A new inference of mantle viscosity based upon joint inversion of convection and glacial isostatic adjustment data

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Abstract

We present new profiles of mantle viscosity derived on the basis of non-linear, Occam-style joint inversions of an extensive set of data associated with mantle convection and glacial isostatic adjustment (GIA). The convection related observables include satellite-derived free-air gravity harmonics, the geodetically inferred excess ellipticity of the CMB and tectonic plate motions. The GIA constraints involve two classes of observables previously shown to be relatively insensitive to errors in the late Pleistocene ice history: the so-called Fennoscandian relaxation spectrum (FRS) and a set of site-specific decay times determined from the postglacial sea-level history in Hudson Bay and Sweden. The inverted viscosity profiles show a significant, three orders of magnitude, increase from the upper mantle (mean value of ∼4×1020 Pa s) to a high-viscosity (>1023 Pa s) peak at 2000 km depth, followed by a reduction toward the core-mantle boundary.

Introduction

The value and variation of mantle viscosity exerts a primary control on the planform and evolution of convective motions within the Earth's mantle (e.g., [1]). Inferences of long-term mantle viscosity have been derived from two principle sources: (1) data related to the isostatic adjustment of the Earth in consequence of the late Pleistocene glacial cycles, or glacial isostatic adjustment (henceforth GIA) (e.g., [2], [3], [4], [5], [6], [7]); and (2) a suite of surface geophysical observables linked to the convection process (e.g., [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19]).The debate over mantle viscosity has a long and rather complex history (e.g., [7]). Analyses of convection observables have consistently concluded that mantle viscosity increases by a factor of 30 or more from the base of the lithosphere to the core-mantle-boundary (CMB) (e.g., [8], [9], [10], [11], [12]). In contrast, inferences based upon GIA data have been subject to a more contentious debate. While the first GIA study of depth-dependent structure argued for a large increase with depth [20], this view was displaced by the global analyses of Cathles [2] and Peltier and colleagues (e.g. [3], [4], [21]), who advocated models in which the viscosity increased only moderately, if at all, with depth. The latter arguments prevailed for over a decade and they motivated efforts to invoke transient long-term viscosity [22], [23], [24] as a means for reconciling the GIA and convection-based inferences.

The necessity of invoking transient viscosity was subsequently weakened by three separate lines of geodynamic research. First, Nakada and Lambeck [6] argued that sea-level data from Australia are best-fit by models with a viscosity increase, from upper to lower mantle, broadly similar to values inferred from convection observables. Second, Mitrovica [7] showed that many previous arguments for a moderate increase in viscosity were biased by a misinterpretation of the so-called Haskell constraint [25] (1021 Pa s) on “average” mantle viscosity. Specifically, these studies assumed that the classic constraint, based on Fennoscandian uplift curves, applied to a region down to the 670 km boundary between the upper and lower mantle; however, resolving power analyses yield averaging kernels for the Fennoscandian data set that extend to ∼twice this depth [7], [26]. Accordingly, the Haskell constraint can be satisfied by compensating any increase in lower mantle viscosity (above 1021 Pa s) by a suitable decrease within the upper mantle (e.g., as in [6]). Third, a series of joint inversions of GIA and convection data [27], [28], [29] have shown that radial viscosity profiles exist that appear capable of simultaneously reconciling subsets of the two data types.

Fig. 1A and B shows radial viscosity profiles we previously derived [28] on the basis of non-linear inversions of the long-wavelength (up to degree and order 8) components of Earth's free air gravity field, the excess ellipticity of the core-mantle-boundary (CMB) and a set of site-specific postglacial decay times (see next section) from sites in Hudson Bay and Fennoscandia. The subset of convection data are insensitive to absolute viscosity; Fig. 1A was obtained by adopting a relative viscosity profile that was known to fit these data [15], and then determining an absolute scaling that provided a best-fit to the GIA data. In contrast, the profile in Fig. 1B was generated by a simultaneous inversion of both data sets.

These profiles provide comparable fits to the two data sets, and both are characterized by a large, ∼2 order of magnitude, increase in viscosity from upper to lower mantle. While the two profiles appear to differ, at least in detail, these differences disappear when comparisons are based on the inherent resolving power of the adopted data sets (rather than the value of individual model layers). As an example, the resolving kernel associated with the decay time of postglacial uplift in central Fennoscandia extends from the base of the lithosphere to a depth of ∼1300 km [7], [27], [28]; when this average is applied to the two profiles, a consistent value near 21 (on the log scale), compatible with the classic Haskell constraint, is obtained in both cases. A similar consistency is evident when the resolving kernel associated with the postglacial decay time of Hudson Bay uplift is applied to the two inverted profiles. In this case, the resolving kernel extends from about 400 to 1800 km depth, peaking near the boundary between the upper and lower mantle [27], [28], and in both cases an average ∼21, as required to satisfy the Hudson Bay constraint [27], [28], is computed.

In this paper, our goal is to report on the result of new non-linear, Occam-style inversions for mantle viscosity generated from a larger data set of GIA and convection observables. In particular, the postglacial decay time data set will be supplemented by a newly derived relaxation spectrum for the postglacial uplift of Fennoscandia [30]. The spectrum represents an estimate of the decay time of the uplift as a function of spherical harmonic degree (or spatial scale), and it resolves viscosity structure on a variety of radial length scales within the top half of the mantle [30]. For example, the spectrum provides a robust constraint (∼5×1020 Pa s) on the average viscosity from the base of the lithosphere to 550 km depth, a region entirely within the upper mantle. Most importantly, the spectrum (as well as the site-specific decay times described above) are relatively insensitive to uncertainties in the space–time history of late Pleistocene (and more recent) ice cover.

The data set of convection-related observables used to generate Fig. 1A,B will be augmented to include shorter wavelength components of the free-air gravity field, up to spherical harmonic degree and order 32. These constraints are further supplemented here by the addition of dynamic surface topography and present-day (horizontal divergence of) tectonic plate motions, where both fields are also represented by harmonic coefficients up to degree and order 32. A description of these different data may be found in a previous related study [31]. The plate velocities are of special importance because, unlike the other convection-related data, they provide sensitivity to the absolute value of mantle viscosity. The convection constraints on absolute viscosity were completely absent in our earlier joint viscosity inversions [27], [28]. The inclusion of a wide range of horizontal wavelengths in the convection data, compared to the previous joint inversions [27], [28], provides greater sensitivity to radial viscosity over a wider range of radial length scales, which span the entire depth of the mantle.

In recent work, we inverted this set of convection (i.e., gravity, CMB topography and plate motion) observables to generate the radial viscosity profiles shown in Fig. 1C and D [31]. The results in the two frames, derived in the complete absence of GIA constraints, are distinguished on the basis of the seismic S-wave heterogeneity model adopted, as input, into the inversions. Both models are characterized by a mean upper mantle viscosity of ∼3×1020 Pa s, and two high viscosity peaks within the lower mantle; one at the top of the lower mantle and the other at 2000 km depth. In originally reporting on these inversions [31], we focussed on the implications of the viscosity peak at 2000 km depth for the thermochemical structure of the deep mantle. For example, the deep viscosity peak was responsible for the observed reddening of the seismic spectrum in the bottom half of the mantle. Furthermore, large-scale seismic anomalies below Africa and the Pacific were found to be thermally buoyant, upwelling “megaplumes”, despite the presence of chemical heterogeneity necessary to reconcile the anti-correlation between bulk sound and S-wave seismic models [32], [33].

A comparison of the viscosity inferences in Fig. 1 reveals that the combined convection data sets (Fig. 1C,D) require a lower-mantle viscosity which is significantly greater than for the previous joint viscosity inversions (Fig. 1A,B). The main convection data used in the previous joint inversions were the long-wavelength free-air gravity anomalies, whereas the more recent inversions (Fig. 1C,D) also included the tectonic plate motions. A reconciliation of the constraints provided by both the plate motions and the gravity data requires a higher average value for the lower mantle viscosity. This requirement for increased average lower-mantle viscosity will also be apparent in the new joint inversions presented below. Another important difference between the two sets of inversions concerns the presence or absence of a low-viscosity notch at 670 km depth. The Occam inversions, which were used to derive the recent profiles (Fig. 1C,D), explicitly penalize the radial roughness of the viscosity variation and therefore cannot resolve the presence of such a sharp, locally defined feature. The profiles with such a low-viscosity layer (Fig. 1A) were obtained by imposing this layer a-priori and not during the inversions. This a-priori parameterization will again be explored in the present series of inversions.

In extending our inversions to consider a larger data base of GIA and convection observables, a number of questions arise. First, will it still be possible to simultaneously reconcile the two data types (GIA, convection) using a single profile of mantle viscosity? That is, can a profile be found that satisfies the suite of constraints (or resolved averages) imposed by the various data subsets? This question is particularly important from the perspective of the absolute viscosity since the tectonic plate motions may perhaps require a rather different absolute viscosity, especially in the upper mantle, than do the GIA data. Second, what will the basic features of this model, if it exists, be? For example, how will the convection-only inferences in Fig. 1C and D be impacted by the introduction of a large set of GIA constraints? A simple example suggests that the impact will be significant. If we apply the resolving kernels for either the Hudson Bay or Fennoscandian decay times to the profiles in Fig. 1C and D, then averages well in excess of observational constraint of ∼1021 Pa s are obtained. We thus conclude, a priori, that these convection-only inferences violate a set of robust GIA constraints, including (via the Fennoscandian kernel) the Haskell inference.

Section snippets

GIA data sets and forward model

Inferences of viscosity based upon GIA data are complicated by uncertainties in the late Pleistocene and Holocene ice history. Within the GIA literature, various data types have been promoted on the basis of their relative insensitivity to this history, including the Fennoscandian relaxation spectrum (FRS) and site-specific decay times associated with the postglacial uplift of previously glaciated regions.

The FRS represents the decay time of postglacial deformation as a function of harmonic

Results and discussion

In representing our inversion results, we have discretized the mantle viscosity structure into a set of 25 layers from the surface to the CMB. Our forward predictions also adopt this parameterization, although the GIA calculations incorporate an elastic lithosphere, of thickness LT. In practise, purely elastic behaviour is established in the viscoelastic GIA models by setting the viscosity to extremely high values in all numerical nodes down to a depth of LT. In this regard, our results are

Final remarks

We have reported on a new set of joint inversions of mantle convection and postglacial rebound observables for the radial profile of mantle viscosity. Relative to our previous joint inversions (Fig. 1A,B), the results described here include data related to shorter wavelength gravity anomalies, the dynamic surface topography, horizontal divergence of plate motions and the Fennoscandian relaxation spectrum. It is important to emphasize that our new inferences are entirely consistent with the

Acknowledgements

The authors wish to thank J. Wahr and M. Nakada for constructive and helpful reviews and the editors (Anny Cazenave and Scott King) for their efforts. This work was supported by NSERC and by the Earth System Evolution Program of the Canadian Institute of Advanced Research. A.M.F. acknowledges support provided by a Canada Research Chair grant and also the computational infrastructure funded by the Canada Foundation for Innovation.

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