of light. It was therefore necessary to use dynamic geometry software in autonomy, build the dynamic Minkowski diagram and deduce results on the chronological order of events according to the reference frames. Most students had not used GeoGebra software in high school. The students completed a first version of their homework assignments, and then worked for two hours in half class in the computer room to finalise their GeoGebra file. Each student also made an MP3 recording to summarize the entire sequence, which totalled five hours. The work of four students was analysed, and they were selected based on their academic performance (success and difficult students). With the MWS extended model we can perform a priori analysis of each of the tasks to be executed by the students. The level of difficulty of each task, can be appreciated by looking at the portion of the genesis instrumental-discursive or semiotic-discursive, more difficult, compared to the genesis semiotic-instrumental. A posteriori analyses allow us to study the tasks assigned to the students, and those carried out by them, considering the rationality frameworks of mathematics and physics. This study focused on the links between epistemological and cognitive planes. It led to the identification of the sets of rationality frameworks when solving this problem of special relativity treated by a diagrammatic approach. The dynamic geometry software GeoGebra allows to perform an additional semiotic genesis with its dynamic aspect. The cursor allows the speed conditions to change from one reference frame to another and the student can see the results directly on the screen. An instrumental genesis different from the paper-pencil activity, is also used during the construction of the Minkowski diagram. Finally, we hypothesise that GeoGebra, by making it easier to make conclusions about the relative chronological order of events, leads to the activation of an original discursive genesis. The audio recording, which the students made to summarise all the tasks performed during this sequence, allows them to communicate on the Minkowski diagram without directly implementing it. It is the sign of semiotic and discursive interactions favouring a conceptualisation of notions of special relativity. Student B's work is presented here. The Minkowski diagram includes the three events, the Ox, Oc.t, Oc.t' axes, the x = c.t line, and projections parallel to the Oc.t' axis cutting an Ox' axis that is not correctly positioned. Also, the cursor does not appear. The notion of event seems to be mobilised as well as that of reference frame since the two reference points appear explicitly even if it remains imperfect. The notion of relative chronological order does not seem to be a well-known one (Figure 5).

The second version which has been reworked in class (personalised help from the teacher to give technical information on the cursor and on the meanings of the lines x = c.t. or x' = c.t') includes the different elements which were missing in the first version. The Ox' axis is well positioned, the cursor for changing the v/c value appears and the c.t coordinates of the different events also. Parallels to the Oc.t' axis or to the Ox' axis passing through the different events are also represented. This work highlights tasks described by semiotic-instrumental interactions, properly performed by Student B and more marginally tasks described by instrumental-discursive interactions. Important confusions are shown in the audio recording, on the construction of the line O.ct' with an incorrect coefficient director (coefficient director of 0.6 instead of 1/0.6 when v = 0.6.c). This recording also shows that Student B is saying results without using the Minkowski diagram or in a basic way. There is confusion about the goal of the activity (comparing speeds from Student B), or about the notion of the speed of a system in a given reference frame (Student B speaks rather of the speed of a reference frame). The explanations of the plot of the line x = c.t. are ambiguous as the positions of the c.t' coordinates of the various events (Figure 6). The black arrows describe a genesis on the extended MWS corresponding to a correctly performed task, the black arrows crossed out an incorrectly performed task and the dotted arrows, a partially successful task.

Conclusion

The extended MWS framework, allowed us to analyse the tasks associated with certain stages of the modelling cycle in a problem involving special relativity. It considers the mobilisation of the epistemological planes of mathematics, and/or physics for each of the tasks requested. The extended MWS framework also led us to show that the GeoGebra software develops specific geneses in relation to a paper-pencil activity. A new semiotic genesis leads to a visualisation of the change in the time coordinates of events as a function of Armineh's speed v with respect to Daniel. A new instrumental genesis corresponds to the manipulation of the dynamic geometry software, with the cursor functionality allowing to simply change the experimental conditions. Finally, a new discursive genesis makes it possible to conclude on the chronological order of events, according to the reference frame of study and the speed v. The extended MWS model leads to an a priori analysis of each of the tasks to be performed by the students and to a successful a posteriori analysis of the work performed by them. We then plan to analyse, using the extended MWS model or one of its evolutions, the tasks implemented at each stage of the global modelling cycle during other sequences using special relativity. Preliminary results tend to show that the genesis as well as the epistemological plans of mathematics and physics are not mobilised in the same way according to the stage of the modelling cycle, and that the teacher often, assumes implicitly an important part of the stages of the modelling cycle (we suppose an important assumption of responsibility for the stages going from the real situation to the real model), which does not help "in our opinion" the students to understand the problems which are given to them. Problem-solving exercises allow this understanding, but it takes time to complete. We plan to carry out the same study from the point of view of the theory of both worlds, that of objects/events and theories/models. It would thus be possible to finely analyse the passage between the two worlds via the simulated world using the extended MWS model.

Figure 1 :

 1 Figure 1: Extended MWS Model

Figure 2 :

 2 Figure 2: The "real model" of the situation

Figure 3 :

 3 Figure 3: The mathematical model of the situation

Figure 4 :

 4 Figure 4: A priori analysis of cursor usage with GeoGebra

Figure 5 :Figure 6 :

 56 Figure 5: Screenshot of the first GeoGebra version of Student B

Acknowledgement

I would like to thank all the members of the LDAR MWS working group.