The instability mechanism in general reaction-diffusion-advection systems is key to understanding flow-driven self-organization. Here we report the general conditions for instability in $d$-dimensional space, formally termed relative advection instability (RAI) given its exclusive dependence on the relative velocity difference. We demonstrate that the critical wave vector at the onset of instability aligns with the relative advection direction, leading to emergent traveling waves, whereas stationary periodic patterns only occur under specific orientations. Unlike the classical Turing paradigm, RAI is governed by a critical advection disparity rather than a diffusion ratio, allowing for patterns even with equal diffusion coefficients. Numerical simulations confirm that these patterns consist of traveling waves with distinct wavelengths, speeds, and amplitudes.

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• Pattern formation
• Patterns in complex systems
• Population dynamics