 Research article
 Open access
 Published:
Thermal modeling of subduction zones with prescribed and evolving 2D and 3D slab geometries
Progress in Earth and Planetary Science volume 11, Article number: 14 (2024)
Abstract
The determination of the temperature in and above the slab in subduction zones, using models where the top of the slab is precisely known, is important to test hypotheses regarding the causes of arc volcanism and intermediatedepth seismicity. While 2D and 3D models can predict the thermal structure with high precision for fixed slab geometries, a number of regions are characterized by relatively large geometrical changes over time. Examples include the flat slab segments in South America that evolved from more steeply dipping geometries to the present day flat slab geometry. We devise, implement, and test a numerical approach to model the thermal evolution of a subduction zone with prescribed changes in slab geometry over time. Our numerical model approximates the subduction zone geometry by employing time dependent deformation of a Bézier spline that is used as the slab interface in a finite element discretization of the Stokes and heat equations. We implement the numerical model using the FEniCS open source finite element suite and describe the means by which we compute approximations of the subduction zone velocity, temperature, and pressure fields. We compute and compare the 3D time evolving numerical model with its 2D analogy at crosssections for slabs that evolve to the presentday structure of a flat segment of the subducting Nazca plate.
1 Introduction
1.1 The importance of the thermal structure of flat slab segments
Upon subduction the oceanic lithosphere warms and undergoes metamorphic phase changes. These can release fluids that may lead to arc volcanism and intermediatedepth seismicity. These regions have significant potential for major natural hazards that include underthrusting seismic events along the plate interface and explosive arc volcanism. The geophysical and geochemical processes that cause such hazards are strongly controlled by temperature (for a recent review see van Keken and Wilson 2023a) and it is of great interest to a broad community of Earth scientists to understand the thermal structure of subduction zones.
Of particular interest to us is the intermediatedepth and deep seismicity that occurs at depths below the brittle–ductile transition and require mechanisms other than brittle failure. Shear heating instabilities (Kelemen and Hirth 2007), dehydration embrittlement (Jung et al. 2004; Raleigh and Paterson 1965), and hydrationrelated embrittlement (e.g., Shiina et al. 2013; Shirey et al. 2021; van Keken et al. 2012) are three such mechanisms. The difference between the last two is that dehydration embrittlement would limit the seismicity to the location of metamorphic dehydration whereas hydrationrelated embrittlement may occur wherever fluids that have been liberated by such dehydration reactions migrate. The wide distribution of these fluids that is predicted by fluid flow modeling (Wilson et al. 2014), observed at locations of intermediatedepth seismicity (Bloch et al. 2018; Shiina et al. 2013, 2017), and seen in petrological studies of exhumed oceanic crust (Bebout and PennistonDorland 2016) provides strong support for the latter hypothesis. Thermal modeling suggests intermediatedepth seismicity is limited to be above major dehydration reactions such as that of blueschistout or antigoriteout phase boundaries (Sippl et al. 2019; van Keken et al. 2012; Wei et al. 2017). Further indications of petrological controls on the locations of seismicity are provided by Abers et al. (2013) who showed that the upper plane of seismicity in cold subduction zones tends to be limited to the oceanic crust whereas the seismicity in warm subduction zones occurs in the mantle portion of the subducting slab. See van Keken and Wilson (2023a) for a broader discussion of the relationship between metamorphic dehydration reactions, fluids, and intermediatedepth seismicity.
The observational evidence, combined with modeling constraints, strongly suggests fluids and intermediatedepth earthquakes are related but how can this relationship be further constrained and quantified? Wagner et al. (2020) lay out an elegant motivation that a number of flat slab regions provide natural experiments to study this question. In a number of regions on Earth, such as below Southern Alaska (Finzel et al. 2011), Colombia (Wagner et al. 2017), Mexico, Peru, and Chile (Manea et al. 2017), subduction zones are characterized by flat slabs where upon subduction the slab top stays flat over significant distances after it reaches a certain depth below the continental lithosphere. Periods of flat slab subduction in the geological past have also been suggested to cause orogenic events and ore deposits far from paleogeographically constrained plate boundaries. These include the late Cretaceous to Paleocene Laramide orogeny (see, e.g., Carrapa et al. 2019; Fan and Carrapa 2014) and the Mesozoic South China fold belt (Li and Li 2007). It is generally understood that these slab segments form by trench rollback with the continental lithosphere overriding the slab at shallow depth causing effective flattening.
Specific modernday flat slab regions that would allow us to further quantify the fluidseismicity relationship include the presentday Pampean slab (beneath Chile and Argentina) and the Peruvian flat slab segment. Both likely evolved from steeper subduction to near flat subduction due to the subduction of more buoyant thickened ridges such as the Juan Fernández and Nazca Ridges (Antonijevic et al. 2015; ContrerasReyes et al. 2019; Gutscher et al. 1999). These regions are of particular interest as the intermediatedepth seismicity varies significantly alongtrench. This may be caused by a variable hydration state of the incoming lithosphere and the thermal evolution of the subducting crust and mantle (see Wagner et al. 2020, for observational evidence and the development of a testable hypothesis). These locations therefore suggest at least a qualitative correlation between fluids and seismicity. In order to further test and quantify this correlation we need a good understanding of the thermal structure of the flat slab as it evolves.
Numerical modeling provides an important complement to observational studies as it can predict the subduction zone thermal structure by computing approximate solutions to the partial differential equations governing the conservation of mass, momentum, and thermal energy. Thermal models of flat slab segments have provided insights into the thermal evolution of the slab and overriding lithosphere but have generally been predicted using 2D crosssections (e.g., Axen et al. 2018; Currie and Copeland 2022; English et al. 2003; Liu et al. 2022; Manea and Manea 2011; Marot et al. 2014). The geological evolution of the South Peruvian and Pampean slabs is influenced by strong temporal changes in 3D geometry over time. This makes reliable predictions of their thermal evolution challenging as it needs to be approached with methods that can prescribe such 3D geometrical evolution in a paleogeographically consistent fashion. A few 3D model simulations exist for flat slab reconstructions (Liu et al. 2008; Schmid et al. 2002) or their geodynamical evolution (Jadamec and Haynie 2017; Jadamec et al. 2013; Taramón et al. 2015) that are useful for their intended comparisons with, for example, seismic tomography or plate motions. The employed numerical resolution is generally low and it is difficult to precisely determine the slab surface, which makes it difficult to use these models for the prediction of the precise thermal structure of the subducting slab needed to understand the relationships between earthquake locations, slab stratigraphy, and water content. For this purpose the slab top location should be precisely known and models should have numerical grid spacings of less than a few km in the thermal boundary layers (van Keken et al. 2002, 2008).
Finite Element (FE) models have the particular advantage of being able to precisely prescribe model interfaces (such as the top of the slab or the Moho of the overriding plate) and to be able to use grid refinement that allows for high resolution near thermal boundary layers with coarse grids where the velocity and temperature solutions have small gradients, allowing for high precision and computational efficiency at the same time (see, e.g., Peacock and Wang 1999; Syracuse et al. 2010; van Keken et al. 2002, 2019; Wada and Wang 2009). In this paper we will lay the computational groundwork for such highresolution FE models that, due to advances in computational methods and software design, can be used to study the thermal structure of subduction zones in both 2D and 3D and with timevarying geometry in a consistent fashion. This modeling approach will allow us (and other researchers) to study the relationship between intermediatedepth seismicity, mineralogy, and water content as laid out in, for example, Wagner et al. (2020). While the examples presented here are specific for models that evolve from intermediate dip to flat slab, the opensource modeling framework we present here is sufficiently general to be used for the thermal modeling of any deforming slab geometry.
1.2 Finite element modeling of subduction zones
Thermal modeling of subduction zones aids in the interpretation of the chemical and physical processes that take place in the descending slab. To study the thermal structure of presentday subduction zones with known geometry and forcing functions (such as age of the oceanic lithosphere at the trench and convergence speed), most existing 2D models (see summary in van Keken and Wilson 2023a) combine a kinematically prescribed slab (or slab surface) and a dynamic mantle wedge (in addition to a dynamic slab if only the slab surface velocity is kinematically prescribed). This approach works well for regions where the slab geometry is fixed (even if the forcing functions may change with time). The use of numerical methods also allows for the exploration of 3D geometries enabling the study of subduction obliquity, alongtrench variations in slab geometry, and interactions between multiple slabs (Bengtson and van Keken 2012; Kneller and van Keken 2012; Rosas et al. 2016; Plunder et al. 2018; Wada and He 2017) that can lead to complicated 3D wedge flow that regionally affect the temperature distribution in the subducting slab.
The kinematicdynamic approach described above has significant limitations when changes in geometry occur over the lifetime of a subduction zone. This occurs, for example, when slabs change from intermediate or steep dip to shallower dip or even to flat slab subduction.
The thermal structure of flat slabs has been investigated with 2D steadystate kinematicdynamic models (e.g., Gutscher and Peacock 2003; Manea et al. 2017) but the steadystate nature of these models may obscure important effects of the geometrical evolution of the slab. An alternative approach is to model subduction evolution with dynamical models (e.g., Gerya et al. 2009; Liu et al. 2022) but the modeled evolution may not conform closely to paleogeographic constraints and models of slab evolution. It may also be difficult to precisely trace out the subducting oceanic crust within the evolving slab. The inherent and complex 3D nature of these regions also suggests the best approach to understanding the thermal evolution is achieved in a framework that allows for 3D timedependent modeling where both geometry and forcing functions can be described.
Developing such a framework lays forth a number of requirements that extend beyond the standard FE discretization scheme. The subduction zone computational model must support: (a) a flexible description of the timedependent slab interface geometry; (b) imposition of arbitrary geometry dependent boundary conditions; and (c) scalable distribution of the discretized problem for solution with parallel linear algebra packages (i.e., support efficient computation of small 2D models on local machines and large 3D models on high performance computers).
To address these requirements we use the components of the FEniCS project for assembly of our FE systems (Logg et al. 2012). The work presented here builds on our extensive experience using FEniCS for flexible solution of the equations governing subduction zone thermal structure and mantle dynamics. This includes the modeling of the thermal structure of the subduction slab with and without shear heating (Abers et al. 2020; van Keken et al. 2019) and the role of fluid transport through the slab and mantle wedge (Cerpa et al. 2017; Wilson et al. 2014, 2017). We have demonstrated the precision of the FEniCS applications by comparison to semianalytical approaches (with comparisons, for example, to solutions from Molnar and England (1990) as in van Keken et al. (2019)), published subduction zone benchmarks (van Keken et al. 2008), and intercode comparisons involving detailed reproductions of published models (as shown in van Keken and Wilson 2023b) that were made using fully independent FE software such as Sepran (van den Berg et al. 2015). Other useful geodynamical applications using FEniCS include studies of oceanic crust formation and recycling in mantle convection models (Jones et al. 2021) and the accurate modeling of buoyancy driven flows in incompressible and slightly compressible media (Sime et al. 2021, 2022).
In this paper we will use FEniCS for evolving subduction zone models where the final subduction zone geometry is defined by the approximation of a seismically determined slab surface position (see, e.g., Fig. 1) by a Bézier spline (Bspline). These Bsplines may be manipulated such that a time evolving geometry may be defined. The specification of the Bspline further provides convenient interfacing with ComputerAided Design (CAD) software such that 2D and 3D volumes of the domain of interest may be generated. These CAD geometries furthermore interface with mesh generators in a straightforward manner yielding the spatial tessellation of the domain necessary for discretization of the model by the FE method. The nature of the flow on the slab interface is handled by employing Nitsche’s method for the weak imposition of boundary data (Nitsche 1971). This provides us with much greater flexibility when prescribing the geometrydependent flow direction along the slab interface. Finally, interfacing with the linear algebra solvers provided by the Portable Extensible Toolkit for Scientific Computation (PETSc) library gives us the ability to tailor scalable solution methods for the underlying discretized FE linear system (Balay et al. 2023).
In the remainder of this paper we will provide the mathematical and technical description for this new modeling approach, describe the numerical implementation, provide examples of modeling the evolution of a flat slab segment loosely based on the Chilean/Argentinian geometry in both 2D and 3D, and provide a detailed comparison of how the 3D models differ from simplified 2D crosssectional models to show that 3D timedependent evolution is important for determining the thermal structure of the subducting crust. In a future article we will use this new modeling ability to specifically test the hypotheses regarding the cause of intermediatedepth seismicity as laid out in Wagner et al. (2020).
2 Methods
2.1 Time evolving domain
Let \(t \in {\mathcal {I}} = [0, t_\text {slab}]\) be the time domain of the model where \(t_\text {slab}\) is the total time for the slab surface to deform from its initial to final state. Let \(\Omega (t) \subset {\mathbb {R}}^D\) be the spatial domain of interest at a given time t, where \(D \in \{ 2, 3 \}\) is the spatial dimension. The domain has boundary \(\partial \Omega (t)\) with outward pointing normal unit vector \(\hat{\varvec{n}}(t)\) and tangential unit vectors \(\hat{\varvec{\tau }}_i(t)\), \(i=1,\ldots ,D1\). For brevity of notation we assume that all quantities deriving from the domain are functions of time and write \(\Omega = \Omega (t)\). Furthermore we uniquely define each point in \(\Omega\) according to the standard Cartesian reference frame with coordinate tuples and orthogonal unit directions \(\varvec{x} = (x, z)\) and \((\hat{\varvec{x}}, \hat{\varvec{z}})\) when \(D=2\) and \(\varvec{x} = (x, y, z)\) and \((\hat{\varvec{x}}, \hat{\varvec{y}}, \hat{\varvec{z}})\) when \(D=3\). We further define the radial distance from the origin \(r = \Vert \varvec{x} \Vert _2\), the unit vector pointing in the radial direction \(\hat{\varvec{r}}\), and given the radius of the Earth \(r_0\) we define the depth \(d = r_0  r\).
The exterior boundary is divided into components \(\partial \Omega _i\) such that \(\partial \Omega = \cup _i \partial \Omega _i\) and no component overlaps \(\cap _i \partial \Omega _i = \emptyset\). On the interior of the geometry we prescribe an interior boundary, \(\Gamma _\text {slab}\), with unit normal \(\hat{\varvec{n}}_\text {slab}\). This interior boundary aligns with an approximation of the subduction zone’s slab interface geometry. This interface defines the surface of bisection of the domain \(\Omega\) into \(\Omega _\text {slab}\) and \(\Omega _\text {wedge} \cup \Omega _\text {plate}\) such that \(\Omega = \Omega _\text {slab} \cup \Omega _\text {wedge} \cup \Omega _\text {plate}\) (see Fig. 2). \(\Omega _\text {slab}\) extends a depth \(d_\text {slab}\) beneath \(\Gamma _\text {slab}\) while \(\Omega _\text {plate}\) occupies a thickness of \(d_\text {plate}\) at the top of the domain and above \(\Gamma _\text {slab}\). Each of these subdomains has a boundary with outward pointing unit normal vector \(\hat{\varvec{n}}_\text {slab}\), \(\hat{\varvec{n}}_\text {wedge}\), and \(\hat{\varvec{n}}_\text {plate}\), respectively.
The interior boundary \(\Gamma _\text {slab}\) is further subdivided into components above a coupling depth, \(d_c\), that is embedded in \(\Gamma _\text {slab}\) (a point when \(D=2\) or a curve when \(D=3\)). This coupling depth is the component of \(\Gamma _\text {slab}\) below which the slab and wedge velocities will become fully coupled and above which a fault discontinuity will be modeled. To facilitate this the slab and wedge domains are separated above \(d_c\) such that the slab interface is labeled \(\Gamma _\text {slab fault}\) from the slab side and \(\Gamma _\text {wedge no slip}\) from the wedge side. The specific boundary conditions to be applied on each of these components will be introduced in Sect. 2.3. A schematic diagram of an abstract representation of the \(D=2\) subduction zone domain is shown in Fig. 2. The extrusion of the shown domain in the \(\hat{\varvec{y}}\) direction yields a \(D=3\) subduction zone domain. In this case we label the near and far faces \(\partial \Omega _\text {near}\) and \(\partial \Omega _\text {far}\), respectively.
2.2 Underlying partial differential equations (PDEs)
The subduction zone evolution is modeled by the incompressible Stokes approximation where we seek velocity \(\varvec{u} : \Omega \rightarrow {\mathbb {R}}^D\), pressure \(p : \Omega \rightarrow {\mathbb {R}}\) and temperature \(T : \Omega \rightarrow {\mathbb {R}}\) that satisfy
with a stress tensor
Here \(\rho\) is the density, \(c_p\) is the heat capacity, and k is the thermal conductivity, which are all considered piecewise constant across the domain. \(\varvec{I} \in {\mathbb {R}}^{D\times D}\) is the identity tensor, \(\eta : \Omega \rightarrow {\mathbb {R}}^+\) is the viscosity, and \(Q : \Omega \rightarrow {\mathbb {R}}^+ \cup \{ 0 \}\) is the volumetric heat production rate. Equations (1) and (2) comprise the Stokes system of equations conserving momentum and mass, respectively. Equation (3) expresses the conservation of energy of the system under our incompressible approximation. In our numerical simulations we solve a rescaled formulation of Eqs. (1) to (3) as described in Appendix A.
The viscosity model employed is that of diffusion creep combined with a nearrigid crust modeled by a constant high viscosity such that
Here \(A_\text {diff} = {1.32043 \times 10^{9}}\,\hbox {Pa}\,\hbox {s}\) is a constant prefactor, \(E_\text {diff} = {335 \times 10^{3}}\,\hbox {kJ}\,\hbox {mol}^{1}\) is the activation energy, \(R = {8.3145}\,\hbox {J}\,\hbox {mol}^{1}\,\hbox {K}^{1}\) is the gas constant, and \(\eta _\text {max} = {10^{26}}\,\hbox {Pa}\,\hbox {s}\) is the maximum viscosity within \(\Omega _\text {slab}\) and \(\Omega _\text {wedge}\). Here we limit ourselves to this temperaturedependent rheology, that is based on diffusion creep in olivine (Karato and Wu 1993). Detailed comparisons have shown that the nearsteady state thermal structure is very similar to that when an olivine dislocation creep rheology is used (van Keken et al. 2008). A full list of assumed constants used in the modeling is provided in Table 1.
2.3 Boundary conditions
The domain boundary \(\partial \Omega\) is subdivided into components as shown in Fig. 2 (plus \(\partial \Omega _\text {near}\) and \(\partial \Omega _\text {far}\) when \(D=3\)). The conditions to be imposed on the velocity and temperature fields are tabulated in Table 2. Here we specify the functions that are imposed.
2.3.1 Slab convergence and deformation direction
The down going slab velocity is decomposed into two components, a convergence speed \(u_\text {conv}\) acting in the direction \(\hat{\varvec{\tau }}_\text {conv}\) and the velocity of the geometry’s time evolving deformation, \(\varvec{u}_\text {slab}\), such that \(\varvec{u} = u_\text {conv} \hat{\varvec{\tau }}_\text {conv} + \varvec{u}_\text {slab}\) on \(\Gamma _\text {slab} \cup \Gamma _\text {slab fault}\). We define \(\hat{\varvec{\tau }}_\text {conv}\) in terms of a prescribed direction, \(\hat{\varvec{d}}_\text {conv}\). The vector \(\hat{\varvec{\tau }}_\text {conv}\) is the unit vector lying tangential to \(\Gamma _\text {slab}\), parallel to \(\hat{\varvec{d}}_\text {conv}\) and in the direction of \(\hat{\varvec{d}}_\text {conv}\). This may also be interpreted as \(\hat{\varvec{\tau }}_\text {conv}\) being the vector pointing in direction \(\hat{\varvec{d}}_\text {conv}\) and tangential to the intersection of the surface \(\Gamma _\text {slab}\) and the surface defined by the normal vector \(\hat{\varvec{r}} \times \hat{\varvec{d}}_\text {conv}\) (see Fig. 3). We therefore define
Furthermore we define the remaining vector lying tangential to \(\Gamma _\text {slab}\) and perpendicular to \(\hat{\varvec{\tau }}_\text {conv}\) and \(\hat{\varvec{n}}_\text {slab}\).
A schematic diagram of these vectors is shown in Fig. 3.
In addition to the prescribed slab convergence velocity we include the slab deformation velocity as a result of timedependent geometry changes. We write \(\varvec{u}_\text {slab}\) and \(\hat{\varvec{d}}_\text {slab}\) to be the slab deformation velocity and direction, respectively. These quantities will be fully defined in Sect. 2.4 where we will also describe the mathematical representation of \(\Gamma _\text {slab}\).
2.3.2 Temperature models
The surface temperature is assumed to be a constant value
The slab inlet temperature is selected from a half space cooling model
where \(T_\text {max} = {1573\,\mathrm{\text {K}}}\) is the maximum temperature, \(\mathop {\textrm{erf}}\limits (\cdot )\) is the error function, \(\kappa _\text {in} = \left. k / (\rho c_p) \right _{\varvec{x} \in \partial \Omega _\text {slab inlet}}\) is the thermal diffusivity at the slab inlet and \(t_{{50}\,\hbox {Myr}}\) is \({50}\,\hbox {Myr}\).
The outlet temperature is
where \(T_\text {1D}(d)\) is the solution of the initial value problem
where \(q_\text {surf}\) is the surface heat flux.
2.4 Discretization and solution
In this section, we introduce the discretization and solution schemes we employ to compute numerical approximations of the evolving subduction zone model. We summarize our procedure in Algorithm 1.
2.5 Mapping slab surface geometries to coordinate data
Seismic readings provide observations of the slab interface geometry. These data consist of coordinate tuples of longitude, latitude, and depth. Our aim is to transform these data into a Cartesian system where a central radial vector aligns with the \(\hat{\varvec{z}}\) direction.
Let the set of \(N_i\) longitude \(\lambda _i\), latitude \(\mu _i\), and depth \(d_i\) data points for a given point on the slab surface be
where longitude and latitude are measured in degrees and depth in kilometers. Initially these data are transformed to align the central radial vector with the \(\hat{\varvec{z}}\) axis of the Cartesian system. We define
such that
with spherical and Cartesian coordinate representations
respectively. The transform to each system is given by
2.6 Surface data to Bspline approximation
We seek a smooth and continuous approximation of the slab interface surface from the seismic observation data \(X_\text {slab}\). To this end, the data \(X_\text {slab}\) are approximated by an \(l_2\) projection to a nonperiodic Bspline of order \({p}\) (see, e.g., Piegl and Tiller 1997).
We define a nonperiodic Bspline by
where \({{p}} = m  n  1\) is the Bspline order, \(\varvec{C}_i\), \(i=0,\ldots ,n\), are the control points, \(\Xi = \{ \xi _i \}_{i=0}^{m}\) is the knot vector where each knot lies in the unit interval \(\xi _i \in [0, 1]\), \(i=0,\ldots ,m\), each knot is ordered such that \(\xi _i \le \xi _{i+1}\), \(i=0,\ldots ,m1\), and \(B_{i,{{p}}}(\xi )\), \(i=0,\ldots ,m\), are the Bspline basis functions. On the unit interval \(\xi \in [0, 1]\) these basis functions are
A Bspline surface of order \(\varvec{{{p}}} = ({{p}}_1, {p}_{2})\) is defined by a tensor product of Bsplines on the orthogonal coordinates \(\varvec{\xi } = (\xi _1, \xi _2) \in [0, 1]^2\) such that
With the definition of the Bspline surface in place we define the evolution of the slab surface with time. Let \(\vartheta (t) : {\mathcal {I}} \rightarrow [0, 1]\) be a parameterization of the evolution period of the slab. We write the slab surface Bspline
where \(S^\text {initial}_{(\varvec{{{p}}}, \varvec{\Xi })}(\varvec{\xi })\) and \(S^\text {final}_{(\varvec{{{p}}}, \varvec{\Xi })}(\varvec{\xi })\) are the initial and final slab geometries, respectively. We emphasize that \(S^\text {initial}\) and \(S^\text {final}\) share a common order, \({{p}}_1 = {{p}}_2\), and knot vector, \(\varvec{\Xi }\). Their individual definition is determined by their distinct control points \(\varvec{C}^\text {initial}_{i,j}\) and \(\varvec{C}^\text {final}_{i,j}\).
With appropriate choices of \(\vartheta (t)\) the putative evolution of the slab may be prescribed in the model. In our experiments we employ a straightforward linear transition such that
This linear transition also favors a simple definition of the deformation path undertaken by the modeled slab surface
along with the velocity of the slab deformation
A schematic of the evolution of \(S^\text {slab}_{(\varvec{{\varvec{p}}}, \varvec{\Xi })}\) is shown in Fig. 4. We highlight that this method may be extended to arbitrary numbers of prescribed initial, intermediate, and final subduction zone geometries yielding more sophisticated evolution.
2.7 Enveloping the slab surface
With the representation of the slab evolution using a Bspline we now require its envelopment in a model volume geometry as described in Fig. 2. A motivating advantage of a Bspline representation of \(\Gamma _\text {slab}\) is its typical compatibility with CAD software such as Open CASCADE (www.opencascade.com) that leverages splines to describe complicated geometries. These splines may be manipulated with a number of geometric operations, of which we employ:

Extrusion: Transform a spline along a path generating a higher dimensional shape from the swept path.

Union, intersection and difference: On a collection of shapes generate a single shape composed of their boolean union, intersection or difference, respectively.
With these operations we describe the process we employ, in a qualitative sense, which provides us with the geometry volumes demonstrated in this work. A diagram of this process is shown in Fig. 5.
To generate the slab volume, \(\Omega _\text {slab}\), the spline \(S^\text {slab}_{(\varvec{{\varvec{p}}}, \varvec{\Xi })}(\varvec{\xi }, t)\) is extruded in the \(\hat{\varvec{z}}\) direction by a distance of \(d_\text {slab}\). To generate the plate and wedge volumes, \(\Omega _\text {plate} \cup \Omega _\text {wedge}\), the spline \(S^\text {slab}_{({\varvec{p}}, \varvec{\Xi })}(\varvec{\xi }, t)\) is extruded in the \(\hat{\varvec{z}}\) direction by a distance greater than the maximum extent of the depth of the spline. This volume is then intersected with a sphere of radius \(r_0\) yielding \(\Omega _\text {plate} \cup \Omega _\text {wedge}\). The distinct volumes \(\Omega _\text {plate}\) and \(\Omega _\text {wedge}\) are then formed by embedding the surface of a sphere of radius \(r_0  d_\text {plate}\). The coupling depth is also embedded in \(S^{\text{slab}}_{(\varvec{{p}}, {\varvec{\Xi }})}\) by finding the intersection with a sphere of radius \(r_0  d_c\).
2.8 Spatial and temporal discretization
The time interval \({\mathcal {I}} = [0, t_\text {slab}]\) is discretized into time steps \({\mathcal {I}}_{\Delta t} = \{t_0, t_1, \ldots , t_\text {slab}\}\) where \(t_0< t_1< \cdots < t_\text {slab}\). We write the time step size \(\Delta t_n = t_{n+1}  t_n\) and we use the superscript index n to denote the evaluation of a function at a particular time step, e.g., \(T^n = T(t_n)\). At each time step, the domain \(\Omega ^{n} = \Omega (t_n)\) is subdivided into a tessellation of simplices (triangles when \(D=2\) and tetrahedra when \(D=3\)) which we call a mesh. Each simplex in the mesh is named a cell and denoted \(\kappa\) such that the tessellation \({\mathcal {T}}^{n} = \{\kappa ^n\}\). The meshing procedure accounts for the internal boundary \(\Gamma _\text {slab}(t_n)\) ensuring facets of cells are aligned with the surface providing an appropriate approximation.
The spatial components of Eqs. (1), (2) and (3) are discretized by the FE method. We employ a P2P1 TaylorHood element pair for the Stokes system’s velocity and pressure approximations (Taylor and Hood 1973) and a standard quadratic continuous Lagrange element for the temperature approximation. The boundary conditions enforced on \(\Gamma _\text {slab fault}\) and \(\Gamma _\text {wedge no slip}\) require a discontinuous velocity solution. Furthermore a discontinuous pressure solution is required on \(\Gamma _\text {slab fault}\), \(\Gamma _\text {wedge no slip}\) and \(\Gamma _\text {slab}\). The jump conditions of the temperature approximation are satisfied by enforcing \(C^0\) continuity across \(\Gamma _\text {slab fault}\), \(\Gamma _\text {wedge no slip}\) and \(\Gamma _\text {slab}\). To this end we define the following spaces:

1
\(\varvec{V}^{h,n} = \{D\)dimensional piecewise polynomials of degree 2 defined in each cell of the mesh \({\mathcal {T}}^n\) and continuous across cell boundaries except those that overlap the interior boundaries \(\Gamma _{\text{slab fault}}\) and \(\Gamma _{\text{wedge no slip}} \}\)

2
\(Q^{h,n} = \{\)scalar piecewise polynomials of degree 1 defined in each cell of the mesh \({\mathcal {T}}^n\) and continuous across cell boundaries except those that overlap the interior boundaries \(\Gamma _\text {slab fault}\), \(\Gamma _\text {wedge no slip}\) and \(\Gamma _\text {slab}\) \(\}\),

3
\(S^{h,n} = \{\)scalar piecewise polynomials of degree 2 defined in each cell of the mesh \({\mathcal {T}}^n\) and continuous across all cell boundaries\(\}\).
On \({\mathcal {I}}_{\Delta t}\) the time derivative in the the heat equation is discretized using a finite difference scheme such that
Using a backward Euler discretization allows us to write the fully discrete FE formulation for the model: find \((\varvec{u}^{n+1}_h, p^{n+1}_h, T^{n+1}_h) \in (\varvec{V}^{h,n+1} \times Q^{h,n+1} \times S^{h,n+1})\) such that
for all \((\varvec{v}_h, q_h, s_h) \in (\varvec{V}^{h,n+1} \times Q^{h,n+1} \times S^{h,n+1})\). Here \((a, b) = \sum _{\kappa \in {\mathcal {T}}^{n+1}} \int _\kappa a : b {\textrm{d}} \varvec{x}\) is the inner product on the mesh and the terms \(A^i_{\partial \Omega }(\cdot , \cdot )\) are the terms arising from the weak imposition of the boundary conditions via Nitsche’s method stated in Sect. 2.3. We refer to Houston and Sime (2018) regarding the formulation of these terms. With this discretization scheme we also define the discretized slab deformation velocity component used in the velocity boundary condition
2.9 Nonmatching mesh interpolation
An operator \({\mathcal {P}}(\cdot )\) is necessary to transfer the temperature field from the previous time step, \(T^{n}_h({\mathcal {T}}^n)\), to the mesh at the subsequent time step, \(T^n_h({\mathcal {T}}^{n+1})\), such that
The choice of \({\mathcal {P}}(\cdot )\) must account for cases where subsequent meshes do not overlap. In this work we design \({\mathcal {P}}(\cdot )\) to be a nearestneighbor interpolation such that \({\mathcal {P}}(T^{n}_h({\mathcal {T}}^{n}))\) interpolates \(T^{n}_h({\mathcal {T}}^n)\) in the overlapping volume \({\mathcal {T}}^{n+1} \cap {\mathcal {T}}^n\). In the remaining volume, \({\mathcal {T}}^{n+1} \setminus {\mathcal {T}}^n\), \({\mathcal {P}}(T^{n}_h({\mathcal {T}}^{n}))\) interpolates the value of \(T^{n}_h({\mathcal {T}}^n)\) that lies closest to the interpolation point. Specifically for each interpolation point \(\varvec{x}_i\) of \(T^{n}_h({\mathcal {T}}^{n+1})\) we have
Our choice of \({\mathcal {P}}(\cdot )\) here is the motivation for selecting the backward Euler finite difference scheme in the temporal discretization. A higher order finite difference scheme would have to carefully account for fields defined on both \({\mathcal {T}}^{n+1}\) and \({\mathcal {T}}^{n}\) in the FE formulation.
2.10 Picard iteration and computational linear algebra solvers
The fully discrete system in Eqs. (27), (28) and (29) is nonlinear. We use a Picard iterative scheme to compute their solutions’ approximations and minimize the residual formulations. This requires us to split the solution of the Stokes system from the energy equation. Therefore we introduce a subscript index, \(\ell\), corresponding to the Picard iteration number. Given an initial guess of the temperature field \(T^{n+1}_{h,\ell =0} = {\mathcal {P}}(T^n_h)\), we compute the sequence as shown in Algorithm 2.
The linear systems that underlie the FE discretization are typically too large to compute in reasonable time with direct factorization due to the spatial fidelity required from the mesh. This is especially pertinent in the \(D=3\) case where computation by direct factorization is unfeasible. We employ an iterative scheme for both the Stokes and heat equation sub problems in each Picard iteration. The Stokes system is solved by full Schur complement reduction using Flexible Generalized Minimal Residual (FGMRES) iterative method (Saad 1993). The velocity block is preconditioned using the algebraic multigrid method with nearnullspace informed smoothed aggregation as provided by PETSc (Balay et al. 2019). The pressure block is preconditioned with the inverse viscosity weighted pressure mass matrix. The heat equation is solved using Generalized Minimal Residual (GMRES) iterative method (Saad and Schultz 1986) and preconditioned with incomplete LU (iLU) factorization. For more details on solving such systems using iterative schemes and devising appropriate preconditioners see, for example, May and Moresi (2008).
2.11 Implementation
In this section, we list the computational tools and libraries that facilitate our computational model. The FE system assembly is enabled by the FEniCS project, this includes:

1
Basix for precomputation of FE bases (Scroggs et al. 2022),

2
Unified Form Language (UFL) for the computational symbolic algebra representation of FE formulations (Alnæs et al. 2014),

3
FEniCS Form Compiler (FFC) for translation to efficient FE kernels (Kirby and Logg 2006),

4
DOLFINx for the data structures and algorithms necessary for computing FE functions, tabulating their degrees of freedom, managing meshes and facilitating the solution of FE linear systems by third party linear algebra packages (Logg and Wells 2010).
The components of the FEniCS project have been demonstrated to be scalable in the context of thermomechanical analysis in Richardson et al. (2019) using the same linear operators that underlie the momentum and energy FE discretizations in this work.
DOLFINxMPC (Dokken 2022) is used in combination with DOLFINx to construct the function spaces \(\varvec{V}^{h,n}\), \(Q^{h,n}\) and \(S^{h,n}\). Specifically DOLFINxMPC facilitates strong imposition of equality of the FE functions’ degrees of freedom at the \(\Gamma _\text {slab}\), \(\Gamma _\text {slab fault}\) and \(\Gamma _\text {wedge no slip}\) boundaries as required by the velocity and temperature boundary conditions.
The Python library NURBSPython (geomdl) (Bingol and Krishnamurthy 2019) is employed for the Bspline approximation of \(\Gamma _\text {slab}\). Its data structures and functions are necessary for Bspline initialization and manipulation along with its facilitation of the \(l_2\) minimization of point cloud positional data to the Bspline surface geometry.
The computational domain is defined using the CAD framework offered by the aforementioned Open CASCADE Technology. These geometries are then interpreted by the meshing library gmsh (Geuzaine and Remacle 2009) for generation of the sequence of simplicial meshes for each time step between the initial and final slab geometry configurations.
The PETSc library (Balay et al. 2019, 2023) is used for its data structures and algorithms facilitating distributed parallel computation of the linear algebra systems’ solutions. This includes the implementations of the FGMRES and GMRES methods, along with construction of iLU factorization and construction of algebraic multigrid preconditioners.
Automatic formulation of the variational forms arising from the weak imposition of Dirichlet boundary data in Eqs. (27) to (29) is provided by dolfin_dg (Houston and Sime 2018). Finally, to build the necessary environment required to run our model on a high performance computer we use the Spack package manager (Gamblin et al. 2015).
3 Results
Our examples derive their geometric definition of \(S^\text {final}_{(\varvec{{{p}}}, \varvec{\Xi })}\) from a flat slab geometry within the subducting Nazca plate shown in Fig. 1. The initial slab geometry, \(S^\text {initial}_{(\varvec{{{p}}}, \varvec{\Xi })}\), is defined by a straight slab dipping at an angle of 30° with the trench aligned with the final state. These initial and final states will describe the evolution of the slab from the reference frame of a stationary trench. The \(D = 3\) volumes for each time step are constructed as described in Sect. 2.7 with \(d_\text {plate} = {50\,\mathrm{\text {k}\text {m}}}\), \(d_c = {75\,\mathrm{\text {k}\text {m}}}\), and \(d_\text {slab} = {200\,\mathrm{\text {k}\text {m}}}\). We choose the slab convergence direction \(\hat{\varvec{d}}_\text {conv} = \hat{\varvec{x}}\) and speed \(u_\text {conv} = {5}\,\hbox {cm}\,\hbox {yr}^{1}\). The total slab deformation time is \(t_\text {slab} = {11}\,\hbox {Myr}\) that is appropriate for the modeled subduction zone (Antonijevic et al. 2015).
From this \(D = 3\) geometry we further form \(D = 2\) slices by taking crosssections along the planes defined by constant \(y={200\,\mathrm{\text {k}\text {m}}}\), \({0\,\mathrm{\text {k}\text {m}}}\), and \({200\,\mathrm{\text {k}\text {m}}}\). We seek to compare the \(D=2\) model solutions with corresponding crosssections of the \(D=3\) results found by post processing.
The splines \(S^\text {initial}_{(\varvec{{{p}}}, \varvec{\Xi })}\), \(S^\text {final}_{(\varvec{{{p}}}, \varvec{\Xi })}\) and implicitly \(S^\text {slab}_{(\varvec{{{p}}}, \varvec{\Xi })}\) are defined with order \({{p}}_i = 2\) and number of control points \(n_i + 1 = 8\), \(i=1,\ldots ,D1\). In each model, the meshes are generated with cell size constraints of 2 km within 25 km of the velocity coupling depth \(d_c\), 5 km along the slab interface \(\Gamma _\text {slab}\), and \({10\,\mathrm{\text {k}\text {m}}}\) in the remaining volume. This means the nodal point spacing varies from 1 to 5 km in the model. The degree 2 piecewise polynomials used for the velocity and temperature function spaces \(\varvec{V}^{h,n}\) and \(S^{h,n}\) yield a distance between FE Degree of Freedom (DoF) coordinates of approximately half the cell size. Furthermore in each model, the initial temperature field \(T^{n=0}_h\) is prescribed from the computation of the steady state solution of the nonlinear model, \((\varvec{u}^{n=0}_h, p^{n=0}_h, T^{n=0}_h)\), on the initial mesh \({\mathcal {T}}^{n=0}\).
3.1 2D slab
The temperature and velocity fields computed on the \(D=2\) crosssections of the \(D=3\) geometry are shown in Figs. (6), (7), and (8), respectively. Each row corresponds to time snapshots taken over the \(t_\text {slab}={11}\,\hbox {Myr}\) model maximum time that has been discretized with 100 time steps such that \(\Delta t = {0.11}\,\hbox {Myr}\). The slab Bspline is overlaid in each plot as a dotted line. Tracers are added in the velocity plots showing pathlines between the shown time snapshots. Should a tracer leave the geometry between each snapshot, it is removed from the visualization leaving only its remaining tail. The geometry deformation is not shown between snapshots and the tracers do not cross over \(\Gamma _\text {slab}\) at any time in the simulation (though their pathlines may appear to do so).
Convergence of the surface temperature as a function of time step size in the temporal discretization is shown in Fig. 9. In Fig. 10 we show the slab surface temperature as a function of depth as computed in the steady state on the initial geometry \(S^\text{initial}_{({{\varvec{p}}}, \varvec{\Xi })}\), as computed in the full time dependent \(D=2\) model at time \(t = t_\text {slab}\) and as the steady state on the final geometry \(S^\text{final}_{({{\varvec{p}}}, \varvec{\Xi })}\). Finally, the temperature as a function of depth along \(\Gamma _\text {slab}\) at the final time \(t=t_\text {slab}\) are shown in Fig. 15.
3.2 3D slab
Snapshots of the \(D = 3\) case temperature and velocity approximations evaluated on the slab interface are shown in Fig. 11. As in the \(D=2\) case, we discretize the temporal domain with 100 time steps such that \(\Delta t = {0.11}\,\hbox {Myr}\). Overlaid on the velocity plot are arrows indicating the horizontal velocity of flow on the surface in the \(\hat{\varvec{x}}\) and \(\hat{\varvec{y}}\) directions along with cross markers with sizes corresponding to speed in the \(\hat{\varvec{z}}\) direction (into the page). Crosssections of the temperature and velocity solution at \(y = {200\,\mathrm{\text {k}\text {m}}}\), \({0\,\mathrm{\text {k}\text {m}}}\), and \({200\,\mathrm{\text {k}\text {m}}}\) are shown in Figs. 12, 13 and 14, respectively. Tracers and their pathlines are not added to these velocity crosssections due to the inability to visualize their \(\hat{\varvec{y}}\) component.
Crosssections of the temperature field taken at constant \(y = {200\,\mathrm{\text {k}\text {m}}}\), \({0\,\mathrm{\text {k}\text {m}}}\), and \({200\,\mathrm{\text {k}\text {m}}}\) as a function of depth along \(\Gamma _\text {slab}\) at final time \(t=t_\text {slab}\) is shown in Fig. 15. These data overlay the slab surface temperatures as a function of depth computed from the corresponding \(D = 2\) models. Furthermore, convergence of the surface temperatures as a function of time step size in the temporal discretization at these crosssections is shown in Fig. 9. The volume of the \(D=3\) model at \(t=t_\text {slab}\) with these crosssections and additional path tracers is shown in Fig. 16.
The model presented in Fig. 16 gives rise to the largest linear systems solved in this work, comprising \(\backsim {14 \times 10^{6}}\) DoFs at initial time \(t=0\) and \(\backsim {20\times 10^{6}}\) DoFs at final time \(t=t_\text {slab}\). The wall time required to complete the simulation was approximately 16 h when distributed over 64 processes on two AMD \(\hbox {EPYC}^\text {TM}\) 7452 processors. This includes 100 time steps, each requiring an average of 6.5 Picard iterations to resolve the nonlinearity of the coupled system (Eqs. (27), (28) and (29)).
4 Discussion
The surface temperatures computed from the \(D = 2\) and \(D=3\) models shown in Fig. 15 indicate a warming of the slab above the coupling point. This appears to be caused by the slab surface transitioning to a shallower angle than the initial condition, pushing the surface into a warmer region of the wedge (see Figs. 6, 7 and 8, 12, 13, and 14). Examining the \(y = {200\,\mathrm{\text {k}\text {m}}}\) crosssection in Fig. 14 the slab surface does not reach as shallow a depth as in the \(y = {200\,\mathrm{\text {k}\text {m}}}\) and \(y = {0\,\mathrm{\text {k}\text {m}}}\) crosssection cases. This leads to less significant warming of the slab surface above \(d_c\). Consider also the slab surface temperatures shown in Fig. 10 that increase as the slab evolves from the initial steady state to \(t = t_\text {slab}\) and that are reduced when evolved to the steady state with no further slab deformation.
In all cases, the steadystate solution used for the initial temperature field \(T_h^{n=0}\) exhibits a diffusive thickening of the plate within the approximate depths of \({50\,\mathrm{\text {k}\text {m}}}\) and \({100\,\mathrm{\text {k}\text {m}}}\). This feature persists through the simulation and is displaced by the slab deformation. Future models may be improved by prescribing the initial temperature field computed from an unsteady simulation run to a time just after transient effects in the slab become negligible.
The configuration of the slab surface temperature in the \(D = 3\) case shown in Fig. 11 is largely dictated by the velocity boundary condition applied to \(\Gamma _\text {slab}\). Choosing \(\hat{\varvec{d}}_\text {conv} = \hat{\varvec{x}}\) restricts the velocity profile to be very similar to the \(D = 2\) cases along \(\Gamma _\text {slab}\). Deviating from this decision, one avenue is to choose \(\hat{\varvec{d}}_\text {conv} = \hat{\varvec{z}}\) that would yield a convergence velocity in the direction of steepest descent. However in this case, the flow above and below \(\Gamma _\text {slab}\) will become unreasonable for the subduction zone model as a result of satisfying mass conversation, \(\nabla \cdot \varvec{u} = 0\). These flows, that are not realistic in a subduction zone model, typically form as velocity fields impinging or jetting out from \(\Gamma _\text {slab}\) in order to account for diverging and converging flows on the \(\Gamma _\text {slab}\) topology, respectively. An approach to alleviate this issue is to solve for some component of the flow, \(\varvec{u}\), on \(\Gamma _\text {slab}\) implicitly. For example, the velocity prescription on \(\Gamma _\text {slab}\) could be changed such that only the \(\hat{\varvec{\tau }}_\text {conv}\) component is imposed allowing the remaining tangential component in the \(\hat{\varvec{\tau }}^\bot _\text {conv}\) direction to be implicit in the model. This, however, introduces an issue where the deformation velocity component, \((\varvec{u}_\text {slab} \cdot \hat{\varvec{\tau }}^\bot _\text {conv}) \hat{\varvec{\tau }}^\bot _\text {conv}\), must be neglected. Another approach would be to impose a convergence velocity \(\varvec{u}_\text {conv}\), such that \(\varvec{u}_{\Gamma _\text {slab}} = \varvec{u}_\text {conv} + \varvec{u}_\text {slab}\), which is computed from a Stokes problem of topological dimension \(D  1\) defined on \(\Gamma _\text {slab}\). The divergence free constraint defined on the topology of the surface would then ensure no regions of converging or diverging flow. However, the complexity of the mathematical formulation of this problem as well as its implementation for parallel computation is challenging.
The slab temperature approximation close to \(\partial \Omega _\text {slab outlet}\) and \(\partial \Omega _\text {wedge outlet}\) is significantly affected by the nonoverlapping component of the interpolation operation \({\mathcal {P}}(\cdot )\) described in Fig. 2.9. This is indicated, for example, in the \(t>0\) cases of Fig. 15 at depths below 215 km (i.e., the component of the \(\Gamma _\text {slab}\) closest to \(\partial \Omega _\text {slab outlet}\) and \(\partial \Omega _\text {wedge outlet}\)). One can see a small downturn in the temperature that arises from interpolation of \(T_\text {out}\) (Eq. 11). This is colder than the material in the volume that is displaced by the moving slab. This issue could be addressed by ensuring all meshes overlap such that the overall volume remains consistent negating the need for nonoverlapping interpolation. However, this would introduce a large computational cost resolving a volume that is largely spatially removed from the domain of interest close to \(\Gamma _\text {slab}\).
The decision to choose the Bspline properties \({{p}}_i = 2\) and \(n_i + 1 = 8\), \(i=1,\ldots ,D1\), was made to balance production of a robust numerical model against the performance of iterative solvers applied to the linear system underlying the Stokes problem. Choosing a greater fidelity in the knot vector led to degradation of the rate of convergence of the FGMRES method for slab geometries that exhibit rapid nonsmooth changes. Future development of the model would investigate methods to retain the robust solution of the velocity and pressure approximations with greater spatial fidelity of \(\Gamma _\text {slab}\). Additionally the geometric operations required to define the volume using CAD as described in Sect. 2.7 become prohibitively expensive as the Bspline approximation order and control point vectors’ cardinality increases.
The modeling described above shares many of the characteristics of those developed in, e.g., Wilson and van Keken (2023). Importantly, we assumed a kinematic slab surface and solve for the Stokes equation in slab and mantle wedge. We ignored buoyancy in the mantle wedge, but the methodology can be extended in a straightforward manner to include density variations caused by temperature differences, variable water content, or composition. Our models do not currently allow for the evolution of water or composition fields and we do not allow for material exchange between the slab and mantle wedge. This makes it difficult to use this approach for more dynamical investigations that are possible with the particleincell methods used in, e.g., Gerya and Yuen (2003).
As in Wilson and van Keken (2023) we made the simplifying assumption that the top of the slab is decoupled from the overriding crust and mantle wedge to a depth of 80 km by using a kinematic boundary condition on the mantle wedge that switches from zero velocity to full slab velocity around this depth. While this is a reasonable approximation at shallow depth, where the slab is likely decoupled over long geological timescales by repeated underthrusting events along the seismogenic zone, it does not fully represent the likely complex rheological decoupling that occurs past the downdip limit of the seismogenic zone. As such these models cannot be easily used to understand dynamical processes along the plate interface. We note that Wada and Wang (2009) used a thin layer of low viscosity material between slab and wedge to a depth of 75 km which provides a similar effective decoupling (see comparison in van Keken and Wilson 2023b). Introducing this rheological separation creates a pathway to study dynamics along the plate interface (e.g., Behr and Becker 2018; Sobolev and Brown 2019) while maintaining a kinematicdynamic approach. Alternatively a more fully dynamic approach could be used (e.g., Gerya and Meilick 2010). We note that the models described below can be easily extended to study the influence of shear heating due to brittle–ductile processes along the shallow plate interface. This merely requires the introduction of a shear heating term in the heat equation as discussed in, for example, van Keken et al. (2019) and Abers et al. (2020).
We finish this discussion on a cautionary note. These models are sensu stricto based on a toy model (if admittedly a complicated one). The results presented here should be interpreted to indicate that precise description of the slab evolving geometry leads to significant differences between 3D models and 2D crosssections, but the temperaturepressure paths should not be used to compare directly to existing slab models or observations of flat slab subduction. In future work we will apply this modeling framework to regions of flat slab subduction with locally adjusted parameters for geometry, coupling point, structure of the overriding plate, etc.
5 Conclusions
We have devised, implemented, and demonstrated a numerical model of a subduction zone that accounts for a kinematic prescription of a geometrically evolving slab surface. We do this by approximating seismic observations of slab geometries with a Bspline. By constructing a deformation path for the Bspline surface from an initial to a final slab geometry, we are able to evolve this prescribed slab surface geometry over time. Enveloping the slab surface spline in a volume using CAD allows us to create a sequence of meshes in which we compute approximations of the velocity, pressure and temperature of a subduction zone model discretized by the FE method.
Data availability
An implementation of the subduction zone model is provided at https://github.com/natesime/mantleconvection. The data generated by the subduction zone model code and presented in this paper is available in a zenodo repository available via Sime et al. (2023). An animation of a selected 3D flow evolution is available in either repository.
Abbreviations
 Bspline:

Bézier spline
 CAD:

Computeraided design
 DoF:

Degree of freedom
 FE:

Finite element
 FFC:

FEniCS form compiler
 FGMRES:

Flexible generalized minimal residual
 GMRES:

Generalized minimal residual
 iLU:

Incomplete LU
 PDE:

Partial differential equation
 PETSc:

Portable extensible toolkit for scientific computation
 UFL:

Unified form language
References
Abers GA, Nakajima J, van Keken PE, Kita S, Hacker BR (2013) Thermalpetrological controls on the location of earthquakes within subducting plates. Earth Planet Sci Lett 369–370:178–187. https://doi.org/10.1016/j.epsl.2013.03.022
Abers GA, van Keken PE, Wilson CR (2020) Deep decoupling in subduction zones: observations and temperature limits. Geosphere 16:1408–1424. https://doi.org/10.1130/GES02278.1
Alnæs MS, Logg A, Ølgaard KB, Rognes ME, Wells GN (2014) Unified form language: a domainspecific language for weak formulations of partial differential equations. ACM Trans Math Softw 40:1–37. https://doi.org/10.1145/2566630
Anderson M, Alvarado P, Zandt G, Beck S (2007) Geometry and brittle deformation of the subducting Nazca Plate, Central Chile, and Argentina. Geophys J Int 171:419–434. https://doi.org/10.1111/j.1365246X.2007.03483.x
Antonijevic SK, Wagner LS, Beck SL, Long MD, Zandt G, Tavera H (2015) The role of ridges in the formation and longevity of flat slabs. Nature 524:212–215. https://doi.org/10.1038/nature14648
Axen GJ, van Wijk JW, Currie CA (2018) Basal continental mantle lithosphere displaced by flatslab subduction. Nat Geosci 11:961–964. https://doi.org/10.1038/s4156101802639
Balay S, Abhyankar S, Adams MF, Brown J, Brune P, Buschelman K, Dalcin L, Dener A, Eijkhout V, Gropp WD, Karpeyev D, Kaushik D, Knepley MG, May DA, McInnes LC, Mills RT, Munson T, Rupp K, Sanan P, Smith BF, Zampini S, Zhang H, Zhang H (2019) PETSc users manual. https://www.mcs.anl.gov/petsc
Balay S, Abhyankar S, Adams MF, Benson S, Brown J, Brune P, Buschelman K, Constantinescu EM, Dalcin L, Dener A, Eijkhout V, Faibussowitsch J, Gropp WD, Hapla V, Isaac T, Jolivet P, Karpeev D, Kaushik D, Knepley MG, Kong F, Kruger S, May DA, McInnes LC, Mills RT, Mitchell L, Munson T, Roman JE, Rupp K, Sanan P, Sarich J, Smith BF, Zampini S, Zhang H, Zhang H, Zhang J (2023) PETSc web page. https://www.mcs.anl.gov/petsc
Bebout GE, PennistonDorland SC (2016) Fluid and mass transfer at subduction interfaces—the field metamorphic record. Lithos 240–243:228–258. https://doi.org/10.1016/j.lithos.2015.10.007
Behr WM, Becker TW (2018) Sediment control on subduction plate speeds. Earth Planet Sci Lett 502:166–172. https://doi.org/10.1016/j.epsl.2018.08.057
Bengtson AK, van Keken PE (2012) Threedimensional thermal structure of subduction zones: effects of obliquity and curvature. Solid Earth 3:365–373. https://doi.org/10.5194/se33652012
Bingol OR, Krishnamurthy A (2019) NURBSPython: an opensource objectoriented NURBS modeling framework in Python. SoftwareX 9:85–94. https://doi.org/10.1016/j.softx.2018.12.005
Bloch W, John T, Kummerow J, Salazar P, Krüger O, Shapiro S (2018) Watching dehydration: seismic indication for transient fluid pathways in the oceanic mantle of the subducting Nazca slab. Geochem Geophys Geosys 19:3189–3207. https://doi.org/10.1029/2018GC007703
Carrapa B, DeCelles PG, Romero M (2019) Eartly inception of the Laramide Orogeny in Southwestern Montana and Northern Wyoming: implications for models of flatslab subduction. J Geophys Res Solid Earth 124:2102–2123. https://doi.org/10.1029/2018JB016888
Cerpa NG, Wada I, Wilson CR (2017) Fluid migration in the mantle wedge: influence of mineral grain size and mantle compaction. J Geoph Res Solid Earth 122:6247–6268. https://doi.org/10.1002/2017JB014046
ContrerasReyes E, MuñozLinford P, CortesRivas V, BelloGonzales JP, Ruiz JA, Krabbenhöft A (2019) Structure of the collision zone between the Nazca Ridge and the Peruvian convergent margin: geodynamic and seismotectonic implications. Tectonics 38:3416–3435. https://doi.org/10.1029/2019TC005637
Currie CA, Copeland P (2022) Numerical models of Farallon plate subduction: creating and removing a flat slab. Geosphere 18:476–502. https://doi.org/10.1130/GES02393.1
Dokken, J.S DOLFINxMPC v0.5.0. https://github.com/jorgensd/dolfinx_mpc
English JM, Johnston ST, Wang K (2003) Thermal modeling of the Laramide orogeny: testing the flat slab subduction hypothesis. Earth Planet Sci Lett 214:619–632. https://doi.org/10.1016/S0012821X(03)003996
Fan M, Carrapa B (2014) Late Cretaceousearly Eocene Laramide uplift, exhumation, and basin subsidence in Wyoming: crustal responses to flat slab subduction. Tectonics 33:509–529. https://doi.org/10.1002/2012TC003221
Finzel ES, Trop JM, Ridgway KD, Enkelmann E (2011) Upper plate proxies for flatslab subduction processes in southern Alaska. Earth Planet Sci Lett 303:348–360. https://doi.org/10.1016/j.epsl.2011.01.014
Gamblin T, LeGendre MP, Collette MR, Lee GL, Moody A, de Supinski BR, Futral WS (2015) The Spack Package Manager: bringing order to HPC software chaos. In: Supercomputing 2015 (SC’15), Austin, Texas. Art No 40. https://doi.org/10.1145/2807591.2807623
Gerya TV, Fossati D, Cantieni C, Seward D (2009) Dynamic effects of aseismic ridge subduction: numerical modeling. Eur J Miner 21:649–661. https://doi.org/10.1127/09351221/2009/00211931
Gerya TV, Meilick FI (2010) Geodynamic regimes of subduction under an active margin: effects of rheological weakening by fluids and melts. J Metamorph Geol 29:7–31. https://doi.org/10.1111/j.15251314.2010.00904.x
Gerya TV, Yuen DA (2003) Rayleigh–Taylor instabilities from hydration and melting propel 'cold plumes' at subduction zones. Earth Planet Sci Lett 212:47–62. https://doi.org/10.1016/S0012821X(03)002656
Geuzaine C, Remacle JF (2009) Gmsh: A 3D finite element mesh generator with builtin pre and postprocessing facilities. Int J Numer Methods Eng 79:1309–1331. https://doi.org/10.1002/nme.2579
Gutscher MA, Peacock SM (2003) Thermal models of flat subduction and the rupture zone of great subduction earthquakes. J Geoph Res Solid Earth 108:2009. https://doi.org/10.1029/2001JB000787
Gutscher MA, Olivet JL, Aslanian D, Eissen JP, Maury R (1999) The "lost Inca Plateau": cause of flat subduction beneath Peru. Earth Planet Sci Lett 14:395–410. https://doi.org/10.1130/GES01537.1
Houston P, Sime N (2018) Automatic symbolic computation for discontinuous Galerkin finite element methods. SIAM J Sci Comput 40:327–357. https://doi.org/10.1137/17M1129751
Jadamec MA, Haynie KL (2017) Tectonic drivers of the Wrangell block: insights on forearc sliver processes from 3D geodynamic models of Alaska. Tectonics 36:1180–1206. https://doi.org/10.1002/2016TC004410
Jadamec MA, Billen MI, Roeske SM (2013) Threedimensional numerical models of flat slab subduction and the Denali fault driving deformation in southcentral Alaska. Earth Planet Sci Lett 376:29–42. https://doi.org/10.1016/j.epsl.2013.06.009
Jones TD, Sime N, van Keken PE (2021) Burying Earth’s primitive mantle in the slab graveyard. Geochem Geophys Geosyst 22:e2020GC009396. https://doi.org/10.1029/2020GC009396
Jung H, Green HW II, Dobrzhinetskaya LF (2004) Intermediatedepth earthquake faulting by dehydration embrittlement with negative volume change. Nature 428:545–549. https://doi.org/10.1038/nature02412
Karato S, Wu P (1993) Rheology of the upper mantle: a synthesis. Science 260:771–778. https://doi.org/10.1126/science.260.5109.771
Kelemen PB, Hirth G (2007) A periodic shearheating mechanism for intermediatedepth earthquakes in the mantle. Nature 446:787–790. https://doi.org/10.1038/nature05717
Kirby RC, Logg A (2006) A compiler for variational forms. ACM Trans Math Softw 32:417–444. https://doi.org/10.1145/1163641.1163644
Kneller EA, van Keken PE (2012) The effects of threedimensional slab geometry on deformation in the mantle wedge: implications for shear wave anisotropy. Geochem Geophys Geosyst 9:Q01006. https://doi.org/10.1029/2008GC002151
Li ZX, Li XH (2007) Formation of the 1300kmwide intracontinental orogen and postorogenic magmatic province in Mesozoic South China. Geology 35:179–182. https://doi.org/10.10130/G23193A.1
Liu L, Spasojevič S, Gurnis M (2008) Reconstructing Farallon plate subduction beneath North America back to the late Cretaceous. Science 322:934–938. https://doi.org/10.1126/science.1162921
Liu X, Currie CA, Wagner LS (2022) Cooling of the continental plate during flatslab subduction. Geosphere 18:49–68. https://doi.org/10.1130/GES02402.1
Logg A, Wells GN (2010) DOLFIN: automated finite element computing. ACM Trans Math Softw 37:1–28. https://doi.org/10.1145/1731022.1731030
Logg A, Mardal KA, Wells GN (eds) (2012) Automated solution of differential equations by the finite element method. Lecture notes in computational science and engineering, vol 84. Springer, Heidelberg
Manea VC, Manea M (2011) Flatslab thermal structure and evolution beneath Central Mexico. Pure Appl Geophys 168:1475–1487. https://doi.org/10.1007/s0002401002079
Manea VC, Manea M, Ferrari L, OrozcoEsquivel T, Valenzuela RW, Husker A, Kostoglodov V (2017) A review of the geodynamic evolution of flat slab evolution in Mexico, Peru, and Chile. Tectonophysics 695:27–52. https://doi.org/10.1016/j.tecto.2016.11.037
Marot M, Monfret T, Gerbault M, Nolet G, Ranalli G, Pardo M (2014) Flat versus normal subduction zones: a comparison based on 3D regional travel time tomography and petrological modeling of central Chile and western Argentia (29°C35°S). Geophys J Int 199:1633–1654. https://doi.org/10.1093/gji/ggu355
May DA, Moresi L (2008) Preconditioned iterative methods for Stokes flow problems arising in computational geodynamics. Phys Earth Planet Int 171:33–47. https://doi.org/10.1016/j.pepi.2008.07.036
Molnar P, England P (1990) Temperature, heat flux, and frictional stress near major thrust faults. J Geophys Res Solid Earth 95:4833–4856. https://doi.org/10.1029/JB095iB04p04833
Nitsche J (1971) Über ein Variationsprinzip zur Lösung von DirichletProblemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind. Abh Math Semin Univ Hambg 36:9–15
Peacock SM, Wang K (1999) Seismic consequences of warm versus cool subduction metamorphisms: examples from southwest and northeast Japan. Science 286:937–939. https://doi.org/10.1126/science.286.5441.937
Piegl L, Tiller W (1997) The NURBS book. Monographs in visual communication, 2nd edn. Springer, Heidelberg
Plunder A, Thieulot C, van Hinsbergen DJJ (2018) The effect of obliquity on temperature in subduction zones: insights from 3D numerical modeling. Solid Earth 9:759–776. https://doi.org/10.5194/se97592018
Raleigh CB, Paterson MS (1965) Experimental deformation of serpentinite and its tectonic implications. J Geophys Res Solid Earth 70:3965–3985. https://doi.org/10.1029/JZ070i016p03965
Richardson CN, Sime N, Wells GN (2019) Scalable computation of thermomechanical turbomachinery problems. Finite Elem Anal Des 155:32–42. https://doi.org/10.1016/j.finel.2018.11.002
Rosas JC, Currie CA, He J (2016) Threedimensional thermal model of the Costa RicaNicaragua subduction zone. Pure Appl Geophys 173:3317–3339. https://doi.org/10.1007/s0002401511974
Saad Y (1993) A flexible innerouter preconditioned GMRES algorithm. SIAM J Sci Comput 14:461–469. https://doi.org/10.1137/0914028
Saad Y, Schultz MH (1986) GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J Sci Stat Comput 7:856–869. https://doi.org/10.1137/0907058
Schmid C, Goes S, van der Lee S, Giardini D (2002) Fate of the Cenozoic Farralon slab from a comparison of kinematic thermal modeling with tomographic images. Earth Planet Sci Lett 204:17–32. https://doi.org/10.1016/S0012821X(02)009858
Scroggs MW, Dokken JS, Richardson CN, Wells GN (2022) Construction of arbitrary order finite element degreeoffreedom maps on polygonal and polyhedral cell meshes. ACM Trans Math Softw 48:18. https://doi.org/10.1145/3524456
Shiina T, Nakajima J, Matsuzawa T (2013) Seismic evidence for high pore pressures in oceanic crust: implications for fluidrelated embrittlement. Geophys Res Lett 40:2006–2010. https://doi.org/10.1002/grl.50468
Shiina T, Nakjima J, Matsuzawa T, Toyokuni G, Kita S (2017) Depth variations in seismic velocity in the subducting crust: implications for fluidrelated embrittled for intermediatedepth earthquakes. Geophys Res Lett 44:810–817. https://doi.org/10.1002/2016GL071798
Shirey SB, Wagner LS, Walter MJ, Pearson DG, van Keken PE (2021) Slab transport of fluids to deep focus earthquake depths—thermal modeling constraints and evidence from diamonds. AGU Adv 2:e2020AV000304. https://doi.org/10.1029/2020AV000304
Sime N, Maljaars JM, Wilson CR, van Keken PE (2021) An exactly mass conserving and pointwise divergence free velocity method: application to compositional buoyancy driven flow problems in geodynamics. Geochem Geophys Geosyst 22:e2020GC0009349. https://doi.org/10.1029/2020GC009349
Sime N, Wilson CR, van Keken PE (2022) A pointwise conservative method for thermochemical convection under the compressible anelastic liquid approximation. Geochem Geophys Geosyst 23:e2021GC009922. https://doi.org/10.1029/2021GC009922
Sime N, Wilson CR, van Keken PE (2023) Thermal modeling of subduction zones with prescribed and evolving 2D and 3D slab geometries data (1.0) [Data set]. Zenodo. https://doi.org/10.5281/zenodo.8350531
Sippl C, Schurr B, John T, Hainzl S (2019) Filling the gap in a double seismic zone: intraslab seismicity in northern Chile. Lithos 346–347:105155. https://doi.org/10.1016/j.lithos.2019.105155
Sobolev SV, Brown M (2019) Surface erosion events controlled the evolution of plate tectonics on Earth. Nature 570:52–57. https://doi.org/10.1038/s4158601912584
Syracuse EM, van Keken PE, Abers GA (2010) The global range of subduction zone thermal models. Phys Earth Planet Int 183:73–90. https://doi.org/10.1016/j.pepi.2010.02.004
Taramón JM, RodríguezGonzález J, Negredo AM, Billen MI (2015) Influence of cratonic lithosphere on the formation and evolution of flat slabs: insights from 3D timedependent modeling. Geochem Geophys Geosyst 16:2933–2948 https://doi.org/10.1002/2015GC005940
Taylor C, Hood P (1973) A numerical solution of the Navier–Stokes equations using the finite element technique. Comput Fluids 1:73–100. https://doi.org/10.1016/00457930(73)900273
van den Berg AP, Segal G, Yuen DA (2015) SEPRAN: a versatile finiteelement package for a wide variety of problems in geosciences. J Earth Sci 26:89–95. https://doi.org/10.1007/s1258301505080
van Keken PE, Wilson CR (2023) An introductory review of the thermal structure of subduction zones: I—motivation and selected examples. Prog Earth Planet Sci 10:42. https://doi.org/10.1186/s4064502300573z
van Keken PE, Wilson CR (2023) An introductory review of the thermal structure of subduction zones: III—comparison between models and observations. Prog Earth Planet Sci 10:57. https://doi.org/10.1186/s40645023005895
van Keken PE, Kiefer B, Peacock SM (2002) Highresolution models of subduction zones: implications for mineral dehydration reactions and the transport of water to the deep mantle. Geochem Geophys Geosyst 3:1056. https://doi.org/10.1029/2001GC000256
van Keken PE, Kita S, Nakajima J (2012) Thermal structure and intermediatedepth seismicity in the TohokuHokkaido subduction zones. Solid Earth 3:355–364. https://doi.org/10.5194/se33552012
van Keken PE, Wada I, Sime N, Abers GA (2019) Thermal structure of the forearc in subduction zones: a comparison of methodologies. Geochem Geophys Geosyst 20:3268–3288. https://doi.org/10.1029/2019GC008334
van Keken PE, Currie C, King SD, Behn MD, Cagnioncle A, He J, Katz RF, Lin SC, Parmentier EM, Spiegelman M, Wang K (2008) A community benchmark for subduction zone modeling. Phys Earth Planet Int 171:187–197. https://doi.org/10.1016/j.pepi.2008.04.015
Wada I, He J (2017) Thermal structure of the Kanto region, Japan. Geoph Res Lett 44:7194–7202. https://doi.org/10.1002/2017GL073597
Wada I, Wang K (2009) Common depth of slabmantle decoupling: Reconciling diversity and uniformity of subduction zones. Geochem Geophys Geosyst 10:Q10009. https://doi.org/10.1029/2009GC002570
Wagner LS, Jaramillo S, RamirezHoyos LF, Monsalve A, Cardona A, Becker TW (2017) Transient slab flattening beneath Colombia. Geophys Res Lett 44:6616–6623. https://doi.org/10.1002/2017GL073981
Wagner LS, Caddick MJ, Kumar A, Beck SL, Long MD (2020) Effects of oceanic crustal thickness on intermediate depth seismicity. Front Earth Sci 8:244. https://doi.org/10.3389/feart.2020.00244
Wei SS, Wiens DA, van Keken PE, Cai C (2017) Slab temperature control on the Tonga double seismic zone and slab mantle dehydration. Sci Adv 3:e1601755. https://doi.org/10.1126/sciadv.1601755
Wilson CR, van Keken PE (2023) An introductory review of the thermal structure of subduction zones: II—numerical approach and validation. Prog Earth Planet Sci 10:68. https://doi.org/10.1186/s40645023005886
Wilson CR, Spiegelman M, van Keken PE (2017) TerraFERMA: the Transparent Finite Element Rapid Model Assembler for multiphysics problems in Earth sciences. Geochem Geophys Geosyst 18:769–810. https://doi.org/10.1002/2016GC006702
Wilson CR, Spiegelman M, van Keken PE, Hacker BR (2014) Fluid flow in subduction zones: the role of solid rheology and compaction pressure. Earth Planet Sci Lett 401:261–274. https://doi.org/10.1016/j.epsl.2014.05.052
Acknowledgements
The authors thank Lara Wagner for providing the slab surface geometry used in Fig. 1 and for discussions. The authors also thank Jørgen Dokken for his advice regarding the use of DOLFINxMPC.
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This work was supported by National Science Foundation Grant 2021027.
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NS developed and tested the FEniCS finite element approach in collaboration with CW. PvK aided in benchmarking the 2D subduction zone models. All contributed to writing the manuscript.
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Appendix A: Model equations rescaling
Appendix A: Model equations rescaling
Using velocity, length and viscosity scales \(u_r\), \(h_r\) and \(\eta _r\), respectively we define the rescaled quantities
Employing these quantities we arrive at the rescaled Stokesenergy formulation of Eqs. (1), (2) and (3) where we seek \(\varvec{u}^\prime\), \(p^\prime\) and T such that
which after simplification reads
The numerical values used in our computations are \(h_r = {1\,\mathrm{\text {k}\text {m}}}\), \(u_r = {5}\,\hbox {cm}\,\hbox {yr}^{1}\) and \(\eta _r = {10^{21}}\,\hbox {Pa}\,\hbox {s}\).
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Sime, N., Wilson, C.R. & van Keken, P.E. Thermal modeling of subduction zones with prescribed and evolving 2D and 3D slab geometries. Prog Earth Planet Sci 11, 14 (2024). https://doi.org/10.1186/s40645024006114
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DOI: https://doi.org/10.1186/s40645024006114