Evolution of a superfluid vortex filament tangle driven by the Gross-Pitaevskii equation

Abstract

The development and decay of a turbulent vortex tangle driven by the Gross-Pitaevskii equation is studied. Using a recently-developed accurate and robust tracking algorithm, all quantised vortices are extracted from the fields. The Vinen's decay law for the total vortex length with a coefficient that is in quantitative agreement with the values measured in Helium II is observed. The topology of the tangle is then studied showing that linked rings may appear during the decay. The tracking also allows for determining the statistics of small-scales quantities of vortex lines, exhibiting large fluctuations of curvature and torsion. Finally, the temporal evolution of the Kelvin wave spectrum is obtained providing evidence of the development of a weak-wave turbulence cascade.The full understanding of turbulence in a fluid is one of the oldest yet still unsolved problems in physics. A fluid is said to be turbulent when it manifests excitations occurring at several length-scales. Due to the large number of degrees of freedom and the nonlinearity of the governing equations of motion, the problem is usually tackled statistically by introducing assumptions and closures in terms of correlators. This is the case in the seminal work of Kolmogorov in 1941 based on the idea of Richardson's energy cascade, where energy in classical fluids is transferred from large to small scales [1].Superfluids form a particular class among fluids characterised essentially by two main ingredients: the lack of dissipation and the evidence that vortex circulation takes only discrete values multiple of the quantum of circulation [2]. Superfluid examples which are routinely created in laboratories are superfluid liquid Helium (He II) and Bose-Einstein condensates (BECs) made of dilute Alkali gases. Here the superfluid phase is usually modelled via a complex field describing the order parameter of the system and quantised vortices appear as topological defects where the superfluid density vanishes.In three spatial dimensions those defects organise themselves into closed lines (or even open lines at the boundaries if confining sides are considered) of different configurations. Any vortex line point induces a velocity field in the superfluid which affects the motion of any object in the system including the vortex line itself. In general, even for a single closed vortex line, the dynamics are chaotic and the problem does not have analytical solutions. Superfluid turbulence regards the study of the evolution of many vortex lines, a tangle, which induce velocity field gradients in the fluid at several length scales. Different mathematical models have been devised to mimic the dynamics of a superfluid. An example is the vortex filament (VF) model based on the Biot-Savart law that relates vorticity and velocity [3]. This model is able to mimic the dynamics of dense vortex tangles due to a relatively fast numerical integration technique [4]. The VF model implicitly assumes that the superfluid density field is constant everywhere and the vortex structure is...

Approximate time-dependent solution of a master equation with full linear birth-death rates

“…), in the model due to the singularity at z=0 in equation (11). Therefore, it is worth to derive approximate analytic functions of the extinction probability and the waiting time distribution as will be shown in the next section.…”

“…As far as the analytical approach based on the generation function (GF) is concerned, (i) the mean and the variance can be estimated by the GF in equation (11); (ii) the steady-state PM in equation (6) can reproduce the probability distributions obtained in the numerical experiments. However, it is difficult to obtain (iii) the time dependent solution of P(n, t), the extinction probability P(0, t) and the waiting time distribution f (τ) for the process, due to the singularity located at z=0 in the GF in equation (11). When the value ò<0 is small enough to give the parameter a>0 in equation (33) where the state variable z in equation (29) takes positive value, the method of our approximation works.…”

“…There are growing interests on dynamics of phase-singularities (PSs) in complex systems such as ventricular fibrillation [1,2,[4][5][6], defect dynamics in fluids [7,8] and liquid crystals [9], living creatures [10], quantum vortex dynamics [11] and so on. A master equation approach on the number n of PS for studying birth-death dynamics of PSs in a 2D Complex Ginzburg-Landau equation is invented first by Gil, Lega and Meunier [12].…”

“…It is shown that their integrals are written under the constraint (A6) and (A7) as Inserting these results (A13) and (A14) into the expression of g(z, t) by noticing the initial condition, the solution in equation(11).…”

“…Appendix A. Derivation of equation (11) The first and the second equation in equation (10) are written explicitly as…”