Fig. 4: Cascade-induced pattern formation across domains.
a–d, Cascade-induced scale selection can occur in systems ranging from Navier–Stokes turbulence (a,b) and magnetohydrodynamics (MHD) turbulence (c) to mass cascades (d). a, Chiral active fluids are an example of fluids with odd viscosity. As demonstrated in Fig. 3, these fluids are expected to exhibit a double arrested cascade at high enough νodd/ν and Reynolds number (see the Methods for a discussion on orders of magnitude). We have interpreted this phenomenon as the result of a decorrelation of wavenumber triads by odd waves. b, In 2D geophysical flows and plasma, an arrested inverse cascade associated with wavelength selection occurs13,14,15,16,17,18,19. It can be seen as the consequence of the decorrelation of triads by Rossby (or drift) waves, which set the characteristic scale kRh known as the Rhines scale. c, A double arrested cascade has been predicted in the solar wind, based on the properties of inviscid invariants of finite Larmor radius MHD20,21,22. This mechanism, known as a helicity barrier, relies on the change of nature of an inviscid invariant, which interpolates between cross-helicity and magnetic helicity (these quantities cascade in opposite directions). d, Scale selection can also occur in mass cascades, ranging from the stationary distribution of raindrop sizes that would occur in steady-state conditions24 to smoke aerosols25. This arises from the balance between coalescence and breakup of the droplets, which effectively have scale-varying rates (\({w}_{k}^{\pm }\), red and blue curves in the schematic). Similar phenomenology arises in active mixtures46,47,48,49,50,51, although not necessarily with a flux across scales. In the Methods, we provide a minimal model of mass cascade exhibiting scale selection.
