The Kelvin- wave cascade in the vortex filament model: Controversy over? Jason Laurie Weizmann of Science, Israel

Transcription

1 The Kelvin- wave cascade in the vortex filament model: Controversy over? Jason Laurie Weizmann of Science, Israel In with: Andrew Baggaley (Glasgow, UK) 20 September 2013, Université Paris Diderot, France

2 Classical vs quantum turbulence Classical theory of turbulence Kolmogorov Energy flows to small scales removed by viscous Constant flux of energy through an range of scales Energy spectrum E = C 2/3 5/3 Quantum Turbulence Super- cooled Bose gases: e.g. helium 4 below 2.17K Consists of normal and superfluid components Bul superfluid flow is irrota@onal Vor@city confined to 1D topological defects Quan@zed vortex lines Vor@city discre@zed in units of apple = h/m 4 QT consists of a tangle of quan@zed vor@ces K41 energy spectrum s@ll observed at large scales Polarized vortex bundles mimic classical vortex tubes Baggaley, JL, Barenghi, PRL, 109, , (2012)

3 Quantum turbulence at small scales Finite temperature quantum turbulence Interloced normal and superfluid components via mutual Normal fluid dissipates energy through viscosity At zero temperature the normal fluid vanishes What happens to energy at zero temperature? Vortex bundles only mae sense at the scales larger than the inter- vortex distance can reconnect Kelvin- waves Kelvin- waves propagate along vortex lines with dispersion = apple2 ln (a)+c 4 Hypothesis: Wealy nonlinear Kelvin- wave interac@ons transfer energy to even smaller scales Energy dissipa@on mechanism High frequency Kelvin- waves excite phonons that dissipate energy in terms of heat How do Kelvin- waves interact?

4 Kelvin- wave turbulence theory of vortex lines are governed by the Biot- Savart law ṡ = apple 4 Hamiltonian descrip@on Svistunov, PRB, 52, 3647, (1995) Wea nonlinearity expansion I L r s r s 3 dr Consider a single, periodic in z, straight vortex line aligned along x = y =0 Parametrise 2D perturba@ons by s =[x(z),y(z),z] Define a complex canonical variable: w(z) = H Z w H = apple2 1 + Re [w 0 (z 1 )w 0 (z 2 )] dz 1 dz 2 4 q(z 1 z 2 ) 2 + w(z 1 ) w(z 2 ) 2 H = H 2 + H 4 + H 6 + O(" 8 ) " = w(z 1) w(z 2 ) z 1 z 2 1

5 ṅ = 6 The six- wave ine@c equa@on There are no nontrivial four- wave resonances Leading order are resonant six- wave interac@ons Z W,1,2 3,4,5 2 n n 1 n 2 n 3 n 4 n 5 n 1 + n n2 1 n3 1 n4 1 n g ( ) (! +! 1 +! 2! 3! 4! 5 ) d 1 d 2 d 3 d 4 d 5 Interac@on Kernel: W,1,2 3,4,5 / G Kozi, Svistunov, PRL, 92, , (2004) Non- equilibrium Kolmogorov- Zaharov spectra 1. Constant flux of energy: n / apple 2/5 1/5 17/5 Kozi- Svistunov spectrum 2. Constant flux of wave ac@on: n / apple 1/5 1/5 1/5 3

6 The four- wave Locality of wave turbulence theory Kolmogorov- Zaharov spectra only exist if wave are local Collision integral must be convergent Collision integral shown to diverge in the limit of two long Kelvin- waves JL, L vov, Rudeno, Nazareno, PRB, 81, , (2010) Local four- wave ine@c equa@on L vov, Nazareno, JETP, 91, 428, (2010) ṅ = 12 Z n V 1,2,3 2 n 1 n 2 n 3 n n 1 n1 1 n2 1 n3 1 ( ) (!! 1! 2! 3 ) +3 V,2,3 1 2 n 1 n 2 n 3 n n 1 1 n 1 n 1 2 n 1 3 ( ) (! 1!! 2! 3 ) Interac@on Kernel: V 1,2,3 / = 1 apple New scaling for constant energy flux Kolmogorov- Zaharov solu@on Z 2 n d o d 1 d 2 d 3 n / 1/3 2/3 11/3 L vov- Nazareno spectrum

7 History of Kelvin- wave Vinen, Tsubota, Mitani, PRL, 91, 13501, (2003) Model Biot- Savart law (VFM) Forcing type Excite vortex line at specific Kelvin- wave frequency type Smoothing of highest harmonic 3

8 History of Kelvin- wave Kozi, Svistunov, PRL, 94, , (2005) Model Biot- Savart Hamiltonian with scale- scheme Forcing type None (decaying) type Periodically set high harmonics to zero

9 History of Kelvin- wave Boué, Dasgupta, JL, L vov, Nazareno, Procaccia, PRB, 84, , (2011) Model Local- nonlinear equa@on (nonlocal limit of Biot- Savart law) Forcing type Addi@ve forcing at large- scales Dissipa@on type Large- scale fric@on and hyper- viscosity A = 11/3 N ~ ~ ~ ~ ~ ~ A = 11/3 N = 3.2 ~ ~ N = -17/ ~ Analy@cal and numerical measurement of spectrum pre- factor C LN =0.304 n = C LN 1/3 2/3 11/3

10 History of Kelvin- wave Krstulović, PRE, 86, , (2012) Model Gross- Pitaevsii Forcing type None (decaying) type None but contains phonon emission b) RUN I RUN II RUN III RUN IV n ± ±

11 Why another None have been universally accepted in the community Because debate Model Decaying Poor and/or What is different? 1. Full Biot- Savart without any nonlocal 2. Localised forcing and range) 3. True non- equilibrium steady state (forced and dissipated) 4. to between predicted spectra

12 Our vortex filament model setup ṡ i = apple 4 ln p`i`i+1 a! s 0 i s 00 i + apple 4 I r s i L 0 r s i 3 dr + F s i+1 s i Local contribu@on (LIA) Nonlocal contribu@on (Biot- Savart) s i 1 s i 2 Ini@al straight vortex line periodic along z Third- order Runge- stepping scheme Re- mesh vortex line onto uniform grid aner step Allows us to exponen@ally filter high and low Fourier harmonics Localized (in Fourier space) addi@ve forcing F =[Re(f), Im(f), 0], f = X A exp (iz + i ) 9apple apple11

13 Results β =11/3 β =17/5 β =3 n β L w (z) t z

14 Conclusions New of wealy Kelvin- waves in the vortex filament model Biot- Savart without Non- equilibrium steady state achieved Local (in Fourier space) forcing and range) Evidence for nonlocal theory Clear between predicted wave spectra Bemer agreement with L vov- Nazareno spectrum Spectrum not everything Measurement of energy flux Es@mate numerical pre- factor of spectrum (,!) - plot Strong wave turbulence

15 Than you