Direct measurement of energy transfer in strongly driven rotating turbulence

TL;DR

This work addresses how energy moves in strongly driven rotating turbulence by perturbing a steady flow with a localized energy injection pulse and tracking the resulting evolution in real space, frequency, and horizontal wavenumber domains using spatio-temporal velocity measurements and spectral analyses. It reveals three energy-transfer channels: rapid inertial-wave–driven homogenization in real space, a direct nonlocal transfer of energy from high-frequency 3D modes to low-frequency quasi-geostrophic modes in frequency space, and an inverse cascade within the quasi-geostrophic manifold carried out in tandem with inertial-wave propagation in z; the cascade is compatible with a Kraichnan-like 2D spectrum $E(k) \propto k^{-5/3}$ but remains inseparable from wave dynamics, as inertial waves mediate the process. The observations challenge weak-wave turbulence predictions for strong driving and emphasize that inertial waves play a central role in energy transfer even near the 2D geostrophic manifold, with implications for geophysical and astrophysical rotating flows. Mathematically, the dynamics involve the inertial-wave dispersion $\omega = \pm 2\Omega \cos(\theta)$ and vertical group velocity $C_{g,z}(k) = \frac{2\Omega \sin^2\theta}{k}$, shaping the three transfer channels and underscoring the coupling between 3D wave motion and 2D geostrophic energy.

Abstract

A short, abrupt increase in energy injection rate into steady strongly-driven rotating turbulent flow is used as a probe for energy transfer in the system. The injected excessive energy is localized in time and space and its spectra differ from those of the steady turbulent flow. This allows measuring energy transfer rates, in three different domains: In real space, the injected energy propagates within the turbulent field, as a wave packet of inertial waves. In the frequency domain, energy is transferred non-locally to the low, quasi-geostrophic modes. In wavenumber space, energy locally cascades toward small wavenumbers, in a rate that is consistent with two-dimensionsal (2D) turbulence models. Surprisingly however, the inverse cascade of energy is mediated by inertial waves that propagate within the flow with small, but non-vanishing frequency. Our observations differ from measurements and theoretical predictions of weakly driven turbulence. Yet, they show that in strongly-driven rotating turbulence, inertial waves play an important role in energy transfer, even at the vicinity of the 2D manifold.

Figures (4)

• Figure 1: Horizontally averaged energy density $E(z,t)$ for an ensemble average of 6 experiments with $\Omega=4\pi\,\text{rad}/\text{s}$. Pulse injection occurs at $t=10.5\text{s}$.
• Figure 2: Time evolution of frequency-filtered energy density. (a-c) Horizontally averaged energy density $E_f$ over time and vertical position $z$, for $\omega=1.5\Omega(a)$, $0.6\Omega(b)$, and $0.1\Omega(c)$. Gray lines represent to the vertical group velocity of an inertial wave packet with the respective central frequency. The data was collected for experiments with $\Omega=3.5\pi$rad/sec $Re\approx2100$ and $Ro\approx0.006$. (d) Spectrogram of fully spatially averaged energy density $\bar{E}_f$ over time and frequency $\omega$. The energy of each frequency is normalized to equal 1 at the peak of the pulse. Gray dashed curve marks the theoretical arrival time of an inertial wave packet with frequency $\omega$ to the lowest measured plane.
• Figure 3: (a) Vertically averaged excess energy density spectrum $\Delta\bar{E}$, as a function of time and horizontal wave-number $k_{2d}$. Inset: $\Delta\bar{E}$ as a function of $k_{2d}$ for $t= 13s$ (dashed line), $23s$ (dot-dashed line) and $33s$ (Solid line) (marked on the main figure). The maximum of the instantaneous $\Delta\bar{E}$, $k^\star_{2d}(t)$, is indicated for $t=13s$. The gradual shift of the excess energy to lower wave-numbers is a manifestation of the inverse energy cascade. (b) $|k^\star_{2d}|^{-2/3}$ as a function of time. The agreement with linear growth beyond $t=17s$ indicates $k^\star_{2d}\sim t^{-2/3}$. The data set is the same as in panel (a).
• Figure 4: (a-c) Excess energy density at fixed wave-number, $\Delta E(z,k_{2d},t)$, as a function of vertical position $z$ and time (showcasing the same data as Figure 3, with vertical resolution). The fixed wave-numbers in each subplot are: $k_{2d}=2.28\text{(a),}\ 0.6\text{(b), and }0.34\text{(c), } \text{rad}/\text{cm}$. Slopes of the dashed lines indicate $C_{g,z}$ for the relevant wavenumber and $\theta=\pi/2$ for waves that propagate upwards (white) and downwards (black). As time increases, energy cascades from large to small wave-numbers while propagating vertically up and down. (d) Estimation of the energy propagation speed for each wave-number $k_{2d}$ (symbols), compared with $C_{g,z}(k_{2d})$ from Eq. (\ref{['eq:group velocity']}) (dashed curve).