To capture rapid ductile shear localization, we consider a system of coupled thermomechanical equations governing the conservation of momentum, mass, and energy: 1∂τij∂xj-∂P∂xi=0,21ρ∂ρ∂t=-∂vi∂xi,3ρCp∂T∂t=∂∂xik∂T∂xi+τijε˙ijvi, where τij is the Cauchy stress deviator, xi denotes the Cartesian coordinates, P is pressure (positive in compression), ρ is density, t is time, vi is the velocity vector, Cp is specific heat capacity, T is temperature, k is thermal conductivity, and ε˙ijvi is the viscous component of the deviatoric strain rate. For simplicity, we neglect the inertial terms and body forces (i.e. gravity) from Eq. () as well as adiabatic and radiogenic heating from Eq. (). The last term of Eq. () describes energy from viscous dissipation and it is entirely partitioned into shear heating. These simplifications are discussed in Sect. .

The conservation equations are augmented by a constitutive relation for bulk compressibility: 41Kb∂P∂t=-∂vi∂xi, where Kb is the bulk modulus. For simplicity, we neglect thermal expansion from Eq. (). Combining Eqs. () and (), and integrating the changes in pressure and density yields the equation of state for density: 5ρ=ρrefexp⁡P-PrefKb, where ρref and Pref are the reference density and pressure at atmospheric conditions respectively p. 26.

The deviatoric strain rate is defined as: 6ε˙ij=12∂vi∂xj+∂vj∂xi-13∂vk∂xkδij, where ε˙ij is the deviatoric strain rate and δij is the Kronecker delta. All subsequent references to stress or strain rate refer to the deviatoric parts of the two tensors.

We use Maxwell visco-elasticity , where the strain rate is the sum of its elastic and viscous components: 7ε˙ij=ε˙ijel+ε˙ijvi=12G∂τij∂t+12ητij, where ε˙ijel is the elastic strain rate component, G is the shear modulus, and η is the effective shear viscosity. Whereas the elastic deformation is governed by the shear modulus, viscous deformation is a combination of diffusion creep, dislocation creep, and low-temperature plasticity. Following the approach of Maxwell, we consider all viscous mechanisms in series, which implies that deformation is dominated by the weakest one and that strain rate components are added : 8ε˙IIvi=ε˙IIdif+ε˙IIdis+ε˙IILTP, where the superscripts dif, dis and LTP denote diffusion creep, dislocation creep, and low-temperature plasticity, respectively. The subscript II denotes the square root of the second invariant of an arbitrary second-order tensor Cij: 9CII=12CijCij.

As a consequence of the Maxwell approach in Eq. (), the effective viscosity η can be expressed as: 10η=1ηdif+1ηdis+1ηLTP-1, where 11ηdif=12(Adif)-1dmexp⁡EdifRT,12ηdis=12(Adis)-1nε˙IIdis1n-1exp⁡EdisnRT,13ηLTP=τLTP2ε˙IILTP. A is a prefactor, E is the activation enthalpy, d is grain size, m is the grain size exponent of diffusion creep, R is the universal gas constant, and n is the powerlaw exponent of dislocation creep. The LTP-stress τLTP is given by: 14τLTP=RTELTPσressinh⁡-1ε˙IILTPALTPexp⁡ELTPRT+σb,15σres=σL+σKd, where σb, σL and σK are material constants . Diffusion creep dominates deformation at low stress/high temperature and dislocation creep at medium stress/temperature. LTP is dominant at large stress/low temperature, and it behaves similarly to perfect plasticity due to brittle deformation. It remains inactive below a critical stress, but accommodates all deformation that would otherwise increase stress beyond this threshold.

We use models with simple shear boundary conditions and central weak inclusions to initialize localization of deformation. Heat flux is zero across all domain boundaries. In 1D, the weak zone is introduced by multiplying the flow law prefactors Adif and Adis (see Sect. ) by a weakening factor ω which follows a Gaussian distribution with a minimum of 1 and a maximum of 2 (Fig. a). The full-width-half-maximum of the distribution is 200 m, and the extent of the entire model is 10 km.

The vertical and horizontal extents of the 2D model are 10 and 60 km, respectively. The weak inclusion is an ellipse with semi-major axes of 375 and 125 m, respectively. Within this anomaly, Adif and Adis are multiplied by 2, and σb is divided by 2. The different implementations of the weak inclusion are discussed in Sect. . The lateral boundary conditions in the 2D model are periodic (Fig. b). Unless stated otherwise, the material parameters used in all models are listed in Table .

In the 1D configuration, the spatial dimensions are reduced to the vertical y-direction, and Eqs. ()–() are simplified. With simple shear boundary conditions, no gravity, and no thermal expansion, the divergence of velocity is inherently zero. The conservation of mass (Eq. ) simplifies to: 16∂ρ∂t=0, and Eq. () simplifies to: 17∂P∂t=0.

This renders the model incompressible, with density and pressure constant in time. Furthermore, the velocity vector is reduced to its horizontal component, which simplifies Eq. () to: 18ε˙xy=12∂vx∂y, with the other components of the strain rate and stress tensor equal to zero. This simplifies the conservation of momentum (Eq. ) to: 19∂τxy∂y=0.

For both the 1D and 2D models, we employ a variable grid, with the smallest vertical cell size in the center of the domain. In the 1D models, the central quarter of the grid consists of uniformly sized cells, while spacing increases linearly towards the model boundaries. The outermost cells are approximately 125 times larger than those at the center, allowing for maximum resolution in the region where thermal runaway is expected to occur. This is common practice when investigating thermal runaway e.g.,. Material properties and most field variables are defined at cell centers, whereas velocity and heat flux are located at cell edges (Fig. a).

In 2D, the grid refinement is limited to a factor of 2 to avoid convergence issues arising from large cell aspect ratios. In the horizontal direction, all cells are the same size. We use 1536 and 256 cells in the horizontal and vertical direction respectively, yielding resolutions of about 39 m (horizontal) and 26–52 m (vertical). We use a staggered grid approach where material properties, temperature, pressure, viscous dissipation, and normal stress components are defined at cell centers. Velocity and heat flux are defined on cell edges, while shear stress components are located at cell corners (Fig. b).

In the APT approach, the conservation equations are solved at each physical time step by introducing a pseudo-time derivative for each equation and iteratively updating the primary variables vi, P, and T until the residuals drop below a given numerical tolerance. Applying this procedure to Eqs. ()–() yields: 20∂vi∂ψ=∂τij∂xj-∂P∂xi,21∂P∂ψ=1Kb∂P∂t+∂vi∂xi,22∂T∂ψ=1ρCp∂∂xik∂T∂xi+τijε˙ijvi-∂T∂t, where ∂/∂ψ denotes the pseudo-time derivative. During each pseudo-time iteration, each primary variable is incremented proportionally to the sum of the current residual and the previous increment : 23Δγ=∂γ∂ψ+1-1ζγΔγprevΔψγ, where γ represents one of the primary variables vi, P, or T, Δγ is the current increment of the respective variable, Δγprev is the increment from the previous iteration, ζγ is the damping parameter (>1). Δψγ is the size of the pseudo-time step given by: 24Δψvi=Δxifvη,25ΔψP=fPηmax⁡(nci),26ΔψT=min⁡min⁡(Δxi)22ndimκ,Δt2, where Δxi is the grid spacing, fv and fP are factors, nci is the number of cells in each dimension, ndim is the number of dimensions, κ=k/(ρCp) and Δt is the physical time step.

The left hand side terms in Eqs. ()–() are equivalent to the residuals of the conservation equations. Once all of them are smaller than a given numerical tolerance of 10-6 after normalization, the solution is converged and is equivalent to a fully implicit, backward Euler solution with converged nonlinearities.

Given the nonlinear nature of dislocation creep and low-temperature plasticity, the strain rate partitioning (Eq. ) cannot be solved analytically but requires a numerical approach. It can be updated and solved alongside the conservation equations (Eqs. –). To stabilize the rheology solver, we use a relaxation approach for the viscosity updates of each mechanism during the pseudo-transient (PT) iterations: 27ηiit=exp⁡(1-ηrel)log⁡(ηiit-1)+ηrellog⁡(ηit), where the superscript it denotes the iteration count, ηrel<1 is the relaxation factor , and ηit is the target viscosity (i.e. the new viscosity without relaxation). We discuss our strategy for solving the strain rate partitioning in Appendix A and Fig. .

To stabilize the model during thermal runaway and mitigate mesh dependency, we test three regularization strategies: (i) viscosity regularization, (ii) gradient regularization, and (iii) inclusion of latent heat of melting. All approaches aim to limit maximum strain rates and prevent viscosities from dropping below a critical threshold. We note that alternative regularization strategies for brittle failure have been proposed in the literature e.g.,. We discuss the strategies' relation to physical mechanisms in Sect. .

The basic model behavior is described in Fig. b and the introduction. Outside the runaway phase, time steps ranging from tens to thousands of years are sufficient. However, resolving the thermal runaway phase requires time steps on the order of milliseconds. While large time steps may adequately capture the long-term stress evolution, they fail to resolve the transient dynamics leading up to runaway (Fig. ). In particular, they significantly underestimate temperature increase and slip velocity (Fig. b and d).

As the spontaneous onset of runaway cannot be predicted a priori, an adaptive time-stepping scheme is critical. Identifying suitable time steps is a well-known challenge in scientific computing, and a number of studies propose different methods e.g.,. As thermal runaway is driven by the conversion of elastic energy to thermal energy e.g., the most indicative parameters for the onset and intensity of runaway are stress and temperature. Therefore, we constrain time steps by limiting the maximum allowable change in these two quantities (Δτ, ΔT). Similar strategies are employed in earthquake modeling studies . We evaluate three methods for implementing adaptive time stepping. In all cases, we define threshold values of Δτmax=50MPa and ΔTmax=5K.

Numerical solvers commonly use internal scaling to center quantities around 1 which minimizes round-off errors due to numerical precision. To do so, a set of scales is created, and all dimensional quantities are divided by an appropriate combination of these scales. As an example, a geodynamic model focused on plate-scale deformation might use a time scale of tsc=1012s and a stress scale of τsc=108Pa which combine to a viscosity scale of ηsc=1020Pas. This means a time step of 100 years is scaled to ΔtND=Δttsc≈0.003156, where ΔtND is the nondimensional time step used inside the solver.

This becomes problematic once adaptive time stepping reduces the dimensional time step to one second, as this is equivalent to 10-12 after scaling. Considering the numerical precision of 10-15, such a low value is prone to round-off errors. To mitigate this, we decrease tsc by one order of magnitude every time the nondimensional time step drops below 10-9. Changing the time scale does not only impact the nondimensional time step but all quantities which carry units of seconds such as strain rates, velocities, and viscosities. All of these have to be rescaled together. This is convenient for viscosities as they also decrease significantly during runaway. It is also beneficial for velocities and strain rates as they increase during runaway and decreasing tsc increases the velocity and strain rate scales.

In Fig. , we demonstrate how rescaling facilitates convergence and reduces the number of iterations by two orders of magnitude for a model that takes time steps as low as 25 µs. Without rescaling, the model requires about 2×109 iterations in total to solve, the majority of them during thermal runaway. Rescaling properties with time scales in their units as soon as the nondimensional time step ΔtND drops below 10-14 reduces the number of iterations by one order of magnitude. Rescaling at ΔtND<10-12 reduces the total number of iterations by another order of magnitude, and only half of them are used during the runaway. Further reduction of the critical ΔtND has only negligible effects (Fig. ) despite the proximity of the values to numerical precision.

During thermal runaway, the viscosity within the shear zone decreases dramatically (more than 10 orders of magnitude) due to the temperature increase. Large contrasts in material properties are challenging for numerical solvers e.g.,, especially for iterative approaches which rely on local conditioning. Elasticity can reduce the stiffness contrast between high and low viscosity areas, but this is not sufficient to guarantee convergence. Even if the solver converges, shear zones often thin to the width of one grid cell. In this case, the mechanical behavior of the model is governed by the numerical resolution instead of the physics of the problem . To alleviate this issue and improve reproducibility, we test three regularization methods: viscosity regularization (see Sect. ), gradient regularization (see Sect. ), and latent heat of melting (see Sect. ). To quantify the impact of viscosity and gradient regularization, we run 60 1D simulations in which we vary between five different numerical resolutions (63–1023 cells), with six different viscosity regularization values ηreg (106–1018Pas), and six different gradient regularization values λreg (1–32 m). As latent heat proved to be unable to regularize the reference model, we did not include it in this parameter study. We use maximum velocity vmax, maximum temperature Tmax, and shear zone width dsz as diagnostic parameters for the analysis.

During the elastic loading phase, the model typically converges within a few (<100) PT iterations. While such fast convergence is computationally efficient, it can introduce numerical errors when using the viscosity relaxation method (Eq. ). In this approach, the viscosity is incrementally updated in each iteration using a relaxation factor, commonly ηrel=0.01, meaning that only 1 % of the computed viscosity update is applied per iteration. Although this under-relaxation stabilizes the solver, it can hinder convergence of the viscosity field for a low PT iteration count.

Figure b shows that after 100 iterations, the viscosity update has only progressed about halfway towards its target value. Converging viscosity relaxation (i.e., reaching the updated steady-state value) typically requires around 500 iterations for ηrel=0.01. Failing to accurately resolve viscosity relaxation may become problematic near the onset of LTP creep, where ηLTP drops rapidly as stress approaches the yield threshold τLTP.

LTP accommodates all deformation that would otherwise increase stress beyond this threshold. If ηLTP and the associated strain rate partitioning are not updated fast enough, stresses can significantly exceed τLTP, requiring corrective adjustments in subsequent time steps (Fig. a). This not only leads to an incorrect stress evolution, but can also trigger spurious slip events that would not occur under properly updated stress conditions.

To mitigate this issue, we monitor the convergence between viscosity ηit and target viscosity ηt (Eq. ). When the relative difference |ηit-ηt|ηt is smaller than the viscosity tolerance tolη, viscosity is considered converged. Once the conservation equations (Sect. ), the viscosity, and the strain rate partitioning (Appendix A) are converged, we accept the solution. This ensures that both rheological and mechanical responses are correctly captured during the elastic-to-LTP transition (Fig. a).

The stress overshoot for insufficient viscosity convergence is more prominent when the steady-state stresses of diffusion and dislocation creep are large. For the model in Fig. a, we increased Edif and Edis to 435 and 670 kJmol-1, respectively, which is equivalent to considering the pressure dependence of the rheology and 10 GPa of background pressure (Table ).

Adaptive time stepping remains critical in 2D. During elastic loading, time steps are typically on the order of decades; they shrink to months at the onset of thermal runaway, to hours during rupture propagation, and to seconds at peak velocities. Setting a lower time step bound can dampen thermal runaway, or, if set too high, cause solver failure. For most of the simulations, the predictive time stepping strategy (Sect. ) suffices. However, when rupture fronts meet across the periodic boundaries, restarting time steps (Sect. ) is required to maintain stability.

Regularization plays a similar role in 2D as in 1D. It enforces a lower bound on viscosity and upper bounds on strain rate and velocity. Due to the more limited spatial resolution in 2D, the shear zone thickness is often constrained by grid size unless a high regularization viscosity (∼1016Pas) is used. If a higher spatial resolution can be achieved through improved refinement or significant increase in grid cells, regularization viscosity will again become the controlling factor. This equally applies to adaptive rescaling (Sect. ), which becomes essential when smaller time steps and higher velocities exacerbate round-off errors. Given its superior performance at fine resolutions, the viscosity regularization is the preferred method in 2D.

Finally, monitoring the convergence of the relaxed viscosity (Sect. ) has minimal impact in 2D. Even before reaching the LTP threshold, the number of iterations per time step increases to ∼5000 to solve the conservation equations, ensuring that the relaxation-based updates are well-converged.

To compare the 1D and 2D models, we ran a 1D model using the same limited refinement as in 2D and compared the results (Fig. ). Both models exhibit very similar trends in stress, maximum temperature, maximum velocity, and minimum viscosity. As long as the thermal runaway fronts remain more than 10 km away from the domain boundaries (Fig. a–f), the 2D and 1D models show nearly identical vmax and Tmax (Fig. b and c). Once the rupture fronts meet due to periodic boundaries, vmax increases by a factor of ∼3 and Tmax by ∼250°C. This surplus is likely caused by the increased stress in front of the rupture tips (yellow lobes in Fig. ). When the tips connect, they can release more stress which is converted into heat, resulting in faster slip.

The most notable difference is the duration of the LTP-dominated phase, which lasts over 10 kyr in the 1D model but only ∼200 years in 2D. This discrepancy stems from differences in how the anomaly is defined. In 1D, only the flow laws of diffusion and dislocation creep are weakened. In 2D, the LTP back stress σb is also reduced. This difference was necessary as reducing σb in 1D prevents stresses in the entire model from reaching values above 1 GPa, while omitting this weakening in 2D hampers localization significantly.

As stated above, we neglect inertial terms from Eq. () for simplicity. To determine to which extent this assumption is justified, we roughly estimate the inertial term ρ∂vi∂t. In 2D, we assume the extreme case that a grid node accelerates to the maximum velocity (5 mms-1) in a single time step (∼15s). The resulting value for ρ∂vi∂t is on the order of 1 kgs-2m-2, whereas the term ∂τij∂xj on the right hand side of the governing two-dimensional momentum equation is about seven orders of magnitude larger. In this case, neglecting the inertial term remains justified. However, in 1D models with the lowest tested values of ηreg or λreg, the inertial term could reach much larger values due to the larger velocities. In this case, inertia could reduce acceleration.

Gravity is neglected from Eq. () because the orientation of the shear zone is arbitrary in reality. Thermal expansion is neglected from Eq. () and adiabatic heating from Eq. () as they did not play a significant role in a previous 2D study on thermal runaway .

Adding grain size evolution could have a significant impact on the rheology and energy balance of our models, depending on how much energy from viscous dissipation is partitioned into it. The partition factor spans several orders of magnitude in the literature with a maximum of 10 % e.g.,. Furthermore, it might be strain-dependent, as experimental studies suggest that only about 10 % of olivine grains recrystallize at a strain of 1 . Most of our models do not even reach a strain of 0.1.

Endothermic phase transformations are another potential sink for thermal energy during runaway. Intermediate-depth earthquakes are commonly associated with the antigorite-olivine transformation , and estimate the enthalpy change of this reaction to be on the order of 2.5×108Jm-3. This is about one quarter of the energy density required for full melting. Consequently, this process would not have a significant effect on the energy balance during thermal runaway. Deep-focus earthquakes are associated with the olivine-ringwoodite transformation , but this reaction is exothermic and cannot act as an energy sink during runaway.

We only show cases with a single perturbation. As the 2D setup uses periodic boundary conditions, it approximates a setup with multiple perturbations on the same vertical coordinate. The results show that temperature and slip velocity peak when two rupture fronts unite (Fig. b and c). A more realistic case could involve perturbations of different size, strength, and location. Comparing the length of the LTP-dominated warm-up period with the runaway phase suggests that once runaway initiates in one location, the rupture would quickly release stress from surrounding perturbations, resulting in a single dominant rupture.

Figure b illustrates that 1D models can reach temperatures that exceed any observed or constrained values for the Earth when using viscosity regularization. Similarly, models using gradient regularization reach slip velocities that are significantly faster than any observed solid deformation, including earthquake slip and seismic waves (Fig. e). These unrealistic values are inherent to 1D models as they imply an infinite shear zone e.g.,. While such models struggle to accurately describe peak runaway conditions, they are still useful in investigating how localization develops in the first place e.g.,.