Main

The presence of seismic anisotropy, the dependence of seismic wave speed on direction of propagation, in the inner core (IC) was proposed over 30 years ago to explain the early arrival times of IC-sensitive seismic body waves (PKPdf) travelling on paths parallel to the Earth’s rotation axis1,2 and anomalous splitting of core-sensitive free oscillations3. This anisotropy is thought to result from the alignment of iron crystals caused by deformation in a flow field induced by the evolution of the core, that is, deformation texturing. Previously, different geodynamic4 and plastic deformation mechanisms5 were explored to explain the variation of PKPdf travel times with the angle of the ray path with respect to the rotation axis. Here we combine geodynamic modelling of flow in the IC, allowing for slow lateral translation, with present knowledge of the mineralogy and deformation mechanisms proposed for the IC to explain spatial patterns of observed seismic travel times in an updated dataset.

Indeed, early models of seismic anisotropy based on measurements of PKPdf travel times featured constant cylindrical anisotropy, with the fast axis parallel to Earth’s rotation axis. Further work on IC structure has revealed increasing complexity. Recent IC models comprise two quasi-hemispheres of differing strengths of anisotropy; ~4.8% on average in the quasi-western hemisphere (WH) and ~1.4% in the quasi-eastern hemisphere (EH)6,7,8. Anisotropy strength increases with depth in the IC up to 8.8% at the centre of the IC9. However, past models have suffered from poor data coverage on polar paths, due to the limited distribution of earthquakes and stations.

Seismic analysis of asymmetric anisotropy

Aiming to address this issue, we have made new differential travel-time measurements of PKPab–df and PKPbc–df from recent seismic deployments (Fig. 1 and Extended Data Fig. 1), increasing sampling of the IC along polar directions at a large range of depths, and added them to our existing global collection (Methods). The updated dataset samples the IC close to Earth’s rotation axis, from the inner core boundary (ICB) to within 35 km of the centre of the Earth.

Fig. 1: Sampling of the inner core with polar PKP rays.
figure 1

a, Ray paths of PKP branches used here for an example source (star) and receiver (triangle). PKPdf samples the IC, while PKPbc and PKPab remain in the outer core. ξ is the angle that the PKPdf path in the IC makes with Earth’s rotation axis. b–d, Only polar paths (ξ < 35°) from source to receiver colour-coded by effective IC velocity anomaly (line colour) and ξ (symbol colour) for paths turning between 5,200 and 5,600 km depth (b), 5,600 and 6,000 km (c) and 6,000 km and Earth’s centre (d); 530 polar paths are shown. Turning points for PKPbc–df and PKPab–df ray pairs are shown as diamonds and circles, respectively. We exclude the South Sandwich Islands (SSI) to Alaska paths. The grey region marks the best-fitting WH boundaries determined in this study and the solid line marks the EH boundary39.

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Differential travel-time anomalies, expressed as the effective P-wave velocity anomaly within the IC (\({{\mathrm{dln}} V} = - \frac{{{\mathrm{d}}T}}{{{\mathrm{Tic}}}}\), where T is the travel time of the wave, dT is the differential travel time anomaly, Tic is the travel time through the IC, and V is the effective velocity) exhibit a strong dependence on ξ, the angle of the path within the IC relative to the rotation axis (Fig. 1a), with residuals of up to 9.9 s at the largest distances for polar paths and ±2 s for more equatorial paths (Extended Data Fig. 2). Furthermore, the residuals depend on both the longitude and depth of the turning point of the ray (Fig. 1b–d). To first order, as has been found previously6,7,8,9, the data exhibit hemispherical differences (Fig. 1 and Extended Data Fig. 3). Assuming a linear dependence of anisotropy on depth in each hemisphere, we determine the best fitting western boundary of the WH to be between 166° W and 159° W (Methods and Extended Data Fig. 4). However, sharp hemispherical boundaries are difficult to reconcile with geodynamic models of IC growth.

Examining the data more closely, we find that the effective velocity anomaly linearly increases with distance, that is, turning point radius in the IC, in both hemispheres (Fig. 2b). The gradient with distance is approximately equal in both hemispheres, but with an offset to larger anomalies in the WH. This gradient is dependent on ξ and is steepest and most robustly defined for polar paths (0 < ξ < 15°) (Extended Data Fig. 5). The largest effective velocity anomalies (≥3.5% dlnV) are recorded in the WH, for rays bottoming at around 400 km radius (distances ≥170°) with longitude ~60° W, not at the centre of the IC. Our travel-time data suggest a depth dependence of anisotropy that, to first order, is smooth and asymmetric with respect to the centre of the Earth, rather than a hemispherical pattern with sharp boundaries between the hemispheres.

Fig. 2: Effective velocity anomaly in the inner core from PKPab–df and PKPbc–df travel time observations as a function of ξ and epicentral distance.
figure 2

a, Effective velocity anomalies as a function of ξ display a hemispherical pattern implying stronger anisotropy in the WH (red) than in the EH (blue). Anisotropy curves from fitting equation S1 for each hemisphere are shown as solid lines. Data from the SSI to Alaska are excluded (Extended Data Fig. 9). b, IC velocity anomaly as a function of epicentral distance, and thus bottoming radius of the ray, in the WH (left) and EH (right), for data with ξ ≤ 15° (shaded region in a). Solid lines mark linear fits as a function of distance in the respective hemispheres with mirror images across the centre of the Earth (180°) shown as broken lines. Moving averages (diamonds) and s.d. at 2.5° increments in distance highlight the robust trends. The EH trend is extended to meet the WH trend (at 400 km radius) with a dotted blue line and the WH trend is extended beyond the data constraints to the rotation axis with a dotted red line.

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Growing anisotropic texture in the inner core

To interpret the seismological observations, we consider that the core probably grows preferentially at the Equator due to Taylor column convection in the outer core, which induces more efficient heat transport in the cylindrically radial direction10,11. Isostatic adjustment would cause the oblate IC to flow inwards from the Equator and up towards the poles10,12. Such a flow would be confined to the uppermost layer if a strong density stratification existed, and would induce deformation at depth if not12. Any asymmetry to the heat extraction from the IC in the plane of the Equator would cause asymmetric growth11,13 resulting in lateral advection of the growing IC and thus slow net translation. Previous studies have attempted to explain the depth dependence of anisotropy by degree 2 flow10,13 on the one hand, and the hemispherical dichotomy by degree 1 flow14,15 on the other. However, these hemispherical studies considered fast convective instabilities resulting in a degree 1 flow that, alone, could not produce the observed seismic anisotropy pattern and strength. Guided by our seismic observations, we combine the processes of preferential equatorial growth and hemispherically asymmetric growth and analytically model the flow pattern in a neutrally stratified IC (Fig. 3; Methods). Advection of strained crystals along the translation axis shifts the pattern of high deformation laterally from the axis of rotation, with lateral offset of the high deformation zone from the rotation axis depending on the chosen translation rate. A key assumption here is that the translation rate is slower than the rate of growth, resulting from differential growth13 and not from simultaneous melting and freezing on opposite hemispheres14,15. Given the limited constraints, the age of the IC and the translation velocity are both free parameters of such a model. The differential growth rate of the IC between the Equator and the poles is described by the parameter S2, which controls the magnitude and pattern of strain experienced. S2 is loosely constrained to between 0 and 1 from dynamical arguments10, while geodynamical models of the outer core11,13 argue for a value of ~0.4. Constraints on S2 are not strong, however additional information can be brought in from mineral physics, which provides constraints on strain-rate-dependent development of anisotropy.

Fig. 3: Inner-core growth, strain and anisotropy.
figure 3

a, Sketch combining preferential equatorial growth (driven by Taylor column convection in the outer core) and asymmetric growth rate with imposed IC growth rate at the boundary and internal flow shown by black and white arrows, respectively, and expected topography (exaggerated for visualization) shown by the orange line. b, Asymmetric growth and movement in from the Equator and towards the poles causing lateral and vertical advection of the strongest deformation. Both a and b show the meridional plane along the axis of translation with the dashed line showing the Equator. The strain field aligns iron grains, producing strong anisotropy in the deep IC that is offset from the rotation (N–S) axis in the equatorial plane (c), and elongated parallel to the rotation axis in the meridional plane (d) along the direction of translation shown by the blue arrow in c. Calculated with IC age of 0.5 Ga, S2 of 0.6, translation rate of 0.3 and translation direction from 120° E to 60° W.

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In our model, the present-day IC seismic anisotropy is a function of the initial single-crystal anisotropy, the slip planes of crystal deformation and the flow field. Crystallographic alignment of a polycrystal is necessary to generate substantial anisotropy on the length scale of the IC. Using visco-plastic self-consistent modelling (VPSC)16, we calculate the anisotropy resulting from dislocation creep in the strain field produced by our geodynamic models for different IC ages, translation rates and single-crystal structures. Despite body-centred cubic (bcc) iron having strong single-crystal anisotropy17, we find that it cannot produce strong polycrystal anisotropy, nor can face-centred cubic iron, as has also been shown previously18. In contrast, plastic deformation of a hexagonally close-packed (hcp) iron–nickel alloy (Fe93.75Ni6.25; ref. 19), compatible with cosmo-chemical constraints20, with slip on the <c + a> pyramidal planes21,22 produces an anisotropic IC with up to 6.6% anisotropy (Fig. 3d) that can fit the seismic data well. In this model, the fast direction of anisotropy becomes aligned with the rotation axis and the slow direction varies with depth (Extended Data Fig. 6), matching observations23. Pure hcp iron does not produce as strong a match to our observations (Extended Data Figs. 7 and 8).

The pattern and strength of the flow field induced by IC growth impacts the strain that crystals experience, and is controlled by the IC age, S2, the translation rate, and the direction of translation. The IC age trades off linearly with strain rate and duration, but the dislocation creep has a nonlinear relationship between stress and strain rate (Methods), implying that the degree and pattern of crystal alignment, and therefore the pattern and strength of the resulting seismic anisotropy, varies with duration, and so with IC age. The total strain is controlled by the parameter S2; thus, the strain rate is controlled by both IC age and S2. We constrain the IC growth history by running models with a range of ages, translation rates and values of S2 and compare predicted anisotropy from these models with our seismic observations (Fig. 4). The data are best fit by models with 0.4 ≤ S2 ≤ 0.8. Within the range of acceptable IC ages22,23,24 (Methods), we find S2 = 0.6 and an IC age of 0.5 Ga to best fit both the seismic observations and geodynamic constraints, although the constraint on age is not strong (Fig. 4d–f; Methods). Translation at a rate of 0.3 IC radii over the 0.5 Ga IC lifetime along an axis oriented in the equatorial plane from 120° E towards 60° W matches the geographic pattern of anisotropy, achieving a 93% variance reduction for the polar data compared with 89% for a model with no translation. Our model shows increasing anisotropy strength with depth. The model also displays weak anisotropy near the ICB that is stronger in the WH than the EH, which is qualitatively compatible with models of hemispherically distinct isotropy in the upper IC from measurements of PKIKP24 and P′P′df25 travel times, and with constraints on the magnitude and distribution of anisotropy from normal modes26,27.

Fig. 4: Predicted versus observed PKP velocity anomalies for hcp Fe93.75Ni6.25 at 5,500 K and 360 GPa and trade-offs between IC age, S2 and translation rate.
figure 4

a–c, Predicted (dark blue and red dots and with mean as grey squares) and observed (light blue and orange dots and with mean as black diamonds) effective velocity anomalies as a function of epicentral distance (a) for data with ξ ≤ 15°, marked by the shaded region in b and c, and as a function of ξ in the western (b) and eastern (c) hemispheres for the IC growth model in Fig. 3. The open circle in a marks the predicted effective velocity anomaly for a path along the rotation axis. Error bars for the data show the mean and one s.d. at 2.5° (a) and 5° increments (b,c). d–f, Variance reduction of the model relative to the data illustrating the trade-offs between IC age and S2 (d), S2 and translation rate (e) and translation rate and IC age (f). Grey circles mark tested values and the green circle marks the best-fitting parameters, corresponding to the model in a–c and at which the 3D space is sampled. The green line tracks the best x value at any given y value. Models in the shaded region have too young IC ages based on core conductivity. The surface is interpolated with a ‘minimum curvature’ spline.

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Remaining discrepancies between observations and predictions may result from contamination of the observations by mantle structure and small-scale structure in the IC. While differential measurements remove some of the effect of upper mantle heterogeneity on the PKP travel times, even modest three-dimensional (3D) velocity structure deeper in the mantle can influence them28. The largest travel-time anomalies that we observe (9.9 s) are for PKPab–df measurements between 170° and 175° distance, where there is large lateral separation between the two ray paths such that they could experience considerably different velocities in the deep mantle. Still, mantle velocity anomalies such as ultra-low–velocity zones (ULVZs) and the large low-shear velocity provinces (LLSVPs) could generate at most 1–2 s travel time delays. Furthermore, the data with large travel-time anomalies pierce the core–mantle boundary at distinctly different locations (Fig. 1b–d), and no ULVZs have yet been reported in these regions29. Thus, mantle structure would mostly introduce scatter and not strongly obscure the first-order IC anisotropic pattern that we model.

Implications for core and mantle evolution

Within the limits of the assumptions made, particularly the assumption of dislocation creep, the proposed model has implications for the physical properties of the IC. Assuming Yoshida-style deformation restricts the range of possible IC viscosities (η > 1018 Pa s) and requires the IC age to be larger than the diffusive timescale, which may range from 0.2 to 1.5 Ga (ref. 4) depending on the chosen core conductivity. This constraint places the viscosity at the upper end of the range recently obtained by density function theory30.

Our model suggests that the seismic structure of the IC records the large-scale pattern of the heat flux at the ICB, which is controlled by the dynamics of the outer core and the heat flux variations at the core–mantle boundary11. Our preferred model has a translation rate of 0.3 and a ratio of polar to equatorial growth (S2) of 0.6. This corresponds to a growth rate that is 40% lower at the poles and 130% larger at the Equator compared with the global average. The growth rate at the Equator varies between the eastern and western hemispheres from 100% to 160% of the global average rate, respectively. This pattern is similar to that obtained when forcing the geodynamo with heat fluxes at the top of the core based on present-day lower-mantle structure11 and suggests that the asymmetry in heat extraction has been stable in the outer core for times similar to the age of the IC. This agrees with indications that the currently observed LLSVPs separated by a ring of high seismic velocities at the base of the mantle may have been stable for at least 200–300 Ma (refs. 31,32,33), and with the potential existence of structures in the mantle stabilizing the convection pattern34. In contrast, geomagnetic observations of outer-core patterns that imply forcing by bottom-up interactions35 may indicate either a recent change in IC dynamics from passive to active dynamics or complex interactions between the inner and outer core that may be described at smaller scales than are considered here. Our modelling supports a relatively high core conductivity, as it favours a young IC age (~0.5 Ga) and requires the absence of convective instabilities. Preventing the development of thermal instabilities with an IC age of 0.5 Ga requires IC thermal conductivity >120 W m−1 K−1 (ref. 4). Better resolving 3D patterns of seismic anisotropy in the IC may help further document the uneven growth history of the IC, providing a record of the global-scale pattern of outer-core dynamics. While our model does not consider the smaller-scale seismic structure of the inner core36, we provide a holistic model of IC growth capable of matching the observed seismic anisotropy and consistent with available palaeomagnetic observations and mineral physics data37,38.

Methods

Seismology

We collect PKPab–df and PKPbc–df differential travel-time measurements to determine inner-core structure (Fig. 1). Differential travel-time anomalies, calculated with respect to a one-dimensional (1D) reference model, can thus be attributed to the IC, at least to a first order.

Our dataset comprises the existing University of California Berkeley PKP travel-time-data collection23,40,41,42 and additional data43. This collection includes 2,944 and 1,170 PKPab–df and PKPbc–df differential travel-time measurements, respectively. Here we have added a total of 614 PKPab–df and 416 PKPbc–df measurements from both recent events in the South Sandwich Islands between 23 October 2015 and 15 September 2017 observed in Alaska and other nearby stations in the northern hemisphere and events of M > 5.5 at latitudes greater than 50° N between 1 January 2008 and 31 June 2017 observed at distances beyond 150° and stations in the southern hemisphere, collected using the Standing Order for Data (SOD) mass-downloader tool and Incorporated Research Institutions for Seismology (IRIS) Wilber3 tool (289 observations from the South Sandwich Islands to Alaska and 741 observations from other high-latitude events observed in Antarctica). These events were recorded at networks YT, ZM, 2C, AI, AU, ER, G, GE, II, IU, PS, SY, C, 9G and ID.

Locations and arrival times for events before and after 2009 are from the EHB44 and International Seismological Centre (ISC) catalogues, respectively. We removed the linear trend and mean from the vertical component data and deconvolved the instrument response to velocity. Data were bandpass filtered between 0.4–2.0 Hz, and the Hilbert transform was applied to take account of the phase shift between PKPab and PKPdf. We manually picked phase onsets relative to predicted times from the 1D reference model ak135 (ref. 45) after applying ellipticity corrections46. PKPdf and PKPbc are picked on the untransformed data, while PKPab is picked on the Hilbert-transformed data. We classified picks based on the clarity of the signal onset and the prominence of the signal in the unfiltered trace. Following picking and classification, we retained 614 and 416 highest-quality differential PKPab–df and PKPbc–df travel times, respectively, measured with respect to ak135 (all data are shown, split by quality, in Extended Data Fig. 2). These new measurements, combined with the existing catalogues mentioned above, yield 3,558 and 1,586 high-quality PKPab–df and PKPbc–df measurements, respectively. We use only the high-quality data for all plots and calculations in this paper.

Attributing the entire travel-time anomaly to structure in the IC, we convert travel times to velocity anomalies relative to 1D model ak13545 as: \(- \frac{{{\mathrm{d}}T}}{{{\mathrm{Tic}}}} = \frac{{{\mathrm{d}}v}}{v}\), where Tic and v are reference travel times and velocities in the IC, respectively. This accounts for the difference in path length between the shallow and more deeply travelling waves. We construct cylindrically symmetric models of anisotropy, in which the perturbation to a spherically symmetric model47 is expressed as:

$$\frac{{\updelta v}}{v} = a + b\cos ^2\xi + c\cos^4\xi$$
(1)

where v and δv represent the reference velocity and velocity perturbations, respectively, and ξ is the angle of the ray-path direction at the bottoming point of the ray with respect to Earth’s rotation axis. We determine the coefficients a, b and c (which can depend on depth and location, depending on the model considered) by fitting our data with an L1 norm to account for outliers. The apparent IC velocity anomaly will be the integrated effect of the velocity anomalies along the ray path in the inner core.

To best illustrate the hemispherical differences, we update hemisphere boundaries by grid searching for the location of the western boundary of the western hemisphere. We hold the eastern boundary fixed to that found previously39, as our dataset has limited coverage in this region while the previous study was designed to sample the eastern boundary. Seeking to test the model of an IC with hemispherical and depth-dependent anisotropy, we split the data into two hemispheres described by the candidate boundaries and fit models of velocity anomaly as a function of distance to the polar data (ξ ≤ 15°). Only the polar data show notable hemisphericity, thus we exclude higher-ξ data to avoid biasing the fit with data with little resolving power. We seek to minimize the combined misfit to the two straight lines (Extended Data Fig. 5). Hemispheres are assigned based on the longitude of the turning point of the ray, an approximation that works well for polar data but for equatorial data leads to smearing of hemispherical differences. However, since equatorial data do not show large differences between hemispheres this approximation is not problematic. The western boundary of the western hemisphere produces equal fits to the data when located between 166° W and 159° W, with a very sharp falloff in R2 at locations <166° W or >153° W. While we simplify the boundary to a line of constant longitude, we cannot rule out a bent western boundary48. When we repeat this test with the less-polar data, ξ < 35°, the best-fitting hemisphere locations are similar with a sharp falloff at <153° W.

To determine the robustness of the resolved gradients of velocity with depth in each hemisphere, we perform an interaction–effect analysis using the data shown in Fig. 2b. We find that to 95% confidence we cannot reject the null hypothesis that the gradient of the two hemispheres is the same, that is, the gradient of the two hemispheres is statistically the same. To determine the robustness of the offset in intercept between the two gradients, we perform a bootstrap resampling of the same data. We find that the second standard deviations (s.d.) about the bootstrapped means do not overlap between the two hemispheres. We conclude that the trends of velocity with depth in the two hemispheres have statistically distinct intercepts but statistically very similar gradients.

South Sandwich Islands to Alaska anomaly

PKPbc–df and PKPab–df data recorded at stations in Alaska show a spread of travel-time anomalies that do not match the global pattern as a function of ξ (Extended Data Fig. 9), as has been previously reported36,49,50. This is especially clear for PKPbc–df measurements that show travel-time anomalies of up to 6 s, in contrast to measurements outside of Alaska of less than 3 s on the most polar paths. This data may be contaminated by the Alaska slab49,51,52. We thus remove data recorded in Alaska from the analysis presented here, but we keep data from events in Alaska, which are not affected by the slab and fit the global trends (Extended Data Fig. 9).

Geodynamics

It has been proposed by different groups that viscous relaxation of topography at the inner core boundary, caused by the differential growth rate of the inner core, may orient crystals in the inner core and explain the inner-core bulk anisotropy (Supplementary Fig. 1)10. Flows in the inner core induced by preferential growth at the Equator have a vertical cylindrical axis of symmetry and tend to align the crystals along this axis close to the centre of the inner core. However, such a model cannot explain the observation of hemispherical differences in IC anisotropy. Meanwhile, lateral translation caused by either simultaneous melting and crystallization on opposite sides of the IC14,15 or an unstable compositional gradient11,13 has been proposed to explain the hemispherical dichotomy in the IC.

Here we consider the flows induced by the differential growth rate at the inner core boundary, where the differential growth rate is the sum of two previously studied patterns: preferential equatorial growth and hemispherical asymmetric growth (Supplementary Fig. 1). We consider neutral density stratification of the inner core, as it is the only regime in which deformation occurs at depth4. This drastically reduces the parameter space where such a flow could be observed, as a slightly unstable density stratification develops large-scale convection53 and a stable density stratification inhibits radial flows and layers of high deformation develop near the inner core boundary12. As discussed before4, preferential growth at the Equator would still develop large-scale flows for stable stratification for large viscosity values (η > 1018 Pa s) and an inner core age larger than the diffusive timescale (0.2–1.5 Ga). The assumption of Yoshida-style convection thus restricts the range of possible viscosities for the IC.

We solve the conservation of momentum equation for an incompressible fluid of constant viscosity η and constant density ρ in a spherical shell whose radius (Ric) increases with time (t) as \(R_{{\mathrm{ic}}}\left( t \right) = R_{{\mathrm{ic}}}\left( {\tau _{{\mathrm{ic}}}} \right)\sqrt {t/\tau _{{\mathrm{ic}}}}\) from time = 0, representing the nucleation of the inner core, to time τic, today. The assumption of neutral stratification allows for a complete analytical solution for the flow for both the equatorial10 and hemispherical patterns.

To determine the trajectory of a particle in the inner core, we fix the position of the particle today (at τic) and integrate the trajectory backward in time using GrowYourIC54. The intersection of the trajectory and the ICB in the past corresponds to the time of crystallization of the material.

We output the positions, velocity components and velocity gradients of the particles with time and use this to calculate crystal orientations. To obtain a first idea of the deformation experienced by the particle, we calculate the von Mises equivalent strain rate and its average over the trajectory for tcrystallization > 0.

In modelling the growth and resulting strain in the inner core, we also test the dependence of S2 on the preferred IC age and translation rate. We explore translation rates between 0 and 0.5 in increments of 0.1 radii of the IC over the age of the IC, IC ages between 0.25 and 1.5 in increments of 0.25, and S2 between 0.2 and 1.0 in increments of 0.2, searching for the model that best matches the observed anisotropy. Such slow translation rates, which are lower than the crystallization rate, require only differential freezing and no melting, unlike models of fast translation14,15. We calculate the core growth and translation rates for nondimensionalized time and IC size. We then scale the model using the radius of the inner core at the present day (1,217.5 km) and the chosen age of the inner core. Thus, we scale the instantaneous strain rate by the inverse of the IC age (1/τic) and the time step (dt) by the IC age (dt × τic). Inner-core age linearly affects the strain rate, but the maximum total accumulated strain for all IC ages is equal to S2; thus, the value of S2 affects the total accumulated strain and the strain rate, while the IC age only affects the strain rate. For each IC age, S2 and translation rate, we use VPSC to calculate the resulting deformation. We model deformation by dislocation creep, thus strain rate and time step have a nonlinear influence on the generation of anisotropy. Comparison of the resultant anisotropy models with our seismic observations suggests best-fitting ages between 0.5 and 1.5 Ga, depending on S2, with a translation rate of 0.3 radii over the age of the IC. Since models with S2 < 0.4 generate too little anisotropy to match the data and models with S2 ≥ 0.8 require an IC age of 0.25 Ga (which is probably too young55,56,57), we determine reasonable bounds for S2 of 0.4 ≤ S2 ≤ 0.6, with the age trading off accordingly between 1.5 and 0.5 Ga, respectively. External constraints on the parameter S2 are poor, but previous work based on outer-core geodynamics considerations11 has preferred S2 = 0.4, which is consistent with our preferred range. The data can be fit by models with ages that are consistent with the range suggested from palaeomagnetic constraints (between 0.5 and 1.3 Ga; refs. 37,38).

Anisotropy strength depends on IC age and S2, thus by matching the observed strength of anisotropy for a given S2 we can estimate IC age. Models with S2 = 0.4 and inner core ages >0.5 Ga generate strong and localized anisotropy capable of matching our observations. Models run with inner core ages <0.5 Ga have maximum anisotropies of less than 6.0%, and the volume of the IC with maximum anisotropy is very small. These models predict weaker anisotropy and lower gradients of anisotropy with depth than what we observe. In contrast, for older inner cores the maximum anisotropy reaches ~7.0% in places and thus achieves nearly full alignment of crystals, for which anisotropy would be 7.5%. For S2 = 0.6 inner core ages ≥0.5 predict strong anisotropy with a maximum anisotropy of ~7.0% in places, equivalent to the models with S2 = 0.4 and IC age ≥1.0 Ga. In fact, for S2 = 0.6 and IC ages >1.0 Ga, the models begin to predict anisotropy that is stronger than the observations. Age of the IC also depends on IC viscosity through its influence on strain rate, but the viscosity is fixed to >1018 Pa s by our assumption of Yoshida-style deformation4. Alternative solutions to fit the large anisotropy could be easier deformation of hcp iron–nickel alloy than in our VPSC simulation, stronger single-crystal anisotropy, or pressure dependence of the single-crystal anisotropy (Extended Data Figure 8). However, both the deformation behaviour and precise anisotropic pattern of iron and iron alloys at high pressures and temperatures are not well constrained.

We rotate the resulting model of anisotropy about the rotation axis and in the equatorial plane through 360° in 10° increments and compare the misfit with the data. We thus find that the best-fitting growth direction is from 120°E towards 60°W, placing the points of fastest and slowest growth under the Banda Sea and Brazil, respectively. Interestingly, these are very similar to the foci of growth and melting modelled by ref. 15, albeit with the opposite direction of growth.

Mineral physics

We calculate the anisotropy that would result from deformation of an IC of a given composition in the presence of the strain field described above. An important component is the composition chosen, as anisotropy of the single crystal controls the anisotropy of the bulk model after deformation. Experimental studies indicate that hcp iron is stable at IC conditions58, but this is complicated by the presence of lighter elements in the IC. The bcc iron phase may also be stable59, depending on the strain field60. First principles calculations estimate the anisotropy of pure, single iron crystals to range from 4.9 to 7.9% for hcp iron (given as the total range from minimum to maximum dlnV), and up to 14.7% for bcc iron17,61,42 and potentially up to 20% near the melting point of hcp iron62, although there is debate over the trends of anisotropy as a function of pressure and temperature61,63,64,65,66. The pattern of anisotropy for iron near its melting point62 is very different from the observed IC anisotropy. Alloys of iron with plausible light elements modify the character of anisotropy, but the limited number of experiments leaves the dependence on pressure and temperature uncertain19,67,68,69,70,71. We select hcp iron–nickel alloy (Fe93.75Ni6.25; ref. 19) as its pattern of single-crystal anisotropy (Supplementary Fig. 2) is most similar to the observed anisotropy (Extended Data Figs. 3 and 9) and it is consistent with cosmo-chemical calculations of the core’s composition20.

We calculate the development of crystal preferred orientation (CPO) in the presence of the strain field resulting from our above inner-core growth models using the VPSC modelling code16. Groups of 1,500 particles, representing crystals of hcp iron–nickel alloy (Fe93.75Ni6.25), are generated at the ICB throughout the growth history of the inner core. Crystal growth at the ICB may cause pre-texuring72. We model particles with an initial solidification pre-texture in which the c axes of the hcp iron crystals are oriented in the plane of the ICB, as in previous work5. The group of particles deforms as it is subjected to the strain along the tracer path. The deformation is controlled by the crystal slip systems, for which we use those of hcp iron. Following from a previous study5, we allow slip along the <c + a> pyramidal planes of hcp iron and lock the remaining slip systems and we set the normalized critically resolved shear stresses to ∞ for the basal <a>, prismatic <a> and pyramidal <a> plane slip systems, and 0.5 for the pyramidal <c + a> plane slip system. We measure the resultant CPO at the present day.

CPO developed at each step of the growth model is combined with its respective elastic tensor to determine the resultant anisotropy. We incorporate estimates of the elastic tensors resulting from ab initio molecular dynamic simulations. For our chosen hcp FeNi alloy, elastic tensors are only available at 0 K and 360 GPa, and 5500 K and 360 GPa (ref. 19). The temperature range of the inner core is probably very small, on the order of 30 K (ref. 73). We thus neglect the pressure and temperature dependence of the elastic constants and calculate the resultant CPO for Fe93.75Ni6.25 alloy at 5500 K and 360 GPa (Supplementary Fig. 11). The discrepancy between the observed and predicted anisotropy in the eastern hemisphere (Fig. 4a) may result from the single-crystal anisotropy being fixed with respect to pressure, and thus not allowing weak enough anisotropy to match the data.

We seek to understand the influence of the physical state with depth in the IC on the elastic tensors. As above, we neglect the temperature dependence given its small impact on elastic tensors but consider the influence of pressure. Given the limited data for FeNi alloys, we assess the effects of the pressure dependence of anisotropy using pure Fe, for which there are data at a range of pressures from ab initio calculations19,61. Pressure as a function of radius was extracted from the Preliminary Reference Earth Model74, where the pressure ranges from 330 GPa at the ICB to 364 GPa at the centre of the Earth. A reference point of 360 GPa and 5500 K was chosen, and the derivatives of pressure at a constant temperature were determined by a middle difference method using results from the above-mentioned studies. Elastic constants from the reference point were then interpolated using a Taylor expansion to the second derivative of pressure from the reference point to the pressure at each location along the geodynamic streamline (Extended Data Fig. 8). We find that at the pressures of the ICB, pure hcp Fe would show weaker anisotropy of 5% and stronger anisotropy of 7% at the centre of the IC.

To predict travel-time anomalies generated by the modelled anisotropy, we trace rays through 1D velocity model ak135 between the ICB piercing points for each of our observations using TauP75, assuming propagation along the theoretical ray path in the 1D model. We interpolate the anisotropy model to a 50 × 50 × 50 km grid spacing and interpolate the ray to increase spatial sampling. For each ray segment, we find the anisotropy at the nearest model location, measure the ξ angle of the ray segment, calculate the velocity anomaly for that ξ angle using Christoffel’s equation and calculate the resulting travel-time anomaly given the length of the ray segment. We sum the travel-time anomalies over the ray to find the total predicted anomaly for each path through the anisotropy model. We calculate the variance reduction between the observed and predicted travel-time anomalies for the most polar data, ξ < 15°, without separating hemispheres.