Kelvin waves from the equatorial Indian Ocean modulate the nonlinear internal waves in the Andaman Sea

In the equatorial Indian Ocean, strong westerly and easterly wind anomaly can drive eastward downwelling and upwelling Kelvin waves, respectively, which play an important role in determining the circulations and thermal structures near the equator. Kelvin waves can propagate into the Andaman Sea, a marginal sea located to the northeast of the Indian Ocean. In the Andaman Sea, nonlinear internal waves (NLIWs) that are crucial in facilitating the mixing in the ocean interior and maintaining the ecosystem are found to be extremely active. Although both equatorial Kelvin waves and NLIWs have been well known in oceanography, the influence of equatorial Kelvin waves on NLIWs in the Andaman Sea remains unclear. In this study, based on long-term mooring measurements in the southern Andaman Sea, it is found that the NLIW amplitude shows remarkable intraseasonal and seasonal variances, and these variances can be mostly explained by the occurrence of equatorial Kelvin waves. Downwelling Kelvin waves can deepen the thermocline depth by tens of meters, so that the NLIW amplitude can be reduced up to 22%. Meanwhile, upwelling Kelvin waves can notably uplift the thermocline depth and the NLIW amplitude can be enhanced up to 32%. These discoveries provide credible evidence that basin-scale waves from the open ocean can remotely modulate small-scale internal waves in marginal seas.


Introduction
Internal waves, or internal gravity waves, are ubiquitous in the interior of the stratified ocean. They can be initiated by a variety of external disturbances, such as wind stress at the sea surface, low-frequency flows in the ocean interior, and barotropic tidal currents at the sea bottom (Garrett and Munk 1979, Jing et al 2018, Whalen et al 2020. According to the theoretical framework followed, internal waves can be roughly classified into linear internal waves and nonlinear internal waves (NLIWs), described by linear and nonlinear theories, respectively. NLIWs, also referred to as internal solitary waves or internal solitons, are mainly generated when barotropic tidal currents pass through abrupt bathymetry, such as sea mounts, underwater ridges and continental shelves (Jackson et al 2012). Compared to linear internal waves, NLIWs have a large vertical isopycnal displacement and a strong current within a short period, and thus NLIWs play a key role in inducing turbulent mixing and redistributing energy and materials in the oceans (Moum et al 2003, Shroyer et al 2010, Huang et al 2016.
NLIWs in the Andaman Sea have been known for a long time (Perry andSchimke 1965, Osborne andBurch 1980). They are generated at the narrow shallow straits in the west and the continental shelves in the east of the Andaman Sea and are widely distributed in the basin (black curves in figure 1(b)). NLIWs are important in maintaining the ecosystem in the Andaman Sea, especially for coral reef development (Roder et al 2010, Wall et al 2015, Liu et al 2022. The measurements from two long-term moorings (Yang et al 2021) revealed evident temporal variations in the NLIWs in the Andaman Sea. The NLIWs have averagely greater amplitudes in the boreal spring and autumn, which is quite different from the seasonal variations in other NLIW hotspots, such as the South China Sea and the Bay of Biscay, where NLIW amplitude peaks in summer (Gerkema 2001, Huang et al 2022. In the equatorial Indian Ocean (EIO), strong westerly and easterly anomaly can lead to downwelling and upwelling Kelvin waves, respectively (Wyrtki 1973, Rao et al 2016. In situ observations in the EIO reveal that Kelvin waves lead to remarkable disturbances in the thermocline and background currents during their propagations McPhaden 2011, Rao et al 2016). When eastward Kelvin waves reach the eastern boundary at Sumatra Island (figure 1(a)), part of their energy is reflected and propagates westward as Rossby waves (Yu et al 1991, Han 2005. Another part of their energy propagates northward and southward along the coast of Sumatra Island in the form of coastal Kelvin waves. The southern branch of the coastal Kelvin waves propagates further southeastward and reaches the passages of the Indonesian throughflow, modulating the magnitude and vertical structures of the throughflow (Qiu et al 1999). The northern branch of the coastal Kelvin waves enters the Andaman Sea after bypassing Sumatra Island and then travels northward along the eastern boundary of the basin (Rao et al 2010). Given that NLIWs are active in the Andaman Sea, we have two questions here: what are the influences of Kelvin waves on NLIW behaviors in the Andaman Sea? What role do Kelvin waves play in determining the particular temporal variations in NLIWs in the Andaman Sea?
It is well-known that equatorial Kelvin waves respond directly to global climate changes. Specifically, the strength and vertical structure of Kelvin waves in the EIO are closely related to the Indian Ocean Dipole (IOD) (Vinayachandran et al 1999, Iskandar et al 2008. By examining the modulations by basin-scale Kelvin waves on small-scale NLIWs, we can further understand the connection between global-scale climate processes and smallscale internal wave behaviors, with Kelvin waves as a bridge. However, the influences of equatorial Kelvin waves on NLIWs have received little attention in the past and remain unclear. Motivated by these unknowns, the data collected from a long-term mooring in the southern Andaman Sea are employed in this study to investigate the modulations by the Kelvin waves from the EIO on the NLIWs in the Andaman Sea. The variation in NLIW amplitude when Kelvin waves prevail is, to the authors' knowledge, accurately estimated for the first time.

Data
The data collected by a subsurface mooring named QB2 deployed in the southern Andaman Sea (96.13 • E, 5.93 • N, depth ∼1340 m; the red pentagram in figure 1(b)) from December 2016 to September 2018 are employed in this study. At the mooring, two 75 kHz RDI Long Ranger acoustic Doppler current profilers (ADCPs) were mounted at a nominal depth of 500 m, with one looking upward and the other looking downward. The ADCPs sampled at 3 min interval, and the bin number and bin size of the ADCPs were 37 and 16 m, respectively. Due to the exhaustion of battery power, the ADCPs stopped working in July 2018. In addition, the mooring was equipped with 21 Sea-Bird Electronics (SBE)-56 temperature loggers and 4 SBE-37 CTD sensors to monitor the temperature over depths of ∼50-1200 m with a sampling interval of 3 min. A detailed configuration of the mooring and preliminary processing of the data have been introduced by Yang et al (2021).
To visualize the horizontal propagation of Kelvin waves, we used the satellite-derived sea surface height anomaly (SSHA) obtained from the E.U. Copernicus Marine Service. Moreover, the daily averaged 10 m wind obtained from the NCEP/NCAR Reanalysis was used to calculate the zonal wind anomaly in the EIO.

Separating intraseasonal and seasonal components
According to the driving mechanisms, the nearequatorial winds in the Indian Ocean drive Kelvin waves mainly in two frequency bands (figure 2). Intraseasonal atmospheric oscillations, so-called Madden-Julian oscillations, are characterized by surface wind anomaly in the EIO with periods ranging from 30 to 90 d (Zhang 2005). Meanwhile, the surface wind anomaly in the EIO also oscillates in a semiannual period, determined by the monsoon prevailing/transition. Accordingly, we separate the zonal wind anomaly in the EIO into intraseasonal and seasonal components with periods of 15-90 d and >90 d, respectively, giving rise to corresponding intraseasonal Kelvin waves (IKWs) and seasonal Kelvin waves (SKWs), respectively.

Time window moving average
The NLIWs in the Andaman Sea occur intermittently, and the observed time series of NLIWs are unevenly spaced (Yang et al 2021). To obtain the intraseasonal  and seasonal variation tendencies of NLIW amplitudes, we have employed the time window moving average method. We denote the unevenly spaced time series of the observed NLIW amplitudes by A (n) = A (t n ) = (A t1 , . . . , A tN ), where n is the serial number, t n = (t 1 , . . . , t N ) is the observation times, and N indicates the number of NLIWs observed over the whole observation period. At time t n , the moving average of NLIW amplitude A within a time window T is calculated throughĀ Here, N T is the number of NLIWs within a time window, and N T consistently varies when the time window 'moves' along the observed time series of NLIW amplitudes. By applying the above time window moving average method with a specific time window T, we can obtain the temporal variation tendency of NLIW amplitudes with periods longer than T. We validated the time window moving average method by applying it to the continuous velocity measurements (figure S1 in the supplementary material). This method has been shown to be a useful tool for analyzing the long-term variation tendency of NLIWs. In this study, to obtain the seasonal variation tendency of NLIW amplitudes, we have set the time window T into 90 d and then employed the time window moving average method introduced above. Meanwhile, by setting the time window T into 15 d, we can obtain the variation tendency of NLIW amplitudes with periods longer than 15 d through employing the time window moving average method, and then the intraseasonal variation tendency of NLIW amplitudes can be obtained by subtracting the 90 day movingaverage results from the 15 day moving-average results.

Modulation by IKWs on nonlinear internal waves
The corresponding relationship between the zonal wind anomaly in the EIO and Kelvin waves for the intraseasonal component is shown in figures 3(a) and (b). The strong westerly anomalies in the EIO (positive values in figure 3(a)) lead to positive SSHAs ( figure 3(b)), which correspond to downwelling IKWs. Similarly, the strong easterly anomalies (negative values in figure 3(a)) induce negative SSHAs and give rise to upwelling IKWs. The generated IKWs then propagate eastward and gradually strengthen. When reaching Sumatra Island (position B), the northern branch of the IKWs propagates along the southern coast of Sumatra Island (B)-(C) and finally enters the Andaman Sea (C)-(D) (figure 1(a) and figure 3(b)). To identify the occurrence of Kelvin waves, we used a threshold of 2 (−2) cm of the SSHAs at [95 • E, 0] to determine the occurrence of a downwelling (upwelling) IKW event. In this way, there were a total of 12 downwelling and 11 upwelling IKW events during the mooring observation period (marked with red and blue triangles in figure 3(b), respectively). The strengths between these Kelvin wave events differ significantly, which is due to the differences in the strengths and fetches of zonal winds when driving Kelvin waves ( figure 3(a)). On average, the downwelling and upwelling IKWs induced an SSHA at [95 • E, 0] by 4.8 and −4.1 cm, respectively.
When the equatorial Kelvin waves entered the Andaman Sea, they gave rise to an evident disturbance in the upper-layer stratification. The measurements at mooring QB2 showed that the downwelling IKWs on average deepened the thermocline depth by 10.0 m, and the upwelling IKWs on average uplifted the thermocline depth by 8.0 m ( figure 3(c)). The evident changes in the background environment in the Andaman Sea induced by equatorial Kelvin waves had a remarkable influence on the behaviors of NLIWs. In figure 4, we compared the NLIWs when typical downwelling (figure 4(b)) and upwelling (figure 4(d)) Kelvin waves occurred. Comparatively, with a shallower thermocline (magenta curve), the NLIW packets during the upwelling Kelvin wave event (blue arrows in figure 4(d)) have more NLIWs and larger wave amplitudes than those during the downwelling Kelvin wave event (red arrows in figure 4(b)). Quantitatively, we calculated the amplitudes of the NLIW packets (i.e., the amplitude of the NLIW that has the greatest amplitude in the packet) during these two periods. The average amplitude of the two NLIW packets during the upwelling Kelvin wave event was 44.4 m, 26.5% larger than that during the downwelling Kelvin wave event with a magnitude of 35.1 m.
The modulation by equatorial Kelvin waves on NLIW amplitudes is clearer by examining the occurrence of Kelvin wave events and the corresponding variations in NLIW amplitudes ( figure 3(d)). In general, a downwelling and an upwelling IKW could notably reduce and enhance the NLIW amplitudes, respectively. Over the whole mooring observation period, the downwelling IKWs (red triangles in figure 3(b)) reduced the NLIW amplitude by  figure 3(b)) enhanced the NLIW amplitude by an average of 4.1 m, accounting for 10.0% of the amplitude averaged over the whole observation period. It is worth noting that the upwelling IKW in September 2017 enhanced the NLIW amplitude by as much as 13.1 m, accounting for 32.0% of the wave amplitude averaged over the whole observation period. The variances in NLIW amplitudes, as well as SSHAs and thermocline depths, induced by each IKW event are shown in detail in figure S4 and tables S1 and S2 in the supplementary material.

Modulation by SKWs on nonlinear internal waves
Although the driving mechanism is different, the SKWs showed similar responses to the zonal wind anomaly in the EIO as the IKWs. During the boreal summer (June-September) and winter (December-March) monsoons, the EIO was dominated by easterly anomalies ( figure 5(a)), which gave rise to upwelling SKWs (blue triangles in figure 5(b)). During the boreal spring (April-May) and autumn (October-November) monsoon transition periods, the westerly anomalies were strong in the EIO and gave rise to downwelling SKWs (red triangles in figure 5(b)). Clearly, there were two pairs of upwelling and downwelling SKWs each year. However, due to the differences in the strengths and fetches of equatorial zonal winds when driving Kelvin waves, the intensities between these SKWs were different. The upwelling SKWs during the winter monsoons (1# and 3# blue triangles) induced an SSHA at [95 • E, 0] by approximately −2 cm and were more obvious than the one generated during the summer monsoon (2# blue triangle). In contrast, the SSHAs induced by the downwelling SKWs during the spring and autumn monsoon transition periods (1# and 2# red triangles) were close to each other, with a magnitude of approximately 14 cm.
The occurrences of SKWs corresponded very well with the variations in thermocline depth ( figure 5(c)). The correlation coefficient between the seasonal components of the SSHAs at [95 • E, 0] and the seasonal components of mooring-measured thermocline depth anomalies reaches as high as 0.82 with the SSHAs leading the thermocline depth by 7 d, which is roughly the time needed by Kelvin waves propagating from [95 • E, 0] to the mooring location. The upwelling SKWs, especially those generated during the winter monsoons, effectively uplifted the thermocline. The mooring measurements at QB2 revealed that the thermocline depth was uplifted by as much as 15.5 m by the upwelling SKW in early 2018. However, the upwelling Kelvin wave generated during the summer monsoon had little impact on the thermocline depth due to its relatively weak strength. In comparison, the downwelling SKWs formed during the spring and autumn monsoon transition periods both effectively deepened the thermocline depth, and the thermocline deepening during the autumn monsoon transition period was relatively more evident, reaching as much as 11.7 m in late 2017.
The changes in the background environments with seasonal periods led to distinct seasonal cycles in the NLIW amplitudes, which were enhanced significantly in March-May and reduced significantly in June-July and November-December (figure 5(d)). The variations in the NLIW amplitudes showed a good correspondence with the occurrences of SKWs. The downwelling SKWs during the monsoon transition periods (red triangles in figure 5(b)) effectively reduced the NLIW amplitudes. Specifically, the NLIW amplitude dropped by 4.4 m during the autumn monsoon transition period (2# red triangle), accounting for 10.7% of the NLIW amplitude averaged over the whole observation period. On the other hand, the upwelling SKWs during monsoon periods, especially during the winter monsoons (1# and 3# blue triangles in figure 5(b)), effectively enhanced the NLIW amplitudes. In early 2018, the upwelling SKW enhanced the NLIW amplitude by 8.5 m, accounting for 20.7% of the NLIW amplitude averaged over the whole observation period. The variances in NLIW amplitudes, as well as SSHAs and thermocline depths, induced by each SKW event are shown in detail in tables S3 and S4 in the supplementary material.

Summary and discussion
In this study, data collected from a long-term subsurface mooring were used to investigate the modulations by Kelvin waves from the EIO on the NLIWs in the Andaman Sea. Strong westerly and easterly anomalies in the EIO drove downwelling and upwelling Kelvin waves, respectively. These waves then propagated eastward and entered the Andaman Sea, deepening or uplifting the thermocline by tens of meters. The occurrences of Kelvin waves showed a good correspondence with the variations in NLIW amplitude in the Andaman Sea for both IKWs and SKWs that were driven by different atmospheric forces. The downwelling IKWs could reduce the NLIW amplitude by up to 22%, and the upwelling IKWs could enhance the NLIW amplitude by up to 32%. The occurrence of IKWs explained most of the intraseasonal variations in the NLIW amplitude with periods of approximately one month ( figure 3). Moreover, the downwelling SKWs could reduce the NLIW amplitude by up to 10.7%, and the upwelling SKWs could enhance the NLIW amplitude by up to 20.7%. The occurrence of SKWs explained most of the seasonal variations in the NLIW amplitude (figure 5).
The results above present a clear picture of the modulation by equatorial Kelvin waves on NLIWs. Kelvin waves from the EIO deepen or uplift the thermocline in the Andaman Sea, which significantly affects the NLIW amplitude in the basin. A deep thermocline can reduce the NLIW amplitude, and a shallow thermocline can enhance the NLIW amplitude. However, the measurements (figures 3 and 5) also indicate that stronger Kelvin waves do not always lead to larger variations in the thermocline depth and NLIW amplitude. This non-correspondence is possibly attributed to two reasons.
First, the Kelvin waves induced disturbances not only on the thermocline but also on the background currents (see figures S2 and S3 in the supplementary material), which had varying intensities and directions around the source region (location S in figure 1(b)) and the propagation path of NLIWs. Previous studies have revealed that strong background currents can influence the generation of NLIWs and redistribute the wave energy along their crests (Li and Farmer 2011, Xie et al 2015, Huang et al 2017. Thus, the varying background currents may contribute to the observed non-correspondence between the Kelvin waves and the NLIW amplitude variations. Second, in addition to equatorial Kelvin waves, there are other dynamic processes in the Andaman Sea, such as monsoons, which can also change the circulations and thermocline structures in the basin (Rizal et al 2012, Liu et al 2018. However, the circulation and thermocline in the Andaman Sea are found to be primarily determined by remote wind forcing from the EIO rather than local wind forcing , Chatterjee et al 2017. Therefore, we suppose that the Kelvin waves from the EIO are the dominant factor that determines the long-period (>15 d) variations in the NLIWs in the Andaman Sea. Other dynamic processes in the Andaman Sea play a relatively small role and contribute to the observed noncorrespondence between the Kelvin waves and the NLIW amplitude variations.
In summary, when analyzing the NLIW behaviors in the Andaman Sea, especially their long-period variations, we must consider the influence of the Kelvin waves from the EIO. The same is true in the Pacific Ocean and Atlantic Ocean since the Kelvin waves from the equator of these oceans can play a nonnegligible role in modulating the NLIWs in marginal seas. Moreover, it is found that the boreal spring downwelling Kelvin wave can be significantly weakened during a positive IOD year (Reppin et al 1999, Rao et al 2016. The downwelling Kelvin wave, as revealed in our study, can reduce the NLIW amplitudes in the Andaman Sea. Therefore, the occurrence of a positive IOD event may lead to fewer reductions in the NLIW amplitude during the boreal spring monsoon transition period. In this way, the connection between global-scale climate changes and small-scale internal wave movements is established, and this connection can help us to better understand the influences of global climate changes on the local ocean environment.
The data cannot be made publicly available upon publication because they are owned by a third party and the terms of use prevent public distribution. The data that support the findings of this study are available upon reasonable request from the authors.