To a large extent, the thermal structure of a subduction zone determines where seismicity occurs through controls on the transition from brittle to ductile deformation and the depth of dehydration reactions. Thermal models of subduction zones can help understand the distribution of seismicity by accurately modelling the thermal structure of the subduction zone. Here, we assess a common simplification in thermal models of subduction zones, i.e. constant values for the thermal parameters. We use temperature-dependent parameterisations, constrained by lab data, for the thermal conductivity, heat capacity, and density to systematically test their effect on the resulting thermal structure of the slab. To isolate this effect, we use the well-defined, thoroughly studied, and highly simplified model setup of the subduction community benchmark by

The thermal structure of subduction zones plays a vital role in controlling many geological and petrological processes, including the dehydration of the subducting plate

From these examples, it becomes clear that it is important to have a thorough understanding of the thermal structure of a slab in order to better understand the distribution of the full spectrum of seismicity associated with the subduction process. However, it is hard to obtain direct observational data on the thermal structure of the slab due to the inaccessibility of subduction zones and the difficulty of obtaining data at great depths (i.e. larger than 10 km).

The dependence of seismic wave speeds on temperature allows seismic tomography studies to give a broad overview of the large-scale thermal structure of the subduction zone as a whole, but such studies typically lack the resolution to infer the thermal structure of the slab itself in great detail

In light of the limited available data on the thermal structure of subduction zones, geodynamic numerical modelling provides a way of investigating the complete temperature field of subduction zones in relation to the thermal and dynamic evolution of the slab

Indeed, numerical models of the temperature structure of subduction zones are subject to a range of simplifications. One, which we seek to address here, is that the thermal parameters in the model, i.e. the thermal conductivity, heat capacity, and density, are commonly assumed to be constant or merely material dependent. In contrast, laboratory experiments have shown that these parameters actually depend on temperature and can differ by as much as a factor of 2 depending on the temperature

Given the sensitivity of the various processes mentioned above to small-scale variations in the temperature evolution of the slab, we therefore seek to quantify the potential impact of the temperature dependence of thermal parameters on subduction zone thermal structure and to build towards their routine incorporation.

In order to assess the effect of temperature-dependent thermal parameters on the resulting thermal structure of the slab, we perform a systematic study by using the well-defined, highly simplified setup of the subduction community benchmark by

We base our models on the subduction zone community benchmark presented by

In the following, we first discuss the governing equations (Sect.

Following

We also solve for temperature

We consider a purely viscous rheology, where we relate the deviatoric stress tensor

For the diffusion creep rheology, we use the simplified diffusion creep viscosity formulation

We combine these formulations for diffusion and dislocation creep into one rheology by assuming two viscous dampers in series

To avoid unrealistically high stresses, we limit the maximum viscosity in the model to

We use the two-dimensional model setup of the community benchmark for subduction zone modelling presented by

Model setup.

We consider a domain that is

We fix the overriding plate by prescribing no slip (i.e. zero velocity in both the

For the conservation of energy, we apply a constant

We first solve the Stokes equations across the entire domain. As we are only interested in the velocity field in the mantle wedge, we overwrite the resulting velocity solution in the slab and overriding plate by our boundary conditions, i.e. no slip in the overriding plate and a constant subduction velocity of 5 cm yr

As the left-hand boundary condition for the conservation of energy, we prescribe the thermal structure of the incoming oceanic plate. In the original community benchmark,

However, the half-space cooling model does not satisfy petrological constraints and fails to satisfy heat flow and bathymetric data for plate ages greater than

We calculate the structure of the incoming oceanic plate in a linked, separate Python script, with the coordinate convention that the

As input parameters, we choose a constant

We solve for the temperature evolution of the incoming oceanic plate with the desired thermal parameters (Sect.

We use temperature-dependent expressions for the thermal conductivity, heat capacity, and density using parameterisations based on observational experimental data for the way in which these values change with temperature.

Temperature dependence of

Following

Like

For the heat capacity

For the dependency of density on temperature (Fig.

Formulations for the temperature dependence of the thermal conductivity, heat capacity, and density other than the ones described here are also available

In our preferred formulations for the temperature dependence of the thermal parameters, the thermal conductivity

To systematically test the effect of temperature-dependent parameters on the thermal structure of subduction zones, we run the suite of simulations outlined in Table

Simulations

To illustrate how the different simulations differ in terms of the temperature dependence of the thermal parameters, we show the thermal diffusivity

The thermal diffusivity

The top of the oceanic plate, where temperatures are low and hence where thermal parameters differ most from the constant values used in

Another important aspect in a subduction zone is the compositional structure of the overriding plate. In our standard set of models, there is no crustal layer within the overriding plate. To assess the effect of the crust of the overriding plate on the thermal structure of the slab, we run another two sets of models based on our main set of 10 models and including the oceanic-crust parameterisation in the slab of the _mc models but with an additional crustal layer in the overriding plate. The first set of models (denoted _op – oceanic plate) incorporates a crustal layer with a thickness of 7 km and half the thermal conductivity values for a given temperature, similarly to our parameterisation of oceanic crust in the slab, as if the overriding plate were oceanic in origin. The second set of models considers a continental upper plate (denoted _cp – continental plate) with a crustal thickness of 35 km, halved thermal conductivity values, and a density that is multiplied by 0.79 to obtain realistic values for crustal densities at low temperatures.

Note that for these three sets of models that include a parameterisation for oceanic crust in the slab, we also include this crustal-layer approximation in the one-dimensional plate model that calculates the left temperature boundary condition. In contrast, the half-space cooling model used by

To illustrate the applicability of our results to the variety of subduction zones observed on Earth, we also run two end-member models with constant and temperature-dependent thermal parameters for a model with a younger (

To assess our models and quantify their differences, we use the three diagnostics defined in the community benchmark by

In addition to the diagnostics previously used in

Postprocessing and visualisation are primarily done using MATLAB scripts (available at

The results from the reference model case2c_PvK with constant thermal parameters are shown in Fig.

Snapshots of different variables for model case2c_PvK with constant values for the thermal parameters based on

The reference model has a combined dislocation and diffusion creep rheology in contrast to the original cases presented in

Model diagnostics

In model case2c_bc, we build upon our reference model and change the initial and boundary temperature condition of the subducting oceanic plate at the left of the model from a half-space cooling model to the plate model. This does not incur major changes in the model diagnostics (Table

Using the temperature-dependent thermal conductivity according to

When using a temperature-dependent heat capacity, the model diagnostics show larger temperatures in the mantle wedge compared to the reference model with a constant heat capacity value. Similarly, the subducting slab is warmer, and isotherms penetrate less deeply into the mantle. For our preferred heat capacity model with 89 % forsterite and values from

Using the values of

When we use a temperature-dependent density in model case2c_rho, the model is cooler than the reference model, case2c_PvK, but the effect is less pronounced than for the thermal conductivity (Table

We show the results for the model case2c_all in Fig.

Snapshots of different variables for model case2c_all with our preferred temperature-dependent functions for all thermal parameters

Change in maximum isotherm depth within the slab for models with different variations of temperature-dependent thermal parameters (Table

Within the slab itself, Fig.

To summarise the effect of using temperature-dependent thermal parameters for all our models with a plate age of 50 Myr, we plot the maximum depth of the 350, 450, and 600

The surface heat flux in the models varies across the surface, with values of

Similarly to the models with a plate age of

Change in maximum isotherm depth within the slab for end-member models with different subducting-plate ages (Table

When we include a parameterisation for a crustal layer at the top of the oceanic plate in the model, the diagnostics show a warmer top of the slab and mantle wedge. Within the slab, temperatures are also warmer, resulting in shallower maximum depths of the isotherms within the slab (Table S1 in the Supplement; Fig. S29 in the Supplement; results are similar to the results of the overriding plate models shown in Fig.

The difference between isotherm depths for models with and without the oceanic-crust parameterisation becomes more pronounced with increasing temperature. So, for example, between models case2c_bc and case2c_bc_mc, the difference in depth of the 350

An exception to these trends is the reference model case2c_PvK_mc, which uses the half-space cooling model as a left-hand-side temperature boundary condition that does not account for the 7 km oceanic-crust layer in the slab. Using this simplified boundary condition actually results in a colder slab with deeper isotherms (Fig. S29 in the Supplement). The large differences between the model case2c_PvK_mc and the other models in the _mc model batch show that the choice of temperature boundary condition and the use of a consistent temperature boundary condition is crucial.

The models that include the parameterisation for an overriding plate also include the oceanic-crust parameterisation in the slab. The models including an oceanic or continental plate all show the same trends as the models that do not include an overriding plate parameterisation but do include the oceanic-crust parameterisation. Typically, the models including a continental overriding plate result in a warmer slab and shallower isotherms compared to the other models. Figure

We find that using temperature-dependent thermal parameters does not significantly change the first-order thermal structure of a subduction zone in our models, with the large-scale features remaining the same. Hence, when considering large-scale subduction dynamics, the use of temperature-dependent thermal parameters is likely not an important factor. However, our results clearly show that the temperature-dependent thermal parameters do affect the thermal structure of the slab on the order of tens of degrees and hence tens of kilometres in these simple models of subduction zones. On a similar scale, our results show that the temperature boundary condition on the left-hand-side of the model influences the temperature field of the slab. On the other hand, the inclusion of an overriding plate does not significantly affect the temperature field of the modelled slab.

Using temperature-dependent thermal parameters also affects the predicted surface heat flux in our models. As surface heat flux is one of the principal observables used in constraining subduction zone thermal models, all regional subduction zone thermal models are therefore affected by the inclusion of temperature-dependent thermal parameters. Our models with different plate ages show that our conclusions are valid regardless of the slab age, but they still lack realism in terms of model geometry and the inclusion of many important processes relevant for the development of a realistic thermal structure of the slab.

The found effect of using temperature-dependent thermal parameters on the order of tens of degrees could be significant when assessing the detailed thermal structures of local slabs for seismological applications. In this section, we first discuss the implications of our results for modelling the thermal structure of subduction zones in light of megathrust, intermediate-depth, and deep seismicity while taking into account the simple and geometrically unrealistic nature of our models (Sect.

The temperature structure of a slab determines to a large extent where seismicity is expected to occur through its effect on both the mode of failure and the onset of ductile behaviour and its control on geochemical transitions within the slab and along the megathrust interface, including dehydration reactions. Here we summarise those effects and highlight how the results presented in Sect.

Although the shallow slab geometry in our model is clearly a simplification, at depths consistent with intermediate-depth seismicity, the slab dip of

Additionally, previous thermal models

The cause of deep earthquakes (

The spatial extent of megathrust seismicity and, in particular, the maximum potential earthquake magnitude for interface events in a subduction zone are determined by the size of the seismogenic zone (and hence the maximum rupture width) bounded by empirically derived up-dip and down-dip limits

The down-dip limit of the seismogenic zone is typically associated with the transition from brittle behaviour at lower temperatures to the onset of ductile behaviour at higher temperatures. The exact isotherms corresponding to this change in deformation style are still debated, with estimates ranging from 250 to 550

Using the constant slab dip of 45

However, our models here are of limited use in assessing the sensitivity of the temperature along the shallow subduction interface to the inclusion of temperature-dependent thermal properties for several reasons. First of all, in our simplified model geometry, the shallow dip of our interface is significantly larger than that typically seen in the interface seismogenic zone of most subduction zones

However, noting the impact that the variation in thermal properties at low temperatures (e.g. Fig.

As the subducting plate descends, it typically undergoes a range of mineralogical transitions relating to the increase in pressure and temperature. These mineralogical changes, particularly the location at which dehydration reactions release fluids into the slab system, play a controlling role in determining the location of intraslab seismicity and also in influencing a range of other geophysical phenomena, from the internal impedance and velocity contrasts within the slab

Whilst the kinematic constraints we impose on the slab in our models mean there is little variation in lithostatic pressure between models, we have shown that including the temperature dependence of thermal parameters in the modelling of slab thermal structures has an impact on the resultant temperature field. Whilst these changes are small relative to the total change in temperature experienced by the slab during subduction, they lead to a slightly different pressure–temperature evolution for the slab material. An additional crustal layer in our models, for which we currently only employ a simple parameterisation, further alters the temperature field. We note that the changes in the temperature evolution of the uppermost

Lastly, the model diagnostics we focus on here centre around the maximum depth of a given isotherm. However, the variation in thermal structure that we study will also impact on the thermal cross section of the slab at any given depth – with marginally colder slabs having a significantly greater volume of material below a given temperature at a given depth and hence potentially altering the volatile fluxes within slabs into the middle mantle.

With the exception of a different rheology in the mantle wedge, where we combine both diffusion and dislocation creep, we use the same model setup as the subduction zone community benchmark presented by

The model setup is greatly simplified, and many complexities that are known to influence the thermal structure of the slab are ignored. As illustrated in the benchmark of

Hence, the effect of using temperature-dependent thermal parameters in thermal models instead of constant values is a secondary effect to rheology when comparing isoviscous and nonlinear rheologies (i.e. compare Figs. S1–S4 to S5–S7 in the Supplement). However, when comparing nonlinear rheologies, using temperature-dependent thermal parameters instead of constant values will likely have a greater effect on the thermal structure of the slab than changing the details of the rheology formulation. Note that these conclusions relate to the thermal structure of the slab; the rheology plays a major role in the thermal structure of the mantle wedge and overriding plate, as is evident from the original benchmarks presented in

Apart from a simplified rheology, we also employ a simplified geometry in our model setup. Although the model serves as a good benchmark and although we can infer some implications for seismicity from this simple setup, a strictly

Although we focus here on the effect of using temperature-dependent thermal parameters, there are numerous other processes relevant to the developing thermal structure of a subduction zone

Although we consider a set of models where we include a simple parameterisation as an approximation of a crustal layer in the slab, our models are still restricted to a single composition. Hence, we do not explicitly include a crustal layer, and we neglect the impact of the mineralogical evolution of the slab on the temperature structure, both through the variation in thermal parameters with evolving mineralogy and through the latent heat of mineralogical transformation. Our results suggest that including compositional heterogeneity, specifically oceanic crust, does not change the observed trends concerning the effect of temperature-dependent thermal parameters on the models. Hence, the main conclusions presented in this work are not affected by compositional heterogeneity.
However, thermal parameters do vary greatly with composition

Lastly, there are numerous functions describing the temperature and pressure dependence of the thermal parameters in the literature, and existing functions are continuously updated with improved values for constants to better fit laboratory data. It is outside of the scope of this work to test all different formulations, and here we follow

In this work, we look at the effect of using temperature-dependent thermal parameters in thermal models of subduction zones compared to using constant values.

We find that the use of temperature-dependent thermal parameters does not significantly affect the first-order thermal structure of a subducting slab, like the choice of rheology or plate age would. However, the local thermal structure of the slab does change on the order of tens of degrees when using temperature-dependent thermal parameters. This second-order effect could be significant for specific modelling applications that consider, for instance, seismicity and phase changes.

Using only a temperature-dependent conductivity decreases the temperature in the slab and results in a larger predicted seismogenic zone width and deeper intermediate-depth seismicity, with the maximum depth of the 600

Solely employing a temperature-dependent heat capacity has the opposite effect and results in a warmer slab with a shallower down-dip limit of the seismogenic zone and predicted depth of dehydration reactions responsible for intermediate-depth seismicity.

Using only a temperature-dependent density has the least effect on the thermal structure of the slab when compared to the reference model with constant values, although the slab is overall colder.

Combining the temperature dependence of the three thermal parameters negates the effect on the thermal structure of the slab slightly, but the strong cooling of the slab produced by both the temperature-dependent thermal conductivity and density dominates. Therefore, the modelled slab is colder than a slab modelled with constant thermal parameters with, e.g. the maximum depth of the 600

Models including a parameterisation of oceanic crust in the slab and an oceanic or overriding plate show the same trends. We find that choosing consistent temperature boundary conditions is crucial and can otherwise lead to large differences in the temperature field of the slab. In contrast, the nature, or existence, of an overriding plate does not significantly affect the temperature field of the slab.

We caution that the conclusions drawn here stem from a highly simplified model of a subduction zone that is not suitable for direct comparisons to nature and does not aim to reproduce observations. Hence, the details of our findings will likely change for more complex and realistic model setups. However, even considering the simplifications in our model setup, our results indicate that the changes in the modelled thermal structure of the slab, as well as the predicted surface heat flux, will have important implications for the estimated size of the seismogenic zone in these kinds of thermal models and for predictions where intermediate-depth seismicity might occur. These implications for the inclusion of temperature-dependent thermal parameters are of a lower order than the established first-order controls on the subduction zone thermal structure, such as the employed rheology and plate age. Nevertheless, we suggest that, for optimal comparison to data and to avoid misinterpretations, temperature-dependent thermal parameters are also an important modelling ingredient that should be taken into account when using thermal(-mechanical) models of subduction zones, especially in studies that have seismological applications.

The models were run with the open-source code xFieldstone (

The supplement related to this article is available online at:

IvZ and TJC conceived the study. IvZ designed and ran the models, analysed the results, and wrote the article. CT and IvZ wrote the code xFieldstone. TJC supervised IvZ and contributed to the analysis of the models. All authors discussed the results and contributed to the model design and the final article. The author order of the second and third authors was decided by a coin flip.

The contact author has declared that none of the authors has any competing interests.

Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

We thank the editor Taras Gerya and three anonymous reviewers at

We thank Peter van Keken for providing the original data from

Elements of this work were undertaken on ARC4, part of the High-Performance Computing facilities at the University of Leeds, UK. We also ran simulations on the Plejades work stations of the German Aerospace Center (DLR), Germany. Iris van Zelst and Timothy J. Craig were funded by the Royal Society (UK) through University Research Fellowship no. URF

This research has been supported by the Royal Society (grant nos. URF

This paper was edited by Taras Gerya and reviewed by three anonymous referees.