**4. The Guiding Role of the Theory of Rossby Wave Energy Dispersion in the Study on the Dynamic Processes of the Variability of the East Asian Winter Monsoon System**

The variability of the winter climate in China is controlled by the East Asian winter monsoon system. The interannual and interdecadal variabilities of the East Asian winter monsoon system is the main cause of winter rain, snow, and ice disasters in China [42]. However, there are few studies on the dynamic processes of interannual and interdecadal variabilities in East Asian winter monsoon system. Under the guidance of the theory of Rossby wave energy dispersion, some studies on the dynamic processes of the East Asian winter monsoon system have been carried out in recent years. Especially, some meteorogists in China have paid a special attention to the impacts of the interannual and interdecadal variations of quasi-stationary planetary waves in the three-dimensional spherical atmosphere on the East Asian winter monsoon system and the low-temperature rain, snow, and ice disasters in China.

#### *4.1. The Guiding Role of the Theory of Rossby Wave Dispersion in the Study on the Dynamical Processes of the Interannual Variability of the East Asian Winter Monsoon System*

The winter monsoon prevails in East Asia, and due to the interannual variability of the winter monsoon system, winter temperature and precipitation in China show the significant interannual variability. Occasionally, severe low-temperature and snow disasters frequently occur in winter, such as the freezing event occurred in January 2008. Due to the anomalously strong East Asian winter monsoon, the severe low-temperature and snow disasters occurred in southwest, central and southern China, which caused economic losses of more than 150 billion yuan [42].

The dynamical processes of interannual variability of the East Asian winter monsoon system are closely related to the interannual oscillations of these two propagating waveguides of quasi-stationary planetary waves during boreal winter. Previous studies [22,23,43] proposed that the variability of these two waveguides exists an opposite oscillation on the interannual time scale. The above studies suggest that the interannual variability of the East Asian winter monsoon system is significantly related to the interannual oscillation of the propagating waveguides of quasi-stationary planetary waves. When the polar waveguide strengthens, then the low-latitude waveguide weakens, and anomalously strong quasistationary planetary waves in the troposphere will propagate towards the stratosphere through the polar waveguide, while the propagation of quasi-stationary planetary waves towards the upper troposphere near the low latitudes through the low-latitude waveguide will be weakened. In contrast, when the polar waveguide weakens, then, the low-latitude waveguide strengthens, and anomalously strong quasi-stationary planetary waves in the troposphere will propagate from middle latitudes towards the upper troposphere at low latitudes through the low-latitude waveguide. In this case, the propagation of anomalously strong quasi-stationary planetary waves in the troposphere from middle to high latitudes towards the top of the troposphere over low-latitude region via the low-latitude waveguide is stronger, while the propagation of quasi-stationary planetary waves to the stratosphere through the polar waveguide is significantly weaker.

Huang et al. [24] studied the influence of the interannual oscillations of the propagating waveguides of quasi-stationary planetary waves in the three-dimensional atmosphere on the interannual variability of the East Asian winter monsoon system and analyzed the relationship between the East Asian climate anomalies and boreal quasi-stationary planetary wave activity during the winters of 2005 and 2006. In the winter of 2005, the temperature from Western Europe through the Urals to Siberia and East Asia was lower than normal, and the temperature in the east side of the Urals and the northwest side of the Mongolian Plateau was more than 2 ◦C lower than normal, which resulted in the cold winter and frequent cold waves outbreaked in China. This in turn led to anomalously strong snow in Northwest and Northeast China as well as strong precipitation in the Yangtze River basin. However, in the winter of 2006, the temperature in the Siberia and East Asia was higher than normal, and the temperature in Europe from the Urals to Baikal Lake was more than 2 ◦C higher than normal, resulting in a warm winter in China.

The obvious difference between the winter climate in Eurasia in 2005 and that in 2006 was closely related to the oscillation of the propagating waveguides of quasi-stationary planetary waves in the Northern Hemisphere during these two winters. Figure 9a,b show the E-P fluxes and the scatterplot distributions of the quasi-stationary planetary waves during the boreal winters of 2005 and 2006, respectively. As shown in Figure 9a, the polar waveguide of the boreal quasi-stationary planetary wave propagation in the winter of 2005 was strong, while the low-latitude waveguide was weak, i.e., the propagation of quasi-stationary planetary wave along the polar waveguide to the stratosphere was enhanced. This caused that the convergence of planetary wave E-P fluxes was enhanced in the upper troposphere at high latitudes and weakened the convergence of planetary wave E-P fluxes in the upper troposphere in the subtropics, i.e., enhanced divergence of the E-P fluxes in this region, which caused the weakening of the boreal polar frontal jet and the strengthening of the subtropical jet. This can facilitate the development of high pressure systems over the Siberia and the strengthening of the East Asian winter monsoon. As shown in Figure 9b, the polar waveguide of quasi-stationary planetary wave propagation in the winter of 2006 was weak, while the low-latitude waveguide was strong, i.e., the propagation of quasi-stationary planetary waves along the low-latitude waveguide was enhanced in the upper troposphere at low latitudes. This caused that the convergence of planetary wave E-P fluxes was enhanced in the upper troposphere in the subtropical region, while the convergence of planetary wave E-P fluxes in the upper troposphere at high latitudes was weakened, i.e., the divergence of the E-P fluxes was enhanced. This caused the strengthening of the boreal polar jet and the weakening of the subtropical jet, which was detrimental to the development of the Siberian high and brought about the weakening of the East Asian winter monsoon. As shown in Figure 9c, the difference between the propagations of quasi-stationary planetary waves in these two boreal winters was also evident.

**Figure 9.** Composite distributions of the E-P fluxes (multiplied by ρ−<sup>1</sup> for displaying purpose) (vectors, units: m<sup>3</sup> s<sup>−</sup>2) of quasi-stationary planetary waves for wavenumbers 1–3 and their divergence (shaded, units: m<sup>3</sup> s−<sup>1</sup> d<sup>−</sup>1; *Y*-axis denotes vertical levels, units: hPa) over the Northern Hemisphere in the winters of (**a**) 2005 (December 2005 to February 2006) and (**b**) 2006 (December 2006 to February 2007), and (**c**) the difference between them. Solid and dashed lines indicate positive (divergence) and negative (convergence) of planetary wave E-P fluxes, respectively. And the divergence/convergence regions of the E-P flux are shaded with red/blue colors (from Huang et al. [24]).

#### *4.2. The Guiding Role of the Theory of Rossby Wave Energy Dispersion in the Study on the Dynamic Processes of the Interdecadal Variability of the East Asian Winter Monsoon System*

Wang et al. [25] have proposed that the East Asian winter monsoon system has not only significant interannual variability but also significant interdecadal variability. From the 1970s to the beginning of the 21st century, winter temperatures in China experienced two significant interdecadal variations. During the period from 1976 to 1987, the East Asian winter monsoon was strong, the winter temperature in China was generally low, and the frequency of cold wave outbreaks in China was high. However, during the period from 1988 to 1998, the East Asian winter monsoon was weak, the winter temperature in China was generally high, the frequency of cold wave outbreaks was significantly low, then, the warm winter frequently occurred. The result studied by Huang et al. [42] showed that during 1999–2010, winter temperatures in China changed significantly again, the colder temperature occurred in north China and warmer temperature appeared in south China. Moreover, temperature changed to a meridional oscillation pattern, and the interannual variability of winter temperatures changed from a 3–4 year cycle to a 2–8 year cycle. Our recent analysis shows that the variability of the East Asian winter monsoon during 2011–2020 was roughly similar to that during 1999–2010, and no a significant interdecadal variability occurred. During this period, several cold waves occurred in China, such as 21–25 January 2016, and 28–31 December 2020, where many regions in China experienced severe cooling and occurred low-temperature and snow disasters. These disasters caused severe economic losses.

The dynamic processes of interdecadal variability of winter climate in China and the East Asian winter monsoon system have been studied [25,42]. The results showed that the interdecadal variability of the East Asian winter monsoon system occurred in the midlate 1980s and late 1990s, which was closely related to the interdecadal oscillation of the propagating waveguides of quasi-stationary planetary waves in boreal winter. Recently, it is analyzed that the interdecadal oscillations of propagating waveguides of quasi-stationary planetary waves in the Northern Hemisphere using the NCEP/NCAR reanalysis data from 1961 to 2020. Figure 10a–c show the composite distributions of the E-P fluxes of quasistationary planetary waves for wavenumber 1–3 and their divergences over the Northern Hemisphere averaged for the winters of 1976–1987, 1988–1998, and 1999–2020, respectively. From Figure 10a, it can be seen that during the period of 1976–1987, the polar waveguide of quasi-stationary planetary waves for the boreal winters was strong, i.e., the propagation of planetary waves was strong along the polar waveguide up to the stratosphere over 60◦ N and weak along the low-latitude waveguide to the upper troposphere over low latitudes. These caused strong convergence of the E-P flux of planetary waves in the troposphere and stratosphere over high latitudes of the Northern Hemisphere and weak divergence of the E-P fluxes in the middle and upper troposphere over the subtropical region near 30◦ N. Moreover, as shown in Figure 10b, during the period of 1988–1998, the propagation of the boreal winter quasi-stationary planetary waves changed. Compared with Figure 10a, the polar waveguide of the boreal winter quasi-stationary planetary waves was weak during this period, while the low-latitude waveguide became strong. In other words, the propagation of planetary waves towards the stratosphere along the polar waveguide over high latitudes became weaker in the winters of 1988–1998, comparing with that in the winters of 1976–1987. And the propagation of planetary waves to the upper troposphere over low latitudes along the low waveguide became stronger, comparing with that in the winters of 1976–1987, which caused the E-P fluxes of the quasi-stationary planetary waves over high latitudes during winters of 1988–1998 were weaker than those during 1976–1987. This means that there was a positive difference of the divergence. But the divergence of the E-P fluxes of quasi-stationary planetary waves in the upper troposphere over the subtropical region became stronger in the winters of 1988–1998 than that in the winters of 1976–1987. In addition, as shown in Figure 10c, the polar waveguide of propagation of the boreal quasi-stationary planetary waves was again stronger and the low-latitude waveguide was weaker in the winters from 1999 to 2020. This means that the propagation

of planetary waves became stronger along the polar waveguide up to the stratosphere at high latitudes and weaker towards the upper troposphere of the subtropical region near 30◦ N via the low-latitude waveguide. The convergence of E-P fluxes of planetary waves in the upper troposphere and stratosphere at high latitudes was stronger than that in the winters of 1988–1998, but the divergence of planetary wave E-P fluxes in the upper troposphere over the subtropical region near 30◦ N was weaker than that in the winters of 1988–1998.

**Figure 10.** Composite distributions of E-P fluxes of quasi-stationary planetary waves for wave numbers 1–3 and their divergences (shaded, units: m3 s−<sup>1</sup> d<sup>−</sup>1; *Y*-axis denotes vertical levels, units: hPa) over the Northern Hemisphere averaged for the winters of (**a**) 1976–1987, (**b**) 1988–1998 and (**c**) 1999–2020. Solid and dashed lines indicate positive (divergence) and negative (convergence) divergence of E-P fluxes. Data of wind fields and temperature are from the NCEP/NCAR reanalysis data [30].

#### *4.3. Dynamic Effect of the Propagating Waveguide Oscillations of Quasi-Stationary Planetary Waves on the Variability of the East Asian Winter Monsoons*

The above results show that the boreal winter quasi-stationary planetary waveguides in the three-dimensional atmosphere have not only significant interannual oscillations but also significant interdecadal oscillations. Moreover, two significant interdecadal oscillations of the propagating waveguides of the boreal winter quasi-stationary planetary waves occurred since the 1970s. These oscillations of the propagating waveguides of quasistationary planetary waves caused the variability of the divergence or convergence of the E-P fluxes of quasi-stationary planetary waves. According to the wave-flow interaction equation for the spherical atmospheric planetary waves derived by Edmon et al. [29], the variation of the divergence of the E-P fluxes of quasi-stationary planetary waves will cause the variation of the zonal mean flow during boreal winter and the variation of the Arctic Oscillation (AO) index. According to previous studies [44,45], if the AO index is negative in a winter, the winter monsoon in East Asia is strong in the winter; conversely, if the AO index is positive in a winter, then the winter monsoon in East Asia is weak in the winter. Therefore, the interannual and interdecadal oscillations of the propagating waveguides of quasi-stationary planetary waves in the boreal winter will affect the interannual and interdecadal variabilities of the East Asian winter monsoon system.

It may see from the above studies that under the guidance of academician Ye's theory of Rossby wave dispersion, some studies on the dynamic processes of the variabilities of the East Asian winter monsoon system have been carried out. In particular, the study on the dynamic processes of the influence on the interannual and interdecadal oscillations of the propagating waveguides of quasi-stationary planetary waves on the East Asian winter monsoon variations has achieved an important progress.

#### **5. Conclusions and Discussion**

The theory of Rossby wave energy dispersion proposed by academician Ye in the 1940s not only improves the study on the characteristics of two-dimensional and threedimensional propagations of quasi-stationary planetary waves in the spherical atmosphere but also provides a scientific basis for the study on the mechanisms related to the anomalies of global atmospheric circulation. And this theory also lays the theoretical basis for the study on the dynamic processes of tropospheric-stratospheric interactions and their mechanisms. This paper reviews the guiding role of the theory of Rossby wave energy dispersion in the studies on the characteristics of three-dimensional propagation of quasi-stationary planetary waves in the spherical atmosphere and their impacts on the dynamic processes of the interannual and interdecadal variabilities of the East Asian summer and winter monsoon systems. In particular, this paper reviews the impacts of the interannual and interdecadal variabilities of the EAP pattern teleconnection wave train propagating along the meridional direction over East Asia and the Silk Road pattern teleconnection wave train propagating along the zonal direction in the subtropical jet from West Asia to East Asia on the variabilities of the East Asian summer monsoon system. This paper also reviews the studies on the dynamical processes of the impacts of the interannual and interdecadal oscillations of the propagating waveguides of quasi-stationary planetary waves on the variabilities of the East Asian winter monsoon system.

The energy dispersion of waves is an important theoretical problem in the fluid dynamics, academician Ye first applied it to the study on the mechanisms of atmospheric circulation variability. His research inspired the study on the dynamics of the two- and three-dimensional spherical propagations of quasi-stationary planetary waves, and guides the study on the dynamical processes of the variabilities of the East Asian winter and summer monsoon systems.

Under the guidance of academician Yeh's theory of Rossby wave energy dispersion, our research group has investigated the dynamical processes of the interannual and interdecadal variabilities of the East Asian winter and summer monsoon systems. The results show that the variabilities of the East Asian winter and summer monsoon systems are the variablities of circulation seeing from their phenomena, but these variabilities are also the variations of quasi-stationary planetary waves according to their mechanisms. Currently, this theory is developing and expanding, and it has wide applications not only in the study on atmospheric circulation dynamics at middle and high latitudes but also in the study on the dynamics of typhoon genesis and evolution in tropical regions [46–51]. Moreover, the nonlinear effects of the energy dispersion of Rossby wave in atmosphere and the nonlinear interactions of different quasi-stationary planetary wave trains also need to be further investigated.

**Author Contributions:** Conceptualization, R.H.; validation, R.H., J.H., Y.L. and R.L.; writing—original draft preparation, R.H. and J.H.; writing—review and editing, R.H., J.H. and Y.L.; visualization, Y.L., J.H. and R.H.; supervision, R.H. and R.L. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work was supported by the National Natural Science Foundation of China (41721004).

**Data Availability Statement:** The NCEP/NCAR reanalysis data can be found in the website: [https://psl.noaa.gov/data/gridded/data.ncep.reanalysis.html (accessed on 5 July 2021)]. The daily observational precipitation datasets in mainland China are provided by the National Meteorological Information Center of Chinese Meteorological Administration, which can be achived in the website: [https://data.cma.cn/ (accessed on 20 March 2021)].

**Acknowledgments:** We are thankful to the NCEP/NCAR and other scientific agencies for providing the datasets for our analysis. The authors are grateful for the comments and suggestions provided by the editor and anonymous reviewers.

**Conflicts of Interest:** The authors declare no conflict of interest.
