Condensate in quasi-two-dimensional turbulence

We investigate the process of formation of large-scale structures in a turbulent flow confined in a thin layer. By means of direct numerical simulations of the Navier-Stokes equations, forced at an intermediate scale $L_f$, we obtain a split of the energy cascade in which one fraction of the input goes to small scales generating the three-dimensional direct cascade. The remaining energy flows to large scales producing the inverse cascade which eventually causes the formation of a quasi-two-dimensional condensed state at the largest horizontal scale. Our results show that the connection between the two actors of the split energy cascade in thin layers is tighter than what was established before: the small-scale three-dimensional turbulence acts as an effective viscosity and dissipates the large-scale energy thus providing a viscosity-independent mechanism for arresting the growth of the condensate. This scenario is supported by quantitative predictions of the saturation energy in the condensate.

Figure 1

Snapshots of the square vertical vorticity for the simulation at $S=1/4$ at times $t=24t_f$ (left panel) and $t=1200t_f$ (right panel). The same logarithmic color scale is used for the two cases, with blue (yellow) representing small (large) values. For clarity, the vertical scale has been stretched by a factor 2.

Figure 2

Temporal evolution of the kinetic energy of the 2D mode $E^{2\text{D}}$ (left panel) and $E^{3\text{D}}$ (right panel) for $S=1/8$ (red, solid line) and $S=1/4$ (blue, dashed line).

Figure 3

Left: Dissipation spectra $D\left(k_h\right)$ as a function of $k_h={(k_x^2+k_y^2)}^{1/2}$ for $S=1/8$ at time $t=4050t_f$ (red, solid line), and for $S=1/4$ at time $t=2650t_f$ (blue, dashed line). Right: Temporal evolution of the energy dissipation rate ${\varepsilon }_{\text{dis}}$ for $S=1/8$ (red, solid line) and $S=1/4$ (blue, dashed line).

Figure 4

Left: Temporal evolution of the kinetic energy rescaled with $E_c$ and $t_c$ estimated by (3), for $S=1/8$ (red, solid line) and $S=1/4$ (blue, dashed line). Right: Radial profile of the mean vorticy of the condensate $\mathrm{\Omega }\left(r\right)$ for $S=1/8$ (red, solid line) and $S=1/4$ (blue, dashed line). The scaling behavior $\mathrm{\Omega }\left(r\right)\sim r^{-1}$ is represented by the black dotted line.

Figure 5

Left: 2D energy spectra $E\left(k_h\right)$ as a function of $k_h={(k_x^2+k_y^2)}^{1/2}$ for $S=1/4$ at times $t/t_f=$ 2.4 (a), 4.8 (b), 7.1 (c), 12 (d), 24 (e), 71 (f), and 2650 (g). The black dashed line represents the scaling $k^{-5/3}$. Right: 2D energy spectrum $E\left(k_h\right)$ as a function of $k_h={(k_x^2+k_y^2)}^{1/2}$ of the 2D mode ${\mathit{u}}^{2\text{D}}$ (red, dashed line), 3D mode ${\mathit{u}}^{3\text{D}}$ (blue, solid line), and total velocity field $\mathit{u}$ (black, dotted line) as a function of the horizontal wave number $k_h$.

Figure 6

Left: Spectral energy flux $\mathrm{\Pi }\left(k\right)$ for $S=1/8$ at time $t=25t_f$ (black, dotted line), $t=4050t_f$ (blue, solid line), and 2D energy flux ${\mathrm{\Pi }}^{2\text{D}}\left(k\right)$ at $t=4050t_f$ (red, dashed line). Right: Spectral energy flux $\mathrm{\Pi }\left(k\right)$ for $S=1/4$ at time $t=25t_f$ (black, dotted line), $t=2650t_f$ (blue, solid line), and 2D energy flux ${\mathrm{\Pi }}^{2\text{D}}\left(k\right)$ at $t=2650t_f$ (red, dashed line).

Figure 7

Spectral energy transfer $T(K,Q)$ (blue, crosses), and 2D spectral energy transfer $T^{2\text{D}}(K,Q)$ (red, times) for $K=1$, averaged in time for $t>t_s$. Left panel: $S=1/8, t_s=3000t_f$. Right panel: $S=1/4, t_s=2500t_f$.