突波為時空間突發的非確定局域超高振幅波 (extreme amplitude wave
event)，廣存於各類非線性失穩波系統，例如：法拉第波、水面波、非線性
介質中的光波和微粒電漿聲波，生活中常見於海洋上的瘋狗浪 (rogue
wave)。過去的研究指出極高振幅偶發突波源於調制非穩性 (modulation
instability) 而產生如孤立子般的超高振幅調制波包，而突波研究為近年來
非線性波動研究的重要議題，突波形成是否具有與如何尋得預警指標，為
爭議性的重要問題。
此篇論文將探討非線性法拉第波因外界上下振動之驅動振幅增加而失
穩後，所形成時空上突發之不穩定突波，找出預警指標 (precursor)，並從
波動與粒子間交互作用 (wave-particle interaction) 的觀點重新了解水流如
何造成時空突波。此研究發現極高振幅偶發突波的產生取決於時空周遭波
型變化，在極高振幅偶發突波發生的前數週，環繞突波的波峰環場稜線
(preceding surrounding water ridge) 積分高度和波峰環場稜線高度的離散程
度為重要的突波生成預警條件。此外更發現在類晶格排列的法拉第波系統
中，極高振幅偶發突波的產生和前數週波峰環場稜線的高階鍵角取向有序
度 (high-order bond orientational order) 並無直接的關聯。透過時空上波形
的演化資訊，波峰環場稜線的積分高度為極高振幅偶發突波生成的重要預
警指標。

The uncertainly occurred and highly localized extreme amplitude rare event,
called rogue wave event, occurs in various nonlinear systems such as Faraday
waves, water surface waves, optical waves, and dust acoustic waves in plasma.
Modulation instability, causing wave-amplitude modulation and the formation
of amplitude soliton-like structure, has been widely accepted as the underlying
mechanism for rogue wave generation. However, how the generation mechanism
for extreme wave events can be understood from the Lagrangian view and
whether the precursors can be identified are still open issues. In this work, the
generation and the precursors of rogue wave events are experimentally
investigated through a Faraday water wave system exhibiting disordered oscillon
pattern. The above issues are addressed from wave-particle interaction view by
using the surrounding waveform information preceding the rogue wave events.
It is found that the oscillon peak height is correlated with high angular
average and the low standard deviation of its prior surrounding waveform, but
uncorrelated with preceding waveform with high-order orientational symmetries.
The preceding water ridge with high angular average and the low standard
deviation of ridge height is the key to determining the strong inward water
focusing, which further leads to the subsequent rogue wave generation. The
angular average normalized by the standard deviation of the surround waveform
serves as a good precursor for rogue wave generation before several periods.

30
Bibliography

[1] C. Kharif, E. Pelinovsky, and A. Slunyaev, Rogue Waves in the Ocean
(Springer, 2009)
[2] N. Akhmediev and E. Pelinovsky, Editorial – Introductory remarks on
“Discussion & Debate: Rogue Waves – Towards a Unifying Concept?”, Eur.
Phys. J. Special Topics 185, 1 (2010)
[3] A. Chabchoub, N. P. Hoffmann, and N. Akhmediev, Rogue Wave
Observation in a Water Wave Tank, Phys. Rev. Lett. 106, 204502 (2011)
[4] M. Onorato, A. R. Osborne, M. Serio, and L. Cavaleri, Modulational
instability and non-Gaussian statistics in experimental random water-wave
trains, Phys. Fluids 17, 078101 (2005)
[5] M. Shats, H. Punzmann, and H. Xia, Capillary Rogue Waves, Phys. Rev.
Lett. 104, 104503 (2010)
[6] H. Xia, T. Maimbourg, H. Punzmann, and M. Shats, Oscillon Dynamics and
Rogue Wave Generation in Faraday Surface Ripples, Phys. Rev. Lett. 109,
114502 (2012)
[7] D. R. Solli, C. Ropers, P. Koonath and B. Jalali, Optical rogue waves, Nature
450, 1054-1057 (2007)
[8] A. Montina, U. Bortolozzo, S. Residori, and F. T. Arecchi, Non-Gaussian
Statistics and Extreme Waves in a Nonlinear Optical Cavity, Phys. Rev. Lett.
103, 173901 (2009)
[9] A. N. Ganshin, V. B. Efimov, G. V. Kolmakov, L. P. Mezhov-Deglin, and P.
V. E. McClintock, Observation of an Inverse Energy Cascade in Developed
Acoustic Turbulence, Phys. Rev. Lett. 101, 065303 (2008)
[10] H. Bailung, S. K. Sharma, and Y. Nakamura, Observation of Peregrine
Solitons in a Multicomponent Plasma with Negative Ions, Phys. Rev. Lett.
107, 255005 (2011)
[11] Y. Y. Tsai, J. Y. Tsai, and L. I, Generation of acoustic rogue waves in dusty
plasmas through three-dimensional particle, Nat. Phys. 12, 573 (2016)
[12] S. Haver, A possible freak wave event measured at the draupner jacket
january 1 1995. Proc Rogue Waves 20-22 October (2004)
[13] P. K. Shukla, I. Kourakis, B. Eliasson, M. Marklund, and L. Stenflo, Instability and Evolution of Nonlinearly Interacting Water Waves, Phys. Rev.
Lett. 97, 094501 (2006)
[14] S. Birkholz, C. Br´ee, A. Demircan, and G. Steinmeyer, Predictability of
Rogue Events, Phys. Rev. Lett. 114, 213901 (2015)
[15] W. Cousins and T. P. Sapsis, Reduced-order precursors of rare events in
unidirectional nonlinear water waves, J. Fluid Mech. 790, 368 (2016)
[16] A. Kudrolli and J. P. Gollub, Patterns and spatiotemporal chaos in
parametrically forced surface, Physica D 97, 133 (1996)
[17] D. Binks and W. van de Water, Nonlinear Pattern Formation of Faraday
Waves, Phys. Rev. Lett. 78, 4043 (1997)
[18] A. V. Kityk, J. Embs, V. V. Mekhonoshin, and C. Wagner, Spatiotemporal
characterization of interfacial Faraday waves by means of a light absorption,
Phys. Rev. E 72, 036209 (2005)
[19] I. Shani, G. Cohen, and J. Fineberg, Localized Instability on the Route to
Disorder in Faraday Waves, Phys. Rev. Lett. 104, 184507 (2010)
[20] A. B. Ezersky, D. A. Ermoshin, and S. V. Kiyashko, Dynamics of defects in
parametrically excited capillary ripples, Phys. Rev. E 51, 4411 (1995)
[21] W. B. Wright, R. Budakian, and S. J. Putterman, Diffusing Light
Photography of Fully Developed Isotropic Ripple Turbulence, Phys. Rev.
Lett. 76, 4528 (1996)
[22] A. von Kameke, F. Huhn, G. Fern´andez-Garc´ıa, A. P. Muuzuri, and V.
P´erez-Muuzuri, Double Cascade Turbulence and Richardson Dispersion in
a Horizontal Fluid Flow, Phys. Rev. Lett. 107, 074502 (2011)
[23] N. Francois, H. Xia, H. Punzmann, S. Ramsden, and M. Shats, Three-
Dimensional Fluid Motion in Faraday Waves Creation of Vorticity and
Generation of Two-Dimensional Turbulence, Phys. Rev. X 4, 021021 (2014)
[24] H. Xia, M. Shats, and H. Punzmann, Modulation instability and capillary
wave turbulence, Europhys. Lett. 91, 14002 (2010)
[25] U. Y. Peng, M. C. Chang, and L. I, Lagrangian-Eulerian dynamics of
breaking shallow water waves through tracer tracking of fluid elements,
Phys. Rev. E 87, 023017 (2013)
[26] H. Y. Chen, C. Y. Liu, and L. I, Identifying Faraday rogue wave precursors
from surrounding waveform information, Phys. Rev. Fluids 3, 064401 (2018)