Confirming Wave Turbulence Predictions in Rotating Turbulence

TL;DR

This work demonstrates that steady rotating turbulence hosts an inertial-wave forward cascade described by weak wave turbulence theory, coexisting with a dominant quasi-2D energy-containing field. By decomposing the flow into 2D and 3D components, the authors show that the 3D residual field consists of inertial waves whose spectrum follows the anisotropic WTT prediction E(k_r, k_z) ~ sqrt(epsilon Omega) k_r^{-5/2} k_z^{-1/2} (for k_z << k_r). They verify the full scaling through k_r and omega dependencies, including E(k_r, omega) ~ sqrt(epsilon/Omega) k_r^{-4} (omega/2 Omega)^{-1/2} and E(omega) ~ A (omega/2 Omega)^{-1/2} with A ~ Omega^{-1/2}, and observe collapse with p^{3/2} scaling of energy input. The results provide a solid basis for studying interactions between the quasi-2D and inertial-wave fields and for advancing experimental and theoretical work on rotating turbulence.

Abstract

Though highly impacting our lives, rotating turbulent flows are not well understood. These anisotropic three-dimensional disordered flows are governed by different nonlinear processes, each of which can be dominant in a different range of parameters. More than 20 years ago, Galtier used weak wave turbulence theory (WTT) to derive explicit predictions for the energy spectrum of rotating turbulence. The spectrum is an outcome of forward energy transfer by inertial waves, the linear modes of rotating fluid systems. This spectrum has not yet been observed in freely evolving flows. In this work, we show that the predicted WTT field does exist in steady rotating turbulence, alongside with the more energetic quasi two-dimensional turbulent field. By removing the 2D component from the steady state velocity field, we show that the remainder three-dimensional field consists of inertial waves and exactly obeys WTT predictions. Our analysis verifies the dependence of the energy spectrum on all four relevant parameters and provides limits, beyond which WTT predictions fail. These results provide a solid basis for new theoretical and experimental works focused on the coexistence of the quasi 2D field and the inertial waves field and on their interactions.

Figures (5)

• Figure 1: A snapshot of the energy density (top row) and $z$ component of vorticity (bottom row) of a rotating turbulent flow field (left), its vertical average $\boldsymbol{v}_{2D}$ (center), and the residual field $\boldsymbol{v}_{3D}$ (right), measured in a cubic domain in an experiment with $\mathrm{Re} \approx 3000$ and $\mathrm{Ro}\approx 0.01$. The full flow field is turbulent and anisotropic, exhibiting structures on a wide range of horizontal scales, and long-range vertical correlations. The 2D field is dominated by disordered, large-scale vortex-like structures that persist for many rotation periods. The 3D field consists of small scales in three dimensions, but with clear anisotropy, showing longer vertical correlation than in the horizontal directions.
• Figure 2: Horizontal energy spectra: $E_\perp$ as a function of the horizontally projected wave number $k_r$ of the full velocity field $\boldsymbol{v}$ (yellow dashed line), the vertically averaged flow $\boldsymbol{v}_{2D}$ (blue squares), and the residual field $\boldsymbol{v}_{3D}$ (red triangles). Energy is injected at $k_{r}=k_\textrm{inj}\approx 1.8 \, \textrm{rad}/\textrm{cm}$. For $k_r<k_\textrm{inj}$, the energy density is dominated by $\boldsymbol{v}_{2D}$ which follows a $k_{r}^{-5/3}$ power law. For $k_r>k_\textrm{inj}$$\boldsymbol{v}_{3D}$ dominates and the spectra follow a $k_{r}^{-5/2}$ power law, consistent with WTT scaling (Eq. \ref{['eq:galtier spectrum k_r k_z']}).
• Figure 3: The energy density $E(\theta,\omega)$ of the residual flow field $\boldsymbol{v}_{3D}$ shown as a function of the normalized frequency and angle $\theta$ between the wave vector $\bm{k}$ and the axis of rotation $\bm{\Omega}$. Energy is concentrated along the inertial wave dispersion relation (dashed lines).
• Figure 4: Energy density $E_{\omega}(k_r)$ of the residual 3D flow, shown as a function of the horizontally projected wavenumber $k_r$ for several fixed values of frequency $\omega$. At low frequencies, the spectra follow the $k_r^{-4}$ power law predicted by WTT, while at higher frequencies, deviations from this scaling become apparent. The inset shows the root mean square error (RMSE) of the $k_r^{-4}$ fit as a function of $\omega/(2\Omega)$, for six different experiments. Each symbol corresponds to a different dataset. The fit error remains low at low frequencies and rises sharply around $\omega/(2\Omega) \approx 0.4$, indicating a breakdown of WTT predictions.
• Figure 5: (a) Temporal energy spectrum of the residual 3D velocity field $\boldsymbol{v}_{3D}$ as a function of the normalized frequency $\omega/(2\Omega)$ for several experiments with different rotation rates $\Omega$ and injection pressure $p$. For $\omega/(2\Omega)\lesssim 0.4$, the data are consistent with $E(\omega)\sim(\omega/ 2\Omega)^{-1/2}$ predicted by the WTT. (b) The same data shown in panel (a) compensated by $\sqrt{\omega\Omega}$ in the main plot, and additionally by a factor $p^{-3/2}$ in the inset. The good data collapse confirms the scaling of the energy spectrum with frequency, angular velocity, and forward energy flux.