LOADING FROM THE FARALLON SLAB
4.4 Data and Methods for Model Input
4.4.2 Seismically Constrained Mantle Thermal Structure
Temperature anomalies create lateral changes in buoyancy that drive mantle flow.
The thermal structure of mantle convection models is often constrained by seismic tomography, assuming the seismic anomalies are thermal in origin. It is com- mon to simply scale the velocity to an effective temperature, a method that several studies have used to constrain the temperature and density structure of the mantle beneath eastern North America (Liu et al., 2008; Spasojevic et al., 2009). The more rigorous approach is to use the full anharmonic and anelastic components of the seismic wave-speed equations and the appropriate elastic moduli values constrained by mineral physics experimental data or theoretical values for a given composition (Karato, 1993; Goes et al., 2000; Cammarano et al., 2003; Goes, 2002; Cammarano and Guerri, 2017). The importance of anelasticity — the viscoelastic behaviour at seismic frequency — in the interpretation of seismic velocities was first demon- strated by Karato (1993). Properly accounting for anelasticity is essential because it significantly reduces the sensitivity of shear velocity to temperature, especially
Figure 4.2: Data-sets used to construct the lithospheric thermal structure used in our geodynamic models. We use the seafloor age grid from Seton et al. (2020) (upper left panel) to compute the oceanic plate thickness (upper right panel) and temperature (middle right panel, 20 km depth) according to the plate cooling model. We combine the oceanic plate thermal model with the continental thermal model of Artemieva (2006) (middle left panel, 80 km depth). A slice of the combined temperature model at 80 km depth is shown in the bottom right panel, and the combined LAB depth, which corresponds to the 1300°C isotherm, is shown in the bottom left panel.
in very hot regions near the solidus. This means temperature anomalies associated with low velocity anomalies in the mantle are significantly smaller than they would be if considering anharmonic effects alone (Karato, 1993; Cammarano and Guerri, 2017). Anelasticity also makes the temperature derivative of velocity strongly tem- perature dependent and hence the conversion of velocity into temperature non-linear (Goes et al., 2000).
With the latter method, compositional effects can largely be neglected in the upper mantle, with uncertainties on estimated temperatures ranging from±100 K in the upper mantle to ±250 K in the shallow lower mantle (Cammarano et al., 2003).
In the upper mantle, many studies (Goes et al., 2000; Goes, 2002; Cammarano et al., 2003) invert the absolute seismic velocity for absolute temperatures. Despite being non-linear, with a single datum (seismic velocity) and a single model pa- rameter (temperature), an inversion scheme can be rather fast and computationally inexpensive. However, due to the limited resolution of seismic tomography, the increase in compositional influence on seismic velocity with depth, and the uncer- tainty on the physical properties of deep mantle minerals (Cammarano et al., 2003), inversion tends to become unstable for the lower mantle and drastically over-predict temperatures in the deep earth. Because of this, we adopt the approach of scaling the velocities to temperature using a numerically derived depth dependent scaling factor that, like the inversion technique, incorporates the effect of anelasticity.
Seismic velocity including both anharmonic and anelastic terms is given by 𝑉(𝑃, 𝑇 , 𝑍 , 𝜔) =𝑉𝑎𝑛 ℎ(𝑃, 𝑇 , 𝑋)
"
1− 𝑄−1(𝜔, 𝑇) 2tan(𝜋 𝑎/2)
#
, (4.9)
where the anharmonic component,𝑉𝑎𝑛 ℎ, is𝑉𝑝 or𝑉𝑠 (Eq. 4.10), and the anelastic component is the term in brackets. Q is the seismic quality factor - the inverse of the seismic attenuation. 𝑄is thought to be weakly frequency dependent, where𝜔𝑎 controls the frequency dependence of attenuation (Eq. 4.11) and𝑎typically ranges from 0.1 to 0.3 (Karato, 1993; Cammarano et al., 2003; Cammarano and Guerri, 2017).
𝑉𝑝=
√︄
𝐾+4𝜇/3
𝜌 (4.10)
𝑉 𝑠=
√︂𝜇 𝜌
Q is defined differently for p- and s-waves and is primarily dependent on the 𝑎 parameter and the activation enthalpy,𝐻:
𝑄𝜇 =𝑄𝑆 =𝐴𝜔𝑎exp(𝑎 𝐻/𝑅𝑇) (4.11) 𝑄𝑃 =(1−𝐿)𝑄−1
𝐾 +𝐿𝑄−1
𝜇 (4.12)
𝐿 =(4/3) (𝑉𝑆/𝑉𝑃)2. (4.13)
The scaling factor of velocity to temperature is typically given as the inverse of the derivative of the natural log of velocity with respect to temperature (Eq. 4.14),
𝛿𝑇 = 𝜕ln𝑉
𝜕𝑇
−1𝛿𝑉
𝑉
, (4.14)
where𝑉 is either𝑉𝑝 or𝑉𝑠. The derivation of 𝜕ln𝑉/𝜕𝑇 is given in Appendix C.1, yielding:
𝜕ln(𝑉)
𝜕𝑇
= 𝜕ln(𝑉𝑜)
𝜕𝑇
− 𝑎 𝐻
𝑅𝑇2
𝑄−1 2tan(𝜋 𝑎/2)
. (4.15)
The first term is the derivative of the anharmonic seismic wave-speed with respect to temperature and is computed as in Eq. 4.16, given that bulk modulus, shear modulus, and density are each a function of temperature (see derivation in Appendix C.1).
𝜕ln(𝑉𝑜)
𝜕𝑇
= 1
2 < 𝜌 > 𝑉2 𝑜
𝜕 < 𝑀 >
𝜕𝑇
−𝑉2
𝑜
𝜕 < 𝜌 >
𝜕𝑇
, (4.16)
where < 𝑀 > and < 𝜌 > are the Voigt-Reuss-Hill averaged elastic moduli and density, where𝑀 is either𝐾+4𝜇/3 for P-waves or𝜇for S-waves, and𝑉𝑜is either 𝑉𝑆or𝑉𝑃, as in Eq 4.9.
We use BurnMan (Myhill et al., 2021; Cottaar et al., 2014), a mineral physics toolbox, to calculate the elastic moduli and density at different temperatures and pressures. We use a pyrolitic composition based on that of Frost (2008) consisting of 56% forsterite, 13% pyrope garnet, 14% clinopyroxene, and 17% orthopyroxene in the upper mantle; 28% wadsleyite, 28% ringwoodite, 40% majorite, and 4% ca- perovskite in the transition zone; and 80% Mg-perovskite, 7% Ca-perovskite, and 13% magnesiowustite in the lower mantle. Pressure and density are computed on the basis of PREM (Dziewonski and Anderson, 1981). BurnMan constructs a composite of these minerals for each defined layer, the elastic properties of which are com- puted using a Voigt-Reuss-Hill averaging scheme and the mineralogical database of Stixrude and Lithgow-Bertelloni (2011), which assumes a Mie-Grueneiesen-Debye equation of state with third order finite strain expansion for the shear modulus.
Temperature derivatives of the elastic moduli are numerically calculated at different depths using BurnMan’s K, G, and𝜌values over a range of temperatures at constant pressure (Figure C.1).
There are a number of different Q models that can be used in the anelastic term in Eq 4.15, either purely seismic Q models (Anderson and Hart, 1978; Dziewonski
and Anderson, 1981) or Q calculated from mineral physics experimental data and calibrated to fit seismic observations (Sobolev et al., 1996; Goes et al., 2000; Cam- marano et al., 2003). As Q is dependent on a number of factors, calculating Q per Eq. 4.11 requires knowledge of the activation enthalpy and frequency dependence of attenuation. We use the SL8 Q model of Anderson and Hart (1978), a precursor to PREM, as in Steinberger and Calderwood (2006) (Figure C.2b). From this Q model, we calculate the corresponding activation enthalpy profile (Figure C.2) to use in the expansion of the temperature derivative of velocity. We use 𝑎 = 0.17 and 𝐴 =0.056 in the upper mantle and 𝑎 = 0.15 and 𝐴 =3.6 in the lower mantle, which yields similar profiles to those used in models Q5 and Q7 in Cammarano et al.
(2003) and Steinberger and Calderwood (2006). The SL8 Q-model of Anderson and Hart (1978) also produces temperature estimates from𝑉𝑝 and𝑉𝑠 that correlate well, as shown in the close to 1-to-1 trend in Figure C.3, suggesting a thermal origin to the anomalies and consistent with the finding that temperatures obtained from P and S waves separately agree well when anelasticity is included (Goes et al., 2000).
With𝑄,𝐻,𝑎, and the derivatives of the moduli computed in BurnMan, we calculate depth dependent scaling factors for converting seismic velocity to temperature (Eq.
4.14) (Figure 4.3b). The computed 𝜕ln(𝑉)/𝜕𝑇 values agree well with those of Steinberger and Calderwood (2006), and the magnitude of the scaling factors is on the order of those used by Spasojevic et al. (2009) and Liu et al. (2008), who find that a scaling of about 2x103°C/km/s produces flow models consistent with plate motions, stratigraphy, and the history of Farallon subduction beneath North America since the Late Cretaceous.
For the global mantle tomography, we use the TX2019 slab model from Lu et al.
(2019). By including a priori 3D slab structure defined by seismicity, this model addresses shortcomings in using seismic tomography to infer density or thermal anomalies that arise from discrepancies between detailed studies of slabs and global tomography models (Lu et al., 2019). In this model, slab locations are defined a priori by seismicity on a 0.1° x 0.1° grid (Lu et al., 2019). The Farallon slab is identifiable as an elongate high S-wave velocity anomaly in both P and S between 600–2200 km in the mid-lower mantle (Lu et al., 2019; Ren et al., 2007) (Fig. 4.4).
P-wave anomalies are determined relative to AK135, and S-wave anomalies are rel- ative to TNA-SNA. We first convert all velocity anomalies to absolute velocity using their respective reference profiles, then recompute 𝛿𝑉𝑝,𝑠 with respect to PREM for consistency with the above derived scaling factors. The velocity anomaly at each
Figure 4.3: Seismic velocity to temperature scaling factors and estimated temperature profiles. a) Vp (dark blue) and Vs (teal) used to construct the scaling factors, predicted from BurnMan for the given composition and consistent with PREM. Shading covers the range of velocities in the TX2019 (Lu et al., 2019) model at a given depth. b) Derivative of the log of viscosity with respect to temperature for the Q models of PREM, QSL8 (Anderson and Hart, 1978), Q5 and Q7 (Cammarano et al., 2003), as discussed in the text. Color for Vp and Vs as in (a). c) The scale factor used to convert the velocity anomalies to temperature anomalies: inverse of the temperature derivative of the natural log of velocity, normalized by the mean radial velocity at that depth. Colors for Vp and Vs as in (a);
line-styles as in (b). d) Resulting temperatures from the velocity to temperature conversion. Black line: geotherm (Steinberger and Calderwood, 2006). Colored shading: full range of temperatures predicted from the velocities of the TX2019 model at a given depth after temperature anomalies are added to the geotherm.
point in the model is multiplied by the scaling factor corresponding to its depth to yield a temperature anomaly for that point. Temperatures are determined from𝑉𝑃 and𝑉𝑆separately and then averaged to get the final temperature anomaly. Absolute temperatures are determined by adding the calculated temperature anomalies to the mantle geotherm (Figure 4.3d). Below tomographic resolution, there is a trade-off between seismic anomaly resolution and grid spacing, which results in a non-unique estimate of absolute seismic velocity and an underestimation of effective temper- ature (Spasojevic et al., 2009). However, because geodynamic models are driven by thermal gradients and changes in buoyancy, exact knowledge of the absolute
temperature is not critical to our set up and the differential temperature computed by scaling the seismic anomalies is sufficient to constrain the thermal buoyancy and flow in the mantle beneath North America.
Figure 4.4: Vertical slices of 3D seismic velocity structure and corresponding mantle thermal structure. Left) Vertical slices of the TX2019 (Lu et al., 2019) velocity field for both𝑑𝑉𝑠and𝑑𝑉𝑝 along two transects. Profile A-A′trends SW-NE across eastern North America and passes through the New Madrid Seismic Zone (marked by the orange inverted triangle) and the Western Quebec and Charlevoix Seismic Zones (red inverted triangle). Profile B-B′trends W-E across the US and passes through the New Madrid Seismic Zone. Right) Vertical slices along the same two transects of the effective temperature field (𝛿𝑇) calculated from the velocity anomalies. Locations of the seismic zones are marked as in the velocity plots; the Farallon slab stands out as an elongate cold thermal anomaly.
The central-eastern North American lithosphere is dominated by high velocities, whereas the eastern Appalachians are characterized by largely average to low ve- locities, including two strong localized low velocity anomalies in the northeast Ap- palachians at around 100 km depth (Schmandt and Lin, 2014; Boyce et al., 2019).
The high velocities of the North American shield terminate abruptly against the Grenville Front to the east, beyond which low velocities dominate in the Grenville Province. Lithospheric thickness is not observed to vary greatly in this region, but there are some abrupt variations from north to south, which are consistent with previous seismic studies of the LAB in eastern North America (Hopper and Fischer, 2018; Artemieva, 2006). These sharp variations could be explained by metasomatic processes that modified the lithospheric mantle composition during subduction along the Laurentian margin (Boyce et al., 2019) and which could have played a role in weakening the crust and lithosphere. This is consistent with the fact that crustal earthquakes in the northern Grenville Province tend to concentrate away from Archean lithosphere and more towards the Appalachian Front and are pervasive throughout the younger altered lithosphere of the Grenville Province in the south (Boyce et al., 2019).