Cross-diffusion waves resulting from multiscale, multi-physics instabilities: theory

Abstract

Abstract. We propose a multiscale approach for coupling multi-physics processes across the scales. The physics is based on discrete phenomena, triggered by local thermo-hydro-mechano-chemical (THMC) instabilities, that cause cross-diffusion (quasi-soliton) acceleration waves. These waves nucleate when the overall stress field is incompatible with accelerations from local feedbacks of generalized THMC thermodynamic forces that trigger generalized thermodynamic fluxes of another kind. Cross-diffusion terms in the 4×4 THMC diffusion matrix are shown to lead to multiple diffusional P and S wave equations as coupled THMC solutions. Uncertainties in the location of meso-scale material instabilities are captured by a wave-scale correlation of probability amplitudes. Cross-diffusional waves have unusual dispersion patterns and, although they assume a solitary state, do not behave like solitons but show complex interactions when they collide. Their characteristic wavenumber and constant speed define mesoscopic internal material time–space relations entirely defined by the coefficients of the coupled THMC reaction–cross-diffusion equations. A companion paper proposes an application of the theory to earthquakes showing that excitation waves triggered by local reactions can, through an extreme effect of a cross-diffusional wave operator, lead to an energy cascade connecting large and small scales and cause solid-state turbulence.

Cross-diffusion waves resulting from multiscale, multiphysics instabilities: application to earthquakes

“…An application of self-diffusing reaction-diffusion equations to mineralizing systems has been proposed recently (Oberst et al, 2018). This contribution investigates geological applications of the possible relation to the reaction-cross-diffusion wave phenomenon due to the new addition of the cross-diffusion term proposed in Part 1 (Regenauer-Lieb et al, 2021). In this contribution we will show why the addition of the cross-diffusion term may be an important element for the earthquake source mechanism.…”

“…Part 1 (Regenauer-Lieb et al, 2021) introduced a (geo-)wave-mechanics-and physics-based formulation for deciphering patterns of multiphase material instabilities from the molecular scale to any larger scale. Although the paper is formulated for Earth sciences the approach constitutes a generic theory for any material.…”

“…1) and many other similar relationships in nature (examples are shown in Fig. 1 of Part 1; Regenauer-Lieb et al, 2021). To simplify the equations and address a frequently overlooked deformation mode we use the Helmholtz decomposition presented in Part 1 (Regenauer-Lieb et al, 2021) and only discuss examples for dissipative pressure (P ) waves.…”

“…1 of Part 1; Regenauer-Lieb et al, 2021). To simplify the equations and address a frequently overlooked deformation mode we use the Helmholtz decomposition presented in Part 1 (Regenauer-Lieb et al, 2021) and only discuss examples for dissipative pressure (P ) waves. The rich dynamics of superposing dissipative pressure (P ) -and shear (S) -waves will be subject of future research.…”

“…The wave-mechanics approach offers just this opportunity as it encompasses all conservation laws. The wave mechanics approach of Part 1 (Regenauer-Lieb et al, 2021) gives, in particular, a fundamental physics-based first guess to investigate these critical domains that are not directly accessible. We thereby identify critical conditions for many important Earth science problems such as the physics of earthquakes, extraction of geothermal energy, safety of nuclear waste disposal, reservoir engineering for oil and gas, the formation of mineral deposits, induced seismicity, natural hazards, groundwater management, and CO 2 sequestration and utilization.…”