Boundary-condition control of reaction–diffusion band spacing in metamorphic rocks

Abstract We develop a reaction--diffusion model of the activator--inhibitor type for metamorphic band spacing, with a slowly diffusing solid phase as activator and a rapidly diffusing fluid phase as inhibitor. We show that the observable spacing is selected not by the critical wavelength $\lambda_c$ alone but by mode locking: on a finite domain, the boundaries quantise the admissible wavenumbers and lock the pattern onto the nearest permitted mode. Linear stability analysis yields $d_c \approx 43.4$ and $\lambda_c = 2\pi (D_u D_v / \det \JJ)^{1/4}$. Neumann boundaries on hand-specimen-sized domains shift the observed spacing by up to $29\,\%$ from $\lambda_c$. We map $d_c(k_3,k_4)$ over three orders of magnitude and provide a calibration recipe linking model parameters to petrological observables. For the East Kimberley zebra rock, the predicted spacing of $0.3$--$12$~mm brackets the observed $3$--$10$~mm. We state explicitly what the model predicts and what it only permits. \bigskip\noindent\textbf{Keywords:} reaction--diffusion; activator--inhibitor kinetics; Turing instability; metamorphic banding; zebra rock; orbicular granite; boundary conditions; finite-difference modelling; Onsager coefficients.