Introduction

The Earth’s rotation rate, or length of day (LOD), is known to vary over a broad spectrum of frequencies, ranging from the slow, secular increase caused by tidal dissipation, to inter-annual variations resulting from the interaction between atmospheric flows and the solid crust1,2. Large amplitude fluctuations are present in measurements of the LOD, ranging from timescales of several decades (~60)3 to sub-decadal (~6 yr)4. These fluctuations are widely believed to be the result of exchanges of angular momentum between the liquid core and the solid mantle4,5,6, however the physical mechanisms responsible for this exchange are not fully understood7. The required axial torques may arise from a variety of sources, including pressure gradients, electromagnetic and viscous stresses at the core mantle boundary (CMB), as well as gravitational torques between the mantle and solid inner core8. Previous scaling arguments7 suggest that viscous drag on the CMB is too small to produce the required torque. The electromagnetic torques can explain LOD if the electrical conductivity of the lower mantle is sufficiently large3. Gravitational coupling between the inner core and mantle is also capable of producing the target torque of 1018N m8,9. Pressure, or topographic, coupling occurs wherever the CMB deviates from spherical symmetry. Topographic coupling between the core and mantle is one of the least understood mechanisms, due in part to the complex techniques required to model it. To the authors’ knowledge, only a single study has used three-dimensional simulations, and this work used an approximate set of boundary conditions over a limited range of parameters10. Towards this end, the present work examines the role of topographic coupling over a broad set of numerical experiments of rotating convection in a spherical shell in which topography is superposed onto the outer boundary. The primary goals are to test the scaling behavior of topographic torques with varying input parameters, including the rotation rate, forcing strength, and topographic amplitude, and extrapolate the findings to the conditions of the outer core.

Seismological studies report CMB topography with peak-to-trough amplitudes near 5 km11,12. There is little overlap in the spatial structure of seismically-derived CMB topography, indicating that it is poorly constrained13. Nevertheless, it is likely that CMB topography is characterized by a broad spectrum of amplitudes and wavelengths, and may therefore have a structure that is similar to the Earth’s surface14,15, albeit with a smaller amplitude. The presence of seismically slow material at the base of the mantle (e.g., the so-called Large Low Shear Velocity Provinces, or LLSVPs) below southern Africa and the southern Pacific16 likely contributes to deformations at the CMB, which may influence flow in the outer core17.

When deviations in spherical symmetry are present on the CMB, non-radial pressure forces will be exerted on the CMB by the outer core. Such forces can then give rise to torques on the mantle that, in turn, cause changes in its angular momentum. This topographic torque is defined by

$${{\mathbf{\Gamma }}}=\oint {{\bf{r}}} {\times} p\,d{{\bf{a}}},$$
(1)

where r is the position vector pointing from the center of the planet to the surface of the CMB, p is pressure, and da is a surface area element of the CMB with the normal chosen to point away from the core. In this study we allow for deviations from spherical symmetry and denote the radial position of the CMB as r = rcmb−ϵ h(θ, ϕ), where rcmb = 20/13 is the non-dimensional radius of the CMB in the absence of topography, ϵ is the topographic amplitude and h(θ, ϕ) defines the topographic shape (see equation (8)). In our non-dimensionalization, ϵ is the ratio of topography depth (in km) to the core depth (2260 km) and is therefore expected to be O(10−3) at the CMB. If we assume that topographic amplitudes are small relative to the depth of the core and some characteristic topographic width, an approximate expression for the axial component (i.e. pointing in the direction of Earth’s rotation axis) of the topographic torque is given by

$${\Gamma }_{z}\approx \epsilon \oint h\left(\theta ,\phi \right){\partial }_{\phi }p\,da,$$
(2)

where ∂ϕp is thee longitudinal pressure gradient within the core. If the pressure variations are induced by the topography, then there is an order O(ϵ) perturbation to the pressure gradient and \({\Gamma }_{z}=O\left({\epsilon }^{2}\right)\)15,18,19,20,21. Alternatively, if the dominant contribution to the pressure gradient is not dependent on the topographic amplitude, the magnitude of the axial torque scales like Γz = O(ϵ). Given that ϵ ~ 10−3 at the CMB, these two scalings can yield vastly different estimates for the topographic torque. We ensure that the topographic amplitude is less than the topographic width in all of our simulations so that these two mechanisms can be distinguished, although we emphasize that the exact relation (equation (1)) is used to calculate the torques. Details on the topographic parameters are reported in the Methods (Table 1). We are unaware of any previous studies that have conclusively demonstrated the existence of either scaling in a fully self-consistent convection-driven flow.

We address the problem of core mantle topographic coupling by employing direct numerical simulations of hydrodynamic convection in a rotating spherical shell in which a single topographic bump with Gaussian profile protrudes into the core; a schematic of the domain is shown in Supplementary Fig. 1. This choice of topography is made both for simplicity, and to ensure a rich interaction between core flows and the mantle given that a Gaussian profile is characterized by a broad spectrum of wavelengths. In the absence of topography, the dynamics of the convection is controlled by the non-dimensional Rayleigh, Ekman, and Prandtl numbers, defined by Ra = gαΔTd3/νκ, Ek = ν/(Ωd2), and Pr = ν/κ, respectively. These first two parameters determine the relative influence of the buoeyancy and Coriolis forces in the system, whereas the Prandtl number determines the ratio of the thermal diffusion time to the viscous diffusion time. For simplicity, we set Pr = 1 in all simulations. We refer the reader to the Methods for a complete description of the mathematical system, but note that d is the dimensional depth of the outer core (~2260 km). An important output parameter is the Reynolds number, Re = Ud/ν, where U is a typical flow speed. The outer core is thought to be turbulent (Re ≫ 1) and yet strongly constrained by rotation so that Ek and the Rossby number, Ro = EkRe, are much less than unity. We refer to this regime as rapidly rotating and note that the parameter \(\widetilde{Ra}=E{k}^{4/3}Ra\) controls the convective state22,23. In the limit of rapid rotation, the critical Rayleigh number scales as Ek−4/3; this implies that Ra must be at least this large to generate convection. It then becomes convenient to characterize the convective dynamics in terms \(\widetilde{Ra}\), which is an order one quantity in the limit of vanishing Ek. Due to computational limitations, numerical simulations cannot employ the same non-dimensional parameters of the core. Rather, as is common practice24,25, our goal is to vary these parameters systematically to identify the appropriate asymptotic regime such that our findings can be extrapolated to core conditions. For the topographic bump, we fix the location and width, and vary the amplitude, ϵ. The viscous Ekman layer that is present along both the inner and outer boundaries has a thickness of δE = O(Ek1/2). A conservative estimate for CMB topography is 1 km, which suggests that ϵ/δE ~ 104 in the core. To preserve the relevant dynamics, it is necessary to maintain the ordering of length scales ϵ ≫ δE. In our simulations the depth of the Ekman layers are greater than 10−3 so we employ values of ϵ that exceed this depth. Thus, three non-dimensional control parameters (\(Ek,\widetilde{Ra},\epsilon\)) are varied in this investigation.

In this study, over our investigated range of parameters, we confirm that topographic torques scale linearly with topographic amplitude. The simulation data also suggests a quadratic scaling with flow speeds, which can be explained via analysis of the governing equations. These findings indicate that small-scale convective turbulence in the outer core is capable of generating topographic torques of sufficient magnitude to explain decadal and sub-decadal LOD variations.

Results

Visualizations of the flow field are shown in Fig. 1 for the most extreme rotation rate (Ek = 10−6) at \(\widetilde{Ra}=60\) and ϵ = 0.2. The meridional slice shows the presence of columnar structures that effectively span the entire local depth of the domain. These structures drift with time and interact with the CMB topography. This view, along with equatorial slices, shows the hallmark of turbulent rotating spherical convection: strong mixing in planes perpendicular to the rotation axis, yet approximately invariant parallel to the rotation axis. The dynamical effects of the topography are present in these snapshots of the flow. In Fig. 1b, we find subdued mixing in the column extending axially from the bump, with locally cold fluid indicated by the predominantly blue color. Further evidence that this bump impacts the dynamics of the core is found in the time- and axially-averaged pressure field shown in Fig. 1c; a steady, cyclonic gyre is observed directly above the topography.

Fig. 1: Flow visualizations.
figure 1

Visualizations for the most turbulent simulation: Ek = 10−6, \(\widetilde{Ra}=60\), and ϵ = 0.2. a meridional cross-section of the instantaneous vertical velocity; b equatorial cross-section of the instantaneous temperature field with the conductive profile and zonal-average removed. c Time- and depth-averaged pressure. Blue (brown) corresponds to pressure lows (highs). The circular arrow in (c) indicates the direction of the cyclonic flow over the bump.

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Time series and corresponding periodograms of the topographic torque are shown in Fig. 2 for ϵ = 0.2, \(\widetilde{Ra}=(20,60)\) and all values of Ek. The asymptotic theory of rapidly rotating convection26,27 predicts that the topographic torque should scale with the Ekman number as Ek−1. Details on the origin of this scaling are provided in Methods D. We refer to the quantity EkΓz as the reduced torque, in analogy to \(\widetilde{Ra}\), and plot it in Fig. 2a, b.

Fig. 2: Temporal variation of the reduced torques.
figure 2

(Top row) (a, b) Time series of the reduced torque and (bottom row) (c, d) the corresponding periodograms of the torque for ϵ = 0.2. Left (a, c) \(\widetilde{Ra}=20\) and right (b, d) \(\widetilde{Ra}=60\) for all Ek. Red and green bars in (b) correspond to decadal and centenary timescales, respectively.

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In the rapidly rotating regime, we expect the temporal dynamics to be dominated by the convective Rossby waves. In our non-dimensonal units, these waves have angular frequencies that vary as ω = O(Ek−2/3) in the limit of vanishing Ekman number28. Thus, in order to meaningfully compare the simulations at different rotation rates (Ekman numbers), the data is plotted against the rescaled time and frequency, \(\widetilde{t}=E{k}^{-2/3}t\) and \(\widetilde{\omega }=E{k}^{2/3}\omega\), respectively. The periodograms are an estimate of the spectral density and are calculated with Welch’s method29 over a Hann window of size \(\Delta \widetilde{t}=10\). The data are normalized by the power in the lowest frequency mode. Here we observe that in these rescaled units the simulations are characterized by similar time scales, showing that the asymptotic state relevant for understanding core dynamics is present in the models. Although this timescale is not commonly reported in previous studies of rotating convection and dynamos, we can rescale previously published temporal data into units of \(\widetilde{t}\). For instance, Schaeffer et al.30 studied the dynamics of rapidly rotating dynamos and reported dominant frequencies in the range 10−4−10−3 when non-dimensionalized by the rotation rate; these frequencies are equivalent to \(\widetilde{t}=O(1)-O(10)\) and are therefore similar to the timescales in our simulations. Estimates for the convective overturn time based on the geomagnetic secular variation also suggest timescales of \(\widetilde{t}=O\left(1\right)\)31. As \(\widetilde{Ra}\) is increased the amplitude of the fluctuations increase, as does the temporal complexity. However, the periodograms do not differ substantially between different \(\widetilde{Ra}\), suggesting the increased complexity at higher \(\widetilde{Ra}\) is restricted to smaller amplitude fluctuations.

Under Earth-like conditions, a decadal timescale (10 years) corresponds to \(\Delta \widetilde{t}\approx 0.2\). The red shaded region in Fig. 2c corresponds to \(\Delta \widetilde{t}=0.2\) while the green shaded region corresponds to \(\Delta \widetilde{t}=2.0\) (100 yrs.). Although our current data suggest that fluctuations take place on a timescale closer to 100 years, the simulations are far less turbulent than the expected conditions of the core; accepted values of flow speeds for the core are at least five orders of magnitude greater than the simulated values. As the forcing is increased and the flow becomes more turbulent, the advective timescales are known to decrease32, and this behavior is also observed in the simulations. Therefore, we believe that the timescales of the fluctuations make topographic torques good candidates for both the  ~6 and  ~60-year LOD fluctuations.

The standard deviation of the axial topographic torque (\({\Gamma }_{z}^{{\prime} }\)) is computed for all simulations and plotted in Fig. 3a. Dashed (solid) lines with open (solid) symbols correspond to ϵ = 0.1 (ϵ = 0.2). For a fixed topographic amplitude we observe an increase in \({\Gamma }_{z}^{{\prime} }\) as Ek is reduced at fixed \(\widetilde{Ra}\), showing that CMB topography has a stronger influence on super-critical core flows as the rotation rate increases. We emphasize that the abscissae are the reduced quantity \(\widetilde{Ra}\). In dimensional terms, this means that as the Coriolis force is increased, so too is the buoyancy force, so that the flow remains super-critical. We also find that \({\Gamma }_{z}^{{\prime} }\) increases as \(\widetilde{Ra}\) is increased, demonstrating that larger pressure gradients develop within the flow as the buoyancy force is increased.

Fig. 3: Axial topographic torque versus reduced Rayleigh number.
figure 3

a Raw (unscaled) fluctuating axial torque and b torque normalized by the predicted asymptotic scaling, ϵ/Ek. The reduced Rayleigh number is defined by \(\widetilde{Ra}=RaE{k}^{4/3}\).

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A plot of the reduced torque is shown in the Supplementary Fig. 2a. The reduced torque is of order unity across the entire range of parameters investigated and is separated into two distinct branches corresponding to ϵ = 0.1 and ϵ = 0.2. To determine how the torque depends on topographic amplitude, we rescale the reduced torque by ϵ as shown in Fig. 3b. Good collapse of the data is observed across the range of \(\widetilde{Ra}\) investigated. Rescaling the torques by Ek/ϵ2, as shown in Supplementary Fig. 2b, yields values that are no longer of order unity. Furthermore, there is notable separation in the data corresponding to different amplitudes which is strongly suggestive that a scaling of \({\Gamma }_{z}^{{\prime} }=O({\epsilon }^{2})\) is not represented by our data.

Further testing of the amplitude dependence is carried out by varying ϵ for Ekman numbers Ek = [10−5, 3 × 10−6] and fixed \(\widetilde{Ra}=40\). Figure 4 shows the computed reduced torque for these cases. A least squares power law fit to the data yielded \({\Gamma }_{z}^{{\prime} }Ek=7.7{\epsilon }^{1.07}\), and is shown by the dashed line. Slopes of ϵ and ϵ2 are shown for reference. These results provide further evidence that the topographic torque scales linearly with topographic amplitude, as would be expected if the primary component of the pressure that contributes to the torque is that associated with the convection (i.e., the pressure variation is, to leading order, not dependent on ϵ).

Fig. 4: Reduced topographic torque versus topographic amplitude.
figure 4

\(\widetilde{Ra}=40\) and Ek = (10−5, 3 × 10−6). The abscissa is ϵ, the non-dimensional topographic amplitude. A least squares fit of the data gives a scaling of \({\Gamma }_{z}^{{\prime} }Ek=7.7{\epsilon }^{1.07}\) shown with the dashed line. Linear and quadratic scalings in ϵ are shown for reference.

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To extrapolate our findings to the conditions of the core, we require the functional dependence \({\Gamma }_{z}^{{\prime} }=f(\widetilde{Ra})\), or, alternatively, \({\Gamma }_{z}^{{\prime} }=g(Re)\), for some functions f and g. The asymptotic theory presented in Methods D predicts that for Ek → 0, and sufficiently large \(\widetilde{Ra}\), the torque should scale as

$${\Gamma }_{z}^{{\prime} }=O\left(\epsilon R{e}^{2}E{k}^{-1/3}\right).$$
(3)

This relationship arises from an approximate balance between the ageostrophic component of the pressure gradient force and advective nonlinearities in the prognostic evolution equation. To test this scaling, we plot the quantity \({\Gamma }_{z}^{{\prime} }E{k}^{1/3}/\epsilon\) as a function of Re in Fig. 5a. With the exception of the simulations with small values of \(\widetilde{Ra}\) in which the flow is very close to the onset of convection and is therefore dominated by a few oscillatory modes, the Ek−1/3ϵ scaling collapses the data well. Only our most extreme simulations are approaching the quadratic flow speed scaling (Re2), as indicated in Fig. 5b, which shows the fully compensated torque, \({\Gamma }_{z}^{{\prime} }E{k}^{1/3}/(\epsilon R{e}^{2})\). Ultimately, we want to be able to predict torques using only the input parameters, rather than an output parameter such as Re. Previous work33,34,35 has found that flow speeds scale approximately as \(Re \sim RaEkP{r}^{-1}=\widetilde{Ra}P{r}^{-1}E{k}^{-1/3}\) when \(\widetilde{Ra}\) becomes large. Omitting the Prandtl number dependence, this leads to \({\Gamma }_{z}^{{\prime} } \sim \epsilon {\widetilde{Ra}}^{2}E{k}^{-1}\). Supplementary Fig. 3 shows the reduced torque rescaled by \(\epsilon {\widetilde{Ra}}^{2}\) and a reasonable collapse of the data is observed, indicating this dependence may be robust at large Rayleigh numbers.

Fig. 5: Dependence of topographic torque on flow speed.
figure 5

a Topographic torque scaled by Ek1/3/ϵ as a function of flow speed (Re); slopes of Re and Re2 are shown for reference. b Compensated topographic torque versus Re. Symbols are as defined in Fig. 3.

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Discussion

The significance that topographic torques play in explaining the large amplitude fluctuations in the LOD has been debated for more than 50 years36. This debate, which is centered on two competing hypotheses, has focused on how the torques depend on the typical amplitude of the topography. It is known that a non-axisymmetric pressure gradient is required to generate non-trivial axial torques on the mantle. The first hypothesis assumes that the flow in the core is predominantly axisymmetric and the topography induces the necessary non-axisymmetric pressure gradients18. Models using this hypothesis yield topographic torques that scale quadratically with the topographic amplitude. The second hypothesis considers the convection to be the dominant source of asymmetry, such that the pressure gradients do not depend on topographic amplitude to leading order, which leads to a linear scaling. Our investigation has shown that this second mechanism dominates when convection is considered in the problem—convection-driven waves with angular frequency ω = O(Ek−2/3), which are inherently non-axisymmetric, yield pressure gradients that interact with the topography to produce a torque that is linearly proportional to the topographic amplitude.

The results also show that the topographic torque scales nearly quadratically with the flow speeds at the most extreme parameter values accessed in the simulations. This scaling is nearly consistent with the classical picture of ‘form drag’ in which a flow traveling with speed U is diverted by an obstacle. In turbulent flows, the pressure near a boundary is expected to scale quadratically with flow speed in order to allow for complete stagnation of the incident flow37. However, in this study, rotation appears to play an additional role via the factor of Ek−1/3. This Ekman number dependence suggests that torque is dependent not only on flow speeds but also on the spatial structure of the underlying convection, which is controlled by the importance of rotation through the Ekman number. We stress that the present study was performed at a fixed topographic width, and we expect different topographic shapes to influence the torque (see equation (2)).

We present a scaling argument based on quasi-geostrophic theory27,38, which recovers the empirical scaling for the topographic torque given by \({\Gamma }_{z}^{{\prime} }=O(\epsilon R{e}^{2}E{k}^{-1/3})\). It is notable that this scaling differs from the Re/Ek scaling originally predicted by Hide36. Hide used the geostrophic scaling (equation (12)) to predict the magnitude of Γz, although he noted that there would be partial cancellation in the surface integral. However, Anufriev and Braginsky18 pointed out that there is total cancellation if only the geotrophic flow is considered. An important consideration is that this argument holds for all geostrophic structures. Therefore, topographic torques arise only from scales where a mechanism to disrupt geostrophy, such as viscosity or magnetism, is present. Our interpretation and data support this conclusion, implying that perturbations to the geostrophic pressure need to be considered.

We report our scaling for \({\Gamma }_{z}^{{\prime} }\) in terms of Re and Ek rather than the control parameters Ra, Ek, and Pr because the data and asymptotic theory are more directly supportive of the former. Regardless, our derived torque scaling should be understood to depend on all of the control parameters. We note that the flow speed, and therefore the Reynolds number, is not independent of Ek. For instance, it is known that when the system is strongly forced by fixed temperature boundary conditions, the Reynolds number scales approximately like Re = O(EkRa/Pr) and the singularity as Ek → 0 is avoided34.

A dimensional torque of ~1018 N m is required to explain observed LOD variations7. In dimensional form, our scaling becomes

$${\Gamma }_{z, \, \dim }^{{\prime} } \sim \epsilon E{k}^{-1/3}\rho {U}^{2}{d}^{3},$$
(4)

where U is a characteristic flow speed and fluid density is denoted by ρ. Using estimates for the outer core, we have ρ ~104 kg m−3, ϵ ~ 10−3 (corresponding to an amplitude of 2.26 km), U = (1−5) × 10−4 m s−1, d = 2260 km, and Ek = 10−15. These values give a dimensional torque of (1.2 × 1017−2.9 × 1018) N m, which is consistent with the required torques.

Equation (3) and the theory provided in Methods D and E require a choice for the scaling of the horizontal length scale ℓ. In this work, we have collapsed the numerical data using the asymptotic scale ℓ = O(Ek1/3), which is well known from linear theory, but also holds in the strongly turbulent regime27. Some studies suggest an inertial scaling, characterized by \(\ell =O\left(R{o}^{1/2}\right)\), holds for the most energetic scale in the system39,40. The most extreme simulations of rotating convective turbulence carried out to date do not show clear evidence of the inertial scaling34, although it has been shown that these two theories are likely indistinguishable in the currently accessible parameter regime41. This is likely attributable to the fact that both scalings can be shown to contain an Ek1/3 factor33,35,42, and since the effect of particular flow scales on topographic torques is not known, we prefer the use of the Ek1/3 scale, which captures all convective length scales.

Several additional physical effects will be investigated in future studies to develop a better understanding of topographic torques. For instance, magnetism and global-scale topography likely also play a role in controlling LOD. However, because the primary flow quantities, such as flow speeds, show similar scaling behavior in simulations of both dynamos and convection25,35,43, the scaling of the topographic torques is unlikely to change fundamentally when magnetism is introduced. We expect the CMB to be characterized by a broad range of topographic wavelengths13,14, and the present work focused on a topographic bump with a wavelength that is small in comparison with the depth of the outer core. Thus, systematically varying the topographic wavelength is necessary to elucidate additional interactions between core flows and the CMB. Earth’s core is liquid metal and is therefore characterized by a thermal Prandtl number, Pr, that is less than unity. Our study is restricted to Pr = 1, although it is known that convective dynamics are strongly influenced by the Prandtl number44,45,46. Zonal flows become more prominent as Pr is reduced below unity47—understanding how this change influences topographic torques is of considerable interest for future investigations. For instance, it is known that when Pr < 1 large amplitude standing Rossby waves can be excited by topography, though their influence on the topographic torque is still uknown48. The possibility of stably stratified layers regionalized near the LLSVP’s has been proposed by previous work, for instance, see Mound et al.49. The existence of such regions would likely have a direct impact on the scaling of topographic torques.

Methods

Governing equations

For simplicity, the present work focuses solely on the hydrodynamic problem of convection of a Boussinesq fluid contained within a rotating spherical shell with inner radius ricb and outer radius rcmb. In the absence of topography, the aspect ratio of the shell is fixed at χ = ricb/rcmb = 0.35. As discussed below, finite amplitude topography is superposed onto the outer shell. The physical properties of the fluid are the kinematic viscosity ν, thermal diffusivity κ and thermal expansion coefficient α. The inner and outer boundaries are held at fixed temperatures Ticb and Tcmb, respectively. Ω is the rotation rate. We assume a uniform distribution of mass within the shell such that the gravitational acceleration varies linearly with radius and has magnitude g at the outer boundary. The governing equations reflect the conservation of momentum, energy, and mass and are non-dimensionalized by the shell gap-width d = 2260 km, viscous diffusion time d2/ν, and temperature difference ΔT = Ticb − Tcmb. The equations then become

$$\frac{\partial {{\bf{u}}}}{\partial t}+{{\bf{u}}}\cdot \nabla {{\bf{u}}}=-\nabla p+{\nabla }^{2}{{\bf{u}}}-\frac{2}{Ek}\hat{{{\bf{z}}}}\times {{\bf{u}}}+\frac{Ra}{Pr}\frac{{{\bf{r}}}}{{r}_{cmb}}\vartheta$$
(5a)
$$\frac{\partial \vartheta }{\partial t}+{{\bf{u}}}\cdot \nabla \vartheta =\frac{1}{Pr}{\nabla }^{2}\vartheta$$
(5b)
$$\nabla \cdot {{\bf{u}}}=0.$$
(5c)

The fluid variables are the velocity u, pressure p and temperature ϑ. The non-dimensional control parameters are the Rayleigh number, Ekman number, and the thermal Prandtl number, defined by, respectively,

$$Ra=\frac{g\alpha \Delta T{d}^{3}}{\nu \kappa },\qquad Ek=\frac{\nu }{\Omega {d}^{2}},\qquad Pr=\frac{\nu }{\kappa }.$$
(6)

The Prandtl number is fixed to unity. No-slip, impenetrable and isothermal boundary conditions are employed on the inner and outer surfaces. The Reynolds number is a diagnostic parameter that characterizes the flow speed. We define it as

$$Re=\overline{\sqrt{ \langle {{\bf{u}}}\cdot {{\bf{u}}} \rangle }},$$
(7)

where the overline denotes a time average and \(\langle \cdot \rangle\) indicates a volume average.

Model setup and numerical details

Given that the CMB is poorly constrained by seismology, we are motivated to use as simple a model as possible for its shape, whilst retaining the ability to easily control essential physical characteristics such as amplitude and width. In this regard we superpose a Gaussian bump on the outer boundary, as shown in Supplementary Fig. 1. Rotating convection in a spherical shell is known to be anisotropic and inhomogeneous, which suggests the location of the topography may also play a key role, at least insofar as the dynamical consequences of the topography on the convective flows are concerned. For the present investigation, we place the bump in the lower hemisphere at a latitude that is intended to ensure a robust interaction with time-dependent convective Rossby waves and the geostrophic, zonal flows that develop in a rotating, spherical geometry. We fix the coordinates of the center of the base of the bump to \(\left({\theta }_{0},{\phi }_{0}\right)\) such that it lies directly beneath the halfway point between inner and outer boundaries, as shown in Supplementary Fig. 1b. The spherical coordinates are thus \(\left({\theta }_{0},{\phi }_{0}\right)=\left(2.40,0\right)\). The topographic shape function is given by

$$h(\theta ,\phi )=\exp \left\{-\frac{1}{\beta }\left[\frac{{\left({{\bf{r}}}-{{{\bf{r}}}}_{0}\right)}^{2}}{{r}_{cmb}^{2}}\right]\right\},$$
(8)

where r0 is the vector pointing from the origin to \(\left({r}_{cmb},{\theta }_{0},{\phi }_{0}\right)\). The topographic amplitude is ϵ and β denotes a measure of the width (lateral extent) of the topography. We fix the width in this study such that β−1 = 40. As mentioned in previous work48, the Gaussian profile is particularly useful since its representation in spectral space requires a broad range of spherical harmonics, and therefore lends the possibility of richer dynamical interactions with the convective flow field. Table 1 lists topographic parameters, including the aspect ratio for the topography, ϵ/w, where w is defined as the full width at 1% maximum. Note that only the topographic amplitude is varied in the present study.

Table 1 Topographic parameters
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The simulations are performed with the computational fluid dynamics code Nek500050, which uses the spectral element method51 and therefore allows for solving the governing equations in domains with irregular boundaries. Nek5000 has been extensively benchmarked and scales efficiently to large numbers of processors. The simulations were run on up to 2048 processors for our most demanding simulations. Nek5000 uses a CFL condition to dynamically update the timestep. Our tests indicate that for identical non-dimensional control parameters the Nek5000 code is approximately a factor of ten slower in comparison to the commonly used spectral code, Rayleigh52, although spectral codes are generally incapabale of handling irregular boundaries. All simulations used a degree 7 polynomial interpolant within the spectral elements. The most demanding simulations required 331,776 spectral elements, corresponding to almost 1.7 × 108 physical grid points. The code was benchmarked against the hydrodynamic case 0 presented in Christensen et al.53 for a spherical geometry, and all values were found to be within the recommended ranges. To our knowledge, no benchmark of convection in a deformed sphere exists; however, extensive tests were performed to ensure that the deformed boundaries were being correctly implemented.

The use of no-slip mechanical boundary conditions implies that viscous torques act along both the inner and outer surfaces. Viscous torques were computed both as a means to check consistency in the numerical method and to compare with the topographic torque. The total viscous torque acting on both the inner boundary (icb) and the outer boundary (cmb) is calculated as

$${\Gamma }_{z}^{v}=-\oint _{icb,cmb}{{\bf{r}}}\times \left[\left(\nabla {{\bf{u}}}+{\left(\nabla {{\bf{u}}}\right)}^{T}\right)\cdot d{{\bf{a}}}\right],$$
(9)

where the superscript T denotes a transpose. Supplementary Fig. 4a shows a representative time series of the pressure torque on the outer boundary and the sum of the viscous torques on the inner and outer boundary, as well as the rate of change of the axial angular momentum of the fluid dLz/dt. We find excellent agreement between  −dLz/dt and \({\Gamma }_{z}^{v}+{\Gamma }_{z}\), providing a verification of the calculation of the torques.

The ratio of the standard deviatieons of the viscous and topographic torque on the outer boundary, denoted by γ, is shown in Supplementary Fig. 4b. This ratio is near unity near the onset of convection, but is generally a decreasing function of \(\widetilde{Ra}\). These trends indicate that the viscous torques are likely negligible in the parameter regime of the core.

Choice of parameter values

Core turbulence is thought to be strongly constrained by the effects of the Coriolis force. Thus, our present goal is to model convection in this rotationally constrained regime while simultaneously allowing for as broad a range of spatiotemporal scales as is numerically feasible. Moreover, as is often done in studies of core dynamics, it is important to understand how the dynamics of the system change as the parameters are made more extreme. We therefore cover a broad range in Rayleigh number and Ekman number in an effort to elucidate the scaling behavior of the dynamics. Our simulations use Ekman numbers of Ek = [10−6, 3 × 10−6, 10−5, 3 × 10−5, 10−4] and reduced Rayleigh numbers of \(\widetilde{Ra}=[3,10,20,40,60]\).

Concerning the choice of topographic parameters, the present work is focused predominantly on how the topographic torque and associated convective dynamics depend on the amplitude, ϵ. Although seismic studies and mantle convection simulations indicate that CMB topography amplitudes are unlikely to exceed ϵ = O(10−3), it makes little sense to use amplitudes this small, given that the investigated range of Ekman numbers in our simulations is so distant from the Earth’s core. The topographic amplitude at the CMB is thought to be much larger than the thickness of the Ekman boundary layer that is present. We therefore choose topographic amplitudes that maintain this ordering of length scales in the system, i.e., ϵ/δE ≫ 1, such that the topographic bump does not reside within the Ekman layer. The majority of the simulations were performed with ϵ = 0.2, 0.1 in which this ordering is well satisfied, and more limited sweeps are carried out for ϵ = [0.005, 0.01, 0.02, 0.03, 0.05, 0.1, 0.2, 0.3]. 

Asymptotic scalings for topographic torques

Insight into the dependence of the topographic torque on both the input parameter, Ek, and the output parameter, Re, can be gained by appealing to the asymptotic theory of rotating convection. Our goal is to provide only a brief summary that allows for understanding the results of the present study—further details are provided in previous work27,35,38. For a fixed value of the Ekman number, the flow is characterized by a geostrophic force balance between the Coriolis and pressure gradient forces, provided that \(\widetilde{Ra}\) is not too large33. The convective dynamics are then the result of perturbations, or small corrections, to this balance. We denote the relevant small parameter by δ ≡ Ek1/3, which characterizes the ratio of gradients in the axial (measured along the rotation axis) and horizontal (measured perpendicular to the rotation axis) directions. Thus, the largest components of the gradient operator scale as \(\nabla =O\left({\delta }^{-1}\right)\) and the velocity field scales as \({{\bf{u}}}=O\left({\delta }^{-1}\right)\). We define rescaled quantities as \(\widetilde{\nabla }\equiv \delta \,\nabla =O(1)\) and \(\widetilde{{{\bf{u}}}}\equiv \delta \,{{\bf{u}}}=O(1)\). A self-consistent set of perturbation equations can then be derived that allows for the possible importance of all terms (buoyancy, inertia, advection, and the viscous force) in the prognostic evolution equation35,38. The relative sizes of these perturbative terms are then determined by the strength of the buoyancy forcing, or \(\widetilde{Ra}\). The requirement of leading order geostrophy implies that the pressure scales as

$$p=O({\delta }^{-3}).$$
(10)

However, this scaling provides the asymptotic size of only the leading order, geostrophic component of the pressure. As alluded to in previous work18,54, the topographic torque is identically zero when the flow is completely geostrophic because it is constrained to follow lines of constant axial height, known as geostrophic contours55. As we show here, it is then necessary to explicitly account for higher-order corrections to the pressure. More specifically, the pressure is expanded in a perturbation series according to

$$p={\delta }^{-3}\left[{\widetilde{p}}_{0}+\delta \,{\widetilde{p}}_{1}+O\left({\delta }^{2}\right)\right],$$
(11)

This expansion, along with an analogous expansion for the velocity field, can then be inserted into the momentum equation and the leading order geostrophic balance arises as

$$2\hat{{{\bf{z}}}}\times {\widetilde{{{\bf{u}}}}}_{0}\approx -\widetilde{\nabla }{\widetilde{p}}_{0},$$
(12)

where p0 is now explicitly identified as the geostrophic component of the pressure. For simplicity, we focus on the approximate form of the torque, equation (2), which can be decomposed according to

$${\Gamma }_{z}=\epsilon \oint h\,{\partial }_{\phi }p\,da=\epsilon {\delta }^{-4}{I}_{0}+\epsilon {\delta }^{-3}{I}_{1}+O\left({\delta }^{-2}\right),$$
(13)

where the two integrals are defined by

$${I}_{0}=\oint h\,{\widetilde{\partial }}_{\phi }{\widetilde{p}}_{0}\,da,\quad \,{{\mbox{and}}}\,\quad {I}_{1}=\oint h\,{\widetilde{\partial }}_{\phi }{\widetilde{p}}_{1}\,da.$$
(14)

The first integral is identically zero since geostrophic flow must follow geostrophic contours, implying linear dependence between h and the geostrophic pressure. Therefore, the asymptotic analysis tells us that the torque integral is of order

$${\Gamma }_{z}=\epsilon {\delta }^{-3}{I}_{1}=O\left(\epsilon E{k}^{-1}\right).$$
(15)

This suggests that the reduced torque, \({\Gamma }_{z}^{{\prime} }Ek\), is an order one quantity in the limit Ek → 0.

If we now consider the effect of flow speed, a second scaling involving the Reynolds number, Re, can be obtained. Since the primary contribution to the torque arises from perturbations to the geostrophic pressure, we suppose that \(\widetilde{\nabla }{\widetilde{p}}_{1}\) must be comparable to other terms appearing in the prognostic equation governing the QG dynamics. For sufficiently large values of \(\widetilde{Ra}\) we assume that nonlinearities become as large as pressure gradients. In the QG theory, \({\widetilde{{{\bf{u}}}}}_{0}=O(Re\delta ),\) so that

$$\widetilde{\nabla }{\widetilde{p}}_{1} \sim {\widetilde{{{\bf{u}}}}}_{0}\cdot \widetilde{\nabla }{\widetilde{{{\bf{u}}}}}_{0}=O\left({\delta }^{2}R{e}^{2}\right),\qquad \Rightarrow \qquad {\widetilde{\partial }}_{\phi }{\widetilde{p}}_{1}=O\left(R{e}^{2}E{k}^{2/3}\right).$$
(16)

Using the above scaling in the integral I1 then suggests that the torque scales as

$${\Gamma }_{z}=O\left(\epsilon R{e}^{2}E{k}^{-1/3}\right).$$
(17)

Equations (15) and (17) can be reconciled by noting that Re = O(Ek−1/3) in the asymptotic theory.

Relating \(\widetilde{t}\) to the length of day

The rescaled time \(\widetilde{t}\) can be related to the length of the day as follows. Denoting dimensional time (in seconds) as τ, we have

$$\widetilde{t}=E{k}^{-2/3}\left(\frac{\nu }{{d}^{2}}\right)\tau =E{k}^{1/3}\left(\frac{\nu }{{d}^{2}}\right)\left(\frac{\Omega {d}^{2}}{\nu }\right)\tau =E{k}^{1/3}\Omega \tau .$$
(18)

We note that \(\widetilde{t}\) should be considered as a small-scale diffusion time. From the linear stability analysis ℓ/d = Ek1/3 where ℓ is the length scale associated with the convection at onset. Even in the turbulent regime, the flow structures, such as the thickness of the Taylor columns, may remain \(O\left(E{k}^{1/3}\right)\)34,35, and convection experiments41 suggest that we are in a regime where the proposed Ek1/3 and Ro1/2 scalings39,56 may be indistinguishable. Therefore,

$$\widetilde{t}=\left(\frac{\nu }{{\ell }^{2}}\right)\tau .$$
(19)

Plugging in Ek = 10−15 and Ω = 7.27 × 10−5 s−1 into equation (18) gives

$$\widetilde{t}=(7.27\times 1{0}^{-10}{{\mbox{s}}}^{-1})\cdot \tau .$$
(20)

A decadal timescale corresponds to Δτ ≈ 3 × 108 s such that

$$\Delta \widetilde{t}\approx 0.2.$$
(21)