Decaying two-dimensional turbulence undergoes statistical heating

An emergent property of decaying two-dimensional turbulence is shown to be a weak but persistent statistical heating, or tendency towards clustering of like-signed vortices. The rate of this heating, which is driven by changes to the vortex population due to vortex mergers and straining events, provides a strong constraint on both vortex decay laws and kinetic theories of vortex interactions. A quasi-equilibrium statistical theory determines the energy spectrum from just the vortex interaction energy and bulk information about the number density of the vortices. The emergent heating rate is shown to determine the upscale transfer of energy, and is a powerful constraint on power-law descriptions of the evolving vortex population.

Figure 1

Left: Time evolution of the vortex interaction energy $\varepsilon \left(t\right)$ in simulations (solid lines: ensemble average; light curves: individual runs) with narrowband initial conditions centered on $k_i=16$ (red), $k_i=32$ (black), and $k_i=64$ (blue). Insets: Snapshots of vorticity. Upper right: Evolution of fit to max. Vortex size $a_m\left(t\right)$ in the three ensembles (dashed line, $1/3$ power law). Lower right: Vortex number density $c\left(t\right)$ (dashed line, $-2/3$ power law).

Figure 2

Ensemble-average normalized vortex interaction energy $\varepsilon \left(t\right)$ in the $k_i=32$ 2DCT simulations, using different numerical schemes: CLAM (black curve), pseudospectral with hyperdiffusion (red curve), pseudospectral Navier-Stokes with time-varying viscosity (blue curve), and standard pseudospectral Navier-Stokes (green curve). See text for details.

Figure 3

Ensemble-average energy spectra from the DNS (points) versus predictions from the equilibrium statistical theory (lines) at $t=50t_{\mathrm{eddy}}$ (left) and $t=250t_{\mathrm{eddy}}$ (right). The dotted lines show a $k^{-5}$ high-wave-number spectrum as predicted by DSMGT. For initial data wave number $k_i=32$, results from both the CLAM (circles) and hyperdiffusion (crosses) DNS are given.

Figure 4

Time evolution of $\varepsilon$ in the DNS (solid black curve ensemble mean, grey area $\pm 1$ standard deviation), versus predictions of Eq. (9) for vortex distributions of the form (8) evolving according to temporal power laws [solid blue curve, DSMGT law; dashed blue curve, fit with $(\mu ,\nu )=(0.30,-0.71)$; dotted blue curves $(\mu ,\nu )=(0.30\pm 0.03,-0.71)$]. Red curves show results from the fluid energy conserving punctuated GVD model with $q=3$ and $s=0.2$ (solid red) $s=0.6$ (dashed red) and $s=1$ (dotted red).

Figure 5

Left: Density of states function $W\left(\varepsilon \right)$ in the GVD system, calculated for vortices with areas distributed according to Eq. (8), with the free parameters $(a_m,c,{\omega }_0)$ chosen as a best fit to the DNS results at $t=50$ for the $k_i=32$ ensemble (solid black curve), the $k_i=64$ ensemble (dashed blue curve) and the $k_i=16$ ensemble (dotted red curve). Right: The corresponding (scaled) inverse temperature $\beta \left(\varepsilon \right)=W^{\prime }\left(\varepsilon \right)/W\left(\varepsilon \right)$.