Inverse cascade behavior in freely decaying two-dimensional fluid turbulence

P. D. Mininni and A. Pouquet

Phys. Rev. E 87, 033002 – Published 6 March 2013

Abstract

We present results from an ensemble of 50 runs of two-dimensional hydrodynamic turbulence with spatial resolution of $2048^{2}$ grid points and from an ensemble of 10 runs with $4096^{2}$ grid points. All runs in each ensemble have random initial conditions with the same initial integral scale, energy, enstrophy, and Reynolds number. When both ensemble and time averaged, an inverse energy cascade behavior is observed, even in the absence of external mechanical forcing: The energy spectrum at scales larger than the characteristic scale of the flow follows a $k^{- 5 / 3}$ law, with negative flux, together with a $k^{- 3}$ law at smaller scales, and a positive flux of enstrophy. The source of energy for this behavior comes from the modal energy around the energy-containing scale at $t = 0$ . The results shed some light on the connections between decaying and forced turbulence and recent controversies in experimental studies of two-dimensional and magnetohydrodynamic turbulent flows.

Received 18 May 2011

DOI:https://doi.org/10.1103/PhysRevE.87.033002

Authors & Affiliations

P. D. Mininni^1,2 and A. Pouquet²

¹Departamento de Física, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires and IFIBA, CONICET, Ciudad Universitaria, 1428 Buenos Aires, Argentina
²NCAR, P.O. Box 3000, Boulder, Colorado 80307-3000, USA

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Vol. 87, Iss. 3 — March 2013

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Images

Figure 1
Time evolution of the (a) energy spectrum, (b) energy flux, and (c) enstrophy flux in a single $2048^{3}$ simulation from $t = 0.5$ (black line) to $t = 6$ (light gray or light blue line). Slopes in the energy spectrum are indicated as references. The curves corresponding to $t = 0.5$ and 6 are indicated in all panels by arrows and the vertical dashed lines indicate the initial energy containing wave number $k_{0}$ . In (b) and (c), note the displacement to smaller wave numbers of the minimum of energy flux and to larger wave numbers of the maximum of enstrophy flux. The inset in (a) shows the time evolution of the energy (solid line) and of the enstrophy (dashed line) in this run, with the color changing with time following the colors used for the different curves in the spectrum and fluxes.Reuse & Permissions
Figure 2
Time evolution of $E^{<} / E$ (solid line) and $E^{>} / E$ (dashed line) (ratio of the energy at large and small scales, respectively, to the total energy) as a function of time for all runs in the set of simulations with (a) $2048^{2}$ grid points and (b) $4096^{2}$ grid points.Reuse & Permissions
Figure 3
(a) Time evolution of the integral scale for all the $2048^{2}$ runs; note the dispersion around the mean evolution. The inset shows the time evolution of the energy for all runs, which are very close. (b) Time evolution of the standard deviation $σ$ in the integral scale of all the $2048^{2}$ runs. The inset shows the standard deviation in the energy in all the $2048^{2}$ runs as a function of time.Reuse & Permissions
Figure 4
Energy spectra for ten $2048^{2}$ runs at $t = 1.5$ (no averaging performed here). Slopes are indicated as references. The vertical dashed line corresponds to the initial energy-containing wave number $k_{0}$ . The inset shows the same ten spectra at $t = 6$ . Note that the large-scale spectra look like $\sim k$ , which could be interpreted as eddy noise in two dimensions (see [35]). However, unlike three-dimensional turbulence, the amplitude of the peak of the energy spectrum is larger than its initial value (at $t = 0$ ) at the same wave number.Reuse & Permissions
Figure 5
(a) Time- and ensemble-averaged energy spectrum, (b) energy flux, and (c) enstrophy flux over the fifty $2048^{2}$ simulations and from $t = 0.5$ to 6. Slopes in (a) are indicated as references. The vertical dashed lines correspond to the initial energy-containing wave number $k_{0}$ .Reuse & Permissions
Figure 6
(a) Time- and ensemble-averaged energy spectrum and (b) energy flux over the ten $4096^{2}$ simulations and from $t = 0.5$ to 6. Slopes in (a) are indicated as references. The vertical dashed lines correspond to the initial energy-containing wave number $k_{0}$ .Reuse & Permissions

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