Bidirectional energy cascade in surface capillary waves

Based on an experiment and simulations, we report that an energy cascade in surface capillary waves can be bidirectional, that is, can simultaneously flow towards large and small wavelength scales from the pumping scales. The bidirectional energy cascade provides an effective global coupling mechanism between the scales. We show that formation of the bidirectional cascade leads to creation of large-scale, large-amplitude waves on the fluid surface.

Figure 1

(a) Schematic of the experimental setup. An optical cryostat containing the cell is not shown. (b) Snapshot of the turbulent surface of superfluid helium. The driving frequency is ${\omega }_d/2\pi =113$ Hz. (c),(d) Formation of large-amplitude waves on the surface at $\omega <{\omega }_d$ by increasing the ac driving voltage from $U_d=4$ V (c) to $U_d=14$ V (d). The driving frequency (arrow) is ${\omega }_d/2\pi =68$ Hz. The wavelength at the driving frequency ${\omega }_d$ is $\simeq 780\mu \mathrm{m}$. The conventional direct Kolmogorov-Zakharov (KZ) spectrum of capillary turbulence $I\left(\omega \right)\propto {\omega }^{-3.5}$ between $2\times {10}^2\text{Hz}<\omega /2\pi <2\times {10}^3\text{Hz}$ is shown by the dashed line. Formation of low-frequency harmonics at $\omega <{\omega }_d$ with amplitudes larger than those at the driving frequency ${\omega }_d$ is clearly visible for $U_d=14$ V.

Figure 2

Buildup of capillary turbulence on the surface of superfluid helium. The spectra are calculated via short-time Fourier transform of the recorded signal. The moments, for which the spectra (a)–(c) are calculated, are labeled on the figure. The driving frequency is ${\omega }_d/2\pi =199$ Hz (arrowed) and the driving amplitude is $U_d=97$ V. Formation of the wave at a frequency equal to half the driving frequency is seen in panel (b), and formation of multiple low-frequency harmonics is seen in (c).

Figure 3

Three- and four-wave interaction rates, ${\gamma }_k^{\left(3\right)}$ (solid curve) and ${\gamma }_k^{\left(4\right)}$ (short-dashed curve) calculated from Eqs. (10), (12), and (18) as functions of the resonance number $n$. It is seen that one has ${\gamma }_k^{\left(3\right)}\gg {\gamma }_k^{\left(4\right)}$ for all resonance numbers $n$; thus, the three-wave processes dominate. For comparison, the distance between two neighboring resonances $\Delta {\omega }_{k_n}\equiv \omega \left(k_{n+1}\right)-\omega \left(k_n\right)$ is also shown by a long-dashed curve. The discreteness of the resonance frequency spectrum is inessential since ${\gamma }_k^{\left(3\right)}>\Delta {\omega }_{k_n}$.

Figure 4

(a) Numerical steady-state spectrum $I\left(\omega \right)$ of sustained surface oscillations in the presence of low- and high-frequency damping (peaks) and in the presence of only high-frequency damping (open squares). The spectra are shown in units of ${\lambda }_c^2{\omega }_c^{-1}$. The spectra are averaged over an interval of ${10}^6{t}_c$, where $t_c={\omega }_c^{-1}$ is a numerical unit of time. The surface is driven at a frequency ${\omega }_d$ of the 50th resonance (arrowed). The power-law spectral behavior $I\left(\omega \right)\propto {\omega }^{-3.5}$ (KZ) and ${\omega }^{-1}$ (the thermal equilibrium) are shown by dashed lines. The radius of the cylindrical cell is $R=15{\lambda }_c$. (b) Absolute value of the energy flux, $\left|\Pi \right|$, in units of $\alpha {\omega }_c$, incoming to the spectral domain $\omega <{\omega }_d$ as a function of the low-frequency damping coefficient ${\gamma }_{\mathrm{LF}}$. The cell radii are $R=15{\lambda }_c$ (circles) and $30{\lambda }_c$ (triangles). Vertical bars show the fluctuations of the flux as a standard deviation about the mean. The dashed lines are shown to guide the eye.

Figure 5

Probability distribution functions of $\text{Re}\left(a\right)$ for the 20th resonant mode at frequency $\omega =9.38{\omega }_c$ (a) and for the 10th mode at frequency $\omega =3.62{\omega }_c$ (b) in the absence of low-frequency damping, ${\gamma }_{\mathrm{LF}}=0$ (open columns), and at ${\gamma }_{\mathrm{LF}}=0.25{\gamma }_{\mathrm{HF}}$ (shaded columns). PDFs are calculated for the spectra shown in Fig. 4. Dashed (green) lines show the Gaussian fit to the PDFs. The line segments connecting the points are shown to guide the eye.