Energy Transfer by Inertial Waves during the Buildup of Turbulence in a Rotating System

We study the transition from fluid at rest to turbulence in a rotating tank. The energy is transported by inertial wave packets through the fluid volume. These high amplitude waves propagate at velocities consistent with those calculated from linearized theory [H. P. Greenspan, The Theory of Rotating Fluids (Cambridge University Press, Cambridge, England, 1968)]. A “front” in the temporal evolution of the energy power spectrum indicates a time scale for energy transport at the linear wave speed. Nonlinear energy transfer between modes is governed by a different, longer, time scale. The observed mechanisms can lead to significant differences between rotating and two-dimensional turbulent flows.

Figure 1

The experimental system. A covered, cylindrical transparent water tank is placed on a rotating table (${\Omega }_{\max }=16\mathrm{rad}/\mathrm{s}$). Water is injected and drawn in through the injection device (see text) at the bottom of the tank. Laser sheets illuminate a cross section at a variable height $h$, and a corotating camera grabs images of the light scattered from $50\mu \mathrm{m}$ diameter polyamide particles suspended within the water.

Figure 2

Five snapshots of the kinetic energy density field $E=v^2+u^2$ measured at different times ($t=1.3$, 3.3, 4.7, 10, and 100 s) for (a)–(e) after the initiation ($t=0$) of energy injection. Energy appears over the entire cross section at $t=3.3\mathrm{s}$ (b). The amount of energy increases, while the typical length scale slightly decreases (c). Some rearrangement of eddies is visible at $t=10\mathrm{s}$ (d), and it is only after a time of order tens of seconds that a steady turbulent field is achieved (e). The measurements were obtained for $h=56\mathrm{cm}$, $\Omega =9.4\mathrm{rad}/\mathrm{s}$, and $P=6\mathrm{W}$. Each cross section is 80 cm in diameter. A linear color bar in $(\mathrm{cm}/\mathrm{s}{)}^2$ is used for (a)–(d), whereas in (e) its range is expanded by a factor of 3.

Figure 3

A typical plot of the temporal evolution of the energy power spectrum. A color map of the energy spectrum $E(k)$ (logarithmic color bar) as a function of the time $t$ from the initiation of energy injection. The spectrum does not contain energy for the first few seconds, after which a linear front is observed (marked with a dotted line), indicating a distinct time ${\tau }_1(k)$ for the “arrival” of each wave number $k$. The inset shows the time evolution of $E(k=0.17)$. The energy increases within 1 s, defining a time $t={\tau }_1(k)$ (marked with dashed line) after which it varies much more slowly. A second front can be observed at small wave numbers (dashed line), defining a second time ${\tau }_2(k)$, which decreases with $k$ and indicates nonlinear energy transfer from small to large scales. The purple triangles mark the times at which the flow fields presented in Figs. 2a, 2b, 2c, 2d were measured. Experimental conditions were as in Fig. 2.

Figure 4

The arrival time of inertial waves. (a) ${\tau }_1(k)$, measured for various heights and rotation rates yielding a set of quasilinear curves. (b) Once ${\tau }_1$ is rescaled by $h/2\Omega$, these data collapse onto a linear line of slope $0.9\pm 0.05$. The front is, thus, consistent with the traveling time of waves with group velocity $v_g=2\Omega /k$ from the injector to the measurement plane. The measurement heights (in cm) and rotation rates (in rad/s) are indicated in (b). Energy injection rate: $P=12\mathrm{W}$.

Figure 5

The effect of energy injection rate on ${\tau }_1$ and ${\tau }_2$. ${\tau }_1$ (open symbols) and ${\tau }_2$ (solid symbols) versus the wave number $k$, measured at $h=32\mathrm{cm}$, $\Omega =11\mathrm{rad}/\mathrm{s}$, and different injection rates $P$ (squares: 1.5 W; circles: 6 W; triangles: 45 W). The higher the power injected, the shorter ${\tau }_2$ is, as expected from a time that is governed by nonlinear interactions. On the other hand, the linear time ${\tau }_1(k)$shows no dependence on $P$.