- FRONTIER LETTER
- Open Access
Testing a toroidal magnetic field imaging method at the core-mantle boundary using a numerical dynamo model
© Takahashi; licensee Springer. 2014
Received: 29 August 2014
Accepted: 13 November 2014
Published: 26 November 2014
I quantitatively test a method of toroidal field imaging at the core-mantle boundary (CMB) using a synthetic magnetic field and core surface flow data from a 3-D self-consistent numerical dynamo model with a thin electrically conducting layer overlying the CMB, like the D ″ layer. With complete knowledge of the core flow, the imaged toroidal field well reproduces the magnitude and pattern of the dynamo model toroidal field. However, quality of the imaging depends strongly on latitude. In particular, the amplitude and correlation between the dynamo model and the imaged toroidal fields decline substantially at low latitude. Such degradation in imaging quality is due to inability to account for the radial derivative of the toroidal field, that is, an effect of magnetic diffusion, which is not incorporated in the method.
The geomagnetic main field and its secular variation measured by orbiting satellites and at magnetic observatories correspond to those of the poloidal constituent, whereas the toroidal counterparts, which are bound to the core, are not observable above the core-mantle boundary (CMB). Constraining the strength, the spatial distribution and secular variation of the toroidal component of the geomagnetic field are essentially important to understand not only the dynamics of the geodynamo but also the electromagnetic (EM) core-mantle coupling, one of the possible mechanisms of decadal variation in the length of day (LOD) (Morrison ).
Finite electric current flows in the mantle. The mantle electrical conductivity σ m ∼ 1 S/m is small relative to that of the core σ c ∼5×105 S/m. In particular, the post-perovskite phase within the D ″ layer above the CMB has greater electrical conductivity (approximately 102 S/m) (Ohta et al. ). Therefore, the electric current or the corresponding toroidal field may leak into the mantle from the core, by which the EM coupling would occur. Some attempts to observationally constrain the toroidal magnetic field at the CMB have been pursued by electric potential measurements over distances larger than 1,000 km (Lanzerotti et al. ; Shimizu et al. ), whereas there are also some discussions on the consistency of such observations with dynamo theory (Levy and Pearce ; Shimizu and Utada ).
A global distribution of the toroidal field at the CMB can be estimated by a method based on a core flow model inverted from the radial components of the geomagnetic field and its secular variation via frozen-flux approximation (Roberts and Scott ). Love and Bloxham () determine the toroidal field at the CMB to account for LOD variation via the EM coupling assuming a steady core flow. However, it is found that only an implausibly strong and spatially complex toroidal field is consistent with flow advection and LOD variation (Love and Bloxham ). Such a difficulty may be alleviated to some extent by taking time-dependence of the core flow into account (Holme ). However, a fact must be kept in mind that the inverted core flows are in principle non-unique (Backus ), and there is no way to know how well the toroidal field is retrieved properly from such a flow model.
Here, I test the method to infer the toroidal field at the CMB using a numerical dynamo model. It is a great advantage to utilize numerical dynamo modeling, because observations are limited, giving the poloidal field and indirectly the flow, while numerical dynamos have it all, including the toroidal field. Therefore, the major concern in this study is not an uncertainty arising from the non-uniqueness of the core flow estimation but that arising from several approximations to derive the toroidal field imaging method as introduced below.
I extend my numerical dynamo model (Takahashi et al. , ; Takahashi and Shimizu ) to implement an electrically conducting mantle overlying the fluid outer core. Thermally driven convection alone is considered for simplicity, although thermo-chemical convection may be more appropriate to the Earth’s core (Takahashi ). The model solves numerically the magnetohydrodynamic equations in a rotating spherical shell filled with an electrically conducting fluid obeying the Boussinesq approximation. The radii of the inner and outer cores are r i and r o , respectively, and the radius ratio r i /r o is 0.35. The solid inner core is assumed to be insulating.
where α is the thermal expansion coefficient, g o is the gravitational acceleration at the CMB, and κ is the thermal diffusivity.
is solved, where P m∗ is the magnetic Prandtl number in the layer. Here, I examine the case at σ∗=1/2,500, which is comparable with the electrical conductivity of the post-perovskite phase (Ohta et al. ).
Spherical harmonic expansion is truncated at degree and order 95. The number of the radial grid points is 80 in the outer core and 20 in the D ″ layer. In the outer core, radial derivatives are evaluated using combined compact finite differencing (Takahashi ), whereas ordinary finite differencing is used in the D ″ layer. At the ICB and CMB, no-slip and fixed heat flux boundary conditions are adopted for the velocity field and temperature, respectively. Continuity of the magnetic field and the tangential electric field is imposed at the CMB. At the top of the D ″ layer, the toroidal field vanishes, while the poloidal field is smoothly connected with the potential field.
In the present study, I set R a=1,500, E=10−4, P m=2, and P r=1. The magnetic Reynolds number and the Elsasser number of the run using the mean values of the velocity and magnetic fields over the volume of the spherical shell are 103 and 1.79, respectively. The model lies in the dipole-dominated, non-reversing regime.
Toroidal field imaging method
Therefore, the toroidal field can, in principle, be retrieved from the knowledge of the radial magnetic field, the core flow, and the electrical conductivity (more precisely conductance, σ m δ) of the D ″ layer.
The discarded term in Equation 11 represents leakage of the electric current due to diffusion, which induces the leakage EM torque on the mantle. The influence of removing it on imaging quality is also examined. It is noted that the expression given in Equation 12 includes uncertainty regarding effective thickness of the viscous boundary layer. Hence, core flows at different depths beneath the CMB are tried for imaging.
Then, the depth dependence of amplitude and correlation coefficient are examined. In the examination, u H at different depths down to r=0.91r o is used for imaging the CMB toroidal field, while for the other quantities such as B r and , those at the CMB are used. Regarding amplitude, the above-mentioned tendency remains unchanged with the depth below the viscous boundary layer in FR and TR cases (Figure 4c). In both cases, the ratio steeply increases from the CMB and reaches the maximum at r∼0.98r o , then gradually declines (it is the reason why I show plots at r=0.984r o as below the Ekman boundary layer of thickness d ν ).
The correlation coefficient behaves differently in FR and TR (Figure 4d). In FR, the correlation coefficient takes the maximum at r∼0.98r o just beneath the boundary layer like the amplitude ratio, whereas the maximum correlation in TR is obtained using a core flow slightly deeper than that in FR, although improvement in correlation is insignificant.
Discussion and concluding remarks
In this study, I have examined the quality of the method to image the toroidal field at the CMB using numerical dynamo modeling. With perfect knowledge of the radial magnetic field, core surface flow, and the electrical conductivity of the D ″ layer, the imaging method can reproduce much of the CMB toroidal field in terms of magnitude and pattern in FR and TR. However, the method fails to well reconstruct the toroidal field in low latitude, where the toroidal field generation is not dominated by the advection of the radial magnetic field alone. Since effects of magnetic diffusion are not taken into account in the present method, the low-latitude toroidal field could be underestimated by as much as 50%. Thus, the present method would provide us with a lower bound of the CMB toroidal field in the low-latitude region. Whether it is also the case in dynamo models at more Earth-like parameters, that is, lower E and Pm, should carefully be examined.
Contrary to the low latitude, the toroidal field in mid and high latitudes is generated by the process of flow advection. The recovered toroidal field tends to be slightly overestimated at FR in these regions. Such an overestimation may be the influence of an approximation that the D ″ layer is a thin sheet, whereby the horizontal diffusion is neglected relative to the radial diffusion. Let l be the horizontal scale of the toroidal field in the D ″. Then, the relative significance of the horizontal diffusion to the radial diffusion scales as D=(δ/l)2. The condition D≪1 for verifying the thin sheet approximation is not met on small scales. Indeed, D∼1 at spherical harmonic degree n=20, given l∼r o /n. It is anticipated that the overestimation is alleviated in a formulation without the thin sheet approximation. The effective conductance of the mantle will decrease with n, whereas Equation 12 includes the mantle conductance fixed at the D ″ conductance. Nevertheless, the overestimation is no more than 20% in the FR case. This indicates the peripheral contribution of flow advection on small scales, which basically agrees with Figure seven in Holme (). Underestimation in TR arises probably from a different cause. The most likely one is that contributions from the small-scale components in B r and u H to the generation processes of the large-scale toroidal field are not properly represented in TR.
As to spatial correlation, the overall imaging quality is fairly well in both FR and TR cases (correlation coefficient is larger than 0.8) and not very sensitive to the depth of the core flows adopted for imaging as long as the core flow beneath the boundary layer is used.
In conclusion, any approximations adopted in the present method do not cause a serious problem. Therefore, the toroidal field imaging method based on Equation 12 may be applicable with some care to real observational data. However, the fact must be kept in mind before applying the method to observational data that the results in the present study are derived from the perfectly known core flows by forward modeling. Thus, effects must be understood on the ability and quality of the toroidal field reconstruction method of using inverted non-unique core flows with different a priori assumptions such as a purely toroidal flow (Whaler ), steady flow (Voorhies and Backus ), tangentially geostrophic flow (LeMouël ), helical flow (Amit and Olson ), and tangentially magnetostrophic flow (Asari and Lesur ). This is obviously the next step of the study, where dynamo modeling would also be a help (Rau et al. ; Amit et al. ; Fournier et al. ; Aubert and Fournier ). Amit and Christensen () find in numerical dynamos that poloidal field diffusion is roughly evenly distributed at all latitudes. The present model is also the case (although not shown). However, if poloidal field diffusion should also be concentrated at low latitude in the geomagnetic field, core flow inversions from geomagnetic secular variation may be affected, in particular, at low latitude (Amit and Christensen ).
Besides, to reliably image the CMB toroidal magnetic field, an accurate electrical conductivity structure in the D ″ layer (conductance) is required, which is to be determined by experimental, theoretical, and observational studies. Then, the magnitude of the toroidal field that may be imaged by the present method could be compared with other estimates based on torsional oscillations (Buffett et al. ; Gillet et al. ).
The author would like to thank Hagay Amit and an anonymous reviewer for their thorough reviews and insightful comments. FT is supported by the Japan Society for the Promotion of Science under a grant-in-aid for young scientists (B) No. 24740303. Numerical simulations were performed on the Earth Simulator at the Earth Simulator Center, Yokohama, Japan.
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