Extended Data Fig. 1: Arrest of turbulent cascades in numerical simulations.

(a) The total energy flux Π(k) is decomposed into heterochiral Π_hete(k) and homochiral Π_homo(k) components for direct cascading cases (i,ii) and inverse cascading cases (iii,iv). The cases without odd viscosity (i,iii) are compared to the cases with odd viscosity ν_odd/ν = 255 (ii) and ν_odd/ν = 212 (iv). Odd viscosity enhances the homochiral manifold that predominantly cascades inversely, which is in turn balanced by an increased heterochiral flux. For the inverse cascading cases this leads to a flux loop condensate state with vanishing net flux. In (iii, iv) hyperdissipation is used to mimic increased scale separation. See also the section Helical decomposition in the SI. (b) The anisotropic kinetic energy spectrum E(k_z, k_⊥) with odd viscosity normalized by the case without odd viscosity E₀(k_z, k_⊥) for the forward cascading case with ν_odd/ν = 255 (top panel) and the inverse cascading case with ν_odd/ν = 212 (bottom panel). Both panels indicate the regions in k-space where the energy condensation due to odd viscosity occurs. (c) In order to determine the characteristic wavelengths in the vorticity field for the direct cascading case, we compute the vorticity spectrum ∥ω(k)∥² as k²E(k). Without odd viscosity, the maximum of the spectrum is close to the dissipative scale. When odd viscosity is present, a stronger peak emerges in the spectrum as a consequence of the spectral condensation at intermediate scales, evidencing the wavelength selection. (d) The lozenges give the value of h ≡ E(k_c)/E₀(k_c) obtained from the numerical simulations, and are compared with the predicted scaling \(h\propto {({\nu }_{{\rm{odd}}}/\nu )}^{1/3}\) (black dashed line). (e) We demonstrate the kinetic energy spectrum E(k) for the case of a sharp transition from a direct cascade at small k to an inverse cascade at large k, modeled as a step function of odd viscosity, stepping at \({\mathop{k}\limits^{ \sim }}_{{\rm{o}}{\rm{d}}{\rm{d}}}\) (orange), compared to the case without odd viscosity (black). Here, Θ is the Heaviside step function. The sharp transition leads to a sharp condensation at \({\mathop{k}\limits^{ \sim }}_{c}\equiv {\mathop{k}\limits^{ \sim }}_{{\rm{o}}{\rm{d}}{\rm{d}}}\) and a diffusive equipartitioned scaling ∝k² to the left of it. The resulting pattern in ω_x is shown in (f), with typical wavelength \({\widetilde{k}}_{c}^{-1}={\widetilde{k}}_{{\rm{odd}}}^{-1}\) in both the horizontal and vertical directions. See also the section Odd hyperviscosity in the SI.