**1. Introduction**

We define equatorial symmetry, antisymmetry, and asymmetry of any tropical climate variable as the exact same, exact opposite, and different states of that variable at the same latitude in the two hemispheres centered at the Equator. Clearly, any equatorially asymmetric variable can be decomposed as the sum of equatorially symmetric and antisymmetric components of that variable.

Matsuno, in his classic paper, obtained a mathematical solution for the eigenmodes of free tropical waves [1]. Based on their equatorially symmetric or antisymmetric nature, these eigenmodes are associated with totally different spatiotemporal structures of wind and pressure. Gill (1980) presented the spatial structure of stationary, forced tropical atmospheric general circulation in response to thermal forcing, which may be either equatorially symmetric or antisymmetric [2]. For instance, in response to an equatorially symmetric forcing with the maximum located at the Equator, stationary Kevin wave response

**Citation:** Gao, Y.; Liu, X.; Lu, J. Tropical Surface Temperature and Atmospheric Latent Heating: A Whole-Tropics Perspective Based on TRMM and ERA5 Datasets. *Remote Sens.* **2023**, *15*, 2746. https://doi.org/ 10.3390/rs15112746

Academic Editor: Itamar Lensky

Received: 19 April 2023 Revised: 16 May 2023 Accepted: 22 May 2023 Published: 25 May 2023

**Copyright:** © 2023 by the authors. 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/).

(Walker-circulation type response) appears at the east side of the forcing, while equatorially symmetric stationary Rossby waves appear at the northwest and southwest sides of the forcing. On the contrary, when the thermal forcing is equatorially antisymmetric, the equatorially antisymmetric stationary mixed Rossby-gravity wave and Rossby wave responses, which may mainly be zonally confined nearby the forcing, lead to a Monsoon-type response with the presence of a cross-equatorial flow and cyclonic response in one hemisphere and anti-cyclonic response in the other hemisphere.

It is well known that tropical sea surface temperature (SST) variability, including the ENSO variability in the Pacific and the modes of SST variability in the tropical Indian Ocean and Atlantic, may affect both the tropical atmospheric and global circulation through atmospheric heating and teleconnections [3–5]. Indeed, various meteorological disasters, such as drought, flooding, cold spells, and heat waves, can be associated with the tropical surface temperature forcing in different tropical ocean basins. While many of the studies have been focused on the modes of thermal forcing in individual ocean basins, some studies have considered the cross-basin interaction in the tropics during recent years [6,7].

Given the importance of atmospheric heating in shaping the response of general atmospheric circulation to tropical SST variability, many studies have paid attention to the linkage between SST and tropical atmospheric convection (or tropical latent heating, TLH). While the SST-convection (also TLH or precipitation) linkage can be local, this local connection is highly nonlinear and depends on the large-scale background of tropical circulation [8–12]. On the other hand, the changes in tropical rainfall patterns as responses to modes of tropical SST variability such as ENSO, Indian Ocean Dipole (IOD), and Atlantic Niño are also extensively investigated [6,7].

Due to the hemispheric asymmetry of land–sea distribution, equatorial asymmetry prevails in both the mean climate and the variability of tropical SST and precipitation/ latent heating. Indeed, various mechanisms responsible for the equatorial asymmetry of tropical SST and ITCZ have been proposed, such as the wind–evaporation-feedback mechanism [13], asymmetric upwelling associated with meridional oceanic heat transport [14,15], and coupled atmosphere–ocean energy balance constraint associated with cross-equatorial ocean transport [16], the asymmetric oceanic transport by north equatorial countercurrent (NECC) [17], among others.

It is interesting to note, however, that the decomposition of an asymmetric SST pattern into the sum of equatorial symmetric and antisymmetric components has rarely been conducted, except in an analysis of the interaction of annual-cycle and interannual variability of SST in the tropical Eastern Pacific [18]. The reason is that the hemispheric asymmetry of land–sea distribution makes it impossible to find an SST pair when the underlying boundary is ocean in one hemisphere but is land in the other. However, this difficulty can be easily overcome by using tropical surface temperature, including both SST and land surface temperature, rather than SST only.

Because of the small heat inertia of the land surface, the variability of tropical land surface temperature (LST), in which there are more short time-scale oscillations, is usually considered as a response to SST forcing. While this is physically reasonable, it should be noted that lower frequency variability of LST also exists, and just as SST, LST renders the atmosphere with underlying boundary conditions of momentum, heat, and moisture. Furthermore, the SST-LST asymmetry itself may well lead to an asymmetric response in atmospheric motions. Therefore, we take a different, whole-tropics (30◦S–30◦N) perspective, which considers tropical SST and LST together, labeled as tropical surface temperature (TST) hereafter.

In this short note, we decompose the TST and TLH into equatorially symmetric and antisymmetric parts and then investigate the relation between the symmetric and antisymmetric components separately for TST and TLH based on correlation analysis. Furthermore, we investigate the relation between interannual TST variability and interannual variability of TLH across the whole-tropics, rather than with the SST confined in a single tropical ocean

basin, by applying standard singular value decomposition (SVD), empirical orthogonal function (EOF), correlation, and regression methods.

The layout of the note is as follows. The data and method are presented in Section 2, while Section 3 presents the analysis of the main results. The summary and discussion are in Section 4.

#### **2. Data and Methods**

#### *2.1. Data*

The Tropical Rainfall Measuring Mission (TRMM) produces tropical and subtropical estimates of precipitation based on remote observations. The TMPA (TRMM Multisatellite Precipitation Analysis) version 7 dataset [19] obtained from the NASA archive (ftp:// disc2.nascom.nasa.gov/ftp/data/s4pa//TRMM\_L3/, accessed on 24 April 2021) and aggregated to a high spatial (0.25◦) and temporal (daily) resolution are employed in this study, with the time domain ranging from 1998 to 2019.

The single-level atmospheric and oceanic variables, named skin temperature and sea surface temperature, are obtained from the European Centre for Medium-Range Weather Forecasts (ECMWF) Reanalysis 5 (ERA5) datasets. These two variables are merged to obtain the TST dataset. They were downloaded with a spatial resolution of 0.25◦ × 0.25◦ and monthly temporal resolution from 1998 to 2019 and then interpolated into a resolution of 1.0◦ × 1.0◦ using bilinear interpolation for this study.

The Oceanic Niño Index (ONI) is one of the ENSO indices, based on the SST in the Niño 3.4 region, which is obtained from https://climatedataguide.ucar.edu/climate-data/ nino-sst-indices-nino-12-3-34-4-oni-and-tni (accessed on 8 November 2022).

#### *2.2. Methods*

The variables, for example, the TST or TLH, are divided into two components: equatorially symmetric and antisymmetric. Assuming the TST over the Northern Hemisphere is A, and the TST over the Southern Hemisphere is B with the latitude reversed. The symmetric component is defined as 1/2 (A + B), and the antisymmetric component is 1/2 (A − B) in the Northern Hemisphere and 1/2 (B − A) in the Southern Hemisphere. Therefore, the original field can be divided into the sum of symmetric and antisymmetric fields. Note such a decomposition on the TST, rather than on SST only, can take the intrinsic asymmetry of land–sea contrast into consideration, and hence it may better reveal the coupling of land, sea, and atmosphere over the tropics.

The singular value decomposition analysis (SVD) method of detecting temporally synchronous spatial patterns is also used in this study. The method is based on a singular value decomposition of the matrix whose elements are covariances between observations made at different grid points in two geophysical fields, for example, TST and TLH, in this study. Here, we briefly describe the SVD method following Wallace et al. (1992) [20] and Hu (1997) [21]. TST and TRMM-based TLH data are denoted as *s*(*x*, *t*) and *z*(*x*, *t*), respectively, where *x* is space location, and *t* is time. The SVD analysis is then a linear transformation:

$$s(\mathbf{x}, t) \approx \sum\_{n=1}^{N} a\_n(t) p\_n(\mathbf{x}) \tag{1}$$

$$z(\mathbf{x}, \mathbf{t}) \approx \sum\_{n=1}^{N} b\_n(\mathbf{t}) q\_n(\mathbf{x}) \tag{2}$$

in which the pairs of coupled spatial patterns, *pn*(*x*) and *qn*(*x*) (also called left and right SVD spatial patterns, respectively), and their temporal expansion coefficients, *an*(*t*) and *bn*(*t*), are identified. Here N is the number of SVD modes. As described by Zhang et al. (2018) [22], the SVD analysis was conducted as follows: First, the cross-covariance matrix of *s*(*x*, *t*) and *z*(*x*, *t*), *Csz*, was calculated. Secondly, the eigenvalues (also called singular values) *σ<sup>n</sup>* of the matrix were obtained by solving |*Csz* − *σI*| = 0 where *I* is the identity matrix. Next, the eigenvectors (that is, the SVD patterns, *pn*(*x*) and *qn*(*x*)) corresponding to each eigenvalue were obtained. *pn*(*x*), *qn*(*x*), *an*(*t*), and *bn*(*t*) are the components of an SVD mode. All modes are arranged so that their *σ<sup>n</sup>* appear in descending order. The first

pair of singular patterns describes the largest fraction of the squire covariance between the two fields, and each succeeding pair describes a maximum fraction of square covariance that is unexplained by the previous pairs. The contribution of the nth mode to the total covariance of the two fields is measured by squared covariance fraction:

$$\text{SCF}\_{\text{ll}} = \sigma\_{\text{n}}^{2} / \sum\_{n}^{N} \sigma\_{n}^{2} \tag{3}$$

To verify the results obtained from the SVD analysis, we further apply the standard empirical orthogonal function (EOF) decomposition to the symmetric and antisymmetric TST fields and then apply the linear regression method onto the TLH field, i.e., by regressing the monthly TLH field upon the principal components (PCs) of the first EOF (EOF1) of symmetric and antisymmetric TST fields. The independent EOF and regression analyses are necessary to verify the robustness of the conclusion obtained from the SVD analysis.

We also calculate the correlation coefficients between symmetric and antisymmetric components of TST (or TLH) locally at each grid point over the tropics. Understandably when the variable at a given location is equatorially asymmetric, the correlation is positive in one hemisphere and negative in another hemisphere; but when the variable at a given location is mainly equatorially symmetric or antisymmetric, then the correlation is weak because one of the symmetric or antisymmetric components is close to zero in this case. As such, the spatial pattern of the correlation may reveal the meridional structure of the variation of TST or TLH.

Figure 1 summarizes the flowchart of methods adopted in this study.

**Figure 1.** Flowchart of the methods used in this study.
