Fig. 3: Odd waves induce wavelength selection and flux loops.
a–i, We perform direct simulations of the Navier–Stokes equation without and with odd viscosity. In a–e, energy is injected at wavenumbers kin < kodd and the direct cascade dominates. In f–i, kin > kodd and the inverse cascade dominates. a,b, Slices of the in-plane component ωx of the vorticity with kin < kodd. Without odd viscosity (a), vortices of all sizes are present. With odd viscosity (b, in which νodd/ν = 255), characteristic horizontal and vertical scales \({k}_{{\rm{c}}}^{-1}\) and \({k}_{{\rm{odd}}}^{-1}\) emerge (black arrows). This wavelength selection originates from the arrest of the direct cascade near kodd. c, Energy spectrum E(k) and flux Π(k) (inset) obtained from simulations, for different values of odd viscosity (legend in e). Energy flows from the injection scale kin (red arrow) towards larger k, as evidenced by the positive energy flux Π(k). The cascade is progressively arrested near kodd and energy piles up, triggering viscous dissipation. d, The relative energetic amplification and/or attenuation due to odd viscosity is measured by the compensated spectrum E(k)/E0(k) (where E0(k) is the energy spectrum without odd viscosity), which peaks at a scale kc (diamonds). The peak position kc decreases as odd viscosity increases (inset), as predicted by scaling arguments (dashed line; see equation (5)). e, Plotting the compensated spectra against k/kodd confirms that condensation begins near kodd (blue arrow) and follows the scaling prediction (dashed line; see equations (2)–(4)). f,g, Slices of the in-plane velocity component vx when kin > kodd. We visualize vx instead of ωx to emphasize the large scales. Without odd viscosity (f), structures of all scales are present, dominated by the injection scale. With odd viscosity (g, in which νodd/ν = 212), secondary features with larger sizes appear because of the arrest of the inverse cascade. h, Energy spectrum E(k) and flux Π(k) obtained from the simulations (diamonds indicate kodd). i, The inverse cascade is arrested by a flux-loop mechanism, as evidenced by a decomposition of the flux in homochiral (blue) and heterochiral (red) channels that correspond, respectively, to triads with different or same signs of helicity. In i, we have used hyperdissipation in the simulations to highlight the flux loop (Extended Data Fig. 1a iv).
