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

This study models striped and orbicular patterns in metamorphic rocks using a Caputo time-fractional Gray--Scott reaction--diffusion system. Simulations for \(α∈\{0.99,0.95,0.90,0.85\}\) and six mafic diffusivities (\(Dᵤ₀∈\{1×10⁻⁴,4×10⁻⁴,5×10⁻⁴,6×10⁻⁴,8×10⁻⁴,1×10⁻³\}\)) reveal a spot-to-stripe transition, with lower \(α\) producing more diffuse patterns. Stability analysis at \(α=0.85\) gives a Matignon angle of \(76.5^∘\), Turing instability for \(μ∈[0.0597,0.1003]\), critical wavenumber \(k_c=17.77\), and dimensionless wavelength \(λ=0.3535\). A parameter scan over 800 values of \(μ\) identifies 456 equilibria, of which 160 are linearly stable and 66 satisfy the fractional Turing condition. At the representative value \(μ=0.05971\), the homogeneous steady state is \((u^*,v^*)=(0.546024,29.814209)\). Convergence tests at \(α=0.95\) confirm second-order spatial accuracy, with grid-refinement \(L^2\) errors \(5.85×10⁻⁵\), \(1.42×10⁻⁵\), and \(6.26×10⁻⁶\), and L1 temporal accuracy with time-step errors \(7.18×10⁻⁶\), \(4.74×10⁻⁶\), and \(2.39×10⁻⁶\). The Lyapunov functional decays monotonically and \(Λ=0.032603>0\), 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.