Shell model of superfluid turbulence

Superfluid helium consists of two interpenetrating fluids, a viscous normal fluid, and an inviscid superfluid, coupled by a mutual friction. We develop a two-fluid shell model to study superfluid turbulence. We investigate the energy spectra and the balance of fluxes between the two fluids as a function of temperature in continuously forced turbulence, and, in the absence of forcing, the decay of turbulence. We furthermore investigate deviations from the $k^{-5/3}$ spectrum caused by the mutual friction force. We compare our results with experiments and existing calculations. We find that, at sufficiently low temperatures a build-up of energy develops at high wave numbers suggesting the need for a further dissipative effect, such as the Kelvin wave cascade and phonon emission.

Figure 1

Top: Main plot: Log-log plot of superfluid (blue diamond) and normal fluid (red circle) energy spectra $E_k$ (${\mathrm{cm}}^2{\mathrm{s}}^{-2}$) vs wave number $k$ (${\mathrm{cm}}^{-1}$) at $T=2.157\mathrm{K}$. The solid line denotes the $k^{-5/3}$ Kolmogorov spectrum. The vertical dotted line marks $k_{\ell }$. Inset: Compensated spectra. As above, but $k^{5/3}{E}_k$ vs $k$. Bottom: Main plot: Log-lin plots of time-averaged scale-by-scale energy budget $d{E}_k/dt$ (${\mathrm{cm}}^2{\mathrm{s}}^{-3}$) vs wave number $k$ (${\mathrm{cm}}^{-1}$) for the two-fluid model at high temperature ($2.157\mathrm{K}$). We show the fluxes for the normal fluid (hollow shapes) and superfluid (solid shapes). The main plot shows the balance of fluxes in the normal fluid between the inertial term $T_m^n$ (red squares) and the viscous term $D_m^n$ (blue triangles), and in the superfluid between the inertial term $T_m^s$ (red squares) and the mutual friction term $M_m^s$ (grey circles). We also show the total flux $d{E}_m/dt$ (black diamonds). Inset: In the inset, we show the fluxes over the entire wave number range, including the flux due to the external forcing ${\epsilon }_{inj}^{n,s}$ (light blue line), which is nonzero only at shell $n=4$. The vertical dotted line denotes the time-averaged wave number $k_{\ell }$ corresponding to the intervortex spacing.

Figure 2

Top: Main plot: Log-log plot of superfluid (blue diamond) and normal fluid (red circle) energy spectra $E_k$ (${\mathrm{cm}}^2{\mathrm{s}}^{-2}$) vs wave number $k$ (${\mathrm{cm}}^{-1}$) as in Fig. 1 (top) but at $T=1.96\mathrm{K}$. The solid line denotes the $k^{-5/3}$ Kolmogorov spectrum. The vertical dotted line marks $k_{\ell }$. Inset: Compensated spectra. As above, but $k^{5/3}{E}_k$ vs $k$. Bottom: Main plot: Log-lin plots of time-averaged scale-by-scale energy budget $d{E}_k/dt$ (${\mathrm{cm}}^2{\mathrm{s}}^{-3}$) vs wave number $k$ (${\mathrm{cm}}^{-1}$) for the two-fluid model as in Fig. 1 (bottom) but at medium temperature ($1.96\mathrm{K}$). We show the fluxes for the normal fluid (hollow shapes) and superfluid (solid shapes). The main plot shows the balance of fluxes in the normal fluid between the inertial term $T_m^n$ (red squares) and the viscous term $D_m^n$ (blue triangles), and in the superfluid between the inertial term $T_m^s$ (red squares) and the mutual friction term $M_m^s$ (grey circles). We also show the total flux $d{E}_m/dt$ (black diamonds). Inset: In the inset, we show the fluxes over the entire wave number range, including the flux due to the external forcing ${\epsilon }_{inj}^{n,s}$ (light blue line), which is nonzero only at shell $n=4$. The vertical dotted line denotes the time-averaged wave number $k_{\ell }$ corresponding to the intervortex spacing.

Figure 3

Top: Main plot: Log-log plot of superfluid (blue diamond) and normal fluid (red circle) energy spectra $E_k$ (${\mathrm{cm}}^2{\mathrm{s}}^{-2}$) vs wave number $k$ (${\mathrm{cm}}^{-1}$) as in Fig. 1 (top) but at $T=1.44\mathrm{K}$. The solid line denotes the $k^{-5/3}$ Kolmogorov spectrum. The vertical dotted line marks $k_{\ell }$. Inset: Compensated spectra. As above, but $k^{5/3}{E}_k$ vs $k$. Bottom: Main plot: Log-lin plots of time-averaged scale-by-scale energy budget $d{E}_k/dt$ (${\mathrm{cm}}^2{\mathrm{s}}^{-3}$) vs wave number $k$ (${\mathrm{cm}}^{-1}$) for the two-fluid model as in Fig. 1 (bottom) but at low temperature ($1.44\mathrm{K}$). We show the fluxes for the normal fluid (hollow shapes) and superfluid (solid shapes). The main plot shows the balance of fluxes in the normal fluid between the mutual friction term $M_m^n$ (red squares) and the viscous term $D_m^n$ (blue triangles), and in the superfluid between the inertial term $T_m^s$ (red squares) and the mutual friction term $M_m^s$ (grey circles). We also show the total flux $d{E}_m/dt$ (black diamonds). Inset: In the inset, we show the fluxes over the entire wave number range, including the flux due to the external forcing ${\epsilon }_{inj}^{n,s}$ (light blue line), which is nonzero only at shell $n=4$. The vertical dotted line denotes the time-averaged wave number $k_{\ell }$ corresponding to the intervortex spacing.

Figure 4

Log-log plot showing dissipation of turbulent energy in normal fluid at low temperature. Coupled normal fluid (red circles) is seen to deviate from uncoupled normal fluid (green triangles) as a result of mutual friction with superfluid (blue diamonds). Coupled normal fluid follows $k^{-5/3}$ for $k<k_{\eta }$, deviates slightly for $k_{\eta }<k<k_{☆}$, and follows a $k^{-17/3}$ power law for $k>k_{☆}$. The vertical dotted line is $k_{\ell }$ and the short- and long-dashed lines are respectively $k_{\eta }$ and $k_{☆}$. For clarity $k_{\eta }<k_{☆}<k_{\ell }$.

Figure 5

Log-log plot of development of build-up of energy in superfluid spectrum. Spectra (bottom to top) at time $t=0$ (red), $0.025$ (green), $0.05$ (dark blue), $0.5$ (pink), 5 (light blue), 50 (black), 500 (orange), 5000 (grey), and $10000$s (red) after lowering the temperature. $k_l$ (not shown) moves from $k\approx {10}^3$ to $k\approx 2·{10}^4$ during this period. The thick black line denotes the $k^{-5/3}$ Kolmogorov spectrum.

Figure 6

Log-log plot of vortex line density$L$ vs Reynolds Number $Re$ at all three temperatures: high (red squares), medium (grey circles), and low (blue triangles). The points from left to right correspond to increasing forcing, the leftmost point using forcing $f=(1+i)·5·{10}^{-6}$, and the rightmost point $f=(1+i)·5·{10}^1$. The Reynolds Number is calculated at the first shell ($n=1$) giving $D=2^4$. The solid black line is $L=R{e}^{3/2}$.

Figure 7

Decay of energy spectra for normal fluid (top) and superfluid (bottom) over 10 realizations. Spectra top to bottom: After 500, 1000, $2500,$ and 5000s. Also shown is the $k^{-5/3}$ spectrum (solid black line).

Figure 8

Decay of total energy over period of 5000s ensemble averaged over 10 realizations. Top to bottom: High (red line), medium (grey line), and low (blue line) temperatures. Shifted to show power law. Also shown is the $t^{-2}$ spectrum (solid black line).

Figure 9

Decay of vortex line density over period of 5000s ensemble averaged over 10 realizations. Top to bottom: Low (blue line), medium (grey line), and high (red line) temperatures. Data is not shifted. Also shown is the $t^{-3/2}$ spectrum (solid black line).

Figure 10

Log-log plot of fully developed, steady power spectrum for superfluid (red circles) in the presence of a stationary normal fluid in ${}^3$He-B. We show the two power-laws, $k^{-3}$ and $k^{-5/3}$. The dashed line is $k_{+}$, the wave number at which the inertial term becomes of the same order as the mutual friction term, and the dotted line is $k_{\ell }$, the intervortex spacing.