GIA Modeling with CitcomSVE

GLACIAL ISOSTATIC ADJUSTMENT

5.2 GIA Modeling with CitcomSVE

low viscosity lithospheric weak zone may amplify GIA stress magnitudes by a factor of 5-10 compared to homogeneous lithosphere (Grollimund and Zoback, 2001; Wu and Mazzotti, 2007) and could produce clockwise rotations of post-glacial rebound stresses in the crust above the weak zone (Wu and Mazzotti, 2007). Inclusion of such weak zones has also yielded better fits to both modeled stress and strain data in the St. Lawrence River Valley (Mazzotti et al., 2005).

We explore the hypothesis that glacial isostatic adjustment promotes intraplate seis- micity in eastern North America via perturbation to the intraplate stress field and reactivation of pre-existing faults. We develop high resolution global models of the solid earth response to glacial loading and unloading with CitcomSVE (Zhong et al., 2022), a spherical finite-element viscoelastic GIA code that implements the sea level equation and ice-loading history of ICE-6G (Peltier et al., 2015) for a fully 3D or 1D Earth viscosity structure. The models use a viscosity structure based on the seismically and geologically constrained thermal structure implemented in the mantle flow models with CitcomS (see Chapter 4). Like in the CitcomS models, we include local-scale, low-viscosity lithospheric weak zones at the locations of the geologically mapped aulacogens (Whitmeyer and Karlstrom, 2007) and other tectonically inherited structures. We calculate the stress tensor, with which we com- pute the 𝑆_{𝐻 𝑚 𝑎𝑥} direction, the deviatoric stress magnitude, and the Coulomb stress on known faults. We compare our results for the present day to stresses of the World Stress Map (Heidbach et al., 2018) and to those obtained from mantle flow models using the same Earth structure (Chapter 4).

pseudo-spectral approach (Mitrovica and Peltier, 1991; Mitrovica et al., 1994; Wu and Wal, 2003; Mitrovica and Milne, 2003; Kendall et al., 2005; Spada and Stocchi, 2007), finite element methods (Zhong et al., 2003; Paulson et al., 2005; Lund, 2005; Wang and Wu, 2006; Dal Forno et al., 2012; A et al., 2013; Zhong et al., 2022), spectral-finite element methods (Martinec, 2000; Tanaka et al., 2011), and finite volume methods (Latychev et al., 2005). Numerical methods such as finite- elements have the advantage of being able to incorporate fully 3D Earth structure spanning several orders of magnitude of variation in lateral viscosity (Lund, 2005;

Zhong et al., 2015). It is ultimately the lateral gradients in viscosity that give rise to gradients in the velocity field that lead to the stress perturbations in the lithosphere in which we are interested (see Chapter 4). Thus, it is essential to utilize fully 3D viscosity structure, rather than compare the stress fields arising from different local 1D viscosity structures.

With respect to glacially induced faulting, most models to date assume a flat-Earth and thus cannot account for the ocean load by means of the sea level equation (Wu and Hasegawa, 1996; Steffen et al., 2021), or they make use of a spectral approach (Wu and Johnston, 2000) that cannot accommodate 3D viscosity. To our knowledge only one previous study on GIA induced seismicity has been published using a spherical finite-element GIA model formulation (Craig et al., 2016; Steffen et al., 2021). Sphericity of the model is important for large diameter loads, such as the Laurentide, especially when considering areas farther from the ice margin, as in the central-eastern United States (Wu and Johnston, 1998), and it has the advantage of incorporating the effect of ocean loading and the distal effects of Fennoscandian and Antarctic glaciations.

We model the impact of GIA on intraplate stress using the fully spherical finite element code CitcomSVE, which models dynamic deformation of a viscoelastic and incompressible planetary mantle in response to surface loading (Zhong et al., 2003; Zhong et al., 2022). CitcomSVE was built from the widely used purely viscous geodynamics finite-element code CitcomS (Zhong et al., 2003; Zhong et al., 2000; Zhong et al., 2008) by implementing viscoelastic deformation within a Lagrangian formulation compatible with either linear or non-linear viscosity (Kang et al., 2022). The code has been extensively benchmarked for time-dependent loading on the surface of a viscoelastic mantle, using both single harmonic loading and the ICE-6G ice-history model (Bellas et al., 2020; A et al., 2013; Paulson et al., 2005; Zhong et al., 2003; Zhong et al., 2022), as well as tidal loading on

either an elastic (Zhong et al., 2012) or viscoelastic (Zhong et al., 2022) mantle. By comparing surface loading, tidal loading, and ice loading model predictions to semi- analytic solutions (A et al., 2013; Han and Wahr, 1995; Paulson et al., 2005), the code has demonstrated successful prediction of deformation rates, displacements, and relative sea level changes (Zhong et al., 2022).

CitcomSVE solves the equations for load induced deformation derived from the laws of conservation of mass and momentum and of gravitation for an incompressible self-gravitating, viscoelastic mantle (Maxwell body) overlying an inviscid fluid core (Zhong et al., 2003; Zhong et al., 2022):

∇ ·𝑢¯ =0 (5.1)

∇ ·𝜎¯¯ +𝜌

0∇𝜙− ∇(𝑢¯·𝜌

0𝑔𝑟ˆ)−𝜌^𝐸

1𝑔+𝜌

0∇𝑉_𝑎=0 (5.2)

∇²𝜙=−4𝜋𝐺 𝜌^𝐸

1, (5.3)

where ¯𝑢is the displacement vector (𝑢_𝑟 being displacement in the radial direction);

𝜙 is the perturbation to the gravitational potential due to deformation; 𝑉_𝑎 is the applied potential, when applicable; ¯¯𝜎 is the stress tensor; 𝜌

0 is the unperturbed mantle density; 𝑔 is the gravitational acceleration; 𝜌^𝐸

1 = −𝑢¯ · ∇𝜌

0 is the Eulerian density perturbation for an incompressible medium; and G is the gravitational constant. Poisson’s equation (Eq. 5.3) states that the density distribution determines the gravitational potential and acceleration. The notion of a self-gravitating Earth arises from the fact that when deforming stresses are applied, movement of masses alters the local gravity and its potential but such that Poisson’s equation remains satisfied (Wu et al., 2021). The applied deformation gives rise to strain 𝜀_{𝑖 𝑗} and stress 𝜎_{𝑖 𝑗} in addition to the perturbed density, all of which must together satisfy conservation of momentum (Eq. 5.2). The terms of this equation, from left-to- right, are the divergence of the stress, the perturbed gravity field (i.e., self-gravity), the advection of the pre-stress, and the buoyancy force from the local perturbation.

Zero-shear force boundary conditions are applied at the surface and core-mantle boundary (CMB) such that both can deform dynamically. With the incompressibility assumption, the three primary sources of mass anomalies are surface topography (i.e., radial displacement), CMB topography, and the surface loads themselves. The surface boundary condition is

𝜎_{𝑖 𝑗}𝑛_𝑗 =−𝜎

0𝑛_𝑖, for𝑟 =𝑟_𝑠, (5.4)

representing the pressure loads at the surface (𝜎

0) from GIA as a function of time and space.

With a Maxwell rheology, the total deformation is the sum of the elastic and viscous strains:

𝜀_{𝑖 𝑗} =𝜀^𝑒

𝑖 𝑗 +𝜀^𝑣

𝑖 𝑗 = 1 2

𝜕 𝑢_𝑖

𝜕 𝑥_𝑗 +

𝜕 𝑢_𝑗

𝜕 𝑥_𝑖

. (5.5)

𝜀^𝑒

𝑖 𝑗 and𝜀¤^𝑣

𝑖 𝑗 are the elastic strain tensor and viscous strain rate tensor, respectively:

𝜀^𝑒

𝑖 𝑗 = 1 2𝜇

(𝜎_{𝑖 𝑗} +𝑃𝛿_{𝑖 𝑗}) (5.6)

¤ 𝜀^𝑣

𝑖 𝑗 = 1 2𝜂

(𝜎_{𝑖 𝑗} +𝑃𝛿_{𝑖 𝑗}). (5.7)

𝑃is the dynamic pressure,𝛿_{𝑖 𝑗} is the Kronecker delta,𝜇is the shear modulus, and𝜂 is the viscosity. The resulting rheological equation is the sum of the time derivatives of equations 5.6 and 5.7:

𝜎_{𝑖 𝑗} + 𝜂 𝜇

¤ 𝜎_{𝑖 𝑗} =−

𝑃+ 𝜂

𝜇 𝑃¤

𝛿_{𝑖 𝑗} +2𝜂𝜀¤_{𝑖 𝑗}. (5.8) Because the deformation is time-dependent due to viscoelasticity, the rheologi- cal equations are discretized in time via an incremental displacement formulation (Zhong et al., 2003; Zhong et al., 2022) such that the stress tensor at time𝑡includes a pre-stress term accounting for the stresses at the previous timestep:

𝜎^𝑛

𝑖 𝑗 =−𝑃^𝑛𝛿_{𝑖 𝑗} +2 ˜𝜂Δ𝜀^𝑛

𝑖 𝑗 + 𝑓 𝜏^Pre

𝑖 𝑗 . (5.9)

The pre-stress is the deviatoric stress at timestep𝑛−1 (𝜏^pre

𝑖 𝑗 =𝜏^𝑛−1

𝑖 𝑗 =𝜎^𝑛−1

𝑖 𝑗 +𝑃^𝑛−1𝛿_{𝑖 𝑗}).

˜

𝜂 =𝜂/(𝛼+Δ𝑡), 𝑓 = (𝛼−Δ𝑡/2)/(𝛼+Δ𝑡/2), and 𝛼= 𝜂/𝜇is the Maxwell time. It is this advection of the pre-stress term in Eq. 5.2 that provides the restoring force of isostasy in the fluid that enables post-glacial rebound (Wu, 1992; Wu et al., 2021).

The stress tensor at a given timestep can in theory be computed from Equation 5.9 after solving for the incremental displacement and effective pressure. However, to avoid needing to calculate dynamic pressure from the effective pressure, only the deviatoric stress𝜏_{𝑖 𝑗} is computed (Equation 5.10).

𝜏^𝑛

𝑖 𝑗 =𝜎^𝑛

𝑖 𝑗 +𝑃^𝑛𝛿_{𝑖 𝑗} =2 ˜𝜂Δ𝜀^𝑛

𝑖 𝑗 + 𝑓 𝜏^pre

𝑖 𝑗 (5.10)

To accurately model GIA, it is important to account for mass exchange and mass redistribution between continental ice and water in the oceans. As such, CitcomSVE fully implements the sea level equation (Eq. 5.11) (Farrell and Clark, 1976), which gives the change in the height of the ocean load since the onset of glaciation (𝐿

0):

𝐿0(𝜃 , 𝜙, 𝑡)= [𝑁(𝜃 , 𝜙, 𝑡) −𝑈(𝜃 , 𝜙, 𝑡) +𝑐(𝑡)]𝑂(𝜃 , 𝜙, 𝑡) (5.11) 𝑐(𝑡)= 1

𝐴0

− 𝑀

ice(𝑡) 𝜌_𝑤

−

∫

(𝑁−𝑈)𝑂 𝑑𝑆

(5.12) The ocean load is a function of the GIA-related geoid anomaly (𝑁) and radial displacement (𝑈), an ocean function (𝑂(𝜃 , 𝜙, 𝑡)) that describes the distribution of ocean versus land (1 for ocean and 0 for land), and the barystatic sea level (𝑐(𝑡), Eq. 5.12), which is a function of the mass of the ice (𝑀

ice), the density of water (𝜌_𝑤), and the geoid anomalies and radial displacements integrated over the area of the oceans (𝐴

0). The mass of the ice is calculated for a given ice-history model.

While the ocean load is dependent on the incremental displacement and the geoid, it also directly impacts them through the equation of motion and Poisson’s equation.

CitcomSVE handles this interdependence through an iterative scheme applied at every time-step, which solves for displacements, gravitational potential, and the sea level height changes self-consistently. With self-gravitation, the code also accounts for the contribution from the gravitational attraction of seawater to the mass of the ice (Austermann and Mitrovica, 2015; Zhong et al., 2008).

CitcomSVE takes either a 1D or 3D viscosity structure as input, along with an incrementally defined ice loading history (Eq. 5.9). Poisson’s equation for the gravitational potential anomaly is solved in the spherical harmonic domain, while displacements are solved on the finite element grid. Upon solving the GIA problem, the program outputs cumulative topography; incremental radial, latitudinal, and longitudinal displacements; and total and incremental gravitational potential at the surface nodal points of the finite element grid; it outputs the second invariant of deviatoric stress and the viscosity for every nodal point in the 3D domain. We added the ability to output all six components of the deviatoric stress tensor for use in our stress analysis.

The ability to impose 3D viscosity structure in our GIA models is key. Glacial rebound at the margins of formerly glaciated areas tends to be sensitive to variations in lithospheric thickness (Zhong et al., 2003; Latychev et al., 2005), suggesting that the effect of GIA on stress perturbations may be enhanced when coupled with

lateral variations in the depth of the lithosphere-asthenosphere boundary (LAB) and lithospheric viscosity. Wu and Mazzotti (2007) have also shown that narrow ductile zones cutting through the full lithosphere have a larger effect on GIA induced deformation than a single mantle weak layer alone. As with our mantle flow models, when assessing the impact of GIA on the intraplate stress field, we likewise consider lateral variations in lithospheric thickness via 3D viscosity structure and how weak- zones may locally amplify stress perturbations (Mazzotti and Townend, 2010; Wu and Mazzotti, 2007). We employ the same geologic and geophysical constraints on the weak-zones as we do using CitcomS (Section 5.3.3).

5.3 Input Ice-Load and Viscosity Structures