*3.1. Marine Heatwaves*

As shown in Figure 2c,d sudden positive or negative SST anomalies may occur where Rossby waves are resonantly forced [1]. Two major positive anomalies occurred during the time of observation, namely from 1 January 2019 to 27 September 2021. The first positive anomaly occurred on 30 May 2019, the second on 23 July 2021. The first is one month ahead of the corresponding negative SSH anomaly, while the second is 2 weeks in advance (Figure 2a,c). Anticipation of SST anomalies means that they occurred while the thermocline was lifting. One major negative SST anomaly occurred on 5 January 2020, one week behind the corresponding positive SSH anomaly, while the thermocline was deepening (Figure 2b,d).

**Figure 2.** Abrupt events highlighted by SSH at 34.125◦ N, 148.125◦ E in (**a**) and at 34.125◦ N, 140.125◦ E in (**b**), and by SST averaged along the parallel 34.125◦ N between 145.625◦ E and 148.125◦ E, and filtered in the band of 1–68 days to emphasize the rapid variations while attenuating the annual variations. This series is used as the time reference in the wavelet analysis of data. (**a**,**c**) are referring to warm events, (**b**,**d**) to cold events. SST data is provided by NOAA https://www.ncei.noaa.gov/data/sea-surface-temperatu re-optimum-interpolation/v2.1/access/avhrr/ (accessed on 27 April 2022).

Each SST anomaly corresponds to an opposite SSH anomaly. The reverse is not true; some SSH anomalies do not produce significant SST anomalies. This suggests strong ocean–atmosphere interactions are required for the Rossby waves to produce coherent SST anomalies, with a threshold effect.

#### 3.1.1. The Marine Heatwave That Occurred on 21 July 2021

The climatic impact of this heatwave was significant. One of the most notable records in July 2021 was registered in Asahikawa 43◦46 N, 142◦22 E [63]. The city registered 36.2 ◦C on 27 July, breaking the previous record of 36 ◦C set on 7 August 1989.

#### 3.1.2. Wavelet Analysis of Climatic State Variables

As shown in Figure 3a,c,e,g,i, the harmonics of SSH are visible along the North-Pacific gyre, mainly between latitudes 25◦ N and 35◦ N, and between longitudes 130◦ E and 180◦. The longitudinal and meridional extensions of Rossby waves increases with period. With regarding to the harmonic 1/6, the SSH anomaly in Figure 3i,j extends over areas of the northwestern Pacific Ocean, including the Yellow Sea, and the entire Sea of Japan.

**Figure 3.** The amplitude (**a**,**c**,**e**,**g**,**i**) and the phase (**b**,**d**,**f**,**h**,**j**) of the harmonics of SSH. The periods are 1/6 yr in (**a**), (**b**) 1/12 yr in (**c**), (**d**) 1/24 yr in (**e**), (**f**) 1/48 yr in (**g**), and (**h**) 1/96 yr in (**i**,**j**). The amplitudes are expressed in 16 classes, each containing the same number of individuals (quantiles). The color of the bar associated with the phase represents an angle varying from −180◦ to + 180◦ [61] (each class corresponds to 20◦). This angle is reflected by a segment of time of one period, hence the coincidence of the colors at the ends. Time lags in (**b**,**d**,**f**,**h**,**j**) are relative to 23 July 2021. The time reference is the SST anomaly averaged along the parallel 34.125◦ N between 145.625◦ E and 148.125◦ E. The SSH anomaly is negative when the time lag is zero (the SSH anomaly is negatively correlated with the SST anomaly and late compared to SST). Only the phase corresponding to the 37.5% quantile of the highest values of the amplitude is displayed. Same data sources as in previous figures.

In Figure 3b,d,f,h,j, the phase of the SSH anomaly clearly shows a succession of ridges and troughs in phase opposition. The momentum equations applied to a quasigeostrophic motion of oceanic Rossby waves show that the meridional geostrophic current *V* is in phase with the forcing while both the zonal current *U* and SSH anomalies are in quadrature. However, the phase is more precise for the zonal current (Figure 4) and the meridional current (Figure 5) than for SSH in Figure 3. Indeed, the estimation of geostrophic current velocities from SSH anomalies has a filtering effect because it involves surrounding measurements of SSH. This has the effect of reducing noise and making the interpolated values more representative than the raw measurements of SSH.

**Figure 4.** Same as Figure 3 for the amplitude (**a**,**c**,**e**,**g**,**i**) and the phase (**b**,**d**,**f**,**h**,**j**) of the zonal geostrophic current *U* oriented to the east when the time lag is zero. Same data sources as in previous figures.

**Figure 5.** Same as Figure 3 for the amplitude (**a**,**c**,**e**,**g**,**i**) and the phase (**b**,**d**,**f**,**h**,**j**) of the meridional geostrophic current *V* oriented to the south when the time lag is zero. Same data sources as in previous figures.

The modulated geostrophic currents change direction every apparent half-wavelength of Rossby waves. Thus, in Figure 4, the zonal current *U* shows a succession of regions in phase opposition whose size corresponds to an apparent half wavelength of Rossby waves. These regions form a mosaic of cells in which the zonal geostrophic currents converge or diverge when the cell is translated longitudinally by half of a wavelength. This alternation is still observable for meridional current *V* (Figure 5), but this time convergence or divergence occur in the North-South direction.

Figures 4 and 5 confirm the previous observations regarding the longitudinal and meridional extensions of Rossby waves along the gyre from SSH as the period increases. This also applies to the speed of modulated geostrophic currents. However, the anomalies of modulated geostrophic currents remain localized along the gyre as the period increases, without stretching to the Yellow Sea, and the Sea of Japan, as does SSH. This difference in

the behavior of Rossby waves in the semi-closed seas suggests that these seas are not large enough to allow the formation of perceptible geostrophic currents.

Anomalies in opposite phase also widen with the period, consistent with the increase in apparent Rossby wavelength. The anomalies of the zonal component of the geostrophic current *U* extend longitudinally with the period while the anomalies of the meridional component *V* extend latitudinally as shown in Figures 4j and 5j.

Downwelling that occurs in convergent cells means that the thermocline lowers, the intake of warm water resulting from geostrophic currents. On the contrary, upwelling that occurs in divergent cells makes the thermocline rise, restoring warm water under the effect of geostrophic currents. The alternation of convergent or divergent cells throughout the gyre at mid-latitudes highlights the determining role of these cells regarding their climatic impacts. These privileged ocean–atmosphere interactions along the gyre occur at all time scales extending from the annual, seasonal cycles to time intervals not exceeding a few days.

These ocean–atmosphere interactions induce atmospheric baroclinic instabilities as suggested by the variations in SST at the rate of the different periods of the Rossby waves, as shown in Figure 6. The transient SST anomalies occur along the gyre from which the Kuroshio leaves the Asian continent to a longitude close to 180◦. Regarding the harmonic 1/6, the SST anomaly in Figure 6i,j is translated over extensive areas of the northwestern Pacific Ocean, including the Yellow Sea, the entire Sea of Japan, and part of the Sea of Okhotsk, as does SSH.

**Figure 6.** Same as Figure 3 for the amplitude (**a**,**c**,**e**,**g**,**i**) and the phase (**b**,**d**,**f**,**h**,**j**) of SST whose anomalies are positive when the time lag is zero. Same data sources as in previous figures.

Compared to SSH, SST anomalies are translated to the north while widening (Figure 6). This translation that appears especially during the first 3 periods is of short duration, which suggests the role of the atmosphere. Highly contrasted during the first 3 periods, the phases of SST anomalies become uniform as the period increases. As shown in the Figure 6j, uniformity of the phase is achieved for the harmonic 1/6, which confirms that the lifetime of the SST anomaly is very short compared to the period close to 2 months.

#### *3.2. The Marine Cold Wave That Accurred on 5 January 2020*

Figure 7 shows the amplitude and the phase of SST anomalies during the cold event. Anomalies are little translated toward the north, which suggests the weakness of the SST response to the meridional component of the wind resulting from a high-pressure system initiated by the negative SST anomaly of the gyre. Here again, the phase shows a mosaic of convergent and divergent cells characterized by the inversion of geostrophic currents (Figure 7d,f). But contrary to what happens for MHWs, the phase does not homogenize when the period increases, reflecting the SSH anomaly. This suggests the weakness of the ocean–atmosphere interactions, hence the weak climatic impact of marine cold waves.

**Figure 7.** Same as Figure 3 for the amplitude (**a**,**c**,**e**,**g**,**i**) and the phase (**b**,**d**,**f**,**h**,**j**) of SST whose anomalies are negative when the time lag is zero. Time lags in (**b**,**d**,**f**,**h**,**j**) are expressed in relation to 5 January 2020. Same data sources as in previous figures.

#### *3.3. Subtropical Cyclones*

Subtropical cyclones develop at mid-latitudes around a stationary front due to an upper-level disturbance, generally an upper-level trough downstream of a strong westerly jet [63,64]. Cyclogenesis results from the combination of vorticity advection and thermal advection created by the latitudinal temperature gradient, a low-pressure center causing upward motion around the low [65]. This rotational flow will push polar air equatorward west of the low via its cold front, and warmer air will push poleward low via the warm front.

#### 3.3.1. An Extreme Precipitation Event, Germany, July 2021

During one week in July 2021, severe flooding occurred across Europe due to dangerous thunderstorms and rain, hitting Germany the hardest. This country experienced up to 182 mm of rain within 72 h. More than 170 people have lost their lives and entire communities have been destroyed. The number of victims of this flood disaster exceeds that of all previous inland floods in Germany since 1900 combined [66].

In mid-July 2021, a pronounced high altitude low shifted from France to the Alps and southern Germany. On its front, very warm and humid air masses were directed to the north and east of Germany, concomitantly with fresher Atlantic air to the south and south-west of Germany, causing record rainfall in parts of North Rhine-Westphalia and Rhineland-Palatinate.

#### 3.3.2. Wavelet Analysis of State Variables

Precipitation height is represented in Figure 8. Here again, the low-pressure system is decomposed into the 5 period bands, the time shift of precipitation areas being relative to the date of occurrence of the extreme rainfall event on 14 July 2021.

**Figure 8.** Same as Figure 3 for the amplitude (**a**,**c**,**e**,**g**,**i**) and the phase (**b**,**d**,**f**,**h**,**j**) of the harmonics of the precipitation height in western Europe. Time lags in (**b**,**d**,**f**,**h**,**j**) are expressed in relation to 14 July 2021. The time reference is the rainfall height in Germany at 47◦ N, 18◦ E. Daily precipitation data is provided by NOAA https://www.ncei.noaa.gov/data/global-precipitation-climatology-project-gp cp-daily/access/ (accessed 27 April 2022).

The two main precipitation areas represented in Figure 8i,j, i.e., within the band centered on the period 1/6 yr, are independent since they are strongly out of phase with each other. In contrast, Figure 8g,h highlights a coherent low-pressure system at the synoptic scale within the band centered on the period 1/12 yr. The phase of the three main rainfall areas over central and western Europe are indeed only slightly shifted.

According to Figure 8e,f, the rotation of the low-pressure system occurs within the period band centered on 1/24 yr. This deduction is based on the presence of two rainfall areas in phase opposition on both sides of the disaster area, which confirms the hypothesis that the different rainfall areas belong to the same dynamic system at a synoptic scale. The cyclonic flow is fed mainly by the Atlantic west of the coasts of western Europe (Figure 9g,h) and the Baltic Sea (Figure 9e,f).

**Figure 9.** Same as Figure 8 for the amplitude (**a**,**c**,**e**,**g**,**i**) and the phase (**b**,**d**,**f**,**h**,**j**) of SST. SST anomalies are positive when the time lag is zero. Same data sources as in previous figures.

An SST anomaly over the Atlantic west of the coasts of southern England, Ireland, and France does indeed occur within the period band centered on 1/12 yr (Figure 9g,h), reaching more than 1 ◦C. The phase of this anomaly is close to zero, showing that ocean–atmosphere interactions are occurring while the SST anomaly is peaking. With regard to the Baltic Sea, the SST anomalies peak within the period bands centered on 1/48 and 1/24 yr, as shown in Figure 9c–f. The phase of the anomaly is close to zero within the period band centered on 1/24 yr, while it is slightly shifted negatively within the period band centered on 1/48 yr, but it nevertheless contributes significantly to the feeding of the low-pressure system by peaking the day before the extreme event occurred.

In the Baltic Sea, SST increases when SSH decreases, i.e., when the thermocline rises. This phenomenon is mainly observable within the period band centered on 1/24 yr (Figures 9e,f and 10e,f) where the phase of the SST anomaly is close to zero. The phases of both SSH and SST anomalies are uniform in seas bordered by coasts, which modifies the apparent wavelength of Rossby waves. It is elongated, in this case, in the absence of a strong current flowing east in which Rossby waves would be embedded. The latter result from the declination of the sun and the variation in solar irradiance during the year, which induces the motion of the thermocline. The westward propagating Rossby waves and their harmonics remain confined in these seas. Convection processes occur in subsurface water, favoring the warming of surface water. These conditions are conducive to the formation of baroclinic instabilities in the atmosphere as a result of increased evaporation.

**Figure 10.** Same as Figure 8 for the amplitude (**a**,**c**,**e**,**g**,**i**) and the phase (**b**,**d**,**f**,**h**,**j**) of SSH. SSH anomalies are negative when the time lag is zero. Same data sources as in previous figures.

With regard to the Atlantic Ocean, SSH anomalies are weak off the coasts of western Europe, whereas the amplitude of SST anomalies is high. This suggests that this temperature anomaly results from atmospheric phenomena that translate the SST anomalies developing along the North Atlantic gyre toward the east, a process that leads to baroclinic instabilities in the atmosphere.

As shown in Figure 8a–f, the size of the cyclonic flow reduces as the mean period decreases from 1/24 to 1/96 yr, remaining centered in Germany while the rotation accelerates. Within the period band centered on 1/96 yr, the precipitation area is concentrated between latitudes 45◦ N and 55◦ N, and longitudes 3◦ E and 20◦ E. East of 20◦ E the precipitation does not contribute to the genesis of the extreme event since it is strongly out of phase (Figure 8b). In addition, the phase is uniform, close to zero, within sampling errors (the step is daily).

Figure 8b,d,f show that several atmospheric layers are rotating simultaneously. They concentrate around the disaster zone as the rotation accelerates. The half-period of rotation passes from the order of 4 days (Figure 8f) to a few hours (Figure 8b). Since the precipitation areas concentrate around the axis of rotation of the low-pressure system while the rotation period decreases, this suggests that the rotation is accelerating as the layer rises, driven by the upward flow of the cyclonic system. In this way, the uppermost layer is fed by the lower layers. Its phase is uniform so that the rotation period is less than the duration of the extreme precipitation event. The water vapor contained in the different atmospheric layers condenses when they rise due to the lowering of the temperature, which leads to heavy precipitation.

However, the concentration of precipitation, which occurs during cycles of shorter periods, cannot be approached using the same data, which is beyond the scope of this article. Here, the spatial and temporal resolution of the rainfall data [60] are suited to highlighting

the various stages leading to the deepening of the low-pressure system, namely the merging of the various low-pressure systems at the synoptic scale, and the feeding of the cyclonic flow from the Atlantic Ocean off the coasts of western Europe and the Baltic Sea.

#### **4. Discussion**

#### *4.1. Marine Heatwaves*

Regarding MHWs, uniformization of the phase as the SST anomaly migrates north only becomes mature in the 1/6 harmonic mode. In the 1/12 mode, the maturation of the SST anomaly is not complete, a time shift of the order of one week remaining within the anomaly (Figure 6h).

With regard to short cycles corresponding to harmonic modes 1/24, 1/48, and 1/96, the northernmost fringe of the SST anomaly whose phase is heterogeneous is transient. Indeed, it disappears completely during long cycles, the SST anomaly concentrating around a zonal midline at approximately 42◦ N (Figure 6j). This suggests that the warm, humid air from the low-pressure system warms the sea surface as it migrates north, inducing convective processes in subsurface water while the SST becomes increasingly cold. This promotes the creation of a vertical profile of convection/evaporation tending toward an equilibrium between the thermocline and the surface of the ocean. But stratification of the subsurface water leading to this vertical profile seems unstable and does not occur systematically, as shown in Figure 2.

This strong ocean–atmosphere interaction which causes the thermocline to rise, could explain the uniformization of the phases of the SST anomalies at latitudes where the northward thermal gradient of surface water is steep. Uniformization of the phases then amounts to assuming an overall movement of the thermocline during the longest cycle, hence the brief but intense heatwave which appeared around 27 July 2021.

The northward translation of the SST anomaly is only significant in the case of heatwaves due to the low-pressure system that forms above the gyre before developing into a synoptic cyclonic system. This enhances the SST response to latent and sensible heat fluxes directed to the north. This sudden SST response to atmospheric transfers has already been observed, which sparked interest in this research [21]. According to the authors, MHW observed at the sea surface in the summer of 2021 was the largest in extent and intensity since the beginning of satellite measurements of global SST in 1982, with a strong societal impact.

Other works reported such MHWs in the northwestern Pacific [9,22]. Ref. [22] reported the positive SST anomaly that occurred in August 2020 in subtropical waters in the surroundings of the gyre 120◦ E–180◦ E, 20◦ N–35◦ N, which was attributed to anthropogenic forcing. Further investigations seem necessary to validate such a hypothesis. Indeed, this positive SST anomaly does not seem distinguishable from internal variability in the context of the present study (Figure 2c).

In [17] the SST of the Oyashio region abruptly increased in the summer of 2010, and a high summertime SST repeated every year until 2016. This was attributed to the strengthening of the Kuroshio water influence. In [9], extreme weather and MHWs are reported; these occurred simultaneously around the Pacific shelf off southeastern Hokkaido, Japan. In these two cases, the influence of the western boundary current was presumably involved, in conjunction with extreme weather. Based on recent works relying on the properties of Rossby waves at mid-latitudes, the present paper proposes a common cause for these intriguing phenomena.

#### *4.2. Subtropical Cyclones*

A low-pressure system is forming at the synoptic scale, the result of the merger of several low-pressure subsystems. To achieve this merger, dew-point fronts have to be formed, separating moist air masses found ahead of the dry line from drier air masses found behind it. The drier air behind dew-point fronts lifts up the moist air ahead, triggering strong moist convection. A barometric trough gradually forms, which creates a convergence zone in the lower layers of the atmosphere and upper-level divergence.

The increasingly rapid rotation of cyclonic flows in the various atmospheric layers as they rise produces an extreme rainfall event. The rapid cycles of cyclogenesis contrast with the slowly maturing phenomena without which the cyclonic system could not have developed with such magnitude. They may lead to SST anomalies concomitant with the extreme rainfall event, which occur within the period bands centered on 1/12 and 1/24 yr. Monitoring these maturation processes could help predict the occurrence of devastating climatic phenomena.

The analysis of the different stages leading to subtropical low-pressure systems makes it possible to address an essential problem that relates to the presumed impact of anthropogenic forcing. One mechanism for the increase in such transient events discussed in the literature is related to the slowing of the predominant westerly wind circulation evident in observational data [66,67], due to a strong warming of the Arctic as a result of global warming [68]. Such a slowdown has been linked to observed increases in the persistence of weather systems [69,70].

By influencing the rapid cycles of cyclogenesis, such a mechanism could contribute to explaining the increase in the frequency of extreme rainfall events observed during the last decades in the northern hemisphere, in particular in the North America. But the ubiquity of the increase in the frequency as well as the intensity of extreme rainfall events also suggest an evolution in the mechanisms favoring the development of cyclonic flows at the synoptic scale. This hypothesis is corroborated by the fact that extreme rainfall events occur in places deemed not to be flood-prone, causing numerous victims, as happened in Germany in July 2021, thus deceiving the vigilance of weather-watch systems.

The development of coherent SST anomalies, the main driver of synoptic-scale subtropical cyclones, is unambiguously linked to the propagation of oceanic Rossby waves. These result from solar forcing, independent of anthropogenic forcing. In contrast, other mechanisms related to global warming appear to be decisive in the context of slow cycles during which the coalescence of low-pressure systems occurs. Such mechanisms are strengthened by a temperature increase of ocean surface water associated with an overall increase in atmospheric humidity, which lowers the dew point and favors the formation of fronts. In return, the extension of the low-pressure system at the synoptic scale centered on a continental low favors the feeding of the cyclonic flow by overlapping over surrounding SST anomalies. Owing to the accumulated latent heat, with regard to their internal energy these low-pressure systems promote upper-level lows, favoring blocks. This may explain the record precipitations observed during the last decades when pouring over regions deemed not to be flood prone, as has occurred in many places in Western and Central Europe.

#### **5. Conclusions**

The wavelet analysis of high temporal and spatial resolution data, namely SSH, geostrophic currents, and SST in the northwestern Pacific, allowed the highlighting of the formation of a mosaic of convergent and divergent cells along the north Pacific gyre from where the Kuroshio leaves the Asian continent to nearly 180◦. Upwelling and downwelling are associated with Rossby waves of short apparent wavelengths embedded in the wind-driven current of the gyre. The driver of the fundamental Rossby wave and the harmonics is the declination of the sun. Sudden SSH anomalies may occur, some of them producing abrupt extensive positive or negative SST anomalies, opposite in sign to SSH anomalies from which they originated. This phenomenon is general and is observable along the subtropical gyres where the western boundary currents move away from the continents.

Regarding MHWs in the northwestern Pacific, a warm SST anomaly associated with the northward component of the wind resulting from the low-pression system induces an SST response to latent and sensible heat transfer where the latitudinal SST gradient is steep. The SST anomaly is then shifted north while the phases become uniform.

The wavelet analysis of high temporal and spatial resolution of SSH, SST, and rainfall height in the North Atlantic, the Baltic Sea, and northwest Europe has made it possible to highlight the evolution of an extratropical cyclone in northwestern Europe, of exceptional intensity, at different time scales. Intensification of subtropical cyclones as well as the increase in their frequency appear to be mainly related to the evolution of conditions favoring the formation of low-pressure systems at the synoptic scale. These conditions are probably exacerbated by anthropogenic warming which promotes the maturation of the mechanisms leading to the coalescence of lows. In these conditions, the interactions between the atmosphere and the coherent positive SST anomalies on the surrounding ocean play a major role in feeding the cyclonic flow centered on a continental low. Owing to the accumulated latent heat, extreme subtropical cyclones induce upper-level lows that favor the persistence of the cyclonic flow.

The innovative nature of this study is based on the dynamics of the various systems implicated in the formation of extreme climatic events. These events are the culmination of exceptional circumstances, some of which are foreseeable due to their relatively long maturation time. Some avenues are developed with the aim of better understanding how anthropogenic warming can modify certain key mechanisms in the evolution of the dynamic system at the interface between the oceans and the atmosphere.

Future work will focus on the role played by the anthropogenic forcing in the formation of extensive MHWs. On the other hand, by taking advantage of high-resolution data on geostrophic currents, a systematic study of short-period Rossby waves developing where the western boundary currents leave the continents to re-enter the subtropical gyres would be rich in teaching how to specify their climatic impacts, including the conditions of formation of MHWs and extreme rainfall events. Using the same method of investigation, other case studies focusing in particular on the southern hemisphere are required with the aim of generalizing these investigations.

**Funding:** This research received no external funding.

**Data Availability Statement:** Only public data duly referenced are used.

**Conflicts of Interest:** The author declares no conflict of interest.
