D Cébron
email: cebron@irphe.univ-mrs.fr

M Le Bars

J Leontini

P Maubert

P Le Gal

A systematic numerical study of the tidal instability in a rotating triaxial ellipsoid

Keywords: tides, tidal/elliptical instability, planetary cores, finite element numerical simulations

The full non-linear evolution of the tidal instability is studied numerically in an ellipsoidal fluid domain relevant for planetary cores applications. Our numerical model, based on a finite element method, is first validated by reproducing some known analytical results. This model is then used to address open questions that were up to now inaccessible using theoretical and experimental approaches. Growth rates and mode selection of the instability are systematically studied as a function of the aspect ratio of the ellipsoid and as a function of the inclination of the rotation axis compared to the deformation plane. We also quantify the saturation amplitude of the flow driven by the instability and calculate the viscous dissipation that it causes. This tidal dissipation can be of major importance for some geophysical situations and we thus derive general scaling laws which are applied to typical planetary cores.

Introduction

Tides are large scale deformations induced by gravitational interactions that affect all the layers of a given planet or star. They play an important role in geo-and astrophysics and so, many studies are devoted to this subject. Their most obvious phenomena are of course the oceanic flows on Earth, but tides are also responsible for the intense volcanism on Io for instance. In stars but also in liquid planetary cores, tidal forcing induces an elliptical deformation of the rotating streamlines that may excite a parametric resonance of inertial waves called the elliptical or tidal instability. This instability could, for instance, be responsible for the surprising magnetic field in Io [START_REF] Kerswell | Tidal instability as the source for Io's magnetic signature[END_REF][START_REF] Lacaze | Magnetic field induced by elliptical instability in a rotating spheroid[END_REF][START_REF] Herreman | On the effects of an imposed magnetic field on the elliptical instability in rotating spheroids[END_REF] and for fluctuations in the Earth's magnetic field on a typical timescale of 10,000 years [START_REF] Aldridge | Elliptical instability of the Earth's fluid core[END_REF]. It may also have a significant influence on the orbital evolution of binary stars [START_REF] Rieutord | Evolution of rotation in binaries: physical processes[END_REF] and moon-planet systems [START_REF] Bars | Tidal instability in stellar and planetary binary system[END_REF].

As described in the review of [START_REF] Kerswell | Elliptical instability[END_REF], the elliptical instability is a three-dimensional instability which may grow as soon as a rotating fluid possesses elliptical streamlines. As previously stated, it is due to a triadic parametric resonance between two inertial waves of the rotating fluid [START_REF] Kelvin | Vibrations of a columnar vortex[END_REF] and the underlying strain field responsible for the elliptic deformation [START_REF] Bayly | Three-dimensional instability of elliptical flow[END_REF][START_REF] Waleffe | On the three-dimensional instability of strained vortices[END_REF]. Such an instability has been found in many different contexts, including those where the strain field is due to vortex interactions or to elliptically deformed boundaries. It has thus been the focus of numerous theoretical and experimental studies, devoted to the dynamics of two-dimensional turbulent flows (e.g. [START_REF] Widnall | The instability of short waves on a vortex ring[END_REF][START_REF] Moore | The instability of a straight vortex filament in a strain field[END_REF] or [START_REF] Haj-Hariri | Three-dimensional instability of viscoelastic elliptic vortices[END_REF] for a viscoelastic fluid), to the stability of wakes [START_REF] Leweke | Three-dimensional instabilities in wake transition[END_REF], to the dynamics of vortex pairs [START_REF] Leweke | Cooperative elliptic instability of a vortex pair[END_REF][START_REF] Dizès | Theoretical predictions for the elliptic instability in a two-vortex flow[END_REF][START_REF] Meunier | A merging criterion for two-dimensional co-rotating vortices[END_REF], and to the flow inside deformed rotating cylinders [START_REF] Gledzer | An experimental and theoretical study of the stability of motion of a liquid in an elliptical cylinder[END_REF][START_REF] Malkus | An experimental study of global instabilities due to tidal (elliptical) distortion of a rotating elastic cylinder[END_REF][START_REF] Eloy | Experimental study of the multipolar vortex instability[END_REF][START_REF] Eloy | Elliptic and triangular instabilities in rotating cylinders[END_REF], ellipsoids [START_REF] Gledzer | Finite dimensional approximation of the motions of incompressible fluid in an ellipsoidal cavity[END_REF][START_REF] Kerswell | Tidal excitation of hydromagnetic waves and their damping in the Earth[END_REF][START_REF] Lacaze | Elliptical instability in a rotating spheroid[END_REF] and shells [START_REF] Aldridge | Elliptical instability of the Earth's fluid core[END_REF][START_REF] Seyed-Mahmoud | Elliptical instability in rotating spherical fluid shells: application to Earth's fluid core[END_REF][START_REF] Lacaze | Elliptical instability of the flow in a rotating shell[END_REF]. More recently, magnetohydrodynamical effects of the elliptical instability have been investigated, with applications in MHD turbulence [START_REF] Thess | Transition from two-dimensional to three-dimensional magnetohydrodynamic turbulence[END_REF] or in induction processes in planets [START_REF] Lacaze | Magnetic field induced by elliptical instability in a rotating spheroid[END_REF][START_REF] Herreman | On the effects of an imposed magnetic field on the elliptical instability in rotating spheroids[END_REF].

From a numerical point of view, most studies have been devoted to the dynamics of deformed two-dimensional vortices in three-dimensional domains [e.g. [START_REF] Lundgren | Transition to turbulence in an elliptic vortex[END_REF][START_REF] Sipp | Elliptic instability in two-dimensional flattened Taylor-Green vortices[END_REF][START_REF] Lacaze | Elliptic instability in a strained Batchelor vortex[END_REF][START_REF] Roy | Stability of a pair of co-rotating vortices with axial flow[END_REF]. For instance, the seminal paper of [START_REF] Pierrehumbert | Universal short-wave instability of two-dimensional eddies in an inviscid fluid[END_REF] considered the growth of a perturbation on the planar velocity field associated with an elliptical vortex inside a box with zero normalflow boundary conditions, assuming periodicity in the axial direction; the linear eigenvalue problem was then solved by spectral methods, and the growth rate of the elliptical instability was found. Numerical studies in rotating containers are less numerous. [START_REF] Mason | Nonlinear evolution of the elliptical instability: an example of inertial wave breakdown[END_REF] used a non-orthogonal elliptico-polar coordinate system in order to solve the flow in an elliptically deformed cylinder by spectral methods; the non-linear temporal evolution of two different modes of the elliptical instability was calculated with no-slip boundary conditions on the sidewalls and stress-free conditions on the top and the bottom of the cylinder. [START_REF] Seyed-Mahmoud | A numerical model for elliptical instability of the Earth's fluid outer core[END_REF] determined numerically the frequencies and the growth rates of the instability in both ellipsoid and ellipsoidal shell geometries using a linear Galerkin method, projecting the flow on a selected number of inertial waves. Finally, [START_REF] Shangli | Further evidence for an elliptical instability in rotating fluid bars and ellipsoidal stars[END_REF] studied the stability of self-gravitating compressible ellipsoidal fluid configurations and pointed out the occurence of the elliptical instability. However, to the best of our knowledge, the full non-linear evolution of the tidal instability in an ellipsoidal geometry has not yet been simulated numerically. This is the purpose of the present work, which aims at completing our previous theoretical and experimental investigations [START_REF] Lacaze | Elliptical instability in a rotating spheroid[END_REF][START_REF] Lacaze | Magnetic field induced by elliptical instability in a rotating spheroid[END_REF][START_REF] Herreman | On the effects of an imposed magnetic field on the elliptical instability in rotating spheroids[END_REF][START_REF] Bars | Tidal instability in stellar and planetary binary system[END_REF] in accessing global quantities of fundamental importance for planetary applications.

In this paper, we focus on hydrodynamics only, leaving thermal and magnetic field interactions with the tidal instability for further investigations [e.g. 6]. In section 2, the system and the numerical method are presented. The numerical model is validated by comparison with the theoretical results found in the literature, regarding the flow in a rotating sphere submitted to a small and fixed elliptical deformation. In section 3, we then study the influence of three additional complications of fundamental importance for geophysical applications: (i) the influence of the length of the ellipsoid along its rotation axis compared to the mean equatorial radius (i.e. the body oblateness in geophysical terms), (ii) the influence of a background rotation, corresponding to the orbital motion of the companion body responsible for the tidal deformation, and (iii) the influence of the inclination of the rotation axis compared to the deformation plane (i.e. the obliquity). In section 4, we quantify systematically two large-scale quantities relevant for planetary applications, namely (i) the amplitude of the instability at saturation, and (ii) the power dissipated by the instability. General laws in terms of dimensionless numbers are derived and applied to typical planetary cores in section 5.

Numerical model and its validation

Definition of the system and description of the numerical method

The present study takes place in direct continuity of our experimental studies of the elliptical instability in a deformed spheroid [START_REF] Lacaze | Elliptical instability in a rotating spheroid[END_REF][START_REF] Lacaze | Magnetic field induced by elliptical instability in a rotating spheroid[END_REF][START_REF] Herreman | On the effects of an imposed magnetic field on the elliptical instability in rotating spheroids[END_REF][START_REF] Bars | Tidal instability in stellar and planetary binary system[END_REF]. In these experiments, a hollow sphere of radius R, molded in a silicone cylinder, is filled with liquid and set in rotation at a constant angular velocity Ω about its axis (Oz), while it is slightly compressed by a quantity s along the axis (Ox), perpendicular to the rotation axis. The geometry is then a triaxial ellipsoid of axes (a, b, c) = (Rs, R + s, R), with an equatorial ellipticity ε = (b 2 -a 2 )/(a 2 + b 2 ) and a constant tangential velocity along the deformed boundaries, equal to ΩR at the equator. Such a configuration is a model for a liquid planetary core with no solid inner core (e.g. like the jovian moon Io or the early Earth), surrounded by a deformable mantle with a constant tangential velocity. Similarly, our numerical model studies the rotating flow inside an ellipsoid of axes (a, b, c) related to the frame (Ox, Oy, Oz), with a constant tangential velocity all along the boundary in each plane perpendicular to the rotation vector Ω (see the sketch in figure 1). In order to extend our previous experimental study, the length c of the ellipsoid can be chosen independently of the other lengths a and b (with b > a). Moreover, the rotation axis of the ellipsoid can be inclined compared to the c-axis. Note however that in this paper, unless otherwise specified, we choose c equal to the mean equatorial radius R eq = (a + b)/2 and a rotation axis along (Oz), as in the experimental setup. In all simulations, the fluid is initially at rest and we suddenly impose at time t = 0 a constant angular rate Ω such that the tangential velocity along the deformed boundaries in each plane perpendicular to the rotation axis, is equal to Ω (a ′ + b ′ )/2, where a ′ and b ′ are the axes of the elliptic boundary in this plane. In the following, results are non-dimensionalised using the mean equatorial radius R eq as the length scale and Ω -1 as the time scale. Then five dimensionless numbers are used to fully describe the system: the Ekman number E = ν/(Ω R 2 eq ), where ν is the kinematic viscosity of the fluid, the ellipticity ε = (b 2 -a 2 )/(a 2 + b 2 ) of the elliptical deformation, the aspect ratio c/b which quantifies the oblateness of the ellipsoid, and finally the inclination θ and declination φ of the rotation axis. The problem numerically solved is then described by the following system of dimensionless equations:

∂u ∂t + u • ∇u = -∇p + E △ u -2 Ω * c × u, (1)
∇ • u = 0. ( 2
)
where the no-slip boundary conditions are used for the fluid. Note that we work is the reference frame where the ellipsoidal shape is at rest, which is the inertial frame of reference in most of the paper. The Coriolis force -2 Ω * c × u is only used in section 3.2 where the whole triaxial ellipsoid is submitted to a global rotation at Ω * c e z . Usually, numerical studies of planetary cores benefit from their spherical geometry to use fast and precise spectral methods. In our case however, there is no simple symmetry. Our computations are thus performed with a standard finite element method widely used in engineering studies, which allows to deal with complex geometries, such as our triaxial ellipsoid, and to simply impose the boundary conditions. Note that a very efficient finite element method was recently introduced by [START_REF] Chan | An EBE finite element method for simulating nonlinear flows in rotating spheroidal cavities[END_REF], but it is up to now restricted to spheroidal geometries (a = b). Using a commercial software 1 , an unstructured mesh with tetrahedral elements was created. The mesh element type employed is the standard Lagrange element P1 -P2, which is linear for the pressure field but quadratic for the velocity field. Note that no stabilization techniques have been used in this work. We use the so-called Implicit Differential-Algebraic solver (IDA solver), based on backward differencing formulas [START_REF] Hindmarsh | SUNDIALS: Suite of Nonlinear and Differential/Algebraic Equation Solvers[END_REF]. At each time step the system is solved with the sparse direct linear solver PARDISO 2 .

The elliptical instability induces a three-dimensional destabilization of the initial two-dimensional elliptical streamlines. To study its global properties, it thus seems natural to introduce the mean value of the vertical velocity W = 1 V V |w| dτ, with w the dimensionless vertical velocity and V the volume of the ellipsoid. The typical evolution of W as a function of time is shown in figure 2 (a) for E = 1/500 and ε = 0.317. At t = 0, the fluid is at rest in the ellipsoid, and the no-slip condition at the boundary proceeds to set the fluid in rotation. The first peak in W, just after t = 0, is due to the Ekman pumping which appears during the spin-up stage, which typically takes place over the Ekman time t E = E -1/2 Ω -1 , much faster than the viscous time scale

t v = R 2 eq /ν = E -1 Ω -1 [3]
. For instance, in the case shown on figure 2 (a), the dimensionless Ekman time gives Ω t E ≈ 22, which agrees with the numerical results. After this initial stage, the fluid is essentially in solid body rotation. From this state, the exponential growth of the instability can be seen, before an overshoot and a stationary saturation. Having defined the growth rate σ of the elliptical instability as the time constant of the exponential growth, a convergence study on this growth rate is given in the figure 2 (b). The number of degrees of freedom (DoF) used in most of the simulations of this work ranges between 4 • 10 4 DoF and 7 • 10 4 DoF, depending on the ellipticity and the Ekman number, in order to reach a compromise between a good convergence (see figure 2 (b)) and a reasonable CPU time.

Validation of our numerical simulations

A first visual validation is done on the shape of the flow in comparing the experimental visualisation on figure 3 (a) with the numerical simulation on figure 3 (b). As can be seen, we recover the classical S-shape of the spin-over mode, which induces an additional solid body rotation of the fluid around the axis of maximum strain, i.e. perpendicular to the imposed rotation axis (Oz) and at an angle of about 45 o compared to the deformation axis (Ox) [START_REF] Lacaze | Elliptical instability in a rotating spheroid[END_REF]. In order to quantitatively validate the numerical model, the evolution of the growth rate of the instability is compared in figure 4 to the linear theory given in [START_REF] Lacaze | Elliptical instability in a rotating spheroid[END_REF] for small ellipticities :

σ ε = 1 2 -K √ E ε , (3)
where K is a constant equal to K = 2.62 in the limit of small ε ( [START_REF] Hollerbach | Oscillatory internal shear layers in rotating and precessing flows[END_REF][START_REF] Lacaze | Elliptical instability in a rotating spheroid[END_REF]). Note that the second term on the right hand side of the equation 3 corresponds to the viscous damping of the growth rate due to the presence of Ekman layers near boundaries [see [START_REF] Kudlick | On the transient motions in a contained rotating fluid[END_REF][START_REF] Hollerbach | Oscillatory internal shear layers in rotating and precessing flows[END_REF]. The expression (3) allows to define a critical Ekman number for the onset of the instability equal to

E c = ε 2 K 2 . ( 4
)
Close to the threshold, the numerical results closely follow the linear analysis, for values of ellipticity as large as ε ≈ 0.5. Note in particular that all curves for various (ε, E) superimpose providing that σ/ε is expressed as a function of √ E/ε. This is especially interesting in order to apply our results to planets, whose very small E are not directly accessible by our numerical tool, but can be compensated for by large ε in simulations.

Finally, for very large ellipticities (ε > 0.5), figure 4 shows that the growth rate decreases toward zero. The elliptical instability finally disappears for ellipticities greater than a critical value which depends on the Ekman number. The position of the maximum for the variation of σ with the ellipticity is around ε = 0.5, whatever the Ekman number is. This indicates that the decrease is probably due to geometrical effects related to the large value of ε, rather than to any viscous attenuation.

Systematic numerical studies of geophysical complexities

Influence of the length of the ellipsoid along the rotation axis

Having validated our numerical code, we can investigate the influence of the aspect ratio c/b on the instability for a given ellipticity in the equatorial plane and a given rotation rate around the axis (Oz). This is directly related to geoand astro-physical flows as oblateness of planets, like the Earth for instance, is most of the time much stronger than its tidal deformation, meaning that the rotation axis is also the smallest one. The situation is even more pronounced in certain stars, as for instance Regulus A whose diameter is about 32% greater at the equator than the distance between its poles [START_REF] Mcalister | First results from the CHARA array. I. An interferometric and spectroscopic study of the fast rotator α Leonis (Regulus)[END_REF].

Early theoretical work on this aspect was done by [START_REF] Kerswell | Tidal excitation of hydromagnetic waves and their damping in the Earth[END_REF], who considered the inertial wave basis of an oblate spheroid (a = b) and calculated the first 60 subharmonic exact resonances and their growth rate for small ellipticities depending on the value of c [see also 23, for a special application to the case of Io]. An explicit theoretical answer for the growth rate has also been found by [START_REF] Gledzer | Instability of bounded flows with elliptical streamlines[END_REF], starting from the base flow

u b = - a b y e x + b a x e y , (5)
and looking for inviscid perturbations that are linear in space variables, corresponding to the classical spin-over mode.

Here, e x and e y are respectively the unit vectors of (Ox) and (Oy). In an open domain, the base flow (5) corresponds to elliptical streamlines with an ellipticity ε = (b 2 -a 2 )/(b 2 + a 2 ) as in our numerical model, but with a variable tangential velocity. As shown in figure 5, this base flow is a very good approximation of our configuration where we impose a constant tangential velocity on the elliptical boundary, outside a small boundary layer close to the external wall where recirculation cells take place. For such a flow, the inviscid growth rate determined by [START_REF] Gledzer | Instability of bounded flows with elliptical streamlines[END_REF] is:

σ = (b 2 -c 2 )(c 2 -a 2 ) (b 2 + c 2 )(a 2 + c 2 ) (6)
Note that this theoretical growth rate is valid for a ≤ c ≤ b only and is zero for c = b or c = a. Then the maximal theoretical growth rate σ max = b-a a+b is obtained for c = √ ab. One can notice that expression (3) leading to σ = ε 2 in the inviscid case is recovered with a = R eq + s and b = R eq -s in the limit ε → 0. Note also that the experimental choice c = (a + b)/2, equivalent to c = √ ab for small ellipticities, results in the growth rate remaining very close to the maximum; for instance, even for an ellipticity of 0.7, the difference between the growth rate calculated for c = a+b 2 and the maximum value is only 2%. The numerical results are presented in figure 6. In order to compare them with the inviscid analytical prediction (6), a viscous damping term -K √ E is added to the expression of the inviscid growth rate, similarly to the classical expression (3). Excellent agreement is then found for a ≤ c ≤ b, again validating our approach. However, a slightly different constant K = 2.5 (instead of 2.62 in (6), valid in the limit of small ε) allows a better fit of our data. For strong deformations, this constant clearly depends on the considered ellipticity; for instance, a similar study performed at ε = 0.42 implies K = 2.78. Nevertheless, one can notice that K always remains about the same order of magnitude. One can also notice that because of the scattering in the numerical results presented in figure 4 (a), such a constant K = 2.5 works equally well. Hence in the following, we systematically use K = 2.5.

In addition to the verification of the theoretical law of [START_REF] Gledzer | Instability of bounded flows with elliptical streamlines[END_REF], we are now in the position of exploring the range outside a ≤ c ≤ b, where other modes may grow that are not necessarily linear in space variables. As shown in figure 6, different modes of the elliptical instability, characterized by their main frequency of oscillation, appear depending on the ratio c/b. In this view, the variation of the oblateness is similar to the variation of the aspect ratio in the case of an elliptically deformed cylinder [START_REF] Eloy | Elliptic and triangular instabilities in rotating cylinders[END_REF]. Since the elliptical instability comes from the parametric resonance of two inertial waves of azimuthal wave number m and m + 2 with the underlying strain field, the corresponding mode is written (m, m + 2) and is related to a pulsation of frequency ω mode = m + 1. For instance, the spin-over mode corresponds to the stationary mode (-1, 1) with half a wavelength along the axis of rotation. According to figure 6 (b), the mode (1,3) can be observed when c < b and the mode (0,2) when c > a. For even larger aspect ratio, the mode (-1,1) with a larger wavenumber than the spin-over takes place. Note however that because of the geometrical confinement, no stationary mode can be excited for c < b.

Influence of a background rotation

In geo-and astrophysics, the tidal deformation of a given body (planet, moon or star) is also in rotation because of the orbital motion of the companion body. The influence of this background rotation on the development of the elliptical instability has been studied theoretically, using short-wavelength analysis [START_REF] Craik | The stability of unbounded two-and three-dimensional flows subject to body forces: some exact solutions[END_REF][START_REF] Leblanc | On the three-dimensional instabilities of plane flows subjected to Coriolis force[END_REF][START_REF] Dizès | Three-dimensional instability of a multipolar vortex in a rotating flow[END_REF] or normal mode analysis [START_REF] Gledzer | Instability of bounded flows with elliptical streamlines[END_REF][START_REF] Kerswell | Tidal excitation of hydromagnetic waves and their damping in the Earth[END_REF], and also numerically for specific vortices such as Stuart vortices [START_REF] Leblanc | Effects of the Coriolis force on the stability of Stuart vortices[END_REF][START_REF] Potylitsin | Three-dimensional destabilization of Stuart vortices: the influence of rotation and ellipticity[END_REF] or Taylor-Green vortices [START_REF] Sipp | Vortices in rotating systems: Centrifugal, elliptic and hyperbolic type instabilities[END_REF]. It has been studied experimentally in deformed cylinders [START_REF] Vladimirov | The stability of the elliptically deformed rotation of an ideal incompressible fluid in a Coriolis force field[END_REF][START_REF] Bars | Coriolis effects on the elliptical instability in cylindrical and spherical rotating containers[END_REF] and ellipsoids [START_REF] Boubnov | Effect of Coriolis force field on the motion of a fluid inside an ellipsoidal cavity[END_REF][START_REF] Bars | Tidal instability in stellar and planetary binary system[END_REF]. All these works show that the background rotation has a stabilizing effect on cyclones and a destabilizing effect on anticyclones, except when the background rotation almost compensates for the flow rotation, in which case the elliptical instability disappears.

The theoretical expression of the growth rate for perturbations that are linear in space variables (i.e. corresponding to the classical spin-over mode) can be readily obtained by following the same method as [START_REF] Gledzer | Instability of bounded flows with elliptical streamlines[END_REF], but taking into account an additional Coriolis force coming from the background rotation at a given angular velocity

Ω c = Ω * c Ω: σ = (b 2 -c 2 + 2 Ω * c a b)(c 2 -a 2 -2 Ω * c a b) (b 2 + c 2 )(a 2 + c 2 ) . (7)
Actually, this is a particular case of the more general stability analysis detailled in [START_REF] Cébron | Tilt-over mode in a precessing triaxial ellipsoid[END_REF] for a precessing triaxial ellipsoid.

The existence range of this mode is now between c = b 1 + 2Ω * c a/b and c = a 1 + 2 Ω * c b/a and the maximal theoretical growth rate is now exactly obtained for

c 2 a b = Ω * c + 1 + Ω * c a/b + b/a + Ω * c .
In our numerical model, the addition of such a Coriolis force is straightforward and numerical results are presented in figure 7 (a). Again, they compare well with the theoretical prediction, providing that the viscous damping term, -2.5 √ E, found in section 3.1 is added to the inviscid expression [START_REF] Cébron | Tilt-over mode in a precessing triaxial ellipsoid[END_REF]. Other modes are selected by the Coriolis force outside of the spin-over mode resonance band, as already observed experimentally by [START_REF] Bars | Tidal instability in stellar and planetary binary system[END_REF].

Finally, we can investigate how the oblateness and the background rotation interact with each others. Considering for instance an oblate ellipsoid with a ratio c/b such as the mode (1, 3) is selected in absence of background rotation, the spin-over mode is recovered when decreasing Ω c , as shown in figure 7 (b). More generally, we find that the (-1, 1) mode is the most unstable one in the anticyclonic domain up to a value Ω c /Ω tot -1 where the flow restabilises, in agreement with the conclusions of [START_REF] Bars | Tidal instability in stellar and planetary binary system[END_REF].

Influence of the obliquity of the ellipsoid

In planetary cores, tidal deformations are aligned with the orbital plane rather than with the equatorial plane, which means that they are not orthogonal to the rotation axis. Previous works do not take into account this phenomenon, even though the obliquity can be significant, e.g. 23 • 26 ′ for the Earth. In this section, we thus investigate the effect of obliquity for the first time and consider that, as a first approximation, the shape of the body remains a triaxial ellipsoid.

In the numerical model, the rotation axis is tilted and oriented along the unit vector

k c = (cos(φ) sin(θ), sin(φ) sin(θ), cos(θ)),
where φ is its azimuth angle and θ its colatitude angle in spherical coordinates (fig. 1). In assuming for example that the rotation axis is tilted in the (Oxz) plane (i.e. φ = 0), we can study how the development of the elliptical instability changes depending on the obliquity θ. For θ = 0, one recovers the usual configuration with ε = (b 2 -a 2 )/(b 2 +a 2 ) and an aspect ratio c/b. For θ = π/2, one recovers results from section 3.1 in exchanging a and c, i.e. ε = |b 2 -c 2 |/(b 2 +c 2 ) and the aspect ratio is now a/b. In between, streamlines in planes perpendicular to the rotation axis are also elliptical, but their centers are not located along the rotation axis anymore, except in the equatorial plane z = 0. Besides the effective ellipticity measured in each plane now depends on z, the maximum being reached in the equatorial plane z = 0. Seen from this equatorial plane, the apparent length of the ellipsoid along the rotation axis is c = ( sin 2 θ a 2 + cos 2 θ c 2 )

-1 2 , whereas the great and small axes of the elliptical streamlines are respectively b = b and ã = ( cos 2 θ a 2 + sin 2 θ c 2 )

-1
2 . One can then estimate the growth rate of the instability using formulas (3) or ( 6) with these apparent lengths. Numerical results for the spin-over mode compared with these approximations are presented in figure 8. Results agree well for obliquity up to θ ∼ 20 • . Then, the numerical growth rate significantly differs, probably because the geometry is very far from an apparent ellipsoid in rotation around one of its principal axes. It remains however between the two expressions proposed. Note that for the Ekman number studied here, we do not see the tidal instability reappearing around θ ∼ 90 • , which will however be the case at smaller Ekman number: as seen in section 3.1, in this case other modes could appear because of the modified oblateness.

Scaling laws for global quantities of geophysical interest

Scaling law for the amplitude of the flow driven by the instability

The amplitude of the flow driven by the instability at saturation remains up to now an open question, mainly because it is determined by strong non-linear interactions. It is however an important quantity for geo-and astrophysical applications, since it allows us to evaluate the influence of elliptical aspects compared to the other relevant ingredients of planetary dynamics, such as convection. In order to study the amplitude of the flow driven by the instability at saturation, we define the amplitude A * by the maximum value over the volume V:

A * = max V ||u -u b || (8)
where u is the dimensionless velocity field and u b is the theoretical base flow defined in section 3.1. The evolution of A * as a function of the Ekman number is shown in figure 9 (a) for various ellipticities: it can be seen that A * is not zero below the threshold of the elliptical instability because of the differences around the outer bound between the theoretical base flow and the flow numerically obtained (see figure 5). Actually, the difference is maximal on the boundary, along the smallest equatorial axis where the theoretical velocity is maximal. There, the corresponding amplitude λ (ε) can be calculated theoretically :

λ (ε) = b a+b 2 -1 = 2 1 + 1-ε 1+ε -1 (9)
which only depends on ε. One then defines the amplitude of the flow driven by the instability by A = A * -λ (ε) , which is equal to 0 below the threshold (fig. 9 (b)). Far from threshold, a secondary instability appears, which induces a secondary dynamics superimposed on the primary state : e.g. typically, for ε = 0.317, E = 1/1500, c = (a + b)/2, the spinover mode is no more stationary, and the flow is slightly oscillating around the spinover mean flow, at a pulsation ω sec ≈ 1.4. Note that this is in agreement with [START_REF] Kerswell | Elliptical instability[END_REF] which predicts that the primary elliptical instability should saturate and be stable only in a small strain window εε c = O(E). Thus, the amplitude A has to be averaged in time to smooth out the small scale fluctuations. Since those small-scale fluctuations take place on a typical time scale comparable to one revolution 2π/Ω, whereas the characteristic time for the tidal instability is the inverse of the growth rate, i.e. (Ω ε/2) -1 , the averaging is performed about a typical time Ω -1 √ 4π/ε corresponding to the geometrical mean of these two extreme values. With this definition, figure 9 (b) shows that all results collapse on the same generic law above the threshold providing that we use the variable E c /E -1. Far from threshold, the amplitude seems to saturate around 1, which means that the velocities generated by the tidal instability are comparable to the imposed boundary rotation.

One can notice that near the threshold, a square root A ≈ 0.6 √ E c /E -1 fits the results, which is in agreement with a pitchfork bifurcation. Actually, this scaling law can be obtained analytically, starting from the simple model used in [START_REF] Lacaze | Elliptical instability in a rotating spheroid[END_REF] to describe the viscous non-linear evolution of the spinover mode. This model, first introduced by [START_REF] Hough | The oscillations of a rotating ellipsoidal shell containing fluid[END_REF] and [START_REF] Poincaré | Sur la précession des corps déformables[END_REF] for an inviscid solid-body rotation in a spheroid, reduces to:

ωx = -α (1 + ω z ) ω y -ν so ω x ( 10
) ωy = -β (1 + ω z ) ω x -ν so ω y (11) ωz = ε ω x ω y -ν ec ω z + ν nl (ω 2 x + ω 2 y ) ( 12
)
where ω is the rotation vector of the spinover mode, α = ε 2-ε and β = ε 2+ε . The damping terms are given by theory (no adjustable parameter): ν so = 2.62 √ E is the linear viscous damping rate of the spinover mode (first calculated by [START_REF] Greenspan | The Theory of Rotating Fluids[END_REF]), ν ec = 2.85 √ E is the linear viscous damping of axial rotation and ν nl = 1.42 √ E is the viscous boundary layer effect on the non-linear interaction of the spinover mode with itself (see [START_REF] Greenspan | The Theory of Rotating Fluids[END_REF]). Even if this model does not take into account all the viscous terms of O( √ E) or the non-linear corrections induced by the internal shear layers, it satisfyingly agrees with experiments, regarding the growth rate as well as the non-linear saturation [START_REF] Lacaze | Elliptical instability in a rotating spheroid[END_REF]. Thus, this model can be used with confidence to describe the viscous non-linear evolution of the spinover mode.

After little algebra, we obtain a non-trivial stationary state for ε > 2ν so , given by:

ω x = ± ν ec [ √ αβ -ν so ] β ε -ν nl [ √ αβ + β 2 / √ αβ] ≈ ± ν ec [ε -2 ν so ] ε 2 -2 ν nl ε ( 13
)
ω y = ∓ ν ec [ √ αβ -ν so ] α ε -ν nl [ √ αβ + α 2 / √ αβ] ≈ ∓ ν ec [ε -2 ν so ] ε 2 -2 ν nl ε ≈ ∓ ω x ( 14
)
ω z = ν so ε √ 4 -ε 2 -1 ≈ 2 ν so ε -1 (15)
where approximations are done assuming ε ≪ 1. Now, the amplitude A corresponds to the norm of the flow ω × r driven by the spinover mode. Then, near the threshold (ε ≈ 2 • 2.62 √ E c and E c /E -1 ≪ 1), we obtain:

A ≈ ν ec 2 (ν so -ν nl ) E c E -1 ≈ 1.1 E c E -1 ( 16
)
which is in good agreement with the numerical fit.

Scaling law for the viscous dissipation by the instability

As explained for instance in [START_REF] Rieutord | Evolution of rotation in binaries: physical processes[END_REF] or in [START_REF] Bars | Tidal instability in stellar and planetary binary system[END_REF], the energy dissipated by tides is a primordial quantity which directly influences the orbital evolution and rotational history of a binary system during its synchronization. It is however poorly known [e.g. [START_REF] Williams | Geological constraints on the Precambrian history of Earth's rotation and the Moon's orbit[END_REF][START_REF] Touma | Evolution of the Earth-Moon system[END_REF]. In most traditional models, fluid dissipation in the planetary core is supposed to be negligible. However this may not be the case when the elliptical instability is excited at a rather large amplitude, inducing important shears between the bulk of the fluid and the boundary. Our purpose here is to systematically quantify the variation of this dissipation with the ellipticity and the Ekman number.

The dissipated power balance of the incompressible flow in the ellipsoid is given by:

V ∂ ∂t ρ ũ2 2 dτ = S σv • n • ũ ds - V σv : ∇ ũ dτ (17)
where σv = η ∇ ũ + t ∇ ũ is the viscous stress tensor of the newtonian fluid, S the surface of the ellipsoid, ũ = Ω R eq u is the dimensionalized velocity field, ρ the volumic mass and η the dynamic viscosity of the fluid. In this section, we focus on the stationary spinover mode and then the two terms on the right side of equation ( 17) balance each others: the first one allows to maintain the rotation of the fluid by the no-slip conditions on the boundary while the second one is the volume dissipation P of the fluid. In the following, we use as a power scale the dissipated power during the spin-down stage of the equivalent sphere of radius R eq and moment of inertia I ∆ (I ∆ = 2/5 MR 2 for an homogeneous sphere of mass M and radius R), i.e. the kinetic energy of rotation E c = 1 2 I ∆ Ω 2 divided by the Ekman time t E . Note that most of the dissipated power comes naturally from the boundary layers: its determination is thus a challenging task from a numerical point of view, and extra care must be taken regarding the convergence of the simulations, as shown for instance in figure 2 (b). Below the threshold, the dissipated power is not zero because of the recirculation patterns of our base flow (see figure 5). This dissipation is due to the flow driven by the ellipticity, which scales as ε ΩR for small ellipticities. Then, the dimensionalized dissipated power below the threshold simply scales as

V σv : ∇ ũ dτ ∼ η ε 2 Ω 2 R 3 (18)
which is confirmed by the dissipated power measured in our numerical simulations below the threshold. Far from threshold, the model proposed by [START_REF] Bars | Tidal instability in stellar and planetary binary system[END_REF] considers that the spin-over mode simply corresponds to a supplementary solid body rotation Ω SO outside the outer viscous boundary layer of thickness h = ξ √ ν/Ω S O , ξ being a constant of order 1. Then, the dissipation is only located in this boundary layer, where the fluid rotation has to match the imposed velocity conditions at the outer boundary. With this simple model, the torque of the container on the fluid is

C m/c = -2 M ν R
h Ω SO , and the power dissipated by the system is 2 , which can also be written

P = -2 M ν R h Ω S O
P = -2 ξ M R √ ν Ω 5/2 A 5/2
, where A is the dimensionless spin-over mode amplitude studied in the previous section. The dimensionless power given by this model can finally be stated as

P dissip = |P| 1 2 I ∆ Ω 3 √ E = 10 ξ A 5/2 , (19)
where, according to the previous section, A is about 1 in the small Ekman number limit studied here. The evolution of P dissip measured in our numerical model is shown in figure 10. It confirms the behavior predicted by [START_REF] Hollerbach | Oscillatory internal shear layers in rotating and precessing flows[END_REF] with a saturation far from threshold and ξ ≈ 1. This behavior is especially interesting for planetary applications, as will be studied in the following section.

Orders of magnitude for planetary applications

From a geophysical point of view, the results from the previous sections can be used to derive the orders of magnitude involved in planetary cores. Regarding the jovian moon Io, which is clearly unstable to the tidal instability and where the strong tidal dissipation is a topical question [START_REF] Lainey | Strong tidal dissipation in Io and Jupiter from astrometric observations[END_REF], our numerical study confirms the first trends given by [START_REF] Bars | Tidal instability in stellar and planetary binary system[END_REF], with however a numerically determined scaling factor of the order 1/ξ ≈ 1 in the power dissipation estimates. Moreover, the most important result comes from the oblateness effects. Indeed, according to the data given in [START_REF] Kerswell | Tidal instability as the source for Io's magnetic signature[END_REF], c/b ∼ 0.995 and a/b > 0.999, which means that the excited mode of the tidal instability in Io cannot be the spinover mode, but has to be oscillatory. Note that this result is not modified when the background rotation is taken into account.

A similar analysis can be done for the early Earth, with a Moon two times closer than today. In this case, the length of the day was around 10 hours according to [START_REF] Touma | Evolution of the Earth-Moon system[END_REF] and the lunar tides were around eight times stronger, whereas the solar tides were about the same. Then, the actual tidal amplitude of 50 cm allows to estimate the ellipticity of the early Earth: ε ∼ 10 -6 . In considering a similar but totally molten core, we estimate E c /E -1 ∼ 70, which means that the early Earth's core was clearly unstable to tidal instability, and that the tidal instability was then at saturation. Moreover, with an orbital period for the Moon (1/2) 3/2 shorter than the actual moon (Kepler's Third Law), the dissipated power given by the model was around P ∼ 10 18 W. This estimation seems huge in comparison with the present dissipation by tidal friction (∼ 3.75 • 10 12 W according to [START_REF] Munk | Abyssal recipes II: energetics of tidal and wind mixing[END_REF]) but actually, it simply suggests that the Earth-Moon system was then in rapid evolution. Once again, the possible mode of the tidal instability was not the spin-over mode neither any stationary (-1, 1) mode, considering that the oblateness of the rapidly rotating early Earth was larger than the actual oblateness. In fact, we expect stationary modes to be only marginally excited in geophysical systems, limited to very peculiar configuration regarding the oblateness and the rotation of the tidal bulge.

Note that computations similar to the ones presented here for an ellipsoidal geometry have been performed for an ellipsoidal shell in order to study the influence of a solid inner planetary core. Similar conclusions are then found regarding the physics of the tidal instability. Only a slightly larger dissipation term due to the presence of an inner viscous boundary layer has to be taken into account.

Conclusion

This paper presents the first systematic numerical study of the tidal instability in a rotating ellipsoid. The numerical approach is a powerful tool to complete the knowledge derived from previous theoretical and experimental studies. The effects of oblateness, background rotation and obliquity, which lead to the selection of various instability modes, have been studied. We have also defined scaling laws regarding the amplitude of the flow driven by the instability and its viscous dissipation, which confirm the primordial role played by tidal effects at a planetary scale.

Another fundamental geophysical issue regarding the elliptical instability is whether or not it can drive a planetary dynamo. The results presented herein show that the numerical approach is able to faithfully capture the physics of the hydrodynamic elliptical instability. Thus, the present work provides a solid basis to study the full MHD problem encountered in planetary core flows undergoing tidal instability. These flows can only be studied numerically due to their inherent complexity. Therefore, a next step will be to introduce the effects of a magnetic field in our model. 2 ) and comparison with the linear theory indicated by a dashed line. The coefficient 2.62 comes from [START_REF] Lacaze | Elliptical instability in a rotating spheroid[END_REF] and is valid in the limit of small ellipticity. Good agreement is found for values of ε up to 0.5. (b) Evolution of the growth rate depending on the ellipticity for two values of the Ekman number E = 1/344 and E = 1/800 (ε ≤ 0.8, c = a+b 2 ). As also seen in (a), the growth rate agrees with the linear analytical analysis close to the threshold and then decreases for large values of the ellipticity. 5), which presents a variable tangential velocity along elliptical streamlines. Good agreement is found with the numerical results, except in the small outer viscous boundary layer, where recirculation cells take place to match the imposed constant velocity along the boundary. Note in particular that the maximum velocity is not reached at the boundary but within this viscous boundary layer.

Figure 1 :Figure 2 :

 12 Figure 1: Sketch of the problem under consideration : a rotating flow inside a triaxial ellipsoid of axes (a, b, c) related to a frame (Ox, Oy, Oz) with a constant tangential velocity all along the boundary in each plane perpendicular to the rotation vector Ω.

Figure 3 :Figure 4 :

 34 Figure 3: Validation of the numerical simulations. (a) Kalliroscopic visualization of the spin-over mode in the meridional plane of maximum shear for E = 1/4000 and ε = 0.16. The typical S shape of the rotation axis is due to the combination of the main rotation imposed by the boundary and the spin-over mode. (b) Slices of the velocity field ||u|| and surface of iso-value ||u|| = 0.15 at saturation of the tidal instability for E = 1/344, ε = 0.317, c = a+b 2 , and 49900 DoF. The classical S-shape of the spin-over mode is recovered.

Figure 5 :

 5 Figure 5: Azimuthal velocity v θ for E = 1/95, ε = 0.1, c = a+b 2 . (a) Slice along (Oy) at x = 0. (b) Slice along (Ox) at y = 0. (c) Slice in the equatorial plane of the vertical component of the vorticity. Dashed lines in (a) and (b) correspond to the theoretical base flow[START_REF] Cadot | Characterization of the low pressure filaments in three-dimensional turbulent shear flow[END_REF], which presents a variable tangential velocity along elliptical streamlines. Good agreement is found with the numerical results, except in the small outer viscous boundary layer, where recirculation cells take place to match the imposed constant velocity along the boundary. Note in particular that the maximum velocity is not reached at the boundary but within this viscous boundary layer.

Figure 6 :Figure 7 : 2 eq= 10 - 3 Figure 8 :Figure 9 : 2 ε

 672103892 Figure 6: Influence of the length of the ellipsoid along the rotation axis on the mode selection of the tidal instability (E = 1/688, ε = 0.317). (a) Variation of the growth rate. (b) Variation of the main frequency of the selected mode determined by Fourier analysis of its saturation state.

Figure 10 :

 10 Figure 10: Viscous dissipation by the tidal instability, as a function of the distance from threshold. The results collapse on a generic law and seem to converge towards a saturation value far from threshold.

COMSOL Multiphysics

www.pardiso-project.org


Dr Jan-Paul Zock
email: jpzock@creal.cat

David Vizcaya

Nicole Le Moual

UPDATE ON ASTHMA AND CLEANERS

Keywords: Cleaning, cleaners, asthma, respiratory

Update on asthma and cleaners.

INTRODUCTION

There is growing evidence that cleaners are at increased risk of asthma and other respiratory disorders. A systematic review published in this journal in 2006 [START_REF] Jaakkola | Professional cleaning and asthma[END_REF] summarised the literature on this topic published in the period 2003 to 2005, and selected key articles published before 2003. The authors concluded that the evidence for an increased risk of asthma in cleaning workers was consistent and accumulating, and that a few specific occupational risk factors (products or tasks) had been suggested. However, they also noted that it was largely unclear which of the potentially important irritant and/or sensitising mechanisms play a predominant role.

The main aim of the present report is to concisely review the recent literature published since 2006 on the topic of respiratory health in cleaning workers. In addition, we wished to broaden the scope by including also relevant studies addressing more generally respiratory effects of cleaning products when used either at work in other occupations such as health care professionals, or in private households.

SEARCH STRATEGY

A Medline search was performed for articles using the key words (cleaning OR cleaners) AND (asthma OR respiratory) published in the period 2006 to 2009. We made a critical selection relevant for the topic, and added important references from the period 2006 to 2009 that were not identified by the Medline search. Finally, 21 papers published in international peer-reviewed journals were included: three reviews on occupational asthma [24], four papers from a population-based longitudinal survey [56,7*,8], three surveillance studies [9,10*,11], eight papers from six workforce-based studies [1219], and three papers on the characterization of exposures in cleaners [20,21*,22].

REVIEWS ON OCCUPATIONAL ASTHMA

The proceedings of the Third Jack Pepys Workshop on Asthma in the Workplace [START_REF] Tarlo | Third Jack Pepys Workshop on Asthma in the Workplace Participants. An official ATS proceedings: asthma in the workplace: the Third Jack Pepys Workshop on Asthma in the Workplace: answered and unanswered questions[END_REF] recognised a role of irritant exposures in asthma among cleaners. The relevance of accidental exposures was also noted, hypothesising that some cleaners may have had an initial accidental exposure and following that may have work-exacerbated asthma. A recent review of agents that can cause occupational asthma [START_REF] Malo | Agents causing occupational asthma[END_REF] included several cleaning agents, mostly disinfectants. Although the issue of multiple exposures in cleaning workers was stressed, the main focus was put on sensitising properties of cleaning agents and hence on asthma with a latency period. Nevertheless, it was recognised that many cleaning agents are respiratory irritants and could interact with sensitisers in the development of occupational asthma. Overall, authors considered that not much was known about the risk factors, exposure levels, clinical features, and pathogenetic mechanisms of asthma related to cleaning agents. A review focusing on mechanisms in occupational asthma considered only one form of irritant-induced occupational asthma: that without a latency period after acute exposures to high concentrations [START_REF] Maestrelli | Mechanisms of occupational asthma[END_REF]. Cleaning agents associated with occupational asthma were classified as low molecular weight agents, inducing asthma following a non-IgE mediated mechanism.

POPULATION-BASED STUDIES

Epidemiological studies based on general population samples have several advantages over workforce-based studies. The most important are the possibility to evaluate similar exposures across occupations, better possibilities to control for health-related selection biases such as the Healthy Worker Effect, and the opportunity to assess the burden of occupational disease in the general population. The previous review [START_REF] Jaakkola | Professional cleaning and asthma[END_REF] reported a relevant number of population-based studies that found higher prevalence rates of asthma or lower respiratory tract symptoms suggestive of asthma in current cleaning workers when compared to non-manual workers or to all other occupations. Interestingly, one of the studies also showed an increased risk of asthma in women who had left the cleaning profession, which was suggestive of an enduring effect after removal from exposure. Importantly, almost all of the reviewed epidemiological studies had a cross-sectional design, some of which following case-control approaches. Thus, the reported excess risks are likely to include both new-onset occupational asthma and work-exacerbated asthma. Evidence from longitudinal studies, indispensable to specifically address new-onset occupational asthma, was very limited. Only one population-based incident case-control study from Finland reported a non-significantly 42% increased risk of new-onset asthma in cleaning women as compared to professionals, clerks and administrative workers. Data for male cleaning workers were not reported.

We identified four recent papers from the longitudinal European Community Respiratory Health Survey (ECRHS II) that associated asthma with cleaning. Regarding occupation, Kogevinas and colleagues [START_REF] Kogevinas | Exposure to substances in the workplace and new-onset asthma: an International prospective population-based study (ECRHS-II)[END_REF] reported a higher risk of new-onset asthma in 358 individuals that had been employed in cleaning and caretaking during the nine-year follow-up period as compared to 4,143 employed in professional, clerical or administrative jobs throughout during the follow-up (Relative Risk (RR) 1.7; 95% Confidence Interval (CI) 0.93.2). In addition, the authors applied an asthma-specific job exposure matrix (JEM) and the incidence of new-onset asthma for individuals with exposure to cleaning products across all occupations in the study was significantly higher than for those unexposed to any agent according to the JEM (RR 1.8; CI 1.03.2). An increased risk for incident asthma was also observed for the group of 291 nurses (RR 2.2; CI 1.34.0). A more specific analysis was reported in a separate paper using information from job-specific questionnaires that had been obtained for nurses in the ECRHS II interviews [START_REF] Mirabelli | Occupational risk factors for asthma among nurses and related healthcare professionals in an International study[END_REF]. The highest risks of incident asthma were found among nurses who used ammonia, bleach and/or cleaning products in spray form at work. This large international study suggested that professional use of cleaning products in various occupations was related to new-onset asthma. It remained unclear to what extent such exposures also contributed to work-exacerbated asthma.

Using the same base population of the ECRHS II, the risk of new-onset asthma was also evaluated in relation to the use of household cleaning products among 3,503 adults who did the cleaning in their own homes and did not have asthma at baseline [7*]. Using three definitions of asthma, this study showed a significant 30% to 50% increased risk of incident asthma related to the use of products in spray-form at least once a week. This association was linked predominantly to the most commonly used air fresheners, glass cleaners, and furniture cleaning sprays, and the risk increased when frequency of use or number of different sprays increased. Among other products, not used in spray applications, weekly use of hypochlorite bleach was non-significantly associated with new-onset asthma (RR between 1.1 and 1.3). An additional cross-sectional analysis using the ECRHS II participants who did the cleaning in their homes showed that frequent use of bleach (that is, on four or more days per week) was associated with the prevalence of lower respiratory tract symptoms and possibly with bronchial hyperresponsiveness [START_REF] Zock | Domestic use of hypochlorite bleach, atopic sensitization and respiratory symptoms in adults[END_REF]. Perhaps paradoxically, this analysis also showed an inverse association of bleach use with the prevalence of atopic sensitisation to common environmental allergens, hay fever and allergic symptoms. The inverse association for atopic sensitisation was present for specific IgE to both indoor and outdoor allergens, was consistent in various subgroups including those without clinically apparent respiratory disease, and the effect was stronger when bleach was used more frequently.

The authors hypothesised that this could be explained either by direct exposure to volatile chlorinated agents released during or shortly after bleach use, or indirectly due to a cleaner home environment.

SURVEILLANCE STUDIES

Registry reports of surveillance schemes are helpful in assessing the incidence of occupational diseases in a defined area in a certain time period, and to evaluate the causal agents. The previous review [START_REF] Jaakkola | Professional cleaning and asthma[END_REF] included several occupational asthma surveillance studies from various countries that had reported cases of occupational asthma in cleaning workers, or related to professional use of cleaning products. We identified three new studies published in or after 2006 analysing cleaning workers or cleaning products using surveillance programmes. A voluntary reporting system of occupational respiratory diseases in the autonomous region of Catalonia (Spain) showed 174 newly diagnosed cases of occupational asthma in the year 2002 (incidence 77 cases per million) accounting for 48% of all occupational respiratory diseases voluntarily reported [START_REF] Orriols | Reported occupational respiratory diseases in Catalonia[END_REF]. For 15 cases (9%), cleaning products were suspected as being the causal agent. Interestingly, this surveillance system also notified 46 cases of acute inhalations (20 cases per million).

Cleaning was the most frequently reported occupation (26%) being associated with acute inhalations. Follow-up on these cases could not be performed, and therefore it remained unclear how many of these cases developed chronic respiratory effects including Reactive Airways Dysfunction Syndrome (RADS).

A study from Alberta province (Canada) [10*] applied an interesting data linkage approach to estimate the extent and distribution of new-onset occupational asthma. The rationale was that using numbers of cases presenting for compensation typically underestimate the true incidence, since claims are not made for particularly less severe diseases, and in addition some important sectors are typically underrepresented. Workers' Compensation Board claims for any reason in the period 1995 to 2004 were identified. Data were linked to physician billing data, and incident cases of asthma were identified as new onset adult asthma in the 12 months prior to a claim without asthma in the previous 3 years. Occupations were linked to the asthmaspecific JEM mentioned earlier [START_REF] Kogevinas | Exposure to substances in the workplace and new-onset asthma: an International prospective population-based study (ECRHS-II)[END_REF], and an incident case-referent analysis was performed. Exposure to low molecular weight cleaning products was associated with an increased risk of asthma in men (Odds Ratio (OR) 1.9; CI 1.32.7) but, however, not in women (OR 0.9; CI 0.81.1). No significantly increased risk for asthma in cleaning workers as such was reported.

The Sentinel Event Notification System for Occupational Risks (SENSOR) performed in four US states identified 2,995 cases of work-related asthma in the period 1993 to 2000, the majority classified as new-onset asthma. Educational services formed the third most frequently reported industry sector (9%) associated with work-related asthma. A specific analysis was performed for the 265 cases from the educational services industry [START_REF] Mazurek | Work-related asthma in the educational services industry: California, Massachusetts, Michigan, and New Jersey, 1993-2000[END_REF]. Janitors, cleaners, and housekeepers accounted for 12% of the cases, with 84% classified as new-onset asthma. Across all occupations, cleaning products were reported as being the causal agents for 20% of all cases of work-related asthma. Most cleaning products were not specified, and among the specified products the most common were formaldehyde, graffiti remover, bleach, carpet cleaners and ammonia solution. This paper using SENSOR data added evidence for work-related asthma provided earlier regarding cleaning workers and cleaning products among health care workers, summarised in the previous review [START_REF] Jaakkola | Professional cleaning and asthma[END_REF].

WORKFORCE-BASED STUDIES

Advantages of workforce-based studies over population-based studies are opportunities for more specific and less biased assessment of occupational exposure, and bigger numbers of individuals with specific exposures. We identified reports of workforce-based studies among cleaning workers and among other occupations that involve exposure to cleaning agents.

Cleaning workers

Obadia and co-workers [START_REF] Obadia | Relationships between asthma and work exposures among non-domestic cleaners in Ontario[END_REF] studied the prevalence of asthma and respiratory symptoms in 566 non-domestic cleaners and 587 other building workers from two union locals in Ontario province (Canada). Men were predominantly school cleaners, and the prevalence rate of physician-diagnosed asthma seemed to be higher than in male "other building workers" (OR 2.1; CI 0.94.8). However, when narrowing the definition to those with asthma onset after starting work, this association disappeared (OR 0.9; CI 0.42.3). Although no association for physician-diagnosed asthma was apparent in women, female cleaners were significantly more likely than other building workers to report any respiratory symptoms in the previous year (OR 2.6; CI 1.64.3). This was more pronounced for and work-related symptoms when defined as improving when away from work (OR 3.9; CI 2.17.4). The authors identified several specific tasks at risk for work-related symptoms among male cleaners, including waxing and wax stripping of floors, spot cleaning of carpets, oiling of furniture, cleaning tiles and cleaning grout. Another paper using the same study population reported that skin symptoms, particularly rash in the last year, were more common in cleaners, and that asthma and respiratory symptoms were more common among male cleaners with rash [START_REF] Lynde | Cutaneous and respiratory symptoms among professional cleaners[END_REF]. The authors concluded that there is a strong link between work-related symptoms of asthma and dermatitis among cleaners.

Massin and colleagues reported a study among 175 cleaning and disinfecting workers in the food industry, predominated by slaughterhouses [START_REF] Massin | Respiratory symptoms and bronchial responsiveness among cleaning and disinfecting workers in the food industry[END_REF]. Personal exposure levels of nitrogen trichloride and aldehydes were determined. The prevalence rates of chronic respiratory symptoms and asthma among the 175 exposed workers were not different from those in the control group comprising 70 non-exposed workers of small manufacturing plants. The prevalence of irritation symptoms (eye, nose, sore throat, dry cough) was significantly higher in exposed, particularly among those with the highest exposure levels or the longest exposure duration in the food industry. The prevalence of bronchial hyperresponsiveness to methacholine was non-significantly higher in exposed workers.

A cross-sectional study among 341 non-domestic cleaners employed by 32 cleaning service companies in Brazil included questionnaires, skin prick testing and pulmonary function tests [START_REF] De Fátima Maçãira | Rhinitis and asthma symptoms in non-domestic cleaners from the Sao Paulo metropolitan area, Brazil[END_REF]. Chlorine bleach was the most common agent related to both upper and lower airway symptoms. Work-related asthma or rhinitis increased monotonously with the number of years employed in cleaning. However, most of the tasks performed and products used showed similar frequencies for the workers with work-related asthma/rhinitis and those without symptoms.

A panel study was reported among 43 Spanish domestic cleaning women with asthma and/or chronic bronchitis [START_REF] Medina-Ramón | Short-term respiratory effects of cleaning exposures in domestic cleaning women[END_REF]. Associations between specific daily cleaning exposures and respiratory symptoms and peak flow were evaluated. Lower respiratory tract symptoms were more common on working days, particularly when involving more than 8 hours of cleaning work. When taking into account both cleaning work in other people's homes and cleaning their own home, it was found that the use of diluted bleach, degreasing sprays and air refreshing sprays were independently associated with a higher probability of having lower respiratory tract symptoms on the day the products were used. In addition, 11 out of 37 cases (30%) with adequate repeated peak flow data were identified as having occupational asthma when analysed using the Occupational Asthma System (OASYS) and/or reviewed by an expert. This study showed that a relevant proportion of cleaning workers with asthma could be diagnosed as having occupational asthma using internationally acknowledged methods, and suggested that occupational exposure to cleaning products can aggravate pre-existing obstructive lung disease.

Other occupations involving exposure to cleaning agents

Another study on short-term respiratory effects of cleaning exposures was done in US homemakers [START_REF] Bernstein | Evaluation of cleaning activities on respiratory symptoms in asthmatic female homemakers[END_REF]. Associations between specific cleaning activities and the use of specific cleaning products and lower respiratory tract symptoms were evaluated in 25 female homemakers with asthma and 19 without asthma. Symptoms and peak flow were recorded daily within 30 minutes before and after home cleaning activities. Details of the cleaning tasks and products were recorded as well. The authors found that cleaning activities were related to short-term lower respiratory tract symptoms among asthmatic women, whereas no effects were apparent in non-asthmatic women. This could not be associated with specific cleaning exposures. In any case, results of this study are suggestive of exacerbations of in principle well-controlled asthma due to household cleaning work.

A study among 2,738 health care professionals from Texas [START_REF] Delclos | Occupational risk factors and asthma among health care professionals[END_REF] evaluated occupational risk factors for asthma in physicians, occupational therapists, nurses and respiratory therapists. After applying an industry-specific job exposure matrix, exposure to cleaning agents was associated with asthma. The prevalence rate of physiciandiagnosed asthma with onset after entry into the health care profession was higher among workers exposed to agents used for instrument cleaning (OR 2.2; CI 1.33.7) and surface cleaners (OR 2.0; CI 1.23.4). In addition, the prevalence of respiratory symptoms consistent with bronchial hyperresponsiveness was higher among those with exposure to surface cleaners (OR 1.6; CI 1.22.1) and also associated with the report of a spill at work (OR 2.0; CI 1.33.2), the latter being suggestive of an acute inhalation to irritants. The prevalence of new-onset asthma was highest among the 448 nursing professionals, and a more detailed analysis within the latter subgroup highlighted the importance of exposure to cleaning and disinfecting products used for instrument and surface cleaning [START_REF] Arif | Occupational exposures and asthma among nursing professionals[END_REF].

EXPOSURE STUDIES

Characterisation of exposure to cleaning chemicals in different occupational settings is important to understand determinants of exposure and hence to develop preventive workplace control measures, to contribute to the hypothesis generation of effect mechanisms, and to improve the quantification of exposure-response relationships in epidemiological studies. Apart from the study in the food industry mentioned earlier [START_REF] Massin | Respiratory symptoms and bronchial responsiveness among cleaning and disinfecting workers in the food industry[END_REF] we found reports of three studies in the period 2006-2009 that aimed at the characterisation of exposure to cleaning products.

Arif and co-workers [START_REF] Arif | Occupational exposures among domestic and industrial professional cleaners[END_REF] performed a qualitative study among 32 domestic and 47 industrial cleaning workers in the US. Domestic cleaners demonstrated significant skills deficit across each of these three themes as compared to industrial cleaners. Domestic cleaners reported more frequent exposure to respiratory irritants and sensitizers and also reported adverse respiratory symptoms as compared to industrial cleaners.

A detailed characterisation of exposures to cleaning products among US hospital cleaners was reported [21*]. Authors followed an integrated approach including workplace interviews, reviews of product Materials Safety Data Sheets and the scientific literature on adverse health effects to humans, reviews of physical/chemical properties of cleaning ingredients, and occupational hygiene observational analyses.

They concluded that a combination of product evaluation and workplace exposure data is necessary to develop strategies for protecting workers from cleaning hazards. The conducted task based assessment allowed classification of cleaning tasks in different exposure categories (Figure 1), a strategy that can be employed by epidemiological investigations of the impact of cleaning on health.

Finally, in an experimental study the airborne concentrations of glycol ethers (regulated toxic air contaminants) and terpenoids (including d-limonene which has sensitising properties) were measured during and after the application of five different cleaning products and air fresheners [START_REF] Singer | Cleaning products and air fresheners: emissions and resulting concentrations of glycol ethers and terpenoids[END_REF]. The authors found that the application of common cleaning products resulted in excessive exposures to potentially harmful volatile pollutants. Airborne levels of these chemical agents remained elevated for many hours after cleaning.

CONCLUSIONS

Studies reported in the last four years have strengthened the evidence of adverse respiratory effects, particularly asthma, in cleaning workers and otherwise in relation to the use of cleaning products (Table 1). The population at risk is particularly large; apart from cleaning workers many others are occupationally exposed to cleaning agents. This indicates a highly important public health issue, particularly when considering also the ubiquitous use of cleaning products in private homes. Cleaning-related asthma is a preventable disease and in order to develop prevention strategies it is indispensable to identify its specific risk factors and underlying mechanisms.

Both new-onset asthma and work-exacerbated asthma due to cleaning exposures may play a role, although more longitudinal surveys and specific studies in asthmatic populations are required. Both in domestic and non-domestic cleaners, the reviewed papers suggested a number of specific tasks and products that apparently increased the risk of asthma. The most important products that have been repeatedly reported include products in spray-form, chlorine bleach and other disinfectants. Many cleaning agents are respiratory irritants and some have sensitising properties. Relevant exposure levels to volatile compounds released from cleaning products have been reported to occur during common cleaning activities.

New studies should include a detailed exposure assessment and characterise effect mechanisms in relation to specific exposures. Both new-onset asthma, with and without latency periods, and work-exacerbated asthma should be considered. Rather than trying to answer the question whether the respiratory effects seen in cleaning workers or otherwise related to cleaning products are "real" or "true" (occupational) asthma, the authors judge it more relevant to characterise asthma phenotypes in order to help evaluating underlying effect mechanisms. The application of respiratory biomarkers and perhaps specific bronchial challenge testing can be helpful.

Figure 1 :

 1 Figure 1: Process flow diagram of the tasks performed for patient room cleaning. The shaded boxes indicate cleaning tasks/steps with higher potential for inhalation exposure. Reproduced from [21*].

Table 1 :

 1 Summary of main findings on the relation between cleaning exposures and respiratory health from recent papers describing original studies

	Type and design of	Study population	Main findings	Reference
	study
	Multi-centre	General population	-Incidence of asthma 2.2	5,6,7*,8
	population-based	samples from Europe,	per 1000 person-years
	longitudinal study	USA and Australia;	-Increased risk for new-onset
		including 358 cleaners,	asthma among:
		291 nurses and 3501	• cleaning workers
		homemakers.	• those with occupational exposure
		Years 19902003.	to cleaning products
			• nurses, particularly those who use
			ammonia, bleach and/or cleaning
			sprays
			• homemakers who use sprays
	Surveillance	174 newly diagnosed	-Incidence of occupational asthma	9
	programme; voluntary	cases of occupational	77 cases per million
	notification of work-	asthma from Catalonia,	-In 15% of the cases causal agents
	related respiratory	Spain.	were cleaning products
	diseases	Year 2002.	-Incidence of acute inhalations 20
			cases per million
			-Cleaning was the most frequently
			reported occupation (26%)
	Data linkage of	12,554 new-onset	-Incidence rate of new-onset	10*
	compensation claims	asthma cases among	asthma 1.6%
	and physician billing	782,908 with claims in	-Increased risk related to cleaning
	data	Alberta, Canada.	agents in men but not in women.
		Years 19952004.
	Surveillance SENSOR	265 educational service	-Cleaning workers accounted for	11
		workers with work-related	12% of the cases of work-related
		asthma from four USA	asthma
		states. Years 19932000.	-Cleaning products were causal
			agents for 20% of all cases
			-Most commonly reported were
			formaldehyde, graffiti remover,
			bleach, carpet cleaners and
			ammonia solution
	Workforce-based cross-	566 non-domestic	-Cleaning women had higher risk	12,13
	sectional study	cleaners and 587 other	of work-related respiratory
		building workers from	symptoms
		Ontario, Canada.	-Main risk factors for work-related
			symptoms among male cleaners
			were waxing and wax stripping of
			floors, spot cleaning of carpets,
			oiling of furniture, cleaning tiles
			and cleaning grout
	Workforce-based cross-	175 cleaning and	Risk of irritative symptoms	14
	sectional study	disinfecting workers in	increased with exposure (either
		the French food industry	intensity or duration).
		and 70 non-exposed.
	Workforce-based cross-	341 non-domestic	-Chlorine bleach was the most
	sectional study	cleaners employed in	common agent related to
		cleaning service	respiratory symptoms.
		companies from Brazil	-Risk of work-related asthma or

ACKNOWLEDGEMENTS Part of the work reported in this review was supported by the Health Institute Carlos III / European Regional Development Fund (01-3058 and PI 06/1378), Afsset (Agence Française de Sécurité Sanitaire de l'Environnement et du Travail; EST-2009/1/15) and the Global Allergy and Asthma European Network (GA 2 LEN; Sixth EU Framework Programme for Research, FOOD-CT-2004-506378).