Spectral condensation in laboratory two-dimensional turbulence

Turbulence in two-dimensional flows is expected to produce a dynamical state in which energy condenses into the largest scale allowed by the system when the scale at which energy is dissipated exceeds the domain size. We study this phenomenon in a laboratory quasi-two-dimensional turbulent flow in a thin layer of electromagnetically driven fluid where the energy is primarily dissipated by bottom friction. By inserting boundaries of different sizes, we fix the driving and damping and vary only the domain size. Although we observe flow patterns that are consistent with previous claims of spectral condensation, we see no signatures in the energy spectrum. An analysis of the scale-to-scale energy flux reveals that small domains weaken the turbulent cascade, even though the bulk forcing and frictional dissipation remain the same. Our results suggest that we lack a robust set of criteria for the existence of spectral condensation, and that claims of condensation in experimental flows with small scale separations must be supported by strong evidence.

Figure 1

Representative instantaneous velocity fields observed for four domain sizes: (a) $20\times 20$ ${\mathrm{cm}}^2$, (b) $15\times 15$ ${\mathrm{cm}}^2$, (c) $10\times 10$ ${\mathrm{cm}}^2$, and (d) $5\times 5$ ${\mathrm{cm}}^2$. The scale is the same for all four panels, and the scale bar shows the magnet length scale $L_m=2.54$ cm.

Figure 2

Energy spectra $E\left(k\right)$ as a function of wave number $k$ for flows in our four restricted domains as well as in the full apparatus with no inserted boundary. The vertical dotted-dashed blue line indicates the injection wave number, and the vertical dashed red line indicates the frictional energy dissipation wave number.

Figure 3

Spatiotemporally averaged spectral energy fluxes ${\mathrm{\Pi }}^{\left(r\right)}$ as a function of filter scale $r$, normalized by the magnet spacing $L_m$. (a) Spectral energy fluxes for different domain sizes, including the full apparatus with no inserted boundaries. Negative values indicate transfer to larger length scales. (b) Spectral energy flux for the $5\times 5$ ${\mathrm{cm}}^2$ domain with the same fluid depth but different Reynolds numbers. The flow fields for each Reynolds number display vortex monopoles, but the spectral energy flux is markedly different.