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Developed quantum turbulence and its decaya)
a)This review is an expanded version of the invited talk given by KRS at the Division of Fluid Dynamics meeting of the American Physical Society in November 2007. A short version will appear in “Ten Chapters in Turbulence” to be published by Cambridge University Press.
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View: Figures


Image of FIG. 1.
FIG. 1.

(Color online) Phase diagrams of the two isotopes of helium. Note that there is no triple point and both isotopes stay in a superfluid liquid phase down to absolute zero (except at high pressures). The region to the left of the dashed line in the upper figure is He II. 3He is a magnetic liquid and its state depends on the externally applied magnetic field (see text for details).

Image of FIG. 2.
FIG. 2.

Temperature dependence of the normal and superfluid density ratio in He II. The inset shows the principle of the Andronikashvili torsional pendulum experiment leading to this measured dependence.

Image of FIG. 3.
FIG. 3.

Simply and multiply connected regions in superfluid helium. The shaded regions lie outside the fluid. The point O lies somewhere inside the loop in A, and inside the shaded circle in B.

Image of FIG. 4.
FIG. 4.

Left: structure of quantized vortex in He II. Right: superfluid velocity and density profiles in the vicinity of the vortex axis. The vertical dashed line is a measure of the radius of the quantized vortex line. It is of the order of an Angstrom (depending weakly on the pressure and the temperature, except very close to Tλ ).

Image of FIG. 5.
FIG. 5.

Photograph of vortex lattice in rotating He II illuminated by laser sheet. The vortices are marked by hydrogen flakes remain trapped by the Magnus force on to cores of quantized vortices.

Image of FIG. 6.
FIG. 6.

Schematic view of the generation of counterflow turbulence of He II and its detection using the second sound attenuation method.

Image of FIG. 7.
FIG. 7.

Experimental arrangement of Awschalom, Milliken and Schwarz.34 S denotes the tritium ion source, G1, G2, G3 are pulsed grids, C1, C2, C3 are collectors, W denotes the channel wall, vs and vn in the counterflow are indicated by arrows. The dotted region indicates a typical ion pulse; the ions are moved by an electric field ɛ.

Image of FIG. 8.
FIG. 8.

The observed velocities of tracking particles plotted versus calculated normal fluid velocity at temperatures as indicated.39

Image of FIG. 9.
FIG. 9.

(Color online) The observed trajectories of tracking particles in counterflowing He II.40 The regular (straight, black) trajectories are those of particles moving with the normal fluid, while irregular trajectories (colored) in the opposite direction are those of particles trapped in vortex cores.

Image of FIG. 10.
FIG. 10.

(Color online) The probability density function of observed horizontal ( x ) and vertical ( z , red) velocities appears very different from the expected conventional PDF of Gaussian form found in classical viscous fluid.40

Image of FIG. 11.
FIG. 11.

Variation with time of the excess second sound attenuation after a heat current 0.14 W/cm2 has been switched off.5 is the attenuation in the steady heat current. The dotted line is a plot of Eq. (24), with τ estimated to be about 0.14 s, based on the geometry of the apparatus.

Image of FIG. 12.
FIG. 12.

Log-log plots of decaying vortex line density L(t), deduced from the relaxing amplitude of the second sound under the assumption of random vortex tangle.47 The early decay period depends on initial counterflow velocity and may even include a period of increasing L(t), but eventually all decay curves closely follow power law decay κL = βt −3/2, where the temperature-dependent prefactor β(T) does not depend on initial conditions but appears to be proportional to the channel size.

Image of FIG. 13.
FIG. 13.

The measured energy spectral density in a flow of liquid helium confined between counterrotating discs at temperatures 2.3 K (a), 2.08 K (b), and 1.4 K (c).6 For further details (see the text).

Image of FIG. 14.
FIG. 14.

Schematic of the experimental apparatus.

Image of FIG. 15.
FIG. 15.

The log-log plot of the decaying vortex line density versus time after grid passes 2 mm above the measuring volume. Each decay curve represents an average of three identical pulls. The decay curves, in order, correspond to mesh Reynolds number ReM  = 2 × 105 (the uppermost one), 1.5 × 105, 105, 5 × 104, 2.5 × 104, 104, 5 × 103, and 2 × 103. For each ReM , the decaying vortex density joins and follows thereafter the universal power decay with exponent −3/2.61

Image of FIG. 16.
FIG. 16.

Experimental arrangement (schematic) to generate QT by oscillating a tightly stretched grid and for measuring its decay times (a). Evolution of the collector signal with time following a burst of grid oscillations66 (b).

Image of FIG. 17.
FIG. 17.

Experimental arrangement of the oscillating grid in 3He-B and associated vorticity detector wires.

Image of FIG. 18.
FIG. 18.

Simulation of quantum turbulence formation due to oscillating grid.74 Each frame shows the vortex configuration at the labeled time. Rings injected from the left quickly collide and recombine producing a vortex tangle which evolves on longer time scales.

Image of FIG. 19.
FIG. 19.

Solid black curves show the inferred vortex line density as a function of time after the cessation of grid motion for initial grid velocities as indicated. Line A is the limiting classical-like behavior as discussed in the text. The halftone data is that for the Oregon towed grid experiments of Skrbek, Niemela, and Donnelly58 in He II, with line B showing the late-time limiting behavior. Line C shows the expected behavior for the data assuming the classical dissipation law in thick normal 3He. Curve D shows the expected (based on Vinen’s equation, as discussed above) behavior for a random tangle in superfluid 3He.75

Image of FIG. 20.
FIG. 20.

Upper row: Cartoon of the vortex configurations produced by spin-down in the experimental cell (side view) at different stages. (1) Regular array of vortex lines during steady rotation before deceleration. (2) Immediately after stopping rotation, turbulence appears at the outer edges but not on the axis of rotation. (3) After about 30 rad of initial rotation, the 3D homogeneous turbulence is everywhere and (4) decays with time. Shaded areas indicate paths of probe ions when sampling the vortex density in the transverse (3) and axial (4) directions. Bottom row: Cartoon of the vortex configurations produced by a pulse of injected ions at T < 0.5 K in the experimental cell (side view) at different stages. (1) up to 1 s: a pulse of charged vortex rings is injected from the left injector. While most make it to the collector as a sharp pulse, some get entangled near the injector. (2) : the tangle spreads into the middle of the cell. (3) : the tangle has occupied all volume; from now on, it is nearly homogeneous (as probed in two directions). (4) Up to 1000 s: the homogeneous tangle is decaying further. The shaded areas indicate the trajectories of ions used to probe the tangle along two orthogonal directions.

Image of FIG. 21.
FIG. 21.

Deduced vortex line density, normalized, for T = 0.15 K (filled symbols) and T = 1.6 K (open symbols). Dashed and solid lines have a slope of −3/2 to guide the eye through the late-time decay at T = 1.6 K and 0.15 K, respectively.

Image of FIG. 22.
FIG. 22.

Left: Temporal decay of a tangle produced by beams of charged vortex rings of different durations and densities at T = 0.15 K. The injection direction and duration and driving fields are indicated. Right: Decay of a tangle produced by a jet of free ions from the bottom injector (•), as well as by an impulsive spin-down to rest from 1.5 rad/s and 0.5 rad/s. All tangles were probed by pulses of free ions in the horizontal direction.

Image of FIG. 23.
FIG. 23.

(Color online) Phase diagram of turbulent superflow in 3He-B found by Finne et al. 68 The principle of the measurements is as follows. (a) The initial state is vortex-free (Landau state) superflow in rotation at Ω, where the normal component is stationary and the superfluid component flows in the rotating frame. (b) A few (ΔN) vortex loops are injected and, after a transient period of loop expansion, the number of rectilinear vortex lines Nf in the final steady state is measured. This state is found to fall in one of two categories. (c) Nf  = Δ N, regular mutual-friction-damped loop expansion. (d) , turbulent loop expansion. This process leads to a total removal of the macroscopic vortex-free superflow as the superfluid component is forced into solid-body-like rotation (on an average) by the formation of a vortex array with the equilibrium number of rectilinear lines, N eq ≈ πR 22Ω/κ ∼ 103. (e) Phase diagram of measured events.

Image of FIG. 24.
FIG. 24.

NMR absorption spectra before and after vortex-loop injection68 (see text for details).

Image of FIG. 25.
FIG. 25.

The general shape of the 3D energy spectrum of QT in the zero temperature limit. For length scales exceeding the quantum length scale (to the left of the thinner vertical arrow), quantum effects are unimportant and a classical Richardson cascade operates, resulting in the inertial range of the Kolmogorov form. At smaller length scales, quantum effects start to dominate and Kelvin-waves cascade transfers energy that eventually becomes dissipated, most likely by acoustic emission (for further details, see text).

Image of FIG. 26.
FIG. 26.

(Color online) The observed temperature dependence of ν eff(T) from various experiments described in the text. The solid (blue) line above the λ-temperature is kinematic viscosity of normal liquid He I.10 The big black dots correspond to our estimate ν eff ≈ κq, as described in the text.

Image of FIG. 27.
FIG. 27.

Left: Log-log plot of the quantity (a 0/a(t) − 1), where a 0 and a(t) denote the initial and decaying second sound amplitude, versus time measured at T = 1.6 K. Under the assumption that the tangle is homogeneous and isotropic, this quantity would be proportional to the vortex line density L. For given experimental conditions, the level a 0/a(t) − 1 = 0.1 would correspond to the vortex line density L ≈ 105 cm−2. The different decay curves correspond to different initial levels of steady-state counterflow turbulence generated at powers 0.5 W (⋄), 0.23 W (squares), 0.18 W (○) and 0.14 W (various triangles), respectively. Three individual decay curves are shown to appreciate the level of reproducibility for the lowest applied power. The inset shows analogous data measured at T = 2.0 K; applied powers: 0.52 W (various triangles), 0.41 W (⋄), and 0.32 W (○). Right: Perpendicular and parallel projections of the initial vortex tangle resulting from the computer simulations clearly show that the vortex tangle is strongly polarized.


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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Developed quantum turbulence and its decaya)