Inner core mantle gravitational locking and the super-rotation of the inner core

Transcription

1 Geophys. J. Int. (2010) 181, do: /j X x Inner core mantle gravtatonal lockng and the super-rotaton of the nner core Matheu Dumberry 1 and Jon Mound 2 1 Department of Physcs, Unversty of Alberta, Edmonton, Canada. E-mal: dumberry@ualberta.ca 2 School of Earth & Envronment, Unversty of Leeds, Leeds, UK Accepted 2010 February 11. Receved 2010 February 11; n orgnal form 2009 August 14 GJI Gravty, geodesy and tdes SUMMARY Sesmologcal observatons suggest that the Earth s sold nner core has been rotatng faster than the mantle over the past several decades, consstent wth the results of some numercal geodynamo models. However, the hemsphercal ansotropy structure of the nner core, also sesmcally observed, may requre the nner core to reman at a relatvely fxed longtudnal algnment wth respect to the mantle, perhaps due to gravtatonal lockng between them. Both of these sesmc observatons may be compatble f the dfferental rotaton of the nner core s oscllatory n nature, wth no mean offset over geologcally long tmescales. In ths work, we nvestgate the possble rates of rotaton of an oscllatng nner core and the dynamcs of couplng wthn the core mantle system from an angular momentum perspectve. We fnd that gravtatonal couplng between the nner core and mantle acts to prevent dfferental rotaton, although the perod at whch lockng occurs dffers dependng on whch of the nner core or mantle s drvng the moton. We also show that for an nternally generated torque, a long perod (longer than 100 yr) oscllaton of the nner core wth a rate equal to 0.25 yr 1, on the hgh end of the rates nferred from sesmc observatons, s possble. However, the mantle oscllatons entraned by gravtatonal couplng n such a scenaro are only margnally compatble wth the observed changes n length of day. We show that, n order to explan the sesmcally nferred rotaton rates, ether the gravtatonal couplng must be lower than prevous estmates, or the electromagnetc couplng at the core mantle boundary must be stronger than typcal estmates. Key words: Earth rotaton varatons; Dynamo: theores and smulatons; Core, outer core and nner core. 1 INTRODUCTION Numercal models of the geodynamo have found that moton of the flud core near the nner core boundary (ICB) can produce electromagnetc (EM) torques on the nner core such that the latter would rotate faster than the mantle (Glatzmaer & Roberts 1996). Sesmc evdence n support of an eastward nner core super-rotaton has been found subsequently usng dfferent technques (see the revews by Tromp 2001; Song 2003). The most recent estmate comes from repeated event/recever confguratons (earthquake doublets) for whch temporal varatons n sesmc waveform are nterpreted to reflect dfferental nner core rotaton (Zhang et al. 2005). Rates of nner core rotaton nferred from ths technque are of the order of yr 1, on the hgh end of the range of varous sesmc estmates. A dfferentally rotatng nner core at a rate of yr 1 s dffcult to reconcle wth the presence of non-sphercal densty structures n the mantle and nner core; gravtatonal nteracton between these should act to prevent relatve rotaton between the two (Buffett 1996). A steady nner core rotaton could exst, but only provded the nner core can deform vscously (Buffett 1997). For steady rates to be as hgh as 0.2 yr 1, vscous deformatons must occur on a relatvely rapd tmescale of 5 10 yr (Dumberry 2007). However, a persstent super-rotaton of the nner core over geologcally long tmescales remans dffcult to reconcle wth another sesmc observaton, the observed longtudnal (hemsphercal) ansotropy structure of the nner core (Tanaka & Hamaguch 1997; Nu & Wen 2001). The orgn of such structure s unknown, though t s lkely connected to longtudnal varatons n outer core dynamcs, ether causng lateral varatons n plastc deformaton or texturng the nner core durng soldfcaton (Sumta & Bergman 2007). In turn, relatvely statonary patterns n core convecton are lkely connected to heat flux varatons at the core mantle boundary (CMB) (Wlls et al. 2007; Sreenvasan & Gubbns 2008) and ndeed may produce lateral varatons n heat flux at the ICB resultng n uneven growth of the nner core (Aubert et al. 2008). Although at present we do not know the mechansm responsble for 806 C 2010 The Authors Journal complaton C 2010RAS

2 Inner core super-rotaton and lockng 807 the nner core hemsphercal ansotropy, f t results from such a dynamc connecton to the mantle or the CMB, t requres the nner core to reman n a relatvely fxed orentaton wth respect to the mantle. A possble scenaro that allows the nner core to be currently n super-rotaton whle remanng n algnment wth the mantle over geologcal tmescales s f ts rotaton s not steady but nstead represents a long perod oscllaton. Snce sesmc observatons suggest an eastward rotaton for the past 30 yr, the perod or typcal tmescale of an oscllatng nner core would have to be at least several decades. Possble rates of rotaton of an oscllatng nner core were consdered by Dumberry (2007) from a geodynamc perspectve, based on a smple dynamc model of angular momentum exchanges between the mantle, flud core and nner core. Ths study showed that the rotaton rate of an oscllatng nner core s constraned by the fluctuatons n mantle rotaton nduced by gravtatonal couplng, whch must not exceed observed length of day (LOD) varatons. Subject to ths constrant, oscllatng nner core rates as hgh as 0.2 yr 1 could be acheved, though only provded the strength of the gravtatonal couplng s lower than prevous estmates [based on the study of Mound & Buffett (2006)] or f the strength of EM couplng at the CMB s hgher than that expected from typcal values of lower mantle conductance and magnetc feld strength. However, an mportant lmtaton of the the model of Dumberry (2007) s that the flud core regon was over smplfed. It was smply dvded n two regons (separated by the tangent cylnder) and no dfferental flud moton was allowed wthn each of these regons (.e. they were consdered as rgd bodes). A more realstc descrpton of the flud core could affect the angular momentum dynamcs and the possble rates of nner core oscllaton. Indeed, decadal tmescale axally nvarant, axsymmetrc, zonal core flows n the form of cocentrc cylnders can be nferred by combnng magnetc feld and LOD observatons (Jault et al. 1988; Jackson et al. 1993). Ths type of flud moton s predcted by theory (Taylor 1963) and may correspond to normal modes of oscllatons between cylnders referred to as torsonal oscllatons (Bragnsky 1970; Zatman & Bloxham 1997). Because of strong EM couplng at the ICB, these rgd zonal flows should effcently entran the nner core n ther moton and may thus possbly affect ts rate of oscllaton. An nternal angular momentum model that allows for such flows may then alter the conclusons reached by Dumberry (2007), n partcular at perods that are close to those of the torsonal oscllatons normal modes where resonance can occur. In ths study, we revst the possble rates of rotaton of an oscllatng nner core. We expand upon the work of Dumberry (2007) by allowng dfferental rotaton of cylnders wthn the flud core. We then nvestgate the possble rates of nner core super-rotaton that can be acheved n ths more complete dynamc scenaro. Our model permts a more general nvestgaton of the dfferental rotaton that can exst wthn the core mantle system. If couplng between two regons s suffcently strong, dfferental moton s small and the two regons are locked together, that s they undergo corotaton. For a gven set of couplng parameters, whether two regons are locked depends on the frequency of the forcng. But, as we llustrate below, t depends also on whch regon s responsble for drvng the oscllatons. Thus, followng the descrpton of our model n the next secton, n Secton 3 we nvestgate lockng between dfferent regons of the core mantle system. Such lockng scenaros are nstructve to elucdate the fundamental dynamcs of the system and to understand the lmts on the rate of nner core dfferental rotaton wth respect to the mantle. The latter ssue, beng the man motvaton for ths study, s nvestgated n Secton 4. 2 ANGULAR MOMENTUM MODEL We consder the problem from an angular momentum perspectve. In the absence of external torques the equatons governng the axal angular momentum of the mantle, nner core and flud core are gven, respectvely, by d dt C m m = Ɣ cmb Ɣ g, (1) d dt C = Ɣ cb + Ɣ g, (2) d c f ω f dv = Ɣ cmb Ɣ cb, (3) dt V where C m and m are, respectvely, the axal moment of nerta and rotaton rate of the mantle and C and are those for the nner core. c f s the axal moment of nerta densty wthn the volume V of the flud core and ω f s the angular velocty of flud parcels. The entre system s assumed to rotate at a steady background rate of 2π per day and the angular veloctes are taken to be departures from ths state. We consder only axal torques and rotatons n ths study, that s torques that act to alter the magntude but not the orentaton of the rotaton vectors. Ɣ cmb and Ɣ cb are the torques from all surface forces actng on the mantle at the CMB and on the nner core at the ICB, respectvely. Ɣ g s the gravtatonal torque exerted by the mantle on the nner core and depends on the rotatonal msalgnment of the nner core and mantle densty felds. For a small angle of msalgnment α,the gravtatonal torque of the mantle on the nner core s gven by Ɣ g = Ɣα, (4) where Ɣ s a proportonalty constant that may be obtaned from a mantle densty model (Buffett 1996). The tme varaton of the msalgnment angle α s related to the dfferental rotaton between the nner core and the mantle. Addtonally, t s also related to the rate at whch vscous relaxaton of the nner core allows ts densty structure to adjust to the msalgned gravtatonal potental mposed by the mantle. We assume that the nner core deforms as a smple Newtonan vscous flud, n whch case ts rate of relaxaton s proportonal to the msalgnment angle. The tme-evoluton of α s thus determned by dα = m α dt τ, (5) where τ s the characterstc tmescale of vscous relaxaton. Usng the mappng provded by Buffett (1997), τ = 1 yr corresponds to a bulk nner core vscosty of η s = Pa s. Let us frst consder the modes of oscllaton that are possble n a smplfed system of a gravtatonally coupled mantle and nner core. For ths ntal exercse, we further assume a perfectly rgd nner core (.e. τ = ) and we neglect couplng at the CMB and ICB (.e. Ɣ cmb = Ɣ cb = 0). Wth these approxmatons, the flud core s decoupled from the nner core and mantle and has to conserve ts own angular momentum. Takng tme-dervatves of (1) and (2) and usng (4) and (5), we can wrte the above system of equatons as d 2 dt 2 C m m = Ɣ( m ), (6) d 2 dt 2 C = Ɣ( m ), (7) Journal complaton C 2010 RAS

3 808 M. Dumberry and J. Mound Table 1. Parameters used n calculatons. Parameter Symbol Value Radus of ICB r m Radus of CMB r f m Core conductvty σ Sm 1 Gravtatonal torque coeffcent Ɣ Nm Inner core vscous relaxaton tme τ yr Conductance at base of mantle G m 10 8 S Axal moment of nerta of Mantle C m kg m 2 Inner core C kg m 2 Flud wthn tangent cylnder C c kg m 2 Flud outsde tangent cylnder C o kg m 2 Radal magnetc feld Rms of axal dpole at CMB Bm d mt Rms of radal feld at CMB Bm m mt Rms of axal dpole at ICB B d 2.0 mt Rms of radal feld at ICB B m 3.0 mt Cylndrcally n flud B s 0.3 mt whch s an egenvalue problem for coupled smple harmonc oscllators. Solutons of ths system gves the natural frequences and ther assocated egenvectors [ m, ] ω 0 = 0 [1, 1] Ɣ(C m + C ) ω 0 = C m C [ 1, C m C The frst soluton s the trval case of non-oscllatory corotaton and the second soluton s the free oscllaton that arses due to mantle-nner core gravtatonal (MICG) couplng. The perod of the MICG mode depends on the strength of the gravtatonal couplng ncorporated n the parameter Ɣ.Takng N m (e.g. Buffett 1996; Mound & Buffett 2006) and usng the moments of nerta (e.g. Stacey 1992) gven n Table 1 gves a MICG perod of 2.8 yr. At perods greater than about 1 yr, strong EM couplng at the ICB locks the flud wthn the tangent cylnder to the nner core (Gubbns 1981; Dumberry & Buffett 1999; Mound & Buffett 2003), such that all materal wthn the tangent cylnder effectvely rotates as a sngle rgd body. If ths s the case, we should replace C n the above equatons wth C + C c,wherec c s the axal moment of nerta of the flud core wthn the tangent cylnder; the perod of the MICG then lengthens to 6.1 yr (Mound & Buffett 2003). Indeed, a perodc 6-yr sgnal s observed n the LOD varatons (Abarca del Ro et al. 2000) and has been nterpreted to represent the MICG mode (Mound & Buffett 2006). If ths nterpretaton s correct, Ɣ cannot depart much from Nmandwehavechosenths value for most of our calculatons n Sectons 3 and 4. In addton, t also suggests that the vscous relaxaton tmescale of the nner core τ cannot be much smaller than 5 yr, otherwse the MICG mode should be rapdly attenuated and not observed. Snce C m (C + C c ), t s possble to approxmate the MICG mode as one n whch the nner core oscllates wth respect to a statonary mantle wth a frequency ω Ɣ/(C + C c ). Ths approxmate result could also be obtaned f one were to assume from the outset that, due to the large dfference n moments of nerta, the mantle should reman relatvely statonary. That s, one could assume that m n eq. (7), leadng to the approxmate soluton descrbed above. Alternatvely, one could suppose a stuaton n whch the nner core was held statonary (by some torque not yet ncluded n the model) and the mantle oscllated about ths fxed nner core. In such a case one would set = 0 n eq. (6) and obtan a frequency of oscllaton ω = Ɣ/C m. Usng the parameters ]. gven n Table 1, ths corresponds to a perod of 97 yr. Ths s not a normal mode of the system descrbed by eqs (6) and (7) as t requres the nner core to be held fxed by some external torque. However, ths perod marks the pont at whch the gravtatonal torque on the mantle from an oscllatng nner core becomes larger than the rotatonal nerta of the mantle. It s thus connected to the ablty of an oscllatng nner core to entran the mantle by gravtatonal couplng, as we nvestgate n more detal n Secton 3. We now return to the complete equatons of axal angular momentum exchange (1) (5) and consder the flud core n more detal. We dvde the flud core nto a large number of separate, co-axal cylnders (1000 for the models presented here). Each cylnder can rotate ndependently and must satsfy ts own angular momentum balance. We gve here a bref descrpton of ths balance; more detals can be found for example n Buffett (1998) and Mound & Buffett (2003, 2005). When the angular velocty of a cylnder f (s) at cylndrcal radus s s proportonal to exp ( ωt), where ω s the frequency of oscllaton, t must obey (Bragnsky 1970) ω 2 m(s) f (s) = d [ T (s) d ] ds ds f (s) + ω [ f (s) + f m (s)], (8) where m(s) = 4πρs 3 (z f z ), (9) s the moment of nerta densty of the cylnders, n whch z f = r 2 f s 2 and z = r 2 s 2 where r f and r are the sphercal radus of the flud core and nner core, respectvely. We note that z = 0whens > r. The parameter T (s) n (8) s a magnetc tenson gven by T (s) = 4π s 3 (z f z )B 2 s μ, (10) o where μ o s the magnetc permeablty and B s represents the (cylndrcally) radal magnetc feld averaged over the cylnder surface. When neghbourng cylnders undergo dfferental rotaton, ths magnetc tenson acts as a restorng force tryng to brng the cylnders back nto algnment. We have chosen a value of B s = 0.3 mt; ths choce leads to natural oscllatons n f (s) (the free modes of torsonal oscllatons) wth longest perods of the order of a few decades. Ths s consstent wth the nterpretaton that observed decadal rgd zonal core flows represent these free modes (e.g. Bragnsky 1970; Zatman & Bloxham 1997; Buffett et al. 2009). The parameters f m (s) and f (s) n (8) capture the couplng of the cylnders to the CMB and ICB, respectvely. They are related to Ɣ cmb and Ɣ cb that appear n (1) (3) by Ɣ cmb = r f 0 f m (s)ds, (11) r Ɣ cb = f (s)ds. (12) 0 Here we are nterested n the part of ths couplng that arses from axal dfferental moton of the flud core wth respect to the mantle and nner core. Consequently, we wrte (Buffett 1998) f m (s) = F m (s)[ f (s) m ], (13) f (s) = F (s)[ f (s) ]. (14) The form of the couplng parameters F m (s)andf (s) depends on the nature of the torque. Couplng at the ICB s lkely to be domnated by Journal complaton C 2010RAS

4 Inner core super-rotaton and lockng 809 EM forces (e.g. Gubbns 1981). A varety of couplng mechansms have been suggested to be mportant at the CMB, though f there s a hgh-conductvty layer n the lower-most mantle (as nferred from nutaton studes, e.g. Mathews et al. 2002), EM forces are also lkely to domnate the couplng. Other couplng mechansms can be ncorporated n our framework, but for reasons of concseness and tractablty, here we restrct our attenton to EM couplng at both the CMB and ICB. EM couplng at a flud sold boundary depends on the morphology of the magnetc feld as well as on the conductvty structure on the sold sde of the boundary. The couplng parameters F m (s)andf (s) ncorporate ths nformaton. Explct calculatons of F m (s) for gven magnetc feld models at the CMB can be found n Dumberry & Mound (2008). The latter study also showed that a very good approxmaton of F m (s) s obtaned when the magnetc feld at the CMB s expressed smply n terms of ts axal dpole part and the rms component of the radal feld. We follow ths prescrpton here and express the couplng at the CMB as (see Dumberry & Mound 2008) F m (s) = 2πs 3 [ (B d m ) 2 ] ( ) + B m 2 r f m G m. (15) z f In ths last equaton, G m s the conductance of the lower-most mantle, for whch we assgn a value of 10 8 S. Ths s compatble wth the conductance requred to explan a part of the observed nutatons of the Earth (e.g. Mathews et al. 2002). We note that n the context of nutatons, G m = 10 8 S represents an upper bound of the conductance that can be sampled on a durnal tmescale. The actual conductance may thus be larger than 10 8 S though we do not have supportng evdence for ths. The quantty Bm m s the rms value of the total radal feld at the CMB (ncludng the axal dpole contrbuton) and Bm d s the axal dpole radal feld at the CMB. The latter s wrtten n terms of the degree one, order zero gauss coeffcent g 0 1, and gven by (Gubbns & Roberts 1987) B d m = 2 ( ra r f ) 3 g 0 1 cos θ. (16) We note that ths expresson s a functon of s through the colattude angle θ. We use values for both the axal dpole and total rms strength that are based on a tme average of the gufm1 model of Jackson et al. (2000) over the perod Our adopted values are gven n Table 1, where we also express the axal dpole ampltude n terms of ts rms strength, that s B d m = 2 3 ( ra r f ) 3 g 0 1. (17) The combnaton of G m = 10 8 S and the values of Bm m and Bd m lsted n Table 1 produce what we refer to here as a typcal EM couplng strength at the CMB. Smlarly, the EM couplng at the ICB can be expressed as (see Dumberry & Mound 2008) F (s) = 2πs 3 [ (B d ) 2 ] ( + B m 2 r G z ), (18) where B d and B m refer to the axal dpole and rms components of the radal magnetc feld at the ICB. B d s related to ts rms strength B d by a relaton equvalent to that of (16) and (17). The conductance factor G ncorporates the detals of the magnetc feld perturbaton created by the shear at the ICB (e.g. Buffett 1998) and s gven by G = 1 4 [1 + sgn(ω)] σ f δ f, (19) where σ f s the conductvty of the flud core (assumed equal to that of the nner core, Sm 1 e.g. Stacey & Anderson 2001), δ f = 2/ωμ o σ f s the skn depth n the flud core, ω s the frequency of oscllaton and sgn (ω) = ω/ ω. Itsusefultowrte the ICB EM couplng n the form F (s) = [1 + sgn(ω)] ω ˆF (s), (20) where σ [ f (B ) ˆF (s) = 2μ πs3 d 2 ] ( ) + B m 2 r. (21) z Due to a lack of drect, observatonal constrants for the geometry of the magnetc feld at the ICB we have adopted feld components (Table 1) havng a smlar geometry to the CMB magnetc feld components, but wth a total ntensty that s greater by approxmately an order of magntude as s typcally observed n geodynamo smulatons (e.g. Chrstensen & Aubert 2006). Our complete model thus comprses the angular momentum equatons for the mantle (1), nner core (2) and the flud cylnders (8), where Ɣ g sspecfedby(4)and(5),andɣ cmb and Ɣ cb are specfed by (11) (21). We assume that, m and f (s) are proportonal to exp ( ωt). If no addtonal forcng s appled to ths system, the solutons are free modes wth (complex) frequences ω and (complex) egenvectors specfyng the relatve ampltudes between, m and f (s). In the followng sectons we apply a forcng to these equatons, also assumed to be proportonal to exp ( ωt). Ths forcng s ether appled drectly on the mantle (added on the rght-hand sde of eq. 1), the nner core (rght-hand sde of eq. 2) or on cylnders nsde the flud core (rght-hand sde of eq. 8). Wth ths added forcng, (complex) solutons of, m and f (s) are found for gven values of ω and are proportonal to the ampltude of the appled torque. Snce the torque between any two regons depends on the product between a couplng strength and ther dfferental rotaton, large couplng strengths lead to small dfferental rotatons. Thus regons that are strongly coupled together tend to be n corotaton, though ther angular veloctes are never exactly dentcal as ths would result n a vanshng torque. 3 THE DYNAMICS OF LOCKING We consder the response of our couplng model to three dfferent forcng scenaros. In the frst, an equal and opposte torque s appled on the flud cylnders that bracket the tangent cylnder. Ths scenaro represents an approxmaton of the nternal forcng that mght occur due to convecton n the flud core and responsble for exctng free modes of torsonal oscllatons. The second scenaro apples a torque solely to the nner core and s not meant to represent any physcal stuaton but s useful for llustratng the range of system dynamcs. The fnal scenaro conssts of a torque actng solely on the mantle, as mght occur due to the tdal acton of the oceans or some other external torque. The model s solved for the complex angular veloctes of the nner core, mantle and flud core cylnders n response to the appled forcng at a gven frequency. At suffcently hgh frequency the choce of one thousand cylnders n the flud core s nsuffcent to accurately resolve the system response; however, the model s well resolved for perods above 1 yr and we shall restrct our attenton to those perods. We are nterested n the effectve lockng of dfferent regons wthn the core mantle system, and n partcular lockng of the nner core to ether the mantle or the flud core wthn the tangent cylnder. For the flud core we defne the average veloctes Journal complaton C 2010 RAS

5 810 M. Dumberry and J. Mound of the regon nsde and outsde the tangent cylnder as c = 1 r m(s) f (s)ds, (22) C c 0 o = 1 r f m(s) f (s)ds, (23) C o r where C c and C o are, respectvely, the moments of nerta of the flud nsde and outsde the tangent cylnder. If two regons wthn the system were perfectly locked, the rato of ther (complex) angular velocty ampltudes would be equal to one; f the regons oscllated wth equal ampltude but exactly out of phase, ths rato would equal mnus one. We explore the behavor of our model for two dfferent choces of nner core vscosty: a stff nner core wth a vscous relaxaton tme of τ = 1000 yr, and a softer nner core wth τ = 10 yr. The results of these two models are presented n Fgs 1 and 2, respectvely. All other model parameters are those of Table 1. We take the nner core as our reference body; n each fgure, we plot the real part of the ratos of the mantle and flud core veloctes wth respect to that of the nner core for a forcng appled at the tangent cylnder (top panel), solely on the nner core (mddle panel) and solely on the mantle (bottom panel). All angular veloctes n our model are lnearly proportonal to the ampltude of the mposed forcng; by consderng angular velocty ratos as shown n Fgs 1 and 2, our results are then ndependent of the forcng ampltude. We begn our analyss wth the case of the stff nner core under the tangent cylnder forcng scenaro (Fg. 1a). We are specfcally nterested n the effcency wth whch the flud part wthn the tangent cylnder can entran the nner core, whch s descrbed by the rato R( c / ) (red lne), and the degree to whch the mantle s then entraned by the nner core, descrbed by R( m / )(green lne). [The notaton R(x) denotes the real part of x.] We fnd that the rato R( c / ) s close to 1 at all perods. At perods of a Fgure 1. Angular veloctes relatve to that of the nner core for a forcng appled to the (a) tangent cylnder (b) nner core and (c) mantle. All panels show the veloctes of the flud nsde the tangent cylnder (red lnes), the flud outsde the tangent cylnder (blue lnes) and the mantle (green lnes). The vscous relaxaton tme of the nner core s τ = 1000 yr, all other parameters are as gven n Table 1. Journal complaton C 2010RAS

6 Inner core super-rotaton and lockng 811 Fgure 2. Angular veloctes relatve to that of the nner core for a forcng appled to the (a) tangent cylnder (b) nner core and (c) mantle. All panels show the veloctes of the flud nsde the tangent cylnder (red lnes), the flud outsde the tangent cylnder (blue lnes) and the mantle (green lnes). The vscous relaxaton tme of the nner core s τ = 10 yr, all other parameters are as gven n Table 1. few decades and smaller, c and dffer by around 5 per cent. Ths s because EM couplng between the nner core and the flud nsde the tangent cylnder s suffcently strong to effectvely lock the two nto corotaton. The perod at whch lockng occurs corresponds to the perod of a free mode of oscllaton between and c where the restorng torque between the two s EM (Gubbns 1981; Dumberry & Buffett 1999; Mound & Buffett 2003). The perod of ths free mode depends on the ampltude of the magnetc feld at the ICB, occurrng at shorter (longer) perod for larger (smaller) feld ampltude. For our values n Table 1, the perod of ths mode s approxmately 0.3 yr. Beyond ths perod, the EM torque at the ICB overcomes the rotatonal nerta of the nner core and the latter s entraned by the flud nsde the tangent cylnder; the rato R( c / ) approaches 1 and the two are effectvely locked together. If the dynamcs of the nner core and the flud nsde the tangent cylnder were only governed by EM couplng between the two, R( c / ) would ndeed be very close to one for all perods exceedng approxmately 1 yr. Ths s clearly not the case, as evdenced by the devaton near perods of 100 yr where R( c / ) ncreases to a peak value of Ths devaton arses as a consequence of gravtatonal couplng between the nner core and the mantle, preventng the nner core to follow the flud part of the tangent cylnder. Indeed, ths s the locaton of the perod of gravtatonal oscllaton of the mantle wth respect to a statonary nner core, the abnormal mode dentfed n the prevous secton, whose perod depends on the quantty Ɣ/C m and equals 97 yr for our choce of parameters. When the perod of the forcng s larger than Ɣ/C m, the rotatonal nerta of the mantle can no longer resst the gravtatonal torque from the rotatonally drven nner core. The nner core entrans the mantle nto moton and the gravtatonal torque from the latter s no longer as effectve to counteract the EM torque at the ICB; the rato R( c / ) decreases back towards 1. We fnd that R( c / ) = 1.35 for a perodc forcng of 10 4 yr. Journal complaton C 2010 RAS

7 812 M. Dumberry and J. Mound The ampltude of the peak n R( c / ) at the locaton of the abnormal mode s determned by the relatve strength between the gravtatonal and ICB couplngs. A larger (smaller) magnetc feld at the ICB would produce a less (more) promnent devaton from 1. Smlarly, for a softer nner core, and thus for a less effectve gravtatonal couplng, the devaton from 1 s also reduced (compare the red lnes of Fgs 1 and 2). The role of the mantle s confrmed by the way n whch the rato R( m / ) changes as a functon of the perod. At short perods, R( m / ) s very small, ndcatng that the mantle s not greatly nvolved n the angular momentum balance. As the perod ncreases, the gravtatonal torque mparts larger acceleratons on the mantle but t s not untl the perod reaches Ɣ/C m that the nner core entrans the mantle nto corotaton. We have verfed that the locaton of ths transton perod s ndeed only a functon of Ɣ/C m and does not vary for dfferent choces of magnetc feld strength at the ICB and CMB, nor for dfferent values of the B s -feld(andhence of the perod of the free modes of torsonal oscllatons). Thus, the restorng torque for ths abnormal mode s entrely gravtatonal. (We note agan that ths s not a free mode of our system. The only gravtatonal free oscllaton that our system supports s the MICG mode dentfed n Secton 2.) At long perods, the mantle and nner core are closely, though not perfectly locked together (R( m / ) = at the longest perods consdered). Fluctuatons n the ampltude of R( o / ) (blue lne) as the frequency of the forcng s vared occur due to the sute of torsonal oscllaton normal modes allowed by the system. For the chosen value of B s the fundamental torsonal oscllaton mode has a perod of 200 yr. Although t s dffcult to see on the scale of the fgure, the ratos R( c / )andr( m / ) also fluctuate as the perod of the forcng comes nto resonance wth the torsonal oscllaton modes. There s no net torque on the system for the tangent cylnder forcng scenaro. The angular momentum of the whole Earth must be conserved whch requres at least one regon to rotate wth opposte phase. Thus, whle the whole tangent cylnder and mantle are movng approxmately together n the long perod lmt, the flud core outsde the tangent cylnder s out of phase. The large ampltude of o s requred n order to balance the larger moment of nerta of the combned mantle and tangent cylnder (compared to C o ) and thus conserve angular momentum. For long perods we fnd that R( o / ) = 7.91, whch s slghtly less than the rato of moments of nerta (C m + C + C c )/C o = We would not expect these ratos to match exactly as the nner core, mantle and flud wthn the tangent cylnder are not n perfect corotaton. We now consder the scenaro n whch a torque s appled solely to the nner core (Fg. 1b). For the entre range of perods consdered the nner core and the flud nsde the tangent cylnder are approxmately locked, although n ths case t s the nner core that s entranng the flud nto moton. The lockng of the mantle wth the forced nner core oscllaton occurs sharply at the 97-yr perod. For the flud outsde the tangent cylnder, the EM couplng wth the mantle exerts a larger nfluence than the couplng wth the tangent cylnder. Thus, motons reman small untl the 97-yr transton perod when the mantle s set nto moton by the nner core. At very long perods, due to the combnaton of the gravtatonal and EM couplng at the CMB, the moton of the entre planet oscllates approxmately as a sngle rgd body n response to the forcng appled solely to the nner core; all ratos n the long perod lmt of Fg. 1(b) dffer from 1 by at most one part n a thousand. For a forcng appled solely to the mantle (Fg. 1c), the response of the core mantle system s dfferent. As s the case for a forcng appled to the nner core, the nner core and flud tangent cylnder are locked for the entre perod range consdered. At short perods, when the gravtatonal torque s not large enough to overcome the nerta of the whole of the tangent cylnder, the nner core moton remans small and as a result the ampltude of R( m / ) s large. The transton to lockng between the mantle and the whole of the tangent cylnder occurs at much shorter perods compared to the nner core forcng scenaro. Ths s because when the mantle s drvng the moton, lockng occurs when the gravtatonal couplng becomes larger than the rotatonal nerta of the whole of the tangent cylnder. Ths perod s Ɣ/(C + C c ) and correspond approxmately wth that of the MICG mode. The breadth of the resonance assocated wth the torsonal oscllaton normal modes prevents the perod of the lockng transton for ths scenaro from beng as sharply defned as that for the nner core forcng scenaro. The flud outsde the tangent cylnder s entraned nto moton by EM couplng wth the mantle, but t does not lock to the rest of the system untl the perod exceeds that of the fundamental mode of torsonal oscllatons of approxmately 200 yr. At perods n excess of 1000 yr the system s effectvely locked, wth all ratos dfferng from 1 by at most one part n a thousand. The perods of the lockng transtons descrbed above do not change when we consder a less vscous nner core (Fg. 2), although they are not as sharp as n the case wth the stffer nner core. For the forcng appled on the nner core and mantle we agan have complete lockng at the longest perods consdered. For the tangent cylnder forcng scenaro, although the lockng transtons as a functon of frequency do not change, the strength of the lockng between the regons s now very dfferent. In partcular, n the long perod lmt, we fnd that R( m / ) = 0.57, and thus that there exsts a sgnfcant dfferental rotaton between the nner core and mantle. Ths s because as the nner core entrans the mantle by gravtatonal couplng, the moton of the latter s restrcted by EM couplng at the CMB wth the dfferentally rotatng flud core (we fnd R( o / ) = 4.55 at the longest perod). By reducng the vscous relaxaton tmescale from 1000 to 10 yr, the strength of the gravtatonal couplng s now sgnfcantly decreased wth respect to the EM couplng at the CMB, reducng the ablty of the nner core to entran the mantle n ts moton. The mportant message to take away from the above analyss s that for an external torque appled drectly on the mantle or the nner core we expect the whole Earth to be corotatng n the long perod lmt. However, n the case of an nternally generated torque, conservaton of angular momentum mples that at least one regon must oscllate wth a reversed phase. Whch nternal regons are corotatng and whch are n dfferental rotaton s determned by the couplng strength between them. In partcular, the nner core can rotate dfferentally wth respect to the mantle f the strength of the couplng by surface forces on the CMB becomes comparable to that of the gravtatonal couplng. 4 DIFFERENTIAL MOTION BETWEEN THE INNER CORE AND MANTLE We now nvestgate n more detal the dfferental rotaton between the nner core and mantle n order to determne the rotaton rates that can be acheved by an oscllatng nner core and to assess whether ths can serve as an explanaton for the sesmcally nferred nner core super-rotaton. The absolute rates of rotaton of every regon are lnearly proportonal to the ampltude of the appled torque. Thus, a dfferental rotaton of the nner core at the sesmcally nferred rate Journal complaton C 2010RAS

8 Inner core super-rotaton and lockng 813 can always be acheved by choosng approprately the ampltude of the appled torque. However, we do not know a pror how large ths torque can be. Moreover, the mantle oscllatons generated n our model cannot be larger than the observed LOD varatons. Therefore, we constran the rates of nner core rotaton, as was done n the study of Dumberry (2007), by renormalzng our results such that the resultng mantle oscllatons equal yr 1, correspondng to a 3 ms change n LOD. Ths represents approxmately the largest observed LOD changes at decadal to mllennal perods and sets a generous upper lmt for the allowed nner core dfferental rotaton. We use the same LOD constrant regardless of the frequency consdered, though at perods close to 1 yr the observed LOD varatons are at least an order of magntude smaller. The allowed nner core dfferental rotaton, subject to the constrant that the observed LOD varatons are not exceeded, can be expressed as dff = (0.005 /yr) 1. (24) m Ths quantty s plotted n Fg. 3. As t was the case for Fgs 1 and 2, the top, mddle and bottom plots correspond to a torque appled at the tangent cylnder, on the nner core and on the mantle, respectvely. We consder the same model parameters as n the prevous secton, as well as two addtonal models: one wth no gravtatonal couplng and one n whch the relaxaton tme of the nner core s only 1 yr. The horzontal dotted lne n each panel of Fg. 3 corresponds to an nner core dfferental rotaton of 0.25 yr 1.Ths s approxmately the largest nner core rotaton rate that s compatble wth a number of dfferent sesmc observatons (e.g. Tromp 2001; Song 2003), though t s possble that the true rate may be sgnfcantly smaller. Not surprsngly, the mantle forcng scenaro (Fg. 3c) s the least effcent n exctng rapd dfferental moton between the mantle and nner core. When constraned to not exceed 3 ms LOD varatons, the mantle forcng scenaro produces a dfferental rotaton much smaller than 0.25 yr 1 at all perods consdered, regardless of the effcency of gravtatonal couplng. Even amplfcaton of the dfferental moton n the vcnty of the MICG mode or the torsonal oscllaton normal modes results n dfferental nner core rotaton that s an order of magntude below the sesmcally nferred rate. In contrast, when the forcng s appled drectly to the nner core (Fg. 3b), rotaton rates of 0.25 yr 1 or hgher are possble. However, ths s only the case for perods shorter than a few decades. Even n the absence of gravtatonal couplng, the nner core forcng scenaro leads to dfferental rotaton rates below 0.25 yr 1 at perods longer than 100 yr. If gravtatonal couplng s present, the dfferental rotaton s reduced sgnfcantly and rates of 0.25 yr 1 are only possble at perods of a few decades or less. We recall that the relatvely large rotaton rates obtaned at perods between 1 and 10 yr are msleadng as subdecadal changes n LOD are at least an order of magntude less than the 3 ms constrant that we have mposed. The most dynamcally relevant case for the Earth s the scenaro wth the forcng across the tangent cylnder (Fg. 3a), as ths mmcs nternal torques that may occur wthn the flud core and the Earth as a whole must conserve ts own angular momentum. In the absence of gravtatonal couplng, a dfferental rotaton n excess of 0.25 yr 1 s acheved at all forcng perods. However, at perods longer than 10 yr, the ncluson of gravtatonal couplng greatly reduces the allowed dfferental rotaton. The stffer the nner core, the smaller s ts dfferental rotaton. At perods between 10 and 100 yr, dfferental rotaton rates as hgh as 0.25 yr 1 cannot be acheved unless the vscous relaxaton tme of the nner core s approxmately 1 yr or shorter, except at specfc frequences where a resonance wth a torsonal oscllaton normal mode amplfes the dfferental rotaton. Once more, the hgh rates at subdecadal perods are msleadng. When the forcng perod s greater than all other tmescales of our system (.e. greater than the torsonal oscllatons modes, τ and Ɣ/C m ), the dfferental rotaton of the nner core reaches an asymptotc rate whch no longer depends on the forcng perod. In ths long perod lmt, for the parameters lsted n Table 1, n order to acheve a dfferental rotaton of 0.25 yr 1, the nner core relaxaton tme must be 0.16 yr. Ths corresponds to an nner core vscosty of Pa s, smlar to the results obtaned n the study of Dumberry (2007). [We note that a dfferent choce of nternal forcng (.e. a torque appled to dfferent cylnders) does affect some of the detals of the solutons presented n Fg. 3(a). However, we have verfed that the general conclusons presented here are not greatly affected by the geometry of the nternal forcng.] The sesmc observatons suggest an eastward super-rotaton of the nner core durng the past 30 yr. Thus, f ths s a reflecton of an oscllatng nner core, ts perod of oscllaton would need to be at least twce as long (.e longer than 60 yr) and close to, or past the pont where, the long perod lmt soluton s applcable. It s possble to construct an analytcal expresson for dff n ths long perod lmt. Ths wll s also reveal how the dfferental rotaton of the nner core depends explctly on the parameters of our model. Combnng eqs (1), (4) and (5) n the lmt of d/dt 0, we obtan Ɣ cmb = Ɣτ( m ). (25) Wrtten n terms of averaged veloctes o and c, the torque Ɣ cmb defned n eqs (11) and (13) can be expressed as Ɣ cmb = F c ( c m ) + F o ( o m ), (26) where, usng (15), F c and F o are defned by r F c = F m ds 0 [ B m 2 = 2πG m r 4 m B d 2 ] f (2 ζ 1 ) + m 3 5 (2 ζ 2) F o = wth r f r F m ds [ B m 2 = 2πG m r 4 m B d 2 ] f ζ 1 + m 3 5 ζ 2 ζ 1 = ( 2 + r 2 o (27) (28) )( 1 r 2 o) 1/2 (29) ζ 2 = ( )( r 2 o 1 r 2 3/2 o) (30) r o = r. (31) r f By assumng a locked tangent cylnder ( c = ) and conservaton of angular momentum from whch ( C + C c o = C o ) C m C o m, (32) Journal complaton C 2010 RAS

9 814 M. Dumberry and J. Mound Fgure 3. Ampltudes of nner core dfferental rotaton for oscllatory moton subject to the constrant that LOD changes do not exceed 3 ms n ampltude at the gven perod. We consder models wth no gravtatonal couplng (red lnes), and gravtatonal couplng wth an nner core relaxaton tme of 1 yr (green lnes), 10 yr (blue lnes) or 1000 yr (black lnes). The dotted horzontal lnes ndcate an nner core dfferental rotaton of 0.25 yr 1. As n prevous fgures, we consder a forcng appled to the (a) tangent cylnder (b) nner core and (c) mantle. the combnaton of (25) and (26) yelds m = ( ) C Ɣτ + F c + F m o C o + 1 Ɣτ + F c F o ( C +C c C o + 1 ). (33) We fnd that substtutng ths approxmaton nto (24) provdes a very good match, n the long perod lmt, to the solutons of our model such as those presented n Fg. 3(a). The analytcal expresson (33) allows us to determne the requred combnaton of parameters such that a rate of 0.25 yr 1 can be Journal complaton C 2010RAS

10 Inner core super-rotaton and lockng 815 Fgure 4. Ampltudes of nner core dfferental rotaton for a forcng appled to the tangent cylnder subject to the constrant that LOD changes do not exceed 3 ms n ampltude at the gven perod. Three dfferent sets of gravtatonal couplng parameters are shown: Ɣ = Nmandτ = yr (red lne); Ɣ = Nmandτ = 10 yr (green lne); Ɣ = Nmandτ = 500 yr (blue lne). Other parameters are as gven n Table 1. The dotted horzontal lnes ndcate an nner core super-rotaton of 0.25 yr 1. acheved n the long perod lmt. If we use the typcal EM couplng parameters gven n Table 1, then F c = Nmsand F o = N m s. In order to get dff = 0.25 yr 1,the product of Ɣ and τ must then be equal to approxmately N m s or, equvalently, to N m yr. Dfferent combnatons of Ɣ and τ affect the dynamcs of gravtatonal couplng, notably the perods of the MICG and abnormal modes, but dff should not change n the long perod lmt. Ths s llustrated n Fg. 4 whereweshow dff computed for the tangent cylnder forcng scenaro for three dfferent combnatons of Ɣ and τ, but each wth a product equal to N m yr. The three cases shown are for: Ɣ = Nmandτ = yr (red lne); Ɣ = N mandτ = 10 yr (green lne); Ɣ = Nmandτ = 500 yr (blue lne). The perods of the MICG and the abnormal modes for each of these three cases are, respectvely: 6.1 and 97 yr; 48 and 751 yr; 336 and 5312 yr. No marked transtons at these perods can be seen on Fg. 4 because the nner core vscous relaxaton tme s smaller than both of these perods for the red and green curves and approxmately equal to the MICG perod for the blue curve. In all cases, the dfferental rotaton of the nner core n the long perod lmt s 0.25 yr 1, n agreement wth our analytcal expresson based on (33). Fg. 4 thus llustrates that t s possble to acheve dff = 0.25 yr 1 wth an oscllatng nner core wthout volatng the observed LOD varatons. For ths to be the case, usng the typcal EM couplng parameters, the condton on the gravtatonal couplng parameters s that the combnaton Ɣτ must be less or equal to N m yr. A larger EM couplng strength at the CMB allows larger values of Ɣτ to be compatble wth dff = 0.25 yr 1. As explaned n Secton 3, ths s because as the nner core entrans the mantle by gravtatonal couplng, the moton of the latter s restrcted by EM couplng wth the flud core at the CMB. Thus, the larger the EM couplng, the lesser the entranment of the mantle by the nner core, resultng n a greater dfferental rotaton between the two. A larger EM couplng can be accomplshed ether by an ncrease n the conductance or an ncrease n the ampltude of the magnetc feld at the CMB. We do not present solutons here for dfferent scenaros of EM couplng, though usng (33), we can determne that, for nstance f Ɣ = Nm,τ = 5yr,and Bm m and Bd m are as gven n Table 1, n order to have dff = 0.25 yr 1, the conductance at the base of the mantle would need to be ncreased to S. Ths s smlar to the concluson reached n the study of Dumberry (2007). 5 DISCUSSION AND CONCLUSION Our nvestgaton has shown how the entranment of the nner core, mantle and flud core nto moton for a gven forcng scenaro depends on the strength of the couplng between these regons relatve to ther rotatonal nerta. EM couplng at the ICB ensures that the nner core and the flud nsde the tangent cylnder are generally locked for perods larger than approxmately 1 yr. Gravtatonal couplng between the mantle and nner core also ensures eventual lockng between the two f the forcng s appled drectly on ether of them. However, because of ther very dfferent moment of nerta, the perod at whch ths lockng occurs depends on whch of the two s drvng the moton. For an external forcng appled to the mantle, the nner core (together wth the flud nsde the tangent cylnder) s expected to be locked to the moton of the mantle at perods longer than Ɣ/(C + C c ), approxmately equal to the perod of the MICG mode. The exact value of ths perod depends on Ɣ;for Ɣ = N m (Mound & Buffett 2006) t s equal to 6.1 yr. Possble physcal sources of mantle forcng nclude angular momentum exchange wth the oceans and atmosphere (e.g. Dckey et al. 2003), or tdal breakng (e.g. Chrstodolouds et al. 1988). We thus expect that f the tmescale of the mantle forcng exceeds a few decades, the small rotatonal nerta of the nner core mples that t cannot resst the gravtatonal torque from the mantle and s forced to rotate n unson wth t. On ths bass, the sesmcally observed nner core super-rotaton cannot represent a fossl rotaton rate, ether from tdal despnnng that has not yet been transmtted to the nner core or from mantle spn changes assocated wth quaternary glacatons (e.g. Blls 1999). If, however, the changes n rotaton are mposed by torques on the nner core, t s only for perods longer than Ɣ/C m that the mantle becomes gravtatonally locked to the nner core. For Ɣ = N m, ths perod s 97 yr. At smaller perods, the rotatonal nerta of the mantle s larger than the gravtatonal torque from the nner core and ths prevents gravtatonal lockng. The forcng scenaro that s the most relevant for the Earth s one where changes n rotaton result from nternal torques wthn the flud core. If the strength of gravtatonal couplng s large, for perods longer than Ɣ/C m the nner core effcently entrans the mantle and the dfferental rotaton between the two s lmted. However, n ths scenaro, the bulk of the flud core outsde the tangent cylnder must rotate n the reverse drecton n order to conserve Journal complaton C 2010 RAS

11 816 M. Dumberry and J. Mound the angular momentum of the whole Earth. Couplng between the flud core and the mantle from surface forces at the CMB can thus prevent the mantle from beng fully gravtatonally entraned by the nner core. A relatvely hgh couplng at the CMB compared to the gravtatonal couplng may thus allow a sgnfcant dfferental rotaton between the nner core and the mantle. Assumng EM couplng at the CMB of typcal strength (based on a lower mantle conductance of 10 8 S and a magnetc feld ampltude consstent wth a downward contnuaton of the feld observed at the Earth s surface), n order to explan a dfferental rotaton as hgh as 0.25 yr 1 (e.g. Zhang et al. 2005) n terms of an oscllatng nner core, the condton on the gravtatonal couplng s Ɣτ N m yr. Larger values of Ɣτ would requre a proportonally larger couplng strength at the CMB. If these constrants are not met, an oscllatng nner core at a rate of 0.25 yr 1 would lead to larger LOD changes than those observed. These constrants are broadly smlar to those derved n the study of Dumberry (2007), where a smplfed model of the flud core was used. Thus, the ncluson of torsonal oscllatons n the flud core, as done n the present study, do not alter sgnfcantly the dynamcs of nternal couplng at long perods. The condton Ɣτ N m yr represents a weaker gravtatonal couplng than prevous estmates. The parameter Ɣ depends on the ampltude of the geod undulatons produced by densty anomales n the mantle (ncludng topography at the CMB) and the densty structure of the nner core (whch should algn wth the mposed mantle geod on long tmescales). It has been proposed that a perodc 6-yr sgnal n the LOD varatons (Abarca del Ro et al. 2000) represents the MICG mode, n whch case Ɣ cannot depart much from N m (Mound & Buffett 2006). As a reference, ths corresponds to degree 2 geod undulatons of approxmately 300 m along the equator of the CMB, compatble wth some geodynamc models (Forte & Pelter 1991; Defragne et al. 1996). For a 6-yr MICG mode to be observed, τ must be at least 5 yr or larger, otherwse t would be rapdly attenuated. Ths leads to Ɣτ N m yr, much larger than the condton we have derved above. Usng the lowest possble value of Ɣτ consstent wth the nterpretaton of the 6-yr LOD sgnal n terms of the MICG mode, Ɣτ = N m yr, together wth our typcal EM couplng strength, the possble nner core oscllaton rates at perods longer than 100 yr cannot exceed 0.01 yr 1. If these parameters are correct, the mplcaton s that the sesmcally nferred rates of nner core dfferental rotaton cannot represent a long perod oscllaton. Larger rates of dfferental nner core rotaton are possble for larger EM couplng at the CMB. Rates as hgh as 0.25 yr 1 can be produced f we ncrease the mantle conductance to S. Whle not completely mplausble, such a hgh conductance would act as an effectve flter on the secular varaton of the core feld observed at the surface. It s thus dffcult to reconcle a conductance ths large wth observed monthly changes n the core feld detected n satellte observatons (Olsen & Mandea 2008). Alternately, snce EM couplng vares wth the square of the rms radal magnetc feld at the CMB (see eq. 15), a value of the latter larger than 0.32 mt based on downward contnuaton of the surface feld would also ncrease the ampltude of EM couplng. Indeed, sgnfcant energy may be present n the small-scale components of the CMB feld, a part of the spectrum that s not accessble from surface observaton because t s masked by the crustal feld. Studes of forced nutatons suggest that the rms value of the magnetc feld at the CMB may be closer to 0.7 mt (Mathews et al. 2002). Adoptng such a radal feld strength, and Ɣτ = N m yr, the mantle conductance requred to get dff = 0.25 yr 1 s reduced to S, less n conflct wth magnetc feld observatons. Such a hgh EM couplng scenaro would thus allow us to explan the sesmc rates of nner core rotaton n terms of a long perod (>100 yr) oscllaton whle not beng n conflct wth a gravtatonal couplng strength based on a 6-yr MICG mode. However, ths would stll requre an nner core vscous relaxaton tme not much larger than 5 yr, correspondng to an nner core vscosty of Pa s. For a weaker CMB couplng than ths hgh EM couplng scenaro, an oscllatng nner core at a dfferental rate of 0.25 yr 1 mples that Ɣτ < N m yr, and thus ether Ɣ < Nm, or τ < 5 yr, or both, n whch case the 6-yr LOD sgnal cannot represent the MICG mode and therefore must be from a dfferent orgn. Ths would also ndcate that the ampltude of the longwavelength geod at the CMB s smaller than 300 m, perhaps n better agreement wth more recent geodynamc models (Smmons et al. 2006). In summary, an explanaton of the dfferental nner core rotaton n terms of a long perod oscllatons s possble, but t requres ether a low gravtatonal couplng strength, a hgher EM couplng strength than our typcal estmate, or both. If the present-day eastward nner core super-rotaton s part of a slow oscllaton, or slow fluctuatons, the tmescale of these remans uncertan. As noted n the ntroducton, a steady rotaton poses a problem f the hemsphercal ansotropy structure of the nner core results from a process controlled by the mantle. One must then make sure that fluctuatons would also not take the nner core too far out of algnment wth the mantle. An nner core oscllatng at a peak rate of 0.2 yr 1 and a perod of 100 yr would lead to a peak-to-peak longtudnal angular dsplacement of 6. If the perod s 1000 yr, the peak-to-peak angular dsplacement s 60. Such longtudnal fluctuatons are not nsgnfcant, but may reman compatble wth an nner core ansotropy mechansm requrng a general algnment wth the mantle. Though our model allows us to calculate the rotaton rates at any perod, t s mportant to note that our calculaton at mllennal tmescales and beyond must be regarded wth suspcon for two reasons. Frst, the results presented here use a model of torsonal oscllatons n whch magnetc dffuson wthn the flud core s not taken nto account; magnetc dffuson should play a role at mllennal tmescales. A more fundamental ssue though s that purely rgd flows may no longer be an approprate descrpton of the flud core dynamcs; varatons n thermal wnds, a type of flow nvolvng axal gradents, may be sgnfcant at mllennal tmescales. Couplng of the nner core to mllennal tmescale varatons n thermal wnd could then serve as an alternate mechansm to explan the sesmcally observed nner core super-rotaton. Indeed, the presence of such varatons has been suggested on the bass of the observed archaeomagnetc secular varaton and mllennal LOD changes (Dumberry & Bloxham 2006; Dumberry & Fnlay 2007). A thermal wnd flow may have opposte flow drectons at the ICB and CMB, and thus have the ablty to apply torques n opposte drectons on the mantle and nner core: ths would naturally promote a hgher dfferental rotaton between the two. ACKNOWLEDGMENTS Matheu Dumberry s currently supported by a NSERC/CRSNG dscovery grant. Jon Mound recognzes support of the WUN-Leeds Fund for Internatonal Research Collaboraton. Journal complaton C 2010RAS

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