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Two scale modeling of superfluid turbulence Tomasz Lipniacki He II: -- normal component, -- superfluid component, -- superfluid vortices. Relative proportion of normal fluid and superfluid as a function of temperature. vortex line core diameter ~ 1Å Solutions of single vortex motion in LIA Fenomenological models of anisotropic turbulence

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Localized Induction Approximation c - curvature - torsion

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Ideal vortex Quasi static solutions: Totally integrable, equivalent to non-linear Schrödinger equation. Countable (infinite) family of invariants:

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Figure1 : 2 Hasimoto soliton on ideal vortex (1972)

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Kida 1981 General solutions of in terms of elliptic integrals including shapes such as circular vortex ring helicoidal and toroidal filament plane sinusoidal filament Euler’s elastica Hasimoto soliton

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Quasi static solutions for quantum vortices ? Quantum vortex shrinks:

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Quasi-static solutions Lipniacki JFM2003 Frenet Seret equations

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In the case of pure translation we get analytic solution where For Solution has no self-crossings

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Quasi-static solution: pure rotation Vortex asymptotically wraps over cones with opening angle for in cylindrical coordinates where

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Quasi-static solution: general case For vortex wraps over paraboloids

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Results in general case I. 4-parametric class of solutions, determined by initial condition II. Each solution corresponds to a specific isometric transformation (t) related analytically to initial condition. III. The asymptotic for is related analytically via transformation (t) to initial condition.

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Shape-preserving solutions Lipniacki, Phys. Fluids Equation is invariant under the transformation If the initial condition is scale-free, than the solution is self-similar In general scale-free curve is a sum of two logarithmic spirals on coaxial cones; there is four parametric class of such curves.

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Shape-preserving solutions solutions with decreasing scale

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Analytic result: shape-preserving solution In the case when transformation is a pure homothety we get analytic solution in implicit form:

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Shape preserving solution: general case Logarithmic spirals on cones

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Results in general case I. 4-parametric class of solutions, determined by initial condition II. Each solution corresponds to a specific similarity transformation III. The asymptotics for is given the initial condition of the original problem (with time).

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Self-similar solution for vortex in Navier-Stokes? N-S is invariant to the same transformation The friction coefficient corresponds to

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Wing tip vortices

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-- Euler equation for superfluid component -- Navier-Stokes equation for normal component Macroscopic description Coupled by mutual friction force where and L is superfluid vortex line density.

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Microscopic description of tangle superfluid vortices Assuming that quantum tangle is “close” to statistical equilibrium derive equations for quantum line length density L and anisotropy parameters of the tangle. Aim Two cases I – rapidly changing counterflow, but uniform in space II – slowly changing counterflow, not uniform in space Statistical equilibrium?

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Model of quantum tangle “Particles” = segments of vortex line of length equal to characteristic radius of curvature. Particles are characterized by their tangent t, normal n, and binormal b vectors. Velocity of each particle is proportional to its binormal (+collective motion). Interactions of particles = reconnections.

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Reconnections 1. Lines lost their identity: two line segments are replaced by two new line segments 2. Introduce new curvature to the system 3. Remove line-length

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Motion of vortex line in the presence of counterflow Evolution of its line-length is

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Evolution of line length density where I average curvature average curvature squared Average binormal vector (normalized)

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Rapidly changing counterflow Generation term: polarization of a tangle by counterflow Degradation term: relaxation due to reconnections

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Generation term: polarization of a tangle by counterflow evolution of vortex ring of radius R and orientation in uniform counterflow unit binormal Statistical equilibrium assumption: distribution of f(b) is the most probable distribution giving right I i.e.

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Degradation term: relaxation due to reconnections Reconnections produce curvature but not net binormal Total curvature of the tangle Reconnection frequency Relaxation time:

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Finally Lipniacki PRB 2001

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Model prediction: there exists a critical frequency of counterflow above which tangle may not be sustained

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Quantum turbulence with high net macroscopic vorticity We assume now Letdenote anisotropy of the tangent Vinen type equation for turbulence with net macroscopic vorticity Vinen-Schwarz equation

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Drift velocity where

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Mutual friction force drag force where finally

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Helium II dynamical equations Lipniacki EJM 2006

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Specific cases: stationary rotating turbulence Pure heat driven turbulence Pure rotation „Sum” Fast rotation Slow rotation

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Plane Couette flow V- normal velocity U-superfluid velocity q- anisotropy l-line length density

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“Hypothetical” vortex configuration

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Summary and Conclusions Self-similar solutions of quantum vortex motion in LIA Dynamic of a quantum tangle in rapidly varying counterflow Dynamic of He II in quasi laminar case. In small scale quantum turbulence, the anisotropy of the tangle in addition to its density L is key to analyze dynamic of both components.

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Non-linear Schrodinger equation (Gross, Pitaevskii ) Madelung Transformation superfluid velocity superfluid density For

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Modified Euler equation With pressure Quantum stress

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Mixed turbulence Kolgomorov spectrum for each fluid - energy flux per unit mass

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Two scale modeling of superfluid turbulence Tomasz Lipniacki He II: -- normal component, -- superfluid component, -- superfluid vortices. Relative proportion of normal fluid and superfluid as a function of temperature. vortex line

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