Kelvin waves from vortex reconnection in superfluid helium at low temperatures

We report on the analysis of the root mean square curvature as a function of the numerical resolution for a single reconnection of two quantized vortex rings in superfluid helium. We find a similar scaling relation as reported in the case of decaying thermal counterflow simulations by L. Kondaurova et al. There the scaling was related to the existence of a Kelvin-wave cascade which was suggested to support the L'vov-Nazarenko spectrum. Here we provide an alternative explanation that does not involve the Kelvin-wave cascade but is due to the sharp cusp generated by a reconnection event in a situation where the maximum curvature is limited by the computational resolution. We also suggest a method for identifying the Kelvin spectrum based on the decay of the rms curvature by mutual friction. Our vortex filament simulation calculations show that the spectrum of Kelvin waves after the reconnection is not simply $n\left(k\right)\propto k^{-\eta }$ with constant $\eta$. At large scales the spectrum seems to be close to the Vinen prediction with $\eta =3$ but becomes steeper at smaller scales.

Figure 1

Decay of $c_{\mathrm{rms}}$ for different initial Kelvin spectra on a straight vortex with periodic boundary conditions and period $L_z=1$ mm. The number of points is 1024 and $\alpha =0.1$. In the main panel the amplitude is $A/L_z=0.01$, see Eq. (10). The different spectra are from top to bottom defined by $\eta =3.0$ (red, Vinen), 3.4 (magenta, Kozik-Svistunov), $3.666...$ (black, L'vov-Nazarenko), 4.0 (blue), and 4.4 (cyan), respectively. The dashed lines are from the analytical model, Eq. (12). The upper inset shows the $c_{\mathrm{rms}}$ for the LN spectrum but for different amplitudes: from top to bottom $A/L_z=0.3$, 0.1, 0.03, 0.01, 0.003, and 0.001, respectively. The lower inset relates the decay exponent $\beta$ in $c_{\mathrm{rms}}\propto t^{-\beta }$ to the exponent $\eta$ in the Kelvin spectrum of Eq. (10). The upper (blue) curve is for small amplitudes with $A/L_z=0.001$, and the lower (red) curve is for rather large amplitudes with $A/L_z=0.1$. The thin dashed line is the exponent appearing in the analytical result [see Eq. (12)].

Figure 2

Time development of the “test-cusp” vortex at $T=0$ that initially hosts the Kozik-Svistunov spectrum. The upper panel illustrates the time development of the local curvature at times 0 (black), 0.5 (red), 1 (blue), 2 (green), 10 (cyan), and 100 ms (magenta). The lower left panel shows the corresponding vortex configurations, and the lower right panel contains the Kelvin spectra where $k=2\pi m/L_z$ is given by the mode number $m$.

Figure 3

Decay of the rms curvature when the initial configuration is a specially arranged sharp cusp with Kozik-Svistunov spectrum, see Fig. 2. The different curves correspond to different values of mutual friction: $\alpha =0.0001$ (blue), 0.001 (cyan), 0.01 (green), 0.1 (magenta), and 1 (red). The inset shows the time development for $c_{\mathrm{rms}}$, and in addition at $\alpha =0$ (black) both for $c_{\mathrm{rms}}$ (solid) and $c_{\mathrm{ave}}$ (dashed), plotted on a linear time scale.

Figure 4

Time development of the rms curvature for different values of the mutual friction parameter $\alpha$ (and ${\alpha }^{\prime }=0$). In the main figure time is measured from the reconnection instant and scaled by $\alpha$, and on the vertical axis the initial curvature $c_{\mathrm{rms}}\left(0\right)=1/R$ is subtracted from the data. Different colors are used for $\alpha =1$ (red), 0.1 (magenta), 0.01 (green), 0.001 (cyan), 0.0001 (blue), and 0.00001 (light green), respectively. The corresponding maximum times simulated are 1, 9, 10, 12, 20, and 30 s, respectively. The black dash-dotted curve is for $\alpha =0.01$ where the initial configuration is taken from $T=0$ simulations, 1 s after the reconnection, and the time (which covers a time window of 5 s) is measured from this instant. The curves are calculated with a resolution of $\mathrm{\Delta }{\xi }_{\min }=0.0025\mathrm{mm}$, while for $\alpha =0.01$ results with $\mathrm{\Delta }{\xi }_{\min }=0.00125\mathrm{mm}$ (dashed, green) and $0.000625\mathrm{mm}$ (dash-dotted, green) are also shown. The power-law behavior is illustrated by plotting the $c_{\mathrm{rms}}\propto {\left(\alpha t\right)}^{-\beta }$ asymptotics with $\beta =0.5$ and 0.38 (dashed) that correspond to $\eta =3$ and 3.48, respectively. In the lower inset linear scales are used without any scaling. The color coding is the same as in the main figure. The case $\alpha =0$ is additionally shown by the black curve. The upper inset is for $\alpha =0.01$ with five different resolutions indicated on the plot, together with the $\beta =0.38$ asymptotic (dashed line).

Figure 5

Time development of the rms curvature at $T=0$ for five different resolutions where the point separation is kept between $\mathrm{\Delta }{\xi }_{\min }...2\mathrm{\Delta }{\xi }_{\min }$. For the two highest resolution runs, times well before the reconnection were omitted and the calculations were continued using the configuration from the run with one step lower resolution where the configuration was still smooth. The left inset shows the detailed behavior near the reconnection event when $\mathrm{\Delta }{\xi }_{\min }=0.000625$ mm. The right inset shows the scaling relation for the root mean square curvature after the reconnection as a function of resolution plus the fit $c_{\mathrm{rms}}\propto \mathrm{\Delta }{\xi }_{\min }^{-0.56}$ (solid line) and the predictions from the L'vov-Nazarenko spectrum (lower dashed) and from the single cusp model (upper dashed).

Figure 6

Time development of the mean curvature for the same five different resolutions which were used to plot the rms curvature in Fig. 5. The inset shows a similar scaling relation as that for $c_{\mathrm{rms}}$. The solid line is a fit, $c_{\mathrm{rms}}\propto \mathrm{\Delta }{\xi }_{\min }^{-0.56}$, to the data points, and the dashed lines are again the predictions from the L'vov-Nazarenko spectrum (lower dashed) and from the single cusp model (upper dashed).

Figure 7

Time development of the local curvature at $T=0$ along the vortex after the reconnection of two rings. The reconnection results in two strong peaks at locations $\xi \approx 0$ and $\xi \approx 0.5L$. The upper panel is for the resolution $\mathrm{\Delta }{\xi }_{\min }=0.00125$ mm, and the lower panel is for twice better resolution $\mathrm{\Delta }{\xi }_{\min }=0.000625$ mm. The black curve is immediately after the reconnection, and other curves are 1 ms (red), 10 ms (blue), and 20 ms (green) after the reconnection. Note the logarithmic vertical axis and that the maximum speed of the spreading Kelvin waves is limited/defined by the resolution.

Figure 8

Decay of the rms curvature in the case of a straight vortex when $\alpha =0.1$ and $L_z=1$ mm. The initial Kelvin spectrum, shown in the inset, is the Vinen spectrum with $\eta =3$ and $A=0.01L_z$ at low $k$'s, while at high $k$'s the spectrum is parametrized by $\eta =4$ and $A=0.1L_z$. The early time behavior (blue curve) is determined using 65 536 points on the vortex, while 8192 points is used to obtain the late time asymptotic (red curve). The dashed lines are the analytical results, Eq. (12), corresponding to both spectra.