Mantle lateral variations and elastogravitational deformations II. Possible effects of a superplume on body tides

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1 Geophys. J. Int. (27) 168, doi: /j X x Mantle lateral variations and elastogravitational deforations II. Possible effects of a superplue on body tides L. Métivier, 1 M. Greff-Lefftz 2 and M. Diaent 2 1 Royal Observatory of Belgiu, Bruxelles, Av. Circulaire, 3, 118 Bruxelles, Belgiu. E-ail: l.etivier@oa.be 2 Institut de Physique du Globe de Paris, case 89, 4 place Jussieu, Paris Cedex 5, France Accepted 26 Noveber 2. Received 26 Noveber 15; in original for 26 January 13 SUMMARY The body tides response (deforation and gravity) of the Earth is generally coputed assuing radial syetry in stratified earth odels, at the hydrostatic equilibriu. We present in this paper nuerical experients with the ai to evaluate the ipact of very large antle heterogeneities of density on body tides. In a copanion paper, we have developed a new earth elasto-gravitational deforation odel able to take into account the heterogeneous structure of the antle. We use this odel to calculate the theoretical induced by three types of spherical heterogeneities in the antle on M2 body tides response. The heterogeneities are: (1) our liit case, a heterogeneity of 1 k of radius in the lower antle; (2) a heterogeneity of 5 k of radius at the botto of the lower antle and (3) a heterogeneity of 285 k of radius in the upper antle. The density variation has been set to 5 kg 3.For each experient, we first calculate the equilibriu state of the Earth when it contains a heterogeneity, including non-hydrostatic prestresses, dynaical topography and lateral variation of density. Then we copute the M2 tidal. We find that the surface tidal displaceent is saller than 1, and that the gravity has a axiu aplitude of 525 nanogal (). Regarding to the present precision in position easureent, the displaceent is too sall to be detected. The gravity should be easurable with superconducting gravieters, which have a instruental precision. In experient 2, the axiu gravity is about 12, and in experient 3, the axiu is about 33. Finally, we investigate the axiu theoretical ipact of the Pacific and the African superplues on the M2 body tide. The superplues have been odelled as two spherical heterogeneities with a radius of 1 k in the lower antle. We find that these superplues induce a axiu in gravity of about 37 with a large part corresponding to a ean variation of gravity. We conclude that we can expect to have a gravity of body tide with an order of agnitude of about hundred of induced by the biggest antle heterogeneities of density. This in gravity should be easurable with superconducting gravieters if all other contributions in the signal could be extracted with a sufficient precision. Key words: body tides, elasto-gravity theory, lateral heterogeneity, antle convection, nuerical ethod, superplues. GJI Geodesy, potential field and applied geophysics 1 INTRODUCTION At periods longer than 1 hr, the ost iportant part of Earth continuous deforation is induced by luni solar tides. The attraction of the Sun and the Moon causes global surface otions and variations in the gravity field, which ay be observed with geodetic eans. The surface displaceent can reach about 5 c and the gravity variations a hundred of. The body tides have been investigated since the 19th century with the work of Lord Kelvin (Sir Willia Thoson 1862). Presently, the ost accepted earth body tide odels deal with an ellipsoidal, rotating Earth, containing a liquid core and an anelastic antle with hydrostatic pre-stresses (Wahr 1981; Wahr & Bergen 1986; Dehant 1987). The inner structure of the Earth in present tides odel is classically based on PREM seisological odel (Dziewonski & Anderson 1981). However, the inner structure of the Earth is ore coplex than in a Spherical Non-Rotating Elastic Isotropic (SNREI) earth odel like PREM. Seisology and fluid dynaic studies show that the antle presents a heterogeneous structure induced by a thero-cheical C 27 The Authors 897 Journal copilation C 27 RAS

2 898 L. Métivier, M. Greff-Lefftz and M. Diaent convection (Davaille 1999; Forte & Mitrovica 21; Gu et al. 21) and departs fro hydrostatic equilibriu state. Large lateral heterogeneities have taken place in a illion year tiescale, as for the two superplues invoked under the Pacific and South Africa superswells (Courtillot et al. 23). These aspects of the antle structure are classically not taken into account in the deforation odels. The elasto-gravitational deforations are presently observed with a very good accuracy. The accuracy of superconducting gravieters and of positioning techniques (GPS, VLBI) has seen a large iproveent the last decade. Moreover, different satellites dedicated to gravity easureent have been launched or will be launched, like the GRACE ission in 22 and GOCE in 27. One of the purpose of this work is to deterine if the present reference body tide odel is sufficiently realistic to correct and to understand the coing deforation and gravity data. Few authors partially investigated the influence of lateral heterogeneities on the Earth tidal deforations. Molodenskiy (1977) was the first to theoretically work on this proble. He investigated a variational approach of the elasto-gravitational equations and their first-order s induced by lateral variations and topographies. Following this way, Wang (1991) coputed a odel of the Earth solid tides with low degree lateral variations of density and of rheological paraeters. Wang (1994) (see also Métivier et al. 25) recalculated also the effect of rotation and ellipticity on the Love nubers. Finally, Dehant et al. (1999) studied the influence of the non-hydrostatical ellipticity of internal boundaries on solid tides. These different works globally showed that the effect of low degree lateral variations on solid tides is sall but not necessarily negligible regarding to present gravieter precision. Yet they did not take into account possible deviatoric pre-stresses whose effects on the Earth s body tides are totally unknown. Taking into account the lateral variations in the calculation is a difficult proble as heterogeneities induce a bias fro hydrostatic equilibriu in the Earth. In a copanion paper (Métivier et al. 26), we have developed a new earth elasto-gravitational deforation odel able to take into account the heterogeneous structure of the antle. The odel has been realized using a spectral eleent ethod. The odel is solved in two steps: we first deterine the solution for a SNREI odel with a hydrostatical state of pre-stress, then lateral variations of density, interface topographies, and deviatoric pre-stresses are introduced as s of the SNREI Earth. The ai of the present article is to evaluate the ipact of antle lateral variations of density on the body tides response of the planet. Experient 1 Experient 2 Experient 3 Dynaical topography Dynaical topography Dynaical topography Geoid topography Geoid topography Geoid topography Figure 1. The three odels of Earth we discuss here. The figure shows transversal representations of the Earth containing a spherical heterogeneity, and, for each experient, the corresponding surface dynaical topography and the geoid topography. C 27 The Authors, GJI, 168, Journal copilation C 27 RAS

3 Mantle lateral variations and deforations: application 899 Sxx Sxy Sxz Syy Syz Pressure Szz Figure 2. The of pre-stress tensor in the lower antle (radius 543 k) due to the presence of a spherical heterogeneity of density in the antle (for the experient 1). M2 tidal response of the SNREI Earth odel c Surface displaceent Surface gravity variation Figure 3. The surface displaceent and the surface gravity variation induced by the M2 tide attraction on the SNREI earth odel at a tie t. In order to understand the relative influence of the different paraeters, we investigate tide s induced by siple spherical heterogeneities of density in the antle. We test three types of heterogeneities of various sizes and positions. Our ai is to provide an idea about tide sensitivity to heterogeneities induced by antle convection like plues and superplues. The first part of the paper is dedicated to the earth reference odel we used. We deterine the coplex equilibriu state of our planet C 27 The Authors, GJI, 168, Journal copilation C 27 RAS

4 9 L. Métivier, M. Greff-Lefftz and M. Diaent when it contains a antle heterogeneity. The presence of lateral heterogeneities in the antle induces a departure fro hydrostatic equilibriu state and dynaical topographies on the interface due to viscous oveent in the antle. In the second part, we study the M2 tide deforation of the planet and deterine the induced by various heterogeneities. Finally, we investigate and discuss the ipact of the African and the Pacifican superplues on body tides. 2 STATIC EQUILIBRIUM STATE OF THE EARTH We present here, the referential earth odel we use in our calculation. A laterally heterogeneous planet presents an equilibriu state ore coplicated to evaluate than a SNREI earth odel. The lateral heterogeneities in the antle (plues, superplues, slabs) are ostly due to a thero-cheical convection. They induce a departure fro hydrostatic equilibriu and a odification of interface shape (the dynaical topography) which have to be known before coputing the tidal. 2.1 The SNREI earth odel Let us define a SNREI earth odel coposed of fluid and solid layers, in which the physical properties continuously and soothly vary with radial position. The layers are deliited by spherical surfaces, called internal boundaries, and the global odel is bounded by an external surface. For siplicity our odel consists of six hoogeneous layers: the solid inner core, the liquid core, the D layer, the lower antle, the upper antle and the crust. The density and the elastic paraeters of the layers are chosen as average of the PREM odel paraeters (Dziewonski & Anderson 1981). 2.2 Mantle spherical heterogeneities Let us define now a new earth odel which contains a spherical heterogeneity of density in the antle rising in the antle on the Experient Surface displaceent Surface tidal gravity Experient 2 Experient Surface tidal gravity Surface tidal gravity Figure 4. On the top: the s of tidal surface displaceent and of tidal surface gravity variation induced by the antle heterogeneity at a tie t (experient 1). On the botto: the of tidal surface gravity variation induced by the antle heterogeneity at a tie t for experient 2 and 3. C 27 The Authors, GJI, 168, Journal copilation C 27 RAS

5 Mantle lateral variations and deforations: application 91 convection tiescale (typically the illion years). We assue an elastic lithosphere and a two-layered viscous antle with a viscosity of 1 21 and Pasfor the upper and lower antle, respectively. Knowing this viscosity profile, we can deterine the new Earth equilibriu state. The heterogeneities in the antle are induced by a thero-cheical convection. They present various shapes and sizes, with a typical density variations of about 1 per cent of the ean density of the Earth. We note particularly that, according to Davaille (1999), the superplues could have a size as large as the lower antle thickness. In order to theoretically evaluate the of the M2 tide induced by the antle heterogeneities, we present three experients (Fig. 1): (1) our liit case, with a heterogeneity of 1 k of radius in the lower antle; (2) with a heterogeneity of 5 k of radius at the botto of the lower antle and (3) with a heterogeneity of 285 k of radius in the upper antle. We believe that these heterogeneities are representative of antle plues and superplues. In the present study, the shape of the heterogeneities is less iportant than the ass involved. The density variation has been set to 5 kg 3.Welocate the centre of heterogeneities on the equator because sei diurnal tides are axiu in this latitude. Fig. 1 shows the surface s of topography and of geoid induced by the heterogeneities. The topography is dynaic on a illion years tiescale, but at tidal tiescale, the topography induced by the heterogeneity can be assued to be static. One can see, in Fig. 1, that the dynaical topography has not the sae sign than the geoid topography when the heterogeneity is located in the lower antle. This is due to the fact that the upper antle is less viscous than the lower antle and the lithosphere (elastic). The rising of the heterogeneity in the lower antle induces in the fluid of the upper antle tangential oveents, which are larger than the local radial otion. Therefore, by coupling, the lithosphere is getting thinner at the top of the heterogeneity and a subsidence can be seen on the surface. The internal boundaries present also topographies, particularly the core antle boundary (CMB). When the heterogeneity is located in the lower antle, the CMB presents a positive topography quite large. In this last case, the geoïd is ore affected by the topography on the CMB than on the surface, leading to a geoid topography positive when the surface dynaical topography is negative. The rising of the heterogeneity in the antle induces in the pre-stress equilibriu of the Earth. Supposing that the spherical heterogeneity is evolving in the antle at the Stoke s velocity, one can deterine the new pre-stress tensor of the Earth. For exaple, we present in Fig. 2 the of pre-stress tensor coponents in the antle (at radius 543 k) induced by the largest heterogeneity. In the present paper, we study only density heterogeneities. However, the antle convection induces also lateral variations in the elastic properties and lateral variations of viscosity. Such aspects of the antle will be address in the future. 3.1 The tidal response of the SNREI earth odel Fig. 3 shows the surface displaceent and the variation of gravity potential induced by the M2 tide on our SNREI odel, when the Moon is located at longitude zero. The displaceent has a axiu of 17 c and the gravity of alost Ipact of lateral heterogeneities Using the algorith described in Métivier et al. (26), we coputed the tidal induced by the heterogeneities (see Fig. 1). The ethod is based on a spectral eleent ethod (Koatitsch & Trop 22; Chaljub et al. 23; Chaljub & Valette 24) and is parallelized. We coputed the solutions using typically eight processors on the parallel coputer server of the Royal Observatory of Belgiu. Fig. 4 shows the surface s of displaceent and of tidal gravity variation induced by the presence of the largest heterogeneity (experient 1). The solutions have been coputed on the defored surface of the Earth. This figure shows the axiu s which correspond to the instant when the Moon is just above of the heterogeneity (or at opposite longitude). One can see that displaceent is saller than 1, and that the gravity has a axiu aplitude of 525 nanogal 3 THE THEORETICAL M2 BODY TIDE RESPONSE OF THE PLANET We now discuss here the s induced by the antle heterogeneities on the M2 body tide of the Earth. The M2 tide is the ajor tide wave affecting the Earth. It is a sei-diurnal wave tide (with a exact period of 12h27) induced by the Moon attraction. We focused our work on this wave tide in order to evaluate the axiu induced by antle heterogeneities on body tides. 18 Figure 5. Siple odel of the Pacifican and the African superplues within the Earth. Botto: the circles show the superplue locations. The radius of the circles corresponds to the radius of the heterogeneities. C 27 The Authors, GJI, 168, Journal copilation C 27 RAS

6 92 L. Me tivier, M. Greff-Lefftz and M. Diaent t= TM2 t= 1/5 TM t= 4/5 TM t= 3/5 TM t= 2/5 TM Figure 6. Evolution with tie of the gravity variation induced by the M2 tide during one tidal period (TM2 = 12h27), for the earth odel shown in Fig. 5. On the top: the tidal gravity variation of the SNREI odel, on the botto: the of tidal gravity variation induced by the two superplues. (). Regarding to the present precision in position easureent, the displaceent is too sall to be detected. The gravity should be easurable with superconducting gravieters which have a instruental precision. For the experients 2 and 3, we only present the gravity s since the displaceent s are too sall to be considered. In the experient 2, the axiu gravity is about 12, and in the experient 3, the axiu is about T H E I M PA C T O F S U P E R P L U M E S O N M 2 B O DY T I D E Many authors proposed that the antle convection is doinated by two very large heterogeneities in the lower antle: the superplues (see e.g. Davaille 1999; Kerr 1999; Courtillot et al. 23). One would be beneath French Polynesia, the other beneath South Africa. These heterogeneities would be the ost iportant ones in the antle. We investigate here what would be the axiu induced by these antle heterogeneities on M2 tide response of the Earth. Fig. 5 shows the superplues location in the Earth. We choose to represent the superplues as two spheres of 1 k of radius. The solution on surface noral gravity is presented on Fig. 6. The solution presents the in evolution with tie during 1 period of the M2 tide. We see that the axiu on gravity is about 37. We note that the solution contains a large degree zero part. Consequently the ajority of the tie signal corresponds to a ean variation of gravity all over the world (about 2 ). This is due to the fact that the heterogeneities are alost at opposite longitude. The sei diurnal tide corresponds to a spherical haronics of degree 2 and order 2, which eans that the tidal attraction is equal at opposite longitude. Therefore, the coupling between the variation of antle density and of the tidal attraction induces a large degree. 5 C O N C LU S I O N We show that the ipact of heterogeneities due to antle convection on body tide is quite sall. The induced by largeheterogeneities in the antle is about 1 per cent, in surface tidal displaceent and in surface tidal gravity. The displaceent pertur- bation is saller than 1 and consequently cannot be detected with present position easureent techniques. The in gravity can reach 5 for the largest plausible heterogeneity in the antle. The order of agnitude is ore than 1 ties larger than the present superconducting gravieter precision. We see particularly that superplues can induce a axiu in gravity of about 37. A large part of the signal corresponds to a ean variation of gravity of about 2 ngal. The superplues odelled in our experient are very large. It is probable that superplues have a saller diension, between the heterogeneity of experient 1 and of experient 2 in Fig. 1. Regarding to solution in the Fig. 4 we can expect to have a gravity with an order of agnitude of about 1 induced by these heterogeneities. This in gravity should be easurable with superconducting gravieters if all other contributions in the signal could be extracted with a sufficient precision. Thus, the body tides should bring inforation on the existence or not of heterogeneities like superplues in the Earth antle. AC K N OW L E D G M E N T S We thank O. de Viron and V. Dehant for fruitful discussions. We thank the two reviewers for their helpful coents on the anuscript. This study is IPGP contribution nuber REFERENCES Chaljub, E. & Valette, B., 24. Spectral eleent odeling of three diensional wave propagation in a self-gravitating Earth with an arbitrarily stratified outer core, Geophys. J. Int., 158, Chaljub, E., Capdeville, Y. & Vilotte, J.-P., 23. Solving elastodynaics in a fluid-solid heterogeneous sphere : a parallel spectral eleent approxiation on non-conforing grids, J. Coput. Phys., 187, Courtillot, V., Davaille, A., Besse, J. & Stock, J., 23. Three distinct types of hotspots in earth s antle, Earth planet. Sci. Lett., 25, Davaille, A., Siultaneous generation of hotspots and superswells by convection in a heterogeneous planetary antle, Nature, 42, Dehant, V., Tidal paraeters for an inelastic Earth, Phys. Earth planet. Inter., 49, Dehant, V., Defraigne, P. & Wahr, J.M., Tides for a convective Earth, J. geophys. Res., 14, C 27 The Authors, GJI, 168, C 27 RAS Journal copilation

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