Fig 1.
Measurements of surface waves by wave buoys operated by Coastal Ocean Monitoring Center (COMC), National Cheng Kung University (NCKU), Taiwan.
(a) The discus buoy is equipped with an GPS receiver, anemometers, radar reflector, temperature sensors, accelerometer-tilt-compass (ATC) sensor and acoustic Doppler current profiler (ADCP). (b) The buoy motions by six degrees of freedom are heave, sway and surge, roll, pitch, and yaw.
Fig 2.
(a) The time series of buoy accelerations and corresponding motions of pitch, roll, and yaw. (b) The power-density spectrum determined by heave acceleration. (c) The time series of surface-wave elevation determined by wave spectrum.
Fig 3.
Histograms of basic wave parameters of measured surface waves.
(a-d) Histograms of rogue wave samples of the significant wave height H1/3, maximum wave height Hmax, peak period Tp and directional spreading σθ, respectively. (e-h) Histograms of non-rogue wave samples of significant wave height H1/3, maximum wave height Hmax, peak period Tp and directional spreading σθ, respectively. There are 663 samples for rogue waves and 600 samples for non-rogue waves, respectively.
Fig 4.
Characteristic wave structures in the spatial-temporal domain and their analytical results from the nonlinear Fourier transform.
(a) Spatial-temporal evolution of the constant envelope. (b) Main spectrum and spines of the constant envelope. The small gap in the spine at zero is a numerical artifact. (c) Hyperelliptic modes of the constant envelope. (d) Spatial-temporal evolution of the fundamental soliton. (e) Main spectrum and spines of the fundamental soliton. (f) Hyperelliptic modes of the fundamental soliton. (g) Spatial-temporal evolution of the A-type doubly periodic solutions of the NLSE with the parameters α3 = 1, ρ = 3, η = 0.4. (h) Main spectrum and spines of the A-type doubly periodic solutions of the NLSE. (i) Hyperelliptic modes of the A-type doubly periodic solutions. The two individual black points in the upper left and right are numerical artifacts. (j) Evolution of the B-type doubly periodic solutions of the NLSE with the parameters α1 = 0.1, α2 = 0.5, α3 = 1. (k) Main spectrum and spines of the B-type doubly periodic solutions of the NLSE. (l) Hyperelliptic modes of the B-type doubly periodic solutions.
Fig 5.
Time series of rogue wave records from Taitung Open Ocean buoy.
The surface-wave elevation is shown as the black line and the magnitude of the complex envelope calculated by Hilbert transform is shown as red line. The blue dashed lines refer to ±H1/3. (a) The rogue-wave data measured from 20:00h on 18 January 2013 with Hmax = 6.67 m, H1/3 = 3.24 m, and AI = 2.06. (b) The rogue-wave data measured from 22:00h on 4 October 2014 with Hmax = 6.96 m, H1/3 = 3.44 m, and AI = 2.02. (c) The rogue-wave data measured from 08:00h on 4 December 2015 with Hmax = 7.53 m, H1/3 = 3.62 m, and AI = 2.08. (d) The rogue-wave data measured from 20:00h on 25 September 2012 with Hmax = 8.05 m, H1/3 = 3.56 m, and AI = 2.26.
Fig 6.
Nonlinear spectra with envelope amplitudes on the vertical axis and carrier wave frequencies on the horizontal axis.
(a) Spectrum of the time series of complex envelope from Fig 5a is classified as type 1: stable mode. (b) Spectrum of the time series of complex envelope from Fig 5b is classified as type 2: small breather. (c) Spectrum of the time series of complex envelope from Fig 5c is classified as type 3: large breather. (d) Spectrum of the time series of complex envelope from Fig 5d is classified as type 4: soliton.
Fig 7.
Classification results of rogue and non-rogue waves based on the four types of nonlinear spectra.
(a) Scatter plot of rogue-wave samples of peak period Tp over maximum wave height Hmax based on the four types of spectra. (b) Scatter plot of non-rogue wave samples of peak period Tp over maximum wave height Hmax based on the four types of spectra. (c) Scatter plot of rogue-wave samples of abnormality index AI over maximum wave height Hmax based on the four types of spectra. (d) Scatter plot of non-rogue wave samples of abnormality index AI over maximum wave height Hmax based on the four types of spectra. The four types of spectra are stable-mode type, small-breather type, large-breather type and soliton type shown as blue circles, green squares, purple triangles and red crosses, respectively.
Fig 8.
Squared peak period vs maximum wave height for nonlinear spectra of the four types.
(a) Time series with rogue waves (b) Time series without rogue waves
Fig 9.
Classification results of rogue and non-rogue waves with low directional spreading.
This figure presents the same information as Fig 7, but only the first quartile of the rogue waves with the lowest directional spreading have been kept for each class.
Fig 10.
Four types of nonlinear spectra and their Benjamin-Feir index.
(a) Scatter plot of rogue-wave samples of one-dimensional Benjamin-Feir index BFI1D over maximum wave height Hmax based on the four types of spectra. (b) Scatter plot of non-rogue wave samples of two-dimensional Benjamin-Feir index BFI2D over maximum wave height Hmax based on the four types of spectra. (c) Scatter plot of rogue-wave samples of one-dimensional Benjamin-Feir index BFI1D over maximum wave height Hmax based on the four types of spectra. (d) Scatter plot of non-rogue wave samples of two-dimensional Benjamin-Feir index BFI2D over maximum wave height Hmax based on the four types of spectra.
Table 1.
Comparison of the mean one- and two-dimensional BFI between time series of rogue waves and non-rogue waves for the four types of nonlinear spectra.
Fig 11.
Four types of nonlinear spectra and their Benjamin-Feir index with low directional spread.
This figure presents the same information as Fig 10, but only the first quartile of the rogue waves with the lowest directional spreading have been kept for each class.
Fig 12.
Relationship of unstable modes to rogue waves.
(a) The rogue-wave data is measured from 06:00h to 06:10h on 29 July 2017 with the maximum wave height Hmax = 20.34 m, the significant wave height Hs = 8.76 m, the peak wave period Tp = 13.64 s, and the abnormality index AI = 2.32. The surface-wave elevation is shown as the black line and the magnitude of the complex envelope calculated by Hilbert transform is shown as red line. The original signal is cut into different time lengths of 600 s, 450 s, 300 s, 150 s, and 75 s, which are labelled from b to f. (b-f) Nonlinear spectra corresponding to the time series from b to f.
Fig 13.
Relationship of the largest nonlinear mode to wave parameters.
(a) Scatter plot of the largest nonlinear mode over maximum wave height. (b) Scatter plot of the largest nonlinear mode over maximum crest height.