**1. Introduction**

Although they seem distant, Marine Heatwaves (MHWs) and extreme subtropical cyclones have a common origin, the resonant forcing of oceanic Rossby waves at midlatitudes. The present research is focused on those Rossby waves whose period varies from a few days to a few months. At mid-latitudes, they form preferentially where the western boundary currents move away from the continents to re-enter the subtropical gyres [1]. These Rossby waves induce very active convergent or divergent geostrophic currents in the formation of positive or negative Sea Surface Temperature (SST) anomalies. They appear as harmonics of an annual fundamental Rossby wave resonantly forced by the declination of the sun.

While the climatic impact of Rossby waves is well known, their interaction with the atmosphere still presents some mysteries. However, behind this natural cause, there

**Citation:** Pinault, J.-L. Morlet Cross-Wavelet Analysis of Climatic State Variables Expressed as a Function of Latitude, Longitude, and Time: New Light on Extreme Events. *Math. Comput. Appl.* **2022**, *27*, 50. https://doi.org/10.3390/mca27030050

Academic Editor: Leonardo Trujillo

Received: 28 April 2022 Accepted: 2 June 2022 Published: 4 June 2022

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2022 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

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is a reality: these extreme events are becoming more and more frequent, as numerous studies show. There is therefore a compelling need to elucidate how anthropogenic warming intervenes in the genesis of these extreme events in order to better understand the ocean–atmosphere interactions involved as well as to better anticipate them.

The proposed method consists in representing, in contiguous bands of periods, the amplitude and the time lag, with respect to the date of occurrence of the extreme event, of each of the climatic state variables. For this, both the amplitude and the phase of the climatic state variables are mapped. The amplitude and the phase are deduced from the cross-wavelet power spectra of these state variables expressed as a function of longitude and latitude, and from a reference time series representative of the evolution of the extreme event. In order to optimize the temporal resolution of the dynamics of the observed phenomena, the cross-wavelet power spectra are both scale-averaged over the bandwidths and time-averaged over a time interval bracketing the date of occurrence of the extreme event, the length of which is equal to the bandwidth.

#### *1.1. Marine Heatwaves*

MHWs are observed in all oceans. They have impacted fishery resources and the occurrence of harmful algal blooms where rich marine ecosystems are at risk [2]. For example, recent MHW events in the Tasman Sea have had dramatic impacts on the ecosystems, fisheries, and aquaculture off Tasmania's east coast [3]. Similar damages have been investigated in the South China Sea where MHWs were strongly regulated by El Niño–Southern Oscillation (ENSO) [4]. The high latitudes are not spared: Alaska was impacted in 2016 [5]. The economic impact of these events, little known until the recent past, has given rise to much research in recent years. However, our understanding of the large-scale drivers and potential predictability of MHW events is still in its infancy.

The dynamic processes related to the initiation of an advective MHW were investigated in continental shelves, namely the Middle Atlantic Bight of the Northwest Atlantic [6], the North West Australia [7], the Indonesian-Australian Basin and areas including the Timor Sea and Kimberley shelf [8], and in the Pacific shelf waters off southeast Hokkaido, Japan [9]. Favorable climatic conditions are mentioned for driving cross-isobath intrusions of warm, salty offshore water onto the continental shelf.

Long-term temperature changes under the influence of human-induced greenhouse gas-forcing drive coastal MHW trends globally. Cross-shore gradients of MHW and SST changes are reported in the Chilean coast region [10], in mid-latitude coasts like the Mediterranean Sea, Japan Sea, and Tasman Sea, as well as in the northeastern coast of the United States [11], along the Australian coastlines [12], in the Tasman Sea [13], in Canada's British Columbia coastal waters, from Queen Charlotte Strait to the Strait of Georgia [14], in the Coastal Zone of Northern Baja California [15], in the Southern California Bight [16], and in the Oyashio region [17–22].

Studies focused on MHWs have reported conditions favoring the warming of surface waters caused by increased solar radiation because of reduced cloud cover, namely in summer MHWs in the South China Sea [18], in the East China Sea, and the South Yellow Sea [19]. The genesis and trend of MHWs in the Indian Ocean and their role in modulating the Indian summer monsoon have been investigated [20], as well as the role of oceanic Rossby waves forced in the interior South Pacific on observed MHW occurrences off southeast Australia [3].

Finally, intense MHWs occurred at the sea surface over extensive areas of the northwestern Pacific Ocean, including the entire Sea of Japan and part of the Sea of Okhotsk [21,22]. An extreme event due to its extension and intensity, occurred in July–August 2021 [21]. In this article, we will attempt to highlight the role played by oceanic Rossby waves in the genesis of this event, the conditions of formation of which have not yet been fully elucidated.

#### *1.2. Extreme Subtropical Cyclones*

The intensity of the heaviest extreme precipitation events is known to increase with global warming [23–27] almost everywhere in the world [28,29]. Particularly impacted are regions subject to subtropical cyclones [30]. At mid-latitudes, these regions are easily identifiable by their precipitation pattern in the 5–10 year band, while they only show a weak seasonality [31,32]. The main areas subject to rainfall oscillation in the 5–10 year band are: (a) Southwest North America, (b) Texas, (c) Southeastern North America, (d) Northeastern North America, (e) Southern Greenland, (f) Europe and Central and Western Asia, (g) the region of the Río de la Plata, (h) Southwestern and Southeastern Australia, and (i) Southeast Asia.

Global warming is projected to lead to a higher intensity of precipitation and longer dry periods in North America [33–35] and Europe [36–41]. Extreme floods during the recent decades in Europe are more frequent compared to the last 500 years [42]. For Germany, the number of people exposed to flood risks could more than triple and damages more than quadruple by the end of the century [43,44]. In summer, an increase is also projected in most parts of Europe, although decreases are projected for some regions in southern and southwestern Europe, partly due to a projected decrease in cyclone frequency in the Mediterranean [45].

In spite of potentially large societal impacts, mechanisms involved in changes in frequency and intensity of heavy precipitation are much less explored. The purpose of this article is to improve techniques for predicting these extreme precipitation events and to advance our knowledge of the possible mechanisms whose incidence and intensity are linked to global warming. For this, we will analyze in detail the different phases of hydroclimatic mechanisms that led to an extreme precipitation event in Germany in July 2021, that is, a region reputed not to be floodable, causing many casualties.

#### *1.3. Oceanic Rossby Waves at Mid-Latitudes*

Oceanic Rossby waves have a well-known effect on the climate. The role of Rossby waves in local air-sea interactions over the tropical Indian Ocean and in remote forcing from the tropical Pacific Ocean has been investigated during El Niño and positive Indian Ocean Dipole years [46,47]. High-resolution subsurface observations have provided insight into equatorial oceanic Rossby wave activity forced by Madden-Julian Oscillation events [48].

However, the role played by the oceanic Rossby waves on the climate is not limited to the tropical oceans. The Rossby waves that develop where the western boundary currents leave the continents to re-enter the subtropical gyres have a strong impact on climate [31]. Located at the same latitudes as the subtropical jet streams, they thus participate in the cyclogenesis of mid-latitude eddy systems (anticyclones and depressions) then moving under these powerful air currents.

Oceanic Rossby waves propagate westward. Being approximately non-dispersive, their phase velocity given by the dispersion relation only depends on the latitude [49]. The phase velocity decreases when the latitude increases. At mid-latitudes, it is lower than the velocity of the eastward propagating wind-driven current of the gyre resulting from Ekman pumping associated with the wind curl. Rossby waves are driven by the circulation of the gyre.

Based on the momentum equations of Rossby waves, these baroclinic waves are forced by changes in solar irradiance induced by solar and orbital cycles [50,51]. This property is specific to Rossby waves at mid-latitudes because, in tropical oceans, they are mainly driven by the wind, in this case the trade winds. Under the effect of radiative forcing, in addition to a Sea Surface Height (SSH) anomaly, the propagation of Rossby waves along the subtropical gyre induces a zonal and a meridional modulated current. The meridional current is in phase with the forcing while the zonal current and the SSH perturbation, i.e., the ridge of the Rossby wave, are in quadrature. During the ascending phase of the zonal ridge, the meridional modulated currents converge toward the ridge.

The convergence causes the thermocline to deepen due to the inflow of warm water from the surface of the ocean. The affected ocean surface extends well beyond the gyre due to the meridional currents. A quarter of a period later, the zonal modulated current reaches its maximum at the same time as the ridge. The zonal currents are in opposite phase on either side of the ridge, causing the zonal propagation of the thermocline wave.

Both the meridional and the zonal modulated current change direction every halfperiod. Note that the speed of the zonal current is expressed in a relative way because the westward propagating Rossby wave is carried by the eastward propagating wind-driven current of the gyre. Its absolute speed is obtained by adding it to that of the steady winddriven current. Thus, the zonal current of the gyre periodically accelerates, slows down, and sometimes even reverses direction.

During its ascending phase, the Rossby wave behaves like a heat sink while, during its recession phase, the upwelling which occurs along the ridge causes the Rossby wave to release heat that has been stored when the thermocline was lowering. This explains the climatic impact of Rossby waves at mid-latitudes; sometimes they favor high-pressure systems, sometimes low-pressure.

#### **2. Materials and Methods**

#### *2.1. The Caldirola–Kanai Oscillator*

Several Rossby waves of different periods overlap along the gyre. Sharing the same zonal and meridional modulated currents, these Rossby waves behave like coupled oscillators with inertia. The equation of the Caldirola–Kanai (CK) oscillator, which is a fundamental model of dissipative systems that is usually used to develop a phenomenological single-particle approach for the damped harmonic oscillator [52], can be expressed by considering the conditions of durability of the dynamic system. For that, the equation of the CK oscillator is formulated to express the mode of coupling between *N* Rossby waves that share the same modulated geostrophic currents [50]:

$$\mathcal{M}\_i \ddot{u}\_i + \gamma \mathcal{M}\_i \dot{u}\_i + \sum\_{j=1}^{N} I\_{ij} \left( u\_i - u\_j \right) = I\_i \cos(\Omega t) \tag{1}$$

where *ui* is the zonal geostrophic current velocity of the *i*th oscillator along the gyre, M*<sup>i</sup>* the mass of water displaced during a cycle resulting from the quasi-geostrophic motion of the *i*th oscillator, *γ* the Rayleigh friction, and *Jij* the measure of the coupling strength between the oscillators *i* and *j*. The right-hand side represents the periodic driving on the *i*th oscillator with frequency Ω and amplitude *Ii*, that is, Coriolis and pressure gradient forces. The restoring force simply depends on the difference in velocity of zonal geostrophic currents between the oscillators. So, it vanishes when the velocities are equal *ui* = *uj* which, in the absence of friction, removes any interaction between the oscillators *i* and *j*. On the other hand, the strength of the interaction increases as the difference in velocities increases. The coupling of Rossby waves is exercised by the fact that the velocities *ui* are common at the convergence of the modulated geostrophic currents of the resonant oscillatory system.

In order to ensure the durability of that dynamic system, the coupled oscillators have to form oscillatory subsystems so that the resonance conditions are defined recursively:

$$
\tau\_i = \frac{1}{n\_i} \tau\_{i-1} \text{ with } \tau\_0 = T \tag{2}
$$

where *ni* = 2 or 3. *T* is the period of the fundamental wave, that is, one year.

The CK oscillators resonate in subharmonic and harmonic modes of the annual fundamental wave. The apparent eastward propagation velocity of this fundamental wave depends on the latitude of the gyre where the western boundary current leaves the continent, and to the velocity of the steady wind-driven current. In the case of a pseudo-periodic forcing, its apparent wavelength is adjusted to the forcing period when the average forcing frequency is present in the frequency spectrum of the dynamic system. Natural frequencies close to the forcing frequency are favored, while those far from it are dampened because

of friction so that the fundamental wave is resonantly forced by the variations in solar irradiance resulting from the declination of the sun. This is by far the primary source of temperature variability in surface and subsurface waters of the oceans at mid-latitudes.

#### *2.2. Data*

Daily gridded data (1/4 degree × 1/4 degree) of SSH, geostrophic currents [53], and SST [54–57] are used. SSH and geostrophic current data begin 15 March 2019. Although starting earlier, SST data is used over the same time interval as SSH. The last update was on 17 October 2021.

Data of precipitation is produced as part of the Global Precipitation Climatology Project (GPCP) Climate Data Record (CDR) Daily analysis, which spans the time period October 1996 to the near present [58,59]. The algorithm to produce the daily 1◦ GPCP product takes inputs from several different sources and merges them to create the most consistent and accurate daily precipitation estimates [60].

#### *2.3. Wavelet Analysis*

#### 2.3.1. Marine Heatwaves

The problem that we are going to tackle, which relates to the genesis of MHWs at mid-latitudes, consists in highlighting the evolution of brief SST anomalies at different time scales, reflecting the driving role of oceanic Rossby waves. A Morlet cross-wavelet analysis is performed to estimate the amplitude of variations in characteristic period bands of four state variables, that is, SSH, modulated geostrophic currents, and SST, as well as their phase compared to a reference time series [61]. Presently, SST averaged along the parallel 34.125◦ N, between 145.625◦ E and 148.125◦ E, is used as the time reference. The average of the SST data over a short segment of the parallel makes it possible to specify the evolution of the heat wave over time by reducing the noise without significantly harming the spatial resolution from which the location of the reference is defined.

As we will see later, SST anomalies observed on 5 January 2020 and on 23 July 2021 are representative of a phenomenon that led to a "marine cold wave" in the first case and a "marine heatwave" in the second. This time reference is chosen so that it unambiguously reflects those two extreme events, both being defined as a sharp surface temperature anomaly (the extremum does not last more than a day), positive or negative as the case may be.

Under these conditions, the square root of the wavelet power applied to the state variable time series, scale-averaged over the period bands, is the regionalized amplitude of anomalies, whatever their date of occurrence. The cross-wavelet power applied to both the state variable time series and the time reference, scale-averaged over the period bands, is the regionalized phase of anomalies. It is the time-lag between the extrema of the anomalies and the date of occurrence of the extreme event, namely the marine cold wave or the MHW [62]. Consequently, for each state variable and for each period band a paired map of the amplitude and the phase of anomalies is obtained.

The wavelet analysis of the state variables is carried out over short periods of time framing the date of occurrence of the extreme events. In this way, for each band, both the amplitude and the phase of the anomalies are time-averaged over a time interval coinciding with the width of the band, centered on the date of occurrence of the extreme events.

The choice of each period band is guided by the properties of the CK oscillator considered as a prototype of coupled Rossby waves. Harmonics of the CK oscillator are identifiable in Figure 1b that represents the Wavelet Fourier spectrum of SSH at 34.125◦ N, 140.125◦ E located on the north Pacific gyre, 0.75◦ south of the Pacific shelf off the southeastern region of Japan. The richness of the Fourier spectrum is probably attributable to the proximity of the coasts of Japan facing the Pacific Ocean. This suggests a local resonance of Rossby waves, which strengthens harmonics. The Fourier spectrum distinctly shows the annual fundamental wave, the amplitude of which is predominant, which gives rise to harmonics whose main periods are 1/3 yr, 1/6 yr, 1/12 yr, and 1/24 yr. Rossby waves are subject to very large fluctuations as attested by the width of the peaks in the Fourier spectrum. Only the amplitudes of the harmonics whose periods are 1/12 yr and 1/24 yr are known with a level of confidence greater than 95% (the lack of precision of the amplitude of the annual Rossby wave results from the short duration of the observation period, which was barely 3 years).

**Figure 1.** The SSH anomaly at 34.125◦ N, 140.125◦ E–(**a**) the raw signal–(**b**) the Wavelet Fourier spectrum (adimentional) and the main harmonics. SSH data is provided by the National Oceanic and Atmospheric Administration (NOAA) https://coastwatch.noaa.gov/pub/socd/lsa/rads/sla/daily/ nrt/ (accessed on 27 April 2022).

In Table 1 the period bands are chosen in accordance with (2). Bandwidths are deduced from the mean period *τ* of harmonics. Lower and upper limits are 0.75 × *τ* and 1.5 × *τ*, respectively, so that the bands are contiguous, since the periods are halved from one harmonic to another. The progression of the bands is continued beyond the periods analyzed (mean periods 1/48 and 1/96 years).


**Table 1.** Properties of observed or presumed harmonics and bandwidths.

### 2.3.2. Subtropical Cyclones

The phenomenological study of climatic phenomena leading to extreme precipitation at mid-latitudes is performed in the same way. Three state variables are jointly analyzed, the precipitation height, i.e., the thickness of the layer of water produced during a day, SST, and SSH. When a positive SST anomaly is locally in phase with the extreme precipitation event while being within the perimeter of the cyclonic low-pressure system, this means that the water vapor evaporated from the ocean is involved in the cycle of cyclogenesis by providing latent heat during the condensation process. This concomitance results from the

fact that the atmospheric phenomena leading to the transport of water vapor within the low-pressure system are very rapid compared to the oceanic processes at the origin of the SST anomalies. As will be justified later, the supply of the cyclonic system from the free surface of the ocean occurs in less than a day, whereas the maturation of a large surface and uniform phase SST anomaly generally takes at least ten days.

#### **3. Results**
