The formation of mineral deposits in mesothermal quartz veins is a complex process that has been the subject of much research. The classical fault-valve hypothesis suggests that mineralization occurs when metamorphic fluids are injected during a brittle event and then locked in to mineralize, but this hypothesis does not fully explain the regular spacing of repeated mineralized patterns that are often observed. This paper proposes a new mechanism for mineralizing systems based on the theory of cnoidal waves in solids. Cnoidal waves are standing waves that can persist for long times in materials under compressive and extensional regimes. We investigate mineral deposits by analytical and numerical methods and show that the cnoidal wave instability theory provides a plausible alternative mechanism for mineralizing systems. This study opens a new avenue for field studies to demonstrate that the mechanism-based cnoidal waves play an essential role in the formation of mineral deposits.

Zebra rocks, characterized by their striking reddish-brown stripes, rods, and spots of hematite (Fe-oxide), showcase complex self-organized patterns formed under far-from-equilibrium conditions. Despite their recognition, the underlying mechanisms remain elusive. We introduce a novel advection-dominated phase-field model that effectively replicates the Liesegang-like patterns observed in Zebra rocks. This model leverages the concept of phase separation, a well-established principle governing Liesegang phenomena. Our findings reveal that initial solute concentration and fluid flow velocity are critical determinants in pattern selection and transition. We quantitatively explain the spacing and width of a specific Liesegang-like pattern category. Furthermore, the model demonstrates that vanishingly low initial concentrations promote the formation of oblique patterns, with inclination angles influenced by rock heterogeneity. Additionally, we establish a quantitative relationship between band thickness and geological parameters for orthogonal bands. This enables the characterization of critical geological parameters based solely on static patterns observed in Zebra rocks, providing valuable insights into their formation environments. The diverse patterns in Zebra rocks share similarities with morphologies observed on early Earth and Mars, such as banded iron formations and hematite spherules. Our model, therefore, offers a plausible explanation for the formation mechanisms of these patterns and presents a powerful tool for deciphering the geochemical environments of their origin.

Here, we extend the Fisher-Kolmogorov-Petrovsky-Piskunov equation to capture the interplay of multiscale and multiphysics coupled processes. We use a minimum of two coupled reaction-diffusion equations with additional nonlocal terms that describe the coupling between scales through mutual cross-diffusivities and regularise the ill-posed reaction-self-diffusion system. Applying bifurcation theory we suggest that geological patterns can be interpreted as physical representations of two classes of well-known instabilities: Turing instability, Hopf bifurcation, and a new class of complex soliton-like waves. The new class appears for small fluid release reactions rates which may, for negligible self-diffusion, lead to an extreme focusing of wave intensity into a short sharp earthquake-like event. We propose a first step approach for detection of these dissipative waves, expected to precede a large scale instability.

Here, we extend the Fisher-Kolmogorov-Petrovsky-Piskunov equation to capture the interplay of multiscale and multiphysics coupled processes. We use a minimum of two coupled reaction-diffusion equations with additional nonlocal terms that describe the coupling between scales through mutual cross-diffusivities. This system of equations incorporates the physics of interaction of thermo-hydro-chemo-mechanical processes and can be used to understand a variety of localisation phenomena in nature. Applying bifurcation theory to the system of equations suggests that geological patterns can be interpreted as physical representation of three classes of well-known instabilities: Turing instability, Hopf bifurcation, and a chaotic regime of complex soliton-like waves. For specific parameters, the proposed system of equations predicts all three classes of instabilities encountered in nature. The third class appears for small fluid release reactions rates as a slow quasi-soliton wave for which our parametric diagram shows possible transition into the Hopf- or Turing-style instability upon dynamic evolution of coefficients.