Fig. 2: Rotating versus odd turbulence.
a–n, We compare turbulence in a fluid rotating with high frequency Ω (a–d and i–k) and a fluid with high odd viscosity (e–h and l–n). a–h, Both fluids are characterized by a rotation direction Ω (along z), making them anisotropic and chiral. The rotation is global in rotating fluids (a). It is induced at microscopic scales in odd fluids, for instance, by particles that all spin in the same direction Ω (e). In both cases, the flow becomes 2D, with column-like structures aligned with Ω, as seen in the kinetic energy (b,f) and the z-averaged vertical vorticity \({\langle {\omega }_{z}\rangle }_{z}\) (c,g) obtained from simulations. The two-dimensionalization originates from the decorrelation by waves in the fluid (inertial waves in d and odd waves in h) of the triads by which energy transfer occurs (d, inset). Modes with kz ≠ 0 have finite frequencies (red lines) and quickly decorrelate, whereas modes with no vertical variation (kz = 0, blue lines) all have ω = 0. i–n, To predict the direction of the cascades (black arrows), we compare the inverse frequency of waves with the time over which energy transfer takes place (the eddy turnover time \({\tau }_{E}^{-1}\propto {k}^{2/3}\)). In rotating fluids (i), the flow is quasi-2D at small wavenumbers (blue region) and isotropic (3D) at large wavenumbers (once \({\tau }_{E}^{-1} > \Omega \), red region). In odd fluids (l), we expect the flow to be quasi-2D at large wavenumbers (blue region) and isotropic at low k (once \({\tau }_{E}^{-1} > {\tau }_{{\rm{o}}{\rm{d}}{\rm{d}}}^{-1}\), red region). The crossover point defines a characteristic scale kodd, in analogy with the Zeman scale kΩ in rotating fluids. We sketch cascades in the energy spectra when the injection scale is smaller (j,m) and larger (k,n) than the characteristic scale. In rotating fluids, there is a direct cascade of energy above the rotation (Zeman) scale (j) and an inverse cascade below (k). This situation is known as a split cascade4. In odd fluids, we expect the situation to be reversed: energy cascades directly for wavenumbers below kodd (n) and inversely above (m), causing a pile-up of energy at the odd viscosity length scale and arresting both cascades. The pile-up is saturated by viscous dissipation, leading to a bump in the energy spectrum at another scale kc.
