Helicity controls the direction of fluxes in rotating turbulence

TL;DR

This work shows that helicity governs the direction of kinetic energy fluxes in rotating turbulence by revealing a dual transfer: fast inertial waves conserve helicity by sign and feed energy inversely into a large-scale 2D condensate, while slower modes exchange helicity with opposite sign and transfer energy forward to small scales. A mean-wave quasi-linear kinetic theory (QLKT) is developed, decomposing fluctuations into helical inertial waves and a 2D mean flow, and separating interactions into a homochiral (Sector H) and heterochiral (Sector A) regime, with a closure provided by the mean-flow balance $\nu {U'}^2 = T_{ m 3D-2D}$. The theory yields analytic expressions for the wave energy flux and the total 3D–2D transfer $T_{ m 3D-2D}$, reproducing the DNS scaling across $Ro$ and $Re$ and predicting a flux-loop state at high $Re$ where $T_{ m 3D-2D}\to 0$ while $U'$ remains finite. To extend validity across rotation rates, the authors introduce wavenumber cutoffs $k_\Omega$, $k_U^{\rm wave}$, $k_U^{\rm eddy}$ and a smooth filtering of near-resonant interactions, achieving quantitative agreement with simulations and clarifying the conditions under which condensates persist or vanish. The framework provides a general approach to sign-changing invariant dynamics in wave-dominated systems, with potential applications to geophysical, stratified, and MHD contexts.

Abstract

Turbulence sustains out-of-equilibrium fluxes that are shaped by conservation laws. Three-dimensional flows conserve energy and sign-indefinite helicity, both being transferred to small scales. Here, we uncover a dual organization of energy fluxes in 3D rotating flows, shaped by helicity. When sufficiently-fast inertial waves interact with a large-scale 2D flow, they conserve their helicity separately by sign. This causes an inverse energy transfer, from 3D to 2D motions, which promotes self-organization and spectral condensation. In contrast, slower modes exchange helicity with modes of opposite helicity sign, similarly to non-rotating 3D turbulence. This generates a forward energy transfer, from the large-scale 2D flow to small 3D scales, coexisting with the inverse transfer. We determine analytically these bi-directional energy transfers to the 2D mean flow via a quasi-linear wave-kinetic theory. The theory captures the main Reynolds number and rotation rate dependence of the mean-flow amplitude in Navier-Stokes simulations from zero to infinite rotation rates.

Figures (10)

• Figure 1: (a-c) Flow visualizations of vertical vorticity in rotating turbulence. (d) Energy transfers to the 2D condensate in DNS, due to waves of same ($\tilde{s}=s$, hollow circles) or opposite helicity signs ($\tilde{s}=-s$, colored circles), as a function of $Ro_{\epsilon} \equiv Ro \times Re^{\frac{1}{2}}$. At large rotation, waves conserve their helicity by sign and energize the 2D flow. At low rotation, modes with opposite helicities extract energy from the 2D flow.
• Figure 2: (a) Energy transfer to the large-scale 2D flow in spectral space $(p_y,p_z)$ from QL theory and (b) schematic of the energy fluxes in the system. Orange line in (a) marks the boundary between Sectors H and A, \ref{['eq:sectors_main']}. Waves in sector H conserve their sign-definite helicity $H_{\boldsymbol{p}}^\pm$, constraining the forward energy flux and causing energy transfer to the 2D flow. In contrast, modes in sector A break this conservation and extract energy from the 2D flow, with resulting energy flux $\Pi> \Pi^*$\ref{['eq:flux_hetero']}, $\Pi^*$ being the energy input from sector H.
• Figure 3: (a) Energy transfer from 3D waves to 2D condensate and (b) rescaled amplitude of the condensate, as measured from the DNS and predicted from the QL theory. Dashed lines show the predictions in each regime: in black, Eq. \ref{['eq:sol_analytic']}; in red, the scalings derived in gome2025waves. Solid lines show a numerical solution of the QL system at $Re=46$, including a wave cutoff $k_\Omega$ and a smooth treatment of homochiral-wave near-resonances. Inset in (b): rescaled amplitude of the 2D flow as a function of $Re$, showing the saturation to a flux-loop state.
• Figure 4: Heatmap of the energy transfer to the 2D flow, $T_{\rm 3D-2D}$, in the $(Ro,Re)$ plane. The colored background corresponds to the prediction from the QL theory. Stars correspond to a scale-selected vortex lattice.
• Figure 5: Reynolds stress in the spectral plane ($p_y,p_z,p_x=8\pi/L_x$), computed when interactions are homochiral (Sector H). This corresponds to the high-rotation regime. Advection by the shear flow generates a decaying forward flux of wave energy, which is transferred to the shear flow (${U'} \langle uv \rangle_{\boldsymbol{p}\boldsymbol{k}} > 0$).