Torque and changes in the LOD<\/p>\n
An axial torque \u0393z acting on the mantle causes a change in its rotation rate (\u0394\u03a9m) according to <\/p>\n
$${C}_{{\\rm{m}}}\\frac{{\\rm{d}}\\Delta {\\varOmega }_{{\\rm{m}}}}{{\\rm{d}}t}={\\varGamma }_{z},$$<\/p>\n
\n (1)\n <\/p>\n
in which Cm\u2009=\u20097.13\u2009\u00d7\u20091037\u2009kg\u2009m2 is the axial moment of inertia of the mantle. \u0394\u03a9m corresponds to a change in the LOD (\u0394LOD) <\/p>\n
$$\\Delta {\\rm{LOD}}=-\\frac{2{\\rm{\\pi }}}{{\\varOmega }_{{\\rm{o}}}^{2}}\\Delta {\\varOmega }_{{\\rm{m}}},$$<\/p>\n
\n (2)\n <\/p>\n
in which \u03a9o\u2009=\u20092\u03c0\/86,400\u2009s\u22121 is the reference mantle rotation rate based on a 24-h day length. The equivalent torque causing the observed \u0394LOD can therefore be reconstructed from <\/p>\n
$${\\varGamma }_{z}=-{C}_{{\\rm{m}}}\\frac{{\\varOmega }_{{\\rm{o}}}^{2}}{2{\\rm{\\pi }}}\\frac{{\\rm{d}}}{{\\rm{d}}t}\\Delta {\\rm{LOD}}.$$<\/p>\n
\n (3)\n <\/p>\n
The decadal \u0394LOD (grey line in Fig.\u20091<\/a>) between 1964 and 2019 are obtained after removing the LOD contributions from the atmosphere, oceans, lunar tidal friction, glacial isostatic adjustment and barystatic processes at Earth\u2019s surface (Extended Data Fig.\u20097<\/a>). The multidecadal \u0394LOD (black line in Fig.\u20091<\/a>) are obtained by applying a third-order low-pass Butterworth filter with a cut-off period of 30\u2009years to the decadal \u0394LOD signal (Extended Data Fig.\u20098<\/a>). More details on these operations are provided in the Supplementary Information<\/a>.<\/p>\n We build predictions of \u0394\u03a9m (and \u0394LOD) from <\/p>\n $${C}_{{\\rm{m}}}\\frac{{\\rm{d}}\\Delta {\\varOmega }_{{\\rm{m}}}}{{\\rm{d}}t}={\\varGamma }_{{\\rm{cmb}}}+{\\varGamma }_{{\\rm{g}}},$$<\/p>\n \n (4)\n <\/p>\n by numerically integrating in time from 1964 to 2019 forward models of the torque from surface forces at the CMB (\u0393cmb) and the gravitational torque from the inner core (\u0393g). We consider two different sources of \u0393cmb, electromagnetic and topographic coupling. Multidecadal \u0394LOD result from fluctuations in the torque on the mantle about a mean balance12<\/a>,42<\/a>,44<\/a>. To focus on these, we subtract a temporal mean from all of our predictions of gravitational, electromagnetic and topographic torques.<\/p>\n Electromagnetic torque<\/p>\n The axial electromagnetic torque can be written as a surface integral over the (assumed spherical) CMB as4<\/a>,5<\/a>,6<\/a><\/p>\n $${\\varGamma }_{{\\rm{em}}}=-\\frac{{r}_{{\\rm{c}}}}{\\mu }{\\int }_{{\\rm{CMB}}}{B}_{{\\rm{r}}}{B}_{\\phi }\\sin \\theta {\\rm{d}}S,$$<\/p>\n \n (5)\n <\/p>\n in which rc\u2009=\u20093,485\u2009km is the radius of the core, \u03bc is the magnetic permeability of free space, Br and B\u03d5 denote the radial and azimuthal components of the magnetic field, respectively, and dS is a surface element, \u03b8 is co-latitude and \u03d5 is longitude. Predictions of the temporal variations of \u0393em depend on models of how Br and B\u03d5 change with time at the CMB.<\/p>\n The magnetic field at the CMB can be decomposed as \\({\\bf{B}}=\\nabla \\times \\nabla \\times {\\mathcal{S}}{\\bf{r}}\\,+\\) \\(\\nabla \\times {\\mathcal{T}}{\\bf{r}}\\), in which r is the radial vector and \\({\\mathcal{S}}\\) and \\({\\mathcal{T}}\\) are poloidal and toroidal scalar fields, respectively. Br only involves \\({\\mathcal{S}}\\), whereas B\u03d5 involves both \\({\\mathcal{S}}\\) and \\({\\mathcal{T}}\\). Consequently, we can separate \u0393em into its poloidal and toroidal contributions to B\u03d5 (refs.\u20095<\/a>,6<\/a>).<\/p>\n The poloidal part of \u0393em can be reconstructed directly from magnetic field models built from observations. Such models provide time-dependent maps of \\({\\mathcal{S}}\\) at the Earth\u2019s surface, which are down-continued to the CMB. For a thin layer of conducting mantle material above the CMB with conductance G, the poloidal torque is5<\/a>,6<\/a><\/p>\n $${\\varGamma }_{{\\rm{em}}}^{{\\rm{S}}}=4{\\rm{\\pi }}G{r}_{{\\rm{e}}}^{4}\\sum _{l,m}{\\left(\\frac{{r}_{{\\rm{e}}}}{{r}_{{\\rm{c}}}}\\right)}^{2l}\\frac{m(l+1)}{l(2l+1)}({g}_{l}^{m}{\\dot{h}}_{l}^{m}-{\\dot{g}}_{l}^{m}{h}_{l}^{m}),$$<\/p>\n \n (6)\n <\/p>\n in which \\({g}_{l}^{m}\\) and \\({h}_{l}^{m}\\) are the Gauss coefficients of the field model (with l and m the spherical harmonic degree and order, respectively), \\({\\dot{g}}_{l}^{m}\\) and \\({\\dot{h}}_{l}^{m}\\) are their time derivatives and re\u2009=\u20096,371\u2009km is Earth\u2019s radius.<\/p>\n The toroidal part of \u0393em is given by <\/p>\n $${\\varGamma }_{{\\rm{e}}{\\rm{m}}}^{{\\rm{T}}}=\\frac{{r}_{{\\rm{c}}}}{\\mu }{\\int }_{{\\rm{C}}{\\rm{M}}{\\rm{B}}}{B}_{{\\rm{r}}}\\frac{\\partial {\\mathcal{T}}}{\\partial \\theta }\\sin \\theta {\\rm{d}}S,$$<\/p>\n \n (7)\n <\/p>\n in which \\({\\mathscr{T}}\\) includes contributions from the diffusion of the toroidal field at the top of the core into the conducting mantle and from the advection of the radial field by the flow v tangential to the CMB. We assume that the latter contribution dominates (see ref.\u20096<\/a>), in which case we can write <\/p>\n $${\\varGamma }_{{\\rm{e}}{\\rm{m}}}^{{\\rm{T}}}=-{r}_{{\\rm{c}}}^{3}G{\\int }_{{\\rm{C}}{\\rm{M}}{\\rm{B}}}{B}_{{\\rm{r}}}\\frac{\\partial }{\\partial \\theta }[{L}^{-2}(\\hat{{\\bf{r}}}\\cdot {\\nabla }_{{\\rm{H}}}\\times {B}_{{\\rm{r}}}{\\bf{v}})]\\sin \\theta {\\rm{d}}S,$$<\/p>\n \n (8)\n <\/p>\n in which \\({L}^{2}=-[\\frac{1}{\\sin \\theta }\\frac{\\partial }{\\partial \\theta }\\left(\\sin \\theta \\frac{\\partial }{\\partial \\theta }\\right)+\\frac{1}{{\\sin }^{2}\\theta }\\frac{{\\partial }^{2}}{\\partial {\\phi }^{2}}]\\) is the angular momentum operator and \u2207H is the surface gradient. Evaluation of this torque requires both a time-dependent model of the magnetic field (to evaluate Br) and a time-dependent model of tangential flow v at the CMB. Models for v, in turn, are built such that they are compatible with the observed secular variation of the magnetic field16<\/a>.<\/p>\n The total electromagnetic torque \u0393em is the sum of \\({\\varGamma }_{{\\rm{em}}}^{{\\rm{S}}}\\) and \\({\\varGamma }_{{\\rm{em}}}^{{\\rm{T}}}\\) and is proportional to the mantle conductance G, which we leave as a free model parameter to be determined. We write G\u2009=\u2009KemGref, in which Gref is a reference conductance set equal to 108\u2009S. For given magnetic field and flow models, the strength of the electromagnetic torque scales with the dimensionless conductance Kem.<\/p>\n Our predictions of \u0393em are based on the magnetic field model COV-OBS.x2 (ref.\u200951<\/a>) and the probabilistic core flow model from ref.\u200917<\/a>. These predictions are shown in Extended Data Fig.\u20092<\/a> and lead to multidecadal \u0394LOD predictions that are broadly reversed compared with that observed (Fig.\u20091<\/a>). The poor \u0394LOD match may be because core flows are not sufficiently well resolved. Indeed, core flow models may be designed such that, as well as matching the observed secular variation, they also generate an electromagnetic torque that matches the observed \u0394LOD (refs.\u20096<\/a>,7<\/a>). The required flow adjustment tends to be small. However, although this small adjustment is not in conflict with the secular variation, it is also not required by it. Synthetic tests using numerical dynamo models show that the large-scale core flows retrieved from the secular variation contribute the most and provide an adequate prediction of the electromagnetic torque48<\/a>. This suggests that the electromagnetic torque prediction built from large-scale core flows should be broadly correct and that the mismatch with the observed \u0394LOD of Fig.\u20091<\/a> indicates instead that electromagnetic coupling at the CMB is not the dominant multidecadal torque on the mantle. Synthetic tests48<\/a> also suggest that the small length scales of the CMB magnetic field (l\u2009>\u200913), inaccessible from observations, may contribute to about a third of the total electromagnetic torque. Our reconstruction of the electromagnetic torque may then be underestimated, contributing to an overestimate in our recovered value of G.<\/p>\n We assume a uniform conductance G in the mantle. Although the bottommost region of the mantle (the D\u2033 region) is most likely heterogeneous, and electrical conductivity is expected to vary with geographic position, the precise way in which it does is unknown, so the simplest approach is to assume a uniform G. Using a non-uniform G can modify the time history of the electromagnetic torque52<\/a> and can thus alter our inversion results for the best-fit torque parameters. However, we consider it unlikely that the geometry of G could reverse the sign of \u0393em and account for most of the multidecadal \u0394LOD on its own.<\/p>\n Topographic torque<\/p>\n The topographic torque \u0393top results from the dynamic pressure associated with core flows acting on the CMB topography. Recent modelling efforts37<\/a>,38<\/a>,39<\/a> focus on the perturbation in the pressure field that is induced by the deflection of a mean tangential flow by CMB bumps in a stratified fluid with buoyancy frequency N and permeated by a magnetic field of strength B0. The axial component of \u0393top can be written as <\/p>\n $${\\varGamma }_{{\\rm{t}}{\\rm{o}}{\\rm{p}}}={r}_{{\\rm{c}}}{\\int }_{{\\rm{C}}{\\rm{M}}{\\rm{B}}}{T}_{\\phi }\\sin \\theta {\\rm{d}}S,$$<\/p>\n \n (9)\n <\/p>\n in which T\u03d5 is the azimuthal stress on the CMB. Idealized models of T\u03d5 can be developed for a steady and uniform azimuthal flow v\u03d5 acting on a localized region of the CMB with a periodic topography of wavelength \u2113 and amplitude h. The model in ref.\u200939<\/a> relates T\u03d5 to v\u03d5 as <\/p>\n $${T}_{\\phi }={\\mathcal{D}}F(\\theta ){\\rm{sign}}({v}_{\\phi })\\sqrt{\\parallel {v}_{\\phi }\\parallel },$$<\/p>\n \n (10)\n <\/p>\n in which \\({\\rm{sign}}({v}_{\\phi })=\\frac{{v}_{\\phi }}{\\parallel {v}_{\\phi }\\parallel }\\), <\/p>\n $${\\mathcal{D}}=\\rho {\\varOmega }_{{\\rm{o}}}\\frac{\\sqrt{\\rho \\mu \\eta }}{{B}_{0}}\\frac{N{h}^{2}}{\\sqrt{{\\ell }}}$$<\/p>\n \n (11)\n <\/p>\n and \u03c1 is the fluid density, \u03b7 is the magnetic diffusivity and F(\u03b8) is a function of co-latitude that takes into account the projection of the Coriolis force with the normal to the boundary (Extended Data Fig.\u20099<\/a>). The parameter \\({\\mathcal{D}}\\) has units of kg\u2009m\u22123\/2\u2009s\u22123\/2 and incorporates a collection of physical parameters that influence the torque. Some of these parameters (\u03c1,\u2009\u03b7,\u2009B0) are reasonably well known, but N is not, and we lack a detailed model of the CMB topography, so h and \u2113 are also not well constrained. To take into account different possible values of \\({\\mathcal{D}}\\), we write it as \\({\\mathcal{D}}={K}_{{\\rm{top}}}{{\\mathcal{D}}}_{{\\rm{ref}}}\\), in which Ktop is a dimensionless parameter and \\({{\\mathcal{D}}}_{{\\rm{ref}}}\\) is a reference value of \\({\\mathcal{D}}\\) set equal to 1\u2009kg\u2009m\u22123\/2\u2009s\u22123\/2. The topographic torque from equation\u2009(9<\/a>) can then be written as <\/p>\n $${\\varGamma }_{{\\rm{t}}{\\rm{o}}{\\rm{p}}}={K}_{{\\rm{t}}{\\rm{o}}{\\rm{p}}}{{\\mathcal{D}}}_{{\\rm{r}}{\\rm{e}}{\\rm{f}}}{r}_{{\\rm{c}}}{\\int }_{{\\rm{C}}{\\rm{M}}{\\rm{B}}}F(\\theta ){\\rm{s}}{\\rm{i}}{\\rm{g}}{\\rm{n}}({v}_{\\phi })\\sqrt{\\parallel {v}_{\\phi }\\parallel }\\sin \\theta {\\rm{d}}S.$$<\/p>\n \n (12)\n <\/p>\n The time history of \u0393top depends on the time history of v\u03d5. For a given flow model at the CMB, Ktop modulates the amplitude of the topographic torque and is a parameter left to be determined. Its numerical value provides a constraint on N for the chosen values of h and \u2113 (Supplementary Information<\/a>).<\/p>\n Predictions of \u0393top based on the flow model in ref.\u200917<\/a> are shown in Extended Data Fig.\u20092<\/a>. By itself, \u0393top does not generate the multidecadal \u0394LOD that match observations (Fig.\u20091<\/a>). Notably, the \u0393top model that we used is highly idealized; it is based on a steady and uniform flow at different points of the CMB, whereas the true flow is time-dependent and laterally varying. Moreover, we assume for simplicity that \\({\\mathcal{D}}\\) is uniform but it probably has lateral variations. Allowing \\({\\mathcal{D}}\\) to vary with position at the CMB would change the time variations of this prediction. However, the general trend of \u0393top follows that of \u0393em and illustrates that, to the first order, both of these torques depend on the fluctuations of the large-scale v\u03d5 flow at the CMB; when v\u03d5 is generally westward (eastward) compared with its mean time average, the tangential stress on the CMB from either electromagnetic or topographic coupling is also westward (eastward).<\/p>\n Gravitational torque<\/p>\n The gravitational torque on the mantle \u0393g is given by11<\/a>: <\/p>\n $${\\varGamma }_{{\\rm{g}}}=\\varGamma \\alpha ,$$<\/p>\n \n (13)\n <\/p>\n in which \u0393 is a strength factor and \u03b1 is the longitudinal angle of the degree\u20092 order\u20092 (long equatorial axis) topography of the inner core with respect to the gravitational potential imposed by mantle mass anomalies. The evolution of \u03b1 depends on the bulk axial angular rotation angle \u03c6 of the inner core (related to the differential inner core angular velocity \u03a9i by \\(\\frac{{\\rm{d}}\\varphi }{{\\rm{d}}t}={\\varOmega }_{{\\rm{i}}}\\)) and the viscous relaxation time \u03c4 for the ICB topography to realign with its equilibrium shape, <\/p>\n $$\\frac{{\\rm{d}}\\alpha }{{\\rm{d}}t}=\\frac{{\\rm{d}}\\varphi }{{\\rm{d}}t}-\\frac{\\alpha }{\\tau }.$$<\/p>\n \n (14)\n <\/p>\n We use a time history model of \u03c6 based on the seismic reconstruction from ref.\u200913<\/a> (Extended Data Fig.\u20091<\/a>). Predictions of \u0393g depend on the time history of \u03b1, which is computed by integrating numerically equation\u2009(14<\/a>) for our model of \u03c6 and a given choice of \u03c4. Note that equation\u2009(14<\/a>) describes the change in \u03b1 with respect to a permanent offset \u03b1o associated with the long-term torque balance on the mantle.<\/p>\n In the limit of \u03c4\u2009\u226a\u2009T\/2\u03c0, when viscous relaxation of the ICB occurs rapidly compared with the nominal multidecadal period of T\u2009\u2248\u200970\u2009years of inner core rotation changes, \\(\\frac{{\\rm{d}}\\alpha }{{\\rm{d}}t}\\ll \\alpha \/\\tau \\), so that, from equation\u2009(14<\/a>), \\(\\alpha \\approx \\tau \\frac{{\\rm{d}}\\varphi }{{\\rm{d}}t}=\\tau {\\varOmega }_{{\\rm{i}}}\\), and \u0393g can be approximated as <\/p>\n $${\\varGamma }_{{\\rm{g}}}\\approx \\varGamma \\tau {\\varOmega }_{{\\rm{i}}}.$$<\/p>\n \n (15)\n <\/p>\n In this limit, \u0393g constrains the product of \u0393 and \u03c4, not their individual values.<\/p>\n The strength factor \u0393 is related to the amplitude of the degree\u20092 order\u20092 geoid at the CMB (specified in terms of a spherical harmonic coefficient \\({U}_{2}^{2}\\)) by45<\/a>,53<\/a><\/p>\n $$\\varGamma =8{g}_{{\\rm{i}}}{r}_{{\\rm{i}}}^{2}({\\rho }_{{\\rm{i}}}-{\\rho }_{{\\rm{f}}}){\\left(\\frac{{r}_{{\\rm{i}}}}{{r}_{{\\rm{c}}}}\\right)}^{2}{|{U}_{2}^{2}|}^{2},$$<\/p>\n \n (16)\n <\/p>\n in which ri\u2009=\u20091,221\u2009km is the ICB radius, gi\u2009=\u20094.4\u2009m\u2009s\u22122 is the gravitational acceleration at the ICB, \u03c1i is the inner core density (assumed uniform) and \u03c1f is the density of the fluid core at the ICB. The peak-to-peak topography of the geoid along the equator of the CMB, \\({h}_{2}^{2}\\), is related to \\({U}_{2}^{2}\\) by \\({h}_{2}^{2}=2\\sqrt{\\frac{45}{96{\\rm{\\pi }}}}{U}_{2}^{2}=0.7725{U}_{2}^{2}\\).<\/p>\n To convert \u0393 to \\({h}_{2}^{2}\\), we use \u03c1i\u2009=\u200912,730\u2009kg\u2009m\u22123 and \u03c1f\u2009=\u200912,160\u2009kg\u2009m\u22123 based on the preliminary reference Earth model (PREM)54<\/a>. However, the density contrast at the ICB remains not well constrained and may be as high as 600\u2013900\u2009kg\u2009m\u22123 (ref.\u200955<\/a>) or as low as 200\u2013300\u2009kg\u2009m\u22123 (ref.\u200956<\/a>). This uncertainty on \u03c1i and \u03c1f maps to an uncertainty on the CMB geoid inferred from \u0393. Using PREM, our range of \u0393 ([0.6\u20134.2]\u2009\u00d7\u20091019\u2009N\u2009m) corresponds to a range of \\({h}_{2}^{2}\\) at the CMB of 31\u201383\u2009m. The gravitational potential anomaly of degree\u20092 varies approximately linearly with radius inside the core45<\/a>,53<\/a>, so the peak-to-peak topography along the equator of the ICB (assumed to coincide with an equipotential surface) is \\(({r}_{{\\rm{i}}}\/{r}_{{\\rm{c}}}){h}_{2}^{2}\\approx 0.35{h}_{2}^{2}\\), corresponding to a range of peak-to-peak ICB topography of 11\u201329\u2009m. Although small, an axial rotation of this topography by about 1\u00b0 (Extended Data Fig.\u20094<\/a>) is sufficient to provide the necessary gravitational torque on the mantle. Our results indicate that viscous relaxation of this ICB topography occurs over a timescale of around 10 years (Extended Data Fig.\u20094<\/a>), amounting to radial motion on the order of 10\u2009m near the top of the inner core. It is unclear whether such a small-amplitude (and low-wavelength) displacement can be detected seismically or whether seismic inferences of viscous deformation near the top of the inner core57<\/a> capture instead a more regional deformation.<\/p>\n Torque balance<\/p>\n Palaeomagnetic observations58<\/a> suggest that the present-day mean westward differential flow at the CMB has persisted for the past 9,000\u2009years, at a mean rate of \u03a9w\u2009=\u20090.09\u00b0\u2009year\u22121. This mean flow produces a mean westward torque on the mantle, from either electromagnetic or topographic coupling or a combination of both. On a long-term average, this torque is balanced by an eastward gravitational torque maintained by a steadily differentially rotating and viscously deforming inner core12<\/a>,40<\/a>,42<\/a>,44<\/a>,59<\/a>. Assuming that the CMB torque is caused by electromagnetic coupling, a measure of the steady westward electromagnetic torque is given by44<\/a>,59<\/a><\/p>\n $${\\varGamma }_{{\\rm{w}}}={K}_{1}{r}_{{\\rm{c}}}^{4}{\\bar{B}}_{{\\rm{r}}}^{2}G{\\varOmega }_{{\\rm{w}}},$$<\/p>\n \n (17)\n <\/p>\n in which \\({\\bar{B}}_{{\\rm{r}}}=0.4\\,{\\rm{mT}}\\) is the root mean square strength of the radial magnetic field at the CMB and K1\u2009=\u20092.3 is a numerical factor. A balance between \u0393w and the gravitational torque (equation\u2009(13<\/a>)) implies a permanent eastward angular offset of <\/p>\n $${\\alpha }_{{\\rm{o}}}=\\frac{{K}_{1}{r}_{{\\rm{c}}}^{4}{\\bar{B}}_{{\\rm{r}}}^{2}G{\\varOmega }_{{\\rm{w}}}}{\\varGamma }.$$<\/p>\n \n (18)\n <\/p>\n Using our best-fit \u0393 (=\u20091.4\u2009\u00d7\u20091019\u2009N\u2009m) and G (=\u20091.08\u2009\u00d7\u2009108\u2009S) gives \u03b1o\u2009=\u20091.19\u00b0. From equation\u2009(14<\/a>), the steady (d\u03b1\/dt\u2009=\u20090) eastward differential inner core rotation associated with this offset is \u03a9i\u2009=\u2009\u03b1o\/\u03c4 and our best-fit estimate of \u03c4 (=\u200910.2\u2009years) gives \u03a9i\u2009=\u20090.12\u00b0\u2009year\u22121, equivalent to that found in dynamo models at Earth conditions18<\/a>. Such a rate would account for a substantial part of the observed differential rotation of the inner core (Extended Data Fig.\u20091<\/a>) and implies that, over the past 9,000\u2009years over which the westward drift has persisted, the inner core has undergone close to three full rotations with respect to the mantle.<\/p>\n Inversion of torque parameters<\/p>\n We retrieve the set of torque parameters (\u0393,\u2009\u03c4,\u2009Kcmb), in which Kcmb is either Kem or Ktop, that best fit the observed \u0394LOD based on a Bayesian framework60<\/a>,61<\/a>. We define the probability of a sample \u03b8\u2009\u2261\u2009\u03b8(\u0393,\u2009\u03c4,\u2009Kcmb) as <\/p>\n $${\\rm{\\pi }}({\\boldsymbol{\\theta }})=\\exp \\left(-\\frac{\\chi {({\\boldsymbol{\\theta }})}^{2}}{2{{\\sigma }}^{2}}\\right),$$<\/p>\n \n (19)\n <\/p>\n in which<\/p>\n $$\\chi ({\\boldsymbol{\\theta }})=\\frac{1}{{N}_{{\\rm{s}}}}\\sqrt{{\\sum }_{i=1}^{{N}_{{\\rm{s}}}}{(\\Delta {{\\rm{L}}{\\rm{O}}{\\rm{D}}}_{{\\rm{p}}{\\rm{r}}{\\rm{e}}}({\\boldsymbol{\\theta }},{t}_{i})-\\Delta {{\\rm{L}}{\\rm{O}}{\\rm{D}}}_{{\\rm{o}}{\\rm{b}}{\\rm{s}}}({t}_{i}))}^{2}},$$<\/p>\n \n (20)\n <\/p>\n is the root mean square misfit between the predicted (\u0394LODpre(\u03b8)) and observed (\u0394LODobs) changes in LOD at times ti over the period 1964\u20132019 using a one-year sampling interval (Ns\u2009=\u200956). \u03c3 is the standard deviation associated with the low-pass-filtered multidecadal \u0394LOD signal. Its numerical value is arbitrary; we use \u03c3\u2009=\u20090.2\u2009ms, which gives a reasonable compromise between ensuring a good fit to the multidecadal \u0394LOD while allowing for some uncertainty in its reconstruction.<\/p>\n To sample the parameter space of \u03b8 and build posterior distributions of \u0393, \u03c4 and Kcmb, we use an adaptive MCMC algorithm60<\/a>,61<\/a>. For a current state of the chain \u03b8n, the distribution of a proposed sample \u03b8* is multivariate normal and specified by <\/p>\n $$q({{\\boldsymbol{\\theta }}}^{\\ast }|{{\\boldsymbol{\\theta }}}_{n}) \\sim {\\mathcal{N}}({{\\boldsymbol{\\theta }}}_{n},{s}^{2}{{\\boldsymbol{\\Sigma }}}_{n}),$$<\/p>\n \n (21)\n <\/p>\n in which \u03a3n is the covariance matrix and s is the step size. For each proposed sample of torque parameters, we randomly select one of the 400 core flow realizations to build a prediction of either \u0393em or \u0393top, which is then multiplied by Kem or Ktop, respectively. We also randomly select a time history of the inner core rotation angle \u03c6 that falls within the bounds of the model of ref.\u200913<\/a> and compute \u0393g from equations\u2009(13<\/a>) and (14<\/a>) based on the values of \u0393 and \u03c4 of the present sample. The \u0394LOD prediction from these is then computed from equation\u2009(4<\/a>). The acceptance of a proposed sample is based on a Metropolis\u2013Hastings algorithm: if \u03c0(\u03b8*)\u2009\u2265\u2009\u03c0(\u03b8n), the proposed sample is accepted; if instead \u03c0(\u03b8*)\u2009<\u2009\u03c0(\u03b8n), a random number u is drawn between 0 and 1, and if u\u2009\u2264\u2009\u03c0(\u03b8*)\/\u03c0(\u03b8n), the proposed sample is accepted; otherwise, it is rejected.<\/p>\n Our adaptive strategy comprises two parts. First, the covariance matrix \u03a3n is iterated over the accepted samples following a recursive scheme based on the Welford algorithm62<\/a>. Second, we adaptively tune the step size s, which controls the exploration efficiency of the Markov chain. We update s every 1,000 iterations based on the recorded acceptance rate (the ratio of accepted to total proposed samples) to maintain it within an empirical optimal range of 0.05\u20130.15. Both \u03a3n and s are adapted during the first 10,000 samples of the burn-in phase (which consists of 20,000 samples). Their values are kept fixed for subsequent samples.<\/p>\n Our final distributions are compiled by combining five independent chains started from a dispersed set of initial guesses (Supplementary Information<\/a>). Each chain consists of 180,000 samples generated after the burn-in phase. To reduce the autocorrelation between draws, we thin the output of every chain by storing only every tenth draw. This results in 18,000 samples per chain and a total of 90,000 posterior samples. The convergence of our distributions were tested on the basis of standard MCMC performance metrics (Supplementary Information<\/a>).<\/p>\n","protected":false},"excerpt":{"rendered":"Torque and changes in the LOD An axial torque \u0393z acting on the mantle causes a change in…\n","protected":false},"author":2,"featured_media":634012,"comment_status":"","ping_status":"","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[24],"tags":[132118,89845,89846,21836,2026,61,60,2027,248,82],"class_list":["post-634011","post","type-post","status-publish","format-standard","has-post-thumbnail","category-physics","tag-core-processes","tag-geodynamics","tag-geomagnetism","tag-geophysics","tag-humanities-and-social-sciences","tag-ie","tag-ireland","tag-multidisciplinary","tag-physics","tag-science"],"_links":{"self":[{"href":"https:\/\/www.newsbeep.com\/ie\/wp-json\/wp\/v2\/posts\/634011","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.newsbeep.com\/ie\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.newsbeep.com\/ie\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.newsbeep.com\/ie\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/www.newsbeep.com\/ie\/wp-json\/wp\/v2\/comments?post=634011"}],"version-history":[{"count":0,"href":"https:\/\/www.newsbeep.com\/ie\/wp-json\/wp\/v2\/posts\/634011\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.newsbeep.com\/ie\/wp-json\/wp\/v2\/media\/634012"}],"wp:attachment":[{"href":"https:\/\/www.newsbeep.com\/ie\/wp-json\/wp\/v2\/media?parent=634011"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.newsbeep.com\/ie\/wp-json\/wp\/v2\/categories?post=634011"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.newsbeep.com\/ie\/wp-json\/wp\/v2\/tags?post=634011"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}