Kolmogorov Spectrum of Superfluid Turbulence: Numerical Analysis of the Gross-Pitaevskii Equation with a Small-Scale Dissipation

The energy spectrum of superfluid turbulence is studied numerically by solving the Gross-Pitaevskii equation. We introduce the dissipation term which works only in the scale smaller than the healing length to remove short wavelength excitations which may hinder the cascade process of quantized vortices in the inertial range. The obtained energy spectrum is consistent with the Kolmogorov law.

Figure 1

How to make the random phase ${\varphi }_0(\mathit{x})$ with $\lambda =4$ (a) and one example of an initial random phase in a cross section ${\varphi }_0(x,y,0)$ (b).

Figure 2

Time development of $E$, $E_{\mathrm{k}\mathrm{i}\mathrm{n}}$, $E_{\mathrm{k}\mathrm{i}\mathrm{n}}^c$, and $E_{\mathrm{k}\mathrm{i}\mathrm{n}}^i$. (a) Case with ${\gamma }_0=0$; (b) case with ${\gamma }_0=1$.

Figure 3

Time dependence of the exponent of the spectrum $\eta$ (a) and the dissipation rate $ϵ$. (a) $\eta$ for ${\gamma }_0=1$. (b) $ϵ$ for ${\gamma }_0=1$. (c) $\eta$ for ${\gamma }_0=0$. (d) $ϵ$ for ${\gamma }_0=0$. In (a) and (c), the error bars are the standard deviations of the data from the fit. The line $\eta =5/3$ is shown to compare the results with the Kolmogorov law.

Figure 4

Vorticity $\mathrm{r}\mathrm{o}\mathrm{t}\mathit{v}(\mathit{x},t)$ and the energy spectrum at $t=6.0$. The contour in (a) is 98% of maximum vorticity. The energy spectrum was obtained by making an ensemble average for 20 different initial states The solid line refers to the Kolmogorov law in Eq. (1).