Experimental quantification of nonlinear time scales in inertial wave rotating turbulence

We study nonlinearities of inertial waves in rotating turbulence. At small Rossby numbers the kinetic energy in the system is contained in helical inertial waves with time dependence amplitudes. In this regime the amplitude variations time scales are slow compared to wave periods, and the spectrum is concentrated along the dispersion relation of the waves. A nonlinear time scale was extracted from the width of the spectrum, which reflects the intensity of nonlinear wave interactions. This nonlinear time scale is found to be proportional to ${\left(\mathit{U}·\mathit{k}\right)}^{-1}$, where $\mathit{k}$ is the wave vector and $\mathit{U}$ is the root-mean-square horizontal velocity, which is dominated by large scales. This correlation, which indicates the existence of turbulence in which inertial waves undergo weak nonlinear interactions, persists only for small Rossby numbers.

Figure 1

(a) Horizontal velocity field energy spectrum $E\left(\theta ,\omega \right)$ for $\mathrm{\Omega }/2\pi =1.5\mathrm{Hz}$ and $\mathrm{Ro}=8\times {10}^{-3}$. The energy in Fourier space is averaged on all $\varphi$ and $k$ (between $k_{\min }=1.33$ and $k_{\max }=3.29\mathrm{rad}/\mathrm{cm}$), and plotted (logarithmic scale) as a function of $\omega /2\mathrm{\Omega }$ and $\theta$. (b) and (c) are the positive and negative helical modes [$E^s(\theta ,\omega )$]. The gray area around $\theta \sim {90}^{\circ }$ is not defined since $u_z\left(\theta \sim {90}^{\circ }\right)$ cannot be calculated.

Figure 2

Fourier energy spectra, $E_k\left(\theta ,\omega \right)$ (logarithmic scale), for $k=1.76\mathrm{rad}/\mathrm{cm}$ and different $\mathrm{Ro}$ ($\mathrm{Ro}=0.004,0.008,0.02,0.06)$. For small $\mathrm{Ro}$, energy is concentrated around the inertial wave dispersion relation. As $\mathrm{Ro}$ increases, the energy spectrum spreads, until for very large $\mathrm{Ro}$ the dispersion relation is no longer a significant feature in the spectrum. See Supplemental Material Fig. S1 for more figures in a larger range of $\mathrm{Ro}$ and $k$ [37].

Figure 3

Spatial Fourier of the velocity field for a single wave vector. (a) Real part of the Fourier transform of the velocity field in the $x$ direction: $\mathrm{Re}\left[u_x\left({\mathit{k}}^{\prime },t\right)\right]$, for $k^{\prime }=1.76\mathrm{rad}/\mathrm{cm},{\theta }^{\prime }={116}^{\circ },{\varphi }^{\prime }={176}^{\circ }$. Rotation frequency is $\mathrm{\Omega }/2\pi =2\mathrm{Hz}$. There is a main frequency governing the signal with amplitude modulations. The inset is a close-up, showing the governing oscillations with the expected period of $0.56\mathrm{s}$ [corresponding to $|{\omega }^{\prime }|=2\mathrm{\Omega }cos\left({\theta }^{\prime }\right)$]. (b) Temporal Fourier spectrum of (a), showing $|u_x\left({\mathit{k}}^{\prime },\omega \right){|}^2$. There are two peaks specifying the positive and negative helical modes. Two vertical lines represent the expected frequency ${\omega }^{\prime }$. (c) Energy spectrum density integrated over $\varphi$ for both $u_x$ and $u_y$ fields. Showing $E_{k^{\prime }}\left({\theta }^{\prime },\omega \right)$ in blue for the positive mode only. The red line is the Gaussian fit with width $\mathrm{\Delta }\omega$ (green line) defined as the standard deviation.

Figure 4

Normalized spectra width as function of normalized interaction time scale for each $k$ and $\theta$ [38], where ${\tau }_U^{-1}=Uksin\theta$. There is a strong correlation between the measured spectra width and ${\tau }_U^{-1}$, indicating nonlinear interactions with mean flow are the main cause of spectrum broadening. The black dashed line represents $y=3x$. The results show a good collapse of the data only for $\mathrm{\Delta }\omega /\omega <1$, where a perfect linear line is observed. For larger values the data is more spread. Thus we see two regimes, where weak nonlinearities with the horizontal modes exist only for ${\left(\omega {\tau }_U\right)}^{-1}<0.3$.