Plate tectonic models of continental growth agree on a critical behaviour of the continental lithosphere, which is characterized by alternating episodes of extension and compression in response to perturbations in the internal energy of the plate(Bak 1996; Hodges 1998; Lister, Forster, and Rawling 2001). The picture that emerges from the AHCZ is that of a thermodynamically structured continental lithosphere evolving around a stable state. This stable state is controlled by the thermochemical configuration of the crust and is modulated by temperature-sensitive dissipative feedback mechanisms in response to plate-boundary forces; any variation in plate-boundary forces would cause a change in the thermodynamic state of the evolving plate and move the system out of equilibrium, i.e. to a state of non-zero internal energy. This simple thermodynamic model enables us to interpret observed crustal thickness variations in the AHCZ as finite-amplitude perturbations from a stable energetic state in equilibrium with active plate-boundary forces. To describe the geodynamics of the continental interiors, it is, therefore, crucial to properly trace the dissipation path and to quantify the magnitudes and rates involved in the process. In doing so, we follow previous studies(Houseman and Houseman 2010) and graphically represent the dissipative evolution via a phase space described in terms of the mean lithosphere temperature and crustal thickness configuration (see Methods Section 4). We find that within this phase space, Cr represents a fixed-point attractor and any external force, i.e. finite-amplitude perturbation, engenders variations in rates of crustal growth (thickening or thinning). This in turn triggers the onset of a dissipative thermo-mechanical feedback loop steering the evolution of the system (see Methods Eqs. 4.1,4.2,& 4.7).
The exact characteristics of the evolutionary path depend on both the amplitude of the initial perturbation, the source of the initial driving energy, and the relaxation time scale of the active dissipative process, whether thermal diffusion and/or viscous deformation. We can quantify the resulting evolution of the plate via a single non-dimensional constant (\({\psi }\)), the ratio of the timescale of thermal diffusion to viscous relaxation, which provides an estimate of the internal energy of the evolving system. Typical ranges of thermal properties and viscosities of the continental lithosphere provide a lower bound to such parameter, that is \({\psi }\ge 1\). In addition, oscillatory cycles around a fixed-point attractor can only arise if the system passes through metastable states, that is states of null total energy but with non-zero thermal and mechanical contributions (Fig. 3–5, see Methods Eq. 4.5–4.7). These states can only occur if the thermal and viscous relaxation time scales differ, thereby imposing a second condition on \({\psi }\) that is, \({\psi }>\)1. In addition, \({\psi }=\)1 implies overly stiff plates which is not representative of Earth featuring mobile-lid plate-tectonics.
If we consider magnitudes of perturbations as found in the AHCZ (Fig. S6), for a linear Newtonian viscosity of the lithosphere, we observe decaying oscillatory behaviour only for a limited range of the energy constant \((1<{\psi }\) < 1000, Fig. S7). Considering a more realistic, non-Newtonian viscosity configuration, the width of the decaying oscillatory domain is additionally limited by the activation energy of the material considered (Fig. 3, 4, & S8). Surface processes (erosion and sedimentation) do not change the general behaviour of the system. The isostatic response of the lithosphere to this ancillary source of deformation acts instead to restore the long-term equilibrium state of the continental plate (Fig. 5 red and blue curves). By decreasing the energy constant \({\psi }\) (e.g. by considering higher effective plate viscosities) we observe further dampening of the oscillations leading to an exponential-like decay in crustal thickness over longer time (Fig. 3, 4 & Fig. S9, S10). As discussed above, the limiting case of \({\psi }\)=1 rules out any metastable state and portrays a path decaying exponentially to the final attractor point. Runaway extension, in the form of exponentially growing oscillations, are only possible at high energy states (e.g., \({\psi }\ge\)1000, Figs. 3 & 4). However, they can be effectively damped out by thermally driven dissipative processes. The main source of dissipation lies in the additional heat source from HPE in the crust (Fig. 5). Neglecting this contribution results in relatively high initial strain-rates, low effective viscosity and in crustal thinning at rates faster than the diffusive time scale of the plate (Fig. 5b). Under these conditions, the system is able to maintain a finite amount of internal energy throughout its evolution, which supports the ongoing crustal thinning finally leading to run away and rift-like extension (Fig. 5, thin yellow, and black and grey lines). The contribution from HPE is to increase the amount of energy being dissipated in the system (see Methods Eq. 4.6: second term in right) thereby lowering the driving force to crustal thinning (Fig. 5b; Methods Eq. 4.5–4.7). As a result, the system undergoes crustal thinning at lower rates (Path A in Fig. 5c). Enhanced dissipation rates from HPE also affect the energetic balance, with the plate gradually losing the driving internal energy upon reaching a first metastable state of null energy (Fig. 5 green open circle; also see Methods Eq. 4.5–4.6). This metastable state is characterized by a balance between extensional forces from viscous creep and increasing compressional forces from thermal cooling. This metastable state is also characterized by a crust that has yet to attain its critical thickness, being thinner and hotter (Fig. 5c&d). Ongoing thermal diffusion leads to plate cooling and crustal thickening associated to the onset of a compressional regime. Cooling and associated crustal thickening continue until reaching a point where heat from the increasing HPE concentration starts to partially compensate for the ongoing cooling (Path B in Fig. 5), leading to lower rates of crustal thickening and a reverse thermal trend. From this point onward, crustal thickening is accompanied by heating at gradually higher rates, and the system passes through a second metastable state of zero internal energy (black circle in Fig. 5). This state is characterized by a crust thicker and colder than Cr, therefore, the system undergoes another cycle of extension-heating followed by cooling-thickening, similar to the one described above. Such cycles of extension and compression are driven by the available energy in the system, i.e., non-zero thermal and/or mechanical energy. As such, their amplitudes are a function of the magnitude of the initial perturbation (Fig. 5a, Alps vs Tibet, and Fig. 3,dashed curves) as well as the non-dimensional energy constant (Fig. S9). The frequency of each oscillation (distance in time between metastable states) depends on the ability to dissipate the available energy that fuels its dynamics. By increasing the HPE concentration (promoting higher dissipation) metastable states lie closer to each other in time, thus lowering the frequency of the oscillations. The limiting case of relatively high HPE content is associated with an „overdamped“ system evolving in time to the final equilibrium crustal thickness without passing through metastable states (i.e., no oscillations) (Figs. 5 & S10), while for the other end-member case of no HPE, the system showcases exponentially growing oscillations and runaway rift-like extension.
Our results show that the high amplitude perturbations to the equilibrium state, as typified in orogens and plateaus in the AHCZ, could be dissipated to attain crustal thickness similar to continental interiors, suggesting a potential mechanism for craton formation, or, depending on the effective role of surface processes, formation of an intracratonic basin(McKenzie and Priestley 2008) (see also Fig. 3–5). A thicker than average continental crust will be subjected to an extensional stress state leading to crustal thinning in response to strain accumulation, and to a time-average dissipation of its potential energy. We additionally found that the time taken to attain such an equilibrium is in the order of a few 100‘s Myrs, a time scale that is consistent with estimated seismogenic strain rates in the AHCZ and other continental interiors(Kreemer et al. 2014; Zoback et al. 2002). The regime of exponentially growing oscillations is associated with a state of extensional forces of the order of plate driving forces (~ 1012N/m) (Fig. 5b & Fig. S10). Such a state is thermodynamically unstable, and therefore it will evolve over time to reach thermodynamic equilibrium via passively thinning the underlying mantle, resembling the dynamics driving passive rifting(Merle 2011).
The lithosphere-asthenosphere system has been shown to be characterised by a marginal stability(Cottrell, Jaupart, and Molnar 2004). This being true, it follows that any perturbation to its initial state from the deeper convective mantle could exert an additional influence on its dissipative evolutionary path. If we consider end-member values for mantle convection velocities and length scales(Ogawa 2008; Tackley 2000), we compute mantle transit time scales in the order of ~ 100 Myrs (Fig. S11). These time scales are in the same order of magnitude as computed relaxation time scales of the lithosphere (Fig. 3–5). We, therefore, suggest that deep mantle perturbations, in the form of plume dynamics or mantle flow, can additionally modulate the system behaviour, potentially leading to runaway extension and the formation of a new oceanic basin.