Presentation on theme: "Two scale modeling of superfluid turbulence Tomasz Lipniacki He II: -- normal component, -- superfluid component, -- superfluid vortices. Relative proportion."— Presentation transcript:
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
Localized Induction Approximation c - curvature - torsion
Ideal vortex Quasi static solutions: Totally integrable, equivalent to non-linear Schrödinger equation. Countable (infinite) family of invariants:
Figure1 : 2 Hasimoto soliton on ideal vortex (1972)
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
Quasi static solutions for quantum vortices ? Quantum vortex shrinks:
In the case of pure translation we get analytic solution where For Solution has no self-crossings
Quasi-static solution: pure rotation Vortex asymptotically wraps over cones with opening angle for in cylindrical coordinates where
Quasi-static solution: general case For vortex wraps over paraboloids
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.
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.
Shape-preserving solutions solutions with decreasing scale
Analytic result: shape-preserving solution In the case when transformation is a pure homothety we get analytic solution in implicit form:
Shape preserving solution: general case Logarithmic spirals on cones
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).
Self-similar solution for vortex in Navier-Stokes? N-S is invariant to the same transformation The friction coefficient corresponds to
Wing tip vortices
-- 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.
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?
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.
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
Motion of vortex line in the presence of counterflow Evolution of its line-length is
Evolution of line length density where I average curvature average curvature squared Average binormal vector (normalized)
Rapidly changing counterflow Generation term: polarization of a tangle by counterflow Degradation term: relaxation due to reconnections
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.
Degradation term: relaxation due to reconnections Reconnections produce curvature but not net binormal Total curvature of the tangle Reconnection frequency Relaxation time:
Finally Lipniacki PRB 2001
Model prediction: there exists a critical frequency of counterflow above which tangle may not be sustained
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
Drift velocity where
Mutual friction force drag force where finally
Helium II dynamical equations Lipniacki EJM 2006
Specific cases: stationary rotating turbulence Pure heat driven turbulence Pure rotation „Sum” Fast rotation Slow rotation
Plane Couette flow V- normal velocity U-superfluid velocity q- anisotropy l-line length density
“Hypothetical” vortex configuration
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.
Non-linear Schrodinger equation (Gross, Pitaevskii ) Madelung Transformation superfluid velocity superfluid density For
Modified Euler equation With pressure Quantum stress
Mixed turbulence Kolgomorov spectrum for each fluid - energy flux per unit mass
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