Mesoscopic physics of finite temperature superfluid turbulence in He4: models, solutions and prospects
Demosthenes Kivotides University of Strathclyde Glasgow
Demosthenes Kivotides Mesoscopic physics of finite temperature superfluid turbulence Prologue
Quantize the Schroedinger equation to obtain a QFT of Galilean-relativistic, spin-0, bosonic, many-particle systems Corresponding field-theoretic Hamiltonian:
2 Z Hˆ = ~ d3x∇ψˆ†(x) · ∇ψˆ(x) + 2m 1 Z d3xd3x0ψˆ†(x)ψˆ†(x0)V (x, x0)ψˆ(x0)ψˆ(x) 2
Quantum Schroedinger field quanta [look also at soliton-like solutions of LSE (Zamboni-Rached, Becami JMP, 2012)] interact with each other via potential V LSE (V = 0) encodes matter inertia and a stress field (quantum potential)
Demosthenes Kivotides Mesoscopic physics of finite temperature superfluid turbulence Prologue
Motion under the quantum potential results in highly nontrivial pathlines Quantum potential effects topological defect formation in the superfluid, which are velocity-field vortices V not required for topological defect formation, but has important fluid dynamic effects, since it is the origin of fluid pressure Strongly-interacting liquid: Slater-Kirkwood potential (R = |x − x0|): V (R) = [770exp(−4.60R) − (1.49/R6)] × 10−12 erg (R in A˚).
Demosthenes Kivotides Mesoscopic physics of finite temperature superfluid turbulence Prologue
Breaking the rigid U(1) symmetry of quantum Schroedinger field leads to superfluidity (Nambu-Goldstone modes) The standard conservation principles lead to normal-fluid hydrodynamics The purpose of the mesoscopic model (MM) is to encode the dynamics of discrete topological defects in the SF, and resolve ALL hydrodynamic motions in the NF Turbulence as a generic name for labeling far-out of equilibrium, chaotic states in nonlinear theories Which are these theories for the superfluid case?
Demosthenes Kivotides Mesoscopic physics of finite temperature superfluid turbulence Model of Physics
Incompressible turbulence in the superfluid is modeled via Vortex Dynamics; in the normal-fluid via the Navier-Stokes equation
Vortex Dynamics of SF vortices include mutual-friction fmf (quantum fluctuations effects), Magnus fM , and reconnection fR effects:
fmf + fM + fR = 0
Inertia is neglected and “reconnection force” induces topological change via pointwise jumps (first introduced in algorithmic calculations by K. Schwarz) Pointwise jumps are just an effective model of quantum-potential action
Demosthenes Kivotides Mesoscopic physics of finite temperature superfluid turbulence Model of Physics
Mutual-friction force fD has (i) a Hall-Vinen (viscous-drag) component (from Quantum Statistical Physics: Thompson and Stamp, PRL 2012) A creeping flow hydrodynamics analysis over cylindical objects gives (Kivotides, PRF 2018)
0 0 0 0 fD = D⊥Xv × (Xv × vvn) + DkXv (Xv · vvn)
0 (ii) An Iordanskii (lift) component: fL = ρnκXv × vvn Here, vvn ≡ X˙ v − Vn, where X˙ v is the vortex velocity, and Vn is the asymptotic normal-fluid velocity at the vortex position
Demosthenes Kivotides Mesoscopic physics of finite temperature superfluid turbulence Model of Physics
The normal-fluid obeys Navier-Stokes dynamics for momentum (Kivotides PRF 2018): ∂Vn(x, t) p Vn · Vn + ∇ + − Vn × (∇ × Vn) − ∂t ρn + ρs 2 Z 2 0 ˙ ν∇ Vn − κ d|XL| [XL × (Vn − XL)]δ(x − XL) − L Z 0 0 ˙ νc d|XL|{XL × [XL × (Vn − XL)]}δ(x − XL) = 0, L
and mass conservation: ∇ · Vn = 0. All following results are direct numerical & algorithmic solutions of the mesoscopic model of physics All results refer to Homogeneous Isotropic (i.e., Grid) Turbulence
Demosthenes Kivotides Mesoscopic physics of finite temperature superfluid turbulence Key results
Two scenarios in homogeneous, isotropic, superfluid turbulence In both cases, MM depicts how mutual-friction tends to equilibrate (at each length scale) the energy content of the two fluids No (dynamically important) superfluid vortex-bundle formation mimicking NF vorticity structure (Kivotides, JFM 2008, POF 2014) Flow instabilities are indicated as key dynamical factors in superfluid physics (what else?)
Demosthenes Kivotides Mesoscopic physics of finite temperature superfluid turbulence Key results
1st case: SF more energetic than NF Mutual-friction forcing of NF from SF excites creeping-flow normal-fluid vorticity (Kivotides, PRF 2018; Kivotides, Barenghi, Samuels, Science 2001)
Demosthenes Kivotides Mesoscopic physics of finite temperature superfluid turbulence Key results
The energy transfer from the SF to the NF smooths small-scale vortex waves on the SF vortices (Samuels and Kivotides, PRL 1999, Kivotides, PLA 2005) Small scale vortex wave damping enforces MM consistency (limiting it to mesoscopic scales) No SF vortex organization Exotic type of NF creeping flow with complicated vorticity field (turbulence without inertia)
Demosthenes Kivotides Mesoscopic physics of finite temperature superfluid turbulence Key results
2nd case: NF more energetic than SF There are two limiting NF vortex geometry cases: curved and quasi-straight vortices Curved NF vortices There are two mechanisms of energy transfer from NF vortices to SF vortex-tangle: Both processes indicate mutual-friction driven inverseSF cascades Contrast with inertia-pressure dominated classical 3D turbulence
Demosthenes Kivotides Mesoscopic physics of finite temperature superfluid turbulence Key results
Process 1: SF rings much smaller than NF rings grow towards the NF vortex scale (no morphology match) Energy transfer from NF Hopf-Link (Re = 1000)
Demosthenes Kivotides Mesoscopic physics of finite temperature superfluid turbulence Key results
Process 2: SF rings of same order or larger size than NF rings develop small-scale, axial NF-flow, instabilities which grow in size to match the scale of the NF structures
Demosthenes Kivotides Mesoscopic physics of finite temperature superfluid turbulence Key results
Remarkably, MM forbids SF vortices to grow beyond energy-transferring NF-vortex sizes It is not about vortex sizes, but about energy levels! (superfluid velocity) No strong correlation between the morphologies of induced SF and NF vorticity structures (Kivotides, PRF 2018)
Demosthenes Kivotides Mesoscopic physics of finite temperature superfluid turbulence Key results
Quasi-straight NF vortices Quasi-straight NF-filaments (Kivotides, PLA 2018) induce a SF complex tangle propagating in the quasi-potential flow between the Rankine-like NF vortex cores (side view)
Demosthenes Kivotides Mesoscopic physics of finite temperature superfluid turbulence Key results
NF vortex rotation effects (top view)
Demosthenes Kivotides Mesoscopic physics of finite temperature superfluid turbulence Key results
There is weak SF vortex-bundle formation within the NF vortex core Two reasons: (a) NF solid-body rotation limited in space, (b) instability The process of NF and SF vorticities matching excites instabilities that destroy the tube
Demosthenes Kivotides Mesoscopic physics of finite temperature superfluid turbulence Numerical & Algorithmic Solutions
NF small scale vorticity damping by mutual-friction force in “cartoon” calculation
Demosthenes Kivotides Mesoscopic physics of finite temperature superfluid turbulence Numerical & Algorithmic Solutions
NF small scale vorticity damping by mutual-friction force in fully developed NF turbulence (compare lack of bundles with Kivotides PRL 2006 scenario)
Demosthenes Kivotides Mesoscopic physics of finite temperature superfluid turbulence Key results
Mutual-friction effects from SF to NF (even when NF more energetic) At sub-Kolmogorov energy scales the NF velocity is small enough for a SF vortex to transfer energy to the NF and create a k−2.2 dissipative NF flow spectrum Effect more pronounced during turbulence decay (Kivotides, JLTP 2015) Hence, large scale, energetic NF eddies create large scale SF vortices, which slowly loose their acquired energy by exciting creeping-flow NF vorticity and creating ”superviscous phenomena” (2% increase in NF viscosity values [Kivotides, JFM 2008]) Organized large scale SF vortex bundles could (in principle) excite large-scale/inertial NF vorticity fields, but SF vortex-bundles do not appear in the solutions
Demosthenes Kivotides Mesoscopic physics of finite temperature superfluid turbulence Mesoscopics to Macroscopics
These suggest the following macroscopic model for mutual-friction action (Kivotides, PLA 2018):
t t fMF = H[ωs − ωs ]fHVBK + (1 − H[ωs − ωs ]) t {H[ωn − ωn][h(Kn)fGM + (1 − h(Kn))fHVBK] + t (1 − H[ωn − ωn])fGM}. “Constitutive modelling” depends on SF vortex intensity and NF vorticity curvature and intensity
Demosthenes Kivotides Mesoscopic physics of finite temperature superfluid turbulence Mesocopics to Macroscopics
Flow points with high ωs values correspond to a structured vortex tangle, and their mutual-friction needs to be handled via fHVBK
In lower ωs regions, the determining factor is ωn:
At flow points with small ωn values, the tangle polarization is negligible, and fGM needs to be employed
At flow points with high ωn values, a mild tangle polarization is expected that depends on the curvature of normal-fluid vorticity-lines, and a mixed model is needed. t max Indicative parameter values are β ≈ 0.95, ωn ≈ 0.8 ωn , and t t ωs ≈ 0.05 ωn.
Demosthenes Kivotides Mesoscopic physics of finite temperature superfluid turbulence Prospects
Mesoscopic SF modelling allows the analysis of SF as a complex fluid with microstructure Exciting era ahead for SF turbulence physics experiments in grid, counterflow and pressure-driven internal flows (what about external?) Crucial role to be played by experiments at the mesoscopic level, i.e., measuring discrete SF vortex positions and Navier-Stokes NF velocities What is required? Investment in new methods for measuring local (in Navier-Stokes sense) vortex and flow quantities Superfluid turbulence is not an iconic nonlinearity problem like NS turbulence; yet, it has a complex phenomenology and should be studied for its own right!
Demosthenes Kivotides Mesoscopic physics of finite temperature superfluid turbulence Ευχαριστώ για την προσοχή σας! Thank you for your attention!
Any questions?
Demosthenes Kivotides Mesoscopic physics of finite temperature superfluid turbulence