The long-term evolution of the mantle is simulated using 2D spherical annulus geometry to examine the effect of heterogeneous thermal conductivity on the stability of reservoirs of primordial material. Often in numerical models, purely depth-dependent profiles emulate mantle conductivity (taking on values between 3 and 9

Tomographic studies of Earth's lower mantle reveal the presence of two large low-velocity provinces (LLVPs) that overlie the core–mantle boundary (CMB). Located below the Pacific Ocean and Africa, LLVPs have a combined coverage of up to 30 % of the CMB with vertical extents as high as 1200 km

Piles' unique chemical composition determines physical properties, most importantly density, viscosity, and enrichment in heat-producing elements (HPEs), which affect their long-term stability. These property values remain uncertain, but numerical simulations that emulate Earth-like piles help constrain their ranges (e.g.

Measured and estimated thermal conductivities of minerals are determined at variable temperature and pressure conditions relevant to regions of the mantle at shallow and great depths. In addition, aggregate compositions with variable fractions of mineral concentrations and elemental inclusions influence the mantle conductivities at depth. Thus, studies that investigate pressure effects (at fixed temperature) (e.g.

Estimates of thermal conductivity at high temperature and high pressure are available, but

In general, measurements from mantle minerals show that thermal conductivity increases with increasing pressure. At a fixed temperature (e.g. at room temperature

The thermal conductivity decreases with increasing temperature and follows a relationship proportional to

Temperature and depth variations in conductivity had been considered previously for Boussinesq or extended Boussinesq fluid simulations. Using the

In this study, we use compressible thermochemical mantle convection models to examine the effect of temperature-, depth-, and composition-dependent conductivity on the stability of thermochemical piles. By refining the conductivity parameters controlling the bottom-to-top conductivity ratio, we try to reproduce conductivity values comparable to lower-mantle measurements. Furthermore, we determine the longevity of systems evolving with Earth-like primordial reservoirs. First, we isolate the effect of purely depth-dependent thermal conductivity to understand the dynamics of primordial reservoirs under different lowermost-mantle conductivity conditions. Next, we introduce the effect of temperature and composition dependence to examine how the heating conditions in conjunction with depth dependence affect pile evolution. We conclude by examining the effect of a fully heterogeneous conductivity (featuring depth dependence based on mantle minerals) on the evolution of primordial reservoirs.

We model compressible thermochemical mantle convection using the finite-volume code StagYY

Thermochemical reservoirs are modelled with a dense primordial material origin. An initial dense layer occupies the bottom 160 km of the lower mantle, corresponding to a volume fraction of approximately 3 %. The buoyancy ratio,

We model a phase change between upper- and lower-mantle materials but neglect the phase change from perovskite (Pv) to post-perovskite (pPv) at the bottom of the mantle.

Heterogeneous conductivity is emulated using a non-dimensional parameterized model that characterizes its variations with non-dimensional depth (

For depth dependence, we performed simulations with two different depth-dependence laws. First, we consider a linear depth dependence given by

The temperature dependence is given by

The composition dependence is given by

Initial temperature profile

Figure

Because conductivity is allowed to vary throughout the system, the governing equation for conservation of energy is given by

Simulations are computed over a non-dimensional diffusion time of 0.0318, corresponding to 11.2

To measure the evolution of the thermochemical structure, we examined our calculations once they became quasi-stationary, as defined by the mean core–mantle boundary heat flow,

First, we isolated the effect of purely depth-dependent conductivity. Relative temperature, primordial material, and conductivity fields are presented in Fig.

From the primordial material fields, we observe that a modest conductivity gradient (

Comparing between each

Temperature (relative to the CMB temperature) (top row), primordial material (centre row), and conductivity (bottom row) fields at

Averaged properties for all cases presented. Supplemental cases are indicated by a letter “S”. All values are computed within in a 2

Next, we examine the combined effects of depth and temperature dependence. Figure

For

For

Temperature fields (relative to the CMB temperature) and conductivity fields at

For cases featuring

The pile configuration and downwelling planform influence one another and evolve in parallel. Specifically, within the first 2

We observed that

The effect of composition dependence is highlighted in Fig. 5 for the cases featuring

Temperature (relative to the CMB temperature) and conductivity fields at

Similarly, for the

For the intermediate depth-dependent case

Finally, we examine the effect of depth dependence, based on measured conductivities of upper- and lower-mantle minerals

Temperature (relative to the CMB temperature) (top row), primordial material (middle row), and conductivity (bottom row) fields at

Evolution of cases featuring

Evolution of the horizontally averaged primordial material density anomalies is illustrated for cases featuring

Figure

Different initial conditions altering the thermal histories of case no. 17 are examined to check the robustness of our findings. We consider lower and higher initial mantle temperatures,

In the reference case, thermochemical convection exhibits a stable two-pile configuration during the entire simulation. The total pile height can be roughly read from the density anomaly plot (i.e. the initial sharp contrast in colour between red and white). For stable piles, the average heights of the pile (

When only

When only

Evolution of cases corresponding to Fig.

Examining the time series for the cases presented,

The distribution of light material (

The evolution of thermochemical reservoirs depends on their temperature and the mean bottom conductivity. The value of bottom conductivity results mainly from the competing thermal and pressure effects. Considering purely depth-dependent conductivity models, a greater lower-mantle conductivity stabilizes thermochemical reservoirs by mitigating any (additional) thermal buoyancy imparted by internal heat sources. When temperature dependence (and, to a lesser extent, composition dependence) is considered, the conductivity of reservoirs is reduced with respect to the surrounding mantle. If depth dependence is insufficient to compensate for this conductivity reduction, a positive feedback loop forms whereby a poorly conducting pile cannot evacuate heat, becomes hotter, and further reduces its conductivity. The imparted thermal buoyancy destabilizes the reservoirs and influences CMB coverage configuration, erosion rate, and the onset of entrainment. The effect of composition dependence is secondary to the thermal effect and quickens the onset of instability (e.g. by approximately 2

Our models are consistent with the two-pile configuration found in lower-resolution tomographic models. Of our simulations that attain a two-pile configuration at approximately

Variations in the physical properties of thermochemical reservoirs, most notably the buoyancy ratio and enrichment in HPEs, have a strong influence on the long-term evolution of these reservoirs. Constraints on these properties are still being determined and subjects of ongoing research. Given the scope of this study, the effect of heterogeneous thermal conductivity on pile stability is difficult to differentiate from variable conditions that affect piles' chemical and thermal buoyancy. For example, greater composition dependence for conductivity is related to pile's enrichment in iron and thus implies density increase requiring an increase in buoyancy ratio. In contrast, amount of enrichment in HPEs will directly influence the thermal buoyancy and destabilize piles in the long term. Furthermore, we do not consider a decaying rate of internal heating. Assuming a mean 3

The core–mantle boundary temperature may also be an important factor in the stability and evolution of thermochemical reservoirs when a heterogeneous conductivity is considered. In our study, the reference state imposes a CMB temperature (

Alternative formulations of the lattice conductivity exist in the literature featuring temperature and density dependences (e.g.

In this study, we investigate the influence of variations in thermal conductivity on thermochemical convection and find that a heterogeneous conductivity strongly influences the long-term evolution of thermochemical reservoirs. The combined influences of temperature and depth variations determine the mean conductivity ratio from bottom to top. In the calculations we present, for the mean conductivity profile to be comparable to the conductivity often assumed in numerical models, the depth-dependent ratio must be at least 9 times the surface conductivity. The composition dependence for conductivity only plays a minor role that augments and behaves similarly to a small conductivity reduction due to temperature. Nevertheless, this effect may be amplified when depth dependence is increased. For the cases we examine, when the mean conductivity in the lowermost mantle is much greater than the surface conductivity, large reservoirs can be maintained until the end of the simulation.

The numerical code is available by reasonable request to Paul James Tackley. A detailed description of the code can be found in

The supplement related to this article is available online at:

JMG: contributed to the study design, formal analysis, visualization, and writing – original draft preparation. FD: contributed to the study design, investigation, visualization, writing – review and editing. YL: contributed to writing – review and editing. WPH: contributed to consultation on the thermal conductivity model and writing – review and editing. PJT: contributed to software by designing the 3D thermochemical convection code and writing – review and editing. All authors collaborated and contributed intellectually to this paper.

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.

This study was funded by the National Science and Technology Council grants NSTC 110-2116-M-001-025 and 111-2116-M-001-025. The authors would like to thank the topical editor Juliane Dannberg and executive editor Susanne Buiter for handling our paper. Joshua Martin Guerrero would like to thank the three anonymous referees for helping improve the clarity of the paper.

This research has been supported by the National Science and Technology Council (NSTC) of Taiwan, Republic of China (grant nos. NTSC 110-2116-M-001-025 and 111-2116-M-001-025).

This paper was edited by Juliane Dannberg and reviewed by three anonymous referees.