Motion of the Mantle in the Translational Modes of the Earth and Mercury

Physics of the Earth and Planetary Interiors 151 (2005) 77–87

Motion of the mantle in the translational modes of the Earth and Mercury

Pavel Grinfeld∗, Jack Wisdom

Department of Earth, Atmospheric, and Planetary Sciences, Massachusetts Institute of Technology, 77 Massachusetts Ave., Cambridge, MA 02139, USA Received 30 July 2004; received in revised form 2 January 2005; accepted 14 January 2005

Abstract

Slichter modes refer to the translational motion of the inner core with respect to the outer core and the mantle [Slichter, L., 1961. The fundamental free mode of the Earth’s inner core. [Proc. Natl. Acad. Sci. U.S.A. 47, 186–190]. The polar Slichter mode is the motion of the inner core along the axis of rotation. Busse [Busse, F.H., 1974. On the free oscillation of the Earth’s inner core. J. Geophys. Res. 79, 753–757] presented an analysis of the polar mode which yielded an expression for its period. Busse’s analysis included the assumption that the mantle was stationary. This approximation is valid for planets with small inner cores, such as the Earth whose inner core is about 1/60 of the total planet mass. On the other hand, many believe that Mercury’s inner core may be enormous. If so, the motion of the mantle should be expected to produce a significant effect. We present a formal framework for including the motion of the mantle in the analysis of the translational motion of the inner core. We analyze the effect of the motion of the mantle on the Slichter modes for a non-rotating planet with an inner core of arbitrary size. We omit the effects of viscosity in the outer core, magnetic effects, and solid tides. Our approach is perturbative and is based on a linearization of Euler’s equations for the motion of the fluid and Newton’s second law for the motion of the inner core. We find an analytical expression for the period of the Slichter mode. Our result agrees with Busse’s in the limiting case of a small inner core. We present the unexpected result that even for Mercury the motion of the mantle does not significantly change the period of the oscillation. © 2005 Elsevier B.V. All rights reserved.

Keywords: Fluid dynamics; Inner core; Mantle; Slichter modes; Perturbation theory

1. Introduction respect to the outer core and the mantle. The first analytical treatment of the problem was performed by Busse The Slichter modes, first studied in Slichter (1961), (1974). In the past 12 years, there has been an explo- refer to the translational motion of the inner core with sion of scientific activity in the research of the normal modes of the Earth primarily due to the emergence of ∗ Corresponding author. superconducting gravimeters capable of detecting the E-mail addresses: [email protected] (P. Grinfeld); relative motion of the inner core by measuring the vari- [email protected] (J. Wisdom). ations in the Earth’s gravitational field. Some authors

0031-9201/$ – see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.pepi.2005.01.003 78 P. Grinfeld, J. Wisdom / Physics of the Earth and Planetary Interiors 151 (2005) 77–87 believe that evidence of Slichter modes can be found in core and the mantle and Euler’s equations for the mo- the existing gravimeter data (Smylie, 1992). However, tion of the liquid core. a definitive detection has proven to be controversial Looking ahead, we find that the motion of the man- (Crossley et al., 1992; Jensen et al., 1995; Hinderer et tle will have little effect on the period of the Slichter al., 1995) and the search for Slichter modes continues modes for the Earth and even for Mercury, for which today (Courtier et al., 2000; Rosat et al., 2003), along it will introduce a correction of only about 0.1%. This with active theoretical research (Smylie and McMillan, is an unexpected and counter-intuitive result given the 2000; Rogister, 2003). significant amplitude of the mantle’s oscillation. Com- The upcoming observations of Mercury (Peale et pare two simple systems, in one a mass m is connected al., 2002; Spohn et al., 2001) will present an exciting to a stationary wall by a spring of stiffness k. This sys- new opportunity for the study of the Slichter modes. tem is analogous√ to a stationary mantle and it frequency Many believe that much like the Earth, Mercury too of oscillation is k/m. In the other system, the mass has a solid inner core in a fluid outer core (Schubert m is connected by the same spring to a mass of 10m et al., 1988). However, unlike the Earth’s inner core which is free to oscillate. This is analogous to a mov- whose mass is 1/60 of the total planet, Mercury’s inner ing mantle of a moving mantle and we explain below core may be enormous (Siegfried and Solomon, 1974) why 10m is appropriate.√ The frequency of oscillation with a radius that is nearly 2/3 of the radius of the planet of the second system is 11k/10m which constitutes and mass that is about 60% of the total planet. A simple an almost 5% difference from the first system. The fact conservation of linear momentum computation shows that we obtain an estimate of 0.1%, rather than 5%, that the amplitude of the mantle’s oscillations equals highlights the effect of the fluid on the dynamics of about 10% of the amplitude of the inner core, compared the system and demonstrates the necessity for a formal to 0.1% for the Earth. The change in the gravitational analytical approach. field due to a displacement of the inner core will be nearly 100 times greater for Mercury than for Earth making the detection of the modes a less challenging 2. Model and methodology proposition. The rotation of the planet leads to a split in the spec- Undoubtedly, an advanced model of the Earth is trum yielding the “Slichter triplet”: a single axial (or needed for a thorough analysis of the Slichter oscilla- polar) mode and two equatorial modes, retrograde and tions and accurate prediction of the eigenperiods. In all prograde. The three eigenperiods are evenly spaced by likelihood, such models will require the use of numer- about a quarter of an hour. Of the three modes, the po- ical methods. We set a more modest goal for ourselves lar mode is best suited for analytical treatment. A first and that is to study the effect of the motion of the man- complete analytical treatment of the polar mode was tle. We therefore choose to study a simple problem that presented by Busse (1974). Busse considered a sim- can be carried through analytically. ple three-layer model of the rotating spherical Earth, in We consider a three layer model of a planet which the outer core was incompressible, inviscid and (Fig. 1) with a rigid inner core Ω1 of density ρ1 and of constant density. As we show below, Busse’s results radius R1, a fluid outer core Ω2 of density ρ2 and ra- are consistent with the assumption that the outer mantle dius R2, and a rigid outer mantle Ω3 of density ρ3 and is stationary. radius R3. We assume that each density is constant. Let We generalize Busse’s analysis to a planet whose Sn be the boundary of domain Ωn. We study the oscil- outer mantle is allowed to move. In the process, we lations of the inner core under the influence of grav- build a formal analytical framework for analyzing a ity and fluid pressure. We assume that the fluid outer three-layer planet, which can be used to incorporate core is incompressible and neglect the effects of viscos- more complicated effects, such as ellipticity of the lay- ity. We exclude from consideration the rotation of the ers, possible phase transformations at the inner core– planet which can affect the frequency of even the polar outer core boundary, and compressibility of the outer Slichter mode since it affects the dynamics of the fluid. core. Our analysis will be performed from “first princi- The fluid plays two important roles. It creates the ples”: Newton’s second law for the motion of the inner restoring gravitational force and is also responsible for P. Grinfeld, J. Wisdom / Physics of the Earth and Planetary Interiors 151 (2005) 77–87 79

We would like most of our intermediate expressions to generalize to the case of multilayer fluid planets as well as to the study of phase transitions. To this end we describe the motion of the constituents of the system by specifying the normal velocities C of the interfaces (see Appendix A) and assuming, for as long as possible, that C is completely arbitrary. We express the three normal velocity fields in the vicinity of the equilibrium configuration as harmonic series C θ, φ = R ω iωt ClmY θ, φ 1( ) 1 i e 1 lm( ) (1a) l,m C θ, φ = R ω iωt ClmY θ, φ 2( ) 2 i e 2 lm( ) (1b) l,m

Fig. 1. The equilibrium configuration. C θ, φ = R ω iωt ClmY θ, φ 3( ) 3i e 3 lm( ) (1c) l,m the effects of pressure. Our approach treats pressure where φ is the longitude, θ the colatitude, and in a formal way as governed by the Euler equations. Ylm(θ, φ) are spherical harmonics normalized to Slichter and Busse provide an insightful decomposi- unity: tion of pressure force into a hydrostatic component and one that is a response to the acceleration of the inner ∗ core. Slichter combined the gravitational force and the Yl m (θ, φ)Y (θ, φ)dS = δl l δm m , 1 1 l2m2 1 2 1 2 hydrostatic pressure into a single expression that van- |r|=1 ishes when the densities of the inner and outer cores match. where ∗ means complex conjugation. Here and in all The spherically symmetric mantle does not exert any subsequent expressions involving complex numbers gravitational forces on the internal bodies. Only its total taking the real part is implied. mass is relevant and so we expect that the final answer The normal boundary velocity of a rigid sphere ρ R ρ R3 − includes 3 and 3 only in the combination 3( 3 can be fully represented by the l = 1 harmonics. R3 = 2). In the absence of the fluid outer core, the inner If the inner core is moving with velocity v ωR iωt AX,AY ,AZ core would be in a state of neutral equilibrium (any i 1 e ( 1 1 1 ) then the resulting normal ve- deviation from our idealization will cause the inner core locity C is given by to “fall” onto the mantle). Consequently, our eventual expression for the oscillation frequency must approach C1(θ, φ) = v · N zero in the limit ρ = 0. 2 = v θ φ + v θ φ + v θ Our approach is perturbative, in which every con- 1 sin cos 2 sin sin 3 cos figuration of the system is treated as a small devia- π = ωR iωt 2 AX − AY Y θ, φ tion from the spherically symmetric stable configura- i 1 e ( 1 i 1 ) 1(−1)( ) tion and all velocities are small. The “unperturbed” 3 spherically symmetric gravitational potential is ψ and 0 π its rate of change ∂ψ/∂t, determined perturbatively, is 4 Z + A Y10(θ, φ) induced by the motion of the system. We solve for the 3 1 fluid velocity field vR, v (v = 0) and pressure p con- π sistent with the translational motion of the inner core + 2 AX + iAY Y θ, φ ( 1 1 ) 11( ) and the mantle. 3 80 P. Grinfeld, J. Wisdom / Physics of the Earth and Planetary Interiors 151 (2005) 77–87

its determinant vanishes. The two equations arise from conservation of momentum for the planet as a whole and Newton’s second law for the inner core. The two C10 C10 unknowns 1 and 2 are, essentially, the amplitudes of oscillation of the inner core and the mantle. We nondimensionalize many of our expressions by using a length scale R∗ and a density ρ∗. The partic- ular choice of R∗ and ρ∗ can be made later. Introduce the dimensionless densities n and dimensionless radii Qn:

ρn Rn n = ; Qn = (3) ρ∗ R∗ and a convenient quantity (G is the gravitational constant) π 4 2 Ψ∗ = Gρ∗R∗ (4) Fig. 2. The velocity of the interface induced by the velocity of the 3 sphere’s center, C = v3 cos θ. that has dimensions of gravitational potential. resulting in 3.1. Conservation of momentum − , ∗ 2π X Y C1( 1) = (C1 1) = (A − iA ) (2a) Conservation of momentum is particularly easy 1 3 1 1 to convert to a linear equation. It is equivalent to 4π Z stating the center of mass of the system remains at C10 = A (2b) 1 3 1 rest: 4π Z ωt Therefore, if the oscillation of the inner core takes place (ρ − ρ )R3iωR A ei R AZ 3 1 2 1 1 1 in the axial direction with amplitude 1 1 , Eq. (2b) π represents the relationship between the amplitude and + 4 ρ R3 − R3 ωR AZ iωt C10 2( 2 1)i 2 2 e 1 . Fig. 2 illustrates this computation for the axial 3 motion of the sphere. 4π Z ωt + ρ (R3 − R3)iωR A ei = 0 3 3 3 2 3 3 3. Overview of the analysis Utilizing (2b), recognizing the fact that for a rigid R AZ = R AZ Q C10 = Q C10 mantle 2 2 3 3 (or 2 2 3 3 ), nondi- 4 We consider the “axial” mode in which the inner mensionalizing by dividing through by ρ∗R∗ and, ﬁ- √ ωt core and the mantle oscillates along the planet’s axis nally, cancelling 4π/3iω ei , we obtain our ﬁrst of rotation. Of course, since we ignore the effects of equation: rotation, the axial direction is in no way preferred ( − )Q4C10 + Q ( Q3 and our oscillation mode is triply degenerate. How- 1 2 1 1 2 2 2 + Q3 − Q3 C10 = ever, the degeneracy is not much of an issue since it 3( 3 2)) 2 0 (5) is removed from the mathematical analysis by a pri- ori stating the direction in which the oscillation takes The ratio of the amplitudes of the motions of the mantle and the inner core is place. The oscillation frequency ω will be determined from 10 3 Q2C (1 − 2)Q the relationship that states that a homogeneous linear 2 = 1 (6) Q C10 Q3 + Q3 − Q3 system of equations has a nonzero solution if and only if 1 1 2 2 3( 3 2) P. Grinfeld, J. Wisdom / Physics of the Earth and Planetary Interiors 151 (2005) 77–87 81

This quantity is small for the Earth (0.1%) but may We represent the time-dependent quantities ψ(t) and be quite substantial for Mercury (10%). The fact that p(t)as the ratio of amplitudes is about 10% explains why we m chose to attach a mass of m to a mass of 10 in our ψ(t) = ψ0 + ψ¯ (t) (9a) spring example above.

p(t) = p0 + p¯ (t) (9b) 3.2. Newton’s second law for the inner core and ψ0 and p0 are time-independent gravitational po- The inner core experiences two forces: the gravita- tential and pressure that correspond to the equilibrium tional force exerted at every point inside the inner core configuration in which the inner core rests at the center and the hydrodynamic force applied at the boundary. of mass of the planet and ψ¯ (t) andp ¯ (t) are small time- The gravitational force is proportional to the density dependent corrections. The equilibrium gravitational of the inner core and the vector gradient of the grav- potential ψ0 and pressure p0 satisfy the hydrostatic itational potential ψ. The hydrostatic force is propor- equation in the outer core: tional to the pressure p at the boundary and points along the boundary normal. Therefore, Newton’s second law ∇p0 + ρ2∇ψ0 = 0. (10) reads An application of Eqs. (8a) and (8b) to Newton’s second M a =− ρ ∇ψ dΩ − pN dS, (7) 1 1 law (7) yields: Ω1 S1 where a is the acceleration of the inner core (a = ∂ψ ∂p Z ωt da −ω2R A ei ) and N is the outward normal—thus the M1 =− ρ1 + N 1 dt S ∂t ∂t minus sign for the pressure contribution. 1 This equation is nonlinear since the domain of + C (ρ ∇ψ + ∇p ) dS (11) integration and the integrand are both time depen- 1 1 0 0 dent. The equation is linearized by taking a time derivative and keeping first order terms. Differ- We make use of the equation of hydrostatic equilibrium entiation of integrals makes use of the following (10) to eliminate p0: formulas for the time dependent volume and surface integrals: ∂ψ ∂p M da =− S ρ + 1 d 1 N dt S ∂t ∂t d 1 f (t, Ω)dΩ dt Ω + C1(ρ1 − ρ2)∇ψ0 (12) ∂f (t, Ω) = dΩ + Cf (t, Ω)dΩ (8a) Ω ∂t ∂Ω Finally, we convert the vector equation into a scalar one by projecting it onto the oscillation axis by dotting the d f (t, S)dS last equation withz ˆ: dt S ␦f (t, S) da ∂ψ ∂p ∂ψ0 = dS − Cκf (t, S)dS, (8b) M1 =− dS ρ1 + + C1(ρ1 − ρ2) S ␦t S dt S ∂t ∂t ∂r 1 where C is the invariant velocity of the interface 4π × Y (θ, φ), (13) introduced above, κ is the mean curvature of the 3 10 interface, and ␦/␦t is the derivative with respect · z = θ ∇ψ · z = ∂ψ /∂z = to the motion of the interface discussed briefly in since N ˆ cos and 0 ˆ √ 0 Appendix A and more thoroughly in Grinfeld (2003) (∂ψ0/∂r)(∂r/∂z) = (∂ψ0/∂r) cos θ and 4π/3Y10 and Grinfeld and Wisdom (2005). (θ, φ) was substituted forcos θ. Therefore, the three 82 P. Grinfeld, J. Wisdom / Physics of the Earth and Planetary Interiors 151 (2005) 77–87

∂ψ ∂ψ ∂p 0 where quantities to be determined are ∂r , ∂t and ∂t . This   is the task that we are turning to now, starting with the −3 Q2 − 3 Q2 − 3 Q2    [ ]1 1 [ ]2 2 [ ]3 3  gravitational potential. A  2 2 2  1      −3 Q2 − 3 Q2   A   [ ]2 2 [ ]3 3   2   2 2   A     3   −3 Q2  4. Gravitational potential   =  [ ]3 3   B2   2     − Q3   B   [ ]1 1  This section contains an outline of how we compute 3   the gravitational potential and its evolution. A detailed  − Q3 − Q3  B4  [ ]1 1 [ ]2 2  description of the method of analysis can be found in − Q3 − Q3 − Q3 Grinfeld and Wisdom (2005). [ ]1 1 [ ]2 2 [ ]3 3 The gravitational potential ψ(r, θ, φ) satisfies the (17) Poisson equation The quantity ∂ψ0/∂r at S1 is given by ∂ψ Ψ ∇2ψ = 4πGρ, (14) 0 ∗ = 1Q1 (18) ∂r R∗ S1 where ρ is taken to be ρ1, ρ2, ρ3 or 0 depending on The equations for the potential perturbation ∂ψ/∂t the region. ψ is finite at the origin, vanishes at infinity, are obtained by differentiating the gravitational system and is continuous along with its derivatives across all (14)–(15b) with respect to time. The differentiation of interfaces. Using the notation [X]n to indicate the jump the bulk Eq. (14) yields ρ = of the quantity X across the interface n (e.g. [ ]1 ∂ψ ρ − ρ ), we write the continuity conditions as ∇2 = 1 2 ∂t 0 (19) indicating that ∂ψ/∂t is harmonic. The boundary con- [ψ] , [ψ] , [ψ] = 0 (15a) 1 2 3 ditions (15a) and (15b) are differentiated in the invariant sense discussed in Grinfeld (2003). We obtain that ∂ψ/∂t N · [∇ψ]1, N · [∇ψ]2, N · [∇ψ]3 = 0 (15b) is continuous across all interfaces, while the normal derivative of ∂ψ/∂t jumps by an amount proportional to the velocity of the interface C and the jump ψ r The equilibrium potential 0( ) is straightforward in the second normal derivative of the unperturbed po- to compute: tential ψ0. We use indicial notation with summation convention: ψ0(r, θ, φ) ∂ψ  = 0 (20a) 2 ∂t  r 1,2,3  1 + A , inner core  R 1  2 ∗  ∂ψ  − Ni ∇ =−C NiNj ∇ ∇ ψ  r 2 r 1 i 1,2,3 [ i j 0]1,2,3  2 + A + B , ∂t , ,  2 2 outer core 1 2 3 2 R∗ R∗ = Ψ , (20b) ∗   r 2 r −1  3 + A + B , ␦/␦t  3 3 mantle These equations are obtained by applying the -  2 R∗ R∗  derivative to the boundary conditons (15a) and (15b)  − ␦/␦t  r 1 and utilizing the algebraic properties of the -  B , 4 R outside derivative outlined in the Appendix A. ∗ Finally, ∂ψ/∂t is finite at the origin and vanishes at (16) infinity. The resulting system is solved by separation P. Grinfeld, J. Wisdom / Physics of the Earth and Planetary Interiors 151 (2005) 77–87 83 of variables: counterpart, to compute the potential inside a slightly ∂ψ ellipsoidal cavity. r, θ, φ ∂t ( ) For simple translational motion, the sole present harmonic is the l = 1, m = 0 term and the expressions  l  lm r  A , inner reduce to  1 R∗  l −l−  r r 1      Alm + Blm , outer A10 2 R∗ 2 R∗ 1 111 = Ψ∗ l −l−      lm r lm r 1  10    l,m  A + B ,  A   011    3 R∗ 3 R∗ mantle 2 10      [] Q1C  −l−  10    1 1  lm r 1  A   001   B , outside 3  10  4 R∗   =−   [] Q2C  (23)  B10   Q3  2 2    1 00 iωt 2 10 |Ylm(θ, φ)iω e , (21)     [] Q3C  B10   Q3 Q3  3 3  3   1 2 0  The remaining sets of six coefficients are determined B10 Q3 Q3 Q3 by satisfying the six boundary conditions (20a) and 4 1 2 3 (20b). Since both sides are expressed as series in spher- C10 C10 C10 ical harmonics, the boundary conditions are met by As discussed above, the scalars 1 , 2 , and 3 are satisfying the identities for each of the spherical har- essentially the nondimensionalized√ amplitudes of os- monics. This leads to a 6 × 6 linear system whose cillations (save for a multiplier of 4π/3). Of particu- solution is lar interest below are the values of ∂ψ/∂t in the domain     Ω − Ω Alm Q−l+1 Q−l+1 Q−l+1 2 1 occupied by the outer core where the fliud  1   1 2 3  equations are solved and the contributions to Newton’s  Alm   Q−l+1 Q−l+1  second law are made. For future reference, we present  2   0 2 3      the expression for ∂ψ/∂t (once again using the fact that  Alm   Q−l+1   3  3  003  Q C10 = Q C10   =−   2 2 3 3 )  lm  l +  l+2   B  2 1  Q 00  2   1   lm   l+ l+  ∂ψ B Q 2 Q 2 0  3   1 2  (r, θ, φ) ∂t Ω −Ω Blm Ql+2 Ql+2 Ql+2 2 1 4 1 2 3   r lm 10 Q C =−Ψ∗ ([]2 + []3)Q2C [ ]1 1 1 2 R   ∗ ×  Q Clm   [ ]2 2 2  (22) r −2 + Q4C10 Y θ, φ ω iωt Q Clm [ ]1 1 1 10( )i e [ ]3 3 3 R∗ We would like to note that the rate of change of poten- (24) tial (21 and 22) applies to arbitrary perturbations of the interfaces, with simple translational motion along the At the inner core boundary, we have axis being a special case for which l = 1 and m = 0. The presented expressions can be used to incorporate ∂ψ the effect of phase transformations at the inner core– θ, φ =−Ψ Q + Q C10 ∂t ( ) ∗ 1(([ ]2 [ ]3) 2 2 outer core boundary, which may result in a very compli- S1 cated evolution of the interface. Further, if the quantity + Q C10 Y θ, φ ω iωt t is not interpreted as time, but simply as a parameter [ ]1 1 1 ) 10( )i e describing the perturbation, (21 and 22) can be used to compute the corrections to the gravitational potential We have computed two of the three unknown quan- for near-spherical geometries. Grinfeld and Wisdom tities in the linearized Newton’s law (13) and now turn (2005) use this formula, along with its second-order our attention to the hydrodynamic pressure p. 84 P. Grinfeld, J. Wisdom / Physics of the Earth and Planetary Interiors 151 (2005) 77–87

5. Motion of the fluid able to make this guess saves us the trouble of solving a system of ODE’s. We assume that the outer core is inviscid and in- We arrive at the following solution for the pressure compressible of constant density ρ2. The velocity and correctionp ¯ and velocities v¯: pressure fields v and p are governed by Euler equations 2 2 lm iωt p¯ (t, r, θ, φ) = ρ2R∗ω P (r)Ylm(θ, φ)e (30a) ∂v 1 + v ·∇v =− ∇p −∇ψ (25a) ∂t ρ R lm iωt 2 v¯ (t, r, θ, φ) = R∗ωF (r)Ylm(θ, φ)e (30b) ∇·v = 0, (25b) 2 R∗ω lm ∂Ylm(θ, φ) ωt We allow slippage at the boundaries—the normal com- v¯ (t, r, θ, φ) = G (r) ei (30c) r2 ∂θ ponent of the velocity of the fluid matches normal velocity of the rigid interface. The linearization procedure starts by introducing the v¯ (t, r, θ, φ) = 0 (30d) small perturbations v¯ andp ¯ to the velocities and the pressure: where r l r −l−1 t r, θ, φ = r, θ, φ + t, r, θ, φ lm lm lm v( ; ) v0( ) v¯( ) (26a) P (r) = d+ + d− (31) R∗ R∗ p t r, θ, φ = p r, θ, φ + p t, r, θ, φ ( ; ) 0( ) ¯ ( ) (26b) lm lm dP (r) F (r) = iR∗ (32) The equilibrium velocities v0 vanish and, if we take a dr time derivative of the Euler equations, the unperturbed lm lm G (r) = P (r) (33) pressure p0 will drop out as well. Therefore, away from lm lm the boundaries the motion of the fluid is governed by and the coefficients d+ and d− are determined by the slippage boundary conditions which lead to the follow- ∂2v¯ ∂v¯ ∂v¯ 1 ∂p¯ ∂ψ + ·∇v¯ + v¯ ·∇ =− ∇ −∇ ing system: ∂t2 ∂t ∂t ρ ∂t ∂t lm −1 lm d+ l −(l + 1) C (27a) = 1 dlm lQl−1 − l + Q−l−2 Clm − 2 ( 1) 2 2 ∇·v¯ = 0 (27b) Alm Ψ∗ 2 v − (34) Since ¯ is considered small, we neglect the quadratic ω2R2 Blm terms 1 2 ∂2 ¯ ∂p ∂ψ The expression for pressure (30a) may be multiplied v =−1 ∇ ¯ −∇ 2 (28a) ∂t ρ2 ∂t ∂t by the normal and integrated over the outer core boundary to yield the total hydrodynamic force acting on the ∇· = . v¯ 0 (28b) inner core. Apply the divergence operator to the Euler equations We now substitute Eqs. (17), (22) and (34) and Q C10 = Q C10 (28a) 2 2 3 3 (the mantle’s boundaries move to- gether) into Newton’s second law (13) to obtain the ∂2∇·v¯ 1 ∂p¯ ∂ψ second algebraic equations for C10 and C10: =− ∇2 −∇2 (29a) 1 2 ∂t2 ρ ∂t ∂t Q3 − Q3 − Ψ 2( 2 1)( 1 2) ∗ Q Since the fluid is incompressible (28b) and ∂ψ/∂t is 2 2 1 2 R∗ω harmonic (19) we observe that ∂p/∂t¯ is harmonic. Therefore, in attempting to solve the linearized − Q3 + Q3 − Q3 − Q C10 hydrodynamic system by separation of variables, we (3 2 2 2( 2 1)( 1 2)) 1 1 know to use form (31) for the radial part ofp ¯ . Being P. Grinfeld, J. Wisdom / Physics of the Earth and Planetary Interiors 151 (2005) 77–87 85 Q3 − Q3 − . Earth Mercury + Q4 − 2( 2 1)( 1 2) ∗ Q C10 3 2 2 2 2 2 2 2 R 6 . R∗ω 1 (10 m) 1 2 1.7 R 6 . = 0 (35) 2 (10 m) 3 5 1.8 6 R3 (10 m) 6.3 2.4 As stated above, the frequency of oscillation ω is ob- ρ 3 −3 . tained by equating to zero the determinant of the system 1 (10 kg m )130 9.5 3 −3 (5 and 35). If the mantle were treated as stationary as ρ2 (10 kg m )12.0 8.0 ω was done by Busse, Eq. (35) alone would yield by ρ (103 kg m−3)4.5 3.0 C10 = C10 3 setting 2 0 and equating the coefficient of 1 to zero. This simple computation would lead directly to Using these values, we compute the intermediate Busse’s formula. quantities: Earth Mercury 6. Expression for the frequency and analysis of M (kg) 6.07 × 1024 3.27 × 1023 results m (1022 kg) 7.24 × 1021 3.09 × 1022

D (1022 kg) 1.72 × 1023 5.77 × 1021 Equating the determinant of the linear system (5), (35) to zero, we can easily solve for ω2. The answer is E (1022 kg) 2.16 × 1024 1.95 × 1023 most easily presented by introducing the total mass of And ﬁnally use the formulas (40) and (41) to arrive the system M: at the following estimates for the eigenperiod, T, π M = 4 ρ R3 + ρ R3 − R3 + ρ R3 − R3 Earth Mercury ( 1 1 2( 2 1) 3( 3 2)) (36) 3 T (h) 4.2361 8.3983

Also introduce three more mass-like quantities Tm M (h) 4.2359 8.3906 4π m = (ρ − ρ )R3 (37) We find that the motion of the mantle introduces 3 1 2 1 a correction for the Earth that is less than one hun- 4π dredth of 1Mercury is about 0.1%. This correction is D = (ρ − ρ )(R3 − R3) (38) 3 1 2 2 1 unexpectedly small given the assumed massiveness of π Mercury’s inner core. In fact, we have not been able E = 4 ρ R3 2 2 (39) to find a simple explanation for the smallness of this 3 correction. Then the natural frequency of oscillation is given by 4π D ω2 = Gρ (40) 8. Conclusions 3 2 3 E + D − m/M 2 (1 ) If we consider a planet with a small inner core, m M, We have presented a general framework for ana- we arrive at Busse’s expression: lyzing the free modes of the Earth that include the rigid motion of the planet’s mantle. This motion is π D 2 4 negligible for the Earth, but for Mercury, whose inner ωm M = Gρ (41) 3 2 3 E + D 2 core may constitute more than half of the planet’s total weight, the motion would be significant. We solved a linearized system of Euler’s equation for the fluid 7. Implications for the Earth and Mercury outer core. The motions of the inner core and mantle are governed by Newton’s second law which includes We make the following assumptions for the Earth the gravitational and hydrostatic effects. For Mercury, and Mercury (Siegfried and Solomon, 1974): the oscillation of the mantle is significant as its am- 86 P. Grinfeld, J. Wisdom / Physics of the Earth and Planetary Interiors 151 (2005) 77–87 plitude is about 10% of that of the inner core. We pendix briefly introduces the main concepts. An in- applied our analysis to compute an analytical expres- depth discussion of the ␦/␦t-derivative and its proper- sion for the period of the Slichter mode for a nonrotat- ties can be found in Grinfeld (2003) and Grinfeld and ing planet. Our expression reduces to Busse’s for the Wisdom (2005). case of a small inner core. We showed the surprising Consider a one parameter family of curves Sτ in- and counter-intuitive result that incorporating the rigid dexed by a time-like parameter τ. The family Sτ can motion of the mantle has a minimal effect on the pe- also be thought of as a time evolution of a single curve riod of Slichter oscillations for Mercury as well as the S. Let Tτ(Sτ) be a scalar field defined on Sτ,soT not Earth. only changes its values with the passing of time but If the translational mode is excited and it is pos- also sees its domain of definition change as well. sible to measure the amplitude of the mantle and the We present a geometric definition of the ␦/␦τ- change in gravitational field on the surface of Mercury, derivative at a point ξ on the surface Sτ at time τ, illus- such measurements may allow a direct determination trated in Fig. 3. Consider two locations of the surface Sτ ∗ of the size of the inner core. Suppose that a gravime- and Sτ∗ at nearby times τ and τ . Draw the straight line ter is placed at the north pole and that the translational orthogonal to Sτ passing through the point ξ mark the ∗ mode is along the axis of rotation. Then if the mantle point ξ where this straight line intersects Sτ∗ . Define: moves up by the amount A then according to Eq. (6) ∗ B = A ρ R3 + ρ R3 − ␦ Tτ Tτ∗ (ξ ) − Tτ(ξ) the mantle moves down by ( 2 2 3( 3 = lim (44) − ␦ τ τ∗→τ τ∗ − τ R3 ρ − ρ R3 1 2))(( 1 2) 1) . Thus, the gravimeter is closer to the inner core by Let 2z be the vector connecting the point ξ to the point ξ∗. Then the velocity of the interface C (also ρ − ρ R3 + ρ − ρ R3 + ρ R3 A + B = A ( 1 2) 1 ( 2 3) 2 3 3 known as the normal velocity) is defined as (ρ − ρ )R3 1 2 1 2z · N C = lim , (45) (42) τ∗→τ τ∗ − τ

The observed change in the gravitational field is due where N is the unit normal to the surface Sτ. Since by ρ − ρ R3 2 2 to the mass ( 1 2) 1 moving closer by the amount construction z is aligned with N, the projection z B. Therefore, the gravitational field, whose magnitude onto the normal is performed largely for the purposes is G(ρ − ρ )R3/r2| , will increase by 2G(ρ − of determining the sign of C.Ifz is the radius vector 1 2 1 r=R3 1 ρ )R3/r3(A + B)| , where G is the gravitational with respect to an arbitrary origin then the definition of 2 1 r=R3 constant. In other words, if g is the accelaration of C (45) can be rewritten as gravity and 2g is amplitude of the observed change ␦ then C = z · ␦τ N 2g = G ρ − ρ Q3 + ρ − ρ R /R 3 + ρ A 2 (( 1 2) 1 ( 2 3)( 2 3) 3) (43)

For Mercury, the coefﬁcient of proportionality between 3 3 2g and A in terms of R1/R3 is 3 × 10 G((R1/R3) + 15.231) which, if measured experimentally, would yield the size of the inner core.

Appendix A. The ␦/␦ t-derivative Fig. 3. Geometic definition of the ␦/␦t-derivative as applied to a This paper relies heavily on the calculus of mov- scalar field Tτ define at time τ on the surface Sτ with surface coor- ing surfaces, which has an illustrious history. This ap- dinates ξ. P. Grinfeld, J. Wisdom / Physics of the Earth and Planetary Interiors 151 (2005) 77–87 87

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