Memory-Driven Pattern Selection in Fractional Reaction--Diffusion Models of Metamorphic Rocks

Abstract This study models striped and orbicular patterns in metamorphic rocks using a Caputo time-fractional Gray--Scott reaction--diffusion system. Simulations for \(\alpha\in\{0.99,0.95,0.90,0.85\}\) and six mafic diffusivities (\(D_{u0}\in\{1\times10^{-4},4\times10^{-4},5\times10^{-4},6\times10^{-4},8\times10^{-4},1\times10^{-3}\}\)) reveal a spot-to-stripe transition, with lower \(\alpha\) producing more diffuse patterns. Stability analysis at \(\alpha=0.85\) gives a Matignon angle of \(76.5^\circ\), Turing instability for \(\mu\in[0.0597,0.1003]\), critical wavenumber \(k_c=17.77\), and dimensionless wavelength \(\lambda=0.3535\). A parameter scan over 800 values of \(\mu\) identifies 456 equilibria, of which 160 are linearly stable and 66 satisfy the fractional Turing condition. At the representative value \(\mu=0.05971\), the homogeneous steady state is \((u^*,v^*)=(0.546024,29.814209)\). Convergence tests at \(\alpha=0.95\) confirm second-order spatial accuracy, with grid-refinement \(L^2\) errors \(5.85\times10^{-5}\), \(1.42\times10^{-5}\), and \(6.26\times10^{-6}\), and L1 temporal accuracy with time-step errors \(7.18\times10^{-6}\), \(4.74\times10^{-6}\), and \(2.39\times10^{-6}\). The Lyapunov functional decays monotonically and \(\Lambda=0.032603>0\), confirming dissipativity. The model qualitatively reproduces the four interface morphologies reported from the Chotanagpur Granite Gneissic Complex, India, and is presented as a complementary framework to thermodynamically based geophysical models for memory-driven transport in metamorphic rocks.