Local Induction Approximation Research Articles

Steady-state counterflow quantum turbulence: Simulation of vortex filaments using the full Biot-Savart law

We perform a numerical simulation of quantum turbulence produced by thermal counterflow in superfluid $^4$He by using the vortex filament model with the full Biot--Savart law. The pioneering work of Schwarz has two shortcomings: it neglects the non-local terms of the Biot--Savart integral (known as the localized induction approximation, LIA) and it employs an unphysical mixing procedure to sustain the statistically steady state of turbulence. For the first time we have succeeded in generating the statistically steady state under periodic boundary conditions without using the LIA or the mixing procedure. This state exhibits the characteristic relation $L=\gamma^2 v_{ns}^2$ between the line-length density $L$ and the counterflow relative velocity $v_{ns}$ and there is quantitative agreement between the coefficient $\gamma$ and some measured values. The parameter $\gamma$ and some anisotropy parameters are calculated as functions of temperature and the counterflow relative velocity. The numerical results obtained using the full Biot--Savart law are compared with those obtained using the LIA. The LIA calculation constructs a layered structure of vortices and does not proceed to a turbulent state but rather to another anisotropic vortex state; thus, the LIA is not suitable for simulations of turbulence.

Motion of a vortex filament in the local induction approximation: Reformulation of the Da Rios–Betchov equations in the extrinsic filament coordinate space

In recognition of the highly non-trivial task of computation of the inverse Hasimoto transformation mapping the intrinsic geometric parameter space onto the extrinsic vortex filament coordinate space a reformulation of the Da Rios–Betchov equations in the latter space is given. The nonlinear localized vortex filament structure solution given by the present formulation is in detailed agreement with the Betchov–Hasimoto solution in the small-amplitude limit and is also in qualitative agreement with laboratory experiment observations of helical-twist solitary waves propagating on concentrated vortices in rotating fluids. The present formulation also provides for a discernible effect of the slipping motion of a vortex filament on the vortex evolution.

A locally induced homoclinic motion of a vortex filament

An exact homoclinic solution of the Da Rios–Betchov equation is derived using the Hirota bilinear equation. This solution describes unsteady motions of a linearly unstable helical or wound closed filament under the localized induction approximation.

Numerical Studies of Counterflow Turbulence

We performed the numerical simulation of quantum turbulence produced by thermal counterflow in superfluid 4He by using the vortex filament model. The pioneering work was made by Schwarz, which has two defects. One is neglecting non-local terms of the Biot-Savart integral (localized induction approximation, LIA), and the other is the unphysical mixing procedure in order to sustain the statistically steady state of turbulence. We succeeded in making the statistically steady state without the LIA and the mixing. This state shows the characteristic relation L=γ 2 v 2 ns between the line-length-density L and the counterflow relative velocity v ns with the quantitative agreement of the coefficient γ with some typical observations. We compare our numerical results with the observation of experiment by Paoletti et al., where thermal counterflow was visualized by solid hydrogen particles.

Theory of Decay of Superfluid Turbulence in the Low-Temperature Limit

We review the theory of relaxational kinetics of superfluid turbulence—a tangle of quantized vortex lines—in the limit of very low temperatures when the motion of vortices is conservative. While certain important aspects of the decay kinetics depend on whether the tangle is non-structured, like the one corresponding to the Kibble-Zurek picture, or essentially polarized, like the one that emulates the Richardson-Kolmogorov regime of classical turbulence, there are common fundamental features. In both cases, there exists an asymptotic range in the wavenumber space where the energy flux is supported by the cascade of Kelvin waves (kelvons)—precessing distortions propagating along the vortex filaments. At large enough wavenumbers, the Kelvin-wave cascade is supported by three-kelvon elastic scattering. At zero temperature, the dissipative cutoff of the Kelvin-wave cascade is due to the emission of phonons, in which an elementary process converts two kelvons with almost opposite momenta into one bulk phonon. Along with the standard set of conservation laws, a crucial role in the theory of low-temperature vortex dynamics is played by the fact of integrability of the local induction approximation (LIA) controlled by the parameter Λ=ln (λ/a 0), with λ the characteristic kelvon wavelength and a 0 the vortex core radius. While excluding a straightforward onset of the pure three-kelvon cascade, the integrability of LIA does not plug the cascade because of the natural availability of the kinetic channels associated with vortex line reconnections. We argue that the crossover from Richardson-Kolmogorov to the Kelvin-wave cascade is due to eventual dominance of local induction of a single line over the collective induction of polarized eddies, which causes the breakdown of classical-fluid regime and gives rise to a reconnection-driven inertial range.

Modeling Kelvin Wave Cascades in Superfluid Helium

We study two different types of simplified models for Kelvin wave turbulence on quantized vortex lines in superfluids near zero temperature. Our first model is obtained from a truncated expansion of the Local Induction Approximation (Truncated-LIA) and it is shown to possess the same scalings and the essential behaviour as the full Biot-Savart model, being much simpler than the latter and, therefore, more amenable to theoretical and numerical investigations. The Truncated-LIA model supports six-wave interactions and dual cascades, which are clearly demonstrated via the direct numerical simulation of this model in the present paper. In particular, our simulations confirm presence of the weak turbulence regime and the theoretically predicted spectra for the direct energy cascade and the inverse wave action cascade. The second type of model we study, the Differential Approximation Model (DAM), takes a further drastic simplification by assuming locality of interactions in $k$-space via a differential closure that preserves the main scalings of the Kelvin wave dynamics. DAMs are even more amenable to study and they form a useful tool by providing simple analytical solutions in the cases when extra physical effects are present, e.g. forcing by reconnections, friction dissipation and phonon radiation. We study these models numerically and test their theoretical predictions, in particular the formation of the stationary spectra, and the closeness of the numerics for the higher-order DAM to the analytical predictions for the lower-order DAM .

A Numerical Study of the Self-Similar Solutions of the Schrödinger Map

We present a numerical study of the self-similar solutions of the localized induction approximation of a vortex filament. These self-similar solutions, which are explicitly known, constitute a one-parameter family and develop a singularity in finite time. We study a number of boundary conditions that allow us to reproduce the mechanism of singularity formation. Some related questions are also considered.

Explicit mean-field radius for nearly parallel vortex filaments in statistical equilibrium with applications to deep ocean convection

Deep ocean convection, under appropriate conditions, gives rise to quasi-2D vortex structures with axes parallel to the rotational axis. These vortex structures appear in axisymmetric arrays that have a characteristic radius or size. This size is dependent, not only on competition between vortex interaction and conservation of angular momentum, but on 3D effects that 2D point vortex models leave out. In this article, we propose a hypothesis that as 3D variations become more significant in these arrays, the process of interaction/angular momentum competition gives way to entropy/angular momentum competition and that this shift results in a reversal of the trend of the radius to decrease with increasing kinetic energy. We derive an explicit, closed-form, mean-field expression for the radius using a quasi-2D model for filaments with a local induction approximation (LIA). We validate the formula with Monte Carlo simulations. Both confirm that there is a reversal in the 2D point vortex contraction trend. We conclude that the proposed shift in competition does happen and that this simple LIA model is sufficient to show it.

Momenta of a vortex tangle by structural complexity analysis

A geometric method based on information from structural complexity is presented to calculate linear and angular momenta of a tangle of vortex filaments in Euler flows. For thin filaments under the so-called localized induction approximation the components of linear momentum admit interpretation in terms of projected area. By computing the signed areas of the projected graph diagrams associated with the vortex tangle, we show how to calculate the two momenta of the system by complexity analysis of tangle diagrams. This method represents a novel technique to extract dynamical information of complex systems from geometric and topological properties and provides a potentially useful tool to test the accuracy of numerical methods and investigate scale distribution of fluid dynamical properties of vortex flows.

A new doubly discrete analogue of smoke ring flow and the real time simulation of fluid flow

Modelling incompressible ideal fluids as a finite collection of vortex filaments is important in physics (super-fluidity, models for the onset of turbulence) as well as for numerical algorithms used in computer graphics for the real time simulation of smoke. Here we introduce a time-discrete evolution equation for arbitrary closed polygons in 3-space that is a discretization of the localized induction approximation of filament motion. This discretization shares with its continuum limit the property that it is a completely integrable system. We apply this polygon evolution to a significant improvement of the numerical algorithms used in computer graphics.

Magnetic vortex filament flows

We exhibit a variational approach to study the magnetic flow associated with a Killing magnetic field in dimension 3. In this context, the solutions of the Lorentz force equation are viewed as Kirchhoff elastic rods and conversely. This provides an amazing connection between two apparently unrelated physical models and, in particular, it ties the classical elastic theory with the Hall effect. Then, these magnetic flows can be regarded as vortex filament flows within the localized induction approximation. The Hasimoto transformation can be used to see the magnetic trajectories as solutions of the cubic nonlinear Schrödinger equation showing the solitonic nature of those.

Elementary Vortex Filament Model of the Discrete Nonlinear Schrödinger Equation

The discretization of a vortex filament is studied. A discrete vortex model and a discrete version of the local induction approximation technique (LIA), which guarantees that the vortex filament obeys the discrete nonlinear Schrodinger equation, is proposed.

Vanishing viscosity limit for an incompressible fluid with concentrated vorticity

We study an incompressible fluid with a sharply concentrated vorticity moving in the whole space. First, we review some known results for the inviscid case, which prove, in particular situations, the local induction approximation. Then, we discuss the vanishing viscosity limit and we state a new result valid for a smoke ring with a very large radius.

Analogy between a vortex-jet filament and the Kirchhoff elastic rod

Motion of a vortex filament with axial flow in the core, i.e. a vortex-jet filament, is governed, in the localized induction approximation, by the first two terms of the localized induction hierarchy. It is shown that permanent form of a steadily translating vortex-jet filament is identical with equilibrium form of an inextensible elastic rod of circular cross-section. A variational principle is presented for the statics of an elastic rod with respect to the director field, which manifests the origin of this analogy.

Self-similar solutions for the 1-D Schrödinger map on the hyperbolic plane

In this paper, we will study self-similar solutions for X t = XẋX ss , the equivalent in the Minkowski 3-space $$\mathbb{R}^{2,1}$$ to the localized induction approximation flow, trying to adapt some results given by Gutierrez, Rivas and Vega. We will show the existence of a one-parameter family of smooth solutions developing a corner in finite time. The main difference with respect to the Euclidean case studied by those authors will be the proof of the boundedness of T, e 1 and e 2, the equivalents of T, b and n in $$\mathbb{R}^{2,1}$$ .

Why the Phase Shifts for Solitons on a Vortex Filament Are So Large: A Theoretical Explanation

The phase "jumps" for solitons interacting on a vortex filament, observed in experiments, have been unaccounted for for more than 20 years. Using explicit formulas describing the interaction of two solitons on a thin vortex filament in the localized induction approximation, we show that an appropriate choice of the parameters of the solitons leads to large phase shifts. This result does not depend on the axial flow along the filament.

Self-similar solutions of the localized induction approximation: singularity formation

We investigate the formation of singularities in a self-similar form of regular solutions of the localized induction approximation (also referred as to the binormal flow). This equation appears as an approximation model for the self-induced motion of a vortex filament in an inviscid incompressible fluid. The solutions behave as 3D-logarithmic spirals at infinity.The proofs of the results are strongly based on the existing connection between the binormal flow and certain Schrödinger equations.

Kelvin-wave cascade and decay of superfluid turbulence.

Kelvin waves (kelvons), the distortion waves on vortex lines, play a key part in the relaxation of superfluid turbulence at low temperatures. We present a weak-turbulence theory of kelvons. We show that nontrivial kinetics arises only beyond the local-induction approximation and is governed by three-kelvon collisions; a corresponding kinetic equation is derived. We prove the existence of Kolmogorov cascade and find its spectrum. The qualitative analysis is corroborated by numeric study of the kinetic equation. The application of the results to the theory of superfluid turbulence is discussed.

Leapfrogging vortex rings: Hamiltonian structure, geometric phases and discrete reduction

We present two interesting features of vortex rings in incompressible, Newtonian fluids that involve their Hamiltonian structure. The first feature is for the Hamiltonian model of dynamically interacting thin-cored, coaxial, circular vortex rings described, for example, in the works of Dyson (Philos. Trans. Roy. Soc. London Ser. A 184 (1893) 1041) and Hicks (Proc. Roy. Soc. London Ser. A 102 (1922) 111). For this model, the symplectic reduced space associated with the translational symmetry is constructed. Using this construction, it is shown that for periodic motions on this reduced space, the reconstructed dynamics on the momentum level set can be split into a dynamic phase and a geometric phase. This splitting is done relative to a cotangent bundle connection defined for abelian isotropy symmetry groups. In this setting, the translational motion of leapfrogging vortex pairs is interpreted as the total phase, which has a dynamic and a geometric component. Second, it is shown that if the rings are modeled as coaxial circular filaments, their dynamics and Hamiltonian structure is derivable from a more general Hamiltonian model for N interacting filament rings of arbitrary shape in R 3 , where the mutual interaction is governed by the Biot–Savart law for filaments and the self-interaction is determined by the local induction approximation. The derivation is done using the fixed point set for the action of the group of rotations about the axis of symmetry using methods of discrete reduction theory.

Shape-preserving solutions for quantum vortex motion under localized induction approximation

The motion of a quantum vortex in superfluid helium is considered in the localized induction approximation. In this approximation the instantaneous velocity of quantum vortex is proportional to the local curvature and is parallel to the vector, which is a linear combination of the local binormal and the principal normal to the vortex line. The motion in the direction of the principal normal is specific for a quantum vortex and implies that the vortex shrinks, in contrast to the classical vortex in an ideal fluid. In the present work we deal with two four-parameter classes of shape-preserving solutions (one with increasing and one with decreasing spatial scale) resulting from equations governing the curvature and the torsion. The solutions describe vortex lines whose motion is equivalent to a transformation being a superposition of a homothety and a rotation. In a particular case when the transformation is a pure homothety, we find analytic solutions for the curvature and the torsion. In the general case, when the transformation is a superposition of a nontrivial rotation and a homothety, the asymptotics of the solutions of the first class are given explicitly and are related to the parameters characterizing the transformation. It is found that the solutions of the second class (with decreasing scale) either have asymptotes or are periodic (when the transformation is a pure homothety) or else exhibit chaotic behavior.