Abstract
The lack of a general solution to the governing Navier–Stokes equations means that there is no fundamental theory of turbulence. In the simpler case of pure quantum turbulence, the tangle of identical singly quantized vortices in superfluids at T∼0 may provide a deeper understanding of turbulence in general. The wellknown Kolmogorov theory^{1} predicts the energy distribution of turbulence and how it decays. In normal systems the turbulent energy is generally only a small perturbation on the total thermal energy of the supporting medium. In quantum turbulence, however, the energy is accessible. A stationary condensate is necessarily in its ground state with zero enthalpy. Thus quantum turbulence accounts for the entire free energy of the superfluid and there are no other contributions. Here, we exploit this property to make the first direct measurement of the energy released by freely decaying quantum turbulence. Our results are consistent with a Kolmogorov energy spectrum with an inferred Kolmogorov constant remarkably similar to those of classical fluids.
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The turbulent energy is an important quantity here. Vortex lines contribute two energy components to the system (see Fig. 1), the loss of condensation energy in the vortex core, where the superfluidity is suppressed (negligible in our case), and the kinetic energy in the flow fields surrounding the cores. In earlier experiments on quantum turbulence the quantity usually inferred from experiment is the average vortexline density L (the vortexline length per unit volume)^{2}. Comparison with the velocityfield characterization of classical turbulence usually relies on a phenomenological model^{3} equating the line density to the mean square vorticity of classical turbulence: 〈ω^{2}〉=κ^{2}L^{2}, with κ the circulation quantum. An effective kinematic viscosity ν^{′} is also introduced, enabling the dissipation per unit mass to be written ε=ν^{′}κ^{2}L^{2} in analogy with the classical turbulence dissipation ε=ν〈ω^{2}〉, where ν is the classical kinematic viscosity^{2,3} (the true viscosity of a pure superfluid being identically zero). Although this model has not been directly tested, it agrees qualitatively with measurements of the vortex line density L(t) for decaying quantum turbulence^{4} in superfluid ^{3}He–B and in superfluid ^{4}He (refs 3, 5), allowing a determination of ν^{′}. In contrast, a direct measurement of the energy dissipated in quantum turbulence allows us to test the standard model of decaying turbulence independently of this phenomenological model.
In the experiment presented here, the energy dissipation is measured directly with a ‘blackbody radiator’ (BBR; ref. 6) the main features of which are shown in Fig. 2. The BBR is a thinwalled box with a small orifice in one side, immersed in superfluid ^{3}He–B, and cooled by nuclear refrigeration. Inside the radiator is a thermometer wire resonator, a heater wire resonator to inject power, and a very lowamplitude resonating grid, which generates the turbulence by the initial production of microscopic vortex rings, as discussed in the Methods section. The grid resonator consists of a 5×5 mm goalpost shaped Ta wire carrying a 5×3.5 mm Cu grid mesh of ∼197 lines cm^{−1}.
The experiments are performed in the lowtemperature ballistic regime below 200 μK, where the thermal quasiparticle excitation mean free path is greater than the container dimension. (All data presented here were taken at a pressure of 3.3 bar, where the superfluid transition temperature, T_{C}, is 1.32 mK.) Heat entering the radiator from any source produces ballistic quasiparticles, which thermalize by scattering off the walls, finally emerging as a beam of excitations from the orifice. At steady state, the power emitted in the beam balances the power entering the radiator. From simple kinetic theory, the power emitted^{6} is given by , where 〈n v_{g}〉 is the thermal flux of excitations of mean energy , Δ is the superfluid energy gap, and A the effective area of the orifice. The damping on the thermometer wire is dominated by quasiparticle scattering, which being well understood^{7} provides very sensitive thermometry^{8}. The thermal flux of excitations can therefore be inferred from the measured damping on the thermometer wire^{9}, and from this quantity we obtain the power leaving the BBR (as described in the Methods section below). When turbulence is present, the excess power leaving the BBR provides a direct measure of the energy being released by the freely decaying turbulence, and this allows us to reconstruct the energy content of the original turbulence.
Figure 3 shows the excess power leaving the BBR as a function of time after switching off the drive to the grid. This data was taken for an initial grid velocity of 4.51 mm s^{−1}. While driven, the grid generates both vortices and large numbers of quasiparticle excitations^{10}. In the absence of vortices, we expect the excess power to decay exponentially with the BBR time constant, δQ̇=Q̇_{0}exp(−t/τ_{i}). This is estimated from simple kinetic theory as τ_{i}≃4V/(〈v_{g}〉A), with V the volume of the BBR and 〈v_{g}〉 the mean excitation group velocity^{8}. The BBR dimensions were designed to give a time constant of order τ_{i}∼0.5 s. A series of similar measurements were made for a range of initial grid velocities. All the data presented here were taken at 3.3 bar.
The green lines in the figure show the initial exponential recovery with an intrinsic time constant of τ_{i}=0.54 s, in good agreement with the design value. At later times the excess power decays much more slowly, revealing a much longerlived source of excitations in the BBR. This is the dissipation of the quantum turbulence produced by the grid. We can subtract the initial thermal recovery, δQ̇=Q̇_{0}exp(−t/τ), to leave only the contribution from the slowly decaying vortices. This is shown in Fig. 4, where we plot the dissipation from the turbulence as a function of time for various initial grid velocities. We note that the thermal recovery is only significant during the first few seconds.
From earlier measurements^{11}, confirmed by computer simulations^{12}, we know that at low grid velocities the generated vortex rings travel ballistically. The rings have a selfpropagation speed^{13} of around ∼10 mm s^{−1}, corresponding to diameters of ∼5 μm. In the current experiment, such rings would collide with the BBR walls after ∼0.5 s. As we expect these rings to dissipate rapidly after colliding with a wall, they will not contribute to the latertime dissipation shown in Fig. 4. Above some critical grid velocity the ring density becomes sufficient for collisions, and reconnecting rings then rapidly evolve into quantum turbulence^{11,12}. The critical velocity for this grid is found to be ∼2 mm s^{−1}, because there is no measurable turbulence dissipation below this velocity.
We now compare the results with expectations from classical turbulence. According to the standard model^{1,2}, the Richardson cascade transfers energy from large to small length scales (low to high wavenumbers, k) over an inertial range where viscosity plays no role. The energy is mostly contained in large eddies of length scale l_{e}, but is dissipated by viscous forces at the smallest scales. The energy per unit wavenumber per unit mass in the inertial range is predicted by the Kolmogorov spectrum E_{k}=C ε^{2/3}k^{−5/3}, with C a constant of order unity. Assuming that this spectrum spans the full range of length scales from the cell size (l_{e}=d≃10 mm) to the smallest (dissipative) scale a, the total turbulent energy is obtained by integrating , where V is the volume of the radiator, k_{d}=2π/d and k_{a}=2π/a. Assuming a≪d, the dissipation Q̇=ρ V ε is independent of the dissipative scale a and given by:
where E_{0} is the initial turbulent energy.
Thus the predicted latetime decay of the dissipation should be proportional to t^{−3}, and independent of all fluid properties other than the density. Thus the only unknown parameter is the Kolmogorov constant C. Therefore a measurement of the dissipated energy gives a particularly direct test of this model.
The behaviour at earlier times depends also on the initial energy E_{0}, which determines t_{0} and introduces a second free parameter. The solid lines in Fig. 4 show the behaviour predicted by equation (1) with a Kolmogorov constant of C=2.0. The data at higher grid velocities are seen to agree very well with the model, with initial energies of order 100 pJ. The Kolmogorov constant extracted from the fits, C≃2.0±0.4, falls within the range of typical values attributed to classical fluids^{14}. The behaviour differs at the lower velocities (3.79 mm s^{−1} and below), where the latetime behaviour is consistent with a t^{−2} dependence, as expected for a random tangle with no large scale structure^{2}. (Here the rings have reconnected to form a tangle, but at too low a density to develop a Kolmogorov spectrum). A similar crossover from a random tangle to classicallike turbulence has also been inferred from measurements of the line density in both superfluid ^{3}He–B and superfluid ^{4}He at low temperatures^{4,5,15}.
Our data at high grid velocities agree well with the standard model of classical turbulence. However, there are caveats. First, in practice the initial energy spectrum will depend on the turbulence–production process. To explain towedgrid measurements in superfluid ^{4}He, a more elaborate model^{3} has been developed incorporating an initial k^{2} energy spectrum at large length scales and a timedependent energycontaining length scale l_{e}(t) that grows until it saturates at the container size d, after which the behaviour follows equation (1). This model does not yield substantially improved fits to our data, suggesting that the initial conditions do not critically influence the latetime behaviour. Second, the initial turbulence is clearly not isotropic or homogeneous^{16}. Nevertheless, we observe very similar behaviour with wiregenerated turbulence, even though the initial turbulence distributions must differ substantially, which suggests that the initial homogeneity does not affect the latertime behaviour. Third, one should question whether the vortexline density in the BBR is large enough to provide an inertial range sufficient to support a Kolmogorov spectrum, but we note that measurements of the line density in other experiments^{4,17} suggest that there are no large deviations from the simple model at very low densities, even when the spatial extent d corresponds to just a few line spacings. Finally, as the initial temperature in the BBR, whilst the turbulence is being generated, is quite high (∼0.25T_{C} or 330 μK for the highest velocity data), mutual friction may influence the production process and the initial behaviour^{18}. Fortunately, this effect rapidly becomes negligible, as the radiator promptly cools in the first few seconds.
Methods
The mechanism for generating quantum turbulence. From earlier work^{11}, we know that when we oscillate an object in the superfluid, the flow fields lead initially to the generation of microscopic vortex rings. This process gives rise to a cloud of similarsized rings leaving the moving object, with the rate of creation increasing with the amplitude of the oscillatory flow. Under our conditions, the rings are a few micrometres in diameter and have velocities of ∼10 mm s^{−1}, which is fast on the scale of the BBR dimensions. The subsequent history of this gas of rings depends on the density. At low densities the rings travel independently and rapidly leave the generation site. At some critical density, however, the ring density is high enough that they collide and reconnect to form larger, slower structures, trapping more rings, and turbulence rapidly develops.
The generation process by the grid is thus very different from that in a classical towedgrid scenario. In the present case, the grid is essentially stationary, as the amplitude of motion at the highest grid velocities is only a few hundreds of nanometres, thus generating no significant largescale motion in the liquid. The specific geometry of the grid essentially plays no role, as the grid is only acting as a coarse surface for assisting the generation of the gas of rings. It is perhaps better to think of this device more as a stationary planar transducer (emitting vortex rings instead of sound) rather than as a moving grid as used in classical flow experiments. Nevertheless, this flat transducer provides us with the closest we can currently get to homogeneous quantum turbulence.
Operation and calibration of the blackbody radiator. The thermal damping on the thermometer wire^{9} is given by Δf_{2}^{T}=B exp(−Δ/k_{B}T). At our temperatures (∼200 μK), Δ≫k_{B}T, the exponential is changing very rapidly, providing an extremely precise value of T (at 170 μK a change of only 1 μK gives rise to a change in the damping of nearly 10%). Δf_{2}^{T} is also proportional to 〈n v_{g}〉. Thus we have the quantities needed to determine the beam power, . The beam power can thus be rewritten as Q̇_{beam}=c W_{T}, where (which we designate the ‘width parameter’ of the thermometer wire), T and have been obtained directly from Δf_{2}^{T}, as above, and c is a calibration constant to be determined^{6}.
On injecting large powers, the bulk superfluid outside the BBR also warms slightly, yielding an influx of excitations back through the orifice from the outside. This is accounted for by subtracting the width parameter W^{′} measured by a similar thermometer wire located outside the BBR, close to the orifice, but shielded from the excitation beam by a paper screen. The net power leaving the radiator is thus given by^{5} Q̇_{out}=c δ W, where δ W=W_{T}−W^{′}. In the absence of additional sources of heating, the power leaving the radiator balances the steadystate heat leak into the BBR, Q̇_{leak}=c δ W_{0}.
The radiator is calibrated by measuring the increase in the width parameter ΔW=δ W−δ W_{0} while a measured power Q̇_{ap} is injected into the radiator with the heater wire. A linear response is observed from which the calibration constant c=Q̇_{ap}/ΔW is deduced. Once calibrated, the width parameter directly measures the net power leaving the radiator. The background heat leak is found to be Q̇_{leak}≃2 pW, resulting in a base temperature inside the BBR of 172 μK (∼0.13T_{C}) whilst the surrounding superfluid cools to ∼145 μK. To determine the energy being dissipated by turbulence in the BBR, we correct for the heat leak and refer to the excess power, leaving the BBR, δQ̇=Q̇_{out}−Q̇_{leak}.
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Acknowledgements
We acknowledge technical support from M. G. Ward and A. Stokes, and funding from the UK EPSRC, the FP7 European MICROKELVIN network and the Royal Society.
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All the authors contributed to the devising of the experiment and the analysis of the data. The experiments were carried out by D.I.B., S.N.F. and D.P. The paper was written by D.I.B., S.N.F. and G.R.P.
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Bradley, D., Fisher, S., Guénault, A. et al. Direct measurement of the energy dissipated by quantum turbulence. Nature Phys 7, 473–476 (2011). https://doi.org/10.1038/nphys1963
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DOI: https://doi.org/10.1038/nphys1963
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