1. Introduction

2. Materials and Methods

3. Wave Development within ETC Stormy Area

3.1. Waves under ETC#1

Modeling the wind wave development starts at the origin of the low pressure area, ending (after eight hours) with the formation of ETC#1 and ETC#2 (see e.g Figure 1 from PART I). This allows to follow the space-time wave development under the ETCs from the very beginning. At the initial stage of ETC#1 evolution, as it moves to the east and acquires a completely cyclonic form, areas with continuous growth of SWH and wavelength are clearly obtained in the right sector of the ETC (Figure 2). After careful inspection of the inverse wave age maps, Figure 2, one may find that the development of wind waves begins at the front boundary of the storm area. Since developing waves are slow compared to the ETC translation velocity, they move backward relative to the moving ETC in the course of their development. In the first half of ETC#1 lifespan, fully developed waves (${\alpha }_{\left|\right|}\sim 0.85$) with direction aligned to the wind direction ($\Delta \phi \sim 0$) can be locally found in the mid of the right sector. In the second half of its lifespan, the area of fully developed waves is shifted to the rear-right sector. Second and third columns in Figure 2 exhibit clear time growth in SWH and wavelength of waves generated in the storm area. This fact corresponds to the satellite observations of linear growth of wave energy in the storm area reported in PART I.

From the maps describing wavelength and direction characteristics, a wave system with $\lambda >300\mathrm{m}$ in front of the ETC#1’s forward sector can be easily recognized. The wave direction for this system largely differs from the wind direction. The associated SWHs are much lower than the ETC generated wave ones. This wave system is not related to ETC#1, and was probably generated prior to ETC#1. Animations Animation 1 and animation 2 illustrate origin of this wave system and its evolution.

Figure 2. 6-hour fields of the wind velocity and the model wave parameters in a moving orthogonal coordinate system with the origin at the eye of ETC#1. Columns from left to right: wind velocity, significant wave height ($H{s}_p$), wavelength (${\lambda }_p$) and wave direction (${\phi }_p$), wave-wind directions difference ($\Delta \phi ={\phi }_p-{\phi }_w$), and local inverse wave age (${\alpha }_{\left|\right|}=ucos({\phi }_p-{\phi }_w)/c_p$). The red arrow in the centre of each map indicates the movement direction of ETC#1. Circle radii start at 200m in 200m increment.

Figure 2. 6-hour fields of the wind velocity and the model wave parameters in a moving orthogonal coordinate system with the origin at the eye of ETC#1. Columns from left to right: wind velocity, significant wave height ($H{s}_p$), wavelength (${\lambda }_p$) and wave direction (${\phi }_p$), wave-wind directions difference ($\Delta \phi ={\phi }_p-{\phi }_w$), and local inverse wave age (${\alpha }_{\left|\right|}=ucos({\phi }_p-{\phi }_w)/c_p$). The red arrow in the centre of each map indicates the movement direction of ETC#1. Circle radii start at 200m in 200m increment.

3.2. Waves under ETC#2

The simulation for ETC#2 generated waves is also considered from the first hours of its formation and illustrated Figure 3. The movement direction of ETC#2, shown by red arrows, is changing from 30°, to 110°, between initial hours and last hours of its life, respectively.

As discussed in PART I, both ETC#1 and ETC#2 are fast-moving atmospheric systems in a sense that their translation velocities are larger than the group velocity of generated wind waves. As a result, the generation of wind waves begins when the front boundary of the storm appears at a given location in the ocean. Then, in the process of their development, the generated waves move backward relative to the ETC, reaching their maximum development at the rear boundary of the storm, and, finally, leave the ETC in the form of swell systems. These peculiar wave developments are clearly exhibited in Figure 3. From the inverse wave age maps shown in the last column, the youngest waves (the largest values of ${\alpha }_{\left|\right|}$) are first located along the frontal boundary of the storm. The inverse wave age then gradually decreases towards the storm core, where the developing waves are propagating. Second, similar to the satellite observations (see PART I), the largest value of SWH and wavelength of generated waves are observed in the right and the rear-right sector. This fact suggests that, in the right sector, waves stay under wind forcing for more time than in other sectors. Waves become fully developed, running out of the storm region as swell systems. This finding coincides with the results of self-similar analyses of SWH measurements, discussed in PART I, which confirms efficiency of the extended fetch/duration mechanism when wind and developing waves directions coincide with the ETC movement direction.

At the last stage of the ETC#2 lifetime, the SWH and wavelength reach 17 m and 500 m, respectively. It corresponds to phenomenal sea conditions, defined by the World Meteorological Organization (WMO) as having a significant wave height larger than 14 m. A remarkable point is that at the final stage of ETC#2 lifespan, the waves with $H{s}_p>9\mathrm{m}$ and $\lambda >350\mathrm{m}$ in the rear-left sector occupy a radial range of $300<r<1200$ km. This indicates the creation of a huge wave front which can cause a very dangerous situation for marine navigation. Another interesting feature shown Figure 3 is the presence of ETC#1 swell after its lifespan. In the up-right corner of wind fields, before 13-Feb-2020 15:00:00, we see the tail of ETC#1. However, swell waves generated by ETC#1, remain in the basin until 14-Feb-2020 09:00:00.

Figure 3. The same as Figure 2, but for ETC#2.

Figure 3. The same as Figure 2, but for ETC#2.

3.3. Model vs. Observations

Fields of wave parameters and discussion of their characteristics, presented in Section 3.1 and Section 3.2, are based on the outputs of the 2D model. Figure 1, shown in Section 2.3, already demonstrates a "general" validity, over all data covering the North Atlantic during the studying period. In this section, more specific satellite sensor tracks, crossing the storm area of ETC#2, are used . Both types of model outputs are considered, namely: SWH of primary waves system ($H{s}_p$), and SWH of mixed seas $H{s}_T$.

Cross sections of $H{s}_p$ and $H{s}_T$, as a function of latitudes along the altimeter tracks crossing ETC#2, are shown Figure 4, together with observed SWH, $H{s}_{obs}$. In Figure 4, the geographical location of the tracks is superimposed on the map with wind speed contours and colored map of $H{s}_T$. The altimeter tracks cover various parts of the ETC#2, Figure 4. Comparing the $H{s}_{obs}$, $H{s}_p$ and $H{s}_T$ profiles, a rather good overall consistency is found between model simulations and observations. However, some differences can be notified, e.g., model underestimation of SWH highest values around 15m on 13:00:00 and 23:00:00. On the other hand, after careful inspection, these underestimations, at least on 23:00:00, are related to spatial shift of maximal model values of SWH from the altimeter track. Notice, the model values of $H{s}_T$ and $H{s}_p$ in the storm area of ETC, are almost the same, but a bit different outside, where the existence of the mixed seas is likely plausible.

Figure 4. The model vs the measurements. First and third rows: altimeter tracks superimposed on maps of wind speed (contour lines) and model SWH, $H{s}_p$ (color) ETC#2 . Second and fourth rows: cross section of $H{s}_{obs}$ (black curves), $H{s}_p$ (blue curves), and $H{s}_T$ (red curves) along the altimeter tracks as a function of latitude.

Figure 4. The model vs the measurements. First and third rows: altimeter tracks superimposed on maps of wind speed (contour lines) and model SWH, $H{s}_p$ (color) ETC#2 . Second and fourth rows: cross section of $H{s}_{obs}$ (black curves), $H{s}_p$ (blue curves), and $H{s}_T$ (red curves) along the altimeter tracks as a function of latitude.

4. Evolution of Swell after the ETC Lifespan

Figure 5. 6-hour fields of the wind velocity and the model wave parameters in geographical coordinates after cessation of ETC#2. Columns from left to right: wind velocity, significant wave height ($H{s}_p$), wavelength (${\lambda }_p$) and wave direction (${\phi }_p$), and local inverse wave age (${\alpha }_{\left|\right|}=ucos({\phi }_p-{\phi }_w)/c_p$). Two white arrows indicate two swell fronts considered in the text.

Figure 5. 6-hour fields of the wind velocity and the model wave parameters in geographical coordinates after cessation of ETC#2. Columns from left to right: wind velocity, significant wave height ($H{s}_p$), wavelength (${\lambda }_p$) and wave direction (${\phi }_p$), and local inverse wave age (${\alpha }_{\left|\right|}=ucos({\phi }_p-{\phi }_w)/c_p$). Two white arrows indicate two swell fronts considered in the text.

4.1. Evolution of the Northeastward Swell

Referring to the last row of Figure 3 and first row of Figure 5, a front of waves is observed on 14-Feb-2020 morning, with $\lambda \ge 300$ m, $Hs\ge 10$ m, and ${\alpha }_{\left|\right|}\le 0.85$. These waves occupy a very large area in the rear-right sector of ETC#2. Travelling eastward, this wave front can reach the Celtic Sea and subarctic seas, such as the Norwegian Sea. On Figure 6 the spatio-temporal evolution of this front is illustrated at times, when the SWIM tracks crosses part of the front. First row of Figure 6 is associated with the final moments of ETC#2 life, shown in the last 2 rows of Figure 3. The SWIM tracks are superimposed on the synchronous field of SWH and wavelength obtained from the 2D model.

Going forward in time (from up to down rows of Figure 5 and Figure 6), the east–northeastward propagation of this wave front is easily recognizable. The measurements of $H{s}_T$, ${\lambda }_p$ and ${\phi }_p$ also confirm this front propagation. Based on simulations and measurements, following Figure 6, the spatio-temporal evolution of swell in the eastern side of the North Atlantic Ocean is thus found to influence the European coasts and subarctic seas for a long time after the disappearance of ETC#2. On the maps, Figure 6, the maximal SWH levels decrease from about 17m on 14-FebT09 to 8m on 15-FebT20, following energy dissipation, while the maximal wavelength is > 450 m.

Figure 6. Spatio-temporal evolution of waves, after disappearance of ETCs, when the CFOSAT-SWIM tracks cross eastward moving swell front. Columns from left to right: Geographical location of SWIM tracks superimposed on color-maps of modeled ${\lambda }_p$, where the arrows show ${\phi }_p$; cross section of ${\lambda }_{p_{obs}}$ (black circles) and ${\lambda }_p$ (blue asteroids); cross section of ${\phi }_{p_{obs}}$ (black circles) and ${\phi }_p$ (blue asteroids); SWIM nadir tracks superimposed on color-maps of modeled $H{s}_T$; cross section of $H_{s_{obs}}$ (black curves) and modeled $H{s}_T$ (blue curves).

Figure 6. Spatio-temporal evolution of waves, after disappearance of ETCs, when the CFOSAT-SWIM tracks cross eastward moving swell front. Columns from left to right: Geographical location of SWIM tracks superimposed on color-maps of modeled ${\lambda }_p$, where the arrows show ${\phi }_p$; cross section of ${\lambda }_{p_{obs}}$ (black circles) and ${\lambda }_p$ (blue asteroids); cross section of ${\phi }_{p_{obs}}$ (black circles) and ${\phi }_p$ (blue asteroids); SWIM nadir tracks superimposed on color-maps of modeled $H{s}_T$; cross section of $H_{s_{obs}}$ (black curves) and modeled $H{s}_T$ (blue curves).

The 2D wavenumber spectra derived from SWIM data in the area of wind waves (box I in Figure 6) and in the area of swell front (box II in Figure 6) are shown Figure 7. The SWIM wave spectra are available from the IFREMER database (ftp.ifremer.fr/ifremer/cersat/projects/iwwoc). The second columns of Figure 7 display the omnidirectional spectra $S\left(k\right)$, scaled by $e/k_p$, i.e. $k_{p}S\left(k\right)/e$, as a function of $k/k_p$, where e is the energy (integral of S over k) and $k_p$ spectral peak wavenumber. Green lines show the JONSWAP [41] spectra with the similar scaling. Measured spectrum for the the wind waves is very similar to the empirical JONSWAP one, except for some differences in the slope of the spectral tail. In the swell front area, combination of dominant swell system and much shorted wind waves can be revealed. The presence of wind waves noticeably changes the omnidirectional spectrum due to the addition of energy in the tail of the spectrum, at frequencies above the peak frequency of the wind waves. Nevertheless, shape of the scaled total spectrum (where swell dominates) is again surprisingly close to the JONSWAP spectrum. This observation is similar to what was reported by [42] and [43] under hurricane conditions, - shape of the dominant wave spectra, regardless of whether they are wind waves or swells, is very close to the shape of the JONSWAP spectrum.

Histograms of the model energy distributions of wave-trains over wavelength and directions shown in the third column, serve as a good proxy of 2D spectra. Comparing the first and third column in Figure 7, model spectral distributions of wave-trains are indeed found consistent with 2D spectra. Travel time histogram (last column) gives an idea of the history of wave packets (wind driven and swell) forming a wave pattern at a given time and at a given point.

Figure 7. 2D directional wave spectra derived from SWIM inside the boxes I and II indicated in Figure 6 (left columns). The second column: Scaled omnidirectional spectra, $k_{p}S\left(k\right)/e$, (red circles) as a function of $k/k_p$ compared to scaled JONSWAP spectra (green curves). The third and fourth columns are model histograms of SWH and travel time distributions over wavelength and directions, purple arrow indicates wind direction.

Figure 7. 2D directional wave spectra derived from SWIM inside the boxes I and II indicated in Figure 6 (left columns). The second column: Scaled omnidirectional spectra, $k_{p}S\left(k\right)/e$, (red circles) as a function of $k/k_p$ compared to scaled JONSWAP spectra (green curves). The third and fourth columns are model histograms of SWH and travel time distributions over wavelength and directions, purple arrow indicates wind direction.

Figure 8 shows time evolution of the peak wavelength (SWIM data) and energy collected from all the altimeter tracks crossing the swell front traveling northeastward. The time countdown starts from 14-Feb-2020 9:00:00 - the end of the life of the ETC#2. Although the wavelength measurements by SWIM are very limited, they nevertheless demonstrate expected stationary behavior or slow growth, as predicted by the 2D model. In contrast, the energy of the swell at the front, measured by altimeters, decreases rather quickly. After 40 hours of travel (which is equivalent to a distance of about 2000 km), the energy drops by about 4 times. This observed energy decay is consistent with model simulations of the total wave energy, co-located with altimeter measurements (open circles), with maximal values at the part of swell front which moves east-north.

Red line in Figure 8 shows swell energy attenuation due to divergence of wave-rays which reads:

$$e/e_0={\left(1+c_g{G}_{n0}t\right)}^{-1}$$

(5)

where $G_{n0}$ is the initial value of cross-ray gradient of swell direction. Relationship (5) represents a straightforward solution of Equation (1), for which the effect of the energy source $S_e$ on swell evolution is ignored, with the swell wavelength kept constant. Full solution for swell energy and wavelength evolution due to effect of dissipation, non-linear waves interactions and wave-rays convergence/divergence can be found in [36,37]. Here we consider only the effect of wave-ray divergence, which seems to be the governing mechanism. Initial value of cross-ray gradient in Equation (5) can be evaluated as $G_{n0}=1/R_f$ with $R_f$ the radius of the swell front curvature. The red curve shown Figure 8 corresponds to $R_f=500km$, which is about the radius of ETC#2. Observed attenuation of swell energy, $e\propto t^{-1}$, is remarkably faster than that predicted by effect of the swell energy dissipation [34]: $e\propto t^{-1/2}$, by non-linear wave-wave interactions [44]: $e\propto t^{-1/3}$, and that reported by [12] due to interaction of swell with the airflow.

Figure 8. (a) Time evolution of swell energy: red circles are altimeter measurements, open circles are the model values at the point of the measurements, black curve shows maximal values of the model energy at the front. (b) Time evolution of swell wavelength: red point are SWIM measurements, black curve is the model.

Figure 8. (a) Time evolution of swell energy: red circles are altimeter measurements, open circles are the model values at the point of the measurements, black curve shows maximal values of the model energy at the front. (b) Time evolution of swell wavelength: red point are SWIM measurements, black curve is the model.

4.2. Southward Swell

According to the wind fields, Figure 5, a strong wind jet formed along Greenland at a speed of about 35 ms⁻¹. This wind jet results from the deformation of the frontal part of ETC#2. It provides a fetch length of about more than 1200 km for wave development for more than 12 hours. The waves, which were developing in the frontal part of the ETC#2, getting fully developed, reach phenomenal SWH and wavelength of 18 m and 600 m, respectively, and consequently further propagate as swell. Although, the wind field in the basin became quite complex, this wave front further radiated into the "open ocean" from Cape Farewell, clearly visible in Figure 5.

Some fragments illustrating the development of the wave system along the Greenland coast, leading to the formation of the radiated swell, are captured with SWIM measurements, shown in Figure 9. Model simulations of $H_s$, ${\lambda }_p$ and ${\phi }_p$ are consistent with these measurements, suggesting the model capability to realistically reproduce the wave development and swell evolution. Referring to Figure 5 and Figure 9, the swell front associated with ${\lambda }_p>500m$ and $Hs>13m$ originated on 14-Feb-2020 21:00 at latitude of 56°N. The swell front then further moves southward achieving to 48°N on 15-Feb-2020 21:00 (see inverse wave age and wavelength plots on the fore-said dates on Figure 9).

A synthetic aperture radar (SAR) image in HH-polarization from Sentinel-1B acquired over Cape Farewell on 14-Feb-2020 evening, when the swell front is reaching the open ocean, is shown Figure 10a. To derive distributions of wavelength/direction of waves over this SAR scene, we apply Fast Fourier Transform (FFT) to each of the image boxes with size of $256\times 256$ pixels (Figure 10b,d), which were spread over the whole SAR image, to obtain a 2D wavenumber ($k_x,k_y$) image spectra; two examples are demonstrated in (Figure 10c,e). Spectral level of the SAR image can potentially be converted to the wave elevation spectrum, but this procedure is not straightforward, and we leave this issue out of the scope of this paper. Instead, focusing on wave kinematics, we consider only the SAR detected wavelength and direction. In order to remove ambiguity, we used wave direction from 2D model outputs.

The resulting field of wavenumber vectors derived from the SAR image is shown in Figure 11a. This field exhibits spectacular swell front moving southward over which the wavelength rapidly changes from 500m to 250 m. In addition, the existence of the western boundary of the swell is confirmed by synchronous SWIM measurements shown in the same figure. Besides, SWIM measurements a day later, see Figure 9 bottom row, revealed the similar rapid drop of the spectral peak wavelength and SWH over this swell front, however already shifted southward. Model field of the swell front, Figure 11, is consistent, in general, with the measurements, providing similar change of wavelengths over the front and its location. However, the existence of an area filled with "shorter" waves in the eastern part of the model scene (coinciding with the low-wind speed area) is not confirmed by the measurements.

Figure 9. The same as Figure 6, but when the CFOSAT-SWIM tracks cross southward radiating swell front.

Figure 9. The same as Figure 6, but when the CFOSAT-SWIM tracks cross southward radiating swell front.

Figure 10. (a) Sentinel-1B Synthetic Aperture Radar (SAR) image on HH polarization on 14-Feb-2020 evening of the south of Greenland. (b,d) zoom on SAR image for areas indicated by red and blue boxes in (a). (c,e) are the SAR image wavenumber ($k_x$, $k_y$) spectra of (b,d) zooms.

Figure 10. (a) Sentinel-1B Synthetic Aperture Radar (SAR) image on HH polarization on 14-Feb-2020 evening of the south of Greenland. (b,d) zoom on SAR image for areas indicated by red and blue boxes in (a). (c,e) are the SAR image wavenumber ($k_x$, $k_y$) spectra of (b,d) zooms.

Figure 11. (a) Spatial distributions of dominant wavelengths and directions derived from the SAR image shown in Figure 11a, with the SWIM off-nadir measurements of ${\lambda }_p$ and ${\phi }_p$. (b) The model fields of wavelength and directions of primary wave system; white lines indicate contour of SAR image.

Figure 11. (a) Spatial distributions of dominant wavelengths and directions derived from the SAR image shown in Figure 11a, with the SWIM off-nadir measurements of ${\lambda }_p$ and ${\phi }_p$. (b) The model fields of wavelength and directions of primary wave system; white lines indicate contour of SAR image.

SWIM derived 2D wavenumber spectra on different side of the swell front (box III and box IV in Figure 9) are shown Figure 12. Spectrum inside the swell area displays the superposition of dominant swell system and system of shorter wind waves. Spectrum outside the swell front exhibits a wave system of remarkably shorter waves (as compared with swell in box III) with a wide angular spread. Similar to the case with east-north swell, the scaled omnidirectional spectra, $k_{p}S\left(k\right)/e$, on different sides of the swell front are very similar and well consistent with the shape of the scaled JONSWAP spectrum, again confirming the experimental finding by [42,43] for waves in hurricanes. The histograms of distribution of the model energy of wave-trains over wavelength and directions shown in the third column, are also well consistent with 2D SWIM spectra. The travel time histogram (last column) gives the history of wave packets (wind driven and swell) forming a wave pattern at a given time and at a given point.

Figure 12. The same as Figure 7 but for boxes III and IV indicated in Figure 9.

Figure 12. The same as Figure 7 but for boxes III and IV indicated in Figure 9.

5. In Situ Data

Figure 13. (a) Wind speed, (b) significant wave height, (c) wavelength at location of buoy "NO TS MO 6400045". The designation of the lines is given in the legends to the plots. Buoy-wavelength, ${\lambda }_z=g{T}_z^2/\left(2\pi \right)$, is derived from the measured wave period $T_z$ using the dispersion relation, and buoy-peak-wavelength is ${\lambda }_p=1.3^2{\lambda }_z$. Wind speed on 10-m height is derived from the measured wind speed (1-m height) using the logarithmic wind profile with $C_{D10}=0.0015$. The animation, which shows the waves passing the buoy location can be found in Animation 3.

Figure 13. (a) Wind speed, (b) significant wave height, (c) wavelength at location of buoy "NO TS MO 6400045". The designation of the lines is given in the legends to the plots. Buoy-wavelength, ${\lambda }_z=g{T}_z^2/\left(2\pi \right)$, is derived from the measured wave period $T_z$ using the dispersion relation, and buoy-peak-wavelength is ${\lambda }_p=1.3^2{\lambda }_z$. Wind speed on 10-m height is derived from the measured wind speed (1-m height) using the logarithmic wind profile with $C_{D10}=0.0015$. The animation, which shows the waves passing the buoy location can be found in Animation 3.

6. Conclusions

Abbreviations

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