**3. Results**

#### *Climatology of Annual TLH, TST, Their Symmetric and Antisymmetric Parts*

Figure 2 shows the climatology of the annual means of TLH and TST and their symmetric and antisymmetric parts, respectively. The annual mean TLH in Figure 2a shows a zonal band of TLH maxima located between 20◦S and 20◦N over the whole-tropics, with the maxima value reaching over 150 W m<sup>−</sup>2, corresponding to the ITCZ and SPCZ over the Pacific and the ITCZ over the equatorial Atlantic and the monsoonal rainfall over the vast region from Africa to Asia and the Western Pacific, and the rainfall maxima over the Amazon Basin. The symmetric component of the TLH (Figure 2b) exhibits three centers: the largest is located at the Equatorial Indian Ocean to the Western Pacific Ocean, which corresponds well to the largest warm pool of SST (Figure 2); the second is located over the northern part of South America, which corresponds to a relatively low (land) surface temperature, but is also nearby warm Equatorial Atlantic SST at its east flank; the third and smallest center is located over Equatorial Africa, which again corresponds to a relatively low land surface temperature. The antisymmetric component of the TLH (Figure 2c) reveals that: (1) The positive atmospheric latent heating regions in the Northern Hemisphere are associated with the ITCZ over the Pacific and Atlantic and monsoonal precipitation over Asia and maritime continent; (2) The positive atmospheric latent heating regions in the Southern Hemisphere are associated with the SPCZ over the Pacific and ITCZ over the Indian Ocean, and also the two broad northwest–southeast-oriented regions over Southern Africa to the Western Indian Ocean and over South America to West Atlantic. By comparing

the corresponding annual-mean TST pattern (Figure 2d) and its symmetric (Figure 2e) and antisymmetric (Figure 2f) components with the TLH pattern, we find that the high TLH (rainfall) zones over the sea surface correspond to high SST, but the high TLH zones over the land surface are associated with relatively lower LST. Indeed, the high LST regions are usually desert areas with little rainfall, but the low LST regions are usually associated with tropical forests. This asymmetry in the TLH-TST relation between the land and sea surfaces may have important implications for tropical atmospheric dynamics.

**Figure 2.** The left panel: (**a**) climatology of the annual-mean TRMM-based tropical latent heating (TLH), and its (**b**) symmetric and (**c**) antisymmetric parts. Unit: W/m2. The right panel: (**d**) climatology of the annual-mean tropical surface temperature (TST) and its (**e**) symmetric and (**f**) antisymmetric parts. Units: ◦C.

To further investigate the relationship between TST and TLH, SVD analysis is employed for the original seasonal TST anomaly and the original seasonal TLH anomaly. Note the anomalies in both fields mean that the mean seasonal cycles of TLH and TST have been removed. It can be observed in Figure 3a,b that the leading SVD mode of the TST and TLH is related to El Niño–Southern Oscillation (ENSO), with anomalously warmer SST over the Equatorial Central and Eastern Pacific, cooler SST over the Western Pacific, and also anomalously warmer SST over the Indian Ocean. Correspondingly, anomalously larger TLH is associated with more precipitation over the Central Pacific (near 180◦E) and the Equatorial Indian Ocean; also, equatorially shifted ITCZ and SPCZ are found over the Pacific, while anomalously smaller TLH dominates over the Western Pacific (Figure 3b).

**Figure 3.** The left (**a**) and right (**b**) spatial pattern of the first mode of SVD using TST and TRMMbased TLH. The squared covariance fraction of the first mode, expressed as a percent, is printed on the upper right-hand corner of each map. (**c**,**d**) and (**e**,**f**): same as (**a**,**b**), but using symmetric and antisymmetric components of TST and TLH, respectively. (**g**,**h**): The time series of the expansion coefficient of the three left patterns and three right patterns, respectively, the blue dotted line is the seasonal-averaged Nino 3.4 index.

To compare the difference between the symmetric and antisymmetric TSTs' connection with the original TLH, SVD analyses are also applied to the symmetric and antisymmetric components of TST paired with the original TRMM-based TLH as shown in Figure 3c,d for the first SVD mode of symmetric TST and original TLH, and Figure 3e,f for the first SVD mode of antisymmetric TST and original TLH. We surprisingly find that the first SVD modes of original TLH corresponding to the first SVD modes of symmetric and antisymmetric components of TST show very high similarities (Figure 3d,f), which resemble the pattern in Figure 3b. This seems at odds with the classic Mastuno–Gill theory, which implies a one-toone correspondence between the (anti) symmetric forcing and (anti)symmetric atmospheric response. To verify the above results, we check the time series of the expansion coefficient of the three left (TST) patterns (Figure 3g) and three right (TLH) patterns (Figure 3h). The high correlation coefficients between each other indicate that the leading symmetric and antisymmetric SVD modes co-vary temporally (Table 1) and are highly consistent with the evolution of El Niño events in the tropical Pacific (Table 2).

**Table 1.** The correlation coefficients between the time series of the expansion coefficient of three left and right SVD patterns.



**Table 2.** The correlation coefficients between the time series of the expansion coefficient of three left and right SVD patterns and the Oceanic Niño Index.

To further prove the results obtained from the SVD analysis, we independently employed the EOF analysis for the original seasonal TST anomaly and its symmetric and antisymmetric components (Figure 4). We find the first EOF (EOF1) mode of the seasonal TST anomaly resembles the first left (TST) SVD model in Figure 3a, both indicating the pattern of El Niño events. Furthermore, the EOF1 mode of the symmetric TST in Figure 4b shows a very high similarity with the EOF1 of the original TST (Figure 4a), with the pattern correlation being 0.99, with a significance level less than 0.01. However, the EOF1 mode of the antisymmetric TST (Figure 4c and Table 3) indicates there is also a weaker but non-negligible antisymmetric TST component across the whole-tropics. In addition, the correlation between the corresponding principle components (PCs) of the EOF1 of symmetric and antisymmetric TSTs is 0.82 (Figure 4c and Table 3), with a significance level of less than 0.01. In short, it can be concluded that while the principal EOF mode of TST interannual variability is dominated by its equatorially symmetric component, there is a non-negligible equatorially antisymmetric component that well co-varies with the symmetric part over most parts of the tropical land and ocean areas. Figure 4d (see also the correlation coefficients in Table 4) shows that the PC1s of original, symmetric, and antisymmetric TST fields are consistent with the evolution of El Niño conditions in the tropical Pacific.

Then we further reveal the link between the symmetric or antisymmetric components of TST and the TRMM-based TLH by calculating the correlation (Figure 5a,e) and regression (Figure 5b,f) of the TLH with/onto the corresponding PCs of the EOF1 of symmetric and antisymmetric TSTs. The regression patterns of TLH in Figure 5b,f are also further separated into equatorially symmetric and antisymmetric components in Figure 5c,d (regressed onto PC1 of the symmetric TST) and in Figure 5g,h (regressed onto PC1 of antisymmetric TST), respectively. It is obvious that the correlation or regression patterns of the TLH associated with the PC1 of symmetric TST (the left panel of Figure 5) are very similar to that associated with the PC1 of antisymmetric TST (the right panel of Figure 5). As such, we confirm the results obtained from the SVD analysis are right.

**Table 3.** The correlation coefficients between the principle components of the EOF1 of TST, TST\_SYM, and TST\_ASYM.


**Figure 4.** The first EOF (EOF1) mode of the seasonal-mean: (**a**) original, (**b**) the symmetric part and (**c**) antisymmetric components of TST over the whole-tropics, and (**d**) the corresponding principle components for (**a**–**c**).

**Table 4.** The correlation coefficients between principle components of the EOF1 of TST, TST\_SYM, TST\_ASYM, and the Oceanic Niño Index.


**Figure 5.** (**a**) The correlation and (**b**) the regression of the TRMM-based TLH with/onto the PC1 of equatorially symmetric TST. (**c**) and (**d**) are the symmetric and antisymmetric parts of (**b**), respectively. (**e**–**h**) are the same as (**a**–**d**), but associated with the PC1 of equatorially antisymmetric TST. The areas with a significance level of less than 0.05 are dotted.

To depict the geographic dependence of the symmetric–antisymmetric connection, the local correlations between the symmetric and antisymmetric components of both TST and TLH are calculated in Figure 6a,b. By definition, the high correlation in Figure 6 implies the co-variation between symmetric and antisymmetric components, in which the regions with positive correlation further indicate the dominant regions in the symmetric–antisymmetric connection. Indeed, this corresponds to the condition of equatorially asymmetric variation, i.e., temporal variation is strong in one hemisphere only while very weak in the other hemisphere. On the other hand, low correlations mean that one of the equatorially symmetric and antisymmetric components is dominant, and the symmetric and antisymmetric components vary independently (i.e., being mathematically orthogonal with each other). The low correlation region for TST mainly extends from the Equatorial West Pacific to the Middle Equatorial Pacific (Figure 6a), consistent with the equatorially symmetric SST variability over this region, while the high positive correlations (Figure 6a) in the tropics are located over the Southeastern Pacific, Northern Equatorial Atlantic, North Africa, Australia, South Asia, Southern South America, clearly being consistent with asymmetric TST variability associated with the equatorially asymmetric land–sea distribution. On the other hand, the low correlation regions for TLH (Figure 6b) are mainly limited in the narrow regions of equatorial Africa, the Indian Ocean, western to middle equatorial Pacific, and equatorial South America, but the regions with a high positive correlation of TLH are consistent with the monsoonal rainfall over Africa, Asia, and North America, and also the ITCZs over the Pacific and Atlantic in the Northern Hemisphere, with the SPCZ over the Pacific, ITCZ over the Indian Ocean, and rainfall region over Southern Africa and Southern America in the Southern Hemisphere.

**Figure 6.** The local correlation between equatorially symmetric and antisymmetric components of (**a**) TST and (**b**) TRMM-based TLH. The areas with a significance level of less than 0.05 are dotted.

Figure 6 indicates that the symmetric and antisymmetric components of both TST and TLH co-vary over vast regions of the tropics, and hence the corresponding Matsuno–Gill modes are intrinsically coupled with each other in the tropical ocean-atmosphere–land system. Untangling their relationship through model simulations and mechanistic analysis is needed for a better theoretical understanding of tropical dynamics.

#### **4. Summary and Discussion**

By utilizing seasonally averaged satellite-based TRMM precipitation data as a proxy of tropical latent heating (TLH) and ERA5-based tropical surface temperature (TST) data from 1998 to 2018, we investigate the cross-hemispheric connection in the TLH and TST variability and their co-variability. The interannual variability of both the TST and the TLH is equatorially asymmetric and can be decomposed as the sum of equatorially symmetric and antisymmetric components. Based on the decomposition, we reveal some new features of variability and co-variability of the TST and TLH. The main results are summarized as follows:


and antisymmetric PCs of TST are both nearly coincident with the ENSO index during the 21 years of 1998–2018.

While these results are obtained for all four seasons with the mean seasonal cycle being removed, we note they basically hold individually for each season. Results on individual season-based analyses will be reported later in more detail.

The above results raise some interesting puzzles in the theoretical understanding of tropical atmospheric dynamics. First is that if they are really at odds with the classic Matsuno–Gill theory [1,2] or not. We might expect from the linear Matsuno–Gill theory a one-to-one correspondence between equatorially (anti) symmetric TST forcing and equatorially (anti) symmetric TLH pattern, but due to the intrinsic nonlinearity in tropical dynamics, say, related to convective precipitation, the one-to-one correspondence may at least partially be broken up. Because of the strong co-variability in the observed TST-TLH relation, we may not obtain mechanistic understanding directly from statistical analysis of the observations. Well-designed modeling experiments and theoretical analysis are needed for a decisive solution to the puzzle.

The second puzzle is to what extent are the equatorially symmetric and antisymmetric components of the joint TST-TLH variability interactive with each other? What is the underlying mechanism responsible for the interaction? Indeed, it is quite reasonable to assume that the Matsuno–Gill theory still holds to some degree for the interannual joint TST-TLH variability, but the departure from the theory due to nonlinear interactions may be essential to better understand and predict the tropical variability.

While the PCs of joint TST-TLH variability are clearly associated with the ENSO cycle in the tropical Pacific, the co-variability over the Indian Ocean, tropical Atlantic, and tropical land areas should not be neglected from our analysis. Recent studies have suggested the importance of pantropical interaction or cross-basin interaction in tropical dynamics [6,7]. We further suggest that a whole-tropics perspective that takes the different but connected nature of equatorially symmetric and antisymmetric modes across the wholetropics into consideration may well be useful in understanding and predicting tropical climate variability.

#### **5. Historical Note**

After graduating from the University of Chicago with his famous thesis on energy dispersion in the atmosphere, T.C. Yeh stayed there and became a member of the team on tropical dynamics, second to H. Riehl, the head of the team. The other two members were J. Malkus (J. Simpson) and N. LaSeur. Before his return to China, he published two classic papers on the intensity of the Hadley cell [23] and on trade inversion [24], among others. Although later on he did not focus on tropical dynamics in his lifelong career, his works on tropical dynamics also clearly reflect his style and character in research, i.e., being thorough, systematic, and insightful. This short note is devoted to Prof. Yeh's contribution to the field of tropical dynamics—J. Lu.

**Author Contributions:** Conceptualization, J.L.; methodology, J.L.; validation, Y.G., X.L. and J.L.; formal analysis, Y.G. and X.L.; resources, J.L.; data curation, Y.G.; writing—original draft preparation, J.L. and X.L.; writing—review and editing, J.L.; visualization, Y.G.; supervision, J.L.; project administration, J.L.; funding acquisition, J.L. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by National Natural Science Foundation of China, grant number 42175070.

**Data Availability Statement:** All of data are available by acquiring from the authors.

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