Dynamics of the forward vortex cascade in two-dimensional quantum turbulence
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] N ov Dynamics of the forward vortex cascade intwo-dimensional quantum turbulence
Andrew Forrester and Gary A. Williams
Department of Physics and Astronomy, University of California, Los Angeles, CA 90095E-mail: [email protected]
Abstract.
The dynamics of the forward vortex cascade in 2D turbulence in a superfluid filmis investigated using analytic techniques. The cascade is formed by injecting pairs with thesame initial separation (the stirring scale) at a constant rate. They move to smaller scales withconstant current under the action of frictional forces, finally reaching the core size separation,where they annihilate and the energy is removed by a thermal bath. On switching off theinjection, the pair distribution first decays starting from the initial stirring scale, with the totalvortex density decreasing linearly in time at a rate equal to the initial injection rate. As pairsat smaller scales decay, the vortex density then falls off as a power law, the same power lawfound in recent exact solutions of quenched 2D superfluids.
Constant-current cascades have long played a role in understanding turbulence in fluids. Inclassical 2D turbulence it is well known that there are two such cascades [1, 2], an inversecascade of energy to large length scales with a k − / (Kolmogorov) energy spectrum, anda forward cascade of enstrophy (vorticity) to small length scales, with a k − (or k − ln( k ))energy spectrum. The dynamics of the free decay of such turbulence has been investigated withnumerical simulations [3, 4, 5, 6], where the decay is dominated by the dynamics of the inversecascade where like-sign regions of vorticity merge, heading toward the final state that is twomacroscopic counter-rotating vortices.The nature of the cascades in 2D quantum turbulence, with its discrete quantized vortices,is considerably less well understood. A forward cascade of vortex pairs was found a number ofyears ago [7] for low vortex densities, comparable to or smaller than the density at the thermalKosterlitz-Thouless transition. More recently two numerical simulations of the Gross-Pitaevskiimodel at high vortex densities found instead a forward energy cascade with a k − / spectrum[8, 9]. Further complicating the picture is another set of simulations which found no directcascade, but did find an inverse energy cascade with a roughly k − / spectrum [10, 11].In this paper we show that analytic techniques can be used to study the dynamics of thegrowth and decay of the forward vortex cascade. We find that the decay of the vortex density isinitially linear in time, but then switches at longer times to a power-law decay with an exponentidentical to that found in our recent exact solutions of quenched 2D superfluids [12]. We believethese are the first analytic solutions for the dynamics of any type of fluid turbulence.We consider an incompressible superfluid film connected to a thermal bath at a temperatureof 0.1 T KT , where T KT is the critical Kosterlitz-Thouless temperature where thermally excitedvortex pairs drive the superfluid density to zero. The vortex dynamics are described by the sameFokker-Planck equation [13] used in describing the distribution of vortex pairs of separation r
12, 14], with the addition of a 2D delta function to inject additional pairs of a fixed separation R into the film at a rate α , ∂ Γ ∂ t = 1 r ∂∂r (cid:18) r ∂ Γ ∂r + 2 πK Γ (cid:19) + α δ ( ~r − ~R ) (1)This equation is set in dimensionless form with lengths in units of the vortex core radius a , thevortex-pair distribution function Γ in units a − , and the time in units of the vortex diffusiontime, τ = a / D with D the vortex diffusion coefficient. K is the dimensionless superfluiddensity, K = ¯ h σ s /m k B T , and is determined from the Kosterlitz recursion relation [15] dKdr = − π r K Γ (2)with σ s the superfluid areal density and m the He atomic mass. The dimensionless injectionrate α is given by α = a ˙ Qτ where ˙ Q is the number of vortex pairs injected per unit area pertime at random positions and orientations across the plane.The steady-state solutions of Eq. 1 where ∂ Γ /∂t = 0 were previously found in [7], but onlyfor r < R , while for dynamics the vortices at r > R also need to be included. In the limit oflow injection rates the vortex densities are well below the densities at T KT , and the superfluidfraction is unaffected by the vortices. The solution for r < R is then a constant valueΓ = Γ = α/ πK (3)where K is the value at r = 1 where σ s equals the ”bare” superfluid density σ s . For r > R thesolution is a quasi-thermal distribution extending from R , Γ = Γ ( r − R ) − πK , and which arisesfrom injected pairs initially at separation R getting a thermal kick to higher separation. Thesteady-state flux of vortex pairs from Eq. 1 is J = − ( r∂ Γ/ ∂r + 2 πK Γ)), and for this solution isa constant, J = − α for r < R , and zero for r > R .The two curves in Fig. 1 for the lowest values of α show the steady-state distribution of pairsfor the low-injection regime calculated directly from Eqs. 1 and 2 using fixed-step Runge-Kuttaiteration with the injection scale R = 400, in agreement with the above form of Eq. 3. Thedelta function is approximated with a strongly peaked Gaussian form with spatial width of 2.The superfluid fraction remains unity for r < R , though at α = 1 × − the vortex density isgetting high enough that there is a very slight reduction for r > R .The upper two curves in Fig. 1 show the distribution in the limit where the vortex densitybecomes comparable to that found at T KT , indicated by the dashed curve in Fig. 1. Thesuperfluid fraction is rapidly driven to zero at a finite length scale r that depends on theinjection rate. This effect on the superfluid density shows that this turbulent state is not justcharacterized by isolated dipole pairs, but in fact is a complex many-body state of smaller pairsscreening the interaction of larger pairs to drive down the superfluid density. To a very goodapproximation the solutions in this regime can be represented asΓ = Γ ( r < R )= Γ (1 + ln( r/r )) ( r < r < R )= Γ (1 + ln( R/r )) ( r > R ) . (4)The steady-state flux for this solution is also J = − α for r < R , and zero for r > R .The time dependence of the cascade growth can be studied by solving Eqs. 1 and 2 as afunction of both time and distance using standard numerical techniques, starting from thermalequilibrium at 0.1 T KT and switching on the injection at t = 0. Figure 2 shows the growth ofthe distribution and the pair current for the very low injection rate α = 1 × − . Initiallyhe distribution just broadens as the pairs get thermal kicks to larger and smaller separations,but the frictional forces on the vortex cores also give rise to a net current of pairs to smallerseparations. After a few hundred diffusion times the decaying pairs reach the core size separationwhere they annihilate, and the energy is pulled out by the thermal bath. At longer times thedistribution becomes constant at r < R and increases uniformly toward the equilibrium value.The current is initially only appreciable near the injection scale, and then finally at very longtimes reaches the value of − α , the cascade state where the rate of pairs being injected at R isequal to the rate of pairs being pulled out by the thermal bath at the scale a . The total vortexdensity is shown in Fig. 3, with initially a linear increase with slope α before any pairs are pulledout at a , and then a leveling off towards equilibrium once pairs begin annihilating.On switching off the injection, the decay of the cascade starts at the injection scale, as shownin Fig. 4, since the current of pairs away from R is no longer being replenished by the injectedpairs. Once the region of diminished pairs begins to reach the scale a the distribution starts touniformly decrease, and in fact in this regime the solution of Eqs. 3 and 4 is the exact solutionfound in [12] for a quenched 2D superfluid with an initial temperature equal to 0.1 T KT . At longtimes, however, it comes to equilibrium at the bath temperature. The decaying vortex densityis shown in Fig. 3, initially a linear decay with slope − α , since this is still the rate at which pairsannihilate, but then at long times falling off as t − ( πK − , the value predicted in [12].In the limit of higher injection rates where the superfluid density is driven to zero at finitescales ( α > × − ) the dynamics of the cascade becomes more complicated. The gradient inthe superfluid density K in Eq. 1 leads to an additional current toward large scales that subtractsfrom the forward current and tends to slow down both the growth and decay of the cascade.This is shown in Fig. 5 for the growth of the cascade at α = 1 × − . As the pair distributiongrows the superfluid fraction is depressed to smaller and smaller length scales. The current inthe cascade region r < R only grows slowly since the pairs being injected are divided, with part S u p e r f l u i d f r a c t i o n Pair separation r inject α = 1e-20 1e-7 1e-9 1e-10 1e-1210 -29 -27 -25 -23 -21 -19 -17 -15 -13 -11 -9 -7 P a i r d i s t r i b u t i o n Γ α = 1e-20 1e-12 1e-10 1e-7 1e-9 thermal distributionat T KT Figure 1.
Steady-state solutions of Eqs. 1and 2 for the vortex-pair distribution functionat different injection rates α . -1.0x10 -20 -0.50.0 F l u x -180 -160 -140 -120 -100 -80 -60 -40 -20 P a i r d i s t r i b u t i o n Γ Pair separation r inject t = 00.1301003001000 4000 t = (cid:176)
Figure 2.
Growth of the pair distributionand current in time for α = 1 × − . -18 V o r t e x p a i r d e n s i t y Time
Time
Figure 3.
Growth (left) and decay (right) of the vortex-pair density, for injection rate α = 1 × − . -42 -40 -38 -36 -34 -32 -30 -28 -26 -24 -22 P a i r d i s t r i b u t i o n Γ Pair separation r inject thermal distribution0.1 T KT t = 00.1 300 1000 2000 3000 10000 5000 -1.0x10 -20 -0.50.0 F l u x t = 03000.12000 10003000 Figure 4.
Decay of the pair distributionfunction and current after switching off theinjection α = 1 × − . -44 -41 -38 -35 -32 -29 -26 -23 -20 -17 -14 -11 -8 P a i r d i s t r i b u t i o n Γ t = 11003001000 300010000 20000 50000 t = (cid:176)thermal equilibrium -1x10 -7 F l u x (cid:176) S u p e r f l u i d f r a c t i o n Pair separation r t = 1t = (cid:176)
Figure 5.
Growth of the pair distribution,current, and superfluid fraction for injection α =1 × − .oing in the forward direction and others increasing their separation in the inverse direction,driven by the gradient in K . The vortex density in the cascade region grows more slowly thanlinear in time due to this division, with fits giving a variation as approximately t / .In this analysis we find no evidence of a forward energy cascade with spectrum k − / asfound in the earlier simulations [8, 9], and this may be due to the relatively dilute vortexdensities considered here, in which there will only be annihilation of same-pair vortices whentheir separation reaches a . The vortex densities in Refs. [8] and [9] were quite high, with vorticeson a large fraction of the lattice sites, and in this situation there will be a much higher probabilityof annihilation of opposite-circulation vortices on different pairs, which might give rise to theobserved forward energy cascade. The densities are so high that these systems are certainly notsuperfluid at length scales much larger than the vortex core size. A further difference from oursystem is that the Gross-Pitaevskii model that is simulated is a compressible superfluid, and thesound waves excited in the turbulence may also contribute to energy flows. And finally, thesesimulations had zero dissipation, whereas our forward cascade requires an arbitrarily small butfinite dissipation to develop. The dissipation only enters into the dynamics through the vortexdiffusion constant that sets the time scale; it does not affect the steady-state solutions of Eqs. 3and 4. Weak dissipation still allows the possibility of ”inertial” cascades, as shown by thesimulations of the classical 2D Navier-Stokes equation with finite viscosity [16].We also do not observe an inverse cascade of energy to large scales. There is an initial outwardcurrent current as pairs build up in the region r > R , but this falls to zero in the steady statefor both the low and high injection rates. An inverse cascade requires a clustering of like-signvortices in a negative-temperature state [10, 11], but our Kosterlitz-Thouless model assumesfrom the start a uniform distribution of vortices with positive and negative circulation. Acknowledgements
This work was supported in part by the U. S. National Science Foundation, Grant No. DMR0906467.
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