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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. Solutions of single vortex motion in LIA Fenomenological models of anisotropic turbulence vortex line core diameter ~ 1Å

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**Localized Induction Approximation**

c - curvature - torsion

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**Quasi static solutions:**

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

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**Hasimoto soliton on ideal vortex (1972)**

: F i g u r e 1 2

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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**

For where 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 G(t) related analytically to initial condition. III. The asymptotic for is related analytically via transformation G(t) to initial condition.

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**Shape-preserving solutions**

Lipniacki, Phys. Fluids. 2003 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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**Macroscopic description**

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

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**Microscopic description of tangle superfluid vortices**

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

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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 average curvature average curvature squared I 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**

Vinen-Schwarz equation We assume now Let denote anisotropy of the tangent Vinen type equation for turbulence with net macroscopic vorticity

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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)**

For Madelung Transformation superfluid velocity superfluid density

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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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Monroe L. Weber-Shirk S chool of Civil and Environmental Engineering Basic Governing Differential Equations CEE 331 June 12, 2015.

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