Mesoscale helicity distinguishes Vinen from Kolmogorov turbulence in helium-II

Experiments and numerical simulations show that quantum turbulence exists in two distinct limiting regimes: Kolmogorov turbulence (which shares with classical turbulence the important property of a cascade of kinetic energy from large eddies to small eddies) and Vinen turbulence (which is more similar to a random flow). In this work, we define a mesoscale helicity for the superfluid, which, tested in numerical experiments, distinguishes the two turbulent regimes, quantifying the amount of nonlocal vortex interactions and the orientation of the vortex lines.

Figure 1

The four simple vortex configurations (described in the main text) used to illustrate the physical meaning of the mesoscale helicity $\mathcal{H}$ defined in Eq. (6). The computed values of $\mathcal{H}$ are as follows: (a) $\mathcal{H}/\kappa =1.72\times {10}^{-6}{\text{cm}}^2/\text{s}$, (b) $\mathcal{H}/\kappa =1.96\times {10}^{-6}{\text{cm}}^2/\text{s}$, (c) $\mathcal{H}/\kappa =1.48\times {10}^{-6}{\text{cm}}^2/\text{s}$, (d) $\mathcal{H}/\kappa =-1.10\times {10}^{-5}{\text{cm}}^2/\text{s}$. Vortices are colored according to the local value of the helicity density $h\left(\xi \right)$. Arrows indicate the orientation of the vortices. The unit of length shown on the axes is ${10}^{-1}\text{cm}$.

Figure 2

Thermally driven turbulence. Snapshot of turbulent vortex tangles driven by uniform normal fluid (Vinen regime) at $t=2400\mathrm{s}$. The vortex lines are color-coded according to the local mesoscale helicity density $h\left(\xi \right)={\mathbf{s}}^{\prime }\left(\xi \right)·{\mathbf{v}}^{\text{non}}(\mathbf{s}\left(\xi \right))$.

Figure 3

Thermally driven turbulence. Temporal evolution of vortex line density $\mathcal{L}\left(t\right)$ (${\text{cm}}^{-2}$, blue curve) and mesoscale helicity $\mathcal{H}$ divided by $\kappa L$ (cm/s, red curve) for thermal counterflow. Time $t$ is indicated in seconds.

Figure 4

Injection-driven turbulence. Snapshot of turbulent vortex tangles driven by constant vortex ring injection at zero temperature at $t=175\mathrm{s}$. The vortex lines are color-coded according to the local mesoscale helicity density $h\left(\xi \right)={\mathbf{s}}^{\prime }\left(\xi \right)·{\mathbf{v}}^{\text{non}}(\mathbf{s}\left(\xi \right))$.

Figure 5

Injection-driven turbulence. Temporal evolution of vortex line density $\mathcal{L}\left(t\right)$ (${\text{cm}}^{-2}$, blue curve) and mesoscale helicity $\mathcal{H}$ divided by $\kappa L$ (cm/s, red curve) for superfluid turbulence driven by constant vortex ring injection at $T=0$. Time $t$ is indicated in seconds.

Figure 6

Thermally driven turbulence. Counterflow superfluid turbulence at $T=1.9\mathrm{K}$: energy spectrum $\widehat{E}$ vs wave number $k$ at $t=2400\mathrm{s}$; the dashed green line indicates the typical $k^{-1}$ isolated vortex scaling, while blue vertical lines indicate wave numbers corresponding to the size of the box $D$ and the average intervortex spacing $\ell$.

Figure 7

Injection-driven turbulence. Superfluid turbulence driven by injection of vortex rings at $T=0$: energy spectrum $\widehat{E}$ vs wave number $k$ at $t=175\mathrm{s}$; the dashed green line shows the slope of the classical Kolmogorov $k^{-5/3}$ scaling, while blue vertical lines indicate wave numbers corresponding to the diameter $2R$ of the injected vortex rings and the average intervortex spacing $\ell$.

Figure 8

Schematic vortex tube with cross sections ${\mathcal{D}}_{\xi }$ of area $\pi a_0^2$. The solid (black) line $\mathbf{s}(\xi ,t)$ is the centerline. The unit tangent vectors ${\mathbf{s}}^{\prime }$ at $\mathbf{s}$ are the (red) arrows. The (blue) vector $\mathit{\sigma }$ is the position vector on each cross section. Within the vortex tube, the vorticity $\mathit{\omega }$ is constant.

Figure 9

(a)–(c) Snapshots of two initially orthogonal vortices undergoing a reconnection at $T=0$. Vortex lines are color-coded according to the local value of the magnitude of the mesoscale helicity density $|{\mathbf{s}}^{\prime }\left(\xi \right)·{\mathbf{v}}_{\text{far}}(\mathbf{s}\left(\xi \right))|$. (d) Evolution of the vortex line density $\mathcal{L}$ in ${\text{cm}}^{-2}$ (blue) and volume-integrated mesoscale helicity $\mathcal{H}/\kappa$ in ${\text{cm}}^2/\text{s}$ (red).

Figure 10

Mechanically driven turbulence. Snapshot of turbulent vortex tangle driven by ABC normal flow at $t=2700\mathrm{s}$. The vortex lines are color-coded according to the local mesoscale helicity density $h\left(\xi \right)={\mathbf{s}}^{\prime }\left(\xi \right)·{\mathbf{v}}^{\text{non}}(\mathbf{s}\left(\xi \right))$.

Figure 11

Mechanically driven turbulence. Temporal evolution of vortex line density $\mathcal{L}\left(t\right)$ (${\text{cm}}^{-2}$, blue curve) and mesoscale helicity $\mathcal{H}$ divided by $\kappa L$ (cm/s, red curve) for superfluid turbulence driven by ABC normal flow. Time $t$ is indicated in seconds.

Figure 12

Mechanically driven turbulence. Superfluid turbulence driven by ABC normal flow: energy spectrum $\widehat{E}$ vs wave number $k$ at $t=2700\mathrm{s}$; the dashed green line shows the slope of the classical Kolmogorov $k^{-5/3}$ scaling, while blue vertical lines indicate wave numbers corresponding to the size of the box $D$ and the average intervortex spacing $\ell$.