Talk at Tsinghua University, Beijing, 27 June 2016

Talk at Tsinghua University, Beijing, 4th International Conference on Nonlinear Waves -- Theory and Applications (27 June 2016).

Transcript

1. Precession resonance mechanism in deep-water surface gravity waves Principal Investigator: Miguel D. Bustamante Team: Brenda Quinn (IRC), Dan Lucas (SFI), Breno Raphaldini (CNPq) PhD students: Brendan Murray (SFI), Shane Walsh (IRC) Complex and Adaptive Systems Laboratory School of Mathematics and Statistics University College Dublin IRELAND June 27th 2016 M D Bustamante (UCD) Nonlinear Waves – 4th Int Conf June 27th 2016 1 / 19
2. Many physical systems consist of nonlinearly interacting oscillations or waves M D Bustamante (UCD) Nonlinear Waves – 4th Int Conf June 27th 2016 2 / 19
3. Many physical systems consist of nonlinearly interacting oscillations or waves Systems: Atmospheric Planetary Waves Ocean Gravity Waves Nonlinear Optics M D Bustamante (UCD) Nonlinear Waves – 4th Int Conf June 27th 2016 2 / 19
4. Many physical systems consist of nonlinearly interacting oscillations or waves Systems: Atmospheric Planetary Waves Ocean Gravity Waves Nonlinear Optics Features: Extreme Events Strong Nonlinear Energy Exchanges Chaos & Turbulence M D Bustamante (UCD) Nonlinear Waves – 4th Int Conf June 27th 2016 2 / 19
5. Nonlinear wave interactions: What is really going on? We have some answers... 1 1 Miguel D. Bustamante, Brenda Quinn, and Dan Lucas, Robust energy transfer mechanism via precession resonance in nonlinear turbulent wave systems, Phys. Rev. Lett. 113 (2014), 084502 M D Bustamante (UCD) Nonlinear Waves – 4th Int Conf June 27th 2016 3 / 19
6. Nonlinear wave interactions: What is really going on? We have some answers... 1 Initial Energy in a set of modes (blob) 1 Miguel D. Bustamante, Brenda Quinn, and Dan Lucas, Robust energy transfer mechanism via precession resonance in nonlinear turbulent wave systems, Phys. Rev. Lett. 113 (2014), 084502 M D Bustamante (UCD) Nonlinear Waves – 4th Int Conf June 27th 2016 3 / 19
7. Nonlinear wave interactions: What is really going on? We have some answers... 1 Initial Energy in a set of modes (blob) Question: Where is this energy more likely to go? 1 Miguel D. Bustamante, Brenda Quinn, and Dan Lucas, Robust energy transfer mechanism via precession resonance in nonlinear turbulent wave systems, Phys. Rev. Lett. 113 (2014), 084502 M D Bustamante (UCD) Nonlinear Waves – 4th Int Conf June 27th 2016 3 / 19
8. Nonlinear wave interactions: What is really going on? We have some answers... 1 Initial Energy in a set of modes (blob) Question: Where is this energy more likely to go? Answer: To precession-resonant triads 1 Miguel D. Bustamante, Brenda Quinn, and Dan Lucas, Robust energy transfer mechanism via precession resonance in nonlinear turbulent wave systems, Phys. Rev. Lett. 113 (2014), 084502 M D Bustamante (UCD) Nonlinear Waves – 4th Int Conf June 27th 2016 3 / 19
9. What is Precession Resonance? (1/3) M D Bustamante (UCD) Nonlinear Waves – 4th Int Conf June 27th 2016 4 / 19
10. What is Precession Resonance? (1/3) Consider Charney-Hasegawa-Mima (CHM) equation, a PDE modelling atmosphere and plasmas: (r2 F ) @ @ t + @ @ x + @ @ x @r2 @ y @ @ y @r2 @ x = 0. M D Bustamante (UCD) Nonlinear Waves – 4th Int Conf June 27th 2016 4 / 19
11. What is Precession Resonance? (1/3) Consider Charney-Hasegawa-Mima (CHM) equation, a PDE modelling atmosphere and plasmas: (r2 F ) @ @ t + @ @ x + @ @ x @r2 @ y @ @ y @r2 @ x = 0. (x, t )(2 R) is the streamfunction. Parameters: F 0, (any sign). Periodic boundary conditions: x 2 [0, 2⇡) ⇥ [0, 2⇡). M D Bustamante (UCD) Nonlinear Waves – 4th Int Conf June 27th 2016 4 / 19
12. What is Precession Resonance? (1/3) Consider Charney-Hasegawa-Mima (CHM) equation, a PDE modelling atmosphere and plasmas: (r2 F ) @ @ t + @ @ x + @ @ x @r2 @ y @ @ y @r2 @ x = 0. (x, t )(2 R) is the streamfunction. Parameters: F 0, (any sign). Periodic boundary conditions: x 2 [0, 2⇡) ⇥ [0, 2⇡). Fourier representation: (x, t ) = P k2Z 2 Ak( t )eik·x+ c. c. Wavevector: k = ( kx , ky ) ˙ Ak + i !k Ak = 1 2 X k 1 ,k 2 2Z 2 Z k k 1 k 2 k 1+ k 2 k Ak 1 Ak 2 . M D Bustamante (UCD) Nonlinear Waves – 4th Int Conf June 27th 2016 4 / 19
13. What is Precession Resonance? (2/3) ˙ Ak + i !k Ak = 1 2 X k 1 ,k 2 2Z 2 Z k k 1 k 2 k 1+ k 2 k Ak 1 Ak 2 . M D Bustamante (UCD) Nonlinear Waves – 4th Int Conf June 27th 2016 5 / 19
14. What is Precession Resonance? (2/3) ˙ Ak + i !k Ak = 1 2 X k 1 ,k 2 2Z 2 Z k k 1 k 2 k 1+ k 2 k Ak 1 Ak 2 . Amplitude-phase representation: Ak = p nk exp( i k). The modes Ak interact in triads: k 1 + k 2 = k 3 ) Dynamical DOF are 'k 3 k 1 k 2 ⌘ k 1 + k 2 k 3 & nk . M D Bustamante (UCD) Nonlinear Waves – 4th Int Conf June 27th 2016 5 / 19
15. What is Precession Resonance? (2/3) ˙ Ak + i !k Ak = 1 2 X k 1 ,k 2 2Z 2 Z k k 1 k 2 k 1+ k 2 k Ak 1 Ak 2 . Amplitude-phase representation: Ak = p nk exp( i k). The modes Ak interact in triads: k 1 + k 2 = k 3 ) Dynamical DOF are 'k 3 k 1 k 2 ⌘ k 1 + k 2 k 3 & nk . ˙ nk = X k 1 ,k 2 Z k k 1 k 2 k k 1 k 2 ( nk nk 1 nk 2 )1 2 cos 'k k 1 k 2 , . . . and another nonlinear equation for ˙ 'k 3 k 1 k 2 (too long, doesn’t fit). M D Bustamante (UCD) Nonlinear Waves – 4th Int Conf June 27th 2016 5 / 19
16. What is Precession Resonance? (2/3) ˙ Ak + i !k Ak = 1 2 X k 1 ,k 2 2Z 2 Z k k 1 k 2 k 1+ k 2 k Ak 1 Ak 2 . Amplitude-phase representation: Ak = p nk exp( i k). The modes Ak interact in triads: k 1 + k 2 = k 3 ) Dynamical DOF are 'k 3 k 1 k 2 ⌘ k 1 + k 2 k 3 & nk . ˙ nk = X k 1 ,k 2 Z k k 1 k 2 k k 1 k 2 ( nk nk 1 nk 2 )1 2 cos 'k k 1 k 2 , . . . and another nonlinear equation for ˙ 'k 3 k 1 k 2 (too long, doesn’t fit). • Time series of ( nk 3 nk 1 nk 2 )1 2 ) : salient frequency peak in spectrum. • Precession frequency: ⌦k 3 k 1 k 2 ⌘ D ˙ 'k 3 k 1 k 2 E (time average). M D Bustamante (UCD) Nonlinear Waves – 4th Int Conf June 27th 2016 5 / 19
17. What is Precession Resonance? (2/3) ˙ Ak + i !k Ak = 1 2 X k 1 ,k 2 2Z 2 Z k k 1 k 2 k 1+ k 2 k Ak 1 Ak 2 . Amplitude-phase representation: Ak = p nk exp( i k). The modes Ak interact in triads: k 1 + k 2 = k 3 ) Dynamical DOF are 'k 3 k 1 k 2 ⌘ k 1 + k 2 k 3 & nk . ˙ nk = X k 1 ,k 2 Z k k 1 k 2 k k 1 k 2 ( nk nk 1 nk 2 )1 2 cos 'k k 1 k 2 , . . . and another nonlinear equation for ˙ 'k 3 k 1 k 2 (too long, doesn’t fit). • Time series of ( nk 3 nk 1 nk 2 )1 2 ) : salient frequency peak in spectrum. • Precession frequency: ⌦k 3 k 1 k 2 ⌘ D ˙ 'k 3 k 1 k 2 E (time average). When these two resonate: ⌦k 3 k 1 k 2 = ) “Unbounded” growth. PRECESSION RESONANCE M D Bustamante (UCD) Nonlinear Waves – 4th Int Conf June 27th 2016 5 / 19
18. (The equation that didn’t fit) ˙ 'k 3 k 1 k 2 = !k 3 !k 1 !k 2 + sin 'k 3 k 1 k 2 ( nk 3 nk 1 nk 2 )1 2 " Z k 1 k 2 k 3 nk 1 + Z k 2 k 3 k 1 nk 2 Z k 3 k 1 k 2 nk 3 # + NNTTk 3 k 1 k 2 . M D Bustamante (UCD) Nonlinear Waves – 4th Int Conf June 27th 2016 6 / 19
19. What is Precession Resonance? (3/3) Simplest Example: Initial condition on a ‘Source’ triad. Frequency known. Compute numerically precession frequency of ‘Target’ triad: ⌦k 4 k 2 k 3 . M D Bustamante (UCD) Nonlinear Waves – 4th Int Conf June 27th 2016 7 / 19
20. What is Precession Resonance? (3/3) Simplest Example: Initial condition on a ‘Source’ triad. Frequency known. Compute numerically precession frequency of ‘Target’ triad: ⌦k 4 k 2 k 3 . 10 14 10 12 10 10 10 8 1 ↵ 2 ↵ 1 5 ↵ 0 20 E 4 E ↵ 0 0.4 0.8 1.2 1.6 ⌦ k 4 k 2 k 3 p E p E 2 p E ⌦k 4 k 2 k 3 = p p 2 Z (Precession Resonance) Compare with e ciency of Energy Transfers to mode k 4 ↵ : initial condition re-scaling parameter. Bustamante, Quinn & Lucas, (2014) Phys. Rev. Lett. 113 , 084502 M D Bustamante (UCD) Nonlinear Waves – 4th Int Conf June 27th 2016 7 / 19
21. Precession Resonance works in Full PDE • Precession resonance in full PDE follows from persistence theorem of invariant manifolds (Fenichel 1971) • ⌦k 4 k 2 k 3 , ⌦k 5 k 3 k 1 , ⌦k 2 k 6 k 1 Competition ) Selection Mechanism: ⌦k c k a k b = p • Full PDE model N = 1282 Pseudospectral, 2/3-rd dealiased: 12 million triads ) Use ‘bins’ instead. M D Bustamante (UCD) Nonlinear Waves – 4th Int Conf June 27th 2016 8 / 19
22. Results for General Initial Conditions (1/2) Large-scale initial condition: nk = ↵ ⇥ 0.0321|k| 2 exp ( |k|/5) for |k|  8 Total enstrophy: E = 0.156↵ Initial phases k: Uniformly distributed on [0, 2⇡) M D Bustamante (UCD) Nonlinear Waves – 4th Int Conf June 27th 2016 9 / 19
23. Results for General Initial Conditions (1/2) Large-scale initial condition: nk = ↵ ⇥ 0.0321|k| 2 exp ( |k|/5) for |k|  8 Total enstrophy: E = 0.156↵ Initial phases k: Uniformly distributed on [0, 2⇡) DNS: pseudospectral method N = 1282, from t = 0 to t = 800/ p E Cascades: Partition the k-space in shell bins defined as follows: Bin1 : 0 < |k|  8, and Binj : 2j+1 < |k|  2j+2 j = 2, 3, . . . Nonlinear interactions lead to successive transfers Bin1 ! Bin2 ! Bin3 ! Bin4 M D Bustamante (UCD) Nonlinear Waves – 4th Int Conf June 27th 2016 9 / 19
24. Results for General Initial Conditions (2/2) Transfer e ciencies have broad peaks Collective synchronisation of precession resonances Strong synchronisation is signalled by minima of standard deviation = p h⌦2 i h⌦i2 / p E averaged over the whole set of triad precessions M D Bustamante (UCD) Nonlinear Waves – 4th Int Conf June 27th 2016 10 / 19
25. Results for General Initial Conditions (2/2) Transfer e ciencies have broad peaks Collective synchronisation of precession resonances Strong synchronisation is signalled by minima of standard deviation = p h⌦2 i h⌦i2 / p E averaged over the whole set of triad precessions M D Bustamante (UCD) Nonlinear Waves – 4th Int Conf June 27th 2016 11 / 19
26. Results for General Initial Conditions (2/2) Transfer e ciencies have broad peaks Collective synchronisation of precession resonances Strong synchronisation is signalled by minima of standard deviation = p h⌦2 i h⌦i2 / p E averaged over the whole set of triad precessions (Go to animations.) M D Bustamante (UCD) Nonlinear Waves – 4th Int Conf June 27th 2016 11 / 19
27. Precession Resonance is a Long-Term Programme Confirmed in deep water waves numerical experiments Ongoing water wave tank experiments in 2016 (Marc Perlin, Michigan) M D Bustamante (UCD) Nonlinear Waves – 4th Int Conf June 27th 2016 12 / 19
28. Deep-water gravity waves on a free surface: Precession-resonant triads (yep, triads!) @ t ⇣ + L + @ x (⇣ @ x ) + L(⇣ L ) + 1 2 @ xx (⇣2L ) + 1 2 L(⇣2@ xx ) +L(⇣ L[⇣ L ]) = 0, @ t + g ⇣ + 1 2 [(@ x )2 (L )2] [L(⇣ L ) + ⇣ @ xx ]L = 0 , where Lˆ k ( t ) = | k | ˆ k ( t ), ˆ k ( t ) = 1 L Z L / 2 L / 2 ( x , t )e ikx dx , see e.g. Choi, W. (1995) J Fluid Mech, 295, 381-394. M D Bustamante (UCD) Nonlinear Waves – 4th Int Conf June 27th 2016 13 / 19
29. Plane-wave scattering scenario (1/3) K1, K2, K3 vs. K1, K2 (5-mode Galerkin truncation) -K2 K1 K3 -K1 K2 K3 = K1 + K2 . 4 interconnected triads. Non-integrable system: 5 DOF = 5 amplitudes + 3 phases 3 conservation laws. M D Bustamante (UCD) Nonlinear Waves – 4th Int Conf June 27th 2016 14 / 19
30. Plane-wave scattering scenario (1/3) K1, K2, K3 vs. K1, K2 (5-mode Galerkin truncation) -K2 K1 K3 -K1 K2 K3 = K1 + K2 . 4 interconnected triads. Non-integrable system: 5 DOF = 5 amplitudes + 3 phases 3 conservation laws. ‘Target’ triad: K3 + ( K1 ) = K2 (modes AK3 , A K1 , AK2 ). Precession frequency: ⌦K2 K3 , K1 = D ˙ K3 + ˙ K1 ˙ K2 E ⇡ ⇣p K1 + K2 + p K1 p K2 ⌘ p g . Time series of | AK3 ( t )|| A K1 ( t )|| AK2 ( t )|: Sharp peak at ⇡ 2 p gK2 PRECESSION RESONANCE: ⌦K2 K3 , K1 = M D Bustamante (UCD) Nonlinear Waves – 4th Int Conf June 27th 2016 14 / 19
31. Plane-wave scattering scenario (1/3) K1, K2, K3 vs. K1, K2 (5-mode Galerkin truncation) -K2 K1 K3 -K1 K2 K3 = K1 + K2 . 4 interconnected triads. Non-integrable system: 5 DOF = 5 amplitudes + 3 phases 3 conservation laws. ‘Target’ triad: K3 + ( K1 ) = K2 (modes AK3 , A K1 , AK2 ). Precession frequency: ⌦K2 K3 , K1 = D ˙ K3 + ˙ K1 ˙ K2 E ⇡ ⇣p K1 + K2 + p K1 p K2 ⌘ p g . Time series of | AK3 ( t )|| A K1 ( t )|| AK2 ( t )|: Sharp peak at ⇡ 2 p gK2 PRECESSION RESONANCE: ⌦K2 K3 , K1 = Solution: K1 = 16 p , K2 = 9 p , p is a constant. M D Bustamante (UCD) Nonlinear Waves – 4th Int Conf June 27th 2016 14 / 19
32. Plane-wave scattering scenario (2/3) K1, K2, K3 vs. K1, K2 (5-mode Galerkin truncation) Numerical Simulations Initial phases: random. Initial amplitudes: | A± K1 | = 6 ⇥ 10 9 Amp , | A± K2 | = 5 ⇥ 10 4 Amp , | AK3 | = 1.5 ⇥ 10 5 Amp . Time series as functions of ↵ = K1 / K2 and dependence on amplitude (resonance expected at ↵ = 16/9 ⇡ 1.778) 10 9 10 8 10 7 10 6 0 500 1000 1500 2000 2500 3000 E k 1 t 1 . 778 1 . 775 1 . 786 10 6 10 5 10 4 10 3 0.01 0.1 1 0 200 400 600 800 1000120014001600 E k 1 t 1 . 775 1 . 767 1 . 778 10 6 10 5 10 4 10 3 0.01 0.1 1 0 100 200 300 400 500 600 700 E k 1 t 1 . 775 1 . 767 1 . 778 Amp = 0.1 Amp = 1 Amp = 1.5 M D Bustamante (UCD) Nonlinear Waves – 4th Int Conf June 27th 2016 15 / 19
33. Plane-wave scattering scenario (3/3) K1, K2, K3 vs. K1, K2 (5-mode Galerkin truncation) Transfer e ciency to A K1 and AK3 Precession resonance check (resonance expected at ↵ = 16/9 ⇡ 1.778) 1e-08 1e-07 1e-06 1e-05 0.0001 0.001 0.01 0.1 1 1.72 1.74 1.76 1.78 1.8 1.82 1.84 1.86 max E k 1 ↵ A = 3 A = 2 A = 1 . 5 A = 1 A = 0 . 2 25 30 35 40 45 1.72 1.74 1.76 1.78 1.8 1.82 1.84 1.86 ⌦k2 k 1 , k3 ↵ 2 !2 A = 3 A = 2 A = 1 . 5 A = 1 A = 0 . 2 0.0001 0.001 0.01 0.1 1 1.72 1.74 1.76 1.78 1.8 1.82 1.84 1.86 max Ek 3 ↵ A = 3 A = 2 A = 1 . 5 A = 1 A = 0 . 2 40 40.5 41 41.5 42 42.5 43 1.72 1.74 1.76 1.78 1.8 1.82 1.84 1.86 ⌦ ↵ 2 ! 2 A = 3 , ⌦ A = 2 , ⌦ A = 1 . 5 , ⌦ A = 3 , A = 2 , A = 1 . 5 , M D Bustamante (UCD) Nonlinear Waves – 4th Int Conf June 27th 2016 16 / 19 30% 40%
34. Full-PDE: 5 wave packets’ scattering scenario (1/2) K1, K2, K3 vs. K1, K2 (resolution N = 8192) M D Bustamante (UCD) Nonlinear Waves – 4th Int Conf June 27th 2016 17 / 19 Probe Probe
35. Full-PDE: 5 wave packets’ scattering scenario (2/2) K1, K2, K3 vs. K1, K2 (resolution N = 8192) 0 . 0005 0 . 001 0 . 0015 0 . 002 0 . 0025 0 . 003 0 . 0035 0 . 004 1 . 72 1 . 74 1 . 76 1 . 78 1 . 8 1 . 82 1 . 84 1 . 86 Ek1 ↵ A = 1 A = 0 . 9 A = 0 . 8 0 2 ⇥ 10 7 4 ⇥ 10 7 6 ⇥ 10 7 8 ⇥ 10 7 1 ⇥ 10 6 1 . 2 ⇥ 10 6 1 . 4 ⇥ 10 6 1 . 72 1 . 74 1 . 76 1 . 78 1 . 8 1 . 82 1 . 84 1 . 86 P!3 ↵ A = 1 A = 0 . 9 A = 0 . 8 30 32 34 36 38 40 42 44 1 . 72 1 . 74 1 . 76 1 . 78 1 . 8 1 . 82 1 . 84 1 . 86 ⌦ ↵ 2 ! 2 A = 1 A = 0 . 9 A = 0 . 8 A = 0 . 9 T = 200 100 110 120 130 140 150 160 170 180 1 . 72 1 . 74 1 . 76 1 . 78 1 . 8 1 . 82 1 . 84 1 . 86 Tr ↵ Figure: Top: Wave-packet energy e ciency E K1 and probe amplitude P! 3 as functions of ↵. Bottom: Precession ⌦ plotted for three values of amplitude rescaling; timescale Tr = 2 ⇡ h| ⌦ 2 ! 2 |i showing a resonant peak. M D Bustamante (UCD) Nonlinear Waves – 4th Int Conf June 27th 2016 18 / 19
36. Precession Resonance is a Long-Term Programme Future work: Optics, Stratified flows, Magneto-hydrodynamics Quartet and higher-order systems (Kelvin waves in superfluids, Kerr nonlinear optics) Including forcing and dissipation – Burgers equations as a test field2 2 Buzzicotti, M., Murray, B. P., Biferale, L., and Bustamante, M. D., Phase and precession evolution in the Burgers equation, Eur. Phys. J. E 39 (2016), no. 3, 34 M D Bustamante (UCD) Nonlinear Waves – 4th Int Conf June 27th 2016 19 / 19
37. Precession Resonance is a Long-Term Programme Future work: Optics, Stratified flows, Magneto-hydrodynamics Quartet and higher-order systems (Kelvin waves in superfluids, Kerr nonlinear optics) Including forcing and dissipation – Burgers equations as a test field2 Hyperlink to Jupiter moons’ precession resonance and Transneptunian objects Hyperlink to animations of precession resonance in Charney-Hasegawa-Mima equation 2 Buzzicotti, M., Murray, B. P., Biferale, L., and Bustamante, M. D., Phase and precession evolution in the Burgers equation, Eur. Phys. J. E 39 (2016), no. 3, 34 M D Bustamante (UCD) Nonlinear Waves – 4th Int Conf June 27th 2016 19 / 19
38. Precession Resonance is a Long-Term Programme Future work: Optics, Stratified flows, Magneto-hydrodynamics Quartet and higher-order systems (Kelvin waves in superfluids, Kerr nonlinear optics) Including forcing and dissipation – Burgers equations as a test field2 Hyperlink to Jupiter moons’ precession resonance and Transneptunian objects Hyperlink to animations of precession resonance in Charney-Hasegawa-Mima equation THANK YOU!!! 2 Buzzicotti, M., Murray, B. P., Biferale, L., and Bustamante, M. D., Phase and precession evolution in the Burgers equation, Eur. Phys. J. E 39 (2016), no. 3, 34 M D Bustamante (UCD) Nonlinear Waves – 4th Int Conf June 27th 2016 19 / 19