# Inference of viscosity jump at 670 km depth and lower mantle viscosity structure from GIA observations

Inference of viscosity jump at 670 km depth and lower mantle viscosity structure from GIA... Summary A viscosity model with an exponential profile described by temperature (T) and pressure (P) distributions and constant activation energy ($$E_{{\rm{um}}}^{\rm{*}}$$ for the upper mantle and $$E_{{\rm{lm}}}^*$$ for the lower mantle) and volume ($$V_{{\rm{um}}}^{\rm{*}}$$ and $$V_{{\rm{lm}}}^*$$) is employed in inferring the viscosity structure of the Earth's mantle from observations of glacial isostatic adjustment (GIA). We first construct standard viscosity models with an average upper-mantle viscosity ($${\bar{\eta }_{{\rm{um}}}}$$) of 2 × 1020 Pa s, a typical value for the oceanic upper-mantle viscosity, satisfying the observationally derived three GIA-related observables, GIA-induced rate of change of the degree-two zonal harmonic of the geopotential, $${\skew5\dot{J}_2}$$, and differential relative sea level (RSL) changes for the Last Glacial Maximum sea levels at Barbados and Bonaparte Gulf in Australia and for RSL changes at 6 kyr BP for Karumba and Halifax Bay in Australia. Standard viscosity models inferred from three GIA-related observables are characterized by a viscosity of ∼1023 Pa s in the deep mantle for an assumed viscosity at 670 km depth, ηlm(670), of (1 − 50) × 1021 Pa s. Postglacial RSL changes at Southport, Bermuda and Everglades in the intermediate region of the North American ice sheet, largely dependent on its gross melting history, have a crucial potential for inference of a viscosity jump at 670 km depth. The analyses of these RSL changes based on the viscosity models with $${\bar{\eta }_{{\rm{um}}}}$$ ≥ 2 × 1020 Pa s and lower-mantle viscosity structures for the standard models yield permissible $${\bar{\eta }_{{\rm{um}}}}$$ and ηlm (670) values, although there is a trade-off between the viscosity and ice history models. Our preferred $${\bar{\eta }_{{\rm{um}}}}$$ and ηlm (670) values are ∼(7 − 9) × 1020 and ∼1022 Pa s, respectively, and the $${\bar{\eta }_{{\rm{um}}}}$$ is higher than that for the typical value of oceanic upper mantle, which may reflect a moderate laterally heterogeneous upper-mantle viscosity. The mantle viscosity structure adopted in this study depends on temperature distribution and activation energy and volume, and it is difficult to discuss the impact of each quantity on the inferred lower-mantle viscosity model. We conclude that models of smooth depth variation in the lower-mantle viscosity following $$\eta ( z )\ \propto {\rm{\ exp}}[ {( {E_{{\rm{lm}}}^* + P( z )V_{{\rm{lm}}}^*} )/{\rm{R}}T( z )} ]$$ with constant $$E_{{\rm{lm}}}^*$$ and $$V_{{\rm{lm}}}^*$$ are consistent with the GIA observations. Earth rotation variations, Sea level change, Rheology: mantle 1 INTRODUCTION Observations related to glacial isostatic adjustment (GIA) due to the last deglaciation such as relative sea level (RSL) variations and the rate of change of low-degree zonal harmonics of Earth's geopotential $${\skew5\dot{J}_n}$$ (n ≥ 2) contain important information about the Earth's viscosity structure (e.g. Peltier 2004; Sabadini et al. 2016). In inferring mantle viscosity from GIA-related observations, many studies have adopted a simple three-layer viscosity model described by elastic lithospheric thickness, upper-mantle viscosity above 670 km depth and lower-mantle viscosity (e.g. Wu & Peltier 1984; Cambiotti et al. 2010; Nakada et al. 2015; Lambeck et al. 2017). For example, Lambeck et al. (2017) inferred the lower-mantle viscosity of ∼1022 Pa s from inversion studies using RSL observations, tilting of palaeo-lake shorelines and present-day crustal displacements for the North American Late Wisconsin ice sheet. Nakada et al. (2015) estimated the GIA-induced $${\skew5\dot{J}_2}$$of −(6.0 − 6.5) × 10−11 yr−1 based on the geodetically derived $${\skew5\dot{J}_2}$$ (Roy & Peltier 2011; Cheng et al. 2013) and modern recent melting history taken from the IPCC 2013 Report (Vaughan et al. 2013) and inferred two permissible lower-mantle viscosities, ∼1022 and (5 − 10) × 1022 Pa s, for the simple three-layer viscosity model. GIA data sets may prefer a more complex lower-mantle viscosity structure (e.g. Ivins et al. 1993; Mitrovica 1996; Vermeersen et al. 1997; Peltier 2004). For example, a viscosity jump in the deep mantle has been inferred from plausible temperature profiles, high-pressure creep in olivine and $${\skew5\dot{J}_2}$$ (Ivins et al. 1993), postglacial decay times in Canada and Scandinavia (Mitrovica 1996) and rotational variations of the Earth (Vermeersen et al. 1997). More recently, Mitrovica et al. (2015) examined the GIA-induced $${\skew5\dot{J}_2}$$ based on a 23-layer lower-mantle viscosity model and argued for a deep mantle viscosity in excess of 1022 Pa s. Moreover, Lau et al. (2016) discretized the radial viscosity profile into 28 layers, 13 in the upper mantle and 15 in the lower mantle, and examined the viscosity structure based on postglacial decay times in Canada and Scandinavia, the Fennoscandian relaxation spectrum, late-Holocene differential sea level highstands in the Australian region and $${\skew5\dot{J}_2}$$. Then, they inferred a mean upper-mantle viscosity of ∼3 × 1020 Pa s, average viscosity of 1021 Pa s from 670 to ∼1500 km depth and 1022 − 1023 Pa in the deep mantle. Also, they insisted that the GIA data sets can be reconciled by a mantle viscosity profile without a viscosity jump at 670 km depth. In fact, their preferred viscosity model has no viscosity jump at 670 km depth. On the other hand, Nakada & Okuno (2016) analysed the GIA-induced $${\skew5\dot{J}_2}$$ based on a two-layer lower-mantle viscosity model with two layers in the lower mantle described by a depth-averaged effective viscosity from 670 to D km depth, η670,D, and a viscosity from D to 2891 km depth (core–mantle boundary, CMB), ηD,2891, with D values of 1191 and 1691 km. Then, they showed that the GIA-induced $${\skew5\dot{J}_2}$$ of −(6.0 − 6.5) × 10−11 yr−1 only requires a viscosity layer higher than (5 − 10) × 1021 Pa s for a depth above the CMB. Moreover, their analyses using the GIA-induced $${\skew5\dot{J}_2}$$, Last Glacial Maximum (LGM) sea levels at Barbados and Bonaparte Gulf in Australia and postglacial sea level highstands at Karumba and Halifax Bay in Australia (see Fig. 1) yielded two permissible viscosity solutions for the lower mantle: η670,1191 > 3 × 1021 and η1191,2891 ∼ (5 − 10) × 1022 Pa s, and η670,1691 > 1022 and η1691,2891 ∼ (5 − 10) × 1022 Pa s (no permissible solution was obtained for the case of η670,D > ηD,2891). The inferred upper-mantle viscosity for both solutions is (1 − 4) × 1020 Pa s. Thus, recent analyses using GIA data sets yield a deep mantle viscosity in excess of 1022 Pa s, while Argus et al. (2014) support a more muted viscosity increase from 5 × 1020 Pa s beneath the lithosphere to 3 × 1021 Pa s above the CMB. Figure 1. View largeDownload slide (a) Spatial distribution of total melted ice thicknesses during the last deglaciation for the North American ice sheet of the IA20 ice model and (a) and (b) relative sea level (RSL) change sites examined in this study. Figure 1. View largeDownload slide (a) Spatial distribution of total melted ice thicknesses during the last deglaciation for the North American ice sheet of the IA20 ice model and (a) and (b) relative sea level (RSL) change sites examined in this study. The viscosity structure inferred from the GIA data sets may be reaching to a general agreement except for the solution by Argus et al. (2014). Here, we examine GIA-related data sets, RSL changes and GIA-induced $${\skew5\dot{J}_n}$$(2 ≤ n ≤ 6), based on the viscosity model with an exponential profile described by pressure (P) and temperature (T) distributions and constant activation energy (E*) and volume (V*) (see Section 2.1). The present study has two main purposes. One is to examine whether the GIA data sets provide a permissible exponential lower-mantle viscosity profile with a viscosity of 1022 − 1023 Pa s in the deep mantle as argued by several GIA studies, and another is to infer the viscosity jump at 670 km depth (Sections 3 and 4). Of course, adopted parameters such as temperature distribution and activation energy and volume are highly uncertain and the results obtained here are preliminary. Inference of lower-mantle rheology based on the experimentally based rheological data in the lower-mantle pressure and temperature conditions may be possible if we assume a homologous temperature scaling viscosity profile using experimentally determined self-diffusion coefficients of silicon in MgSiO3 perovskite under shallow lower-mantle conditions and melting temperatures of lower-mantle minerals (Yamazaki & Karato 2001; Karato 2008). More recently, Marquardt & Miyagi (2015) showed an increase of viscosity throughout the upper 900 km of the lower mantle based on the high-pressure (up to 96 GPa) and room-temperature deformation data for ferropericlase of the main phase of the lower mantle. On the other hand, Rudolph et al. (2015) inferred an increase in viscosity at 800–1200 km depth from the flow model using the long-wavelength non-hydrostatic geoid and seismic tomography data and argued that rheological properties inferred from the deformation experiments by Marquardt & Miyagi (2015) may provide a possible rheological interpretation for their inference on viscosity increase in the mid-mantle. However, it would be difficult to discuss the rheological properties at high temperature in the lower mantle based on the experimental data for room-temperature deformation experiments (e.g. Karato 2008). More recently, Girard et al. (2016) succeeded quantitative deformation experiments for a mixture of bridgmanite and magnesiowüstite, main phases of the lower mantle, at pressures at 24–27.5 GPa and temperatures up to 2000–2150 K. Although deformation mechanisms are not clearly identified from their experimental data, it would be possible to discuss the rheological properties based on the experimentally based rheological data for the lower-mantle conditions in the near future. Then, the present study evaluating quantitative impacts of activation energy and volume on the GIA-based viscosity model may provide some useful constraint on the mineral physics of the lower-mantle rheology. The method for conducting discussions on an exponential mantle viscosity profile has mainly two steps: (i) inference of the mantle viscosity structure from the GIA-induced $${\skew5\dot{J}_2}$$, differential RSL change for LGM sea levels at Barbados and Bonaparte Gulf and that for RSL changes at ∼6 kyr BP for Karumba and Halifax Bay, referred to as three GIA-related observables here, and (ii) inference of mantle viscosity considering the inference obtained in step (i), particularly for the viscosity jump at 670 km depth, based on the GIA-induced $${\skew5\dot{J}_n}$$(2 < n ≤ 6) and postglacial RSL changes at Southport, Bermuda and Everglades in the intermediate region of the North American ice sheet (Fig. 1). The paper is organized as follows. In Section 2, we explain the viscosity and the used ice history models. In Section 3, we examine the sensitivities of three GIA-related observables to viscosity and GIA ice models and infer the viscosity structure from three GIA-related observables. In Section 4, we infer the lower-mantle viscosity structure, particularly a potential viscosity jump at 670 km depth, based on the results in Section 3 and observationally derived GIA-induced zonal secular rates for n = 4 and 6 and postglacial RSL changes at Southport, Bermuda and Everglades. The results obtained in this study are discussed and summarized in Section 5. 2 MODEL ADOPTED IN THIS STUDY 2.1 Earth model We adopt the seismological Preliminary Reference Earth Model (PREM, Dziewonski & Anderson 1981) for density and elastic constants. The viscosity structure is described by elastic lithospheric thickness (H), upper-mantle viscosity, ηum(z) for H < z < 670 km depth, and lower-mantle viscosity, ηlm(z) for 670 ≤ z ≤ 2891 km depth. The depth-dependent viscosity structures, ηum(z) and ηlm(z), are described as follows (e.g. Karato 2008):   $${\eta _{{\rm{um}}}}\left( z \right)\ = {{\rm{A}}_{{\rm{um}}}}{\rm{\ exp}}\left[ {\frac{{E_{{\rm{um}}}^* + P\left( z \right)V_{{\rm{um}}}^*}}{{{\rm{R}}T\left( z \right)}}} \right]\,\,{\rm{for}}\,\,{\rm{H}}\, < z < 670$$ (1)  $${\eta _{{\rm{lm}}}}\left( z \right)\ = {{\rm{A}}_{{\rm{lm}}}}{\rm{\ exp}}\left[ {\frac{{E_{{\rm{lm}}}^* + P\left( z \right)V_{{\rm{lm}}}^*}}{{{\rm{R}}T\left( z \right)}}} \right]\,\,{\rm{for}}\,\, 670\, \le z \le 2891$$ (2)R is the gas constant, and Aum and Alm are constants determined using assumed average upper-mantle viscosity and ηlm(670) value, respectively (see below). E* and V* are activation energy and activation volume (‘um’ for the upper mantle and ‘lm’ for the lower mantle), respectively, and these quantities are assumed to be constant in this study. P(z) and T(z) are pressure and temperature distributions, respectively, and the pressure distribution is derived from the density and gravity values of PREM. Here, we adopt a simple upper-mantle temperature distribution considering a cooling model of oceanic lithosphere and adiabatic temperature gradient in the upper mantle (Γ, e.g. Turcotte & Schubert 2002):   $${T_{{\rm{um}}}}\left( {z,t} \right) = {T_0}\ + \left( {{T_\infty } - {T_0}} \right){\rm{erf}}\left( {\frac{z}{{2\sqrt {\kappa t} }}} \right) + \Gamma z$$ (3)where T0 is the surface temperature, T∞is the temperature at z = ∞ for the cooling model, κ is thermal diffusivity (10−6 m2 s−1 in this study) and t is the plate age. The values of T0 and T∞ are assumed to be 273 and 1600 K, respectively, corresponding to T∞ − T0 = 1300 K by Turcotte & Schubert (2002). The temperature at 670 km depth is 1873 K estimated by Ito & Katsura (1989) based on the high-pressure phase equilibrium in the system Mg2SiO4–Fe2SiO4 and an assumption that the spinel dissociation is responsible for the sharp seismic discontinuity. The temperature distribution for the lower mantle is assumed to be adiabatic and given by:   $${T_{{\rm{lm}}}}\left( z \right) = \ {\rm{B\ exp}}\left( {\frac{{\alpha gz}}{{{C_p}}}} \right)$$ (4)In eq. (4), B is a constant (see below), and α, g and Cp are coefficient of thermal expansion, gravity and specific heat at constant pressure, and assumed to be 2.5 × 10−5 K−1, 10 m s−2 and 1000 J K−1 kg−1, respectively (αg/Cp = 2.5 × 10−7 m−1). The gravity of the lower mantle is assumed to be constant in estimating the lower-mantle temperature. However, this assumption does not change the results at all in the following discussion. The values of Γ(in eq. 3) and B (in eq. 4) are obtained by using Tum(670) = Tlm(670) = 1873 K. In this study, we treat an average upper-mantle viscosity, $${\bar{\eta }_{{\rm{um}}}}$$, and viscosity at 670 km depth (viscosity at the top of the lower mantle), ηlm(670), as key parameters describing the viscosity structure in the mantle. Then, the coefficient Aum in eq. (1) is obtained by taking a weighted harmonic average for the upper-mantle viscosity, ηum(z), and the coefficient of Alm in eq. (2) for ηlm(z) is determined from an assumed ηlm(670) value. In this study, we restrict to the lower-mantle viscosity structure with no low-viscosity D″ layer by considering that the effective lower-mantle viscosity inferred from the GIA-induced $${\skew5\dot{J}_2}$$ is relatively insensitive to the viscosity structure of the D″ layer (Nakada et al. 2015) and that the GIA-related postglacial RSL changes such as those at Barbados, Bermuda and Karumba examined here are insensitive to the low-viscosity D″ layer as well (Nakada & Karato 2012, see also Lau et al. 2016). Our model is based on the classical normal mode approach, and thus deals appropriately with the issues raised by Cambiotti & Sabadini (2010) when dealing with discontinuously or continuously varying density and bulk modulus. We briefly comment out an inferred lower-mantle viscosity structure satisfying the GIA data sets adopted in this study. The inferred lower-mantle viscosity profile is, of course, non-unique. Also, it would be difficult to adequately evaluate the impacts of uncertainties of temperature distribution, $$E_{{\rm{lm}}}^*$$ and $$V_{{\rm{lm}}}^*$$ on the inferred viscosity structure. It is so meaningful that GIA data sets only prefer a depth-dependent quantity of $$( {E_{{\rm{lm}}}^* + P( z )V_{{\rm{lm}}}^*} )/RT( z )$$ for the inferred lower-mantle viscosity structure. We also comment out the plate age t. The age of the seafloor is laterally varying, but its effect on the plate thickness (e.g. Turcotte & Schubert 2002) is not considered in the modeling herein. That is, this parameter must be considered as an average value for the plate thickness throughout the globe. 2.2 Ice history model We adopt six simplified ice history models of the last Pleistocene glaciation, IA ice models (IA10, IA20 and IA30) and IR ice models (IR10, IR20 and IR30) by Nakada & Okuno (2017), constructed based on the method by Lambeck (1993) using ice sheet dynamics (see fig. 1 in Nakada & Okuno 2017). The areas of all ice sheets for the LGM at ∼21 kyr BP are the same as those for the global ice history model ICE-5G (Peltier 2004), and those at an arbitrary time are proportional to the ice volume derived from equivalent sea level (ESL) component of each ice sheet. The ESL is defined as the change in meltwater volume divided by the surface area of the ocean at the present day. Fig. 1(a) shows the spatial distribution of total melted ice thicknesses during the last deglaciation for the North American ice sheet of IA20. These ice models are inferred from the LGM sea levels at Barbados (Fairbanks 1989; Peltier & Fairbanks 2006) and Bonaparte Gulf (Yokoyama et al. 2000), RSL change at Tahiti (Bard et al. 1996; Deschamps et al. 2012) and RSL changes after ∼6 kyr BP at Karumba and Halifax Bay (Chappell et al. 1983; Nakada & Lambeck 1989; Nakada et al. 2016). These models have an identical ESL history with a total ESL component of 127.9 m that is 10 m larger than that of ICE-5G (∼118 m). The overarching difference between the IA and IR ice models is simply the assumed timing history. All IA* ice sheets melt synchronously based on the ESL history. The Antarctic ESL component (ESLSH) is assumed to be 9.3 m (∼10 m) for IA10, 18.7 m (∼20 m) for IA20 and 28.0 m (∼30 m) for IA30 because of uncertainties in the melting history of the Antarctic ice sheet (e.g. Nakada & Lambeck 1988; Nakada et al. 2000; Peltier 2004; Whitehouse et al. 2012; Ivins et al. 2013; Lambeck et al. 2014). The total Antarctic ESL components for the IR* models are the same as those for the IA* models, respectively. The Antarctic ESL history before 6 kyr BP for IR20 is the same as that for ICE-5G with ESLSH∼20 m due to the melting mainly from 12 to 6 kyr BP (see also Nakada et al. 2000, for the inference using RSL data in Antarctica), and the melting ice thicknesses at all sites for the Antarctic IR20 model are proportional to the ESL history. IR20 has a gradual Antarctic melting of ∼2.5 m after 6 kyr BP as was assumed by Nakada & Lambeck (1988). The ESL components of the Laurentide, Fennoscandian and Greenland ice sheets are estimated considering their ESL components for ICE-5G, and these ice sheets melt synchronously based on the ESL history for the Northern Hemisphere (NH) ice sheets. The melted ice thicknesses at an arbitrary time for the Antarctic IR10 ice model are half the ones of IR20. Those for IR30 before 6 kyr BP are three times the ones of IR10, and those after 6 kyr BP are the same as those of IR20. The ESL components for the North American ice sheets including the Greenland ice sheet, which would significantly affect the postglacial RSL changes at Southport, Bermuda and Everglades examined in Section 4, are 94.4, 87.0 and 79.5 m for IA10 (IR10), IA20 (IR20) and IA30 (IR30), respectively. 3 RESULTS FOR THREE GIA-RELATED OBSERVABLES 3.1 Case for constant upper-mantle viscosity model GIA-induced $${\skew5\dot{J}_2}$$ provides an important constraint on the lower-mantle viscosity (e.g. Sabadini et al. 1982; Wu & Peltier 1984; Devoti et al. 2001; Mitrovica et al. 2015; Nakada et al. 2015). Also, differential RSL change for Barbados and Bonaparte Gulf at the LGM, $${\rm{\Delta RSL}}_{\rm{L}}^{{\rm{Bar}},{\rm{Bon}}}$$, and that for Karumba and Halifax Bay at ∼6 kyr BP, $${\rm{\Delta RSL}}_6^{{\rm{Kar}},{\rm{Hal}}}$$, are sensitive to the lower-mantle viscosity (Nakada & Okuno 2016; Nakada et al. 2016). If we denote the LGM sea levels at Barbados and Bonaparte Gulf by $${\rm{RSL}}_{\rm{L}}^{{\rm{Bar}}}$$ and $${\rm{RSL}}_{\rm{L}}^{{\rm{Bon}}}$$, respectively, then the differential RSL change for these sites is given by $${\rm{\Delta RSL}}_{\rm{L}}^{{\rm{Bar}},{\rm{Bon}}}$$ = $${\rm{RSL}}_{\rm{L}}^{{\rm{Bar}}}$$ − $${\rm{RSL}}_{\rm{L}}^{{\rm{Bon}}}$$. Similarly, that for Karumba and Halifax Bay is given by $${\rm{\Delta RSL}}_6^{{\rm{Kar}},{\rm{Hal}}}$$ = $${\rm{RSL}}_6^{{\rm{Kar}}}$$ − $${\rm{RSL}}_6^{{\rm{Hal}}}$$. Here, we discuss the GIA-induced $${\skew5\dot{J}_2}$$ and differential RSL changes for these sites by employing viscosity models with a constant upper-mantle viscosity (ηum) of 2 × 1020 Pa s, H = 65 km, ηlm(670) = 1021 Pa s and $$E_{{\rm{lm}}}^*$$ = 250 kJ mol−1. We hereafter refer $${\skew5\dot{J}_2}$$, $${\rm{\Delta RSL}}_{\rm{L}}^{{\rm{Bar}},{\rm{Bon}}}$$ and $${\rm{\Delta RSL}}_6^{{\rm{Kar}},{\rm{Hal}}}$$ as three GIA-related observables. The observationally derived differential RSL change at the LGM for Barbados and Bonaparte Gulf is 0–25 m from the LGM sea levels of −120 to −98 m for Barbados (Fairbanks 1989; Peltier & Fairbanks 2006) and −123 to −120 m for Bonaparte Gulf (Yokoyama et al. 2000). That for Karumba and Halifax Bay is 0.9–1.9 m from $${\rm{RSL}}_6^{{\rm{Kar}}}$$ ∼ (2.4 ± 0.3) and $${\rm{RSL}}_6^{{\rm{Hal}}}$$ ∼ (1.0 ± 0.2) m (Chappell et al. 1983; see also Lambeck & Nakada 1990). Fig. 2(a) shows the viscosity structures of U1–U6 models with $$V_{{\rm{lm}}}^*$$ = (2.9, 3.05, 3.2, 3.35, 3.5, 3.65) × 10−6 m3 mol−1, respectively (Table 1). The effective lower-mantle viscosity increases with increasing $$V_{{\rm{lm}}}^*$$ value. Table 1 shows the three GIA-related observables based on the IA20 ice model and these viscosity models. With increasing $$V_{{\rm{lm}}}^*$$ value, the $${\skew5\dot{J}_2}$$ magnitude increases in a range of (2.9 − 3.2) × 10−6 m3 mol−1 and decreases in a range of (3.2 − 3.65) × 10−6 m3 mol−1. Fig. 2(b) shows the $${\skew5\dot{J}_2}$$ for IA20 as a function of lower-mantle viscosity (ηlm) based on the simple three-layer viscosity model with ηum = 2 × 1020 Pa s and H = 65 km usually used in the $${\skew5\dot{J}_2}$$ study (e.g. Wu & Peltier 1984). The rates for the viscosity models with $$V_{{\rm{lm}}}^*$$ = 2.9 × 10−6 and 3.50 × 10−6 m3 mol−1, satisfying the observationally derived estimate of −(6.0 − 6.5) × 10−11 yr−1, correspond to the rates for sections ‘B’ and ‘F’ of the black curve in Fig. 2(b), respectively. The rates for the viscosity models with 3.05 × 10−6, 3.2 × 10−6 and 3.35 × 10−6 m3 mol−1 correspond to the rates for sections ‘C’, ‘D’ and ‘E’ of the black curve in Fig. 2(b), respectively. These comparisons indicate that the permissible viscosity solutions with $$V_{{\rm{lm}}}^*$$ ∼ 2.9 × 10−6 and 3.35 × 10−6 < $$V_{{\rm{lm}}}^*$$ ≤ 3. 5 × 10−6 m3 mol−1 correspond to the viscosity solutions of ηlm ∼ (1–2) × 1022 and ηlm ∼(5–10) × 1022 Pa s for the simple three-layer viscosity model in Fig. 2(b), respectively. On the other hand, the $${\rm{\Delta RSL}}_{\rm{L}}^{{\rm{Bar}},{\rm{Bon}}}$$ values are negative for the viscosity models with $$V_{{\rm{lm}}}^*$$ ≤ 3. 2 × 10−6 m3 mol−1, and consequently, differential RSL change for Barbados and Bonaparte Gulf prefers the lower-mantle viscosity structure with $$V_{{\rm{lm}}}^*$$ > 3. 2 × 10−6 m3 mol−1. That is, the GIA-induced $${\skew5\dot{J}_2}$$ and $${\rm{\Delta RSL}}_{\rm{L}}^{{\rm{Bar}},{\rm{Bon}}}$$ prefer the lower-mantle viscosity structure with 3.35 × 10−6<$$V_{{\rm{lm}}}^*$$ ≤ 3. 5 × 10−6 m3 mol−1, which is not inconsistent with the inference from the differential RSL change at ∼6 kyr for Karumba and Halifax Bay (Table 1). This solution corresponds to the lower-mantle viscosity of (5–10) × 1022 Pa s for the simple three-layer viscosity model in Fig. 2(b). Figure 2. View largeDownload slide (a) Viscosity structures of the U1–U6 models with $$V_{{\rm{lm}}}^*$$ = (2.9, 3.05, 3.2, 3.35, 3.5, 3.65) × 10−6 m3 mol−1, respectively (see Table 1) and (b) $${\skew5\dot{J}_2}$$ for the IA20 ice model as a function of lower-mantle viscosity (ηlm) based on the simple three-layer viscosity model with ηum = 2 × 1020 Pa s and H = 65 km usually used in the $${\skew5\dot{J}_2}$$ study. The shaded region shows the range for GIA-induced $${\skew5\dot{J}_2}$$ of −(6.0 − 6.5) × 10−11 yr−1. Figure 2. View largeDownload slide (a) Viscosity structures of the U1–U6 models with $$V_{{\rm{lm}}}^*$$ = (2.9, 3.05, 3.2, 3.35, 3.5, 3.65) × 10−6 m3 mol−1, respectively (see Table 1) and (b) $${\skew5\dot{J}_2}$$ for the IA20 ice model as a function of lower-mantle viscosity (ηlm) based on the simple three-layer viscosity model with ηum = 2 × 1020 Pa s and H = 65 km usually used in the $${\skew5\dot{J}_2}$$ study. The shaded region shows the range for GIA-induced $${\skew5\dot{J}_2}$$ of −(6.0 − 6.5) × 10−11 yr−1. Table 1. Values of $$V_{{\rm{lm}}}^*$$ (×10−6 m3mol−1), Alm (Pa s),$${\skew5\dot{J}_2}$$(×10−11yr−1), $${\rm{RSL}}_{\rm{L}}^{{\rm{Bar}}}$$ (m), $${\rm{RSL}}_{\rm{L}}^{{\rm{Bon}}}$$ (m), $${\rm{\Delta RSL}}_{\rm{L}}^{{\rm{Bar}},{\rm{Bon}}}$$ (m),$${\rm{RSL}}_6^{{\rm{Kar}}}$$ (m), $${\rm{RSL}}_6^{{\rm{Hal}}}$$ (m) and $${\rm{\Delta RSL}}_6^{{\rm{Kar}},{\rm{Hal}}}$$ (m) predicted for the IA20 ice model and viscosity models with a constant upper-mantle viscosity of 2 × 1020 Pa s and elastic lithospheric thickness (H) of 65 km. Model name  $$V_{{\rm{lm}}}^*$$  Alm  $${\skew5\dot{J}_2}$$  $${\rm{RSL}}_{\rm{L}}^{{\rm{Bar}}}$$  $${\rm{RSL}}_{\rm{L}}^{{\rm{Bon}}}$$  $${\rm{\Delta RSL}}_{\rm{L}}^{{\rm{Bar}},{\rm{Bon}}}$$  $${\rm{RSL}}_6^{{\rm{Kar}}}$$  $${\rm{RSL}}_6^{{\rm{Hal}}}$$  $${\rm{\Delta RSL}}_6^{{\rm{Kar}},{\rm{Hal}}}$$  U1  2.90  1.23 × 1012  −6.29  −124.2  −117.8  −6.4  2.20  1.37  0.83  U2  3.05  9.76 × 1011  −6.97  −122.3  −118.2  −4.1  2.33  1.47  0.86  U3  3.20  7.75 × 1011  −7.07  −120.4  −118.7  −1.7  2.37  1.50  0.87  U4  3.35  6.15 × 1011  −6.66  −116.6  −119.9  3.3  2.35  1.43  0.92  U5  3.50  4.88 × 1011  −6.04  −117.6  −119.6  2.0  2.38  1.47  0.91  U6  3.65  3.88 × 1011  −5.45  −118.9  −119.2  0.3  2.38  1.49  0.89  Model name  $$V_{{\rm{lm}}}^*$$  Alm  $${\skew5\dot{J}_2}$$  $${\rm{RSL}}_{\rm{L}}^{{\rm{Bar}}}$$  $${\rm{RSL}}_{\rm{L}}^{{\rm{Bon}}}$$  $${\rm{\Delta RSL}}_{\rm{L}}^{{\rm{Bar}},{\rm{Bon}}}$$  $${\rm{RSL}}_6^{{\rm{Kar}}}$$  $${\rm{RSL}}_6^{{\rm{Hal}}}$$  $${\rm{\Delta RSL}}_6^{{\rm{Kar}},{\rm{Hal}}}$$  U1  2.90  1.23 × 1012  −6.29  −124.2  −117.8  −6.4  2.20  1.37  0.83  U2  3.05  9.76 × 1011  −6.97  −122.3  −118.2  −4.1  2.33  1.47  0.86  U3  3.20  7.75 × 1011  −7.07  −120.4  −118.7  −1.7  2.37  1.50  0.87  U4  3.35  6.15 × 1011  −6.66  −116.6  −119.9  3.3  2.35  1.43  0.92  U5  3.50  4.88 × 1011  −6.04  −117.6  −119.6  2.0  2.38  1.47  0.91  U6  3.65  3.88 × 1011  −5.45  −118.9  −119.2  0.3  2.38  1.49  0.89  Notes: The viscosity at 670 km depth, ηlm(670), is 1021 Pa s and the activation energy for the lower mantle, $$E_{{\rm{lm}}}^*$$, is 250 kJ mol−1. View Large 3.2 Effect of upper-mantle activation energy and volume Here, we examine the effects of activation energy ($$E_{{\rm{um}}}^{\rm{*}}$$) and volume ($$V_{{\rm{um}}}^{\rm{*}}$$) of the upper mantle on the three GIA-related observables by using viscosity models of U5 (Table 1) and M1–M4 shown in Fig. 3 (see also Table 2). The $$E_{{\rm{um}}}^{\rm{*}}$$ value is 250 kJ mol−1 for M1–M3 and 300 kJ mol−1 for M4, corresponding to the experimentally derived estimates for diffusion creep by Karato et al. (1986): 250 kJ mol−1 for wet olivine and 290 kJ mol−1 for dry olivine (see also table 19.1 in Karato 2008). The upper-mantle viscosity for the U5 model is constant and 2 × 1020 Pa s, and that for M1–M4 is a weighted average viscosity ($${\bar{\eta }_{{\rm{um}}}}$$) of 2 × 1020 Pa s. The adopted t value, t = 60 Myr, corresponds to the mean age of the seafloor (e.g. Turcotte & Schubert 2002). The viscosity at ∼65 km depth just beneath the elastic lithosphere is ∼1024 Pa s for M1–M4, corresponding to a Maxwell relaxation time of ∼0.3 Myr. The parameter values describing the lower-mantle viscosity are the same as those for U5. Figure 3. View largeDownload slide Viscosity models U5 with constant upper-mantle viscosity of 2 × 1020 Pa s (Table 1) and M1–M4 with $${\bar{\eta }_{{\rm{um}}}}$$ = 2 × 1020 Pa s (Table 2). The $$E_{{\rm{um}}}^{\rm{*}}$$ value is 250 kJ mol−1 for M1–M3 and 300 kJ mol−1 for M4. Figure 3. View largeDownload slide Viscosity models U5 with constant upper-mantle viscosity of 2 × 1020 Pa s (Table 1) and M1–M4 with $${\bar{\eta }_{{\rm{um}}}}$$ = 2 × 1020 Pa s (Table 2). The $$E_{{\rm{um}}}^{\rm{*}}$$ value is 250 kJ mol−1 for M1–M3 and 300 kJ mol−1 for M4. Table 2. Viscosity models adopted in this study and three GIA-related observables predicted for these viscosity models. Model name  $$E_{{\rm{um}}}^{\rm{*}}$$  $$V_{{\rm{um}}}^{\rm{*}}$$  Aum  $$E_{{\rm{lm}}}^*$$  $$V_{{\rm{lm}}}^*$$  ηlm(670)  Alm  t  H  $${\skew5\dot{J}_2}$$  $${\rm{\Delta RSL}}_{\rm{L}}^{{\rm{Bar}},{\rm{Bon}}}$$  $${\rm{\Delta RSL}}_6^{{\rm{Kar}},{\rm{Hal}}}$$  M1  250  2.0  1.09 × 1012  250  3.5  1021  4.88 × 1011  60  65  –6.0  1.8  0.77  M2  250  3.0  4.78 × 1011  250  3.5  1021  4.88 × 1011  60  65  −6.09  3.0  0.96  M3  250  4.0  2.35 × 1011  250  3.5  1021  4.88 × 1011  60  65  −6.13  4.3  1.18  M4  300  4.0  6.94 × 109  250  3.5  1021  4.88 × 1011  60  65  −6.09  3.8  1.10  M5  250  4.0  2.35 × 1011  200  3.05  1021  2.42 × 1013  60  65  −6.51  3.2  1.17  M6  250  4.0  2.35 × 1011  250  3.425  1021  5.48 × 1011  60  65  −6.53  3.4  1.17  M7  250  4.0  2.35 × 1011  300  3.80  1021  1.24 × 1010  60  65  −6.54  3.5  1.17  M8  250  4.0  2.35 × 1011  250  3.475  1021  5.08 × 1011  60  65  −6.26  4.0  1.18  M9  250  4.0  2.35 × 1011  250  3.2  2 × 1021  1.55 × 1012  60  65  −6.24  5.5  1.23  M10  250  4.0  2.35 × 1011  250  2.85  5 × 1021  6.64 × 1012  60  65  −6.28  6.4  1.25  M11  250  4.0  2.35 × 1011  250  2.6  1022  1.95 × 1013  60  65  −6.23  6.9  1.25  M12  250  4.0  2.35 × 1011  250  2.33  2 × 1022  5.91 × 1013  60  65  −6.27  6.9  1.24  M13  250  4.0  2.35 × 1011  250  1.95  5 × 1022  2.65 × 1014  60  65  −6.26  6.4  1.23  M14  250  4.0  2.35 × 1011  250  3.515  1021  4.77 × 1011  60  65  −6.24  4.2  1.17  M15  250  4.0  2.35 × 1011  250  3.225  2 × 1021  1.49 × 1012  60  65  −6.27  5.5  1.23  M16  250  4.0  2.35 × 1011  250  2.88  5 × 1021  6.34 × 1012  60  65  −6.24  6.5  1.24  M17  250  4.0  2.35 × 1011  250  2.62  1022  1.89 × 1013  60  65  −6.25  6.9  1.24  M18  250  4.0  2.35 × 1011  250  2.36  2 × 1022  5.64 × 1013  60  65  −6.22  6.9  1.23  M19  250  4.0  2.35 × 1011  250  1.985  5 × 1022  2.51 × 1014  60  65  −6.21  6.6  1.22  M20  250  4.0  2.35 × 1011  250  3.2  2 × 1021  1.55 × 1012  100  65  −6.08  4.6  1.23  M21  250  4.0  2.35 × 1011  250  3.17  2 × 1021  1.623 × 1012  100  65  −6.26  4.2  1.23  M22  250  4.0  2.35 × 1011  250  3.2  2 × 1021  1.55 × 1012  60  100  −5.88  5.6  1.61  M23  250  4.0  2.35 × 1011  250  3.15  2 × 1021  1.674 × 1012  60  100  −6.19  4.9  1.61  M24  250  4.0  2.35 × 1011  250  3.125  2 × 1021  1.739 × 1012  60  100  −6.35  4.6  1.61  M25  250  4.0  2.35 × 1011  250  3.2  2 × 1021  1.55 × 1012  200  100  −5.66  5.0  1.4  M26  250  4.0  2.35 × 1011  250  3.12  2 × 1021  1.753 × 1012  200  100  −6.14  4.0  1.4  M27  250  4.0  2.35 × 1011  250  3.08  2 × 1021  1.864 × 1012  200  100  −6.36  3.4  1.4  M28  250  4.0  2.35 × 1011  250  3.05  2 × 1021  1.95 × 1012  60  65  −7.07  3.5  1.23  M29  250  4.0  2.35 × 1011  250  3.35  2 × 1021  1.23 × 1012  60  65  −5.40  7.1  1.23  M30  250  2.0  1.09 × 1012  250  3.20  2 × 1021  1.55 × 1012  60  65  −6.21  2.6  0.84  M31  250  2.0  1.15 × 1012  250  3.20  2 × 1021  1.55 × 1012  60  100  −5.67  3.8  1.35  Model name  $$E_{{\rm{um}}}^{\rm{*}}$$  $$V_{{\rm{um}}}^{\rm{*}}$$  Aum  $$E_{{\rm{lm}}}^*$$  $$V_{{\rm{lm}}}^*$$  ηlm(670)  Alm  t  H  $${\skew5\dot{J}_2}$$  $${\rm{\Delta RSL}}_{\rm{L}}^{{\rm{Bar}},{\rm{Bon}}}$$  $${\rm{\Delta RSL}}_6^{{\rm{Kar}},{\rm{Hal}}}$$  M1  250  2.0  1.09 × 1012  250  3.5  1021  4.88 × 1011  60  65  –6.0  1.8  0.77  M2  250  3.0  4.78 × 1011  250  3.5  1021  4.88 × 1011  60  65  −6.09  3.0  0.96  M3  250  4.0  2.35 × 1011  250  3.5  1021  4.88 × 1011  60  65  −6.13  4.3  1.18  M4  300  4.0  6.94 × 109  250  3.5  1021  4.88 × 1011  60  65  −6.09  3.8  1.10  M5  250  4.0  2.35 × 1011  200  3.05  1021  2.42 × 1013  60  65  −6.51  3.2  1.17  M6  250  4.0  2.35 × 1011  250  3.425  1021  5.48 × 1011  60  65  −6.53  3.4  1.17  M7  250  4.0  2.35 × 1011  300  3.80  1021  1.24 × 1010  60  65  −6.54  3.5  1.17  M8  250  4.0  2.35 × 1011  250  3.475  1021  5.08 × 1011  60  65  −6.26  4.0  1.18  M9  250  4.0  2.35 × 1011  250  3.2  2 × 1021  1.55 × 1012  60  65  −6.24  5.5  1.23  M10  250  4.0  2.35 × 1011  250  2.85  5 × 1021  6.64 × 1012  60  65  −6.28  6.4  1.25  M11  250  4.0  2.35 × 1011  250  2.6  1022  1.95 × 1013  60  65  −6.23  6.9  1.25  M12  250  4.0  2.35 × 1011  250  2.33  2 × 1022  5.91 × 1013  60  65  −6.27  6.9  1.24  M13  250  4.0  2.35 × 1011  250  1.95  5 × 1022  2.65 × 1014  60  65  −6.26  6.4  1.23  M14  250  4.0  2.35 × 1011  250  3.515  1021  4.77 × 1011  60  65  −6.24  4.2  1.17  M15  250  4.0  2.35 × 1011  250  3.225  2 × 1021  1.49 × 1012  60  65  −6.27  5.5  1.23  M16  250  4.0  2.35 × 1011  250  2.88  5 × 1021  6.34 × 1012  60  65  −6.24  6.5  1.24  M17  250  4.0  2.35 × 1011  250  2.62  1022  1.89 × 1013  60  65  −6.25  6.9  1.24  M18  250  4.0  2.35 × 1011  250  2.36  2 × 1022  5.64 × 1013  60  65  −6.22  6.9  1.23  M19  250  4.0  2.35 × 1011  250  1.985  5 × 1022  2.51 × 1014  60  65  −6.21  6.6  1.22  M20  250  4.0  2.35 × 1011  250  3.2  2 × 1021  1.55 × 1012  100  65  −6.08  4.6  1.23  M21  250  4.0  2.35 × 1011  250  3.17  2 × 1021  1.623 × 1012  100  65  −6.26  4.2  1.23  M22  250  4.0  2.35 × 1011  250  3.2  2 × 1021  1.55 × 1012  60  100  −5.88  5.6  1.61  M23  250  4.0  2.35 × 1011  250  3.15  2 × 1021  1.674 × 1012  60  100  −6.19  4.9  1.61  M24  250  4.0  2.35 × 1011  250  3.125  2 × 1021  1.739 × 1012  60  100  −6.35  4.6  1.61  M25  250  4.0  2.35 × 1011  250  3.2  2 × 1021  1.55 × 1012  200  100  −5.66  5.0  1.4  M26  250  4.0  2.35 × 1011  250  3.12  2 × 1021  1.753 × 1012  200  100  −6.14  4.0  1.4  M27  250  4.0  2.35 × 1011  250  3.08  2 × 1021  1.864 × 1012  200  100  −6.36  3.4  1.4  M28  250  4.0  2.35 × 1011  250  3.05  2 × 1021  1.95 × 1012  60  65  −7.07  3.5  1.23  M29  250  4.0  2.35 × 1011  250  3.35  2 × 1021  1.23 × 1012  60  65  −5.40  7.1  1.23  M30  250  2.0  1.09 × 1012  250  3.20  2 × 1021  1.55 × 1012  60  65  −6.21  2.6  0.84  M31  250  2.0  1.15 × 1012  250  3.20  2 × 1021  1.55 × 1012  60  100  −5.67  3.8  1.35  Notes: The GIA-related observables predicted for viscosity models M14–M19 are based on the IR20 ice model and all others are based on the IA20 ice model. The viscosity models M8–M13 are standard viscosity models for IA20, and those for M14–M19 are the ones for IR20. The average upper-mantle viscosity, $${\bar{\eta }_{{\rm{um}}}}$$, for these models is 2 × 1020 Pa s. The parameter values describing viscosity models are: $$E_{{\rm{um}}}^{\rm{*}}$$ (kJ mol−1), $$V_{{\rm{um}}}^{\rm{*}}$$(×10−6 m3mol−1), Aum (Pa s), $$E_{{\rm{lm}}}^*$$ (kJ mol−1), $$V_{{\rm{lm}}}^*$$(×10−6 m3mol−1), ηlm(670) (Pa s), Alm (Pa s), plate age t (Myr) and lithospheric thickness H (km). The three GIA-related observables are: $${\skew5\dot{J}_2}$$(×10−11yr−1), $${\rm{\Delta RSL}}_{\rm{L}}^{{\rm{Bar}},{\rm{Bon}}}$$ (m) and $${\rm{\Delta RSL}}_6^{{\rm{Kar}},{\rm{Hal}}}$$ (m)(see the text). View Large We examine the effect of the activation volume using the viscosity models of M1–M3 with $$V_{{\rm{um}}}^{\rm{*}}$$ = 2 × 10−6, 3 × 10−6 and 4 × 10−6 m3 mol−1, respectively. Variation in upper-mantle viscosity structure increases with increasing $$V_{{\rm{um}}}^{\rm{*}}$$ value (Fig. 3). Table 2 shows the three GIA-related observables based on these viscosity models. The $${\skew5\dot{J}_2}$$ values are approximately the same as those for a constant upper-mantle viscosity model U5, indicating that $${\skew5\dot{J}_2}$$ is less sensitive to the upper-mantle viscosity structure with $${\bar{\eta }_{{\rm{um}}}}$$ = 2 × 1020 Pa s. The $${\rm{\Delta RSL}}_{\rm{L}}^{{\rm{Bar}},{\rm{Bon}}}$$ and $${\rm{\Delta RSL}}_6^{{\rm{Kar}},{\rm{Hal}}}$$ values are sensitive to the upper-mantle viscosity structure and increase with increasing $$V_{{\rm{um}}}^{\rm{*}}$$ value. Consequently, the observationally derived three GIA-related observables prefer the viscosity models with $$V_{{\rm{um}}}^{\rm{*}}$$ = 3 × 10−6 and 4 × 10−6 m3 mol−1. Fig. 3 also shows the viscosity model with $$V_{{\rm{um}}}^{\rm{*}}$$ = 4 × 10−6 m3 mol−1 and $$E_{{\rm{um}}}^{\rm{*}}$$ = 300 kJ mol−1 (M4 in Table 2) used to examine the effect of the activation energy. The difference between the viscosity structures of M3 and M4 is small compared with that between M1 and M3. In fact, the three GIA-related observables for M4 are approximately the same as those for M3 with $$E_{{\rm{um}}}^{\rm{*}}$$ = 250 kJ mol−1. We therefore adopt $$E_{{\rm{um}}}^{\rm{*}}$$ = 250 kJ mol−1 in the following discussion. 3.3 Trade-off between activation energy and volume of the lower mantle We discuss the trade-off between the activation energy ($$E_{{\rm{lm}}}^*$$) and volume ($$V_{{\rm{lm}}}^*$$) of the lower mantle for the three GIA-related observables by employing the viscosity models M5, M6 and M7 with $$E_{{\rm{lm}}}^*$$ = 200, 250 and 300 kJ mol−1, respectively (Table 2). These viscosity models are constructed in such a way as to have almost an identical lower-mantle viscosity structure by considering the trade-off between the activation energy and volume. The trade-off is related to the effect of activation energy (temperature effect) and activation volume (pressure effect) on the lower-mantle viscosity structure that the viscosity decreases with increasing $$E_{{\rm{lm}}}^*$$ value and increases with increasing $$V_{{\rm{lm}}}^*$$ value. Consequently, $$V_{{\rm{lm}}}^*$$ values for M5, M6 and M7 are 3.05 × 10−6, 3.425 × 10−6 and 3.80 × 10−6 m3 mol−1, respectively (Table 2), and $$V_{{\rm{lm}}}^*$$ and $$E_{{\rm{lm}}}^*$$ values for these models give a simple relation of $$V_{{\rm{lm}}}^*$$ (m3 mol−1) ≈ 1.55 × 10−6 + 7.5 × 10−9$$E_{{\rm{lm}}}^*$$ (kJ mol−1). In fact, the three GIA-related observables predicted for these models are almost identical as shown in Table 2. In this study, we adopt $$E_{{\rm{lm}}}^*$$ = 250 kJ mol−1 and treat $$V_{{\rm{lm}}}^*$$ value as a variable on the lower-mantle viscosity structure. 3.4 Standard viscosity models and impact of seafloor age, t We first explain the standard viscosity models with H = 65 km, $${\bar{\eta }_{{\rm{um}}}}$$ = 2 × 1020 Pa s, ηlm(670) = (1, 2, 5, 10, 20, 50) × 1021 Pa s, $$E_{{\rm{um}}}^{\rm{*}}$$ = $$E_{{\rm{lm}}}^*$$ = 250 kJ mol−1, $$V_{{\rm{um}}}^{\rm{*}}$$ = 4 × 10−6 m3 mol−1 and t = 60 Myr for the IA20 and IR20 ice models, and discuss the impact of the plate age, t, on the three GIA-related observables. The $${\bar{\eta }_{{\rm{um}}}}$$ value may be a typical value for the oceanic upper-mantle viscosity (Lambeck et al.2014, 2017). The upper-mantle viscosity structure for these models is the same as that for M3, and the $$V_{{\rm{lm}}}^*$$ value for each standard viscosity model is determined in such a way as to explain the three GIA-related observables and to give the $${\skew5\dot{J}_2}$$ value of ∼ −6.25 × 10−11 yr−1. Fig. 4 shows the standard viscosity models, M8–M13 for the IA20 ice model and M14–M19 for IR20 and $$V_{{\rm{lm}}}^*$$ values and three GIA-related observables for these viscosity models are shown in Table 2. In Table 2, the GIA-related observables for M14–M19 are based on the IR20 ice model and all others are based on IA20. The $${\rm{\Delta RSL}}_{\rm{L}}^{{\rm{Bar}},{\rm{Bon}}}$$ and $${\rm{\Delta RSL}}_6^{{\rm{Kar}},{\rm{Hal}}}$$ values for these viscosity models are consistent with the observationally derived estimates. The lower-mantle viscosity structures for these models correspond to the permissible viscosity solutions inferred from the analyses with the two-layer lower-mantle viscosity model by Nakada & Okuno (2016). The inferred lower-mantle viscosity structure by Lau et al. (2016) may correspond to viscosity model M8 (M14) from the viewpoint of no viscosity jump at 670 km depth and 1022 − 1023 Pa s in the deep mantle. Figure 4. View largeDownload slide Standard viscosity models for the (a) IA20 and (b) IR20 ice models with H = 65 km, $${\bar{\eta }_{{\rm{um}}}}$$ = 2 × 1020 Pa s, ηlm(670) = (1, 2, 5, 10, 20, 50) × 1021 Pa s, $$E_{{\rm{um}}}^{\rm{*}}$$ = $$E_{{\rm{lm}}}^*$$ = 250 kJ mol−1, $$V_{{\rm{um}}}^{\rm{*}}$$ = 4 × 10−6 m3 mol−1 and t = 60 Myr (see Table 2). Figure 4. View largeDownload slide Standard viscosity models for the (a) IA20 and (b) IR20 ice models with H = 65 km, $${\bar{\eta }_{{\rm{um}}}}$$ = 2 × 1020 Pa s, ηlm(670) = (1, 2, 5, 10, 20, 50) × 1021 Pa s, $$E_{{\rm{um}}}^{\rm{*}}$$ = $$E_{{\rm{lm}}}^*$$ = 250 kJ mol−1, $$V_{{\rm{um}}}^{\rm{*}}$$ = 4 × 10−6 m3 mol−1 and t = 60 Myr (see Table 2). We discuss the impact of the plate age, t, by using viscosity models with ηlm(670) = 2 × 1021 Pa s. The following results are also true for other viscosity models with ηlm(670) = (1, 5, 10, 20, 50) × 1021 Pa s, although we do not show the results here. Fig. 5(a) shows the viscosity models of M9 (65, 60), M20 (65, 100), M22 (100, 60) and M25 (100, 200) with $$V_{{\rm{lm}}}^*$$ = 3.2 × 10−6 m3 mol−1. The former and latter numbers within the parenthesis show the lithospheric thickness (H) and plate age (t), respectively. The temperature distributions for M9, M20 and M25 are also shown in Fig. 5(b), in which the temperature at 2891 km depth (CMB) is 3264 K. We adopt viscosity models M20 (65, 100) and M22 (100, 60) to examine the differences between the three GIA-related observables predicted for these two models and those for a standard viscosity model M9 with H = 65 km and t = 60 Myr. Of course, we know that the two models M20 and M22 may contradict each other, since the older plate corresponds to the thinner one in M20 compared to M22. The $${\skew5\dot{J}_2}$$ value for M20 with H = 65 km and t = 100 Myr is −6.08 × 10−11 yr−1 and the magnitude is slightly smaller than that for a standard viscosity model of M9 with H = 65 km and t = 60 Myr. To obtain a value of $${\skew5\dot{J}_2}$$ ∼ −6.25 × 10−11 yr−1, we need $$V_{{\rm{lm}}}^*$$ ∼ 3.17 × 10−6 m3 mol−1 (M21 in Table 2). That is, change in $${\skew5\dot{J}_2}$$ of ∼0.18 × 10−11 yr−1, $$\Delta {\skew5\dot{J}_2}$$∼ 0.18 × 10−11 yr−1, requires change in $$V_{{\rm{lm}}}^*$$ of ∼0.03 × 10−6 m3 mol−1, $${\rm{\Delta }}V_{{\rm{lm}}}^{\rm{*}}$$∼0.03 × 10−6 m3 mol−1. Figure 5. View largeDownload slide (a) Viscosity models M9, M20, M22 and M25 with $$V_{{\rm{lm}}}^*$$ = 3.2 × 10−6 m3 mol−1, (b) temperature distributions for M9, M20 and M25 and (c) lower-mantle viscosity structures for M22, M23 and M24 (see Table 2). The lithospheric thickness is 65 km for M9 and M20 and 100 km for M22 and M25. The plate age (t) is 60 Myr for M9 and M22, 100 Myr for M20 and 200 Myr for M25. Figure 5. View largeDownload slide (a) Viscosity models M9, M20, M22 and M25 with $$V_{{\rm{lm}}}^*$$ = 3.2 × 10−6 m3 mol−1, (b) temperature distributions for M9, M20 and M25 and (c) lower-mantle viscosity structures for M22, M23 and M24 (see Table 2). The lithospheric thickness is 65 km for M9 and M20 and 100 km for M22 and M25. The plate age (t) is 60 Myr for M9 and M22, 100 Myr for M20 and 200 Myr for M25. We next discuss the case for viscosity models with H = 100 km and t = 60 Myr by employing viscosity models M22, M23 and M24 (see Table 2 for the $$V_{{\rm{lm}}}^*$$ values and Fig. 5(c) for the lower-mantle viscosity structures). The $${\skew5\dot{J}_2}$$ value for M22 with $$V_{{\rm{lm}}}^*$$ = 3.2 × 10−6 m3 mol−1 is −5.88 × 10−11 yr−1, and we need $$V_{{\rm{lm}}}^*$$ ∼ (3.15 − 3.125) × 10−6 m3 mol−1 (M23 and M24) to obtain the rate of ∼ −6.25 × 10−11 yr−1, which is related to the reduction of lower-mantle viscosity from sections ‘G’ to ‘F’ of the black curve in Fig. 2(b). This yields $${\rm{\Delta }}V_{{\rm{lm}}}^{\rm{*}}$$ ∼ (0.05 − 0.075) × 10−6 m3 mol−1 for $$\Delta {\skew5\dot{J}_2}$$ ∼ 0.36 × 10−11 yr−1. In the case of H = 100 km and t = 200 Myr, the rate for M25 is −5.66 × 10−11 yr−1 and we require $$V_{{\rm{lm}}}^*$$ ∼ (3.12 − 3.08) × 10−6 m3 mol−1 (M26 and M27) for the rate of ∼ −6.25 × 10−11 yr−1, and $${\rm{\Delta }}V_{{\rm{lm}}}^{\rm{*}}$$ ∼ (0.08 − 0.12) × 10−6 m3 mol−1 for $$\Delta {\skew5\dot{J}_2}$$∼ 0.59 × 10−11 yr−1. The $${\rm{\Delta RSL}}_{\rm{L}}^{{\rm{Bar}},{\rm{Bon}}}$$ and $${\rm{\Delta RSL}}_6^{{\rm{Kar}},{\rm{Hal}}}$$ values for viscosity models M20–M27 satisfy the observationally derived estimates as shown in Table 2. The $${\rm{\Delta RSL}}_{\rm{L}}^{{\rm{Bar}},{\rm{Bon}}}$$ value is slightly sensitive to the $${\rm{\Delta }}V_{{\rm{lm}}}^{\rm{*}}$$ as for $${\skew5\dot{J}_2}$$. On the other hand, the $${\rm{\Delta RSL}}_6^{{\rm{Kar}},{\rm{Hal}}}$$ is insensitive to $${\rm{\Delta }}V_{{\rm{lm}}}^{\rm{*}}$$ for a fixed upper-mantle rheological structure, while the value is sensitive to the upper-mantle viscosity structure and lithospheric thickness. This is consistent with the result by Nakada & Okuno (2016) that the differential RSL change for Karumba and Halifax Bay only yields an effective lower-mantle viscosity higher than (2 − 3) × 1021 Pa s. These numerical experiments show that the upper-mantle viscosity structure depending on the plate age, t, has an impact on the three GIA-related observables. However, change in $${\skew5\dot{J}_2}$$ ($$\Delta {\skew5\dot{J}_2}$$) for an adopted upper-mantle viscosity model is ∼0.6 × 10−11 yr−1 at most, which is adjusted by $${\rm{\Delta }}V_{{\rm{lm}}}^{\rm{*}}$$ ∼ 0.1 × 10−6 m3 mol−1 corresponding to variations in the lower-mantle viscosity structure shown in Fig. 5(c). Also, the observationally derived estimates for $${\rm{\Delta RSL}}_{\rm{L}}^{{\rm{Bar}},{\rm{Bon}}}$$ and $${\rm{\Delta RSL}}_6^{{\rm{Kar}},{\rm{Hal}}}$$ are consistent with the values predicted for the viscosity models M20–M27. These results are also applicable to those for other viscosity models with ηlm(670) = (1, 5, 10, 20, 50) × 1021 Pa s. Moreover, the $${\skew5\dot{J}_2}$$ sensitivity to lower-mantle viscosity takes a parabolic form shown in Fig. 2(b) for the simple three-layer viscosity model with upper-mantle rheological parameters (H and ηum) of 50 ≤ H ≤ 200 km and 1020 ≤ ηum ≤ 1021 Pa s (e.g. Mitrovica & Milne 1998; Nakada & Okuno 2003; Nakada et al. 2015). In that case, the impact of $${\rm{\Delta }}V_{{\rm{lm}}}^{\rm{*}}$$ on $$\Delta {\skew5\dot{J}_2}$$ would be nearly equal to that examined in this section. That is, the upper-mantle temperature distribution depending on the plate age t would have little impact on the inference of the lower-mantle viscosity structure described by eqs (2) –(4). We therefore adopt t = 60 Myr in the following discussion. 3.5 Dependence on the chosen ice history model We examine the impacts of adopted ice history models on the three GIA-related observables by using standard viscosity models for the IA20 ice model (M8–M13 in Table 2) and those for the IR20 ice model (M14–M19 in Table 2). Table 3 shows the GIA-related observables based on the IA10, IA30 and IR20 ice models and viscosity models M8–M13, and those based on the IR10, IR30 and IA20 ice models and viscosity models M14–M19. The $${\skew5\dot{J}_2}$$value for IA10, $$\skew5\dot{J}_2^{{\rm{IA}}10}$$, is approximated using the value for IA20, $$\skew5\dot{J}_2^{{\rm{IA}}20}$$, by $$\skew5\dot{J}_2^{{\rm{IA}}10}$$ ∼ $$\skew5\dot{J}_2^{{\rm{IA}}20}$$ + 0.3 × 10−11 yr−1 (see Table 2). Also, we get $$\skew5\dot{J}_2^{{\rm{IA}}30}$$ ∼$$\skew5\dot{J}_2^{{\rm{IA}}20}$$ − 0.3 × 10−11 yr−1. To obtain the rate of ∼ −6.25 × 10−11 yr−1 for IA10, we require a change in $$V_{{\rm{lm}}}^*$$ ($${\rm{\Delta V}}_{{\rm{lm}}}^{\rm{*}}$$) of ∼ −0.05 × 10−6 m3 mol−1 for the standard viscosity models for IA20 (relation from sections ‘G’ to ‘F’ of the black curve in Fig. 2b). The $${\rm{\Delta }}V_{{\rm{lm}}}^{\rm{*}}$$ value for IA30 is about 0.05 × 10−6 m3 mol−1 (relation from sections ‘E’ to ‘F’ of the black curve in Fig. 2b). The value of $${\rm{\Delta RSL}}_6^{{\rm{Kar}},{\rm{Hal}}}$$ is almost insensitive to the ice model and about 1.25 m. On the other hand, the $${\rm{\Delta RSL}}_{\rm{L}}^{{\rm{Bar}},{\rm{Bon}}}$$ value increases with a decreasing Antarctic ESL component (ESLSH), and it is larger than 6 m for IA10 with ESLSH∼10 m (ESLNH∼120 m) and smaller than 5 m for IA30 with ESLSH∼30 m (ESLNH∼100 m). However, the $${\rm{\Delta RSL}}_{\rm{L}}^{{\rm{Bar}},{\rm{Bon}}}$$ values for IA10 and IA30 satisfy the observationally derived estimate. For IR10 and IR30, we get similar sensitivities of the GIA-related observables to the ice model. For example, the relations between$${\skew5\dot{J}_2}$$ and an ice model are $$\skew5\dot{J}_2^{{\rm{IR}}10}$$ ∼$$\skew5\dot{J}_2^{{\rm{IR}}20}$$ + 0.35 × 10−11 yr−1 and $$\skew5\dot{J}_2^{{\rm{IR}}30}$$ ∼$$\skew5\dot{J}_2^{{\rm{IR}}20}$$ − 0.35 × 10−11 yr−1. Table 3. Three GIA-related observables, $${\skew5\dot{J}_2}$$ (×10−11yr−1), $${\rm{\Delta RSL}}_{\rm{L}}^{{\rm{Bar}},{\rm{Bon}}}$$ (m) and $${\rm{\Delta RSL}}_6^{{\rm{Kar}},{\rm{Hal}}}$$ (m), based on the IA10, IA30 and IR20 ice models and standard viscosity models M8–M13, and those based on the IR10, IR30 and IA20 ice models and standard viscosity models M14–M19. Model name  $${\skew5\dot{J}_2}$$  $${\rm{\Delta RSL}}_{\rm{L}}^{{\rm{Bar}},{\rm{Bon}}}$$  $${\rm{\Delta RSL}}_6^{{\rm{Kar}},{\rm{Hal}}}$$  $${\skew5\dot{J}_2}$$  $${\rm{\Delta RSL}}_{\rm{L}}^{{\rm{Bar}},{\rm{Bon}}}$$  $${\rm{\Delta RSL}}_6^{{\rm{Kar}},{\rm{Hal}}}$$  $${\skew5\dot{J}_2}$$  $${\rm{\Delta RSL}}_{\rm{L}}^{{\rm{Bar}},{\rm{Bon}}}$$  $${\rm{\Delta RSL}}_6^{{\rm{Kar}},{\rm{Hal}}}$$    IA10  IA10  IA10  IA30  IA30  IA30  IR20  IR20  IR20  M8  −5.98  6.2  1.17  −6.56  1.8  1.18  −6.46  3.7  1.17  M9  −5.96  7.7  1.23  −6.54  3.2  1.25  −6.42  5.2  1.23  M10  −6.0  8.6  1.25  −6.57  4.1  1.26  −6.43  6.2  1.25  M11  −5.96  9.0  1.24  −6.53  4.6  1.26  −6.39  6.6  1.24  M12  −5.99  9.0  1.23  −6.57  4.7  1.25  −6.42  6.6  1.23  M13  −5.98  8.4  1.22  −6.56  4.3  1.25  −6.41  6.1  1.23  Model name  $${\skew5\dot{J}_2}$$  $${\rm{\Delta RSL}}_{\rm{L}}^{{\rm{Bar}},{\rm{Bon}}}$$  $${\rm{\Delta RSL}}_6^{{\rm{Kar}},{\rm{Hal}}}$$  $${\skew5\dot{J}_2}$$  $${\rm{\Delta RSL}}_{\rm{L}}^{{\rm{Bar}},{\rm{Bon}}}$$  $${\rm{\Delta RSL}}_6^{{\rm{Kar}},{\rm{Hal}}}$$  $${\skew5\dot{J}_2}$$  $${\rm{\Delta RSL}}_{\rm{L}}^{{\rm{Bar}},{\rm{Bon}}}$$  $${\rm{\Delta RSL}}_6^{{\rm{Kar}},{\rm{Hal}}}$$    IR10  IR10  IR10  IR30  IR30  IR30  IA20  IA20  IA20  M14  −5.87  6.6  1.17  −6.61  1.8  1.19  −6.06  4.5  1.18  M15  −5.90  7.8  1.22  −6.63  3.1  1.25  −6.10  5.8  1.24  M16  −5.90  8.8  1.24  −6.58  4.1  1.26  −6.08  6.8  1.25  M17  −5.90  9.1  1.23  −6.60  4.5  1.26  −6.10  7.0  1.25  M18  −5.87  9.2  1.23  −6.56  4.6  1.25  −6.08  7.1  1.24  M19  −5.87  8.8  1.21  −6.56  4.3  1.24  −6.07  6.8  1.23  Model name  $${\skew5\dot{J}_2}$$  $${\rm{\Delta RSL}}_{\rm{L}}^{{\rm{Bar}},{\rm{Bon}}}$$  $${\rm{\Delta RSL}}_6^{{\rm{Kar}},{\rm{Hal}}}$$  $${\skew5\dot{J}_2}$$  $${\rm{\Delta RSL}}_{\rm{L}}^{{\rm{Bar}},{\rm{Bon}}}$$  $${\rm{\Delta RSL}}_6^{{\rm{Kar}},{\rm{Hal}}}$$  $${\skew5\dot{J}_2}$$  $${\rm{\Delta RSL}}_{\rm{L}}^{{\rm{Bar}},{\rm{Bon}}}$$  $${\rm{\Delta RSL}}_6^{{\rm{Kar}},{\rm{Hal}}}$$    IA10  IA10  IA10  IA30  IA30  IA30  IR20  IR20  IR20  M8  −5.98  6.2  1.17  −6.56  1.8  1.18  −6.46  3.7  1.17  M9  −5.96  7.7  1.23  −6.54  3.2  1.25  −6.42  5.2  1.23  M10  −6.0  8.6  1.25  −6.57  4.1  1.26  −6.43  6.2  1.25  M11  −5.96  9.0  1.24  −6.53  4.6  1.26  −6.39  6.6  1.24  M12  −5.99  9.0  1.23  −6.57  4.7  1.25  −6.42  6.6  1.23  M13  −5.98  8.4  1.22  −6.56  4.3  1.25  −6.41  6.1  1.23  Model name  $${\skew5\dot{J}_2}$$  $${\rm{\Delta RSL}}_{\rm{L}}^{{\rm{Bar}},{\rm{Bon}}}$$  $${\rm{\Delta RSL}}_6^{{\rm{Kar}},{\rm{Hal}}}$$  $${\skew5\dot{J}_2}$$  $${\rm{\Delta RSL}}_{\rm{L}}^{{\rm{Bar}},{\rm{Bon}}}$$  $${\rm{\Delta RSL}}_6^{{\rm{Kar}},{\rm{Hal}}}$$  $${\skew5\dot{J}_2}$$  $${\rm{\Delta RSL}}_{\rm{L}}^{{\rm{Bar}},{\rm{Bon}}}$$  $${\rm{\Delta RSL}}_6^{{\rm{Kar}},{\rm{Hal}}}$$    IR10  IR10  IR10  IR30  IR30  IR30  IA20  IA20  IA20  M14  −5.87  6.6  1.17  −6.61  1.8  1.19  −6.06  4.5  1.18  M15  −5.90  7.8  1.22  −6.63  3.1  1.25  −6.10  5.8  1.24  M16  −5.90  8.8  1.24  −6.58  4.1  1.26  −6.08  6.8  1.25  M17  −5.90  9.1  1.23  −6.60  4.5  1.26  −6.10  7.0  1.25  M18  −5.87  9.2  1.23  −6.56  4.6  1.25  −6.08  7.1  1.24  M19  −5.87  8.8  1.21  −6.56  4.3  1.24  −6.07  6.8  1.23  View Large We next examine the three GIA-related observables based on the IR20 ice model and standard viscosity models M8–M13 (Table 3). The differential RSL changes at the LGM and 6 kyr BP are almost the same as those for the IA20 ice model as inferred from the difference of (0.02–0.04) × 10−6 m3 mol−1 between the $$V_{{\rm{lm}}}^*$$ values for the standard viscosity models for IA20 and IR20 (Tables 2 and 3). Also, the $$\skew5\dot{J}_2^{{\rm{IR}}20}$$ value is given by $$\skew5\dot{J}_2^{{\rm{IR}}20}$$∼$$\skew5\dot{J}_2^{{\rm{IA}}20}$$ − (0.15–0.2) × 10−11 yr−1. Summarizing these numerical experiments, we may conclude that the GIA-induced $${\skew5\dot{J}_2}$$ is less sensitive to the ice model although a change in $$V_{{\rm{lm}}}^*$$ ($${\rm{\Delta }}V_{{\rm{lm}}}^{\rm{*}}$$) of ∼0.05 × 10−6 m3 mol−1 is required for a change in the $${\skew5\dot{J}_2}$$ magnitude ($$\Delta {\skew5\dot{J}_2}$$) of ∼0.3 × 10−11 yr−1. The differential RSL change for Karumba and Halifax Bay is almost insensitive to the ice model. The value of $${\rm{\Delta RSL}}_{\rm{L}}^{{\rm{Bar}},{\rm{Bon}}}$$ increases with an increasing Antarctic ESL component, but the observationally derived estimate of (0–25) m is obtained for the ice and viscosity models adopted in this section. 4 INFERENCE OF VISCOSITY AT 670 KM DEPTH, ηlm(670) 4.1 $${\skew5\dot{J}_n}$$ for n = 3–6 In inferring mantle viscosity from the GIA-induced $${\skew5\dot{J}_n}$$ for n > 2, it would be desirable to evaluate the rates based on the permissible viscosity solution satisfying the GIA-induced $${\skew5\dot{J}_2}$$ of −(6.0 − 6.5) × 10−11 yr−1, which depends on the ice model as inferred from the M8 to M13 viscosity models for IA20 and M14 to M19 for IR20, respectively. We first discuss this point based on the rates for n ≥ 2 predicted for IA20. Fig. 6 shows the rates as a function of $$V_{{\rm{lm}}}^*$$ value for viscosity models with ηlm(670) = (1, 2, 5, 10, 20, 50) × 1021 Pa s, H = 65 km,$${\bar{\eta }_{{\rm{um}}}}$$ = 2 × 1020 Pa s and t = 60 Myr. The standard viscosity model with ηlm (670) = 2 × 1021 Pa s for IA20 (M9), for example, corresponds to the model with $$V_{{\rm{lm}}}^*$$ = 3.2 × 10−6 m3 mol−1 in Fig. 6(b). These figures indicate that the rates for n > 2 are almost insensitive to the $$V_{{\rm{lm}}}^*$$ value for viscosity models satisfying the GIA-induced $${\skew5\dot{J}_2}$$ of −(6.0 − 6.5) × 10−11 yr−1. For example, we consider the rates for ηlm(670) = 1021 Pa s (Fig. 6a). The viscosity models with $$V_{{\rm{lm}}}^*$$ ∼ (3.424 − 3.55) × 10−6 m3 mol−1 explain the observationally derived $${\skew5\dot{J}_2}$$ value, and the $${\skew5\dot{J}_n}$$ (n > 2) for such a range of $$V_{{\rm{lm}}}^*$$ is nearly constant. This is also true for the models with ηlm(670) = (2, 5, 10, 20, 50) × 1021 Pa s. Figure 6. View largeDownload slide GIA-induced $${\skew5\dot{J}_n}$$ (n ≥ 2) as a function of $$V_{{\rm{lm}}}^*$$ value for viscosity models with $$E_{{\rm{um}}}^{\rm{*}}$$ = 250 kJ mol−1, ηlm(670) = (1, 2, 5, 10, 20, 50) × 1021 Pa s, H = 65 km,$${\bar{\eta }_{{\rm{um}}}}$$ = 2 × 1020 Pa s and t = 60 Myr and the IA20 ice model. The shaded region shows the range for GIA-induced $${\skew5\dot{J}_2}$$ of −(6.0 − 6.5) × 10−11 yr−1. Figure 6. View largeDownload slide GIA-induced $${\skew5\dot{J}_n}$$ (n ≥ 2) as a function of $$V_{{\rm{lm}}}^*$$ value for viscosity models with $$E_{{\rm{um}}}^{\rm{*}}$$ = 250 kJ mol−1, ηlm(670) = (1, 2, 5, 10, 20, 50) × 1021 Pa s, H = 65 km,$${\bar{\eta }_{{\rm{um}}}}$$ = 2 × 1020 Pa s and t = 60 Myr and the IA20 ice model. The shaded region shows the range for GIA-induced $${\skew5\dot{J}_2}$$ of −(6.0 − 6.5) × 10−11 yr−1. We next examine the sensitivity of the $${\skew5\dot{J}_n}$$ (n > 2) to the ice model. Table 4 shows the $${\skew5\dot{J}_n}$$ (n > 2) based on the IA20 and IR20 ice models (ESLSH∼20 m) and standard viscosity models for IA20 (M8–M13) and IR20 (M14–M19), respectively. We consider the rates for M8 and M14 with ηlm(670) = 1021 Pa s. The $$\skew5\dot{J}_n^{{\rm{IA}}20}$$ value for M8 is almost the same as the $$\skew5\dot{J}_n^{{\rm{IA}}20}$$ for M14. For example, the $$\skew5\dot{J}_3^{{\rm{IA}}20}$$ is −1.25 × 10−11 yr−1 for M8 and −1.21 × 10−11 yr−1 for M14. However, $$\skew5\dot{J}_n^{{\rm{IA}}20}$$ is significantly different from $$\skew5\dot{J}_n^{{\rm{IR}}20}$$. For example, the $${\skew5\dot{J}_4}$$ value for M8 is −2.67 × 10−11 yr−1 for IA20 and −3.16 × 10−11 yr−1 for IR20. Table 4. GIA-induced $${\skew5\dot{J}_n}$$ of n > 2 (×10−11 yr−1) based on the IA20 and IR20 ice models and standard viscosity models M8 to M13 (for IA20) and M14 to M19 (for IR20). Model name  $${\skew5\dot{J}_3}$$ (IA20)  $${\skew5\dot{J}_4}$$ (IA20)  $${\skew5\dot{J}_5}$$ (IA20)  $${\skew5\dot{J}_6}$$ (IA20)  $${\skew5\dot{J}_3}$$ (IR20)  $${\skew5\dot{J}_4}$$ (IR20)  $${\skew5\dot{J}_5}$$ (IR20)  $${\skew5\dot{J}_6}$$ (IR20)  M8  –1.25  –2.67  4.17  1.40  –0.55  –3.16  4.66  0.56  M9  –1.17  –2.49  3.92  1.35  –0.55  –2.90  4.33  0.66  M10  –1.08  –2.27  3.56  1.25  –0.54  –2.60  3.88  0.70  M11  –1.03  –2.12  3.31  1.18  –0.53  –2.43  3.60  0.69  M12  –1.01  –2.04  3.15  1.13  –0.52  –2.32  3.42  0.67  M13  –1.01  –1.98  3.02  1.09  –0.52  –2.26  3.28  0.66  Model name  $${\skew5\dot{J}_3}$$ (IA20)  $${\skew5\dot{J}_4}$$ (IA20)  $${\skew5\dot{J}_5}$$ (IA20)  $${\skew5\dot{J}_6}$$ (IA20)  $${\skew5\dot{J}_3}$$ (IR20)  $${\skew5\dot{J}_4}$$ (IR20)  $${\skew5\dot{J}_5}$$ (IR20)  $${\skew5\dot{J}_6}$$ (IR20)  M14  –1.21  –2.63  4.14  1.40  –0.54  –3.11  4.63  0.57  M15  –1.15  –2.47  3.90  1.35  –0.54  –2.87  4.30  0.67  M16  –1.05  –2.24  3.53  1.24  –0.53  –2.56  3.84  0.70  M17  –1.01  –2.10  3.29  1.17  –0.52  –2.40  3.58  0.69  M18  –0.98  –2.00  3.12  1.12  –0.51  –2.28  3.38  0.67  M19  –0.97  –1.95  2.99  1.08  –0.50  –2.22  3.24  0.66  Model name  $${\skew5\dot{J}_3}$$ (IA20)  $${\skew5\dot{J}_4}$$ (IA20)  $${\skew5\dot{J}_5}$$ (IA20)  $${\skew5\dot{J}_6}$$ (IA20)  $${\skew5\dot{J}_3}$$ (IR20)  $${\skew5\dot{J}_4}$$ (IR20)  $${\skew5\dot{J}_5}$$ (IR20)  $${\skew5\dot{J}_6}$$ (IR20)  M8  –1.25  –2.67  4.17  1.40  –0.55  –3.16  4.66  0.56  M9  –1.17  –2.49  3.92  1.35  –0.55  –2.90  4.33  0.66  M10  –1.08  –2.27  3.56  1.25  –0.54  –2.60  3.88  0.70  M11  –1.03  –2.12  3.31  1.18  –0.53  –2.43  3.60  0.69  M12  –1.01  –2.04  3.15  1.13  –0.52  –2.32  3.42  0.67  M13  –1.01  –1.98  3.02  1.09  –0.52  –2.26  3.28  0.66  Model name  $${\skew5\dot{J}_3}$$ (IA20)  $${\skew5\dot{J}_4}$$ (IA20)  $${\skew5\dot{J}_5}$$ (IA20)  $${\skew5\dot{J}_6}$$ (IA20)  $${\skew5\dot{J}_3}$$ (IR20)  $${\skew5\dot{J}_4}$$ (IR20)  $${\skew5\dot{J}_5}$$ (IR20)  $${\skew5\dot{J}_6}$$ (IR20)  M14  –1.21  –2.63  4.14  1.40  –0.54  –3.11  4.63  0.57  M15  –1.15  –2.47  3.90  1.35  –0.54  –2.87  4.30  0.67  M16  –1.05  –2.24  3.53  1.24  –0.53  –2.56  3.84  0.70  M17  –1.01  –2.10  3.29  1.17  –0.52  –2.40  3.58  0.69  M18  –0.98  –2.00  3.12  1.12  –0.51  –2.28  3.38  0.67  M19  –0.97  –1.95  2.99  1.08  –0.50  –2.22  3.24  0.66  View Large We further consider the sensitivity using ice models IA20 (ESLSH∼20 m) and IA30 (ESLSH∼30 m) and viscosity models with $$V_{{\rm{lm}}}^*$$ = 2.85 × 10−6 (M10) and 2.9 × 10−6 m3 mol−1 for ηlm (670) = 5 × 1021 Pa s (see Fig. 6c). The $$\skew5\dot{J}_2^{{\rm{IA}}20}$$ and $$\skew5\dot{J}_4^{{\rm{IA}}20}$$ values for M10 are −6.28 × 10−11 and −2.27 × 10−11 yr−1 and those for $$V_{{\rm{lm}}}^*$$ = 2.9 × 10−6 m3 mol−1 are −5.96 × 10−11 and −2.21 × 10−11 yr−1, respectively. That is, the difference between the $$\skew5\dot{J}_4^{{\rm{IA}}20}$$ for both viscosity models is significantly smaller than that for $$\skew5\dot{J}_2^{{\rm{IA}}20}$$. On the other hand, the $$\skew5\dot{J}_2^{{\rm{IA}}30}$$ and $$\skew5\dot{J}_4^{{\rm{IA}}30}$$ values are −6.57 × 10−11 and −2.93 × 10−11 yr−1 for M10, and −6.24 × 10−11 and −2.85 × 10−11 yr−1 for $$V_{{\rm{lm}}}^*$$ = 2.9 × 10−6 m3 mol−1, respectively. That is, the $$\skew5\dot{J}_4^{{\rm{IA}}30}$$ value is nearly similar for both viscosity models as obtained for the IA20 ice model. However, $$\skew5\dot{J}_4^{{\rm{IA}}30}$$ is significantly different from $$\skew5\dot{J}_4^{{\rm{IA}}20}$$. The results for the IA20 and IR20 ice models (Table 4) and those for the IA20 and IA30 ice models indicate that it would be possible to discuss preferred ice and viscosity models satisfying observationally derived $${\skew5\dot{J}_n}$$ (n > 2) by employing standard viscosity models M8–M13 for IA20. More recently, Nakada & Okuno (2017) estimated the GIA-induced $${\skew5\dot{J}_n}$$ (n = 3–6) by considering the geodetically derived $${\skew5\dot{J}_n}$$ by Cheng et al. (1997) and Cox & Chao (2002) and recent (after ∼1900) melting of glaciers and the Greenland and Antarctic ice sheets (Vaughan et al. 2013). The approach by Nakada & Okuno (2017) is essentially the same as that by Tosi et al. (2005) except for recent melting models adopted in both studies (see detailed discussion by Nakada & Okuno 2017). We should also point out that Devoti et al. (2001) inferred a value of the order of 1020 Pa s for the upper-mantle viscosity from the analyses using the geodetically derived $${\skew5\dot{J}_n}$$ (n = 2–6) and suggested an ongoing mass redistribution associated with mass instabilities in Greenland and Antarctica. In inferring mantle viscosity from the GIA-induced $${\skew5\dot{J}_n}$$ (n > 2), Nakada & Okuno (2017) evaluated uncertainties on the rates based on geodetic data and recent melting for the period of 1900–1990, and also by considering the GIA-induced $${\skew5\dot{J}_2}$$ of−(6.0 − 6.5) × 10−11yr−1 inferred from observationally derived $${\skew5\dot{J}_2}$$ for the periods of 1976–1990 and 2002–2001 (Roy & Peltier 2011; Cheng et al. 2013) and the melting rates for both periods (Vaughan et al. 2013, see table 2 in Nakada & Okuno 2017). In this study, we examine the GIA-induced $${\skew5\dot{J}_n}$$ (n = 3–6) considering the uncertainties obtained by Nakada & Okuno (2017; see Fig. 7). Figure 7. View largeDownload slide GIA-induced $${\skew5\dot{J}_n}$$ (n ≥ 2) based on the IA* ice models and standard viscosity models for IA20 (M8–M13), and GIA-induced $${\skew5\dot{J}_4}$$ and $${\skew5\dot{J}_6}$$ based on the geodetically derived $${\skew5\dot{J}_n}$$ by Cheng et al. (1997) and recent (after ∼1900) melting of glaciers and the Greenland and Antarctic ice sheets (Vaughan et al. 2013, see the text for uncertainties on the rates). Figure 7. View largeDownload slide GIA-induced $${\skew5\dot{J}_n}$$ (n ≥ 2) based on the IA* ice models and standard viscosity models for IA20 (M8–M13), and GIA-induced $${\skew5\dot{J}_4}$$ and $${\skew5\dot{J}_6}$$ based on the geodetically derived $${\skew5\dot{J}_n}$$ by Cheng et al. (1997) and recent (after ∼1900) melting of glaciers and the Greenland and Antarctic ice sheets (Vaughan et al. 2013, see the text for uncertainties on the rates). Fig. 7 shows the rates based on the IA10, IA20 and IA30 ice models and standard viscosity models for IA20 (M8–M13). The $${\skew5\dot{J}_4}$$ for IA20, for example, is denoted by (4, IA20). Here, we do not discuss the rates for n = 3 and 5 in inferring mantle viscosity because the odd zonal rates are uncertain due to weakness in the orbital geometry (Cheng et al. 1997 and see also Nakada & Okuno 2017). Fig. 7(a) shows the predicted GIA-induced $${\skew5\dot{J}_n}$$ for n = 2, 4 and 6 and observationally derived estimates for n = 4 and 6 obtained by Nakada & Okuno (2017) using geodetically derived data by Cheng et al. (1997) and recent melting by Vaughan et al. (2013). The observationally derived rate for n = 4 is consistent with the predicted rate regardless of chosen viscosity and ice model. On the other hand, the rate for n = 6 is consistent with the rates predicted for IA30 with ESLSH∼30 m. The inference of an ice model only from $${\skew5\dot{J}_6}$$ is, however, inconclusive if we consider the statement by Cheng et al. (1997) that the accuracy of the estimates of the zonal secular rates from long time-series multisatellite laser ranging data has been difficult to verify and only $${\skew5\dot{J}_2}$$ has been evaluated with confidence. That is, it would be safe to say that the present analysis for the GIA-induced $${\skew5\dot{J}_n}$$ (n > 2) cannot provide useful constraints on the ice model and the viscosity at 670 km depth, ηlm(670) (see also Devoti et al. 2001; Tosi et al. 2005). 4.2 Postglacial RSL changes at Southport, Bermuda and Everglades based on the standard viscosity models We examine postglacial RSL changes at Southport, Bermuda and Everglades in the intermediate region for the North American ice sheet (see Fig. 1a) based on the standard viscosity models for the IA20 and IR20 ice models in inferring the viscosity at 670 km depth, ηlm(670). These RSL changes would be sensitive to the ice model as well as the viscosity structure of the mantle, and therefore inference of the ηlm(670) value depends on an adopted ice model. In that case, our method employed in this study requires that we should discuss these RSL changes based on the standard viscosity models corresponding to a specific ice model, for example, IA10 [steps (i) and (ii) explained in Introduction]. The standard viscosity models, which should be defined for each ice model separately, are required to explain the three GIA-related observables and to give the $${\skew5\dot{J}_2}$$ value of ∼ −6.25 × 10−11 yr−1 (Section 3.4). Although we examined this point for the three GIA-related observables in Section 3.4, we briefly discuss the $$V_{{\rm{lm}}}^*$$ dependence of RSL changes at these sites and $${\skew5\dot{J}_2}$$ by employing viscosity models M28, M9 and M29 for ηlm(670) = 2 × 1021 Pa s with $$V_{{\rm{lm}}}^*$$ values of 3.05 × 10−6, 3.20 × 10−6 and 3.35 × 10−6 m3 mol−1, respectively (Table 2 and see also Fig. 8(a) for their lower-mantle viscosity structures). Figure 8. View largeDownload slide (a) Lower-mantle viscosity structures M9, M28 and M29 with $$V_{{\rm{lm}}}^*$$ values of 3.20 × 10−6, 3.05 × 10−6 and 3.35 × 10−6 m3 mol−1, respectively (see Table 1), and (b)–(d) observed postglacial RSL changes at Southport, Bermuda and Everglades and predicted RSL changes for the IA20 ice model and viscosity models M9, M28 and M29. The data sources for observed RSL changes are: https://www.ncdc.noaa.gov/data-access/paleoclimatology-data/datasets/paleoceanography for the three sites and recent data by Engelhart et al. (2011) for Southport. Figure 8. View largeDownload slide (a) Lower-mantle viscosity structures M9, M28 and M29 with $$V_{{\rm{lm}}}^*$$ values of 3.20 × 10−6, 3.05 × 10−6 and 3.35 × 10−6 m3 mol−1, respectively (see Table 1), and (b)–(d) observed postglacial RSL changes at Southport, Bermuda and Everglades and predicted RSL changes for the IA20 ice model and viscosity models M9, M28 and M29. The data sources for observed RSL changes are: https://www.ncdc.noaa.gov/data-access/paleoclimatology-data/datasets/paleoceanography for the three sites and recent data by Engelhart et al. (2011) for Southport. The $$\skew5\dot{J}_2^{{\rm{IA}}20}$$ ($${\skew5\dot{J}_2}$$ for IA20) values for M28 and M29 are −7.07 × 10−11 and −5.40 × 10−11 yr−1, respectively (−6.24 × 10−11 yr−1 for M9 with $$V_{{\rm{lm}}}^*$$ = 3.20 × 10−6 m3 mol−1); namely, $$\Delta {\skew5\dot{J}_2}$$ ∼ 1.7 × 10−11 yr−1 for $${\rm{\Delta V}}_{{\rm{lm}}}^{\rm{*}}$$ ∼ 0.3 × 10−6 m3 mol−1. By considering the relation for $$\Delta {\skew5\dot{J}_2}$$ and $${\rm{\Delta }}V_{{\rm{lm}}}^{\rm{*}}$$ and $$\skew5\dot{J}_2^{{\rm{IA}}10}$$ = −5.96 × 10−11 yr−1 for M9 (Table 3), we obtain the $$\skew5\dot{J}_2^{{\rm{IA}}10}$$ value of ∼−6.25 × 10−11 yr−1 for a viscosity model with $$V_{{\rm{lm}}}^*$$ ∼ 3.15 × 10−6 m3 mol−1 (see also Fig. 2b). Also, we obtain the $$\skew5\dot{J}_2^{{\rm{IA}}30}$$ of ∼−6.25 × 10−11yr−1 for a viscosity model with $$V_{{\rm{lm}}}^*$$ ∼ 3.25 × 10−6 m3 mol−1 ($$\skew5\dot{J}_2^{{\rm{IA}}30}$$ = −6.54 × 10−11yr−1 for M9). On the other hand, such a change in $$V_{{\rm{lm}}}^*$$ (∼0.3 × 10−6 m3 mol−1) has negligible impacts on postglacial RSL changes at Southport, Everglades and Bermuda as shown in Fig. 8 (the data sources for observed RSL changes are: https://www.ncdc.noaa.gov/data-access/paleoclimatology-data/datasets/paleoceanography for three sites and recent data by Engelhart et al. 2011 for Southport). This is true for other standard viscosity models for IA20 although we do not show the results here. That is, it is possible to discuss preferred viscosity and IA* ice models satisfying observed RSL changes at these sites by employing standard viscosity models M8–M13 for IA20. This is also true for the standard viscosity models for IR20. Fig. 9 shows the RSL changes at these sites based on the IA10, IA20 and IA30 ice models and viscosity models M8–M13 with $${\bar{\eta }_{{\rm{um}}}}$$ = 2 × 1020 Pa s. Here, we mainly discuss the RSL change at 6 kyr BP to examine the sensitivity of postglacial RSL change to the viscosity at 670 km depth, ηlm(670), and an ice model. The magnitude at 6 kyr BP for Southport slightly decreases with increasing ηlm(670) value (total change of ∼2 m) and decreases with decreasing ESL component of the NH ice sheets (ESLNH) (total change of ∼3.5 m). That is, the misfit between the observed and predicted RSL changes becomes slightly smaller for the model with higher ηlm(670) value and smaller ESLNH (larger ESLSH). However, the predicted RSL change is distinctly different from the observed one regardless of the viscosity and ice models. The ice model sensitivity of the RSL change is also true for RSL changes at Bermuda and Everglades. The RSL change at Bermuda is, however, more sensitive to the ηlm(670) value than that at Southport. The RSL change at Everglades is highly sensitive to the ηlm(670) value, and the observed RSL change is consistent with the predicted one for the models with ηlm(670) ≥ 2 × 1022 Pa s (viscosity models M12 and M13) and IA30 (ESLNH ∼ 100 m and ESLSH ∼ 30 m). These numerical experiments for the postglacial RSL change in the intermediate region for the North American ice sheet point out that the sensitivity to the ηlm(670) increases with distance of observation site from the edge of the ice sheet (see Fig. 1a) in the case of $${\bar{\eta }_{{\rm{um}}}}$$ = 2 × 1020 Pa s (we discuss this point based on the results in Fig. 12). Fig. 10 also shows the RSL changes based on the IR* ice models and standard viscosity models for IR20. The RSL changes for these models are almost the same as those for the standard viscosity models for IA20 (see also the $$V_{{\rm{lm}}}^*$$ values for the standard viscosity models with an identical ηlm(670) value, for example, M8 and M14). Figure 9. View largeDownload slide Observed RSL changes at Southport, Bermuda and Everglades and predicted ones based on the IA* ice models and viscosity models M8–M13 with $${\bar{\eta }_{{\rm{um}}}}$$ = 2 × 1020 Pa s. Figure 9. View largeDownload slide Observed RSL changes at Southport, Bermuda and Everglades and predicted ones based on the IA* ice models and viscosity models M8–M13 with $${\bar{\eta }_{{\rm{um}}}}$$ = 2 × 1020 Pa s. Figure 10. View largeDownload slide As in Fig. 7, except for the IR* ice models. Figure 10. View largeDownload slide As in Fig. 7, except for the IR* ice models. The results obtained in this section are shortly summarized as follows: the observed postglacial RSL changes at Southport, Bermuda and Everglades cannot be simultaneously explained by the RSL changes for the standard viscosity models with $${\bar{\eta }_{{\rm{um}}}}$$ = 2 × 1020 Pa s regardless of the chosen ice models adopted here, particularly for RSL changes at Southport and Bermuda. In the next section, we analyse these RSL changes using the viscosity models with $${\bar{\eta }_{{\rm{um}}}}$$ > 2 × 1020 Pa s. 4.3 Postglacial RSL changes at Southport, Bermuda and Everglades based on the viscosity models with $${\bar{\eta }_{{\rm{um}}}}$$ > 2 × 1020 Pa s In inferring the viscosity at 670 km depth from the RSL changes at these sites, we assume that the upper-mantle viscosity is smaller than 1021 Pa s and that the viscosity at 670 km depth is higher than 1021 Pa s, ηlm(670) ≥ 1021 Pa s. Of course, this assumption has no physical evidence. For the $$V_{{\rm{um}}}^{\rm{*}}$$ value of 4.0 × 10−6 m3 mol−1 adopted for standard viscosity models, however, the upper-mantle viscosity around ∼670 km depth becomes higher than 1021 Pa s in the case of $${\bar{\eta }_{{\rm{um}}}}$$ > 2 × 1020 Pa s. We therefore adopt $$V_{{\rm{um}}}^{\rm{*}}$$ = 2.0 × 10−6 m3 mol−1 producing nearly constant upper-mantle viscosity and examine the impact of the upper-mantle viscosity structure on the RSL change. We first examine its impact by employing viscosity models M9, M30 and M31 with ηlm(670) = 2 × 1021 Pa s and $${\bar{\eta }_{{\rm{um}}}}$$ = 2 × 1020 Pa s (Fig. 11a and Table 2). The $$V_{{\rm{um}}}^{\rm{*}}$$ value for M9 is 4.0 × 10−6 m3 mol−1 and that for M30 and M31 is 2.0 × 10−6 m3 mol−1. The elastic lithospheric thickness (H) is 65 km for M9 and M30 and 100 km for M31. The RSL changes at these sites may prefer the elastic lithospheric thickness of 100 km for a viscosity model with upper-mantle viscosity of ∼1021 Pa s and lower-mantle viscosity higher than 1022 Pa s (see figs 17 and 18 in Nakada & Okuno 2016). Fig. 11 shows the RSL changes based on the IA20 ice model and these viscosity models. The following results change insignificantly even if we adopt another viscosity model with ηlm(670) = (1, 5, 10, 20, 50) × 1021 Pa s. The difference between the RSL changes at 6 kyr BP for M9 and M30 is ∼1.5 m for Southport and nearly zero for Bermuda. The difference between those for M30 with H = 65 km and M31 with H = 100 km is also ∼1.5 m for Southport and are nearly zero for Bermuda and Everglades. Thus, the $$V_{{\rm{um}}}^{\rm{*}}$$ value and elastic lithospheric thickness affect the RSL changes at these sites, but these impacts are rather small compared with that for the viscosity jump at 670 km depth depending on the $${\bar{\eta }_{{\rm{um}}}}$$ value (Fig. 12). We therefore examine these RSL changes based on the viscosity models with $$V_{{\rm{um}}}^{\rm{*}}$$ = 2.0 × 10−6 m3 mol−1 and H = 100 km to discuss the viscosity jump at 670 km depth. Figure 11. View largeDownload slide (a) Upper-mantle viscosity structures M9, M30 and M31 with ηlm(670) = 2 × 1021 Pa s and $${\bar{\eta }_{{\rm{um}}}}$$ = 2 × 1020 Pa s, and (b)–(d) observed postglacial RSL changes at Southport, Bermuda and Everglades and predicted ones for the IA20 ice model and viscosity models M9, M30 and M31. The $$V_{{\rm{um}}}^{\rm{*}}$$ value for M9 is 4.0 × 10−6 m3 mol−1 and that for M30 and M31 is 2.0 × 10−6 m3 mol−1 (Table 2). Figure 11. View largeDownload slide (a) Upper-mantle viscosity structures M9, M30 and M31 with ηlm(670) = 2 × 1021 Pa s and $${\bar{\eta }_{{\rm{um}}}}$$ = 2 × 1020 Pa s, and (b)–(d) observed postglacial RSL changes at Southport, Bermuda and Everglades and predicted ones for the IA20 ice model and viscosity models M9, M30 and M31. The $$V_{{\rm{um}}}^{\rm{*}}$$ value for M9 is 4.0 × 10−6 m3 mol−1 and that for M30 and M31 is 2.0 × 10−6 m3 mol−1 (Table 2). Figure 12. View largeDownload slide RSL changes at 6 kyr BP based on the viscosity models with ηlm(670) = (1, 2, 5, 10, 20, 50) × 1021 Pa s and $${\bar{\eta }_{{\rm{um}}}}$$ = (2, 3, 4, 5, 7, 9) × 1020 Pa s and the IA* ice models. The lower-mantle viscosity structures are the same as those for the standard viscosity models for IA20 (M8–M13). The shaded regions show the observational constraints for RSL changes at 6 kyr BP: −8.0 ± 1.5, −6.0 ± 2.0 and −4.0 ± 1.0 m for Southport, Bermuda and Everglades, respectively. Figure 12. View largeDownload slide RSL changes at 6 kyr BP based on the viscosity models with ηlm(670) = (1, 2, 5, 10, 20, 50) × 1021 Pa s and $${\bar{\eta }_{{\rm{um}}}}$$ = (2, 3, 4, 5, 7, 9) × 1020 Pa s and the IA* ice models. The lower-mantle viscosity structures are the same as those for the standard viscosity models for IA20 (M8–M13). The shaded regions show the observational constraints for RSL changes at 6 kyr BP: −8.0 ± 1.5, −6.0 ± 2.0 and −4.0 ± 1.0 m for Southport, Bermuda and Everglades, respectively. Fig. 12 shows the RSL changes at 6 kyr BP based on the viscosity models with ηlm(670) = (1, 2, 5, 10, 20, 50) × 1021 Pa s and $${\bar{\eta }_{{\rm{um}}}}$$ = (2, 3, 4, 5, 7, 9) × 1020 Pa s and ice models IA10 (ESLNH∼120 m, ESLSH∼10 m), IA20 (ESLNH∼110 m, ESLSH∼20 m) and IA30 (ESLNH∼100 m, ESLSH∼30 m). The lower-mantle viscosity structures are the same as those for the standard viscosity models for IA20 (M8–M13). We adopt−8.0 ± 1.5,−6.0 ± 2.0 and−4.0 ± 1.0 m for RSL changes at 6 kyr BP for Southport, Bermuda and Everglades, respectively. The RSL changes at these sites increase with increasing ηlm(670) and $${\bar{\eta }_{{\rm{um}}}}$$ values. We consider the ηlm(670) dependency for a specific $${\bar{\eta }_{{\rm{um}}}}$$ value. The RSL changes at Southport and Bermuda for a higher $${\bar{\eta }_{{\rm{um}}}}$$ model are more dependent on the ηlm(670) value compared with those for a lower one. For example, the difference between the RSL changes at Southport for ηlm(670) = 1021 and 5 × 1022 Pa s is less than 1 m for the model with $${\bar{\eta }_{{\rm{um}}}}$$ = 2 × 1020 Pa s, but larger than 7 m for the model with 9 × 1020 Pa s. Of course, the dependency for $${\bar{\eta }_{{\rm{um}}}}$$ = 2 × 1020 Pa s is consistent with that for the standard viscosity models shown in Fig. 9. On the other hand, the ηlm(670) dependency for RSL change at Everglades is almost independent of $${\bar{\eta }_{{\rm{um}}}}$$ value and the difference between the RSL changes for ηlm(670) = 1021 and 5 × 1022 Pa s is about ∼4 m for all $${\bar{\eta }_{{\rm{um}}}}$$ values (see also Fig. 9). We summarize the permissible solutions for each ice model. The solutions for IA10 are as follows: $${\bar{\eta }_{{\rm{um}}}}$$ ∼ 9 × 1020 and ηlm(670)∼ 5 × 1022 Pa s for Southport, $${\bar{\eta }_{{\rm{um}}}}$$ ∼ (7 − 9) × 1020 and ηlm(670) ≥ 1022 Pa s for Bermuda, and $${\bar{\eta }_{{\rm{um}}}}$$ ≥ 4 × 1020 and ηlm(670) > 5 × 1021 Pa s for Everglades. Those for IA20 are: $${\bar{\eta }_{{\rm{um}}}}$$ ∼ 9 × 1020 and ηlm(670) ≥ 2 × 1022 Pa s for Southport, $${\bar{\eta }_{{\rm{um}}}}$$ ∼ (7 − 9) × 1020 and ηlm(670) > 6 × 1021 Pa s for Bermuda, and $${\bar{\eta }_{{\rm{um}}}}$$ ≥ 4 × 1020 and ηlm(670) > 3 × 1021 Pa s, and $${\bar{\eta }_{{\rm{um}}}}$$ ∼ 3 × 1020 and ηlm(670) ≥ 2 × 1022 Pa s for Everglades. The solutions for IA30 are nearly similar to those for IA20 and as follows: $${\bar{\eta }_{{\rm{um}}}}$$ ∼ (7 − 9) × 1020 and ηlm(670) ≥ 1022 Pa s for Southport, $${\bar{\eta }_{{\rm{um}}}}$$ ∼ (7 − 9) × 1020 and ηlm(670) ≥ 5 × 1021 Pa s, and $${\bar{\eta }_{{\rm{um}}}}$$ ∼ 5 × 1020 and ηlm(670) ≥ 2 × 1022 Pa s for Bermuda, and $${\bar{\eta }_{{\rm{um}}}}$$ ∼ (4 − 9) × 1020 and ηlm(670) ∼ (2 − 20) × 1021 Pa s, and $${\bar{\eta }_{{\rm{um}}}}$$ ∼ 3 × 1020 and ηlm(670) ≥ 7 × 1021 Pa s for Everglades. These results clearly indicate the trade-off between the ice and viscosity models, and the permissible ηlm(670) value decreases with increasing (decreasing) ESLSH (ESLNH) value. Here, we assume a laterally homogeneous viscosity structure although the permissible $${\bar{\eta }_{{\rm{um}}}}$$ value may be related to the distance of observation site from the edge of the ice sheet (Fig. 1a) and approximately decrease with increasing distance. Then, the permissible $${\bar{\eta }_{{\rm{um}}}}$$ and ηlm(670) values are as follows: (i) $${\bar{\eta }_{{\rm{um}}}}$$ ∼ 9 × 1020 and ηlm(670)∼ 5 × 1022 Pa s for IA10, (ii) $${\bar{\eta }_{{\rm{um}}}}$$ ∼ 9 × 1020 and ηlm(670) ∼ 2 × 1022 Pa s for IA20 and (iii) $${\bar{\eta }_{{\rm{um}}}}$$ ∼ (7 − 9) × 1020 and ηlm(670) ∼ 1022 Pa s for IA30. Fig. 13 shows the observed RSL changes and predicted ones based on the IA30 ice model and viscosity models with ηlm(670) = (2, 5, 10, 20) × 1021, $${\bar{\eta }_{{\rm{um}}}}$$ = (2, 3, 4, 5, 7, 9) × 1020 Pa s and H = 100 km. The observed RSL changes are generally consistent with the predicted ones for viscosity models with $${\bar{\eta }_{{\rm{um}}}}$$ ∼ (7 − 9) × 1020 and ηlm(670) ∼ 1022 Pa s. Although the inferred $${\bar{\eta }_{{\rm{um}}}}$$ and ηlm(670) values from postglacial RSL changes at only three sites may be preliminary, there is no doubt that the RSL changes in the intermediate region of the North American ice sheet have a crucial potential for inference of a viscosity jump at 670 km depth. Figure 13. View largeDownload slide Observed postglacial RSL changes at Southport, Bermuda and Everglades and predicted ones based on the IA30 ice model and viscosity models with ηlm(670) = (2, 5, 10, 20) × 1021, $${\bar{\eta }_{{\rm{um}}}}$$ = (2, 3, 4, 5, 7, 9) × 1020 Pa s and H = 100 km. Figure 13. View largeDownload slide Observed postglacial RSL changes at Southport, Bermuda and Everglades and predicted ones based on the IA30 ice model and viscosity models with ηlm(670) = (2, 5, 10, 20) × 1021, $${\bar{\eta }_{{\rm{um}}}}$$ = (2, 3, 4, 5, 7, 9) × 1020 Pa s and H = 100 km. 5 CONCLUDING REMARKS A variety of viscosity models described by different temperature and pressure distributions and activation energy and volume were employed in inferring a reliable mantle viscosity structure from selected GIA data sets. We first constructed standard viscosity models with $${\bar{\eta }_{{\rm{um}}}}$$ = 2 × 1020 Pa s (M8–M13 for the IA20 ice model and M14–M19 for IR20) satisfying the observationally derived three GIA-related observables: GIA-induced $${\skew5\dot{J}_2}$$, and differential RSL changes for the LGM sea levels at Barbados and Bonaparte Gulf and RSL changes at 6 kyr BP for Karumba and Halifax Bay (see Table 2 and Fig. 4). The $${\bar{\eta }_{{\rm{um}}}}$$ value for standard viscosity models may be a typical value for the oceanic upper-mantle viscosity (Lambeck et al.2014, 2017). These viscosity models, characterized by a viscosity of ∼1023 Pa s in the deep mantle, correspond to the permissible viscosity solutions for the two-layer lower-mantle viscosity model using the same GIA data sets by Nakada & Okuno (2016). Such a viscosity model with 1022 − 1023 Pa s in the deep mantle has already been reported by several GIA studies (Ivins et al. 1993; Mitrovica 1996; Vermeersen et al. 1997; Lau et al. 2016), and a more recently reported lower-mantle viscosity structure by Lau et al. (2016) may correspond to the M8 (M14) viscosity model from the viewpoint of no viscosity jump at 670 km depth. It would be fair to note that lower-mantle viscosity structures M8–M13 (M14–M19) may correspond to other representing structures for the lower-mantle viscosity of ηlm ≥ 1022 Pa s inferred using the simple three-layer viscosity model. Such viscosity models with ηlm ≥ 1022 Pa s have been inferred from secular rates of geopotential coefficients up to degree 8 (Tosi et al. 2005), far-field sea level data (Lambeck et al. 2014), GIA-induced $${\skew5\dot{J}_2}$$ (Nakada et al. 2015), GIA-induced $${\skew5\dot{J}_2}$$ and LGM sea levels at Barbados and Bonaparte Gulf (Nakada et al. 2016) and RSL observations, tilting of palaeo-lake shorelines and present-day crustal displacements for the North American Late Wisconsin ice sheet (Lambeck et al. 2017). To constrain the viscosity at 670 km depth, ηlm(670), we examined the GIA-induced $${\skew5\dot{J}_n}$$ (n = 4 and 6) and postglacial RSL changes at Southport, Bermuda and Everglades in the intermediate region for the North American ice sheet. GIA-induced $${\skew5\dot{J}_4}$$ and $${\skew5\dot{J}_6}$$ based on the geodetic data by Cheng et al. (1997) and the recent melting by Vaughan et al. (2013, see Nakada & Okuno 2017) may constrain the ice model, but cannot provide a constraint on the viscosity at 670 km depth (see Fig. 7). However, the inference of the ice model from the GIA-induced $${\skew5\dot{J}_n}$$ for n > 2 would be inconclusive at this time if we consider the accuracy of the estimates of the zonal secular rates for n > 2 (Cheng et al. 1997). The postglacial RSL changes at this region would be little influenced by the detailed melting histories of the North American ice sheet and largely dependent on their gross melting history (e.g. Nakada & Lambeck 1987). In fact, the RSL changes for ice models with an equal ESLNH (ESLSH) value, for example, IA30 and IR30 with ESLNH∼100 m (ESLSH∼30 m), are nearly identical as shown in Figs 9 and 10. However, the standard viscosity models with $${\bar{\eta }_{{\rm{um}}}}$$ = 2 × 1020 Pa s cannot simultaneously explain the observed RSL changes at these sites. That is, the observed RSL changes at Southport and Bermuda are distinctly different from predicted ones regardless of viscosity and ice models, but the observed RSL change at Everglades is consistent with the predicted one for viscosity models with ηlm(670) ≥ 2 × 1022 Pa s (M12 and M13) and the IA30 ice model. On the other hand, the analyses of these RSL changes based on the viscosity models with $${\bar{\eta }_{{\rm{um}}}}$$ > 2 × 1020 Pa s and lower-mantle viscosity structures M8–M13 (M14–M19) yield permissible $${\bar{\eta }_{{\rm{um}}}}$$ and ηlm(670) values although there is a trade-off between the ice and viscosity models: $${\bar{\eta }_{{\rm{um}}}}$$ ∼ (7 − 9) × 1020 and ηlm(670) ∼ (1 − 2) × 1022 Pa s for an ice model with ESLSH∼20 or ∼30 m, and $${\bar{\eta }_{{\rm{um}}}}$$ ∼ 9 × 1020 and ηlm(670) ∼ 5 × 1022 Pa s for an ice model with ESLSH∼10 m (Figs 9, 10, 12 and 13). However, the permissible $${\bar{\eta }_{{\rm{um}}}}$$ for each site appears to decrease with increasing distance from the edge of the ice sheet (Fig. 1a), which may reflect a weak (moderate) laterally heterogeneous upper-mantle viscosity (for example, see Fig. 12 for the IA30 ice model). The viscosity at 670 km depth higher than 1022 Pa s is also supported from the analyses by Nakada & Okuno (2016) using the two-layer lower-mantle viscosity model described by depth-averaged effective viscosities of η670,Dand ηD,2891 (D = 1191 and 1691 km). That is, they showed that the observed postglacial RSL changes at Southport and Bermuda are consistent with the predicted ones for viscosity models with ηum > 6 × 1020, η670,1191 > 1022 (η670,1691 > 2 × 1022) and ηD,2891∼ (5 − 10) × 1022 Pa s and an ice model with ESLSH∼20 or ∼30 m. However, our inferred viscosity model is distinctly different from the viscosity model preferred by Lau et al. (2016) except for the deep mantle viscosity of 1022 − 1023 Pa s. That is, their preferred lower-mantle viscosity structure is as follows (fig. 4 by Lau et al. 2016): an average viscosity of 1021 Pa s from 670 to ∼1500 km depth (no viscosity jump at 670 km depth), a significant viscosity increase in the bottom half of the lower mantle and 1022 − 1023 Pa s in the deep mantle. Although we cannot provide a persuasive explanation for the difference between the lower-mantle viscosity structures (in the top half of the lower mantle) by Lau et al. (2016) and this study, the difference may be attributed to different GIA data sets used in both studies. This is an issue in the future. The inferred upper- and lower-mantle viscosities using postglacial RSL changes at Southport, Bermuda and Everglades and lower-mantle viscosity models satisfying three GIA-related observables are wholly consistent with the inference by Lambeck et al. (2017) using GIA-related geological and geophysical data in the glaciated region for the North American Late Wisconsin ice sheet. Their preferred upper- and lower-mantle viscosities using the simple three-layer viscosity model are (3.5 − 7.5) × 1020 and (0.8 − 2.8) × 1022 Pa s, respectively. The upper-mantle viscosity for both studies is significantly different from the value of (1 − 2) × 1020 Pa s obtained from analyses of far-field ocean islands and continental margin data (Lambeck et al. 2014) largely influenced by isostatic response due to the oceanic upper-mantle viscosity (Lambeck et al. 2017). Then, Lambeck et al. (2017) indicated a lateral variation of upper-mantle viscosity by considering the upper-mantle viscosities inferred from analyses of far-field continental margin data and cratonic continental data, which was first suggested by Nakada & Lambeck (1991). This study, using RSL changes in the intermediate region for the North American ice sheet, also suggests a lateral variation of upper-mantle viscosity. Also, Lambeck et al. (2017) prefer the ESL component of ∼80 m for the North American Late Wisconsin ice sheet (their ice sheet model LW-6). The ESL component for the North American ice sheet of IA30 and IR30 in our study is ∼80 m (see Section 2.2) and almost identical to the value of LW-6. Mantle viscosity structure adopted in this study depends on temperature distribution and activation energy and volume, and it is difficult to discuss the impact of each quantity on the inferred viscosity model. Our preferred viscosity model has a significant viscosity jump at 670 km depth and requires an order of gradual increase in viscosity within the lower mantle. That is, models of smooth depth variation in the lower-mantle viscosity following $$\eta ( z )\ \propto {\rm{\ exp}}[ {( {E_{{\rm{lm}}}^* + P( z )V_{{\rm{lm}}}^*} )/{\rm{R}}T( z )} ]$$ with constant $$E_{{\rm{lm}}}^*$$ and $$V_{{\rm{lm}}}^*$$ are consistent with the GIA observations. Such an increase is not inconsistent with the inference by Mitrovica & Forte (2004) using GIA and convection data sets. The smooth viscosity variation in the lower mantle is consistent with the inference based on the theory of diffusion in ionic solids by Karato (1981), but inconsistent with the inference of a strong increase in viscosity throughout the upper 900 km of the lower mantle from high pressure (at room temperature) deformation data for ferropericlase by Marquardt & Miyagi (2015, see also Rudolph et al. 2015). In this study, the $$V_{{\rm{lm}}}^*$$ value for the preferred lower-mantle viscosity structure is ∼(2 − 3) × 10−6 m3 mol−1 (Table 2) for an adopted temperature distribution and lower-mantle activation energy of $$E_{{\rm{lm}}}^*$$ = 250 kJ mol−1. We hope that the lower-mantle viscosity structure with an exponential profile inferred from GIA data sets may be useful in discussing the mineral physics of the lower-mantle rheology and the extrapolation of laboratory observations to the geological environments. This would be possible in the near future if we consider the recent deformation experiments for a mixture of bridgmanite and magnesiowüstite at lower-mantle conditions by Girard et al. (2016). ACKNOWLEDGEMENTS We thank S. Karato and T. Kubo for constructive suggestions and discussions, and two anonymous reviewers and the editor (J. C. Afonso) for their constructive comments. This work was partly supported by the Japanese Ministry of Education, Science and Culture (Grand-in-Aid for Scientific Research No. 16K05543). REFERENCES Argus D.F., Peltier W.R., Drummond R., Moore A.W., 2014. 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# Inference of viscosity jump at 670 km depth and lower mantle viscosity structure from GIA observations

, Volume 212 (3) – Mar 1, 2018
20 pages

/lp/ou_press/inference-of-viscosity-jump-at-670-km-depth-and-lower-mantle-viscosity-unT5Y1TGe3
Publisher
The Royal Astronomical Society
ISSN
0956-540X
eISSN
1365-246X
D.O.I.
10.1093/gji/ggx519
Publisher site
See Article on Publisher Site

### Abstract

Summary A viscosity model with an exponential profile described by temperature (T) and pressure (P) distributions and constant activation energy ($$E_{{\rm{um}}}^{\rm{*}}$$ for the upper mantle and $$E_{{\rm{lm}}}^*$$ for the lower mantle) and volume ($$V_{{\rm{um}}}^{\rm{*}}$$ and $$V_{{\rm{lm}}}^*$$) is employed in inferring the viscosity structure of the Earth's mantle from observations of glacial isostatic adjustment (GIA). We first construct standard viscosity models with an average upper-mantle viscosity ($${\bar{\eta }_{{\rm{um}}}}$$) of 2 × 1020 Pa s, a typical value for the oceanic upper-mantle viscosity, satisfying the observationally derived three GIA-related observables, GIA-induced rate of change of the degree-two zonal harmonic of the geopotential, $${\skew5\dot{J}_2}$$, and differential relative sea level (RSL) changes for the Last Glacial Maximum sea levels at Barbados and Bonaparte Gulf in Australia and for RSL changes at 6 kyr BP for Karumba and Halifax Bay in Australia. Standard viscosity models inferred from three GIA-related observables are characterized by a viscosity of ∼1023 Pa s in the deep mantle for an assumed viscosity at 670 km depth, ηlm(670), of (1 − 50) × 1021 Pa s. Postglacial RSL changes at Southport, Bermuda and Everglades in the intermediate region of the North American ice sheet, largely dependent on its gross melting history, have a crucial potential for inference of a viscosity jump at 670 km depth. The analyses of these RSL changes based on the viscosity models with $${\bar{\eta }_{{\rm{um}}}}$$ ≥ 2 × 1020 Pa s and lower-mantle viscosity structures for the standard models yield permissible $${\bar{\eta }_{{\rm{um}}}}$$ and ηlm (670) values, although there is a trade-off between the viscosity and ice history models. Our preferred $${\bar{\eta }_{{\rm{um}}}}$$ and ηlm (670) values are ∼(7 − 9) × 1020 and ∼1022 Pa s, respectively, and the $${\bar{\eta }_{{\rm{um}}}}$$ is higher than that for the typical value of oceanic upper mantle, which may reflect a moderate laterally heterogeneous upper-mantle viscosity. The mantle viscosity structure adopted in this study depends on temperature distribution and activation energy and volume, and it is difficult to discuss the impact of each quantity on the inferred lower-mantle viscosity model. We conclude that models of smooth depth variation in the lower-mantle viscosity following $$\eta ( z )\ \propto {\rm{\ exp}}[ {( {E_{{\rm{lm}}}^* + P( z )V_{{\rm{lm}}}^*} )/{\rm{R}}T( z )} ]$$ with constant $$E_{{\rm{lm}}}^*$$ and $$V_{{\rm{lm}}}^*$$ are consistent with the GIA observations. Earth rotation variations, Sea level change, Rheology: mantle 1 INTRODUCTION Observations related to glacial isostatic adjustment (GIA) due to the last deglaciation such as relative sea level (RSL) variations and the rate of change of low-degree zonal harmonics of Earth's geopotential $${\skew5\dot{J}_n}$$ (n ≥ 2) contain important information about the Earth's viscosity structure (e.g. Peltier 2004; Sabadini et al. 2016). In inferring mantle viscosity from GIA-related observations, many studies have adopted a simple three-layer viscosity model described by elastic lithospheric thickness, upper-mantle viscosity above 670 km depth and lower-mantle viscosity (e.g. Wu & Peltier 1984; Cambiotti et al. 2010; Nakada et al. 2015; Lambeck et al. 2017). For example, Lambeck et al. (2017) inferred the lower-mantle viscosity of ∼1022 Pa s from inversion studies using RSL observations, tilting of palaeo-lake shorelines and present-day crustal displacements for the North American Late Wisconsin ice sheet. Nakada et al. (2015) estimated the GIA-induced $${\skew5\dot{J}_2}$$of −(6.0 − 6.5) × 10−11 yr−1 based on the geodetically derived $${\skew5\dot{J}_2}$$ (Roy & Peltier 2011; Cheng et al. 2013) and modern recent melting history taken from the IPCC 2013 Report (Vaughan et al. 2013) and inferred two permissible lower-mantle viscosities, ∼1022 and (5 − 10) × 1022 Pa s, for the simple three-layer viscosity model. GIA data sets may prefer a more complex lower-mantle viscosity structure (e.g. Ivins et al. 1993; Mitrovica 1996; Vermeersen et al. 1997; Peltier 2004). For example, a viscosity jump in the deep mantle has been inferred from plausible temperature profiles, high-pressure creep in olivine and $${\skew5\dot{J}_2}$$ (Ivins et al. 1993), postglacial decay times in Canada and Scandinavia (Mitrovica 1996) and rotational variations of the Earth (Vermeersen et al. 1997). More recently, Mitrovica et al. (2015) examined the GIA-induced $${\skew5\dot{J}_2}$$ based on a 23-layer lower-mantle viscosity model and argued for a deep mantle viscosity in excess of 1022 Pa s. Moreover, Lau et al. (2016) discretized the radial viscosity profile into 28 layers, 13 in the upper mantle and 15 in the lower mantle, and examined the viscosity structure based on postglacial decay times in Canada and Scandinavia, the Fennoscandian relaxation spectrum, late-Holocene differential sea level highstands in the Australian region and $${\skew5\dot{J}_2}$$. Then, they inferred a mean upper-mantle viscosity of ∼3 × 1020 Pa s, average viscosity of 1021 Pa s from 670 to ∼1500 km depth and 1022 − 1023 Pa in the deep mantle. Also, they insisted that the GIA data sets can be reconciled by a mantle viscosity profile without a viscosity jump at 670 km depth. In fact, their preferred viscosity model has no viscosity jump at 670 km depth. On the other hand, Nakada & Okuno (2016) analysed the GIA-induced $${\skew5\dot{J}_2}$$ based on a two-layer lower-mantle viscosity model with two layers in the lower mantle described by a depth-averaged effective viscosity from 670 to D km depth, η670,D, and a viscosity from D to 2891 km depth (core–mantle boundary, CMB), ηD,2891, with D values of 1191 and 1691 km. Then, they showed that the GIA-induced $${\skew5\dot{J}_2}$$ of −(6.0 − 6.5) × 10−11 yr−1 only requires a viscosity layer higher than (5 − 10) × 1021 Pa s for a depth above the CMB. Moreover, their analyses using the GIA-induced $${\skew5\dot{J}_2}$$, Last Glacial Maximum (LGM) sea levels at Barbados and Bonaparte Gulf in Australia and postglacial sea level highstands at Karumba and Halifax Bay in Australia (see Fig. 1) yielded two permissible viscosity solutions for the lower mantle: η670,1191 > 3 × 1021 and η1191,2891 ∼ (5 − 10) × 1022 Pa s, and η670,1691 > 1022 and η1691,2891 ∼ (5 − 10) × 1022 Pa s (no permissible solution was obtained for the case of η670,D > ηD,2891). The inferred upper-mantle viscosity for both solutions is (1 − 4) × 1020 Pa s. Thus, recent analyses using GIA data sets yield a deep mantle viscosity in excess of 1022 Pa s, while Argus et al. (2014) support a more muted viscosity increase from 5 × 1020 Pa s beneath the lithosphere to 3 × 1021 Pa s above the CMB. Figure 1. View largeDownload slide (a) Spatial distribution of total melted ice thicknesses during the last deglaciation for the North American ice sheet of the IA20 ice model and (a) and (b) relative sea level (RSL) change sites examined in this study. Figure 1. View largeDownload slide (a) Spatial distribution of total melted ice thicknesses during the last deglaciation for the North American ice sheet of the IA20 ice model and (a) and (b) relative sea level (RSL) change sites examined in this study. The viscosity structure inferred from the GIA data sets may be reaching to a general agreement except for the solution by Argus et al. (2014). Here, we examine GIA-related data sets, RSL changes and GIA-induced $${\skew5\dot{J}_n}$$(2 ≤ n ≤ 6), based on the viscosity model with an exponential profile described by pressure (P) and temperature (T) distributions and constant activation energy (E*) and volume (V*) (see Section 2.1). The present study has two main purposes. One is to examine whether the GIA data sets provide a permissible exponential lower-mantle viscosity profile with a viscosity of 1022 − 1023 Pa s in the deep mantle as argued by several GIA studies, and another is to infer the viscosity jump at 670 km depth (Sections 3 and 4). Of course, adopted parameters such as temperature distribution and activation energy and volume are highly uncertain and the results obtained here are preliminary. Inference of lower-mantle rheology based on the experimentally based rheological data in the lower-mantle pressure and temperature conditions may be possible if we assume a homologous temperature scaling viscosity profile using experimentally determined self-diffusion coefficients of silicon in MgSiO3 perovskite under shallow lower-mantle conditions and melting temperatures of lower-mantle minerals (Yamazaki & Karato 2001; Karato 2008). More recently, Marquardt & Miyagi (2015) showed an increase of viscosity throughout the upper 900 km of the lower mantle based on the high-pressure (up to 96 GPa) and room-temperature deformation data for ferropericlase of the main phase of the lower mantle. On the other hand, Rudolph et al. (2015) inferred an increase in viscosity at 800–1200 km depth from the flow model using the long-wavelength non-hydrostatic geoid and seismic tomography data and argued that rheological properties inferred from the deformation experiments by Marquardt & Miyagi (2015) may provide a possible rheological interpretation for their inference on viscosity increase in the mid-mantle. However, it would be difficult to discuss the rheological properties at high temperature in the lower mantle based on the experimental data for room-temperature deformation experiments (e.g. Karato 2008). More recently, Girard et al. (2016) succeeded quantitative deformation experiments for a mixture of bridgmanite and magnesiowüstite, main phases of the lower mantle, at pressures at 24–27.5 GPa and temperatures up to 2000–2150 K. Although deformation mechanisms are not clearly identified from their experimental data, it would be possible to discuss the rheological properties based on the experimentally based rheological data for the lower-mantle conditions in the near future. Then, the present study evaluating quantitative impacts of activation energy and volume on the GIA-based viscosity model may provide some useful constraint on the mineral physics of the lower-mantle rheology. The method for conducting discussions on an exponential mantle viscosity profile has mainly two steps: (i) inference of the mantle viscosity structure from the GIA-induced $${\skew5\dot{J}_2}$$, differential RSL change for LGM sea levels at Barbados and Bonaparte Gulf and that for RSL changes at ∼6 kyr BP for Karumba and Halifax Bay, referred to as three GIA-related observables here, and (ii) inference of mantle viscosity considering the inference obtained in step (i), particularly for the viscosity jump at 670 km depth, based on the GIA-induced $${\skew5\dot{J}_n}$$(2 < n ≤ 6) and postglacial RSL changes at Southport, Bermuda and Everglades in the intermediate region of the North American ice sheet (Fig. 1). The paper is organized as follows. In Section 2, we explain the viscosity and the used ice history models. In Section 3, we examine the sensitivities of three GIA-related observables to viscosity and GIA ice models and infer the viscosity structure from three GIA-related observables. In Section 4, we infer the lower-mantle viscosity structure, particularly a potential viscosity jump at 670 km depth, based on the results in Section 3 and observationally derived GIA-induced zonal secular rates for n = 4 and 6 and postglacial RSL changes at Southport, Bermuda and Everglades. The results obtained in this study are discussed and summarized in Section 5. 2 MODEL ADOPTED IN THIS STUDY 2.1 Earth model We adopt the seismological Preliminary Reference Earth Model (PREM, Dziewonski & Anderson 1981) for density and elastic constants. The viscosity structure is described by elastic lithospheric thickness (H), upper-mantle viscosity, ηum(z) for H < z < 670 km depth, and lower-mantle viscosity, ηlm(z) for 670 ≤ z ≤ 2891 km depth. The depth-dependent viscosity structures, ηum(z) and ηlm(z), are described as follows (e.g. Karato 2008):   $${\eta _{{\rm{um}}}}\left( z \right)\ = {{\rm{A}}_{{\rm{um}}}}{\rm{\ exp}}\left[ {\frac{{E_{{\rm{um}}}^* + P\left( z \right)V_{{\rm{um}}}^*}}{{{\rm{R}}T\left( z \right)}}} \right]\,\,{\rm{for}}\,\,{\rm{H}}\, < z < 670$$ (1)  $${\eta _{{\rm{lm}}}}\left( z \right)\ = {{\rm{A}}_{{\rm{lm}}}}{\rm{\ exp}}\left[ {\frac{{E_{{\rm{lm}}}^* + P\left( z \right)V_{{\rm{lm}}}^*}}{{{\rm{R}}T\left( z \right)}}} \right]\,\,{\rm{for}}\,\, 670\, \le z \le 2891$$ (2)R is the gas constant, and Aum and Alm are constants determined using assumed average upper-mantle viscosity and ηlm(670) value, respectively (see below). E* and V* are activation energy and activation volume (‘um’ for the upper mantle and ‘lm’ for the lower mantle), respectively, and these quantities are assumed to be constant in this study. P(z) and T(z) are pressure and temperature distributions, respectively, and the pressure distribution is derived from the density and gravity values of PREM. Here, we adopt a simple upper-mantle temperature distribution considering a cooling model of oceanic lithosphere and adiabatic temperature gradient in the upper mantle (Γ, e.g. Turcotte & Schubert 2002):   $${T_{{\rm{um}}}}\left( {z,t} \right) = {T_0}\ + \left( {{T_\infty } - {T_0}} \right){\rm{erf}}\left( {\frac{z}{{2\sqrt {\kappa t} }}} \right) + \Gamma z$$ (3)where T0 is the surface temperature, T∞is the temperature at z = ∞ for the cooling model, κ is thermal diffusivity (10−6 m2 s−1 in this study) and t is the plate age. The values of T0 and T∞ are assumed to be 273 and 1600 K, respectively, corresponding to T∞ − T0 = 1300 K by Turcotte & Schubert (2002). The temperature at 670 km depth is 1873 K estimated by Ito & Katsura (1989) based on the high-pressure phase equilibrium in the system Mg2SiO4–Fe2SiO4 and an assumption that the spinel dissociation is responsible for the sharp seismic discontinuity. The temperature distribution for the lower mantle is assumed to be adiabatic and given by:   $${T_{{\rm{lm}}}}\left( z \right) = \ {\rm{B\ exp}}\left( {\frac{{\alpha gz}}{{{C_p}}}} \right)$$ (4)In eq. (4), B is a constant (see below), and α, g and Cp are coefficient of thermal expansion, gravity and specific heat at constant pressure, and assumed to be 2.5 × 10−5 K−1, 10 m s−2 and 1000 J K−1 kg−1, respectively (αg/Cp = 2.5 × 10−7 m−1). The gravity of the lower mantle is assumed to be constant in estimating the lower-mantle temperature. However, this assumption does not change the results at all in the following discussion. The values of Γ(in eq. 3) and B (in eq. 4) are obtained by using Tum(670) = Tlm(670) = 1873 K. In this study, we treat an average upper-mantle viscosity, $${\bar{\eta }_{{\rm{um}}}}$$, and viscosity at 670 km depth (viscosity at the top of the lower mantle), ηlm(670), as key parameters describing the viscosity structure in the mantle. Then, the coefficient Aum in eq. (1) is obtained by taking a weighted harmonic average for the upper-mantle viscosity, ηum(z), and the coefficient of Alm in eq. (2) for ηlm(z) is determined from an assumed ηlm(670) value. In this study, we restrict to the lower-mantle viscosity structure with no low-viscosity D″ layer by considering that the effective lower-mantle viscosity inferred from the GIA-induced $${\skew5\dot{J}_2}$$ is relatively insensitive to the viscosity structure of the D″ layer (Nakada et al. 2015) and that the GIA-related postglacial RSL changes such as those at Barbados, Bermuda and Karumba examined here are insensitive to the low-viscosity D″ layer as well (Nakada & Karato 2012, see also Lau et al. 2016). Our model is based on the classical normal mode approach, and thus deals appropriately with the issues raised by Cambiotti & Sabadini (2010) when dealing with discontinuously or continuously varying density and bulk modulus. We briefly comment out an inferred lower-mantle viscosity structure satisfying the GIA data sets adopted in this study. The inferred lower-mantle viscosity profile is, of course, non-unique. Also, it would be difficult to adequately evaluate the impacts of uncertainties of temperature distribution, $$E_{{\rm{lm}}}^*$$ and $$V_{{\rm{lm}}}^*$$ on the inferred viscosity structure. It is so meaningful that GIA data sets only prefer a depth-dependent quantity of $$( {E_{{\rm{lm}}}^* + P( z )V_{{\rm{lm}}}^*} )/RT( z )$$ for the inferred lower-mantle viscosity structure. We also comment out the plate age t. The age of the seafloor is laterally varying, but its effect on the plate thickness (e.g. Turcotte & Schubert 2002) is not considered in the modeling herein. That is, this parameter must be considered as an average value for the plate thickness throughout the globe. 2.2 Ice history model We adopt six simplified ice history models of the last Pleistocene glaciation, IA ice models (IA10, IA20 and IA30) and IR ice models (IR10, IR20 and IR30) by Nakada & Okuno (2017), constructed based on the method by Lambeck (1993) using ice sheet dynamics (see fig. 1 in Nakada & Okuno 2017). The areas of all ice sheets for the LGM at ∼21 kyr BP are the same as those for the global ice history model ICE-5G (Peltier 2004), and those at an arbitrary time are proportional to the ice volume derived from equivalent sea level (ESL) component of each ice sheet. The ESL is defined as the change in meltwater volume divided by the surface area of the ocean at the present day. Fig. 1(a) shows the spatial distribution of total melted ice thicknesses during the last deglaciation for the North American ice sheet of IA20. These ice models are inferred from the LGM sea levels at Barbados (Fairbanks 1989; Peltier & Fairbanks 2006) and Bonaparte Gulf (Yokoyama et al. 2000), RSL change at Tahiti (Bard et al. 1996; Deschamps et al. 2012) and RSL changes after ∼6 kyr BP at Karumba and Halifax Bay (Chappell et al. 1983; Nakada & Lambeck 1989; Nakada et al. 2016). These models have an identical ESL history with a total ESL component of 127.9 m that is 10 m larger than that of ICE-5G (∼118 m). The overarching difference between the IA and IR ice models is simply the assumed timing history. All IA* ice sheets melt synchronously based on the ESL history. The Antarctic ESL component (ESLSH) is assumed to be 9.3 m (∼10 m) for IA10, 18.7 m (∼20 m) for IA20 and 28.0 m (∼30 m) for IA30 because of uncertainties in the melting history of the Antarctic ice sheet (e.g. Nakada & Lambeck 1988; Nakada et al. 2000; Peltier 2004; Whitehouse et al. 2012; Ivins et al. 2013; Lambeck et al. 2014). The total Antarctic ESL components for the IR* models are the same as those for the IA* models, respectively. The Antarctic ESL history before 6 kyr BP for IR20 is the same as that for ICE-5G with ESLSH∼20 m due to the melting mainly from 12 to 6 kyr BP (see also Nakada et al. 2000, for the inference using RSL data in Antarctica), and the melting ice thicknesses at all sites for the Antarctic IR20 model are proportional to the ESL history. IR20 has a gradual Antarctic melting of ∼2.5 m after 6 kyr BP as was assumed by Nakada & Lambeck (1988). The ESL components of the Laurentide, Fennoscandian and Greenland ice sheets are estimated considering their ESL components for ICE-5G, and these ice sheets melt synchronously based on the ESL history for the Northern Hemisphere (NH) ice sheets. The melted ice thicknesses at an arbitrary time for the Antarctic IR10 ice model are half the ones of IR20. Those for IR30 before 6 kyr BP are three times the ones of IR10, and those after 6 kyr BP are the same as those of IR20. The ESL components for the North American ice sheets including the Greenland ice sheet, which would significantly affect the postglacial RSL changes at Southport, Bermuda and Everglades examined in Section 4, are 94.4, 87.0 and 79.5 m for IA10 (IR10), IA20 (IR20) and IA30 (IR30), respectively. 3 RESULTS FOR THREE GIA-RELATED OBSERVABLES 3.1 Case for constant upper-mantle viscosity model GIA-induced $${\skew5\dot{J}_2}$$ provides an important constraint on the lower-mantle viscosity (e.g. Sabadini et al. 1982; Wu & Peltier 1984; Devoti et al. 2001; Mitrovica et al. 2015; Nakada et al. 2015). Also, differential RSL change for Barbados and Bonaparte Gulf at the LGM, $${\rm{\Delta RSL}}_{\rm{L}}^{{\rm{Bar}},{\rm{Bon}}}$$, and that for Karumba and Halifax Bay at ∼6 kyr BP, $${\rm{\Delta RSL}}_6^{{\rm{Kar}},{\rm{Hal}}}$$, are sensitive to the lower-mantle viscosity (Nakada & Okuno 2016; Nakada et al. 2016). If we denote the LGM sea levels at Barbados and Bonaparte Gulf by $${\rm{RSL}}_{\rm{L}}^{{\rm{Bar}}}$$ and $${\rm{RSL}}_{\rm{L}}^{{\rm{Bon}}}$$, respectively, then the differential RSL change for these sites is given by $${\rm{\Delta RSL}}_{\rm{L}}^{{\rm{Bar}},{\rm{Bon}}}$$ = $${\rm{RSL}}_{\rm{L}}^{{\rm{Bar}}}$$ − $${\rm{RSL}}_{\rm{L}}^{{\rm{Bon}}}$$. Similarly, that for Karumba and Halifax Bay is given by $${\rm{\Delta RSL}}_6^{{\rm{Kar}},{\rm{Hal}}}$$ = $${\rm{RSL}}_6^{{\rm{Kar}}}$$ − $${\rm{RSL}}_6^{{\rm{Hal}}}$$. Here, we discuss the GIA-induced $${\skew5\dot{J}_2}$$ and differential RSL changes for these sites by employing viscosity models with a constant upper-mantle viscosity (ηum) of 2 × 1020 Pa s, H = 65 km, ηlm(670) = 1021 Pa s and $$E_{{\rm{lm}}}^*$$ = 250 kJ mol−1. We hereafter refer $${\skew5\dot{J}_2}$$, $${\rm{\Delta RSL}}_{\rm{L}}^{{\rm{Bar}},{\rm{Bon}}}$$ and $${\rm{\Delta RSL}}_6^{{\rm{Kar}},{\rm{Hal}}}$$ as three GIA-related observables. The observationally derived differential RSL change at the LGM for Barbados and Bonaparte Gulf is 0–25 m from the LGM sea levels of −120 to −98 m for Barbados (Fairbanks 1989; Peltier & Fairbanks 2006) and −123 to −120 m for Bonaparte Gulf (Yokoyama et al. 2000). That for Karumba and Halifax Bay is 0.9–1.9 m from $${\rm{RSL}}_6^{{\rm{Kar}}}$$ ∼ (2.4 ± 0.3) and $${\rm{RSL}}_6^{{\rm{Hal}}}$$ ∼ (1.0 ± 0.2) m (Chappell et al. 1983; see also Lambeck & Nakada 1990). Fig. 2(a) shows the viscosity structures of U1–U6 models with $$V_{{\rm{lm}}}^*$$ = (2.9, 3.05, 3.2, 3.35, 3.5, 3.65) × 10−6 m3 mol−1, respectively (Table 1). The effective lower-mantle viscosity increases with increasing $$V_{{\rm{lm}}}^*$$ value. Table 1 shows the three GIA-related observables based on the IA20 ice model and these viscosity models. With increasing $$V_{{\rm{lm}}}^*$$ value, the $${\skew5\dot{J}_2}$$ magnitude increases in a range of (2.9 − 3.2) × 10−6 m3 mol−1 and decreases in a range of (3.2 − 3.65) × 10−6 m3 mol−1. Fig. 2(b) shows the $${\skew5\dot{J}_2}$$ for IA20 as a function of lower-mantle viscosity (ηlm) based on the simple three-layer viscosity model with ηum = 2 × 1020 Pa s and H = 65 km usually used in the $${\skew5\dot{J}_2}$$ study (e.g. Wu & Peltier 1984). The rates for the viscosity models with $$V_{{\rm{lm}}}^*$$ = 2.9 × 10−6 and 3.50 × 10−6 m3 mol−1, satisfying the observationally derived estimate of −(6.0 − 6.5) × 10−11 yr−1, correspond to the rates for sections ‘B’ and ‘F’ of the black curve in Fig. 2(b), respectively. The rates for the viscosity models with 3.05 × 10−6, 3.2 × 10−6 and 3.35 × 10−6 m3 mol−1 correspond to the rates for sections ‘C’, ‘D’ and ‘E’ of the black curve in Fig. 2(b), respectively. These comparisons indicate that the permissible viscosity solutions with $$V_{{\rm{lm}}}^*$$ ∼ 2.9 × 10−6 and 3.35 × 10−6 < $$V_{{\rm{lm}}}^*$$ ≤ 3. 5 × 10−6 m3 mol−1 correspond to the viscosity solutions of ηlm ∼ (1–2) × 1022 and ηlm ∼(5–10) × 1022 Pa s for the simple three-layer viscosity model in Fig. 2(b), respectively. On the other hand, the $${\rm{\Delta RSL}}_{\rm{L}}^{{\rm{Bar}},{\rm{Bon}}}$$ values are negative for the viscosity models with $$V_{{\rm{lm}}}^*$$ ≤ 3. 2 × 10−6 m3 mol−1, and consequently, differential RSL change for Barbados and Bonaparte Gulf prefers the lower-mantle viscosity structure with $$V_{{\rm{lm}}}^*$$ > 3. 2 × 10−6 m3 mol−1. That is, the GIA-induced $${\skew5\dot{J}_2}$$ and $${\rm{\Delta RSL}}_{\rm{L}}^{{\rm{Bar}},{\rm{Bon}}}$$ prefer the lower-mantle viscosity structure with 3.35 × 10−6<$$V_{{\rm{lm}}}^*$$ ≤ 3. 5 × 10−6 m3 mol−1, which is not inconsistent with the inference from the differential RSL change at ∼6 kyr for Karumba and Halifax Bay (Table 1). This solution corresponds to the lower-mantle viscosity of (5–10) × 1022 Pa s for the simple three-layer viscosity model in Fig. 2(b). Figure 2. View largeDownload slide (a) Viscosity structures of the U1–U6 models with $$V_{{\rm{lm}}}^*$$ = (2.9, 3.05, 3.2, 3.35, 3.5, 3.65) × 10−6 m3 mol−1, respectively (see Table 1) and (b) $${\skew5\dot{J}_2}$$ for the IA20 ice model as a function of lower-mantle viscosity (ηlm) based on the simple three-layer viscosity model with ηum = 2 × 1020 Pa s and H = 65 km usually used in the $${\skew5\dot{J}_2}$$ study. The shaded region shows the range for GIA-induced $${\skew5\dot{J}_2}$$ of −(6.0 − 6.5) × 10−11 yr−1. Figure 2. View largeDownload slide (a) Viscosity structures of the U1–U6 models with $$V_{{\rm{lm}}}^*$$ = (2.9, 3.05, 3.2, 3.35, 3.5, 3.65) × 10−6 m3 mol−1, respectively (see Table 1) and (b) $${\skew5\dot{J}_2}$$ for the IA20 ice model as a function of lower-mantle viscosity (ηlm) based on the simple three-layer viscosity model with ηum = 2 × 1020 Pa s and H = 65 km usually used in the $${\skew5\dot{J}_2}$$ study. The shaded region shows the range for GIA-induced $${\skew5\dot{J}_2}$$ of −(6.0 − 6.5) × 10−11 yr−1. Table 1. Values of $$V_{{\rm{lm}}}^*$$ (×10−6 m3mol−1), Alm (Pa s),$${\skew5\dot{J}_2}$$(×10−11yr−1), $${\rm{RSL}}_{\rm{L}}^{{\rm{Bar}}}$$ (m), $${\rm{RSL}}_{\rm{L}}^{{\rm{Bon}}}$$ (m), $${\rm{\Delta RSL}}_{\rm{L}}^{{\rm{Bar}},{\rm{Bon}}}$$ (m),$${\rm{RSL}}_6^{{\rm{Kar}}}$$ (m), $${\rm{RSL}}_6^{{\rm{Hal}}}$$ (m) and $${\rm{\Delta RSL}}_6^{{\rm{Kar}},{\rm{Hal}}}$$ (m) predicted for the IA20 ice model and viscosity models with a constant upper-mantle viscosity of 2 × 1020 Pa s and elastic lithospheric thickness (H) of 65 km. Model name  $$V_{{\rm{lm}}}^*$$  Alm  $${\skew5\dot{J}_2}$$  $${\rm{RSL}}_{\rm{L}}^{{\rm{Bar}}}$$  $${\rm{RSL}}_{\rm{L}}^{{\rm{Bon}}}$$  $${\rm{\Delta RSL}}_{\rm{L}}^{{\rm{Bar}},{\rm{Bon}}}$$  $${\rm{RSL}}_6^{{\rm{Kar}}}$$  $${\rm{RSL}}_6^{{\rm{Hal}}}$$  $${\rm{\Delta RSL}}_6^{{\rm{Kar}},{\rm{Hal}}}$$  U1  2.90  1.23 × 1012  −6.29  −124.2  −117.8  −6.4  2.20  1.37  0.83  U2  3.05  9.76 × 1011  −6.97  −122.3  −118.2  −4.1  2.33  1.47  0.86  U3  3.20  7.75 × 1011  −7.07  −120.4  −118.7  −1.7  2.37  1.50  0.87  U4  3.35  6.15 × 1011  −6.66  −116.6  −119.9  3.3  2.35  1.43  0.92  U5  3.50  4.88 × 1011  −6.04  −117.6  −119.6  2.0  2.38  1.47  0.91  U6  3.65  3.88 × 1011  −5.45  −118.9  −119.2  0.3  2.38  1.49  0.89  Model name  $$V_{{\rm{lm}}}^*$$  Alm  $${\skew5\dot{J}_2}$$  $${\rm{RSL}}_{\rm{L}}^{{\rm{Bar}}}$$  $${\rm{RSL}}_{\rm{L}}^{{\rm{Bon}}}$$  $${\rm{\Delta RSL}}_{\rm{L}}^{{\rm{Bar}},{\rm{Bon}}}$$  $${\rm{RSL}}_6^{{\rm{Kar}}}$$  $${\rm{RSL}}_6^{{\rm{Hal}}}$$  $${\rm{\Delta RSL}}_6^{{\rm{Kar}},{\rm{Hal}}}$$  U1  2.90  1.23 × 1012  −6.29  −124.2  −117.8  −6.4  2.20  1.37  0.83  U2  3.05  9.76 × 1011  −6.97  −122.3  −118.2  −4.1  2.33  1.47  0.86  U3  3.20  7.75 × 1011  −7.07  −120.4  −118.7  −1.7  2.37  1.50  0.87  U4  3.35  6.15 × 1011  −6.66  −116.6  −119.9  3.3  2.35  1.43  0.92  U5  3.50  4.88 × 1011  −6.04  −117.6  −119.6  2.0  2.38  1.47  0.91  U6  3.65  3.88 × 1011  −5.45  −118.9  −119.2  0.3  2.38  1.49  0.89  Notes: The viscosity at 670 km depth, ηlm(670), is 1021 Pa s and the activation energy for the lower mantle, $$E_{{\rm{lm}}}^*$$, is 250 kJ mol−1. View Large 3.2 Effect of upper-mantle activation energy and volume Here, we examine the effects of activation energy ($$E_{{\rm{um}}}^{\rm{*}}$$) and volume ($$V_{{\rm{um}}}^{\rm{*}}$$) of the upper mantle on the three GIA-related observables by using viscosity models of U5 (Table 1) and M1–M4 shown in Fig. 3 (see also Table 2). The $$E_{{\rm{um}}}^{\rm{*}}$$ value is 250 kJ mol−1 for M1–M3 and 300 kJ mol−1 for M4, corresponding to the experimentally derived estimates for diffusion creep by Karato et al. (1986): 250 kJ mol−1 for wet olivine and 290 kJ mol−1 for dry olivine (see also table 19.1 in Karato 2008). The upper-mantle viscosity for the U5 model is constant and 2 × 1020 Pa s, and that for M1–M4 is a weighted average viscosity ($${\bar{\eta }_{{\rm{um}}}}$$) of 2 × 1020 Pa s. The adopted t value, t = 60 Myr, corresponds to the mean age of the seafloor (e.g. Turcotte & Schubert 2002). The viscosity at ∼65 km depth just beneath the elastic lithosphere is ∼1024 Pa s for M1–M4, corresponding to a Maxwell relaxation time of ∼0.3 Myr. The parameter values describing the lower-mantle viscosity are the same as those for U5. Figure 3. View largeDownload slide Viscosity models U5 with constant upper-mantle viscosity of 2 × 1020 Pa s (Table 1) and M1–M4 with $${\bar{\eta }_{{\rm{um}}}}$$ = 2 × 1020 Pa s (Table 2). The $$E_{{\rm{um}}}^{\rm{*}}$$ value is 250 kJ mol−1 for M1–M3 and 300 kJ mol−1 for M4. Figure 3. View largeDownload slide Viscosity models U5 with constant upper-mantle viscosity of 2 × 1020 Pa s (Table 1) and M1–M4 with $${\bar{\eta }_{{\rm{um}}}}$$ = 2 × 1020 Pa s (Table 2). The $$E_{{\rm{um}}}^{\rm{*}}$$ value is 250 kJ mol−1 for M1–M3 and 300 kJ mol−1 for M4. Table 2. Viscosity models adopted in this study and three GIA-related observables predicted for these viscosity models. Model name  $$E_{{\rm{um}}}^{\rm{*}}$$  $$V_{{\rm{um}}}^{\rm{*}}$$  Aum  $$E_{{\rm{lm}}}^*$$  $$V_{{\rm{lm}}}^*$$  ηlm(670)  Alm  t  H  $${\skew5\dot{J}_2}$$  $${\rm{\Delta RSL}}_{\rm{L}}^{{\rm{Bar}},{\rm{Bon}}}$$  $${\rm{\Delta RSL}}_6^{{\rm{Kar}},{\rm{Hal}}}$$  M1  250  2.0  1.09 × 1012  250  3.5  1021  4.88 × 1011  60  65  –6.0  1.8  0.77  M2  250  3.0  4.78 × 1011  250  3.5  1021  4.88 × 1011  60  65  −6.09  3.0  0.96  M3  250  4.0  2.35 × 1011  250  3.5  1021  4.88 × 1011  60  65  −6.13  4.3  1.18  M4  300  4.0  6.94 × 109  250  3.5  1021  4.88 × 1011  60  65  −6.09  3.8  1.10  M5  250  4.0  2.35 × 1011  200  3.05  1021  2.42 × 1013  60  65  −6.51  3.2  1.17  M6  250  4.0  2.35 × 1011  250  3.425  1021  5.48 × 1011  60  65  −6.53  3.4  1.17  M7  250  4.0  2.35 × 1011  300  3.80  1021  1.24 × 1010  60  65  −6.54  3.5  1.17  M8  250  4.0  2.35 × 1011  250  3.475  1021  5.08 × 1011  60  65  −6.26  4.0  1.18  M9  250  4.0  2.35 × 1011  250  3.2  2 × 1021  1.55 × 1012  60  65  −6.24  5.5  1.23  M10  250  4.0  2.35 × 1011  250  2.85  5 × 1021  6.64 × 1012  60  65  −6.28  6.4  1.25  M11  250  4.0  2.35 × 1011  250  2.6  1022  1.95 × 1013  60  65  −6.23  6.9  1.25  M12  250  4.0  2.35 × 1011  250  2.33  2 × 1022  5.91 × 1013  60  65  −6.27  6.9  1.24  M13  250  4.0  2.35 × 1011  250  1.95  5 × 1022  2.65 × 1014  60  65  −6.26  6.4  1.23  M14  250  4.0  2.35 × 1011  250  3.515  1021  4.77 × 1011  60  65  −6.24  4.2  1.17  M15  250  4.0  2.35 × 1011  250  3.225  2 × 1021  1.49 × 1012  60  65  −6.27  5.5  1.23  M16  250  4.0  2.35 × 1011  250  2.88  5 × 1021  6.34 × 1012  60  65  −6.24  6.5  1.24  M17  250  4.0  2.35 × 1011  250  2.62  1022  1.89 × 1013  60  65  −6.25  6.9  1.24  M18  250  4.0  2.35 × 1011  250  2.36  2 × 1022  5.64 × 1013  60  65  −6.22  6.9  1.23  M19  250  4.0  2.35 × 1011  250  1.985  5 × 1022  2.51 × 1014  60  65  −6.21  6.6  1.22  M20  250  4.0  2.35 × 1011  250  3.2  2 × 1021  1.55 × 1012  100  65  −6.08  4.6  1.23  M21  250  4.0  2.35 × 1011  250  3.17  2 × 1021  1.623 × 1012  100  65  −6.26  4.2  1.23  M22  250  4.0  2.35 × 1011  250  3.2  2 × 1021  1.55 × 1012  60  100  −5.88  5.6  1.61  M23  250  4.0  2.35 × 1011  250  3.15  2 × 1021  1.674 × 1012  60  100  −6.19  4.9  1.61  M24  250  4.0  2.35 × 1011  250  3.125  2 × 1021  1.739 × 1012  60  100  −6.35  4.6  1.61  M25  250  4.0  2.35 × 1011  250  3.2  2 × 1021  1.55 × 1012  200  100  −5.66  5.0  1.4  M26  250  4.0  2.35 × 1011  250  3.12  2 × 1021  1.753 × 1012  200  100  −6.14  4.0  1.4  M27  250  4.0  2.35 × 1011  250  3.08  2 × 1021  1.864 × 1012  200  100  −6.36  3.4  1.4  M28  250  4.0  2.35 × 1011  250  3.05  2 × 1021  1.95 × 1012  60  65  −7.07  3.5  1.23  M29  250  4.0  2.35 × 1011  250  3.35  2 × 1021  1.23 × 1012  60  65  −5.40  7.1  1.23  M30  250  2.0  1.09 × 1012  250  3.20  2 × 1021  1.55 × 1012  60  65  −6.21  2.6  0.84  M31  250  2.0  1.15 × 1012  250  3.20  2 × 1021  1.55 × 1012  60  100  −5.67  3.8  1.35  Model name  $$E_{{\rm{um}}}^{\rm{*}}$$  $$V_{{\rm{um}}}^{\rm{*}}$$  Aum  $$E_{{\rm{lm}}}^*$$  $$V_{{\rm{lm}}}^*$$  ηlm(670)  Alm  t  H  $${\skew5\dot{J}_2}$$  $${\rm{\Delta RSL}}_{\rm{L}}^{{\rm{Bar}},{\rm{Bon}}}$$  $${\rm{\Delta RSL}}_6^{{\rm{Kar}},{\rm{Hal}}}$$  M1  250  2.0  1.09 × 1012  250  3.5  1021  4.88 × 1011  60  65  –6.0  1.8  0.77  M2  250  3.0  4.78 × 1011  250  3.5  1021  4.88 × 1011  60  65  −6.09  3.0  0.96  M3  250  4.0  2.35 × 1011  250  3.5  1021  4.88 × 1011  60  65  −6.13  4.3  1.18  M4  300  4.0  6.94 × 109  250  3.5  1021  4.88 × 1011  60  65  −6.09  3.8  1.10  M5  250  4.0  2.35 × 1011  200  3.05  1021  2.42 × 1013  60  65  −6.51  3.2  1.17  M6  250  4.0  2.35 × 1011  250  3.425  1021  5.48 × 1011  60  65  −6.53  3.4  1.17  M7  250  4.0  2.35 × 1011  300  3.80  1021  1.24 × 1010  60  65  −6.54  3.5  1.17  M8  250  4.0  2.35 × 1011  250  3.475  1021  5.08 × 1011  60  65  −6.26  4.0  1.18  M9  250  4.0  2.35 × 1011  250  3.2  2 × 1021  1.55 × 1012  60  65  −6.24  5.5  1.23  M10  250  4.0  2.35 × 1011  250  2.85  5 × 1021  6.64 × 1012  60  65  −6.28  6.4  1.25  M11  250  4.0  2.35 × 1011  250  2.6  1022  1.95 × 1013  60  65  −6.23  6.9  1.25  M12  250  4.0  2.35 × 1011  250  2.33  2 × 1022  5.91 × 1013  60  65  −6.27  6.9  1.24  M13  250  4.0  2.35 × 1011  250  1.95  5 × 1022  2.65 × 1014  60  65  −6.26  6.4  1.23  M14  250  4.0  2.35 × 1011  250  3.515  1021  4.77 × 1011  60  65  −6.24  4.2  1.17  M15  250  4.0  2.35 × 1011  250  3.225  2 × 1021  1.49 × 1012  60  65  −6.27  5.5  1.23  M16  250  4.0  2.35 × 1011  250  2.88  5 × 1021  6.34 × 1012  60  65  −6.24  6.5  1.24  M17  250  4.0  2.35 × 1011  250  2.62  1022  1.89 × 1013  60  65  −6.25  6.9  1.24  M18  250  4.0  2.35 × 1011  250  2.36  2 × 1022  5.64 × 1013  60  65  −6.22  6.9  1.23  M19  250  4.0  2.35 × 1011  250  1.985  5 × 1022  2.51 × 1014  60  65  −6.21  6.6  1.22  M20  250  4.0  2.35 × 1011  250  3.2  2 × 1021  1.55 × 1012  100  65  −6.08  4.6  1.23  M21  250  4.0  2.35 × 1011  250  3.17  2 × 1021  1.623 × 1012  100  65  −6.26  4.2  1.23  M22  250  4.0  2.35 × 1011  250  3.2  2 × 1021  1.55 × 1012  60  100  −5.88  5.6  1.61  M23  250  4.0  2.35 × 1011  250  3.15  2 × 1021  1.674 × 1012  60  100  −6.19  4.9  1.61  M24  250  4.0  2.35 × 1011  250  3.125  2 × 1021  1.739 × 1012  60  100  −6.35  4.6  1.61  M25  250  4.0  2.35 × 1011  250  3.2  2 × 1021  1.55 × 1012  200  100  −5.66  5.0  1.4  M26  250  4.0  2.35 × 1011  250  3.12  2 × 1021  1.753 × 1012  200  100  −6.14  4.0  1.4  M27  250  4.0  2.35 × 1011  250  3.08  2 × 1021  1.864 × 1012  200  100  −6.36  3.4  1.4  M28  250  4.0  2.35 × 1011  250  3.05  2 × 1021  1.95 × 1012  60  65  −7.07  3.5  1.23  M29  250  4.0  2.35 × 1011  250  3.35  2 × 1021  1.23 × 1012  60  65  −5.40  7.1  1.23  M30  250  2.0  1.09 × 1012  250  3.20  2 × 1021  1.55 × 1012  60  65  −6.21  2.6  0.84  M31  250  2.0  1.15 × 1012  250  3.20  2 × 1021  1.55 × 1012  60  100  −5.67  3.8  1.35  Notes: The GIA-related observables predicted for viscosity models M14–M19 are based on the IR20 ice model and all others are based on the IA20 ice model. The viscosity models M8–M13 are standard viscosity models for IA20, and those for M14–M19 are the ones for IR20. The average upper-mantle viscosity, $${\bar{\eta }_{{\rm{um}}}}$$, for these models is 2 × 1020 Pa s. The parameter values describing viscosity models are: $$E_{{\rm{um}}}^{\rm{*}}$$ (kJ mol−1), $$V_{{\rm{um}}}^{\rm{*}}$$(×10−6 m3mol−1), Aum (Pa s), $$E_{{\rm{lm}}}^*$$ (kJ mol−1), $$V_{{\rm{lm}}}^*$$(×10−6 m3mol−1), ηlm(670) (Pa s), Alm (Pa s), plate age t (Myr) and lithospheric thickness H (km). The three GIA-related observables are: $${\skew5\dot{J}_2}$$(×10−11yr−1), $${\rm{\Delta RSL}}_{\rm{L}}^{{\rm{Bar}},{\rm{Bon}}}$$ (m) and $${\rm{\Delta RSL}}_6^{{\rm{Kar}},{\rm{Hal}}}$$ (m)(see the text). View Large We examine the effect of the activation volume using the viscosity models of M1–M3 with $$V_{{\rm{um}}}^{\rm{*}}$$ = 2 × 10−6, 3 × 10−6 and 4 × 10−6 m3 mol−1, respectively. Variation in upper-mantle viscosity structure increases with increasing $$V_{{\rm{um}}}^{\rm{*}}$$ value (Fig. 3). Table 2 shows the three GIA-related observables based on these viscosity models. The $${\skew5\dot{J}_2}$$ values are approximately the same as those for a constant upper-mantle viscosity model U5, indicating that $${\skew5\dot{J}_2}$$ is less sensitive to the upper-mantle viscosity structure with $${\bar{\eta }_{{\rm{um}}}}$$ = 2 × 1020 Pa s. The $${\rm{\Delta RSL}}_{\rm{L}}^{{\rm{Bar}},{\rm{Bon}}}$$ and $${\rm{\Delta RSL}}_6^{{\rm{Kar}},{\rm{Hal}}}$$ values are sensitive to the upper-mantle viscosity structure and increase with increasing $$V_{{\rm{um}}}^{\rm{*}}$$ value. Consequently, the observationally derived three GIA-related observables prefer the viscosity models with $$V_{{\rm{um}}}^{\rm{*}}$$ = 3 × 10−6 and 4 × 10−6 m3 mol−1. Fig. 3 also shows the viscosity model with $$V_{{\rm{um}}}^{\rm{*}}$$ = 4 × 10−6 m3 mol−1 and $$E_{{\rm{um}}}^{\rm{*}}$$ = 300 kJ mol−1 (M4 in Table 2) used to examine the effect of the activation energy. The difference between the viscosity structures of M3 and M4 is small compared with that between M1 and M3. In fact, the three GIA-related observables for M4 are approximately the same as those for M3 with $$E_{{\rm{um}}}^{\rm{*}}$$ = 250 kJ mol−1. We therefore adopt $$E_{{\rm{um}}}^{\rm{*}}$$ = 250 kJ mol−1 in the following discussion. 3.3 Trade-off between activation energy and volume of the lower mantle We discuss the trade-off between the activation energy ($$E_{{\rm{lm}}}^*$$) and volume ($$V_{{\rm{lm}}}^*$$) of the lower mantle for the three GIA-related observables by employing the viscosity models M5, M6 and M7 with $$E_{{\rm{lm}}}^*$$ = 200, 250 and 300 kJ mol−1, respectively (Table 2). These viscosity models are constructed in such a way as to have almost an identical lower-mantle viscosity structure by considering the trade-off between the activation energy and volume. The trade-off is related to the effect of activation energy (temperature effect) and activation volume (pressure effect) on the lower-mantle viscosity structure that the viscosity decreases with increasing $$E_{{\rm{lm}}}^*$$ value and increases with increasing $$V_{{\rm{lm}}}^*$$ value. Consequently, $$V_{{\rm{lm}}}^*$$ values for M5, M6 and M7 are 3.05 × 10−6, 3.425 × 10−6 and 3.80 × 10−6 m3 mol−1, respectively (Table 2), and $$V_{{\rm{lm}}}^*$$ and $$E_{{\rm{lm}}}^*$$ values for these models give a simple relation of $$V_{{\rm{lm}}}^*$$ (m3 mol−1) ≈ 1.55 × 10−6 + 7.5 × 10−9$$E_{{\rm{lm}}}^*$$ (kJ mol−1). In fact, the three GIA-related observables predicted for these models are almost identical as shown in Table 2. In this study, we adopt $$E_{{\rm{lm}}}^*$$ = 250 kJ mol−1 and treat $$V_{{\rm{lm}}}^*$$ value as a variable on the lower-mantle viscosity structure. 3.4 Standard viscosity models and impact of seafloor age, t We first explain the standard viscosity models with H = 65 km, $${\bar{\eta }_{{\rm{um}}}}$$ = 2 × 1020 Pa s, ηlm(670) = (1, 2, 5, 10, 20, 50) × 1021 Pa s, $$E_{{\rm{um}}}^{\rm{*}}$$ = $$E_{{\rm{lm}}}^*$$ = 250 kJ mol−1, $$V_{{\rm{um}}}^{\rm{*}}$$ = 4 × 10−6 m3 mol−1 and t = 60 Myr for the IA20 and IR20 ice models, and discuss the impact of the plate age, t, on the three GIA-related observables. The $${\bar{\eta }_{{\rm{um}}}}$$ value may be a typical value for the oceanic upper-mantle viscosity (Lambeck et al.2014, 2017). The upper-mantle viscosity structure for these models is the same as that for M3, and the $$V_{{\rm{lm}}}^*$$ value for each standard viscosity model is determined in such a way as to explain the three GIA-related observables and to give the $${\skew5\dot{J}_2}$$ value of ∼ −6.25 × 10−11 yr−1. Fig. 4 shows the standard viscosity models, M8–M13 for the IA20 ice model and M14–M19 for IR20 and $$V_{{\rm{lm}}}^*$$ values and three GIA-related observables for these viscosity models are shown in Table 2. In Table 2, the GIA-related observables for M14–M19 are based on the IR20 ice model and all others are based on IA20. The $${\rm{\Delta RSL}}_{\rm{L}}^{{\rm{Bar}},{\rm{Bon}}}$$ and $${\rm{\Delta RSL}}_6^{{\rm{Kar}},{\rm{Hal}}}$$ values for these viscosity models are consistent with the observationally derived estimates. The lower-mantle viscosity structures for these models correspond to the permissible viscosity solutions inferred from the analyses with the two-layer lower-mantle viscosity model by Nakada & Okuno (2016). The inferred lower-mantle viscosity structure by Lau et al. (2016) may correspond to viscosity model M8 (M14) from the viewpoint of no viscosity jump at 670 km depth and 1022 − 1023 Pa s in the deep mantle. Figure 4. View largeDownload slide Standard viscosity models for the (a) IA20 and (b) IR20 ice models with H = 65 km, $${\bar{\eta }_{{\rm{um}}}}$$ = 2 × 1020 Pa s, ηlm(670) = (1, 2, 5, 10, 20, 50) × 1021 Pa s, $$E_{{\rm{um}}}^{\rm{*}}$$ = $$E_{{\rm{lm}}}^*$$ = 250 kJ mol−1, $$V_{{\rm{um}}}^{\rm{*}}$$ = 4 × 10−6 m3 mol−1 and t = 60 Myr (see Table 2). Figure 4. View largeDownload slide Standard viscosity models for the (a) IA20 and (b) IR20 ice models with H = 65 km, $${\bar{\eta }_{{\rm{um}}}}$$ = 2 × 1020 Pa s, ηlm(670) = (1, 2, 5, 10, 20, 50) × 1021 Pa s, $$E_{{\rm{um}}}^{\rm{*}}$$ = $$E_{{\rm{lm}}}^*$$ = 250 kJ mol−1, $$V_{{\rm{um}}}^{\rm{*}}$$ = 4 × 10−6 m3 mol−1 and t = 60 Myr (see Table 2). We discuss the impact of the plate age, t, by using viscosity models with ηlm(670) = 2 × 1021 Pa s. The following results are also true for other viscosity models with ηlm(670) = (1, 5, 10, 20, 50) × 1021 Pa s, although we do not show the results here. Fig. 5(a) shows the viscosity models of M9 (65, 60), M20 (65, 100), M22 (100, 60) and M25 (100, 200) with $$V_{{\rm{lm}}}^*$$ = 3.2 × 10−6 m3 mol−1. The former and latter numbers within the parenthesis show the lithospheric thickness (H) and plate age (t), respectively. The temperature distributions for M9, M20 and M25 are also shown in Fig. 5(b), in which the temperature at 2891 km depth (CMB) is 3264 K. We adopt viscosity models M20 (65, 100) and M22 (100, 60) to examine the differences between the three GIA-related observables predicted for these two models and those for a standard viscosity model M9 with H = 65 km and t = 60 Myr. Of course, we know that the two models M20 and M22 may contradict each other, since the older plate corresponds to the thinner one in M20 compared to M22. The $${\skew5\dot{J}_2}$$ value for M20 with H = 65 km and t = 100 Myr is −6.08 × 10−11 yr−1 and the magnitude is slightly smaller than that for a standard viscosity model of M9 with H = 65 km and t = 60 Myr. To obtain a value of $${\skew5\dot{J}_2}$$ ∼ −6.25 × 10−11 yr−1, we need $$V_{{\rm{lm}}}^*$$ ∼ 3.17 × 10−6 m3 mol−1 (M21 in Table 2). That is, change in $${\skew5\dot{J}_2}$$ of ∼0.18 × 10−11 yr−1, $$\Delta {\skew5\dot{J}_2}$$∼ 0.18 × 10−11 yr−1, requires change in $$V_{{\rm{lm}}}^*$$ of ∼0.03 × 10−6 m3 mol−1, $${\rm{\Delta }}V_{{\rm{lm}}}^{\rm{*}}$$∼0.03 × 10−6 m3 mol−1. Figure 5. View largeDownload slide (a) Viscosity models M9, M20, M22 and M25 with $$V_{{\rm{lm}}}^*$$ = 3.2 × 10−6 m3 mol−1, (b) temperature distributions for M9, M20 and M25 and (c) lower-mantle viscosity structures for M22, M23 and M24 (see Table 2). The lithospheric thickness is 65 km for M9 and M20 and 100 km for M22 and M25. The plate age (t) is 60 Myr for M9 and M22, 100 Myr for M20 and 200 Myr for M25. Figure 5. View largeDownload slide (a) Viscosity models M9, M20, M22 and M25 with $$V_{{\rm{lm}}}^*$$ = 3.2 × 10−6 m3 mol−1, (b) temperature distributions for M9, M20 and M25 and (c) lower-mantle viscosity structures for M22, M23 and M24 (see Table 2). The lithospheric thickness is 65 km for M9 and M20 and 100 km for M22 and M25. The plate age (t) is 60 Myr for M9 and M22, 100 Myr for M20 and 200 Myr for M25. We next discuss the case for viscosity models with H = 100 km and t = 60 Myr by employing viscosity models M22, M23 and M24 (see Table 2 for the $$V_{{\rm{lm}}}^*$$ values and Fig. 5(c) for the lower-mantle viscosity structures). The $${\skew5\dot{J}_2}$$ value for M22 with $$V_{{\rm{lm}}}^*$$ = 3.2 × 10−6 m3 mol−1 is −5.88 × 10−11 yr−1, and we need $$V_{{\rm{lm}}}^*$$ ∼ (3.15 − 3.125) × 10−6 m3 mol−1 (M23 and M24) to obtain the rate of ∼ −6.25 × 10−11 yr−1, which is related to the reduction of lower-mantle viscosity from sections ‘G’ to ‘F’ of the black curve in Fig. 2(b). This yields $${\rm{\Delta }}V_{{\rm{lm}}}^{\rm{*}}$$ ∼ (0.05 − 0.075) × 10−6 m3 mol−1 for $$\Delta {\skew5\dot{J}_2}$$ ∼ 0.36 × 10−11 yr−1. In the case of H = 100 km and t = 200 Myr, the rate for M25 is −5.66 × 10−11 yr−1 and we require $$V_{{\rm{lm}}}^*$$ ∼ (3.12 − 3.08) × 10−6 m3 mol−1 (M26 and M27) for the rate of ∼ −6.25 × 10−11 yr−1, and $${\rm{\Delta }}V_{{\rm{lm}}}^{\rm{*}}$$ ∼ (0.08 − 0.12) × 10−6 m3 mol−1 for $$\Delta {\skew5\dot{J}_2}$$∼ 0.59 × 10−11 yr−1. The $${\rm{\Delta RSL}}_{\rm{L}}^{{\rm{Bar}},{\rm{Bon}}}$$ and $${\rm{\Delta RSL}}_6^{{\rm{Kar}},{\rm{Hal}}}$$ values for viscosity models M20–M27 satisfy the observationally derived estimates as shown in Table 2. The $${\rm{\Delta RSL}}_{\rm{L}}^{{\rm{Bar}},{\rm{Bon}}}$$ value is slightly sensitive to the $${\rm{\Delta }}V_{{\rm{lm}}}^{\rm{*}}$$ as for $${\skew5\dot{J}_2}$$. On the other hand, the $${\rm{\Delta RSL}}_6^{{\rm{Kar}},{\rm{Hal}}}$$ is insensitive to $${\rm{\Delta }}V_{{\rm{lm}}}^{\rm{*}}$$ for a fixed upper-mantle rheological structure, while the value is sensitive to the upper-mantle viscosity structure and lithospheric thickness. This is consistent with the result by Nakada & Okuno (2016) that the differential RSL change for Karumba and Halifax Bay only yields an effective lower-mantle viscosity higher than (2 − 3) × 1021 Pa s. These numerical experiments show that the upper-mantle viscosity structure depending on the plate age, t, has an impact on the three GIA-related observables. However, change in $${\skew5\dot{J}_2}$$ ($$\Delta {\skew5\dot{J}_2}$$) for an adopted upper-mantle viscosity model is ∼0.6 × 10−11 yr−1 at most, which is adjusted by $${\rm{\Delta }}V_{{\rm{lm}}}^{\rm{*}}$$ ∼ 0.1 × 10−6 m3 mol−1 corresponding to variations in the lower-mantle viscosity structure shown in Fig. 5(c). Also, the observationally derived estimates for $${\rm{\Delta RSL}}_{\rm{L}}^{{\rm{Bar}},{\rm{Bon}}}$$ and $${\rm{\Delta RSL}}_6^{{\rm{Kar}},{\rm{Hal}}}$$ are consistent with the values predicted for the viscosity models M20–M27. These results are also applicable to those for other viscosity models with ηlm(670) = (1, 5, 10, 20, 50) × 1021 Pa s. Moreover, the $${\skew5\dot{J}_2}$$ sensitivity to lower-mantle viscosity takes a parabolic form shown in Fig. 2(b) for the simple three-layer viscosity model with upper-mantle rheological parameters (H and ηum) of 50 ≤ H ≤ 200 km and 1020 ≤ ηum ≤ 1021 Pa s (e.g. Mitrovica & Milne 1998; Nakada & Okuno 2003; Nakada et al. 2015). In that case, the impact of $${\rm{\Delta }}V_{{\rm{lm}}}^{\rm{*}}$$ on $$\Delta {\skew5\dot{J}_2}$$ would be nearly equal to that examined in this section. That is, the upper-mantle temperature distribution depending on the plate age t would have little impact on the inference of the lower-mantle viscosity structure described by eqs (2) –(4). We therefore adopt t = 60 Myr in the following discussion. 3.5 Dependence on the chosen ice history model We examine the impacts of adopted ice history models on the three GIA-related observables by using standard viscosity models for the IA20 ice model (M8–M13 in Table 2) and those for the IR20 ice model (M14–M19 in Table 2). Table 3 shows the GIA-related observables based on the IA10, IA30 and IR20 ice models and viscosity models M8–M13, and those based on the IR10, IR30 and IA20 ice models and viscosity models M14–M19. The $${\skew5\dot{J}_2}$$value for IA10, $$\skew5\dot{J}_2^{{\rm{IA}}10}$$, is approximated using the value for IA20, $$\skew5\dot{J}_2^{{\rm{IA}}20}$$, by $$\skew5\dot{J}_2^{{\rm{IA}}10}$$ ∼ $$\skew5\dot{J}_2^{{\rm{IA}}20}$$ + 0.3 × 10−11 yr−1 (see Table 2). Also, we get $$\skew5\dot{J}_2^{{\rm{IA}}30}$$ ∼$$\skew5\dot{J}_2^{{\rm{IA}}20}$$ − 0.3 × 10−11 yr−1. To obtain the rate of ∼ −6.25 × 10−11 yr−1 for IA10, we require a change in $$V_{{\rm{lm}}}^*$$ ($${\rm{\Delta V}}_{{\rm{lm}}}^{\rm{*}}$$) of ∼ −0.05 × 10−6 m3 mol−1 for the standard viscosity models for IA20 (relation from sections ‘G’ to ‘F’ of the black curve in Fig. 2b). The $${\rm{\Delta }}V_{{\rm{lm}}}^{\rm{*}}$$ value for IA30 is about 0.05 × 10−6 m3 mol−1 (relation from sections ‘E’ to ‘F’ of the black curve in Fig. 2b). The value of $${\rm{\Delta RSL}}_6^{{\rm{Kar}},{\rm{Hal}}}$$ is almost insensitive to the ice model and about 1.25 m. On the other hand, the $${\rm{\Delta RSL}}_{\rm{L}}^{{\rm{Bar}},{\rm{Bon}}}$$ value increases with a decreasing Antarctic ESL component (ESLSH), and it is larger than 6 m for IA10 with ESLSH∼10 m (ESLNH∼120 m) and smaller than 5 m for IA30 with ESLSH∼30 m (ESLNH∼100 m). However, the $${\rm{\Delta RSL}}_{\rm{L}}^{{\rm{Bar}},{\rm{Bon}}}$$ values for IA10 and IA30 satisfy the observationally derived estimate. For IR10 and IR30, we get similar sensitivities of the GIA-related observables to the ice model. For example, the relations between$${\skew5\dot{J}_2}$$ and an ice model are $$\skew5\dot{J}_2^{{\rm{IR}}10}$$ ∼$$\skew5\dot{J}_2^{{\rm{IR}}20}$$ + 0.35 × 10−11 yr−1 and $$\skew5\dot{J}_2^{{\rm{IR}}30}$$ ∼$$\skew5\dot{J}_2^{{\rm{IR}}20}$$ − 0.35 × 10−11 yr−1. Table 3. Three GIA-related observables, $${\skew5\dot{J}_2}$$ (×10−11yr−1), $${\rm{\Delta RSL}}_{\rm{L}}^{{\rm{Bar}},{\rm{Bon}}}$$ (m) and $${\rm{\Delta RSL}}_6^{{\rm{Kar}},{\rm{Hal}}}$$ (m), based on the IA10, IA30 and IR20 ice models and standard viscosity models M8–M13, and those based on the IR10, IR30 and IA20 ice models and standard viscosity models M14–M19. Model name  $${\skew5\dot{J}_2}$$  $${\rm{\Delta RSL}}_{\rm{L}}^{{\rm{Bar}},{\rm{Bon}}}$$  $${\rm{\Delta RSL}}_6^{{\rm{Kar}},{\rm{Hal}}}$$  $${\skew5\dot{J}_2}$$  $${\rm{\Delta RSL}}_{\rm{L}}^{{\rm{Bar}},{\rm{Bon}}}$$  $${\rm{\Delta RSL}}_6^{{\rm{Kar}},{\rm{Hal}}}$$  $${\skew5\dot{J}_2}$$  $${\rm{\Delta RSL}}_{\rm{L}}^{{\rm{Bar}},{\rm{Bon}}}$$  $${\rm{\Delta RSL}}_6^{{\rm{Kar}},{\rm{Hal}}}$$    IA10  IA10  IA10  IA30  IA30  IA30  IR20  IR20  IR20  M8  −5.98  6.2  1.17  −6.56  1.8  1.18  −6.46  3.7  1.17  M9  −5.96  7.7  1.23  −6.54  3.2  1.25  −6.42  5.2  1.23  M10  −6.0  8.6  1.25  −6.57  4.1  1.26  −6.43  6.2  1.25  M11  −5.96  9.0  1.24  −6.53  4.6  1.26  −6.39  6.6  1.24  M12  −5.99  9.0  1.23  −6.57  4.7  1.25  −6.42  6.6  1.23  M13  −5.98  8.4  1.22  −6.56  4.3  1.25  −6.41  6.1  1.23  Model name  $${\skew5\dot{J}_2}$$  $${\rm{\Delta RSL}}_{\rm{L}}^{{\rm{Bar}},{\rm{Bon}}}$$  $${\rm{\Delta RSL}}_6^{{\rm{Kar}},{\rm{Hal}}}$$  $${\skew5\dot{J}_2}$$  $${\rm{\Delta RSL}}_{\rm{L}}^{{\rm{Bar}},{\rm{Bon}}}$$  $${\rm{\Delta RSL}}_6^{{\rm{Kar}},{\rm{Hal}}}$$  $${\skew5\dot{J}_2}$$  $${\rm{\Delta RSL}}_{\rm{L}}^{{\rm{Bar}},{\rm{Bon}}}$$  $${\rm{\Delta RSL}}_6^{{\rm{Kar}},{\rm{Hal}}}$$    IR10  IR10  IR10  IR30  IR30  IR30  IA20  IA20  IA20  M14  −5.87  6.6  1.17  −6.61  1.8  1.19  −6.06  4.5  1.18  M15  −5.90  7.8  1.22  −6.63  3.1  1.25  −6.10  5.8  1.24  M16  −5.90  8.8  1.24  −6.58  4.1  1.26  −6.08  6.8  1.25  M17  −5.90  9.1  1.23  −6.60  4.5  1.26  −6.10  7.0  1.25  M18  −5.87  9.2  1.23  −6.56  4.6  1.25  −6.08  7.1  1.24  M19  −5.87  8.8  1.21  −6.56  4.3  1.24  −6.07  6.8  1.23  Model name  $${\skew5\dot{J}_2}$$  $${\rm{\Delta RSL}}_{\rm{L}}^{{\rm{Bar}},{\rm{Bon}}}$$  $${\rm{\Delta RSL}}_6^{{\rm{Kar}},{\rm{Hal}}}$$  $${\skew5\dot{J}_2}$$  $${\rm{\Delta RSL}}_{\rm{L}}^{{\rm{Bar}},{\rm{Bon}}}$$  $${\rm{\Delta RSL}}_6^{{\rm{Kar}},{\rm{Hal}}}$$  $${\skew5\dot{J}_2}$$  $${\rm{\Delta RSL}}_{\rm{L}}^{{\rm{Bar}},{\rm{Bon}}}$$  $${\rm{\Delta RSL}}_6^{{\rm{Kar}},{\rm{Hal}}}$$    IA10  IA10  IA10  IA30  IA30  IA30  IR20  IR20  IR20  M8  −5.98  6.2  1.17  −6.56  1.8  1.18  −6.46  3.7  1.17  M9  −5.96  7.7  1.23  −6.54  3.2  1.25  −6.42  5.2  1.23  M10  −6.0  8.6  1.25  −6.57  4.1  1.26  −6.43  6.2  1.25  M11  −5.96  9.0  1.24  −6.53  4.6  1.26  −6.39  6.6  1.24  M12  −5.99  9.0  1.23  −6.57  4.7  1.25  −6.42  6.6  1.23  M13  −5.98  8.4  1.22  −6.56  4.3  1.25  −6.41  6.1  1.23  Model name  $${\skew5\dot{J}_2}$$  $${\rm{\Delta RSL}}_{\rm{L}}^{{\rm{Bar}},{\rm{Bon}}}$$  $${\rm{\Delta RSL}}_6^{{\rm{Kar}},{\rm{Hal}}}$$  $${\skew5\dot{J}_2}$$  $${\rm{\Delta RSL}}_{\rm{L}}^{{\rm{Bar}},{\rm{Bon}}}$$  $${\rm{\Delta RSL}}_6^{{\rm{Kar}},{\rm{Hal}}}$$  $${\skew5\dot{J}_2}$$  $${\rm{\Delta RSL}}_{\rm{L}}^{{\rm{Bar}},{\rm{Bon}}}$$  $${\rm{\Delta RSL}}_6^{{\rm{Kar}},{\rm{Hal}}}$$    IR10  IR10  IR10  IR30  IR30  IR30  IA20  IA20  IA20  M14  −5.87  6.6  1.17  −6.61  1.8  1.19  −6.06  4.5  1.18  M15  −5.90  7.8  1.22  −6.63  3.1  1.25  −6.10  5.8  1.24  M16  −5.90  8.8  1.24  −6.58  4.1  1.26  −6.08  6.8  1.25  M17  −5.90  9.1  1.23  −6.60  4.5  1.26  −6.10  7.0  1.25  M18  −5.87  9.2  1.23  −6.56  4.6  1.25  −6.08  7.1  1.24  M19  −5.87  8.8  1.21  −6.56  4.3  1.24  −6.07  6.8  1.23  View Large We next examine the three GIA-related observables based on the IR20 ice model and standard viscosity models M8–M13 (Table 3). The differential RSL changes at the LGM and 6 kyr BP are almost the same as those for the IA20 ice model as inferred from the difference of (0.02–0.04) × 10−6 m3 mol−1 between the $$V_{{\rm{lm}}}^*$$ values for the standard viscosity models for IA20 and IR20 (Tables 2 and 3). Also, the $$\skew5\dot{J}_2^{{\rm{IR}}20}$$ value is given by $$\skew5\dot{J}_2^{{\rm{IR}}20}$$∼$$\skew5\dot{J}_2^{{\rm{IA}}20}$$ − (0.15–0.2) × 10−11 yr−1. Summarizing these numerical experiments, we may conclude that the GIA-induced $${\skew5\dot{J}_2}$$ is less sensitive to the ice model although a change in $$V_{{\rm{lm}}}^*$$ ($${\rm{\Delta }}V_{{\rm{lm}}}^{\rm{*}}$$) of ∼0.05 × 10−6 m3 mol−1 is required for a change in the $${\skew5\dot{J}_2}$$ magnitude ($$\Delta {\skew5\dot{J}_2}$$) of ∼0.3 × 10−11 yr−1. The differential RSL change for Karumba and Halifax Bay is almost insensitive to the ice model. The value of $${\rm{\Delta RSL}}_{\rm{L}}^{{\rm{Bar}},{\rm{Bon}}}$$ increases with an increasing Antarctic ESL component, but the observationally derived estimate of (0–25) m is obtained for the ice and viscosity models adopted in this section. 4 INFERENCE OF VISCOSITY AT 670 KM DEPTH, ηlm(670) 4.1 $${\skew5\dot{J}_n}$$ for n = 3–6 In inferring mantle viscosity from the GIA-induced $${\skew5\dot{J}_n}$$ for n > 2, it would be desirable to evaluate the rates based on the permissible viscosity solution satisfying the GIA-induced $${\skew5\dot{J}_2}$$ of −(6.0 − 6.5) × 10−11 yr−1, which depends on the ice model as inferred from the M8 to M13 viscosity models for IA20 and M14 to M19 for IR20, respectively. We first discuss this point based on the rates for n ≥ 2 predicted for IA20. Fig. 6 shows the rates as a function of $$V_{{\rm{lm}}}^*$$ value for viscosity models with ηlm(670) = (1, 2, 5, 10, 20, 50) × 1021 Pa s, H = 65 km,$${\bar{\eta }_{{\rm{um}}}}$$ = 2 × 1020 Pa s and t = 60 Myr. The standard viscosity model with ηlm (670) = 2 × 1021 Pa s for IA20 (M9), for example, corresponds to the model with $$V_{{\rm{lm}}}^*$$ = 3.2 × 10−6 m3 mol−1 in Fig. 6(b). These figures indicate that the rates for n > 2 are almost insensitive to the $$V_{{\rm{lm}}}^*$$ value for viscosity models satisfying the GIA-induced $${\skew5\dot{J}_2}$$ of −(6.0 − 6.5) × 10−11 yr−1. For example, we consider the rates for ηlm(670) = 1021 Pa s (Fig. 6a). The viscosity models with $$V_{{\rm{lm}}}^*$$ ∼ (3.424 − 3.55) × 10−6 m3 mol−1 explain the observationally derived $${\skew5\dot{J}_2}$$ value, and the $${\skew5\dot{J}_n}$$ (n > 2) for such a range of $$V_{{\rm{lm}}}^*$$ is nearly constant. This is also true for the models with ηlm(670) = (2, 5, 10, 20, 50) × 1021 Pa s. Figure 6. View largeDownload slide GIA-induced $${\skew5\dot{J}_n}$$ (n ≥ 2) as a function of $$V_{{\rm{lm}}}^*$$ value for viscosity models with $$E_{{\rm{um}}}^{\rm{*}}$$ = 250 kJ mol−1, ηlm(670) = (1, 2, 5, 10, 20, 50) × 1021 Pa s, H = 65 km,$${\bar{\eta }_{{\rm{um}}}}$$ = 2 × 1020 Pa s and t = 60 Myr and the IA20 ice model. The shaded region shows the range for GIA-induced $${\skew5\dot{J}_2}$$ of −(6.0 − 6.5) × 10−11 yr−1. Figure 6. View largeDownload slide GIA-induced $${\skew5\dot{J}_n}$$ (n ≥ 2) as a function of $$V_{{\rm{lm}}}^*$$ value for viscosity models with $$E_{{\rm{um}}}^{\rm{*}}$$ = 250 kJ mol−1, ηlm(670) = (1, 2, 5, 10, 20, 50) × 1021 Pa s, H = 65 km,$${\bar{\eta }_{{\rm{um}}}}$$ = 2 × 1020 Pa s and t = 60 Myr and the IA20 ice model. The shaded region shows the range for GIA-induced $${\skew5\dot{J}_2}$$ of −(6.0 − 6.5) × 10−11 yr−1. We next examine the sensitivity of the $${\skew5\dot{J}_n}$$ (n > 2) to the ice model. Table 4 shows the $${\skew5\dot{J}_n}$$ (n > 2) based on the IA20 and IR20 ice models (ESLSH∼20 m) and standard viscosity models for IA20 (M8–M13) and IR20 (M14–M19), respectively. We consider the rates for M8 and M14 with ηlm(670) = 1021 Pa s. The $$\skew5\dot{J}_n^{{\rm{IA}}20}$$ value for M8 is almost the same as the $$\skew5\dot{J}_n^{{\rm{IA}}20}$$ for M14. For example, the $$\skew5\dot{J}_3^{{\rm{IA}}20}$$ is −1.25 × 10−11 yr−1 for M8 and −1.21 × 10−11 yr−1 for M14. However, $$\skew5\dot{J}_n^{{\rm{IA}}20}$$ is significantly different from $$\skew5\dot{J}_n^{{\rm{IR}}20}$$. For example, the $${\skew5\dot{J}_4}$$ value for M8 is −2.67 × 10−11 yr−1 for IA20 and −3.16 × 10−11 yr−1 for IR20. Table 4. GIA-induced $${\skew5\dot{J}_n}$$ of n > 2 (×10−11 yr−1) based on the IA20 and IR20 ice models and standard viscosity models M8 to M13 (for IA20) and M14 to M19 (for IR20). Model name  $${\skew5\dot{J}_3}$$ (IA20)  $${\skew5\dot{J}_4}$$ (IA20)  $${\skew5\dot{J}_5}$$ (IA20)  $${\skew5\dot{J}_6}$$ (IA20)  $${\skew5\dot{J}_3}$$ (IR20)  $${\skew5\dot{J}_4}$$ (IR20)  $${\skew5\dot{J}_5}$$ (IR20)  $${\skew5\dot{J}_6}$$ (IR20)  M8  –1.25  –2.67  4.17  1.40  –0.55  –3.16  4.66  0.56  M9  –1.17  –2.49  3.92  1.35  –0.55  –2.90  4.33  0.66  M10  –1.08  –2.27  3.56  1.25  –0.54  –2.60  3.88  0.70  M11  –1.03  –2.12  3.31  1.18  –0.53  –2.43  3.60  0.69  M12  –1.01  –2.04  3.15  1.13  –0.52  –2.32  3.42  0.67  M13  –1.01  –1.98  3.02  1.09  –0.52  –2.26  3.28  0.66  Model name  $${\skew5\dot{J}_3}$$ (IA20)  $${\skew5\dot{J}_4}$$ (IA20)  $${\skew5\dot{J}_5}$$ (IA20)  $${\skew5\dot{J}_6}$$ (IA20)  $${\skew5\dot{J}_3}$$ (IR20)  $${\skew5\dot{J}_4}$$ (IR20)  $${\skew5\dot{J}_5}$$ (IR20)  $${\skew5\dot{J}_6}$$ (IR20)  M14  –1.21  –2.63  4.14  1.40  –0.54  –3.11  4.63  0.57  M15  –1.15  –2.47  3.90  1.35  –0.54  –2.87  4.30  0.67  M16  –1.05  –2.24  3.53  1.24  –0.53  –2.56  3.84  0.70  M17  –1.01  –2.10  3.29  1.17  –0.52  –2.40  3.58  0.69  M18  –0.98  –2.00  3.12  1.12  –0.51  –2.28  3.38  0.67  M19  –0.97  –1.95  2.99  1.08  –0.50  –2.22  3.24  0.66  Model name  $${\skew5\dot{J}_3}$$ (IA20)  $${\skew5\dot{J}_4}$$ (IA20)  $${\skew5\dot{J}_5}$$ (IA20)  $${\skew5\dot{J}_6}$$ (IA20)  $${\skew5\dot{J}_3}$$ (IR20)  $${\skew5\dot{J}_4}$$ (IR20)  $${\skew5\dot{J}_5}$$ (IR20)  $${\skew5\dot{J}_6}$$ (IR20)  M8  –1.25  –2.67  4.17  1.40  –0.55  –3.16  4.66  0.56  M9  –1.17  –2.49  3.92  1.35  –0.55  –2.90  4.33  0.66  M10  –1.08  –2.27  3.56  1.25  –0.54  –2.60  3.88  0.70  M11  –1.03  –2.12  3.31  1.18  –0.53  –2.43  3.60  0.69  M12  –1.01  –2.04  3.15  1.13  –0.52  –2.32  3.42  0.67  M13  –1.01  –1.98  3.02  1.09  –0.52  –2.26  3.28  0.66  Model name  $${\skew5\dot{J}_3}$$ (IA20)  $${\skew5\dot{J}_4}$$ (IA20)  $${\skew5\dot{J}_5}$$ (IA20)  $${\skew5\dot{J}_6}$$ (IA20)  $${\skew5\dot{J}_3}$$ (IR20)  $${\skew5\dot{J}_4}$$ (IR20)  $${\skew5\dot{J}_5}$$ (IR20)  $${\skew5\dot{J}_6}$$ (IR20)  M14  –1.21  –2.63  4.14  1.40  –0.54  –3.11  4.63  0.57  M15  –1.15  –2.47  3.90  1.35  –0.54  –2.87  4.30  0.67  M16  –1.05  –2.24  3.53  1.24  –0.53  –2.56  3.84  0.70  M17  –1.01  –2.10  3.29  1.17  –0.52  –2.40  3.58  0.69  M18  –0.98  –2.00  3.12  1.12  –0.51  –2.28  3.38  0.67  M19  –0.97  –1.95  2.99  1.08  –0.50  –2.22  3.24  0.66  View Large We further consider the sensitivity using ice models IA20 (ESLSH∼20 m) and IA30 (ESLSH∼30 m) and viscosity models with $$V_{{\rm{lm}}}^*$$ = 2.85 × 10−6 (M10) and 2.9 × 10−6 m3 mol−1 for ηlm (670) = 5 × 1021 Pa s (see Fig. 6c). The $$\skew5\dot{J}_2^{{\rm{IA}}20}$$ and $$\skew5\dot{J}_4^{{\rm{IA}}20}$$ values for M10 are −6.28 × 10−11 and −2.27 × 10−11 yr−1 and those for $$V_{{\rm{lm}}}^*$$ = 2.9 × 10−6 m3 mol−1 are −5.96 × 10−11 and −2.21 × 10−11 yr−1, respectively. That is, the difference between the $$\skew5\dot{J}_4^{{\rm{IA}}20}$$ for both viscosity models is significantly smaller than that for $$\skew5\dot{J}_2^{{\rm{IA}}20}$$. On the other hand, the $$\skew5\dot{J}_2^{{\rm{IA}}30}$$ and $$\skew5\dot{J}_4^{{\rm{IA}}30}$$ values are −6.57 × 10−11 and −2.93 × 10−11 yr−1 for M10, and −6.24 × 10−11 and −2.85 × 10−11 yr−1 for $$V_{{\rm{lm}}}^*$$ = 2.9 × 10−6 m3 mol−1, respectively. That is, the $$\skew5\dot{J}_4^{{\rm{IA}}30}$$ value is nearly similar for both viscosity models as obtained for the IA20 ice model. However, $$\skew5\dot{J}_4^{{\rm{IA}}30}$$ is significantly different from $$\skew5\dot{J}_4^{{\rm{IA}}20}$$. The results for the IA20 and IR20 ice models (Table 4) and those for the IA20 and IA30 ice models indicate that it would be possible to discuss preferred ice and viscosity models satisfying observationally derived $${\skew5\dot{J}_n}$$ (n > 2) by employing standard viscosity models M8–M13 for IA20. More recently, Nakada & Okuno (2017) estimated the GIA-induced $${\skew5\dot{J}_n}$$ (n = 3–6) by considering the geodetically derived $${\skew5\dot{J}_n}$$ by Cheng et al. (1997) and Cox & Chao (2002) and recent (after ∼1900) melting of glaciers and the Greenland and Antarctic ice sheets (Vaughan et al. 2013). The approach by Nakada & Okuno (2017) is essentially the same as that by Tosi et al. (2005) except for recent melting models adopted in both studies (see detailed discussion by Nakada & Okuno 2017). We should also point out that Devoti et al. (2001) inferred a value of the order of 1020 Pa s for the upper-mantle viscosity from the analyses using the geodetically derived $${\skew5\dot{J}_n}$$ (n = 2–6) and suggested an ongoing mass redistribution associated with mass instabilities in Greenland and Antarctica. In inferring mantle viscosity from the GIA-induced $${\skew5\dot{J}_n}$$ (n > 2), Nakada & Okuno (2017) evaluated uncertainties on the rates based on geodetic data and recent melting for the period of 1900–1990, and also by considering the GIA-induced $${\skew5\dot{J}_2}$$ of−(6.0 − 6.5) × 10−11yr−1 inferred from observationally derived $${\skew5\dot{J}_2}$$ for the periods of 1976–1990 and 2002–2001 (Roy & Peltier 2011; Cheng et al. 2013) and the melting rates for both periods (Vaughan et al. 2013, see table 2 in Nakada & Okuno 2017). In this study, we examine the GIA-induced $${\skew5\dot{J}_n}$$ (n = 3–6) considering the uncertainties obtained by Nakada & Okuno (2017; see Fig. 7). Figure 7. View largeDownload slide GIA-induced $${\skew5\dot{J}_n}$$ (n ≥ 2) based on the IA* ice models and standard viscosity models for IA20 (M8–M13), and GIA-induced $${\skew5\dot{J}_4}$$ and $${\skew5\dot{J}_6}$$ based on the geodetically derived $${\skew5\dot{J}_n}$$ by Cheng et al. (1997) and recent (after ∼1900) melting of glaciers and the Greenland and Antarctic ice sheets (Vaughan et al. 2013, see the text for uncertainties on the rates). Figure 7. View largeDownload slide GIA-induced $${\skew5\dot{J}_n}$$ (n ≥ 2) based on the IA* ice models and standard viscosity models for IA20 (M8–M13), and GIA-induced $${\skew5\dot{J}_4}$$ and $${\skew5\dot{J}_6}$$ based on the geodetically derived $${\skew5\dot{J}_n}$$ by Cheng et al. (1997) and recent (after ∼1900) melting of glaciers and the Greenland and Antarctic ice sheets (Vaughan et al. 2013, see the text for uncertainties on the rates). Fig. 7 shows the rates based on the IA10, IA20 and IA30 ice models and standard viscosity models for IA20 (M8–M13). The $${\skew5\dot{J}_4}$$ for IA20, for example, is denoted by (4, IA20). Here, we do not discuss the rates for n = 3 and 5 in inferring mantle viscosity because the odd zonal rates are uncertain due to weakness in the orbital geometry (Cheng et al. 1997 and see also Nakada & Okuno 2017). Fig. 7(a) shows the predicted GIA-induced $${\skew5\dot{J}_n}$$ for n = 2, 4 and 6 and observationally derived estimates for n = 4 and 6 obtained by Nakada & Okuno (2017) using geodetically derived data by Cheng et al. (1997) and recent melting by Vaughan et al. (2013). The observationally derived rate for n = 4 is consistent with the predicted rate regardless of chosen viscosity and ice model. On the other hand, the rate for n = 6 is consistent with the rates predicted for IA30 with ESLSH∼30 m. The inference of an ice model only from $${\skew5\dot{J}_6}$$ is, however, inconclusive if we consider the statement by Cheng et al. (1997) that the accuracy of the estimates of the zonal secular rates from long time-series multisatellite laser ranging data has been difficult to verify and only $${\skew5\dot{J}_2}$$ has been evaluated with confidence. That is, it would be safe to say that the present analysis for the GIA-induced $${\skew5\dot{J}_n}$$ (n > 2) cannot provide useful constraints on the ice model and the viscosity at 670 km depth, ηlm(670) (see also Devoti et al. 2001; Tosi et al. 2005). 4.2 Postglacial RSL changes at Southport, Bermuda and Everglades based on the standard viscosity models We examine postglacial RSL changes at Southport, Bermuda and Everglades in the intermediate region for the North American ice sheet (see Fig. 1a) based on the standard viscosity models for the IA20 and IR20 ice models in inferring the viscosity at 670 km depth, ηlm(670). These RSL changes would be sensitive to the ice model as well as the viscosity structure of the mantle, and therefore inference of the ηlm(670) value depends on an adopted ice model. In that case, our method employed in this study requires that we should discuss these RSL changes based on the standard viscosity models corresponding to a specific ice model, for example, IA10 [steps (i) and (ii) explained in Introduction]. The standard viscosity models, which should be defined for each ice model separately, are required to explain the three GIA-related observables and to give the $${\skew5\dot{J}_2}$$ value of ∼ −6.25 × 10−11 yr−1 (Section 3.4). Although we examined this point for the three GIA-related observables in Section 3.4, we briefly discuss the $$V_{{\rm{lm}}}^*$$ dependence of RSL changes at these sites and $${\skew5\dot{J}_2}$$ by employing viscosity models M28, M9 and M29 for ηlm(670) = 2 × 1021 Pa s with $$V_{{\rm{lm}}}^*$$ values of 3.05 × 10−6, 3.20 × 10−6 and 3.35 × 10−6 m3 mol−1, respectively (Table 2 and see also Fig. 8(a) for their lower-mantle viscosity structures). Figure 8. View largeDownload slide (a) Lower-mantle viscosity structures M9, M28 and M29 with $$V_{{\rm{lm}}}^*$$ values of 3.20 × 10−6, 3.05 × 10−6 and 3.35 × 10−6 m3 mol−1, respectively (see Table 1), and (b)–(d) observed postglacial RSL changes at Southport, Bermuda and Everglades and predicted RSL changes for the IA20 ice model and viscosity models M9, M28 and M29. The data sources for observed RSL changes are: https://www.ncdc.noaa.gov/data-access/paleoclimatology-data/datasets/paleoceanography for the three sites and recent data by Engelhart et al. (2011) for Southport. Figure 8. View largeDownload slide (a) Lower-mantle viscosity structures M9, M28 and M29 with $$V_{{\rm{lm}}}^*$$ values of 3.20 × 10−6, 3.05 × 10−6 and 3.35 × 10−6 m3 mol−1, respectively (see Table 1), and (b)–(d) observed postglacial RSL changes at Southport, Bermuda and Everglades and predicted RSL changes for the IA20 ice model and viscosity models M9, M28 and M29. The data sources for observed RSL changes are: https://www.ncdc.noaa.gov/data-access/paleoclimatology-data/datasets/paleoceanography for the three sites and recent data by Engelhart et al. (2011) for Southport. The $$\skew5\dot{J}_2^{{\rm{IA}}20}$$ ($${\skew5\dot{J}_2}$$ for IA20) values for M28 and M29 are −7.07 × 10−11 and −5.40 × 10−11 yr−1, respectively (−6.24 × 10−11 yr−1 for M9 with $$V_{{\rm{lm}}}^*$$ = 3.20 × 10−6 m3 mol−1); namely, $$\Delta {\skew5\dot{J}_2}$$ ∼ 1.7 × 10−11 yr−1 for $${\rm{\Delta V}}_{{\rm{lm}}}^{\rm{*}}$$ ∼ 0.3 × 10−6 m3 mol−1. By considering the relation for $$\Delta {\skew5\dot{J}_2}$$ and $${\rm{\Delta }}V_{{\rm{lm}}}^{\rm{*}}$$ and $$\skew5\dot{J}_2^{{\rm{IA}}10}$$ = −5.96 × 10−11 yr−1 for M9 (Table 3), we obtain the $$\skew5\dot{J}_2^{{\rm{IA}}10}$$ value of ∼−6.25 × 10−11 yr−1 for a viscosity model with $$V_{{\rm{lm}}}^*$$ ∼ 3.15 × 10−6 m3 mol−1 (see also Fig. 2b). Also, we obtain the $$\skew5\dot{J}_2^{{\rm{IA}}30}$$ of ∼−6.25 × 10−11yr−1 for a viscosity model with $$V_{{\rm{lm}}}^*$$ ∼ 3.25 × 10−6 m3 mol−1 ($$\skew5\dot{J}_2^{{\rm{IA}}30}$$ = −6.54 × 10−11yr−1 for M9). On the other hand, such a change in $$V_{{\rm{lm}}}^*$$ (∼0.3 × 10−6 m3 mol−1) has negligible impacts on postglacial RSL changes at Southport, Everglades and Bermuda as shown in Fig. 8 (the data sources for observed RSL changes are: https://www.ncdc.noaa.gov/data-access/paleoclimatology-data/datasets/paleoceanography for three sites and recent data by Engelhart et al. 2011 for Southport). This is true for other standard viscosity models for IA20 although we do not show the results here. That is, it is possible to discuss preferred viscosity and IA* ice models satisfying observed RSL changes at these sites by employing standard viscosity models M8–M13 for IA20. This is also true for the standard viscosity models for IR20. Fig. 9 shows the RSL changes at these sites based on the IA10, IA20 and IA30 ice models and viscosity models M8–M13 with $${\bar{\eta }_{{\rm{um}}}}$$ = 2 × 1020 Pa s. Here, we mainly discuss the RSL change at 6 kyr BP to examine the sensitivity of postglacial RSL change to the viscosity at 670 km depth, ηlm(670), and an ice model. The magnitude at 6 kyr BP for Southport slightly decreases with increasing ηlm(670) value (total change of ∼2 m) and decreases with decreasing ESL component of the NH ice sheets (ESLNH) (total change of ∼3.5 m). That is, the misfit between the observed and predicted RSL changes becomes slightly smaller for the model with higher ηlm(670) value and smaller ESLNH (larger ESLSH). However, the predicted RSL change is distinctly different from the observed one regardless of the viscosity and ice models. The ice model sensitivity of the RSL change is also true for RSL changes at Bermuda and Everglades. The RSL change at Bermuda is, however, more sensitive to the ηlm(670) value than that at Southport. The RSL change at Everglades is highly sensitive to the ηlm(670) value, and the observed RSL change is consistent with the predicted one for the models with ηlm(670) ≥ 2 × 1022 Pa s (viscosity models M12 and M13) and IA30 (ESLNH ∼ 100 m and ESLSH ∼ 30 m). These numerical experiments for the postglacial RSL change in the intermediate region for the North American ice sheet point out that the sensitivity to the ηlm(670) increases with distance of observation site from the edge of the ice sheet (see Fig. 1a) in the case of $${\bar{\eta }_{{\rm{um}}}}$$ = 2 × 1020 Pa s (we discuss this point based on the results in Fig. 12). Fig. 10 also shows the RSL changes based on the IR* ice models and standard viscosity models for IR20. The RSL changes for these models are almost the same as those for the standard viscosity models for IA20 (see also the $$V_{{\rm{lm}}}^*$$ values for the standard viscosity models with an identical ηlm(670) value, for example, M8 and M14). Figure 9. View largeDownload slide Observed RSL changes at Southport, Bermuda and Everglades and predicted ones based on the IA* ice models and viscosity models M8–M13 with $${\bar{\eta }_{{\rm{um}}}}$$ = 2 × 1020 Pa s. Figure 9. View largeDownload slide Observed RSL changes at Southport, Bermuda and Everglades and predicted ones based on the IA* ice models and viscosity models M8–M13 with $${\bar{\eta }_{{\rm{um}}}}$$ = 2 × 1020 Pa s. Figure 10. View largeDownload slide As in Fig. 7, except for the IR* ice models. Figure 10. View largeDownload slide As in Fig. 7, except for the IR* ice models. The results obtained in this section are shortly summarized as follows: the observed postglacial RSL changes at Southport, Bermuda and Everglades cannot be simultaneously explained by the RSL changes for the standard viscosity models with $${\bar{\eta }_{{\rm{um}}}}$$ = 2 × 1020 Pa s regardless of the chosen ice models adopted here, particularly for RSL changes at Southport and Bermuda. In the next section, we analyse these RSL changes using the viscosity models with $${\bar{\eta }_{{\rm{um}}}}$$ > 2 × 1020 Pa s. 4.3 Postglacial RSL changes at Southport, Bermuda and Everglades based on the viscosity models with $${\bar{\eta }_{{\rm{um}}}}$$ > 2 × 1020 Pa s In inferring the viscosity at 670 km depth from the RSL changes at these sites, we assume that the upper-mantle viscosity is smaller than 1021 Pa s and that the viscosity at 670 km depth is higher than 1021 Pa s, ηlm(670) ≥ 1021 Pa s. Of course, this assumption has no physical evidence. For the $$V_{{\rm{um}}}^{\rm{*}}$$ value of 4.0 × 10−6 m3 mol−1 adopted for standard viscosity models, however, the upper-mantle viscosity around ∼670 km depth becomes higher than 1021 Pa s in the case of $${\bar{\eta }_{{\rm{um}}}}$$ > 2 × 1020 Pa s. We therefore adopt $$V_{{\rm{um}}}^{\rm{*}}$$ = 2.0 × 10−6 m3 mol−1 producing nearly constant upper-mantle viscosity and examine the impact of the upper-mantle viscosity structure on the RSL change. We first examine its impact by employing viscosity models M9, M30 and M31 with ηlm(670) = 2 × 1021 Pa s and $${\bar{\eta }_{{\rm{um}}}}$$ = 2 × 1020 Pa s (Fig. 11a and Table 2). The $$V_{{\rm{um}}}^{\rm{*}}$$ value for M9 is 4.0 × 10−6 m3 mol−1 and that for M30 and M31 is 2.0 × 10−6 m3 mol−1. The elastic lithospheric thickness (H) is 65 km for M9 and M30 and 100 km for M31. The RSL changes at these sites may prefer the elastic lithospheric thickness of 100 km for a viscosity model with upper-mantle viscosity of ∼1021 Pa s and lower-mantle viscosity higher than 1022 Pa s (see figs 17 and 18 in Nakada & Okuno 2016). Fig. 11 shows the RSL changes based on the IA20 ice model and these viscosity models. The following results change insignificantly even if we adopt another viscosity model with ηlm(670) = (1, 5, 10, 20, 50) × 1021 Pa s. The difference between the RSL changes at 6 kyr BP for M9 and M30 is ∼1.5 m for Southport and nearly zero for Bermuda. The difference between those for M30 with H = 65 km and M31 with H = 100 km is also ∼1.5 m for Southport and are nearly zero for Bermuda and Everglades. Thus, the $$V_{{\rm{um}}}^{\rm{*}}$$ value and elastic lithospheric thickness affect the RSL changes at these sites, but these impacts are rather small compared with that for the viscosity jump at 670 km depth depending on the $${\bar{\eta }_{{\rm{um}}}}$$ value (Fig. 12). We therefore examine these RSL changes based on the viscosity models with $$V_{{\rm{um}}}^{\rm{*}}$$ = 2.0 × 10−6 m3 mol−1 and H = 100 km to discuss the viscosity jump at 670 km depth. Figure 11. View largeDownload slide (a) Upper-mantle viscosity structures M9, M30 and M31 with ηlm(670) = 2 × 1021 Pa s and $${\bar{\eta }_{{\rm{um}}}}$$ = 2 × 1020 Pa s, and (b)–(d) observed postglacial RSL changes at Southport, Bermuda and Everglades and predicted ones for the IA20 ice model and viscosity models M9, M30 and M31. The $$V_{{\rm{um}}}^{\rm{*}}$$ value for M9 is 4.0 × 10−6 m3 mol−1 and that for M30 and M31 is 2.0 × 10−6 m3 mol−1 (Table 2). Figure 11. View largeDownload slide (a) Upper-mantle viscosity structures M9, M30 and M31 with ηlm(670) = 2 × 1021 Pa s and $${\bar{\eta }_{{\rm{um}}}}$$ = 2 × 1020 Pa s, and (b)–(d) observed postglacial RSL changes at Southport, Bermuda and Everglades and predicted ones for the IA20 ice model and viscosity models M9, M30 and M31. The $$V_{{\rm{um}}}^{\rm{*}}$$ value for M9 is 4.0 × 10−6 m3 mol−1 and that for M30 and M31 is 2.0 × 10−6 m3 mol−1 (Table 2). Figure 12. View largeDownload slide RSL changes at 6 kyr BP based on the viscosity models with ηlm(670) = (1, 2, 5, 10, 20, 50) × 1021 Pa s and $${\bar{\eta }_{{\rm{um}}}}$$ = (2, 3, 4, 5, 7, 9) × 1020 Pa s and the IA* ice models. The lower-mantle viscosity structures are the same as those for the standard viscosity models for IA20 (M8–M13). The shaded regions show the observational constraints for RSL changes at 6 kyr BP: −8.0 ± 1.5, −6.0 ± 2.0 and −4.0 ± 1.0 m for Southport, Bermuda and Everglades, respectively. Figure 12. View largeDownload slide RSL changes at 6 kyr BP based on the viscosity models with ηlm(670) = (1, 2, 5, 10, 20, 50) × 1021 Pa s and $${\bar{\eta }_{{\rm{um}}}}$$ = (2, 3, 4, 5, 7, 9) × 1020 Pa s and the IA* ice models. The lower-mantle viscosity structures are the same as those for the standard viscosity models for IA20 (M8–M13). The shaded regions show the observational constraints for RSL changes at 6 kyr BP: −8.0 ± 1.5, −6.0 ± 2.0 and −4.0 ± 1.0 m for Southport, Bermuda and Everglades, respectively. Fig. 12 shows the RSL changes at 6 kyr BP based on the viscosity models with ηlm(670) = (1, 2, 5, 10, 20, 50) × 1021 Pa s and $${\bar{\eta }_{{\rm{um}}}}$$ = (2, 3, 4, 5, 7, 9) × 1020 Pa s and ice models IA10 (ESLNH∼120 m, ESLSH∼10 m), IA20 (ESLNH∼110 m, ESLSH∼20 m) and IA30 (ESLNH∼100 m, ESLSH∼30 m). The lower-mantle viscosity structures are the same as those for the standard viscosity models for IA20 (M8–M13). We adopt−8.0 ± 1.5,−6.0 ± 2.0 and−4.0 ± 1.0 m for RSL changes at 6 kyr BP for Southport, Bermuda and Everglades, respectively. The RSL changes at these sites increase with increasing ηlm(670) and $${\bar{\eta }_{{\rm{um}}}}$$ values. We consider the ηlm(670) dependency for a specific $${\bar{\eta }_{{\rm{um}}}}$$ value. The RSL changes at Southport and Bermuda for a higher $${\bar{\eta }_{{\rm{um}}}}$$ model are more dependent on the ηlm(670) value compared with those for a lower one. For example, the difference between the RSL changes at Southport for ηlm(670) = 1021 and 5 × 1022 Pa s is less than 1 m for the model with $${\bar{\eta }_{{\rm{um}}}}$$ = 2 × 1020 Pa s, but larger than 7 m for the model with 9 × 1020 Pa s. Of course, the dependency for $${\bar{\eta }_{{\rm{um}}}}$$ = 2 × 1020 Pa s is consistent with that for the standard viscosity models shown in Fig. 9. On the other hand, the ηlm(670) dependency for RSL change at Everglades is almost independent of $${\bar{\eta }_{{\rm{um}}}}$$ value and the difference between the RSL changes for ηlm(670) = 1021 and 5 × 1022 Pa s is about ∼4 m for all $${\bar{\eta }_{{\rm{um}}}}$$ values (see also Fig. 9). We summarize the permissible solutions for each ice model. The solutions for IA10 are as follows: $${\bar{\eta }_{{\rm{um}}}}$$ ∼ 9 × 1020 and ηlm(670)∼ 5 × 1022 Pa s for Southport, $${\bar{\eta }_{{\rm{um}}}}$$ ∼ (7 − 9) × 1020 and ηlm(670) ≥ 1022 Pa s for Bermuda, and $${\bar{\eta }_{{\rm{um}}}}$$ ≥ 4 × 1020 and ηlm(670) > 5 × 1021 Pa s for Everglades. Those for IA20 are: $${\bar{\eta }_{{\rm{um}}}}$$ ∼ 9 × 1020 and ηlm(670) ≥ 2 × 1022 Pa s for Southport, $${\bar{\eta }_{{\rm{um}}}}$$ ∼ (7 − 9) × 1020 and ηlm(670) > 6 × 1021 Pa s for Bermuda, and $${\bar{\eta }_{{\rm{um}}}}$$ ≥ 4 × 1020 and ηlm(670) > 3 × 1021 Pa s, and $${\bar{\eta }_{{\rm{um}}}}$$ ∼ 3 × 1020 and ηlm(670) ≥ 2 × 1022 Pa s for Everglades. The solutions for IA30 are nearly similar to those for IA20 and as follows: $${\bar{\eta }_{{\rm{um}}}}$$ ∼ (7 − 9) × 1020 and ηlm(670) ≥ 1022 Pa s for Southport, $${\bar{\eta }_{{\rm{um}}}}$$ ∼ (7 − 9) × 1020 and ηlm(670) ≥ 5 × 1021 Pa s, and $${\bar{\eta }_{{\rm{um}}}}$$ ∼ 5 × 1020 and ηlm(670) ≥ 2 × 1022 Pa s for Bermuda, and $${\bar{\eta }_{{\rm{um}}}}$$ ∼ (4 − 9) × 1020 and ηlm(670) ∼ (2 − 20) × 1021 Pa s, and $${\bar{\eta }_{{\rm{um}}}}$$ ∼ 3 × 1020 and ηlm(670) ≥ 7 × 1021 Pa s for Everglades. These results clearly indicate the trade-off between the ice and viscosity models, and the permissible ηlm(670) value decreases with increasing (decreasing) ESLSH (ESLNH) value. Here, we assume a laterally homogeneous viscosity structure although the permissible $${\bar{\eta }_{{\rm{um}}}}$$ value may be related to the distance of observation site from the edge of the ice sheet (Fig. 1a) and approximately decrease with increasing distance. Then, the permissible $${\bar{\eta }_{{\rm{um}}}}$$ and ηlm(670) values are as follows: (i) $${\bar{\eta }_{{\rm{um}}}}$$ ∼ 9 × 1020 and ηlm(670)∼ 5 × 1022 Pa s for IA10, (ii) $${\bar{\eta }_{{\rm{um}}}}$$ ∼ 9 × 1020 and ηlm(670) ∼ 2 × 1022 Pa s for IA20 and (iii) $${\bar{\eta }_{{\rm{um}}}}$$ ∼ (7 − 9) × 1020 and ηlm(670) ∼ 1022 Pa s for IA30. Fig. 13 shows the observed RSL changes and predicted ones based on the IA30 ice model and viscosity models with ηlm(670) = (2, 5, 10, 20) × 1021, $${\bar{\eta }_{{\rm{um}}}}$$ = (2, 3, 4, 5, 7, 9) × 1020 Pa s and H = 100 km. The observed RSL changes are generally consistent with the predicted ones for viscosity models with $${\bar{\eta }_{{\rm{um}}}}$$ ∼ (7 − 9) × 1020 and ηlm(670) ∼ 1022 Pa s. Although the inferred $${\bar{\eta }_{{\rm{um}}}}$$ and ηlm(670) values from postglacial RSL changes at only three sites may be preliminary, there is no doubt that the RSL changes in the intermediate region of the North American ice sheet have a crucial potential for inference of a viscosity jump at 670 km depth. Figure 13. View largeDownload slide Observed postglacial RSL changes at Southport, Bermuda and Everglades and predicted ones based on the IA30 ice model and viscosity models with ηlm(670) = (2, 5, 10, 20) × 1021, $${\bar{\eta }_{{\rm{um}}}}$$ = (2, 3, 4, 5, 7, 9) × 1020 Pa s and H = 100 km. Figure 13. View largeDownload slide Observed postglacial RSL changes at Southport, Bermuda and Everglades and predicted ones based on the IA30 ice model and viscosity models with ηlm(670) = (2, 5, 10, 20) × 1021, $${\bar{\eta }_{{\rm{um}}}}$$ = (2, 3, 4, 5, 7, 9) × 1020 Pa s and H = 100 km. 5 CONCLUDING REMARKS A variety of viscosity models described by different temperature and pressure distributions and activation energy and volume were employed in inferring a reliable mantle viscosity structure from selected GIA data sets. We first constructed standard viscosity models with $${\bar{\eta }_{{\rm{um}}}}$$ = 2 × 1020 Pa s (M8–M13 for the IA20 ice model and M14–M19 for IR20) satisfying the observationally derived three GIA-related observables: GIA-induced $${\skew5\dot{J}_2}$$, and differential RSL changes for the LGM sea levels at Barbados and Bonaparte Gulf and RSL changes at 6 kyr BP for Karumba and Halifax Bay (see Table 2 and Fig. 4). The $${\bar{\eta }_{{\rm{um}}}}$$ value for standard viscosity models may be a typical value for the oceanic upper-mantle viscosity (Lambeck et al.2014, 2017). These viscosity models, characterized by a viscosity of ∼1023 Pa s in the deep mantle, correspond to the permissible viscosity solutions for the two-layer lower-mantle viscosity model using the same GIA data sets by Nakada & Okuno (2016). Such a viscosity model with 1022 − 1023 Pa s in the deep mantle has already been reported by several GIA studies (Ivins et al. 1993; Mitrovica 1996; Vermeersen et al. 1997; Lau et al. 2016), and a more recently reported lower-mantle viscosity structure by Lau et al. (2016) may correspond to the M8 (M14) viscosity model from the viewpoint of no viscosity jump at 670 km depth. It would be fair to note that lower-mantle viscosity structures M8–M13 (M14–M19) may correspond to other representing structures for the lower-mantle viscosity of ηlm ≥ 1022 Pa s inferred using the simple three-layer viscosity model. Such viscosity models with ηlm ≥ 1022 Pa s have been inferred from secular rates of geopotential coefficients up to degree 8 (Tosi et al. 2005), far-field sea level data (Lambeck et al. 2014), GIA-induced $${\skew5\dot{J}_2}$$ (Nakada et al. 2015), GIA-induced $${\skew5\dot{J}_2}$$ and LGM sea levels at Barbados and Bonaparte Gulf (Nakada et al. 2016) and RSL observations, tilting of palaeo-lake shorelines and present-day crustal displacements for the North American Late Wisconsin ice sheet (Lambeck et al. 2017). To constrain the viscosity at 670 km depth, ηlm(670), we examined the GIA-induced $${\skew5\dot{J}_n}$$ (n = 4 and 6) and postglacial RSL changes at Southport, Bermuda and Everglades in the intermediate region for the North American ice sheet. GIA-induced $${\skew5\dot{J}_4}$$ and $${\skew5\dot{J}_6}$$ based on the geodetic data by Cheng et al. (1997) and the recent melting by Vaughan et al. (2013, see Nakada & Okuno 2017) may constrain the ice model, but cannot provide a constraint on the viscosity at 670 km depth (see Fig. 7). However, the inference of the ice model from the GIA-induced $${\skew5\dot{J}_n}$$ for n > 2 would be inconclusive at this time if we consider the accuracy of the estimates of the zonal secular rates for n > 2 (Cheng et al. 1997). The postglacial RSL changes at this region would be little influenced by the detailed melting histories of the North American ice sheet and largely dependent on their gross melting history (e.g. Nakada & Lambeck 1987). In fact, the RSL changes for ice models with an equal ESLNH (ESLSH) value, for example, IA30 and IR30 with ESLNH∼100 m (ESLSH∼30 m), are nearly identical as shown in Figs 9 and 10. However, the standard viscosity models with $${\bar{\eta }_{{\rm{um}}}}$$ = 2 × 1020 Pa s cannot simultaneously explain the observed RSL changes at these sites. That is, the observed RSL changes at Southport and Bermuda are distinctly different from predicted ones regardless of viscosity and ice models, but the observed RSL change at Everglades is consistent with the predicted one for viscosity models with ηlm(670) ≥ 2 × 1022 Pa s (M12 and M13) and the IA30 ice model. On the other hand, the analyses of these RSL changes based on the viscosity models with $${\bar{\eta }_{{\rm{um}}}}$$ > 2 × 1020 Pa s and lower-mantle viscosity structures M8–M13 (M14–M19) yield permissible $${\bar{\eta }_{{\rm{um}}}}$$ and ηlm(670) values although there is a trade-off between the ice and viscosity models: $${\bar{\eta }_{{\rm{um}}}}$$ ∼ (7 − 9) × 1020 and ηlm(670) ∼ (1 − 2) × 1022 Pa s for an ice model with ESLSH∼20 or ∼30 m, and $${\bar{\eta }_{{\rm{um}}}}$$ ∼ 9 × 1020 and ηlm(670) ∼ 5 × 1022 Pa s for an ice model with ESLSH∼10 m (Figs 9, 10, 12 and 13). However, the permissible $${\bar{\eta }_{{\rm{um}}}}$$ for each site appears to decrease with increasing distance from the edge of the ice sheet (Fig. 1a), which may reflect a weak (moderate) laterally heterogeneous upper-mantle viscosity (for example, see Fig. 12 for the IA30 ice model). The viscosity at 670 km depth higher than 1022 Pa s is also supported from the analyses by Nakada & Okuno (2016) using the two-layer lower-mantle viscosity model described by depth-averaged effective viscosities of η670,Dand ηD,2891 (D = 1191 and 1691 km). That is, they showed that the observed postglacial RSL changes at Southport and Bermuda are consistent with the predicted ones for viscosity models with ηum > 6 × 1020, η670,1191 > 1022 (η670,1691 > 2 × 1022) and ηD,2891∼ (5 − 10) × 1022 Pa s and an ice model with ESLSH∼20 or ∼30 m. However, our inferred viscosity model is distinctly different from the viscosity model preferred by Lau et al. (2016) except for the deep mantle viscosity of 1022 − 1023 Pa s. That is, their preferred lower-mantle viscosity structure is as follows (fig. 4 by Lau et al. 2016): an average viscosity of 1021 Pa s from 670 to ∼1500 km depth (no viscosity jump at 670 km depth), a significant viscosity increase in the bottom half of the lower mantle and 1022 − 1023 Pa s in the deep mantle. Although we cannot provide a persuasive explanation for the difference between the lower-mantle viscosity structures (in the top half of the lower mantle) by Lau et al. (2016) and this study, the difference may be attributed to different GIA data sets used in both studies. This is an issue in the future. The inferred upper- and lower-mantle viscosities using postglacial RSL changes at Southport, Bermuda and Everglades and lower-mantle viscosity models satisfying three GIA-related observables are wholly consistent with the inference by Lambeck et al. (2017) using GIA-related geological and geophysical data in the glaciated region for the North American Late Wisconsin ice sheet. Their preferred upper- and lower-mantle viscosities using the simple three-layer viscosity model are (3.5 − 7.5) × 1020 and (0.8 − 2.8) × 1022 Pa s, respectively. The upper-mantle viscosity for both studies is significantly different from the value of (1 − 2) × 1020 Pa s obtained from analyses of far-field ocean islands and continental margin data (Lambeck et al. 2014) largely influenced by isostatic response due to the oceanic upper-mantle viscosity (Lambeck et al. 2017). Then, Lambeck et al. (2017) indicated a lateral variation of upper-mantle viscosity by considering the upper-mantle viscosities inferred from analyses of far-field continental margin data and cratonic continental data, which was first suggested by Nakada & Lambeck (1991). This study, using RSL changes in the intermediate region for the North American ice sheet, also suggests a lateral variation of upper-mantle viscosity. Also, Lambeck et al. (2017) prefer the ESL component of ∼80 m for the North American Late Wisconsin ice sheet (their ice sheet model LW-6). The ESL component for the North American ice sheet of IA30 and IR30 in our study is ∼80 m (see Section 2.2) and almost identical to the value of LW-6. Mantle viscosity structure adopted in this study depends on temperature distribution and activation energy and volume, and it is difficult to discuss the impact of each quantity on the inferred viscosity model. Our preferred viscosity model has a significant viscosity jump at 670 km depth and requires an order of gradual increase in viscosity within the lower mantle. That is, models of smooth depth variation in the lower-mantle viscosity following $$\eta ( z )\ \propto {\rm{\ exp}}[ {( {E_{{\rm{lm}}}^* + P( z )V_{{\rm{lm}}}^*} )/{\rm{R}}T( z )} ]$$ with constant $$E_{{\rm{lm}}}^*$$ and $$V_{{\rm{lm}}}^*$$ are consistent with the GIA observations. Such an increase is not inconsistent with the inference by Mitrovica & Forte (2004) using GIA and convection data sets. The smooth viscosity variation in the lower mantle is consistent with the inference based on the theory of diffusion in ionic solids by Karato (1981), but inconsistent with the inference of a strong increase in viscosity throughout the upper 900 km of the lower mantle from high pressure (at room temperature) deformation data for ferropericlase by Marquardt & Miyagi (2015, see also Rudolph et al. 2015). In this study, the $$V_{{\rm{lm}}}^*$$ value for the preferred lower-mantle viscosity structure is ∼(2 − 3) × 10−6 m3 mol−1 (Table 2) for an adopted temperature distribution and lower-mantle activation energy of $$E_{{\rm{lm}}}^*$$ = 250 kJ mol−1. We hope that the lower-mantle viscosity structure with an exponential profile inferred from GIA data sets may be useful in discussing the mineral physics of the lower-mantle rheology and the extrapolation of laboratory observations to the geological environments. This would be possible in the near future if we consider the recent deformation experiments for a mixture of bridgmanite and magnesiowüstite at lower-mantle conditions by Girard et al. (2016). ACKNOWLEDGEMENTS We thank S. Karato and T. Kubo for constructive suggestions and discussions, and two anonymous reviewers and the editor (J. C. Afonso) for their constructive comments. This work was partly supported by the Japanese Ministry of Education, Science and Culture (Grand-in-Aid for Scientific Research No. 16K05543). REFERENCES Argus D.F., Peltier W.R., Drummond R., Moore A.W., 2014. 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Geophysical Journal InternationalOxford University Press

Published: Mar 1, 2018

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