On symmetry and anisotropy of Earth‐core flows

Abstract

[1] Quasi‐geostrophic (QG) flows are a recently developed and very promising paradigm for modeling decadal secular variation (SV). Here we examine the effects of allowing anisotropy and departures of the flow from quasigeostrophy. We perform dedicated numerical experiments of the flow dynamics and magnetic induction inside the Earth's liquid core at time scales characteristic of secular variation of the geomagnetic field. Obtained results motivate new flow inversion regularization featuring an equatorially anti‐symmetric component superimposed to quasi‐geostrophic columns, and stronger latitudinal than longitudinal flow gradients. Applying these constraints allows to explain the observed SV for the whole period 1840–2010, and most significantly, provides a clearly improvement in prediction for decadal length‐of‐day variations for the period 1980–2000. Furthermore, the trace of the inner‐core appears clearly without any assumption for the 1997–2010 period covered by satellite geomagnetic data. Our results support QG being the appropriate description of the force balance within the core on decadal time scales and large spatial scales.

1. Introduction

[2] The secular variation of the magnetic field of the Earth is due to the flow of liquid metal advecting the magnetic field inside the Earth's core. A large number of studies have focused on inference of the core surface flow from geomagnetic field data [Holme, 2007]. Recently, it has been advocated that Quasi‐Geostrophic (QG) flows should give a good description of the flow in the core [Jault, 2008], and kinematic properties of this type of flow have already been used to constrain the core surface flow models inferred from magnetic field data to be symmetric with respect to the equator and purely azimuthal at the rim of the tangent cylinder [Pais and Jault, 2008; Gillet et al., 2009].

[3] Still, some unclear points persist, and our study tries to shed some light on three of them: (i) The QG hypothesis has been justified for asymptotically small slopes [e.g., Jones, 2007], and the question arises as to its validity in the vicinity of the equator of both inner core and core‐mantle boundary. For example, a flow crossing the equator line, like under the Indian ocean and Brazil in models where the tangential geostrophy of flows was not imposed [Olsen and Mandea, 2008], could not be captured by a QG model, which imposes a purely azimuthal flow at the equator. (ii) Even though the global amount of equatorial symmetry of core surface flow models has been in general increasing with time during the period 1840–2010 [Gillet et al., 2011], no analysis has been made to identify which are the spatial features that can be associated to QG dynamics. (iii) Recent efforts have been made to also take into account the uncertainties due to the advection of an unknown small scale geomagnetic field by an unresolvable small scale flow [Eymin and Hulot, 2005; Pais and Jault, 2008]. However, these small scales are poorly constrained [Gillet et al., 2009] and no useful information has been obtained from them. Underparameterization of core flow models due to regularizations penalizing too strongly small length scales may be the cause of the overestimation of decade length of day variations (ΔLOD) seen in most studies. We investigate the possibility that this may be due to aliasing of small scales (which do not carry angular momentum in the model we use [Jault et al., 1988]) to large scales, by comparing estimations using different regularizations.

[4] The paper is organized as follows. Section 2 presents results of direct numerical simulations which serve as motivation to propose new core surface constraints in section 3. Specific features of our flow inversion are also stated there. The resulting core surface flow models are examined in section 4 and compared with previously published models by testing their ability in estimating ΔLOD. The mean flow model computed from CHAOS‐3 is examined. A summary of our findings is presented in the Conclusion.

2. Direct Numerical Simulations

[5] In order to gain insight into the kinematics of the flow inside the Earth core, we run full three dimensional numerical simulations in a spherical shell of radius a, rotating at period T = 2π/Ω, and with a solid inner core of radius 0.35a. The conducting fluid of density ρ, viscosity ν and magnetic diffusivity η = ν is permeated by a constant, toroidal and axisymmetric magnetic field B₀ = B₀∇ × (4r(a − r)r/a³), with B₀ such that the Lehnert number λ = B₀(aΩ)⁻¹ (the ratio of the Alfvén wave to the inertial wave speeds) is set to 10⁻³, close to its value inside the Earth's core. A small λ is important to resolve propagating inertial and Alfvén waves, both having a crucial role on SV dynamics [Jault, 2008]. We solve the Navier‐Stokes equation (including the Coriolis and Lorentz force) together with the induction equation. The Ekman number E = ν(a²Ω)⁻¹ is set to 10⁻⁶ and we use stress‐free boundaries. The mantle and the inner‐core are insulators in these calculations, although the boundary condition has no visible impact on the geometry of the flow. Details about the code are given by Gillet et al. [2011].

[6] The simulation is performed in two steps. First, from a state of rest, we impose a localized bulk force field F^±(r) = F₀∇ × [r exp(−∣r − r₊∣² / δ²) ± r exp(−∣r − r₋∣² / δ²)], where r_± = (0.675a, π/4, ±a/2) in cylindrical coordinates (s, ϕ, z) with origin O in the center of the sphere, and Oz parallel to the global rotation axis. The forcing length‐scale is set to δ = 0.045a, while its amplitude F₀ is chosen so that the resulting flow v is of low amplitude: v ≪ aΩ. After a duration Δt = 6T allowing for the formation of Taylor columns through the propagation of inertial waves, we stop the forcing and let the flow freely decay during the second step of the simulation.

[7] Figure 1 shows the flow after 110 periods of free decay, in the two forcing cases: either symmetric (F⁺) or anti‐symmetric (F⁻) with respect to the equator. We observe that these flows have undistinguishable decay rates. As expected for low values of λ, the symmetric forcing leads to a flow v⁺ that is nearly QG: v_s⁺, v_ϕ⁺ are independent of z, satisfying the Taylor‐Proudman (TP) constraint ∂_zv = 0, while v_z⁺ ∝ z ensures the impermeability of the boundaries. The anti‐symmetric flow v⁻ is certainly less known: v_z⁻ satisfies the TP constraint but induces v_s⁻ and v_ϕ⁻ that do not, in order to ensure the impenetrability of the boundaries. Furthermore, it appears that v⁺ is dominated by v_ϕ⁺ and a tendency to zonation, while v⁻ has higher values at large s or low latitude, where it is dominated by v_z⁻ (or v_θ⁻). The increasingly large boundary slope when s → 1 reduces the amount of v_s⁻ produced by v_z⁻ at the boundaries. Finally, in spite of a localized forcing, both v⁺ and v⁻ tend to spread over the whole volume outside the cylinder tangent to the inner‐core equator (TC). We obtain similar results when replacing the toroidal forcing F^± with a poloidal one, except for v_z⁻ which is weaker and has a more complex dependence on z.

Figure 1

Flows resulting from a (top) symmetric or (bottom) anti‐symmetric forcing followed by 110T of free decay. From left to right: the cylindrical components (v_s, v_ϕ, v_z) in the meridional plane containing r_±, and the spherical components (v_θ, v_ϕ) at the core‐surface (note on the right the use of a 4‐fold symmetry in the simulation). The circles indicate the location of the applied forcing in the meridian plots, and its projection along the z‐axis on the surface plots. The surface projection of the equator of the inner‐core is also drawn. Velocities are expressed in units of aΩ.

3. Proposed Constraints for Inverting the Core Flow and Methodology

[8] We propose to translate the results obtained for v in the previous section to surface core flows u which can be computed from geomagnetic field models. The v⁺ QG flow (also referred to as a columnar flow) leads to the following relation at the core surface [Amit and Olson, 2004]:

The axial component of the v⁻ flow comprises, besides a z‐invariant term, other anti‐symmetric contributions that cannot be separated at the core surface. This precludes the derivation of a kinematic constraint for u⁻. We will thus invert for flows that, besides a symmetric component u⁺ satisfying (1), will also include an antisymmetric component u⁻ which has no particular kinematical constraint imposed.

[9] In the Earth core, due to the very small viscosity and high electrical conductivity, we do not expect any significant damping effect on the flow at the scales that can be probed by magnetic field models. However, observations and numerical results for different natural rotating flow systems, with or without magnetic fields, show a tendency for the flow to develop preferably along parallels, in the form of thin zonal jets. This leads to anisotropic structures elongated in the azimuthal direction but showing small scales in the radial direction. The most emblematic case is the banded atmosphere of Jupiter, but there is also evidence for alternating jets in the Ocean [Maximenko et al., 2005]. Numerical simulations of thermal convection also exhibit this kind of flow [Christensen, 2002] and recent geodynamo simulations by Miyagoshi et al. [2010] show the formation of low‐latitude zonal flows at low Ekman numbers. Inspired by these results, we propose a penalization of the azimuthal gradients, by minimizing the integral over the core‐mantle surface (CMB):

This anisotropic norm can be shown to depend both on spherical harmonic degree ℓ and order m as ℓm². Because it does not restrain zonal flows (and hence torsional oscillations), we superimpose an isotropic penalization of the mean squared velocities over the core surface (∫_CMBu²dS, referred to as ℓ¹ norm).

[10] To cover a time interval as large as possible and treat different geomagnetic field models, we invert the gufm1 [Jackson et al., 2000], the comprehensive CM4 [Sabaka et al., 2004] and the satellite derived CHAOS‐3 [Olsen et al., 2010] geomagnetic field models up to degree ℓ = 13. We obtain snapshots of core surface fluid flow up to ℓ = 26 that can, by advecting the field, explain the secular variation models (frozen‐flux hypothesis). The adopted methodology is described by Holme [2007]. We look for a regularized weighted least squares flow solution that explains snapshots of the geomagnetic field model up to some degree of confidence, which comprises both the information on the noise level of the SV data (σ^d(ℓ)) and an estimation of the SV signal produced by a non‐parameterized contribution of the magnetic field small scales (representativity error, σ^r(ℓ)) [Gillet et al., 2009]. Assuming these two kinds of errors are uncorrelated, we then construct a diagonal predictive data covariance matrix with elements σ^d(ℓ)² + σ^r(ℓ)². The term representing the confidence on the SV model, σ^d(ℓ)², is given by η(2ℓ + 1)⁻¹ (ℓ + 1)⁻¹, where we use as noise level η the value 0.4 (nT/yr)² for gufm1 and CM4 models (assuming that SV degrees ℓ < 10 contain relevant information in both these models [Gillet et al., 2010]), and 0.01 (nT/yr)² for the satellite derived model CHAOS‐3 [Olsen et al., 2010]. The representativity errors are assumed to be independent of time and we use the law σ^r(ℓ)² = 36 exp(−ℓ), close to those derived by Pais and Jault [2008] and Gillet et al. [2009].

[11] The inversion is stabilized using regularization presented above. Condition (1) is converted into a quadratic norm that is introduced in the global objective functional to be minimized by the flow solution. This condition constrains only but u⁺ using a very high Lagrange multiplier. We test the relevance of u⁻ by also inverting for purely symmetric flows that obey condition (1). Norm R_A (equation (2)) is used to introduce anisotropy, in conjunction with the isotropic ℓ¹ norm. To compare with more standard procedures where stronger regularizations are used, we also invert for flows using only an ℓ³ norm, which penalizes the mean squared velocity gradients over the CMB [Gillet et al., 2009].

[12] In order to test the sensitivity of geomagnetic field models to the boundary of the TC, we use the same regularizations over the whole core surface.

4. Inverted Flow Models

[13] The new regularizing norms are tested for a possible improvement of estimates of ΔLOD. We concentrate on the 1960–2002 epoch covered by the CM4 model, for which the imposed anisotropy is responsible for the most clear improvement. It is known that standard flow predictions do recurrently introduce a stronger decaying trend of ΔLOD than observed during this period [Gillet et al., 2009]. We test the hypothesis that this poor estimation may be due to aliasing caused by underparameterization of small scales. This would deteriorate estimation of large scale coefficients, in particular t₁⁰ and t₃⁰ (which carry the core angular momentum) and more importantly deteriorate the accuracy of their time variations. In Figure 2 we show estimates produced by u⁺ flows and flows supporting an equatorial antisymmetric component besides the QG symmetric one (u^± flows), with different small scale penalization strength. The ΔLOD predicted by a large scale u⁺ flow model produces a too strong negative trend (normalized squared misfit χ² = 1.2). When using a weaker regularization the results are clearly improved (χ² = 0.7). But the best result is obtained when allowing for u⁻ and imposing lower longitudinal gradients (χ² = 0.55).

Figure 2

Observed (thick black line) and estimated ΔLOD. (top) Estimates for gufm1 (thin black lines), CM4 (red) and CHAOS‐3 (blue), for 3 different degrees of flow anisotropy. (bottom) Blow up of Figure 2 (top) for the most recent period. Also shown, for CM4, estimates from large‐scale u⁺ (green line with circles), small‐scale u⁺ (green line) and large scale u^± (red line with circles).

[14] We further check the importance of norm R_A in improving the estimations. In Figure 2 we show estimations for different degrees of anisotropy, quantified by different values of the regularizing parameter λ_A weighting R_A: from λ_A = 0 (only the ℓ¹ norm is used) to the maximum degree of anisotropy achieved while still explaining the SV model to a reasonable level. We note that using solely the isotropic regularization ℓ¹ produces too low amplitude estimations, and that increasing the anisotropy constraint (and decreasing the isotropic one) improves the results. This is because the flow energy can be more freely distributed among different length‐scales of the zonal flow coefficients, including those contributing to the core angular momentum.

[15] We quantify the maximum anisotropy imposed on flow models computing α = (R_A⁰ − R*_A)/R_A⁰, where R_A⁰ and R*_A are values of R_A computed respectively for flows inverted with λ_A = 0 (no anisotropy) and for the maximum achieved λ_A. We find α increasing from 0.8 to 0.9 during the 1840–2010 period. The good recovery of the trend and the amplitude of oscillations in ΔLOD is not specific to inversions of CM4, and we show in Figure 2 the good results obtained for gufm1 (1840–1990) and CHAOS‐3 (1997–2010).

[16] We proceed to identify the core surface regions where the u⁻ component is more important, as accounted by SV models in the frozen‐flux approximation. We concentrate on the more recent CHAOS‐3 model (derived from very recent satellite data and revised observatory monthly mean values) and compute the corresponding mean anisotropic flow model ^±, represented in Figure 3. Its rms velocity is 9.7 km/yr during the 1997–2010 period. We note the important contribution of u_θ⁻ in the equatorial region, where the crossing of the equator under Indonesia is quite obvious. However, only small scales are affected by the breakdown of QG near the equator, while the large scale flow remains highly symmetric outside the TC (see Table 1). Most striking is the sharpness with which the ⁺ flow model perceives the inner‐core, as shown by its streamfunction (ξ, Figure 3, right).

Figure 3

Inverted flow ^± using λ_A = 8 × 10⁶ averaged over the CHAOS‐3 era. Components (left) u_θ and (middle) u_ϕ in km/yr, and (right) the stream‐function ξ of the symmetric part ⁺ viewed from north‐pole defined by ⁺ = (cos θ)⁻²r × ∇ξ. The parallel corresponding to the trace of the TC is drawn (as well as ±20°, related to Table 1).

Table 1. Ratio of Symmetric Kinetic Energy Over Total Energy E⁺/(E⁺ + E⁻) for the Flow ^± (Figure 3) Truncated at Different Spherical Harmonic Degree ℓ_tr and Integrated Over the Whole Core Surface, Inside the TC, and Also Over a Band Centered on the Equator and Spanning 40° of Latitude

ℓtr	Core Surface	Inside TC	Equator
6	0.84	0.63	0.91
10	0.80	0.79	0.91
13	0.76	0.38	0.83
18	0.70	0.27	0.67
26	0.66	0.22	0.65

5. Conclusion

[17] Our direct numerical simulations show that for short time‐scale symmetric forcing, the flow response is quasi‐geostrophic in Earth's core conditions. For an anti‐symmetric forcing, an anti‐symmetric counterpart of the QG flow could account for flows crossing the core‐mantle boundary equator. As this component seems to decay at the same rate as the QG component, we argue that the presence of anti‐symmetric flows at the core surface depends mainly on the amount of anti‐symmetric forcing (whether buoyant, turbulent or magnetic) in the Earth core. Our simulations also show the development of longitudinal structures for the symmetric part, and a clear separation between the regions inside and outside the tangent cylinder. These results have inspired new constraints that we used for inverting the flow at the core surface from geomagnetic field models.

[18] Our flow models, with improved estimations of ΔLOD and the natural emergence of symmetric large scale features including the trace of the tangent cylinder, support three main implications: (i) the large scale flow in the Earth core responsible for SV is very likely dominated by QG motions; (ii) significant deviations to QG are expected for smaller scales and in the equatorial region; (iii) SV models seem to detect anisotropic flows in the Earth core, with zonation and strong latitudinal gradients.

Acknowledgments

[19] We would like to thank Dominique Jault and Nicolas Gillet for fruitful discussions, and Nils Olsen for kindly sending us the CHAOS‐3 main field coefficients. This research work was made possible by a joint FCT/CNRS grant. The simulations were run at the Service Commun de Calcul Intensif de l'Observatoire de Grenoble (SCCI).

[20] The Editor thanks Hagay Amit and an anonymous reviewer for their assistance in evaluating this paper.