Integrated analysis of energy transfers in elastic-wave turbulence

In elastic-wave turbulence, strong turbulence appears in small wave numbers while weak turbulence does in large wave numbers. Energy transfers in the coexistence of these turbulent states are numerically investigated in both the Fourier space and the real space. An analytical expression of a detailed energy balance reveals from which mode to which mode energy is transferred in the triad interaction. Stretching energy excited by external force is transferred nonlocally and intermittently to large wave numbers as the kinetic energy in the strong turbulence. In the weak turbulence, the resonant interactions according to the weak turbulence theory produce cascading net energy transfer to large wave numbers. Because the system's nonlinearity shows strong temporal intermittency, the energy transfers are investigated at active and moderate phases separately. The nonlocal interactions in the Fourier space are characterized by the intermittent bundles of fibrous structures in the real space.

Figure 1

Detailed energy balance due to triad interactions: $T_{\mathit{k}{\mathit{k}}_1{\mathit{k}}_2}+T_{{\mathit{k}}_1{\mathit{k}}_2\mathit{k}}+T_{{\mathit{k}}_2\mathit{k}{\mathit{k}}_1}=0$.

Figure 2

Energy-transfer functions. (a) and (b) to $\left|\mathit{k}\right|=8\pi$, (c) and (d) to $\left|\mathit{k}\right|=26\pi$, (e) and (f) to $\left|\mathit{k}\right|=144\pi$, and (g) and (h) to $\left|\mathit{k}\right|=384\pi$. (a), (c), (e), and (g) to kinetic energy ${\mathcal{T}}_{\mathrm{K}\mathit{k}}^{\left(4\right)}(k_1,k_2)$, where $k_1$ and $k_2$ are, respectively, the wave numbers for $\zeta$ and $\chi$ as defined in Eq. (23a), and (b), (d), (f), and (h) to stretching energy ${\mathcal{T}}_{\mathrm{S}\mathit{k}}(k_1,k_2)$, where $k_1$ and $k_2$ are, respectively, the wave numbers for $p$ and $\zeta$ as defined in Eq. (23b).

Figure 3

Binary energy-transfer functions ${\mathcal{T}}_{\mathrm{K}\mathit{k}}^{\left(4\right)}\left(k_2\right)$ and ${\mathcal{T}}_{\mathrm{S}\mathit{k}}\left(k_1\right)$ from wave numbers $k_1$ or $k_2$ (a) to $\left|\mathit{k}\right|=8\pi$, (b) to $\left|\mathit{k}\right|=26\pi$, (c) to $\left|\mathit{k}\right|=144\pi$, and (d) to $\left|\mathit{k}\right|=384\pi$. The vertical lines show the representative wave numbers $\left|\mathit{k}\right|$. The abscissa is scaled logarithmically.

Figure 4

Schematic diagram of net energy transfers due to the local interactions (solid arrows) and the nonlocal interactions (dotted arrows). (a) $T_{\mathrm{K}\mathit{k}{\mathit{k}}_1{\mathit{k}}_2}^{\left(4\right)}$ and (b) $T_{\mathrm{S}\mathit{k}{\mathit{k}}_1{\mathit{k}}_2}$.

Figure 5

Times series of nonlinearity $\eta$ (left axis), and linear energy $E^{\left(2\right)}$ and nonlinear energy $V_{\mathrm{S}}$. The linear energy is divided by 10 for visibility. The mean of $\eta$ and $\pm 1$-sigma levels are represented by dotted lines. The two arrows at the bottom indicate the representative times of the moderate phase and the active phase for later analysis.

Figure 6

(a) Total-energy spectra $\mathcal{E}$ and stretching-energy spectra ${\mathcal{V}}_{\mathrm{S}}$, and (b) energy transfer for the overall average and those for the active and moderate phases.

Figure 7

Binary energy transfers at active and moderate phases. See the figure caption of Fig. 3.

Figure 8

Real-space structures in stretching-energy fields at moderate phase (a) and at active phase (b) pointed out by arrows in Fig. 5.

Figure 9

Energy transfers in real space. (a) and (b) at moderate phase, and (c) and (d) at active phase. (a) and (c) to kinetic energy $T_{\mathrm{K}}^{\left(4\right)}\left(\mathit{x}\right)$, and (b) and (d) to stretching energy $T_{\mathrm{S}}\left(\mathit{x}\right)$.

Figure 10

Coarse-grained energy transfers in real space at active phase. (a) and (b) $r_{\mathrm{CG}}=1/4$, i.e., $k_{\mathrm{CG}}=8\pi$. (c) and (d) $r_{\mathrm{CG}}=1/16$, i.e., $k_{\mathrm{CG}}=32\pi$. (e) and (f) $r_{\mathrm{CG}}=1/64$, i.e., $k_{\mathrm{CG}}=128\pi$. (a), (c), and (e) to kinetic energy $T_{\mathcal{K}}^{\left(4\right)}\left(\mathit{x}\right)$, and (b), (d), and (f) to stretching energy $T_{\mathcal{S}}\left(\mathit{x}\right)$.

Figure 11

Scale dependence of the correlations between energy transfers.

Figure 12

Integrated view of energy transfers.