Fig. 1: Cascade-induced pattern formation.
a, Direct energy cascade: in a turbulent 3D fluid, energy injected at large scales (red arrow) is transferred to smaller and smaller length scales (black arrows) to microscopic length scales in which dissipation occurs (blue arrow), as captured by the so-called energy spectrum E(k), which describes how much kinetic energy is contained in modes with wavenumber k. The energy transfer across scales can be traced to vortices breaking up into smaller and smaller vortices up to dissipative scales. This mechanism is intrinsically nonlinear: it relies on triadic couplings between the modes of the system. b, Inverse energy cascade: in a turbulent 2D fluid, or in a rotating 3D fluid, there is instead a transfer of energy from the scale in which energy is injected (red arrow) to larger and larger scales, and the energy is either dissipated or piles up at the largest scale available (blue arrow), the size of the system. Correspondingly, vortices merge together until only a single positive vortex and a single negative vortex remain, both of which have approximately half the size L of the system. Inverse cascades can also arise in 3D from mirror symmetry breaking4,55,56 or by imposing large-scale shear57. c, In a hypothetical situation in which a direct cascade and an inverse cascade can be put together in the right order (black arrows in the figure), energy will be transferred to an intermediate length scale \({k}_{{\rm{c}}}^{-1}\), leading to the appearance of structures with a characteristic size \({k}_{{\rm{c}}}^{-1}\) independent of the size L of the system. This nonlinear wavelength selection mechanism relying on combined turbulent cascades can be seen as an instance of pattern formation. d, Standard pattern formation from a linear instability: the wavelength \({k}_{{\rm{c}}}^{-1}\) corresponding to the most unstable linear mode (that is, the one with the largest growth rate σ(k)) is selected. As an example, we have shown the coat pattern of a cat.
