R Mehaddi

F Candelier

B Mehlig

Inertial drag on a sphere settling in a stratied uid

We compute the drag force on a sphere settling slowly in a quiescent, linearly stratied uid. Stratication can signicantly enhance the drag experienced by the settling particle. The magnitude of this eect depends on whether uid-density transport around the settling particle is due to diusion, to advection by the disturbance ow caused by the particle, or due to both. It therefore matters how eciently the uid disturbance is convected away from the particle by uid-inertial terms. When these terms dominate, the Oseen drag force must be recovered. We compute by perturbation theory how the Oseen drag is modied by diusion and stratication. Our results are in good agreement with recent direct-numerical simulation studies of the problem.

Introduction

The settling of small solid particles in either gaseous or liquid ows with density stratication is a topic of great interest in uid dynamics. Such multi-phase ows are widely encountered in nature. An example is the settling of small aggregates of organic matter, `marine snow', in the ocean [START_REF] Van Aartrijk | Vertical dispersion of light inertial particles in stably stratied turbulence: The inuence of the basset force[END_REF][START_REF] Guasto | Fluid mechanics of planktonic microorganisms[END_REF][START_REF] Sozza | Largescale connement and small-scale clustering of oating particles in stratied turbulence[END_REF][START_REF] De Pietro | Clustering of vertically constrained passive particles in homogeneous isotropic turbulence[END_REF]. Here density stratication is due to either saltconcentration or temperature gradients. A second example is the dynamics of pollutants in the atmosphere which is often stably-stratied turbulence [START_REF] Jacobson | Fundamentals of atmospheric modeling[END_REF][START_REF] Leel®ssy | Dispersion modeling of air pollutants in the atmosphere: a review[END_REF]. It is important to understand the settling speed of particles in such ows, because it determines their residence times and deposition rates. Also, density-stratied uids occur in industrial processes that involve heated uids [START_REF] Linden | The uid mechanics of natural ventilation[END_REF], or the mixing of uids of dierent densities [START_REF] Turner | Buoyancy eects in uids[END_REF].

To compute, from rst principles, the drag force experienced by the particles as they fall through the ow is a dicult problem, in particular if there is turbulence. Many authors have therefore concentrated on particle dynamics in quiescent stratied uids. Settling through a sharp density interface was measured in experiments [START_REF] Srdi¢-Mitrovi¢ | Gravitational settling of particles through density interfaces[END_REF][START_REF] Abaid | An internal splash: Levitation of falling spheres in stratied uids[END_REF][START_REF] Camassa | Prolonged residence times for particles settling through stratied miscible uids in the stokes regime[END_REF] and in simulation [START_REF] Blanchette | Drops settling in sharp stratication with and without marangoni eects[END_REF][START_REF] Ardekani | Transport of particles, drops, and small organisms in density stratied uids[END_REF][START_REF] Pierson | Inertial settling of a sphere through an interface. part 2. sphere and tail dynamics[END_REF]. Direct-numerical simulation (DNS) studies of particles settling in quiescent, linearly stratied uids were reported by [START_REF] Torres | Flow past a sphere moving vertically in a stratied diusive uid[END_REF]; [START_REF] Yick | Enhanced drag of a sphere settling in a stratied uid at small Reynolds numbers[END_REF]; [START_REF] Ardekani | Stratlets: Low Reynolds number point-force solutions in a stratied uid[END_REF]. Stratication can signicantly slow down the settling particle by enhancing the drag it experiences [START_REF] Yick | Enhanced drag of a sphere settling in a stratied uid at small Reynolds numbers[END_REF]). The reason is that buoyancy dierences due to the stratication tend to prevent the vertical motion of the uid that the particle stirs up as it settles. As a consequence, the disturbance ow remains conned around the particle [START_REF] Ardekani | Stratlets: Low Reynolds number point-force solutions in a stratied uid[END_REF]. How much the particle is slowed down depends on the mechanisms that govern the dynamics of the uid density: diusion of concentration or temperature, or their advection by the disturbance ow, or a combination of diusion and advection.

Which of these mechanisms is most important depends on the physical system in question. In salt water, for example, the diusion coecient of salt is much smaller than the kinematic viscosity of the uid. Therefore salt water is often considered non diusive. But when temperature comes into play this may not be a good approximation, because the diusion coecient of temperature in water is roughly of the same order as the kinematic viscosity. This is even more important in gases where the temperature diusion coecient may exceed the kinematic viscosity of the uid [START_REF] Salazar | On thermal diusivity[END_REF].

The nature of the disturbance ow caused by the settling particle depends on how eciently the uid disturbance is convected away. This is an inertial eect. So stratication, diusion, and convective uid inertia compete to determine the drag force on the particle. When the convective uid-inertia terms dominate so that stratication and diusion do not matter the Oseen drag force must be recovered. The question is how the Oseen drag on the settling sphere is modied by diusion and stratication.

To answer this question we compute the drag force on a sphere settling slowly in a quiescent linearly stratied uid. The density gradient points in the direction of gravity, so that the heavier uid is at the bottom. The importance of convective uid inertia is measured by the particle Reynolds number, Re. The relative importance of advection and diusion is characterised by the Péclet number Pe. The importance of stratication is often quantied by the viscous Richardson number Ri, the ratio of buoyancy and viscous forces [START_REF] Yick | Enhanced drag of a sphere settling in a stratied uid at small Reynolds numbers[END_REF]). Recent direct-numerical simulation studies of the problem [START_REF] Yick | Enhanced drag of a sphere settling in a stratied uid at small Reynolds numbers[END_REF][START_REF] Zhang | Wake of a vertically moving sphere in a linearly stratied uid. 16th Eur. Turbulence Conf[END_REF] explored how the drag depends on the importance of diusivity versus advection, and upon the degree of density stratication. Our goal is to explain their results by perturbation theory, assuming that both Re and Ri are small but nite. Chadwick & Zvirin (1974b,a) analysed this question, but for a sphere moving horizontally in a quiescent non-diusive stratied uid, along surfaces of constant uid density.

Here we study the settling problem, where the particle settles vertically along the uiddensity gradient, so that it crosses the surfaces of constant density. The two problems are quite dierent: when the particle moves horizontally, the streamlines of the ow tend to encircle the sphere in the horizontal plane. When the sphere moves vertically, by contrast, light uid is pushed down into regions of larger uid density, giving rise to complex disturbance-ow patterns [START_REF] Ardekani | Stratlets: Low Reynolds number point-force solutions in a stratied uid[END_REF].

Neglecting eects of convective uid inertia, the dierence between horizontal and vertical motion was compared earlier. When density transport is entirely diusive, the additional drag due to stratication is ve times larger in the vertical than in the horizontal direction [START_REF] Candelier | The history force on a small particle in a linearly stratied uid[END_REF]. When density advection dominates, the vertical drag is seven times larger than the horizontal one [START_REF] Zvirin | Settling of an axially symmetric body in a viscous stratied uid[END_REF].

Despite these qualitative and quantitative physical dierences, the horizontal and vertical problems share an important mathematical property: regular perturbation expansions fail to describe the eects of convective uid inertia and buoyancy due to stratication even if these perturbations are weak. Therefore so-called `singular-perturbation' methods are required to solve the problem. We use the standard method of asymptotic matching (Saman 1965), where inner and outer solutions of the disturbance problem are matched, describing the disturbance ow close to and far from the particle. where gravity g points in the negative ê3-direction (we denote the basis in the lab frame by ê1, ê2, and ê3, and x3 is the third component of the coordinate vector x in the lab frame. The particle settles with velocity u along gravity.

We parametrise the eect of convective inertia and stratication in terms of length scales: the particle radius a, the Oseen length o = a/Re, and the stratication length s = (νκ/N 2 ) 1/4 [START_REF] Ardekani | Stratlets: Low Reynolds number point-force solutions in a stratied uid[END_REF]. Here ν is the kinematic uid viscosity, κ is the diusivity, and N is the Brunt-Vaisala frequency. Diusive changes of the uid density are either due to concentration or temperature diusion. These dierent physical eects can be treated in the same model if changes in the uid density are linearly related to concentration and temperature changes [START_REF] Ardekani | Transport of particles, drops, and small organisms in density stratied uids[END_REF]. This is assured if these changes are small. In the case of temperature diusion, the importance of the diusivity κ is characterised by the Prandtl number Pr = ν/κ = Pe/Re. For concentration diusion, this dimensionless number is usually referred to as the Schmidt number Sc = ν/κ [START_REF] Candelier | The history force on a small particle in a linearly stratied uid[END_REF][START_REF] Doostmohammadi | A numerical study of the dynamics of a particle settling at moderate Reynolds numbers in a linearly stratied uid[END_REF][START_REF] Zhang | Wake of a vertically moving sphere in a linearly stratied uid. 16th Eur. Turbulence Conf[END_REF]. To simplify the notation we do not distinguish between these two conventions here. In the following we refer to this dimensionless parameter as the Prandtl number.

We obtain a uniformly valid perturbation theory to rst order in = a/ s and show that analysing the results in terms of the dimensionless parameter s / o reveals three distinct regimes where density diusion, density advection, and convective uid inertia dominate. Fluid inertia begins to matter when s / o is of the order of or larger than Pr -1/4 . This condition is met in recent (DNS) of the problem [START_REF] Yick | Enhanced drag of a sphere settling in a stratied uid at small Reynolds numbers[END_REF][START_REF] Zhang | Wake of a vertically moving sphere in a linearly stratied uid. 16th Eur. Turbulence Conf[END_REF], and our results are in good agreement with the simulations at small Re and small (but not too small) values of . When is so small that stratication eects are almost entirely negligible, we ascribe deviations between our theory and the DNS of [START_REF] Zhang | Wake of a vertically moving sphere in a linearly stratied uid. 16th Eur. Turbulence Conf[END_REF] to nite-size eects: while our theory applies to an unbounded system, the DNS were performed for a bounded system. The comparison shows that nite-size eects are larger in the homogeneous system.

Formulation of the problem

We consider a spherical particle of radius a and of material density ρ p settling with velocity u in a quiescent stratied uid. The diusivity of the stratifying agent (salt or temperature) is denoted by κ, and the kinematic viscosity of the uid is denoted by ν. The ambient density of the uid is assumed to vary linearly with height

x 3 ρ 0 = ρ ∞ -γx 3 , (2.1)
where γ is the density gradient, and ρ ∞ is a reference density. The problem is illustrated in Fig. 1. The settling particle causes disturbances in the uid, in the local density ρ, and in the pressure p. To determine these disturbances one must solve the Navier-Stokes equation for the uid velocity and the diusion-advection equation for the density. Following [START_REF] Maxey | Equation of motion for a small rigid sphere in a nonuniform ow[END_REF] we express these equations in a frame of reference that translates with the particle position x p (t),

ρ ∂w ∂t r + (w • ∇) w = -∇p + µ∆w + ρg -ρ ∂u ∂t and ∇ • w = 0 , (2.2a) ∂ρ ∂t + (w • ∇) ρ = κ∆ρ . (2.2b) w = 0 , ∂ r ρ = 0 at r = a and w → -u , ρ → ρ 0 as r → ∞ .
(2.2c)

Here r ≡ xx p , the spatial gradients ∇ are w.r.t. r, and the uid velocity is a function of r, w(r, t). The boundary condition for ρ on the surface of the particle is derived from the surface condition ∇ρ • n = 0. This means that the particle surface is impermeable. We dene the disturbance velocity, density, and pressure as:

w = w + u , ρ = ρ -ρ 0 and p = p -p 0 . (2.3)
Here p 0 is the hydrostatic pressure, ρ 0 is the ambient density at height x 3 , and -u is the undisturbed uid velocity in the frame translating with the particle. We de-dimensionalise the problem in the usual fashion (Alias & Page 2017), using the particle radius a for lengths, the typical time τ of variation of the disturbance velocity induced by the boundary condition (2.4c) for times, the terminal Stokes velocity in a homogeneous uid u t = [2a 2 /(9ν)](ρ p /ρ ∞ -1)g for the uid velocity, ρ ∞ νu t /a for the pressure, and γa for the density. All equations below are written in these dimensionless variables.

We further assume that quadratic combinations of the density and pressure disturbances are negligible, and that γx 3 /ρ ∞ 1 in the region of interest. This allows us to ignore density gradients except when multiplied by the gravitational acceleration [START_REF] Gray | The validity of the Boussinesq approximation for liquids and gases[END_REF]. This `Boussinesq' approximation was used in the DNS of the problem by [START_REF] Yick | Enhanced drag of a sphere settling in a stratied uid at small Reynolds numbers[END_REF] and Zhang et al. ( 2017) that we compare with in Section 5, see also [START_REF] Doostmohammadi | A numerical study of the dynamics of a particle settling at moderate Reynolds numbers in a linearly stratied uid[END_REF]. Under these assumptions the non-dimensional equations for the disturbance velocity, density and pressure take the form

ReSl ∂w ∂t + Re (w • ∇) w -(u • ∇) w + Ri ρ ê3 = -∇p + ∆w and ∇ • w = 0 , (2.4a) PeSl ∂ρ ∂t + Pe (w • ∇) ρ -(u • ∇) ρ -w • ê3 = ∆ρ , (2.4b) w = u , ∂ r ρ = cos θ at r = 1 and w → 0 , ρ → 0 as r → ∞ . (2.4c)
Here θ is the angle between the outward unit normal n of the sphere and the vertical direction ê3 . The dimensionless parameters in Eqs. (2.4a) and (2.4b) are the particle Reynolds number, the Péclet number, the Richardson number, and the Strouhal number:

Re = au t /ν , Pe = au t /κ , Ri = a 3 N 2 /(u t ν) , and Sl = a/(u t τ ) .

(2.5)

Here N is the Brunt-Vaisala frequency

N = g γ/ρ ∞ , (2.6)
the frequency at which a perturbation describing a vertically displaced parcel of uid oscillates in a stably-stratied environment [START_REF] Mowbray | A theoretical and experimental investigation of the phase conguration of internal waves of small amplitude in a density stratied liquid[END_REF] (2.4) in a frame of reference translating with the particle. It is convenient to use such moving coordinates, because the problem simplies. For a sphere settling in a homogeneous ow, for example, the problem becomes steady. After a short transient, the buoyancy force balances the viscous drag so that the sphere reaches a steady terminal velocity. In a stratied uid the situation is slightly more complicated. The problem appears to remain unsteady, because the settling sphere experiences a time-dependent buoyancy force since the ambient density ρ 0 varies as a function of height x 3 . We now show under which circumstances this residual time-dependence is negligible. The time scale τ of variation of the disturbance velocity can be estimated as τ -1 ∼ ut /u t where now u t ∼ (2/9)(a 2 g/ν)[ρ p /ρ 0 (t) -1], and ρ 0 (t) is the ambient density evaluated at the time-dependent particle position. We conclude that

Sl ∼ Ri ρ p ρ ∞ (2.7)
In this paper we obtain the drag force on the sphere assuming that convective uid inertia and density stratication matter, but that they are weak enough so that their eects can be treated in perturbation theory. We assume that 0 < Re 1 and 0 < Ri 1 (2.8)

In this limit the unsteady terms in Eqs. (2.4a) and (2.4b) are negligible, so that we can consider the steady problem.

3. Earlier results for Re = 0

For Re = 0 the drag on a sphere settling in a stratied uid was studied theoretically by [START_REF] Zvirin | Settling of an axially symmetric body in a viscous stratied uid[END_REF] and [START_REF] Candelier | The history force on a small particle in a linearly stratied uid[END_REF]. These authors made dierent assumptions concerning the relative importance of advection and diusion in Eq. (2.4b). [START_REF] Zvirin | Settling of an axially symmetric body in a viscous stratied uid[END_REF] assumed that advection is more important than diusion. When advection dominates, the density disturbance ρ scales as z/r near the particle (Chadwick & Zvirin 1974b), in the `inner region' of the problem. As a consequence, the buoyancy term in Eq. (2.4a) balances the viscous Laplacian term at

r ∼ Ri -1/3 . (3.1)
At this distance inner and outer solutions of the disturbance problem must be matched. This implies that advection is more important than diusion in Eq. (2.4b) if Pe > Ri 1/3 . Second, at r ∼ Ri -1/3 the dominant convective inertial term in Eq. (2.4a) is estimated as Re(u • ∇)w (0) ∼ Re Ri 2/3 , where w (0) ∼ 1/r is the solution of the disturbance problem in the Stokes limit. So convective inertial terms are negligible if Re Ri 1/3 . Under these conditions, Pe > Ri 1/3 and Re Ri 1/3 , (3.2) [START_REF] Zvirin | Settling of an axially symmetric body in a viscous stratied uid[END_REF] derived the following expression for the drag force

f 3 = -6πu 3 1 + B Ri 1/3 /Pe Ri 1/3 . (3.3)
Here B(•) is a function given in integral form. In the limit of a non-diusive uid, Pe → ∞, the above expression simplies to:

f 3 = -6πu 3 (1 + 1.060 Ri 1/3 ) . ( 3
f 3 = -6πu 3 (1 + 0.6621 ) . (3.8)
Using Ri = 4 /Pe, we see that the condition Pe corresponds to Pe Ri 1/3 . Comparing with the condition (3.2) it seems that the results (3.4) and (3.8) apply in the opposite limits of large and small Péclet numbers. Below we show, however, that the two approaches are in fact equivalent, although they seem to apply in distinct limits.

Method

We consider the same problem as Candelier et al. ( 2014), but we do not neglect the uid-inertia terms and the eect of advection of the uid density by the disturbance ow. The relative importance of stratication and inertial eects is determined by the magnitude of the length scales s and o in relation to the particle size a. Therefore we use = a/ s [Eq. (3.5)] and s / o as dimensionless parameters. The third parameter is the Prandtl number. In summary, we solve Eqs. (2.4) to rst order in the parameter using the method of asymptotic matching (Saman 1965). Inner and outer solutions of the disturbance problem are matched at r ∼ -1 in the limit 1 , s / o -1 , and Pr arbitrary but xed. (4.1)

In this way we obtain an expression for drag force that is valid regardless of whether diusion or advection dominates: our solution is valid in both limits considered by Candelier et al. ( 2014) and [START_REF] Zvirin | Settling of an axially symmetric body in a viscous stratied uid[END_REF], as well as uniformly in between.

Previous arguments, summarised in 3, appeal to dierent behaviours of the density disturbance to show that the non-linear convective terms Re(w (0) •∇)w (0) and Pe (w (0) • ∇)ρ in Eq. (2.4) can be disregarded. A weakness of these arguments is that the limits of large and small Pe are considered separately. This is not necessary in our formulation.

A general property of the method of asymptotic matching is that it is the magnitude of the dierent terms in the matching region that matters: all terms that are sub-leading in this region can be entirely neglected. As already mentioned in 3, when Re and Ri are small, the disturbance velocity close to the particle is well approximated by the Stokes solution w (0) ∼ 1/r. Assuming this dependence, we can estimate the magnitude of the non-linear convective term Re(w (0) • ∇)w (0) in the matching region. Setting r ∼ -1 we conclude that Re(w (0) • ∇)w (0) is small in this region compared with all other terms in Eq. (2.4a), when is small. The orders of magnitude in Eq. (2.4b) are more dicult to determine because the r-dependence of the density disturbance is not known unless Pe is either small [START_REF] Candelier | The history force on a small particle in a linearly stratied uid[END_REF] or large [START_REF] Zvirin | Settling of an axially symmetric body in a viscous stratied uid[END_REF]. However, since w (0) ∼ in the matching region, we can conclude that the non-linear term Pe (w (0) •∇)ρ is negligible compared with Pe (u • ∇)ρ . As a result, Eqs. (2.4) take the form:

- s o (u • ∇) w = -∇p -4 ρê 3 + ∆w and ∇ • w = 0 , (4.2a) -Pr s o (u • ∇)ρ -w • ê3 = ∆ρ , (4.2b)
with boundary conditions corresponding to (2.4c), and ρ = Pe ρ. The inner solution of Eqs. (4.2) is obtained by a regular perturbation expansion in . To obtain the outer solution one replaces the boundary condition on the particle surface by a singular source term (Saman 1965). To lowest order in this term takes the form 6πu δ(r), corresponding to the leading-order force exerted by the sphere on the uid. Higher -orders in the source term contribute to the force at order O( 2 ), and need not be considered to order . Since the non-linear convective terms are negligible, Eq. ( 4.2) is linear, so that the outer solution can be obtained by Fourier transform, for arbitrary values of . We dene:

f (k) = dx f (x)e -ik•x and f (x) = dk (2π) 3 f (k)e ik•x . (4.3)
We expand the Fourier transform ŵ out (k) of the outer solution in , in terms of gener- alised functions [START_REF] Candelier | Note on the method of matchedasymptotic expansions for determining the force acting on a particle[END_REF][START_REF] Meibohm | Angular velocity of a spheroid log rolling in a simple shear at small Reynolds number[END_REF]:

ŵ out = T (0) + T (1) + 2 T (2) + . . . . (4.4)
This method diers slightly from the standard approach (Saman 1965) that formulates the outer problem in terms of strained coordinates r = r. The advantage of the present approach is that it does not refer to any particular matching length scale for instance the length scale at which the Laplacian is balanced by the buoyancy term in Eq. (4.2a). The only requirement is that is small. For certain cases this approach is equivalent to using the reciprocal theorem to compute inertial corrections [START_REF] Meibohm | Angular velocity of a spheroid log rolling in a simple shear at small Reynolds number[END_REF].

The rst two terms in the expansion (4.4) are obtained from ŵ out as:

T (0) = lim →0
ŵ out and

T (1) = lim →0 1 ( ŵ out - T (0)
) .

(4.5)

The rst term, T (0) (k), is the Fourier transform of the solution of the outer problem at = 0. The next term in the expansion takes the form

T (1) (k) = lim →0 1 3 ŵ out k - T (0) k . (4.6)
We evaluate this limit using the homogeneity properties of the functions ŵ out (k) and

T (0)
(k) [START_REF] Candelier | Note on the method of matchedasymptotic expansions for determining the force acting on a particle[END_REF][START_REF] Meibohm | Angular velocity of a spheroid log rolling in a simple shear at small Reynolds number[END_REF]. This yields

T (1) (k) = δ(k) dk ŵ out | =1 (k ) - T (0) (k ) . (4.7)
Here δ is the three-dimensional Dirac delta function. The functions T (0) (k) and T (1) (k) are readily transformed back from k-to obtain the outer solution in conguration space. In particular, T (1) (r) is found to be r-independent. Since T (1) (r) is constant, the problem to order is equivalent to determining the force on a particle kept xed in a uniform ow (Saman 1965;[START_REF] Meibohm | Angular velocity of a spheroid log rolling in a simple shear at small Reynolds number[END_REF]. It follows that the drag force is

given by

f = -6π u + 8π 3 dk ŵ out (k )| =1 - T (0) (k ) . (4.8)
We note that the force is determined entirely by the solution of the outer problem, as rst shown by Saman (1965) for the lift force on a small sphere in a shear ow.

Results

For = 1 the Fourier transforms ŵ out and ρout of the outer solution read:

ŵ out ρout = -6π k 2 s o ik • u I + A -1 • G • u 0 .
(5.1)

Here I is the 4 × 4 unit tensor, and

A =        -k 2 0 0 -k1k3 k 2 0 -k 2 0 -k2k3 k 2 0 0 -k 2 - (k 2 -k 2 3 ) k 2 0 0 1 Pr -k 2 Pr        , G =        k 2 -k 2 1 k 4 -k1k2 k 4 -k1k3 k 4 0 -k2k1 k 4 k 2 -k 2 2 k 4 -k2k3 k 4 0 -k3k1 k 4 -k3k2 k 4 k 2 -k 2 3 k 4 0 0 0 0 0        . (5.2)
We set u = u 3 ê3 in Eq. (4.8) to nd the drag force on the settling sphere:

f 3 = -6πu 3 (1 + M 33 ) , (5.3a)
M 33 = 3 2π ∞ 0 dk π 0 dθ sin(θ) 3 1-Pr s o 2 k 2 +1 cos (θ) 2 -i cos (θ) s o k 3 Pr s o 2 k 2 +1 cos (θ) 2 + i s o k 3 (Pr+1) cos (θ)-k 4 -1 . (5.3b)
The imaginary part in Eq. (5.3b) vanishes upon integration. Fig. 2 shows how M 33 depends on the ratio s / o for dierent values of Pr, namely, 0.7 (temperature-stratied gas), 7 (temperature-stratied water at 20 o C) and 700 (saltstratied water). When the ratio s / o is very small, the curves collapse onto a horizontal 1 ( ). Coloured solid lines show Eq. (5.3b) for < 0, dashed lines for > 0, for a small value of 0. We take 0 = 0.3. Also shown are power laws in Fr, black solid lines. The vertical dashed line corresponds to Fr=1/Re. (b) Same data as in panel (a) but now with linear axes to emphasise the deviations between theory and DNS at large Fr. The black dashed line shows the drag for the nite homogeneous system obtained by [START_REF] Zhang | Wake of a vertically moving sphere in a linearly stratied uid. 16th Eur. Turbulence Conf[END_REF]. The solid black line shows the theoretical value for the homogeneous innite system. line, Eq. (3.8). In this limit diusion dominates. When s / o reaches Pr -1 , a second regime emerges: diusion and advection in Eq. (2.4b) become of the same order, resulting in a change in the behaviour of the density disturbance. As a result, M 33 in Fig. 2 turns downwards as s / o increases (this does not mean that the drag -6π(1 + M 33 ) increases at xed particle radius a and Reynolds number Re, because = a/ s ). A further transition occurs at s / o ∼ 1/Pr 1/4 , caused by the formation of an Oseen wake behind the particle [START_REF] Lovalenti | The force on a bubble, drop or particle in arbitrary time-dependent motion at small Reynolds number[END_REF]. When s / o 1/Pr 1/4 the curves approach M 33 ≈ (3/8)( s / o ), the Oseen correction [START_REF] Oseen | über die Stokes'sche Formel und über eine verwandte Aufgabe in der Hydrodynamik[END_REF][START_REF] Proudman | Expansions at small Reynolds numbers for the ow past a sphere and circular cylinder[END_REF][START_REF] Lovalenti | The force on a bubble, drop or particle in arbitrary time-dependent motion at small Reynolds number[END_REF]. In this regime stratication and diusion do not matter, the settling particle experiences the uid as if it were homogeneous. For small Pr, only the rst and third regimes are realised.

Eq. (5.3b) is uniformly valid in the limit (4.1), regardless of the value of ( s / o )Pr = (Ri 1/3 /Pe) -3/4 . It is not necessary to assume that Pe Ri 1/3 , the expression holds also when Pe Ri 1/3 . In particular, we can see that Eq. (5.3b) reduces to Eq. (3.3) when convective inertia is negligible, by taking the limit s / o → 0 at xed ( s / o )Pr:

lim s/ o →0 M 33 = 3 Ri 1/3 π ∞ 0 dk π 2 0 dθ sin(θ) 5 sin(θ) 2 + (Ri 1/3 /Pe)k 4 sin(θ) 2 + (Ri 1/3 /Pe)k 4 2 + cos(θ) 2 k 6 .
(5.4) This is precisely the function B(•) in Eq. (3.3) [Eq. ( 29) in [START_REF] Zvirin | Settling of an axially symmetric body in a viscous stratied uid[END_REF]], computed assuming that convective inertia is negligible, and that Pe > Ri 1/3 . Since our solution is uniformly valid, we can conclude that Eq. (3.3) must be valid also for Pe Ri 1/3 , well outside the region of validity stated by [START_REF] Zvirin | Settling of an axially symmetric body in a viscous stratied uid[END_REF]. Closer inspection of their calculation shows that it corresponds to asymptotic matching at r ∼ Ri -1/3 in the limit Ri → 0 keeping Ri 1/3 /Pe constant. The two dierent matching scales r ∼ Ri -1/3 and r ∼ -1 are equivalent in the limits stated, because the ratio of matching scales Ri 1/3 / = (Ri 1/3 /Pe)(Pe/ ) = (Ri 1/3 /Pe)[( s / o )Pr] remains constant.

In summary, Eq. (5.3b) is a uniform approximation comprising three distinct regimes

f 3 ∼ -6πu 3        1+0.6621
for s / o Pr -1 regime 1 (diusion), 1+1.060 Ri 1/3 for Pr -1 s / o Pr -1/4 regime 2 (advection), 1+ 3 8 Re for s / o Pr -1/4 regime 3 (uid inertia).

(5.5)

The dierent regimes are shown in Fig. 2. In the limit of small Pr, the advective regime disappears, as mentioned above.

We now compare the full result, Eq. (5.3b), with DNS by [START_REF] Zhang | Wake of a vertically moving sphere in a linearly stratied uid. 16th Eur. Turbulence Conf[END_REF] and [START_REF] Yick | Enhanced drag of a sphere settling in a stratied uid at small Reynolds numbers[END_REF], at their smallest Re. These authors use slightly dierent dimensionless parameters, namely Re, Pr, and the Froude number Fr Fr = u t /(a N ) .

(5.6)

In terms of Fr, the dimensionless parameters , s / o , and Ri are given by: = (Re/Fr) 1/2 Pr 1/4 , s / o = (Re Fr) 1/2 /Pr 1/4 , and Ri 1/3 = Re 1/3 /Fr = 12/Re is the Stokes drag coecient for an unbounded system. Since Eq. (5.3b) was obtained for small , we plot it as a solid line when < 0.3, and dashed for > 0.3. For 1 < Fr < 10 the data for Pr = 0.7 are in the diusive regime, where the correction to the drag scales as Fr -1/2 . For Pr = 700, the data approach the advection regime where the theory predicts that the drag correction scales Ri 1/3 ∝ Fr -2/3 . Fig. 3(a) shows that the DNS yield a larger drag coecient than our theory when Fr is small. The likely reason is that the non-linear convective terms matter in this regime, because is not small enough (dashed lines in Fig. 3(a) indicate that > 0.3, as mentioned above). For Pr = 700, for example, becomes larger than 0.3 when Fr falls below 14.7 at Re = 0.05.

When do convective uid-inertia eects dominate? The condition s / o = 1/Pr 1/4 corresponds to Fr = 1/Re, independent of Prandtl number. For Re = 0.05 the smallest value used in the DNS this crossover occurs at Fr = 20, indicated by the vertical black dashed line in Fig. 3(a). Eq. (5.3b) allows us to determine the relative importance of convective uid inertia at this value of Fr. For Pr = 0.7 the correction is substantial, 13.5 %. For larger Péclet numbers the correction is smaller, 1.4% at Pr = 7, and 2 % at Pr = 700. That the correction is largest for small Pr can be inferred from Fig. 3(a).

We observe deviations between DNS and theory not only at small values of Fr, discussed above, but also at large values of Fr [Fig. 3(b)]. These deviations at large Fr may be due to nite-size eects. In this limit, the homogeneous Oseen correction dominates, and it is known to be quite sensitive to the size of the simulation domain. [START_REF] Yick | Enhanced drag of a sphere settling in a stratied uid at small Reynolds numbers[END_REF] chose an elliptical simulation domain, with a smallest size L that gives L/(2a) = 40. The domain used by [START_REF] Zhang | Wake of a vertically moving sphere in a linearly stratied uid. 16th Eur. Turbulence Conf[END_REF] was spherical and larger [diameter/(2a) = 80], but even in that case a theory for cylindrical domains [START_REF] Happel | Low Reynolds number hydrodynamics[END_REF] indicates that the drag correction is expected to be larger than the Oseen expression 3 8 Re. This is consistent with Fig. 3(b). We also see that the asymptotic value for the homogeneous uid is reached more quickly in the nite system. Finite-size eects matter less for smaller Fr, because the wake is smaller, of order s .

Conclusions

We calculated how convective uid inertia modies the drag on a sphere slowly settling in a density-stratied uid, at small Richardson and Reynolds numbers. Plotting the results as a function of the dimensionless parameter s / o reveals three distinct regimes, Eq. (5.5). In the rst regime, the drag is determined by diusion of the disturbance density. In the second regime, advection of the disturbance density determines the drag. In the third regime, convection of the disturbance density by uid-inertia terms dominates. Our main result, Eq. (5.3b), is uniformly valid, independently of whether the density dynamics is diusive or advective. This allowed us to show that a result by [START_REF] Zvirin | Settling of an axially symmetric body in a viscous stratied uid[END_REF] is more generally valid than the authors stated.

We compared with recent DNS at small Re and found that convective uid-inertia eects matter for the larger Froude numbers simulated. Since nite-size eects appear to be important at large Fr and small Re, it would be of interest to take these corrections into account in the theory.

The results derived in this paper were obtained in the steady limit. But when a particle is released from above the water surface and plunges into the uid with a given velocity, then unsteady eects must matter, at least initially. DNS of the problem [START_REF] Doostmohammadi | A numerical study of the dynamics of a particle settling at moderate Reynolds numbers in a linearly stratied uid[END_REF] at Re of order unity reveal unsteady eects that depend on the dimensionless numbers of the problem in intricate ways.

A further motivation for considering unsteady eects concerns the unsteady swimming of micro-organism in stratied uids [START_REF] Doostmohammadi | Low-reynoldsnumber swimming at pycnoclines[END_REF][START_REF] Jephson | Species-and stratication-dependent diel vertical migration behaviour of three dinoagellate species in a laboratory study[END_REF][START_REF] Bergström | Behavioural dierences in relation to pycnoclines during vertical migration of the euphausiids meganyctiphanes norvegica (m. sars) and thysanoessa raschii (m. sars)[END_REF]. In oceans or in lakes the surface layers are known to shelter substantial biological activity. For very small organisms (much smaller than 1mm in size in typical ocean conditions) the dynamics of swimming microorganisms is well understood. Buoyancy (Franks & Jae 2008), density or drag asymmetries of the body [START_REF] Roberts | Geotaxis in motile micro-organisms[END_REF][START_REF] Jonsson | Vertical distributions of planktonic ciliates -an experimental analysis of swimming behaviour[END_REF][START_REF] Kessler | Hydrodynamic focusing of motile algal cells[END_REF], and turbulence [START_REF] Durham | Turbulence drives microscale patches of motile phytoplankton[END_REF][START_REF] Gustavsson | Preferential sampling and small-scale clustering of gyrotactic microswimmers in turbulence[END_REF] determine the spatial distribution of these organisms, their encounter rates, and thus their population ecology [START_REF] Guasto | Fluid mechanics of planktonic microorganisms[END_REF]. For larger organisms less is known. The problem becomes considerably more dicult because inertial eects begin to matter (Wang & Ardekani 2012b,a). The method described here allows to take inertial eects into account in perturbation theory. Finally, an important problem is how uid shears aect the dynamics of motile microorganisms. The approach described by [START_REF] Candelier | Time-dependent lift and drag on a rigid body in a viscous steady linear ow[END_REF] makes it possible to address this question.

Figure 1 .

 1 Figure 1. Settling of a sphere in a linearly stratied uid with ambient density ρ0 = ρ∞ -γx3

  .4) Now consider the opposite limit, where the diusive term in Eq. (2.4b) dominates over the advective term. In this case[START_REF] Candelier | The history force on a small particle in a linearly stratied uid[END_REF] showed that the spatial dependence of the disturbance density ρ is of the form ρ ∼ Pe r in the inner region, so that the buoyancy term in Eq. (2.4a) balances the Laplacian viscous term at r ∼ -1 with = a/ s . (3.5) Here s is the stratication length (Ardekani & Stocker 2010) s = νκ/N 2 1/4 . (3.6) It characterises the eect of stratication on the particle dynamics. Under the condition Pe 1 (3.7) Candelier et al. (2014) found

Figure 2 .

 2 Figure 2. Shows Eq. (5.3b) as a function of s/ o for dierent Pr. Also shown are the three dierent regimes in Eq. (5.5), black solid lines.

Figure 3 .

 3 Figure 3. (a) Comparison between Eq. and DNS results for Re= 0.05 by Yick et al. (2009) for Pr= 7 (•), Pr= 700 (•), and by Zhang et al. (2017) for Pr= 0.7 ( ) and Pr= 700

  2/3 . (5.7) Zhang et al. (2017) and Yick et al. (2009) computed the drag coecient C S D of the stratied system. In Fig. 3(a) we plot their result for C S D /C Stokes D -1 versus Fr, and compare it with our result for M 33 . Here C Stokes D

  . The Brunt-Vaisala frequency is an important physical parameter in problems involving stratied uids, such as fountains(Bloomeld & Kerr 1998;[START_REF] Mehaddi | Analytical solutions for turbulent boussinesq fountains in a linearly stratied environment[END_REF]), plumes (McDougall 1978;[START_REF] Woods | Turbulent plumes in nature[END_REF][START_REF] Mehaddi | Naturally bounded plumes[END_REF] or gravity currents[START_REF] Maxworthy | The propagation of a gravity current into a linearly stratied uid[END_REF], to name but a few.We wrote Eqs.

We thank J. Magnaudet and J. Zhang for providing some of the numerical data discussed in Zhang et al. (2017). BM was supported by Vetenskapsrådet [grant number 017-03865], Formas [grant number 2014-585], and by the grant `Bottlenecks for particle growth in turbulent aerosols' from the Knut and Alice Wallenberg Foundation, Dnr. KAW 2014.0048.


EUREQua Nicolas Houy
email: houy@univ-paris1.fr

Lucie Menager
email: menager@univ-paris1.fr

Lucie Ménager

Communication, consensus and order. Who wants to speak first ? *

Keywords: JEL Classification: D70, D82 Common knowledge, Consensus, Communication Protocol

Parikh and Krasucki [1990] showed that if rational agents communicate the value of a function f according to a protocol upon which they have agreed beforehand, they will eventually reach a consensus about the value of f , provided a fairness condition on the protocol and a convexity condition on the function f . In this article, we address the issue of how agents agree on a communication protocol in the case where they communicate in order to learn information. We show that if it is common knowledge among a group of agents that some of them disagree about two protocols, then the consensus value of f must be the same according to the two protocols.

Introduction

Alice and Bob are sitting in front of each other, both wearing either a red hat or a white hat. Suppose that the two hats are red. The teacher tells the children that there is at least one red hat, and asks them whether they know the color of their hat. The two children observe that the other's hat is red, but cannot infer the color of their own hat. The only way for them to answer the teacher is to communicate with each other. Suppose that Alice tells Bob that she does not know the color of her hat. Bob understands that his own hat is red, for if it had been white, Alice would have known that her hat was red. Now Bob knows the color of his hat. But then if he tells Alice that he knows the color of his hat, Alice will not learn anything, for the message of Bob would have been the same regardless of the color of her hat.

Therefore, Alice has no interest to be the first to say whether she knows the color of her hat. This story illustrates the following fact. From the moment that people communicate in order to be better informed, who gets to talk when is important: the communication process is not commutative, for different orders of speech may lead to different outcomes.

It is well known since Geanakoplos and Polemarchakis [1982] that in a group of rational agents, a process of simultaneous communication of posterior beliefs for an event leads to equality of all individual beliefs. [START_REF] Cave | Learning To Agree[END_REF] and [START_REF] Bacharach | Some Extensions of a Claim of Aumann in an Axiomatic Model of Knowledge[END_REF] extended this result to simultaneous communication of decisions, assuming that the decision rule followed by agents satisfies a union consistency property. Yet in most economic situations where agents have to speak together, communication is not simultaneous. It is common sense that each individual speaks one after the other according to a given protocol. Parikh and Krasucki [1990] considered the case where agents of a group communicate with each other, according to a pairwise protocol upon which they have agreed beforehand. They investigated what conditions on the type of messages and on the protocol guarantee that agents eventually reach a consensus, i.e. that from some stage on all the communicated values will be the same. They show that if the protocol is fair, that is if every participant receives information directly or indirectly from every other participant, and if the function f is convex, that is for all pair of disjoint events X, X , there exists a ∈ [0, 1] such that f (X ∪X ) = af (X)+(1-a)f (X ), then communication will eventually lead to a consensus about the value of f . We show that different protocols may lead to different outcomes, in terms of consensus values of f as well as of information learned by the agents during the communication process. This non-commutativity of the order of speech, as well as the fact that agents communicate so as to be better informed, imply that they may have strategic considerations concerning the order of speech. Depending on the state of the world, Alice and Bob may prefer to speak first or second, or may be indifferent. If neither Alice nor Bob wants to speak first, communication can not take place. However, can we conclude that they will not learn anything from each other? The fact that Alice does not want to speak first is informative for Bob. Bob knows that if Alice knew the color of her hat, she wouldn't mind speaking first and saying that she knows the color of her hat. In this paper, we investigate what inferences can be made by rational agents from the common knowledge that some of them disagree about the order of speech.

We show that the following situations are both possible. First, it can be common knowledge in a group of agents that some of them prefer the same order of speech. Second, it can be common knowledge in a group of agents that some of them prefer different orders of speech.

However, we show the surprising result that if it is the case, then the consensus value of f must be the same whatever the order of speech. For instance, if it is common knowledge among Alice and Bob that they both want to speak first, then what they will communicate at the end of the day will be the same, whether Alice or Bob speaks first.

The paper is organized as follows. In Section 2 we describe the model and the basic result of Parikh and Krasucki [1990]. Section 3 defines preferences over protocols and develops the result. The proof of the theorem is given in the Appendix.

Preliminary notions

Let Ω be the finite set of states of the world, and 2 Ω the set of possible events. There are N agents, each agent i being endowed with a partition Π i of Ω. When the state ω ∈ Ω occurs, agent i just knows that the true state of the world belongs to Π i (ω), which is the cell of i's partition that contains ω. We say that a partition Π is finer than a partition Π if and only if for all ω, Π(ω) ⊆ Π (ω) and there exists ω such that Π(ω ) ⊂ Π (ω ). A partition Π is coarser than a partition Π if and only if Π is finer than Π . The partition Π i represents the ability of agent i to distinguish between the states of the world. The coarser her partition is, the less precise her information is, in the sense that she distinguishes among fewer states of the world. As usual, we say that an agent i endowed with a partition Π i knows the event E at state ω if and only if Π i (ω) ⊆ E. We define the meet of the partitions Π 1 , Π 2 , . . . , Π N as the finest common coarsening of these partitions, that is the finest partition M such that for all ω ∈ Ω and for all i = 1, . . . , N , Π i (ω) ⊆ M (ω).

Common knowledge of an event E at a state ω is the situation that occurs when each agent knows E at ω, each agent knows that each of them knows E at ω, each agent knows that each agent knows that each agent knows... etc. Aumann [1976] showed that, given a set of N agents, the meet M of their N partitions is the partition of common knowledge among these N agents. Hence we say that an event E is common knowledge at state ω iff M (ω) ⊆ E.

Before communicating, agents have to agree on a communication protocol that will be applied throughout the debate. The protocol determines which agents are allowed to speak at each date.

Definition 1 A protocol α is a pair of functions (s, r) from N to 2 {1,...,N } . If s(t) = S and r(t) = R, then we interpret S and R as, respectively, the set of senders and the set of receivers of the communication which takes place at time t.

We note Γ the set of protocols. Note that the type of protocols we consider are more general than the ones in PK, for we allow for more than one agent to be senders and receivers of the communication at the same time.

Along the debate, agents communicate by sending messages, which we assume to be delivered instantaneously, that is at time t, messages are simultaneously sent by every i ∈ s(t) and heard by every j ∈ r(t). We assume that the message sent is the private value of some function f defined from the set of subsets of Ω to R. The private value of f for an agent i at state ω is f (Π i (ω)).

Finally, the set of states of the world Ω, the individual partitions (Π i ) i , and the message rule f define an information model I = Ω, (Π i ) i , f . Two assumptions are made on the protocol and on the function f to guarantee that iterated communication of the value of f leads to a consensus about f . As in PK, we assume that the protocol is fair. We adapt PK's definition in our setting, but the meaning remains the same: a protocol is fair if and only if every participant in this protocol communicates directly or indirectly with every other participant infinitely many times. This condition is necessary so that nobody is excluded from communication.

Assumption 1 (A1) The protocol α is fair, that is for all pair of players (i, j), i = j, there exists an infinite number of finite sequences t 1 , . . . , t K , with t k ∈ N for all k ∈ {1, . . . , K},

such that i ∈ s(t 1 ) and j ∈ r(t K ). Assumption 2 (A2) f is convex, that is for all subsets E, E ⊆ Ω such that E ∩ E = ∅, there exists α ∈]0, 1[ such that f (E ∪ E ) = αf (E) + (1 -α)f (E ).
Note that we will have

f (E 1 ∪ E 2 ∪ • • • ∪ E k ) = k i=1 α i f (E i ), with α i ∈ ]0, 1[ ∀ i and k i=1 α i = 1
provided that the E i are pairwise disjoint events. This condition is obeyed by conditional probabilities and implies union consistency 1 à la [START_REF] Cave | Learning To Agree[END_REF].

We now describe how information is aggregated during the debate. At a given date t, the senders s(t) selected by the protocol (s, r) send a message heard by the receivers r(t). Then each individual infers the set of states of the world that are compatible with the messages possibly sent, and updates her partition accordingly. Given an information model Ω, (Π i ) i , f and a communication protocol α, we define by induction on t the set Π α i (ω, t) of possible states for an agent i at time t, given that the state of the world is ω:

Π α i (ω, 0) = Π i (ω)
and for all t ≥ 1,

Π α i (ω, t + 1) = Π α i (ω, t) ∩ {ω ∈ Ω | f (Π α j (ω , t)) = f (Π α j (ω, t)) ∀ j ∈ s(t)} if i ∈ r(t), Π α i (ω, t + 1) = Π α i (ω, t) otherwise.
The next result states that for all i, for all ω, f (Π α i (ω, t)) has a limiting value, and that this value does not depend on i. Under assumptions A1 and A2, participants in the protocol converge to a consensus about the value of f . Proposition 1 (Parikh and Krasucki (1990)) Let Ω, (Π i ) i , f be an information model, and α a communication protocol. Under assumptions A1 and A2, there exists a date T such that for all ω, for all i, j, and all t, t ≥ T , f (Π α i (ω, t)) = f (Π α j (ω, t )).

In the sequel, we will denote Π α i (ω) the limiting value of Π α i (ω, t), and Π α i will be called i's partition of information at consensus. f (Π α (ω)) will denote the limiting value of f (Π α i (ω, t), which does not depend on i, and will be called the consensus value of f at state ω, given that the protocol is α.

3 Who wants to speak first?

We know from Parikh and Krasucki [1990] that given any protocol α, under assumptions A1 and A2, iterated communication of the private value of f eventually leads to a consensus

1 f is union consistent if for all E, E such that E ∩ E = ∅, f (E) = f (E ) ⇒ f (E ∪ E ) = f (E) = f (E ).
about the value of f . The next proposition states that this value may vary according to the protocol.

Proposition 2 There exist an information model Ω, (Π i ) i , f with f convex and two fair protocols α, β for which there exists ω such that f (Π α (ω)) = f (Π β (ω)).

This result can be proved easily for some union consistent functions f . However, to the best of our knowledge, it was not proved for conditional probabilities. As the posterior probabilities of an event are particular union consistent function, it could have been possible that there exist no information model with posterior probabilities such that order matters. We exhibit an example where it does. 2 Example 1 Let Ω = {1, . . . , 13} be the set of states of the world. Suppose that Alice and Bob have a uniform prior P on Ω. They communicate in turn the private value of the function f (.) = P ({2, 3, 4, 8, 12} | .), which is convex, and are endowed with the following partitions of

Ω 3 : Π A = {1, 3, 7, 8} 1/2 {2, 6, 11, 12} 1/2 {4, 5, 10} 1/3 {9} 0 {13} 0 Π B = {1, 3, 5} 1/3 {2} 1 {4, 7, 9, 10, 12, 13} 1/3 {6, 8} 1/2 {11} 0

If Alice speaks first (protocol α), individual partitions at consensus are:

Π α A = {1, 3, 7, 8} 1/2 {2} 1 {11} 0 {6, 12} 1/2 {4, 10} 1/2 {5} 0 {9} 0 {13} 0 Π α B = {1, 3} 1/2 {5} 0 {2} 1 {4, 10} 1/2 {7, 12} 1/2 {9, 13} 0 {6, 8} 1/2 {11} 0
If Bob speaks first (protocol β), individual partitions at consensus are:

Π β A = {1, 3, 7} 1/3 {8} 1 {2} 1 {6} 0 {11} 0 {12} 1 {4, 5, 10} 1/3 {9} 0 {13} 0 Π β B = {1, 3, 5} 1/3 {2} 1 {4, 7, 10, } 1/3 {12} 1 {9, 13} 0 {6} 0 {8} 1 {11} 0 2
We found it by a numerical search. If somebody has a similar example with less than 13 states of the world, please tell us! 3 The subscript reflects the posterior belief in each cell.

6

At state 1, the consensus value of

f is f ({1, 3, 7, 8}) = f ({1, 3}) = 1/2 if Alice speaks first, whereas it is f ({1, 3, 7}) = f ({1, 3, 5}) = 1/3 if Bob speaks first.
We assume that agents communicate in order to be better informed. As a consequence, they prefer protocols that lead them to be better informed at the end of the day. A more precise information is represented by a finer partition. Yet two partitions may not be rankable in the sense of refinement, so we may not be able to say with which partition an agent is better informed. For instance, we cannot say whether an individual is better with Π = {1}{2, 3, 4}

or with Π = {1, 2, 3}{4}. However, we can say that one is better informed with Π than with Π at state 1, and better informed with Π than with Π at state 4.

Definition 2

We say that an agent is better informed with the partition Π α than with the partition Π β at state ω if and only if Π α (ω) ⊂ Π β (ω).

Before communication takes place, the information model Ω, (Π i ) i , f is common knowledge among individuals. Therefore, given any protocol α, consensus partitions (Π α i ) i are also common knowledge. As a consequence, each agent knows ex interim which protocol she prefers among any two orders if she's not indifferent. Definition 3 (Preferences) Let I := Ω, (Π i ) i , f be an information model, and α, β two protocols. The set of states of the world where agent i prefers α to β is denoted B I i (α, β) and is defined by

B I i (α, β) = {ω ∈ Ω | ∀ω ∈ Π i (ω), Π α i (ω ) ⊆ Π β i (ω ) and ∃ ω ∈ Π i (ω) s.t. Π α i (ω ) ⊂ Π β i (ω )}
In Example 1, Alice and Bob are both better informed with the protocol α at state 4 and better informed with the protocol β at state 8. Hence at states 4 and 8, they agree on the protocol they prefer. On the contrary, at state 1, Alice and Bob end up strictly better informed when they speak in second. What happens in that case? Suppose that state 1 occurs, and that Alice and Bob stand in front of each other waiting for the other to speak. Alice knows that the state of the world belongs to {1, 3, 7, 8}. She understands that the state of the world can not be 7 nor 8, for Bob would have spoken first at state 8 and would have been indifferent at state 7. Bob knows that the state of the world belongs to {1, 3, 5}. He understands that the state of the world can not be 5, for he knows that Alice prefers to speak first at state 5.

Hence knowing that the other does not want to speak first makes Alice and Bob understand that the state of the world is in {1, 3}. From now, they have the same private information at state 1. As they cannot learn information from the communication process, they become indifferent between speaking first or second. This example addresses the question of whether it can be common knowledge among two persons that they disagree about the order of speech.

More generally, what inferences can be made by rational agents of a group from the common knowledge that some of them disagree about the order of speech? Our main result states that if it is the case, then the consensus message is the same according to any protocol. 2) and (3) cannot be true simultaneously.

Theorem 1 Let I = Ω, (Π i ) i , f
(1) ∃ i, j such that B I i (a 1 , a 2 ) and B I j (b 1 , b 2 ) are common knowledge at ω.

(2) ω ∈ B I i (a 1 , a 2 ) ∩ B I j (b 1 , b 2 ) and a 1 = b 2 . (3) f (Π α (ω)) = f (Π β (ω)).
The meaning of this result is the following. We prove point (i) with the following example, which describes a situation where it is common knowledge between Alice and Bob that both of them prefer to speak in second. The fact that both prefer to speak in second in order to be better informed is quite intuitive.

When an agent is the second to speak, the first message she hears contains purely private information of the other. When she is the first to speak, the first message she will hear will be a join of the other's private information and her private information, so she may not learn anything. However, we found another example which shows that there exist situations where both agents prefer to speak first.4

Example 2 The set of states of the world be Ω = {1, 2, 3, 4, 5, 6, 7} and Alice and Bob are endowed with a uniform prior P on Ω. They communicate in turn the private value of the function f (.) = P ({1, 2, 7} | .) and are endowed with the following partitions:

Π A = {1, 2} 1 {3, 4} 0 {5, 6, 7} 1/3 Π B = {1, 7} 1 {2, 3, 6} 1/3 {4, 5} 0
If Alice speaks first (protocol α), individual partitions at consensus are:

   Π α A = {1, 2} 1 {3, 4} 0 {5, 6} 0 {7} 1 Π α B = {1} 1 {2} 1 {3} 0 {4} 0 {5} 0 {6} 0 {7} 1 If Bob speaks first (protocol β), individual partitions at consensus are:    Π β A = {1} 1 {2} 1 {3} 0 {4} 0 {5} 0 {6} 0 {7} 1 Π β B = {1, 7} 1 {2} 1 {3, 6} 0 {4, 5} 0
At every state of the world, Alice and Bob both prefer to speak in second: B A (β, α) = Ω and B B (α, β) = Ω, hence at every state of the world, it is common knowledge among Alice and Bob that Alice prefers the order β and Bob the order α. However, it does not contradict Theorem 2 as for all ω, f (Π α (ω)) = f (Π β (ω)).

[4] Geanakoplos J., Polemarchakis H., [1982], We Can't Disagree Forever, Journal Of Economic Theory, 26, 363-390.

[5] Parikh R., Krasucki P., [1990], Communication, Consensus and Knowledge, Journal of Economic Theory, 52, 178-189.

Proof of [Theorem 1]

Consider an information model I = Ω, (Π i ) i , f , and α, β two protocols such that α = β.

Let us show that if points 1) and 2) of theorem 1 are true, then point 3) is false. We show that if there exist two agents i, j and a state ω such that B I i (α, β) and B I j (β, α) are common knowledge at ω, then f (Π α (ω)) = f (Π β (ω)). Clearly, the proof still holds if we invert α and β.

Recall that M (ω) denotes the meet of individual partitions before communication takes place: M = n i=1 Π i . We note Π α the meet of the individual partitions at consensus, given that the protocol is α:

Π α = n i=1 Π α i .
If B I i (α, β) and B I j (β, α) are common knowledge at ω, then we have

M (ω) ⊆ B i (α, β) ∩ B j (β, α)
As Π α (ω) ⊆ M (ω) and Π β (ω) ⊆ M (ω) ∀ ω, we have Π α (ω) ∩ Π β (ω) ⊆ M (ω) ∀ ω. Hence we have

Π α (ω) ∩ Π β (ω) ⊆ B I i (α, β) ∩ B I j (β, α) (1)
Consider some ω ∈ Π α (ω) ∩ Π β (ω) (which is not empty as ω ∈ Π α (ω) ∩ Π β (ω)). By definition of the meet, we have Π α i (ω ) ⊆ Π α (ω ) and Π β i (ω ) ⊆ Π β (ω ). As ω ∈ Π α (ω)∩Π β (ω), we have Π α (ω ) = Π α (ω) and Π β (ω ) = Π β (ω). Then we have

Π α i (ω ) ⊆ Π α (ω) and Π β i (ω ) ⊆ Π β (ω) (2)
By (1), ω ∈ B I i (α, β). It implies that Π α i (ω ) ⊆ Π β i (ω ). Yet Π β i (ω ) ⊆ Π β (ω) by ( 2). Then we have

Π α i (ω ) ⊆ Π α (ω) ∩ Π β (ω)

  be an information model such that A1 and A2 are satisfied, and α, β two protocols such that α = β. Consider a 1 , a 2 , b 1 , b 2 ∈ {α, β}, with a 1 = a 2 and b 1 = b 2 . Assertions (1), (

•

  [START_REF] Aumann | Agreeing to Disagree[END_REF] and[START_REF] Bacharach | Some Extensions of a Claim of Aumann in an Axiomatic Model of Knowledge[END_REF] are true, namely if it is common knowledge at some state ω that Alice and Bob prefer to speak first, then (3) is false, i.e the consensus value of f at ω is the same regardless of the person who speaks first.• If (1) and (3) are true, namely if it is common knowledge at ω that Alice prefers a 1 ∈ {α, β} and Bob prefers b 1 ∈ {α, β}, and if the consensus value of f differs according to whether the protocol is α or β, then (2) is false, i.e Alice and Bob prefer the same protocol (a 1 = b 1 ).• If (2) and (3) are true, namely if Alice and Bob prefer different orders of speech at ω, then (1) is false, i.e these preferences are not common knowledge among them at ω. This proposition states that (i) it can be common knowledge among them that Alice and Bob prefer different orders of speech, (ii) it can be common knowledge among them that Alice and Bob prefer the same order of speech, and (iii) it is possible that Alice and Bob prefer different orders of speech which lead to different consensus values of f .

	The result of Theorem 2 is not due to the fact that propositions (1) and (2) or (1) and (3)
	or (2) and (3) are never true simultaneously.
	Proposition 3 (i) Propositions (1) and (2) of Theorem 2 can be true simultaneously.
	(ii) Propositions (1) and (3) of Theorem 2 can be true simultaneously.
	(iii) Propositions (2) and (3) of Theorem 2 can be true simultaneously.

Maybe because it is less intuitive, it is also pretty tedious (there are 288 states of the world), so it is available from the authors upon request.

We prove point (ii) with the following example, which shows that it is possible that both agents prefer the same order of speech.

Example 3

The set of states of the world be Ω = {1, 2, 3, 4, 5, 6, 7, 8, 9} and Alice and Bob are endowed with a uniform prior P on Ω. They communicate in turn the private value of the function f (.) = P ({1, 6, 7, 9} | .) and are endowed with the following partitions:

If Alice speaks first (protocol α), individual partitions at consensus are:

If Bob speaks first (protocol β), individual partitions at consensus are:

At every state of the world, Alice and Bob prefer that Alice speaks first: B A (α, β) = B B (α, β) = Ω, hence it is common knowledge at any state that both prefer the order α.

Finally, we prove point (iii) with Example 1 in section 2. The partition of common knowledge is M = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13}. At state 1, Alice and Bob prefer to speak second, and f (Π α (1)) = 1/3 = f (Π β (1) = 1/2. However, this is not common knowledge,

for Bob prefers to speak first at states 6 and 8.

By Proposition 1 of Parikh and Krasucki [1990], ∀ i, j, f (Π α i (ω)) = f (Π α j (ω)) for all ω. By definition of the meet, it implies that

The same reasoning applied to Π β j (ω) boils down to f (Π α (ω) ∩ Π β (ω)) = f (Π β (ω)). Hence f (Π α (ω)) = f (Π α (ω))


Julien Paccou
email: julienpaccou@yahoo.fr

Laetitia Michou³

Sami Kolta

Françoise Debiais

Bernrard Cortet

Pascal Guggenbuhl

Les hyperdensités osseuses de l'adulte

Keywords: Densité minérale osseuse, hyperdensité osseuse, ostéocondensation, ostéopétrose, sclérostine, LRP5

Lors de la mesure de densité minérale osseuse (DMO) par DXA (Dual-energy X-ray absorptiometry), il n'est pas rare de retrouver des valeurs augmentées témoignant d'une hyperdensité osseuse. La définition de l'hyperdensité osseuse n'est pas consensuelle (T-scores et/ou Z-scores ≥ +2,5 ou +4 selon les auteurs). Une grande variété de pathologies est associée avec une hyperdensité osseuse avec une classification habituellement selon le caractère localisé ou généralisé ou encore selon le caractère acquis ou constitutionnel de cette hyperdensité osseuse. Une interprétation attentive du compte-rendu et des images de DMO permet dans plus de la moitié des cas de retrouver une étiologie artéfactuelle (arthrose rachidienne, calcifications vasculaires, syndesmophytose…) ou localisée (métastases condensantes, localisation pagétique…). Les causes d'hyperdensité osseuse diffuses et acquises sont nombreuses et variées (fluorose, ostéosclérose diffuse liée à l'ostéodystrophie rénale, hémopathies, hépatite C…) et nécessitent parfois des investigations cliniques et paracliniques supplémentaires pour parvenir à un diagnostic (imagerie, tryptase sérique, sérologie hépatite C…). Enfin, il existe de nombreuses maladies génétiques entraînant une hyperdensité osseuse. Les principales sont les ostéopétroses et les maladies liées à une mutation du gène SOST (sclérostéose et maladie de van Buchem) ou du corécepteur de Wnt, LRP5 (LDL-R-Related Protein 5).

A c c e p t e d M a n u s c r i p t

Introduction

Une hyperdensité osseuse ou ostéocondensation peut être suspectée devant une ostéocondensation sur des radiographies standard mais la densité minérale osseuse (DMO) par absorptiométrie biphotonique à rayons X (DXA) est beaucoup plus précise pour confirmer ce diagnostic en quantifiant cette augmentation de la DMO. En pratique, il n'est pas exceptionnel de trouver une hyperdensité osseuse lors de la mesure de DMO par DXA. S'il s'agit le plus souvent d'une origine artéfactuelle au rachis chez le sujet âgé (arthrose ± associée à une scoliose), il ne faut pas méconnaitre les causes locales ou généralisées (acquises ou génétiques) d'hyperdensité osseuse. Enfin, un certain nombre d'hyperdensités osseuses reste actuellement inexpliqué même si la génétique a permis d'identifier certaines maladies monogéniques (mutations des gènes codant pour la protéine SOST ou encore pour le corécepteur LRP5) responsables de ces hyperdensités osseuses (High Bone Mass pour les anglo-saxons) [1-3].

Définition et prévalence de l'hyperdensité osseuse

Causes artéfactuelles et localisées

Les hyperdensités osseuses diffuses et acquises

Les hyperdensités osseuses génétiques

Conclusion

La

A

  c c e p t e d M a n u s c r i p t proposés. En 2005, Michael Whyte [4] a proposé la définition suivante : Z-score > +2,5. Jusqu'alors, les cas cliniques et les séries de cas publiés, qui correspondent à la majorité des publications sur ce sujet, ont utilisé des seuils différents pour définir l'hyperdensité osseuse. Récemment, une étude britannique [5] regroupant 15 centres, avec un total de 335 115 DMO, a évalué la prévalence de l'hyperdensité osseuse en proposant comme seuil diagnostique un T-score et/ou un Z-score ≥ +4 que ce soit au rachis et/ou à la hanche. Avec cette définition, les auteurs de cette étude ont trouvé une prévalence de l'hyperdensité osseuse de 5/1000 [5]. La moitié des cas d'hyperdensité osseuse était liée à une arthrose rachidienne. Après avoir éliminé les causes évidentes d'hyperdensité osseuse en utilisant le seuil de +4 (arthrose, calcifications vasculaires, hyperostose, syndesmophytose…), ils ont au sein de la population présélectionnée utilisé des seuils différents pour retenir le diagnostic d'hyperdensité osseuse diffuse et inexpliquée : (a) L1 Z-score ≥ +3,2 et hanche totale Z-score ≥ +1,2 OU (b) hanche totale Z-score ≥ +3,2 et L1 Z-score ≥ +1,2 [5]. Le seuil de +3,2 a été utilisé avec comme rationnel la distinction possible entre remaniements artéfactuels et hyperdensités osseuses « vraies » grâce à ce seuil. La vertèbre L1 a été choisie car moins exposée aux remaniements dégénératifs que les vertèbres sous-jacentes. Une densité élevée était nécessaire à la fois à la hanche totale et au rachis pour ne retenir que les cas d'hyperdensités osseuses diffuses. La nécessité d'avoir des valeurs densitométriques élevées à la fois au rachis et à la hanche est vraisemblablement la démarche à adopter pour le diagnostic des hyperdensités osseuses diffuses permettant ainsi de s'affranchir des seules valeurs élevées au rachis en rapport avec des remaniements dégénératifs. Un Z-score ≥ +2 à la fois au rachis et à la hanche nous semble être la valeur seuil la plus appropriée mais nécessite d'être validée au cours de futures études.

A c c e p t e d M a n u s c r i p t 5 La

 5 lecture du compte-rendu de DMO associée aux images permet dans un bon nombre de cas de retrouver une étiologie artéfactuelle à l'hyperdensité osseuse que ce soit en raison de remaniements dégénératifs (arthrose rachidienne ± scoliose), de calcifications vasculaires (aorte abdominale) ou encore en lien avec des syndesmophytes, une hyperostose vertébrale ou du matériel (prothèse vasculaire, vertébroplastie…) [6,7]. Dans l'étude britannique multicentrique, environ 50 % des cas d'hyperdensité osseuse étaient en rapport avec des remaniements dégénératifs [5]. D'autres causes localisées d'hyperdensité osseuse sont à évoquer en fonction du contexte clinique en cas de métastases condensantes (cancers de la prostate, du sein…), d'une maladie de Paget ou encore d'une dysplasie fibreuse [8-10]. Les différentes causes artéfactuelles et localisées sont résumées dans le tableau 1.

  Les causes d'hyperdensité osseuse diffuses et acquises sont nombreuses et variées (tableau 1). Elles peuvent être liées à des anomalies nutritionnelles comme les fluoroses (consommation excessive de thé, de pâte dentifrice, ingestion chronique d'eau de Vichy Saint Yorre, prise de fluor à visée thérapeutique autrefois…) [11], métaboliques comme l'ostéosclérose diffuse à prédominance axiale en cas d'ostéodystrophie rénale avec parfois un aspect typique en maillot de rugby (The rugger jersey spine sign) sur les vertèbres [12,13], endocriniennes comme dans l'hypoparathyroïdie chronique et les pseudo-hypoparathyroïdies [14], hématologiques comme pour la mastocytose et les syndromes myéloprolifératifs (figure 1) [15,16], néoplasiques pour les leucémies, les lymphomes, les ostéocancéroses diffuses (atteinte osseuse métastatique diffuse) de néoplasme solide et les exceptionnels myélomes condensants [17-20] ou encore infectieuses comme dans l'ostéosclérose diffuse liée à l'hépatite C [21,22]. Il y a d'autres causes exceptionnelles d'hyperdensités osseuses diffuses et acquises comme l'ostéomalacie axiale (~20 cas rapportés) [23,24]. Enfin, pour certains auteurs, l'obésité morbide (IMC ≥40) A c c e p t e d M a n u s c r i p t 6 pourrait être en cause dans certains cas d'hyperdensité osseuse [25]. En effet, l'obésité surestime la DMO et certains cas d'hyperdensité osseuse pourraient être expliqués, au moins en partie, par l'obésité morbide.

5 . 2

 52 Il existe différentes maladies génétiques entraînant une hyperdensité osseuse. Les principales sont les ostéopétroses et les maladies liées à une mutation du gène SOST ou du corécepteur de Wnt, LRP5 (LDL-R-Related Protein 5). Cependant, l'atteinte osseuse (quantitative et qualitative) de ces différentes maladies a un retentissement variable sur la résistance osseuse et le risque fracturaire alors que la DMO est en général très élevée (Z-scores pouvant dépasser les +6 à la hanche et au rachis lombaire). La distinction entre ces différentes maladies peut se faire selon le risque fracturaire (augmenté ou diminué) mais aussi en fonction des perturbations du remodelage osseux (distinction parfois artificielle par ailleurs) [3,9]. Il y a également certaines pathologies condensantes associées à la fois avec une perturbation de la formation et de la résorption osseuse comme (i) la maladie de Camurati-Engelmann (plus de 300 cas rapportés) caractérisée par une hyperostose des os longs, du crâne, de la colonne vertébrale et du pelvis (mutations du gène TGFß1) [26] et (ii) la dysplasie hématodiaphysaire de Ghosal (DHDG, prévalence <1/1 000 000, mutations du gène TBXAS1 codant pour la thromboxane synthase) caractérisée par une hyperdensité osseuse et une anémie arégénérative corticosensible [27]. La DHDG a été décrite dans plusieurs familles non apparentées d'origine indienne et du Moyen-Orient. On citera également (iii) l'ostéomésopycnose extrêmement rare (moins de 5 familles ont été rapportées, principalement en France) qui se manifeste par une sclérose en aires de l'axe squelettique (diagnostic A c c e p t e d M a n u s c r i p t 7 différentiel avec la maladie d'Albers-Schönberg) et dont le mécanisme génétique demeure inconnu [28] et (iv) l'ostéopathie striée -sclérose crânienne (OSSC) liée à l'X [29,30]. L'OSSC est caractérisée par des striations longitudinales des métaphyses des os longs, une sclérose des os crânio-faciaux, une macrocéphalie, une fente palatine et une perte auditive (moins de 100 cas rapportés). L'OSSC est liée à des mutations sur le gène WT2 du chromosome X (AMER1), un répresseur de la signalisation Wnt (voie ß-caténine impliquée dans le contrôle des gènes ciblés dans le noyau) [29].5.1 Diminution de la résorption osseuse : les ostéopétrosesLes ostéopétroses correspondent à un phénotype partagé par de nombreuses anomalies génétiques (perte de fonction de CLCN7, de PLEHKM1, de RANKL, de TCIGR1…) (tableau 2) ce qui permet d'expliquer les présentations cliniques extra-osseuses très variables (troubles de la croissance, perturbations hématologiques, troubles de l'audition, de la vision, de la dentition, compressions neurologiques, diminution de l'espérance de vie…) [31,32]. 5.1.1. Données cliniques, épidémiologiques et génétiques Les types d'ostéopétrose sont en général classés en fonction de leur sévérité clinique. Cela va donc des formes sévères ou malignes (« formes infantiles » parfois d'évolution fatale) à des formes modérées (« formes intermédiaires ») voire asymptomatiques ou bénignes (« formes adultes »). Il y a une vaste hétérogénéité de présentation clinique de ces maladies parfois même au sein de la même famille. La présence de signes extra-osseux tels qu'une atteinte neurodégénérative, un retard mental, une atteinte cutanée, des anomalies de l'immunité, une acidose tubulaire rénale doit être recherchée car ces signes orientent vers des formes cliniques particulières d'ostéopétrose (tableau 2) [31]. La plupart des gènes en cause dans l'ostéopétrose codent pour des protéines impliquées dans le contrôle du pH intra et A c c e p t e d M a n u s c r i p t 8 extracellulaire de l'ostéoclaste, jouant un rôle déterminant dans le processus d'acidification des lacunes de résorption [9]. La génétique permet de classer les ostéopétroses en 3 catégories : (i) la forme autosomique récessive ou Autosomal Recessive Osteopetrosis (ARO) (incidence dans la population générale de 1 pour 250 000) qui est la forme la plus sévère avec quelques formes de sévérité intermédiaire (IRO) beaucoup plus rares [9], le type 1 est la plus fréquente des formes malignes (environ 50 % des cas) (tableau 2), (ii) la forme autosomique dominante ou Autosomal Dominant Osteopetrosis (ADO) qui est la plus fréquente (incidence dans la population générale de 5 pour 100 000) [33], (iii) la forme liée à l'X ou X-linked Osteopetrosis (XLO) avec seulement quelques cas décrits dans la littérature associés à une dysplasie ectodermique avec lymphoedème et immunodépression [34,35].5.1.2. Les ostéopétroses de transmission autosomique dominanteLorsque le diagnostic est porté à l'âge adulte, il s'agit des ostéopétroses de transmission autosomique dominante (ADO) historiquement divisées en type 1 (ADO1) et type 2 (ADO2).Il s'est avéré que l'ostéopétrose de type 1 (environ 30 cas décrits dans la littérature) était liée à une mutation gain-de-fonction du gène du corécepteur LRP5 et n'était donc pas liée à une diminution de la résorption osseuse. Cette maladie n'est pas associée à une augmentation du risque fracturaire et il est légitime de ne plus la considérer comme une ostéopétrose[START_REF] Del Fattore | Genetics, pathogenesis and complications of osteopetrosis[END_REF].L'ostéopétrose de type 2 (ou maladie d'Albers-Schönberg) est l'ostéopétrose la plus fréquente chez l'adulte. Elle est causée par une mutation de CLCN7 (codant pour une protéine de transport du chlore vers la lacune de résorption ostéoclastique) avec une pénétrance variable et un spectre de présentation clinique étendu allant de la découverte fortuite d'anomalies radiographiques chez un patient asymptomatique jusqu'à des formes létales en passant par des A c c e p t e d M a n u s c r i p t 9 patients présentant des fractures multiples [36]. Cette ostéopétrose de type 2 est caractérisée radiologiquement par l'épaississement des plateaux vertébraux avec l'aspect classique de colonne vertébrale en « maillot de rugby » ou « vertèbres sandwich » et du bassin « d'os dans l'os » (figure 2). 5.1.3. Un exemple d'ostéopétrose de transmission autosomique récessive : le déficit en anhydrase carbonique de type 2 Le déficit en anhydrase carbonique de type 2 associe des anomalies dentaires, une ostéopétrose, des calcifications des noyaux gris centraux ainsi qu'une acidose tubulaire d'origine rénale [37]. Les présentations cliniques sont très variables avec parfois des fractures des os longs périphériques. Il n'y a pas toujours de retentissement sur la survie des patients atteints de ce type d'ostéopétrose [38]. Augmentation de la formation osseuse 5.2.1. Sclérostéose et maladie de van Buchem Il s'agit de maladies rares dont les premières descriptions remontent aux années 50. Le phénotype de ces 2 maladies est assez similaire mais avec quelques différences sur lesquelles nous reviendrons [39]. Il s'agit de maladies à transmission autosomique récessive. Dans la sclérostéose, il existe une mutation dans le gène codant pour la sclérostine (protéine SOST, inhibitrice de la formation osseuse) entraînant ainsi une perte de fonction alors que dans la maladie de van Buchem, il ne s'agit pas d'une mutation ponctuelle intra-génique mais d'une délétion homozygote de 52 kb en aval du gène codant pour la protéine SOST [40,41]. Il y a un peu moins de 100 cas décrits de sclérostéose (majoritairement population Afrikaner descendant de colons néerlandais) et une trentaine de cas de maladie de van Buchem

A c c e p t e d M a n u s c r i p t 10 ( 6 .

 106 essentiellement localisés dans un village dans le nord des Pays-Bas) [39]. Ces 2 maladies sont l'illustration parfaite que la découverte de maladies monogéniques permet une meilleure compréhension des mécanismes de régulation du métabolisme osseux avec la possibilité à terme de développer de nouvelles voies thérapeutiques [42]. La condensation osseuse est généralisée à l'ensemble du squelette. Elles sont caractérisées par une augmentation de la formation osseuse, une augmentation de la résistance osseuse et une nette diminution du risque fracturaire [39]. Les DMO de ces patients sont particulièrement élevées : Z-scores au rachis lombaire entre +5,4 et +14,4 et Z-scores au col fémoral entre +5,2 et +12,1 [39,43].L'hypertrophie mandibulaire patente qui apparait au cours de la puberté en est un signe caractéristique tout comme l'hypertrophie du crâne pouvant entrainer des compressions neurologiques[START_REF] Van Hul | Van Buchem disease (hyperostosis corticalis generalisata) maps to chromosome 17q12-q21[END_REF]. Les manifestations de la maladie de van Buchem sont considérées comme moins graves que la sclérostéose (Figure3). De plus, la sclérostéose se caractérise par une grande taille et la présence de syndactylie ce qui n'est pas le cas dans la maladie de van Buchem [39].5.2.2. Mutations gain-de-fonction de LRP5Il s'agit de mutations des gènes codant pour les protéines LRP5 entraînant un gain-defonction. LRP5 est un corécepteur de la voie de signalisation Wnt impliqué dans la formation osseuse[START_REF] Boyden | High bone density due to a mutation in LDLreceptor-related protein 5[END_REF]. Les mutations de ce gène entravent l'action de SOST et conduisent à des anomalies phénotypiques proches de celles retrouvées au cours de la sclérostéose. L'hypertrophie mandibulaire est caractéristique, une hypertrophie du crâne et des complications neurologiques ont également été décrites [45]. Dans ces maladies monogéniques, les mesures répétées de la densité osseuse permettent d'évaluer l'évolution de la maladie ostéocondensante vers une éventuelle aggravation dans le temps [46,47]. Le syndrome de Worth est dû à une mutation du gène LRP5 (transmission autosomiqueA c c e p t e d M a n u s c r i p t 11 dominante, description de moins de 10 familles). Les anomalies cranio-faciales se développent durant l'adolescence : front proéminent, hypertrophie mandibulaire, racine du nez élargie, torus palatin [48]. A contrario, les mutations inactivatrices de LRP5 sont en cause dans le syndrome ostéoporose-pseudogliome (prévalence estimée à un cas pour 2 000 000), maladie très rare qui a été à l'origine de la découverte de ce système fondamental dans la formation osseuse [49]. Des mutations inactivatrices du gène LRP4 entraînant également une hyperdensité osseuse ont également été décrites. Ce corécepteur est un facilitateur de l'action de la sclérostine [50,51]. 5.3 Autres maladies génétiques associées avec une hyperdensité osseuse Parmi les autres anomalies génétiques entraînant une hyperdensité osseuse par diminution de la résorption osseuse, on peut citer la pycnodysostose avec plus de 100 cas rapportés depuis 1962 (maladie génétique lysosomale de transmission autosomique récessive caractérisée par une ostéosclérose du squelette, une petite taille et une fragilité osseuse due à une mutation du gène codant pour la cathepsine K) [52]. Citons enfin quelques cas rares d'hyperdensités osseuses génétiques non diffuses comme la mélorhéostose et l'ostéopoecilie. La mélorhéostose est caractérisée par une atteinte monomélique avec ostéocondensation linéaire en « coulée de bougie » (mutation du gène LEMD3 uniquement en cas d'association avec l'ostéopoecilie) [53] et l'ostéopoecilie (caractère bénin de cette affection, de transmission autosomique dominante, mutation du gène LEMD3) est habituellement de découverte fortuite devant des petits foyers bien définis, arrondis ou ovales, d'ostéocondensation [54]. 5.4. Les hyperdensités osseuses actuellement inexpliquées (unexplained high bone mass) A c c e p t e d M a n u s c r i p t patients ayant une hyperdensité osseuse diffuse actuellement inexpliquée mais avec parfois des caractéristiques cliniques suggérant une dysplasie osseuse (hypertrophie de la mandibule, ossifications à l'insertion des tendons et des ligaments, augmentation de l'IMC et de la pointure…). Il s'agit vraisemblablement d'hyperdensités osseuses liées à des atteintes monogéniques ou polygéniques pour le moment inconnues. Des études sur la microarchitecture osseuse de cette population bien particulière ont été réalisées tout comme des études sur les modifications de la composition corporelle et du lien entre hyperdensité osseuse et arthrose [55-57]. Conduite à tenir devant la découverte d'une hyperdensité osseuse L'analyse de l'ostéodensitométrie doit permettre de distinguer les hyperdensités osseuses locales ou généralisées (fémur total, rachis lombaire ou les deux). En cas de cause locale, on doit rechercher une atteinte dégénérative évidente (notamment au rachis) qui ne nécessiterait pas de réaliser d'autres explorations. En dehors de cette situation, nous recommandons dans un premier temps de réaliser des radiographies standard soit du rachis lombaire face et profil ± du bassin de face (si hyperdensité osseuse localisée du rachis) soit des hanches face et profil ± du bassin de face (si hyperdensité osseuse localisée de la hanche). Dans un second temps, d'autres examens d'imagerie peuvent être demandées pour le diagnostic (scanner et/ou IRM et/ou scintigraphie osseuse) associés à une calcémie, une albuminémie et une CRP tout particulièrement en cas de doute sur un processus néoplasique. Les causes généralisées affectant à la fois le rachis et la hanche sont plus rares mais les diagnostics à évoquer sont plus nombreux. Cela nécessite un interrogatoire et un examen clinique approfondis recherchant des facteurs de risque d'hépatite C, des apports excessifs en fluor, des douleurs osseuses ou une atteinte des nerfs crâniens (par un mécanisme compressif), A c c e p t e d M a n u s c r i p t 13 une splénomégalie, une altération de l'état général, des lésions cutanées (en faveur d'une mastocytose systémique), des antécédents familiaux génétiques ou encore un dysmorphisme (tout particulièrement de la face). La recherche d'antécédents de fracture à la fois personnels et familiaux est également indispensable. La mesure de la taille est indispensable. Nous recommandons les explorations biologiques suivantes en l'absence de cause évidente : bilan hépatique et sérologie de l'hépatite C, numération formule sanguine, électrophorèse des protéines sériques, fonction rénale et exploration du métabolisme phosphocalcique avec parathormonémie et 25-OH-vitamine D. Les dosages du fluor sanguin, de la tryptase sérique et des marqueurs du remodelage osseux sont également nécessaires. Concernant l'imagerie, nous recommandons en plus des radiographies du rachis dorsolombaire, des fémurs et du bassin, des radiographies du crâne de face et de profil. Si certains éléments cliniques et paracliniques orientent vers une atteinte monogénique en faveur d'une ostéopétrose ou d'une anomalie de la formation osseuse, nous recommandons d'adresser le patient vers une consultation de génétique. Les investigations à réaliser en génétique dépendront de la sévérité de l'atteinte et de l'enquête familiale. Les orientations diagnostiques en cas de découverte d'une hyperdensité osseuse sont résumées dans la figure 4. Dans ces maladies monogéniques, les mesures répétées de densité osseuse permettent d'évaluer l'évolution de la maladie ostéocondensante vers une éventuelle aggravation au cours du temps. Des investigations et un suivi sont nécessaires chez ces patients concernant par un mécanisme compressif (paralysie faciale) avec surveillance auditive et visuelle notamment (audiogramme, champ visuel, scanner du crâne…) [39]. Chez l'adulte, il faut être particulièrement vigilant concernant les céphalées récurrentes évocatrices d'une hypertension intracrânienne survenant majoritairement à l'âge adulte au cours de la A c c e p t e d M a n u s c r i p t 14 sclérostéose et responsable de morts subites ou de complications post-opératoires en cas de décompression chirurgicale [58].

  Dysplasie ectodermique anhydrotique, déficit immunitaire, lymphoedème, anomalies dentaires, infections NEMO TCIRG1: T-cell, immuneregulator1, ATPase, H+ transporting, lysosomale V0 subunit A3 ; RANKL ou TNFSF11: tumor necrosis factor (ligand) superfamily, member 11 ; CAII: carbonic anhydrase II ; CLCN7: chloride channel7 ; OSTM1: osteopetrosis associated transmembrane protein1 ; PLEKHM1: Pleckstrin homology domain containing, family M (with RUN domain) member1 ; RANK ou TNFRSF11A: tumor necrosis factor receptor superfamily, member 11a, NFKB activator ; LRP5: low density lipoprotein receptor-related protein 5 ; NEMO ou IkBkG: inhibitor of kappa light polypeptide gene enhancer in B-cells, kinase gamma.

Figure 3 :

 3 Figure 3 : photographies de patients atteints de la maladie de van Buchem présentant des caractéristiques cliniques typiques avec élargissement de la mandibule et une hypertrophie des os du crâne. Le patient C présente une compression neurologique (paralysie faciale). Adapté de [40]

Figure 2 :

 2 Figure 2 : Aspect typique d'ostéopétrose autosomique dominante de type 2 ou maladie d'Albers-Schönberg, vertèbres en « maillot de rugby » aussi appelées « vertèbres sandwich » et aspect « d'os dans l'os » au bassin. A noter chez ce patient la survenue d'une fracture de hanche à droite avec remplacement prothétique dans ce cadre pathologique.

  découverte d'une hyperdensité osseuse lors de la réalisation d'une DMO n'est pas exceptionnelle. L'analyse de l'ostéodensitométrie (plus ou moins associées à des radiographies) permet dans un grand nombre de cas de retrouver une origine artéfactuelle à cette hyperdensité osseuse. Les autres situations nécessitent, après un interrogatoire et un examen clinique soigneux, de réaliser des explorations complémentaires (imagerie et

	Arthrose rachidienne (zygapophysaire) ± scoliose associée
	Spondylarthrite ankylosante (syndesmophytes)
	Hyperostose vertébrale engainante (maladie de Forestier)
	Calcification vasculaire (aorte abdominale)
	Cimentoplastie
	Causes acquises (locales)
	Vertèbre(s) ivoire(s) : maladie de Paget, SAPHO, sarcoïdose, lymphome…
	M a n u s c r i p t M a n u s c r i p t M a n u s c r i p t M a n u s c r i p t M a n u s c r i p t Métastases osseuses condensantes (prostate, sein…) Syndrome POEMS Causes acquises (diffuses) Fluorose Ostéodystrophie rénale Acromégalie Mastocytose Myélofibrose Hépatite C Myélome condensant Ostéomalacie axiale a n u s c r i p t Obésité morbide M Tableau 2 : classification des principales formes humaines d'ostéopétrose
	les explorations complémentaires doivent permettre d'évoquer certaines maladies génétiques responsables d'une diminution de la résorption osseuse (ostéopétrose autosomique dominante de type 2) ou d'une augmentation de la formation osseuse (mutations inactivatrices de SOST et mutations gain-de-fonction du corécepteur LRP5). Enfin, un certain nombre d'hyperdensités osseuses reste inexpliqué, il s'agit des syndromes de masse osseuse élevée inexpliqués. A c c e p t e d A c c e p t e d A c c e p t e d A c c e p t e d A c c e p t e d Type Phénotype Gènes impliqués Ostéopétrose autosomique récessive 1 2 3 4 6 7 -Forme maligne néonatale ou infantile, ostéocondensation, fractures, pancytopénie, infections, hépatosplénomégalie, atteinte neurologique -Forme intermédiaire, pauvre en ostéoclastes, petite taille, fractures -Forme intermédiaire associée à une acidose tubulaire rénale, petite taille, retard mental ou forme intermédiaire -Forme infantile maligne, fractures, atteinte médullaire TCIRG1 RANKL CAII CLCN7 OSTM1 PLEKHM1 A c c e p t e d 5 -Forme infantile maligne, fractures, atteinte médullaire, RANK
		-Forme de gravité variable, volontiers intermédiaire
	-Forme sévère pauvre en ostéoclastes associée à une Les auteurs ne déclarent aucun conflit d'intérêt hypogammaglobulinémie
	Ostéopétrose
	autosomique	Ostéocondensation généralisée plus marquée à la voûte	LRP5
	dominante	crânienne, souvent
	1	asymptomatique, douleurs, hypoacousie, pas de	CLCN7
		fractures
	2	Aspect de vertèbres sandwich, aspect d'os dans l'os,
			21

biologie). Ceux-ci permettent le diagnostic des causes acquises d'hyperdensité osseuse qu'elles soient locales ou généralisées. Il est fondamental de rechercher des examens radiologiques antérieurs qui permettent parfois de trancher entre cause génétique ou acquise d'hyperdensité osseuse. Bien entendu, la présentation clinique, le phénotype des patients et/ou

Causes locales

(métastases condensantes, localisation pagétique…)

Hyperdensités osseuses diffuses

Causes acquises

(fluorose, ostéosclérose diffuse liée à l'ostéodystrophie rénale, hémopathies, hépatite C…)

Causes génétiques

Résorption osseuse

Les ostéopétroses

Formation osseuse Sclérostéose, van Buchem et LRP5

Hyperdensités osseuses actuellement inexpliquées