Energy transmission pathways of equatorial waves within the Maritime Continent: a view with the wave energy flux

Abstract

Detailed pathway of wave energy exchange between the Pacific and Indian Oceans through the Indonesian archipelago for a wide range of wave frequency is investigated using a reduced gravity model with realistic coastline. The wave energy flux analysis that can be applicable for all latitudes in a linear shallow water system is adopted. The energy fluxes diagnosed from the model outputs for the incoming Rossby waves from the Pacific clearly indicate two major energy pathways to the Indian Ocean: one turning southward in the Halmahera Sea and reaches the Indian Ocean via the Banda Sea and the Timor Passage, and the other passing through the Makassar and Lombok Straits. The former route, however, is shifted to the western side of the island chain within the Banda Sea due to energy trapping around the island chain. It is also found that strong energy dissipation occurs along the northern coast of New Guinea when the incoming wave period is shorter than 1.5 years. In the case of the Kelvin waves from the Indian Ocean, it is found that the major energy pathway is through the Lombok and Makassar Straits to the Pacific Ocean. However, there appears another pathway along the eastern side of the Sulawesi Island in the Banda Sea to exit through the Molucca Sea only when the wave period is shorter than about one month. This secondary pathway makes it easier for the wave energy from the Indian Ocean to reach the western Pacific Ocean for the short-period waves.

1 Introduction

The Indonesian Seas connects the tropical Pacific and Indian Oceans, providing a low-latitude conduit of water and energy between the two basins (Fig. 1). Transport of water from the Pacific to the Indian Ocean through this region is known as the Indonesian throughflow (ITF) (e.g., Wyrtki 1987; Gordon 2005). Since the ITF constitutes a part of the global thermohaline circulation (Gordon 1986; Sloyan and Rintoul 2001), magnitude of the ITF and its driving mechanism have been investigated extensively (e.g., Wyrtki 1987; Clarke and Liu 1994; Godfrey 1996; Meyers 1996; England and Huang 2005; Hu et al. 2015; Liu et al. 2015). In addition, the Indonesian archipelago plays an important role as a wave guide between the two basins and transmits a part of wave energy to the other basin, by which ocean and climate conditions within the Indonesian archipelago and the surrounding area are affected at various time scales (Sprintall and Révelard 2014).

Fig. 1

Topography in the Indonesian archipelago. Depths are given in meter. Gray shades indicate areas with depth shallower than 300 ms, which is considered as land masses in the reduced gravity model used in this study

Reflection of the equatorial Rossby waves at the entrance of the Indonesian archipelago has been studied as an important process in the delayed action oscillator theory of the El Niño–Southern Oscillation (ENSO) (Suarez and Schopf 1988). There are several theoretical studies investigating impacts of the reflection of equatorial waves at the leaky Pacific western boundary on the signal reaching the Indian Ocean, which is dynamically associated with the ENSO phenomenon (Clarke 1991; Du Penhoat and Cane 1991). Clarke (1991) assumes the land masses in the western Pacific Ocean and the Indonesian archipelago as thin meridional walls located at representative longitude of each island and suggests that 10% of the energy of the first meridional mode Rossby wave coming from the equatorial Pacific penetrates to the Indian Ocean through the Indonesian archipelago at interannual timescales. To estimate a degree of wave reflection and signal penetration quantitatively with the realistic geometry in the archipelago, reduced gravity models are frequently utilized in the previous studies. Potemra (2001) suggests that energy from the central equatorial Pacific does affect not only the ITF transport but also variability in the southeastern Indian Ocean with significant amplitude at semi-annual and longer time scales. Further numerical study by Spall and Pedlosky (2005) shows that 23% of the energy from the equatorial Rossby wave is reflected into the equatorial Kelvin wave at the leaky western boundary of the Pacific Ocean and 10% of the energy reaching the Indian Ocean.

Tide gauge observations and remotely sensed satellite observations of the sea level variability have been utilized to detect such propagating wave signals at the interannual time scale within the Indonesian archipelago and southeastern Indian Ocean. It has been shown that the interannual variability in these variables is mostly related to wind forcing associated with the ENSO in the equatorial Pacific Ocean and that oceanic waves propagate from the equatorial Pacific to the Indian Ocean through the Indonesian archipelago (e.g., Potemra 2001; Feng et al. 2003; Li and Clarke 2004; Wijffels and Meyers 2004). The temperature variations observed by sensors attached on moorings in the Indonesian archipelago also detect interannual variability associated with the ENSO events (e.g., Gordon et al. 1999; Ffield et al. 2000; Susanto et al. 2012). Due to the oceanic waves propagating from the Pacific Ocean, the ITF transport and the Leeuwin Current along the western coast of Australia show significant interannual variability including the ENSO related signals (Gordon et al. 1999; Feng et al. 2003; Susanto and Gordon 2005; Li et al. 2020a). It is also suggested that the oceanic waves from the Pacific Ocean can generate the Ningaloo Niño/Niña events appeared along the northwestern coast of Australia (Kataoka et al. 2014).

In addition, previous studies suggest that significant non-ENSO signals in the ITF transport come from the tropical Indian Ocean (Murtugudde et al. 1998; Qiu et al. 1999; Sprintall et al. 2000; Molcard et al. 2001; Pujiana et al. 2019). For example, Sprintall et al. (2000) observed that a semi-annual Kelvin wave, excited in the equatorial Indian Ocean, propagates southeastward along the Sumatra/Java coasts, through the Lombok Strait, and then northward to the Makassar Strait. In addition to the semi-annual Kelvin waves, the intraseasonal Kelvin waves driven by Madden–Juilan oscillation (MJO) (Madden and Julian 1994) events in the tropical Indian Ocean are suggested to propagate through the Lombok Strait to the Makassar Strait (Shinoda et al. 2016; Tamasiunas et al. 2021). The Ombai Strait is also considered to be an important pathway for the coastally trapped Kelvin waves originated from the equatorial Indian Ocean to flow into the Indonesian archipelago (Durland and Qiu 2003; Wijffels and Meyers 2004; Syamsudin et al. 2004; Schiller et al. 2010). Furthermore, the simple model experiments of Yuan et al. (2018) suggest the possibility of Kelvin wave penetration into the western Pacific from the eastern Indian Ocean through both eastern and western parts of the Indonesian archipelago.

Most of the above-observed studies adopt statistical approaches to discuss signals of the wave propagation, which demonstrate indirect evidence of the wave propagations. In addition, it is rather difficult to detect detailed wave pathways within the Indonesian archipelago and their dependency on the wave frequencies only from the observed data since the mooring data are obtained at specific locations for limited durations and the accuracy of altimeter data within the Indonesian archipelago is still uncertain due to influences of the complex coastal boundaries. On the other hand, the theoretical studies provide only the gross values of reflection and transmission rates with simplified geometry. Numerical models with realistic geometry within the Indonesian archipelago can provide information on characteristics of wave signals, which may contribute to bridge the observational and theoretical studies. However, a systematic study on transmission of planetary waves at various time scales from both the Pacific and Indian Oceans focusing on detailed pathways within the archipelago has not been conducted yet.

More direct expression of the planetary wave propagation can be obtained with the form of wave energy flux, which has been used for studies on wave propagation in the mid-latitude regions (e.g., Orlanski and Sheldon 1993; Harr and Dea 2009; Wang et al. 2018). However, due to the assumption of geostrophy in a typical wave energy flux form, it is difficult to apply this expression to the equatorial waves and their propagation within the complex geometry. Recently, Aiki et al. (2017) developed a new analysis scheme of wave energy flux for the planetary waves, which can be applies for all the latitudinal regions, including the equatorial oceans. In this study, therefore, we examine detailed pathways of wave propagation both from the Pacific to Indian Oceans and processes affecting them within the Indonesian archipelago based on quantitative evaluation by use of the wave energy flux proposed by Aiki et al. (2017). For this purpose, we adopt the simplest possible configuration of the linear reduced gravity model with idealized wind forcing. Note that several studies suggest the importance of nonlinear effects on reflection of the equatorial Rossby waves at the western boundary (Yuan 2005; Yuan et al. 2004, 2019) and on intrusion of wave signal into the Indonesian archipelago (Hu et al. 2022). It has also been suggested that the intraseasonal wave signal from the Indian Ocean may not penetrate into the Indonesian Seas through a shallow channel due to energy subduction (Drushka et al. 2010). However, the present study focuses on examining the horizontal propagation of linear waves to show the basis of detailed wave pathways within the Indonesian archipelago from the viewpoint of wave energy flux covering a wide range of wave frequency. This study sheds light on a new approach to evaluate the equatorial wave propagation and strengthens our understanding based on the existing studies.

This paper is organized as follows. A numerical model and an analysis method utilized in the present study are briefly described in Sect. 2. Section 3 shows results for the cases, in which waves come from the equatorial Pacific Ocean. The wave energy pathways in the Indonesian archipelago and their frequency dependences are discussed. Results for incoming waves from the equatorial Indian Ocean are described in Sect. 4. Summary and discussion are presented in Sect. 5.

2 Model and method

2.1 Numerical model

We adopt a linear reduced gravity model with one active layer to explore paths of wave energy exchange between the eastern Indian and western Pacific Oceans through the Indonesian archipelago in the simplest possible system. The model employs realistic representation of the complex geometry of the Indonesian archipelago. The equations for this model are written as:

$$\begin{aligned}&\frac{\partial u}{\partial t} - fv = -g'\frac{\partial \eta }{\partial x} + \frac{\tau _x}{H} + \nu \nabla ^2u,\\&\frac{\partial v}{\partial t} + fu = -g'\frac{\partial \eta }{\partial y} + \frac{\tau _y}{H} + \nu \nabla ^2v,\\&\frac{\partial \eta }{\partial t} + H\left( \frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}\right) =0, \end{aligned}$$

where u and v are zonal and meridional velocities, respectively, $\eta$ is upper layer thickness anomaly, f is the Coriolis parameter, $g'$ is the reduced gravity and $\tau _x$ and $\tau _y$ correspond to zonal and meridional wind stress, with $\tau _y = 0$ for all the experiments in this study. The mean thickness of the active upper layer H is set to 300 m as in Potemra (2001), and the coefficient of horizontal viscosity $\nu$ has a value of $1\times 10^{3}$ m$^2$ s$^{-1}$. The model has a realistic land geometry in and around the Indonesian archipelago based on the contour of 300 m isobath from ETOPO1 (Amante and Eakins 2009), and is forced by idealized zonal winds with a prescribed period of variation. This model is discretized into a spherical coordinate system with a grid spacing of 0.1$^\circ$ in both zonal and meridional directions on the Arakawa-C grid system. At each position of the respective variables, the model integrated the above equations for zonal and meridional currents as well as upper layer thickness anomaly.

A series of experiments is conducted with two model settings to focus on equatorial waves coming from the equatorial Pacific Ocean or from the equatorial Indian Ocean, respectively. For the Pacific experiment, the model domain extends from 80$^\circ$ E to 60$^\circ$ W and from 30$^\circ$ S to 30$^\circ$ N (Fig. 2a). Sponge layers with the zonal width of 10$^\circ$ for the artificial meridional boundaries at 80$^\circ$ E and 60$^\circ$ W and the meridional width of 5$^\circ$ for the zonal boundaries at 30$^\circ$ S and 30$^\circ$ N are applied to absorb the wave energy and eliminate unexpected reflection and propagation of the waves along the artificial boundaries. Note that the results shown below are robust with a wider model domain to include the whole Indian Ocean, since the energy absorption within the sponge layer is quite effective. The gravity wave speed $c=\sqrt{g'H}$ was set equal to $c=2.62$ m s$^{-1}$ as in Potemra (2001), assuming the first baroclinic mode waves in the equatorial Pacific Ocean.

Fig. 2

Model domains for a the Pacific Ocean experiment and b the Indian Ocean experiment, with the amplitude of the idealized zonal wind forcing in N m$^{-2}$ (color shading). The model boundaries based on the 300 m isobaths (gray shade) are also shown (color figure online)

Idealized wind forcing for the Pacific experiment is given as

$$\begin{aligned} \tau _x=A_0\sin (\omega t)\exp \left[ -\left( \frac{x-x_0}{L_x}\right) ^2-\left( \frac{y-y_0}{L_y}\right) ^2\right] , \end{aligned}$$

where $(x_0,y_0)$ is at 140$^\circ$ W on the equator, $L_x$ and $L_y$ are zonal and meridional widths, respectively, with 4$^\circ$ in both directions, $A_0$ is forcing amplitude of 0.2 N m$^{-2}$, and $\omega$ is the forcing frequency. Since the meridional decay scale $L_y$ is larger than the equatorial deformation radius ($\sim$ 330 km), it is expected that the Rossby wave of the first meridional mode is mainly excited (Spall and Pedlosky 2005). We apply various forcing period from 90 days to 10 years in this study. The model is integrated for 10 forcing cycles, and the last cycle of the forcing period is used for the following analyses.

The Indian experiment is set similar to the Pacific experiment, but the domain extends from 0$^\circ$ to 180$^\circ$ and from 30$^\circ$ S to 30$^\circ$ N (Fig. 2b). The gravity wave speed c is given as 2.99 m s$^{-1}$ for the first baroclinic mode used in Li and Aiki (2020), and the same formulation for the external wind stress is applied, with the different values of center location, i.e., $(x_0,y_0)$ is at 40$^\circ$ E on the equator in this case. For the Indian Ocean experiment, we adopt the forcing period from 10 days to 4 years. The other parameters and settings are the same as in the Pacific experiment.

2.2 Analysis method

Energy flux associated with planetary scale waves is a good indicator for pathways of the wave signals connecting the two basins through the Indonesian archipelago. To obtain wave energy flux in the above numerical model, we utilize a new formulation proposed by Aiki et al. (2017) (hereafter AGC17 scheme), which can be applicable at all latitudes, including the equatorial region, while satisfying coastal boundary conditions. See Appendix 1 for derivation and detailed explanation of the formula. Note that the AGC17 scheme can represent the wave propagation even in the regions where the contribution of viscosity term is large, such as the Indonesian archipelago.

3 Rossby waves from the Pacific to the Indian Oceans

3.1 Energy flux pathways

First, we examine results from experiments with wind forcing in the Pacific Ocean to see the pathways of wave energy transmitted through the Indonesian archipelago. The outputs from the reduced gravity model clearly shows that the wind forcing centering at 140$^\circ$ W excites westward propagating equatorial Rossby waves and eastward propagating equatorial Kelvin waves (Fig. 3a).

Fig. 3

a Hovmoller diagram of the sea level anomaly along 2$^\circ$ N for the 2-year period forcing case. The unit is cm. b Zonal component of energy flux along the equator for the 2-year period forcing case. The unit is W m$^{-1}$

The maximum amplitude of the sea level anomaly associated with these equatorial waves is about 5 cm, which is consistent with the satellite observations (Busalacchi et al. 1994; Boulanger and Menkes 1995; Boulanger and Fu 1996; Boulanger and Menkès 1999). Zonal energy flux along the equator clearly shows the eastward propagation of the equatorial Kelvin waves and the westward propagation of the equatorial Rossby waves (Fig. 3b). Meridional distributions of the zonal energy flux across the international date line from the experiments with the forcing of 4-year and semi-annual periods are shown in the boxes on the right side of Fig. 4. The meridional structure of the energy flux indicates that almost all incoming waves are meridional mode 1 Rossby waves as expected from the model results of Spall and Pedlosky (2005). Moreover, the horizontal distributions of the wave energy fluxes (left panels in Fig. 4) show the signals penetrating from the Pacific to the Indian Ocean, and then waves continue to propagate southward along the western coast of Australia. The maximum amplitude of the sea level along the western coast of Australia is about 3.5 cm, which is consistent with the observed sea level variations (Clarke and Liu 1994; Feng et al. 2003). Therefore, the present simple model and its results capture a realistic situation of ocean wave propagation and are worth investigating the detailed processes.

Fig. 4

(Left panels) Horizontal distribution of the direction of energy flux vectors (arrows) and magnitude of the energy flux in W m$^{-1}$ (color). The energy flux vectors are shown only for those with their magnitude larger than 400 W m$^{-1}$. (Right panels) Meridional distribution of the zonal energy flux of pure incoming wave energy (solid line) and analytical values for the first meridional mode 1 Rossby wave (dashed line) in W m$^{-1}$. The forcing period is a 4 years and b 0.5 years (color figure online)

3.1.1 Interannual time scale

The left panels of Fig. 4 show horizontal distributions of energy flux vectors within and around the Indonesian archipelago, indicating very complex pathways of wave energy from the Pacific to the Indian Oceans. Note that these vectors only indicate the direction of the energy fluxes and their magnitude is shown in color shades to see the pathways clearly. Tables 1 and 2 summarize the energy flux through several key passages at the northern entrance and the southern exit for the archipelago.

Table 1 Southward energy flux crossing the equator through each passage

Table 2 Wave energy flux into the Indian Ocean from the Indonesian seas through each passage with a unit of 10$^9$ W  =  1 GW

In general, for the forcing period of interannual time scales, most of the wave energy propagates through the Indonesian archipelago within its eastern part, through the Halmahera Sea, the Banda Sea, and the Timor Sea, before reaching the northwestern coast of Australia (Fig. 4a). This eastern route of the wave energy pathway is consistent with the previously mentioned wave pathway suggested from the observed sea surface height variability (e.g., Wijffels and Meyers 2004). However, there can be seen several notable details in Fig. 4a, which have not been mentioned in the previous literatures. One such feature is that the major route of the wave energy occupies the western side of island chain in the eastern Banda Sea. Considering the Kelvin wave propagation along the coasts in the southern hemisphere, major part of the energy flux would be expected through the passage between the Seram and New Guinea Islands. However, most of the energy propagates into the Banda Sea along the west coast of Buru Island and almost no energy propagates along the west coast of New Guinea. We will explore a possible reason for this curious wave energy flux distribution in the following subsection.

After reaching the southern part of the Banda Sea, most of this southward energy flux continue to the Indian Ocean via the Timor Sea. In addition, weak southward energy flux appears along the east coast of Sulawesi Island within the Banda Sea from 3$^\circ$ S to 7$^\circ$ S. Note that this poleward energy flux along the east coast of Sulawesi Island is in the opposite direction to the energy propagation due to the coastal Kelvin wave in the southern hemisphere. It is suggested, therefore, that this poleward energy flux is associated with the diffusive boundary layer as mentioned in Spall and Pedlosky (2005). In fact, the width of the southward energy flux along the east coast of Sulawesi Island in the simulated result is about 80 km at 4$^\circ$ S, which is consistent with the representative width of diffusive boundary layer with the viscosity coefficient used in this study ($\nu =1\times 10^3$ m$^2$ s$^{-1}$).

A part of the energy coming to the region south of Halmahera Island returns northward through the Molucca Sea (Table 1), then it merges to the westward energy flux from the northern tip of Halmahera Island to form rather broad westward energy flux to the north of Sulawesi Island. Figure 4a also shows that about 60% of this wave energy flows into the Makassar Strait and then reaches the Lombok Strait. Besides, the remaining 40% of the wave energy propagates northward along the eastern coast of Borneo Island. This poleward energy propagation is not consistent with the propagation of coastal Kelvin wave, and width of the northward energy flux is about 100 km, suggesting again the energy redistribution in the diffusive boundary layer similar to the eastern coast of Sulawesi Island. It is suggested that not only the wave energy from the Makassar Strait but also the wave energy from the Halmahera Sea and the Banda Sea proceed to the Lombok Strait. Indeed Fig. 4a shows westward energy flux in the Flores Sea, which is due to the Rossby waves off the southern coast of Sulawesi and coastal Kelvin waves along the northern coast of the Lesser Sunda Islands.

Finally, most of the energy propagating southward in the Indonesian archipelago continue to the Indian Ocean mainly via the Timor Sea and Lombok Strait, with the remaining southward energy transfer through the Ombai Strait (Table 2). Although the width of the Ombai Strait is relatively wide compared to the deformation radius at the location of the strait, it is the westward energy fluxes in the Flores Sea that transport a part of energy to the Lombok Strait. Note that the same wave energy pathways as described above are reproduced in experiments with a horizontal viscosity of $1\times 10^2$ m$^2$ s$^{-1}$, i.e., ten times larger than the standard case.

Recent mooring observation suggests that the wave propagation through the Halmahera Sea is not as large as that in the previous studies (Li et al. 2020b). The wave energy flux through the Halmahera Sea in the present model may overestimate the wave transmission due to insufficient resolution ($\sim$ 10 km) to represent many small islands in the Halmahera Sea. However, it should be noted that there is no change in the wave energy pathway in the experiment which applied high horizontal viscosity ($5\times 10^3$ m$^2$ s$^{-1}$) only in the Halmahera Sea (not shown).

3.1.2 Role of Halmahera Island and islands in Banda Sea on the energy pathways

In the previous subsection, the importance of Halmahera Island and islands in Banda Sea on the pathways of wave energy impinging from the equatorial Pacific Ocean is suggested. Here, we try to explore their roles in more detail with additional experiments removing these islands in the model. To clarify the role of Halmahera Island, we first conduct simple experiment only with New Guinea and Australia, without Halmahera Island, and with annual wind forcing as in the main experiments. Figure 5a shows the energy fluxes in this experiment. It is clearly indicated that most of the incoming energy continues to propagate westward and a small part of the energy propagates along the west coast of New Guinea as coastal Kelvin wave. Slight westward energy fluxes around 4$^\circ$ S, shown as light color shades without arrows, corresponding to Rossby wave emitted from the coastal Kelvin wave can also be seen on the western side of New Guinea and Australia.

Fig. 5

Horizontal distributions of a energy flux from a sensitivity experiment only with New Guinea and Australia, b energy flux from a sensitivity experiment to which Halmahera Island is added and c energy flux differences between the two sensitivity experiments. Arrows indicate energy flux vectors with magnitude larger than 400 W m$^{-1}$ and red color shading indicates magnitude of energy flux in W m$^{-1}$. d Horizontal distribution of the meridional component of energy flux differences in W m$^{-1}$. e Horizontal distribution of the viscous dissipation near Halmahera Island and New Guinea for with Halmahera case in W m$^{-1}$

As a second step, Halmahera Island is added to the experiment (Fig. 5b). The difference in the energy flux between the two experiments (Fig. 5c, d) can be considered as the contribution of Halmahera Island to the wave energy propagation. Figure 5c clearly shows that the Halmahera Island has a barrier effect on the westward equatorial Rossby wave, and the blocked Rossby wave energy continues to propagate southward through the Halmahera Sea, resulting in enhancement of the Rossby waves in southern hemisphere and the southward coastal Kelvin wave. Figure 5d shows strong southward and northward energy fluxes in the Halmahera Sea, and strong energy dissipation along the eastern coast of the Halmahera Island is shown in Fig. 5e. All these results suggest the importance of the diffusive boundary layer along the east coast of the Halmahera Island to the wave energy propagation.

To investigate the reasons why the wave energy propagates along the west coast of the island chain (Fig. 4) instead of the western coast of New Guinea as shown in Fig. 5b, we conducted an additional sensitivity experiment with Buru and Seram Islands and the island chain in the eastern Banda Sea. Figure 6 shows the contribution of these islands to the energy transport. The island chain has only a small effect on wave energy to reflect back to the Pacific Ocean, and most of the wave energy flows into the Indian Ocean as in the case with Halmahera Island (Fig. 5b). In addition, an anticlockwise energy circulation around the island chain is clearly seen in Fig. 6, which is superposed on the southward energy flux along the west coast of New Guinea in the case with no island chain in the Banda Sea, providing almost no energy flux between the island chain and the New Guinea in the total fields. Thus, the absence of the easternmost energy path can be attributed to the cancelation of Kelvin wave along the west coast of New Guinea by the energy circulation trapped around the islands in the Banda Sea. It takes only about 10 days for Kelvin waves to bypass the islands of the Banda Sea and develop the energy circulation with the group velocity of the first baroclinic mode coastal Kelvin wave ($\sim$ 2.6 m s$^{-1}$), suggesting that it is difficult to detect signals that has a period longer than 10 days from the Pacific Ocean using mooring observations on the west coast of New Guinea.

Fig. 6

Same as Fig. 5c but for the energy flux differences between experiments with and without Buru and Seram Islands and the island chain in the eastern Banda Sea. (with island chain case minus no island chain case)

3.1.3 Semi-annual time scale

The energy pathways shown for the incident waves of different interannual time scales (2- to 10-year period) are very similar to those obtained for the 4-year period case discussed above. The approach taken in this study, numerical experiments with realistic boundaries and AGC17 scheme, also enables us to investigate the energy propagation of higher frequency waves, which has not been discussed much in the previous literatures. It is important to evaluate energy pathways for such shorter time-scale variations since the semi-annual variations are observed in the western tropical Pacific (e.g., Qu et al. 2008). In addition, atmospheric intraseasonal oscillations, such as the MJO, can generate equatorial Kelvin and Rossby waves through surface zonal winds over the Pacific Ocean (Hendon et al. 1998; Zhang et al. 2001) and may affects the circulations in the Indonesian archipelago.

The results of semi-annual forcing case (Fig. 4b) also show two major energy pathways; one through Halmahera Sea to Banda Sea and the other through Makassar Strait and Lombok Strait as in the case of low frequencies. However, the magnitude of the energy flux is clearly smaller than that of the low-frequency case. In particular, the decrease in magnitude is significant in the pathway through the Makassar Strait and Lombok Strait (see Table 2). This discrepancy is attributed to the difference in the amount of energy dissipation at the western boundary of the Pacific Ocean, which is discussed in the next subsection on the energy budget. The decrease in the energy flux through the Lombok Strait may also be due to the reduction of the westward propagating Rossby wave in the Flores and Banda Seas at high frequencies.

3.2 Energy budget

3.2.1 Energy budget for a larger domain

To evaluate the energy budget in a larger domain, a box covering the western equatorial Pacific and the Indonesian archipelago is considered (Fig. 2a) and the amount of wave energy crossing the boundaries is calculated (Fig. 8). It should be noted here that wave energy flux across the eastern section at 180$^\circ$ E averaged over one forcing period, $E'_\textrm{in}$ includes both the incoming wave energy flux and the flux due to reflected waves. To extract pure incoming wave energy crossing the international date line, we conducted an additional experiment, in which a simple meridional western boundary with a sponge layer is incorporated to erase the wave energy associated with the reflected equatorial Kelvin wave. The energy flux across the international date line for this experiment can be considered as the pure westward incoming wave energy, $E_{\textrm{in}}$, and therefore eastward reflected wave energy, $E_{\textrm{ref}}$, can be defined as

$$\begin{aligned} E_{\textrm{ref}}=E_{\textrm{in}}-E'_{\textrm{in}}. \end{aligned}$$

The energy flux that comes out from the inside region of Indonesian Seas to the Indian Ocean, $E_{\textrm{out}}$, is defined as southward energy flux across 8$^\circ$ S between 115$^\circ$–135$^\circ$ E (green line in Fig. 2a). On the other hand, the wave energy dissipated within the Indonesian Seas, $E_{\textrm{disp}}$, is obtained by integrating the convergence of the wave energy flux within the box, assuming a quasi-steady condition for the wave energy fluxes under the average of one wave period.

Fig. 7

Ratios of major terms in the wave energy budget in the box shown in Fig. 2a to the pure incoming wave energy flux across the international date line

Figure 7 shows results of the energy budget as a function of the forcing period, standardized by the incoming energy across the date line for each forcing period. It is clearly shown that, for the period longer than 1.5 years, most of the incoming energy (about 60%) is dissipated within the box, while about 30% of the incoming energy is reflected back to the east of the date line. Then, the remaining 10% of the incoming energy flows into the Indian Ocean. Since the wind forcing in our experiment excites the meridional mode 1 equatorial Rossby wave (see Fig. 4), this result is in good agreement with the result of the analytical investigation by Clarke (1991) and the model calculations by Spall and Pedlosky (2005).

Horizontal distributions of energy dissipation rate are shown in Fig. 8. The dissipation rate is obtained by calculating a divergence of horizontal energy flux. Comparing high frequency and low-frequency cases, there is a common feature that strong energy dissipation occurs in the western boundary layer regions of the Pacific Ocean, such as zonally narrow regions off the east coast of Mindanao, Borneo, Halmahera and New Guinea, as well as regions within the narrow channel such as the Halmahera Sea and the northern entrance of the Banda Sea. In addition, there is strong energy dissipation in the northern coast of Buru Island, as well as the narrow part of the Makassar Strait and the Lombok Strait. Note that northward and southward leakage of the energy across 10$^\circ$ N and 10$^\circ$ S are almost negligible for all frequencies.

Fig. 8

Horizontal distribution of dissipation rate obtained from experiments with the wind forcing of a 4-year and b 180-day periods. The unit is W m$^{-2}$

For the forcing period less than 1.5 years, unlike the low-frequency case, the ratio of dissipated energy within the box increases and that of reflected energy decreases as the period becomes shorter (Fig. 7). This tendency is consistent with the result of Spall and Pedlosky (2005), but their result only shows the decreases of reflected and transmitted energy qualitatively. Thus, detailed quantitative understanding of why the reflection and the transmission are suppressed at higher frequency is necessary. In the high-frequency case (Fig. 8b), the dissipation rate is significantly large along the northern coast of New Guinea between 130$^\circ$ E and 140$^\circ$ E, which cannot be seen in the low-frequency case (Fig. 8a). The forcing period of 1.5 years seems to provide a key time scale for dividing the results into two regimes: one with the weaker dissipation along the northern coast of New Guinea (i.e., the low-frequency cases) and the other with the stronger dissipation there (i.e., the high-frequency cases). It is worth noting that the 1.5-year corresponds to a wave period, for which half of the zonal wavelength of the incoming meridional mode 1 Rossby wave is comparable to the zonal width of the inclined western boundary (New Guinea Island) in the present case. We will discuss a possible mechanism responsible for this difference in the dissipation magnitude in detail in the next subsection.

3.2.2 Dissipation along the northern coast of New Guinea

In the previous studies of the reflection of the equatorial Rossby waves at inclined western boundary in the linear shallow water system (e.g., Cane and Gent 1984; McCalpin 1987), the reflection at the western boundary is considered from the budget of mass flux across the interface between the western boundary layer and the interior ocean, and the difference between incoming wave energy and outgoing wave energy for the western boundary layer is considered to be dissipated in the western boundary layer. However, since we consider the energy penetration further to the west for a wide range of wave spectra including shorter periods, we need to examine the energy dissipation on the inclined western boundary in detail.

To evaluate the energy dissipation on the western boundary in a simpler case, experiments without Solomon Islands and New Ireland are conducted. In these additional experiments, energy transmission and reflection rates show the same dependencies on the forcing period as in the main experiments (Fig. 7). Furthermore, strong energy dissipation along the northern coast of New Guinea can be seen only when forcing period is shorter than 1.5 years. Figure 9 shows time evolutions of meridional currents along the equator when the forcing periods are 4 years and 0.5 years. In both cases, the meridional velocity is significant in the region near the western boundary of the Pacific Ocean (130–145$^\circ$ E). However, the amplitudes of the meridional velocity show very different values: it is about 2 cm s$^{-1}$ for the 4-year period case while it becomes well over 5 cm s$^{-1}$ for the 0.5-year period case.

Fig. 9

Hovmoller diagrams of the meridional current along the equator for a the 4-year period forcing case and b the semi-annual forcing case in cm s$^{-1}$. Contours indicate meridional velocity with amplitudes above 5 cm s$^{-1}$ and contour intervals are 5 cm s$^{-1}$

To clarify the processes that generate the different amplitude in meridional currents across the equator between the short-period and long-period cases, we consider the mass flux entering the inclined west boundary layer as in Cane and Gent (1984) and McCalpin (1987). When the forcing period is long enough, i.e., the zonal wavelength of the incoming Rossby waves is sufficiently longer than the zonal extent of the inclined western boundary, incoming Rossby waves reach the western boundary layer at almost the same phase at any latitude, and the meridionally symmetric mass fluxes enter the western boundary layer. In this case, most of the off-equatorial incoming mass fluxes of the Rossby waves generate equatorward currents in the western boundary layer for redistributing the masses toward the equator and emitting them eastward as the equatorial Kelvin waves. The direction of the mass redistribution in the western boundary layer is opposite in the northern and southern hemispheres; thus, meridional current across the equator is less likely to be formed (Fig. 9a). On the other hand, when the forcing period is rather short, i.e., the zonal wavelength of the incoming Rossby waves is comparable to or less than the zonal extent of the inclined western boundary, incoming Rossby waves reach the western boundary layer at different phases at each latitude, and the meridionally asymmetric mass fluxes are generated in the western boundary layer. For example, a positive mass flux enters in the southern hemisphere while a negative mass flux enters in the northern hemisphere. In this case, the total mass flux entering the western boundary layer, capable of constructing the reflected Kelvin waves, is very small. Therefore, strong meridional current across the equator is formed to connect the asymmetric mass distribution in the western boundary layer as shown in Fig. 9b (see Appendix 2 for details). This strong current across the equator and in the western boundary layer generates large horizontal velocity shear, inducing the strong energy dissipation particularly along the northern coast of New Guinea (Fig. 8b).

4 Kelvin waves from the Indian to the Pacific Oceans

4.1 Responses within the Indian Ocean

For the Indian Ocean experiment, we consider whether and how the equatorial Kelvin waves propagates through the Indonesian archipelago and eventually penetrate into the Pacific Ocean. We first investigate briefly the behavior of waves before they reach the Indonesian archipelago. For a period shorter than 30.7 days, below which no equatorial Rossby waves satisfy the dispersion relation under our model settings, most of the wave energy propagates southeastward along the coast of Sumatra Island after reaching the eastern boundary of the Indian Ocean (Fig. 10a). Only a small part of the incoming wave energy propagates to the north into the Andaman Sea along the eastern boundary of the basin.

Fig. 10

Horizontal distributions of the direction of energy flux vectors and magnitude of the energy flux for the forcing period of a 20 days and b 180 days. Only the vectors with their magnitude larger than 200 W m$^{-1}$ are shown. c Northward energy fluxes across 4$^\circ$ N (blue), 4$^\circ$ S (orange), 12$^\circ$ N (light blue) and 12$^\circ$ S (pale orange) between 90$^\circ$ E and the eastern boundary coast of the Indian Ocean for each forcing period. The sections calculating the energy fluxes are shown with black lines. Negative values indicate southward energy fluxes (color figure online)

On the other hand, for the Kelvin waves with longer periods, the wave energy bifurcates northward and southward almost evenly off Sumatra Island (Fig. 10b). Figure 10c shows northward energy fluxes across 4$^\circ$ and 12$^\circ$ latitude sections near the eastern boundary in both hemispheres as a function of the forcing period. The results indicate that more than a half of the energy crossing 4$^\circ$ N reaches 12$^\circ$ N and propagate further north into the Bay of Bengal, while the energy reaching 12$^\circ$ S is almost zero. This north–south difference suggests that the Bay of Bengal is more sensitive to equatorial waves originating from the equatorial Indian Ocean than the Southeastern Indian Ocean (McCreary et al. 1993; Schott and McCreary 2001).

Figure 10c also shows dependences of energy flux magnitude on the forcing period. While the wave energy across 4$^\circ$ N or 4$^\circ$ S is evenly distributed for the longer periods, we can find asymmetric energy partition between the two hemispheres for sufficiently short period, with more energy in the southern hemisphere. In addition, northward wave energy across 4$^\circ$ N peaks at about a period of one month, consistent with the observed short-period westward propagating Rossby wave near 5$^\circ$ N (Chen et al. 2017).

Unlike the low-frequency case, when the high-frequency Kelvin wave excited in the equatorial Indian Ocean reaches the eastern boundary, more energy is distributed to the south, and the distribution ratio to the south increases with reduction of the frequency. This asymmetric characteristic of the north–south distribution for the waves with short wavelength may be caused by the absence of westward reflecting Rossby waves at high frequencies (Fig. 10a) and also affected by the inclination of the eastern boundary of the Indian Ocean from the northsourth direction

4.2 Energy flux pathways within the archipelago

The wave energy flux vectors in the Indonesian archipelago for the Indian experiments are shown in Fig. 11. An important common feature in all the results with various forcing periods is that most of the incident energy enters the Indonesian archipelago through the Lombok Strait and then, flows into the western Pacific via the Makassar Strait. This waveguide is consistent with the route of wave signal predicted by Clarke and Liu (1994), suggested from observed data by Sprintall et al. (2000) and Pujiana et al. (2013), for example, and simulated in numerical models fo Syamsudin et al. (2004), Schiller et al. (2010) and Yuan et al. (2018), for example. It is confirmed for the first time with direct estimation of the energy fluxes that the same routes can also be seen as the dominant energy pathways.

Fig. 11

Horizontal distributions of the direction of energy flux vectors and magnitude of the energy flux within the Indonesian archipelago for the forcing period of a 20 days b 80 days, and c 360 days. Only the vectors with their magnitude larger than 200 W m$^{-1}$ are shown

As in the case of the Pacific experiments, the result of Indian experiments demonstrate different characteristics between the low- and high-frequency cases. For the low-frequency case, the Kelvin wave signals enter the Indonesian archipelago mainly via the Lombok Strait and slightly via the Ombai Strait. The energy entered the Indonesian seas via the Ombai Strait propagates westward along the northern coast of the Lesser Sunda Islands around 8$^\circ$ S and merges with energy from the Lombok Strait (see Fig. 11c).

On the other hand, in the high-frequency case, there appears a new route passing through the Indonesian seas from the Lombok Strait to the Pacific Ocean. The wave energy coming into the archipelago via the Lombok Strait tends to follow the northward wave guide through the Makassar Strait. However, in the high-frequency case (Fig. 11a), a part of this northward energy separates from the northward waveguide and propagates northeastward along the coast of Sulawesi Island and through the Molucca or Halmahera Seas to reach the Pacific Ocean. The energy fluxes along this additional waveguide decrease with increasing period, due to excitation of the Rossby waves, which transport the energy westward in the Flores Sea to the Makassar Strait at sufficiently long period. In addition, the westward energy propagation from the Ombai Strait is not as strong as the low-frequency case, although the wave energy entering the Indonesian archipelago through the Ombai Strait is larger in the high-frequency case compared to the low-frequency case.

From the above results, two major wave energy pathways from the Indian to the Pacific Oceans can be found:

• 1. The wave propagates northward through the Lombok and Makassar Straits and then across the Celebes Sea to the Pacific Ocean,

• 2. After passing through the Lombok Strait, the wave propagates northeastward through the Flores Sea, and then reaches the Pacific Ocean via the Banda Sea and the Molucca or Halmahera Seas.

The former pathway, which is consistent with the observed MJO related sea level variation (Shinoda et al. 2016) and ITF transport anomaly in the Makassar Strait (e.g., Sprintall et al. 2000), is found in all the experiments with various forcing periods. In contrast, the latter is found only in the experiments with the period of zonal wind forcing shorter than 2 months.

Theoretical and observational studies have reported that the coastal Kelvin waves propagating along the southern coast of Java Island can reach the Ombai Strait and that the associated wave energy enters the Indonesian archipelago through the Ombai Strait, as well as the Lombok Strait (Sprintall et al. 2000; Durland and Qiu 2003; Syamsudin et al. 2004). It is noted that, in the present study, about 65% of the Kelvin wave energy crossing 110$^\circ$ E (green lines in Fig. 10a, b) passes through the Lombok Strait into the Indonesian archipelago except for the case with 10-day period forcing. This value is in good agreement with the results of Syamsudin et al. (2004), indicating 55.6 ± 13.9% from the altimeter data and 65% from the model designed for the first baroclinic mode waves. In the case of 10-day period forcing, only about 30% of the incoming energy passes through the Lombok Strait, which may be due to the strong energy dissipation caused by short wavelength.

4.3 Energy transmission rate

The differences in the properties of short-period and long-period waves also appear in the wave energy transmission rate. Figure 12 shows the incoming wave energy from Indian Ocean, $E_{\textrm{IO}}$, defined as the eastward energy flux across 110$^\circ$ E between 12$^\circ$ S and southern coast of Java Island (green line in Fig. 10a), the wave energy reaching the Pacific Ocean, $E_{\textrm{PA}}$, defined as the eastward energy flux across 140$^\circ$ E between 10$^\circ$ N and the northern coast of New Guinea Island (pink line in Fig. 10a), and transmission rate, $E_{\textrm{PA}}/E_{\textrm{IO}}$, for various period of the incoming waves. It is clearly shown that $E_{\textrm{IO}}$ decreases as the forcing period increases, except for the shortest period of 10 days. This is consistent with the meridional energy partition of the equatorial Kelvin wave off the coast of Sumatra Island shown in Fig. 10c. The energy transmitted to the Pacific Ocean, $E_{\textrm{PA}}$, also decreases as the period increases, but is almost constant for the periods longer than 3 months. Thus, the transmission rate increases slightly with the forcing period for the periods longer than 3 months. In the experiments with the shorter period forcing, the transmission rate has a minimum value of about 12% at the 50-day period, while the maximum of about 27% appears at the 20-day period. It can be said that this maximum transmission rate corresponds to the existence of the additional pathway for the shorter forcing period shown in Fig. 11a. Figure 12 also shows the transmission rate from the experiments with $c=1.69$ m s$^{-1}$ for the second baroclinic mode. The transmission rate of second baroclinic mode wave shows qualitatively similar dependency on the period, and relatively high transmission rate for high-frequency wave coincides with the appearance of the additional eastern pathway (not shown). Note that the lower transmission rate compared to the first mode case is related to larger energy dissipation for the second mode through the viscosity term due to smaller deformation radius.

Fig. 12

Incoming wave energy flux from the Indian Ocean $E_{\textrm{IO}}$ (green), the wave energy reaching the Pacific Ocean $E_{\textrm{PA}}$ (pink), and transmission rate $E_\textrm{PA}/E_{\textrm{IO}}$ (black solid line) as a function of the forcing period. Transmission rates for the 2nd baroclinic mode (gray dashed line) are also shown (color figure online)

The energy transmission rate at high frequency in Fig. 12 does not agree with the result of one-dimensional wave interference problem through an ideal strait by Durland and Qiu (2003). They showed that the energy transmission rate increases monotonically as the wave period increases in a narrow channel-like passage similar to the Lombok Strait. The transmission rate only for the realistic configuration of the passages, however, does not increase monotonically with the period. This is because the energy fluxes through the Lombok Strait are caused by superposition of the northward propagating energy directly from the Indian Ocean and the southward propagating energy returning to the Indian Ocean, which enters the Indonesian archipelago through the Ombai Strait and the Timor Sea and propagates back to the west as the coastal Kelvin waves. Therefore, our results do not represent a pure amount of the northward energy propagation through the Lombok Strait. In addition, the resolution of our experiment, 1$^\circ$/10$^\circ$, may be too coarse to consider the very narrow part of the straits.

Fig. 13

The ratio of the northward energy flux across the equator through the Makassar Strait ($E_{\textrm{Makassar}}$), the Molucca Sea ($E_{\textrm{Molucca}}$), and the Halmahera Sea ($E_\textrm{Halmahera}$) relative to the incoming wave energy from the Indian Ocean ($E_{\textrm{IO}}$) as a function of the forcing period (left axis). Theoretical wave energy transmission rates for each Kelvin wave period are based on the Kelvin wave transmission theory (Durland and Qiu 2003) (dashed line, right axis). The theoretical transmission rates are calculated for the narrowest part of the Makassar strait at 3$^\circ$ N, 50 km wide and 100 km long

Since it is difficult to discuss the transmission rate for a particular passage, for example via the Lombok Strait as mentioned above, we consider the energy which goes across the equator to the north through the Makassar Strait ($E_{\textrm{Makassar}}$), the Molucca Sea ($E_{\textrm{Molucca}}$) and the Halmahera Sea ($E_{\textrm{Halmahera}}$). It is noted that the coastal Kelvin waves trapped around the islands, which are superimposed on the pure incoming waves, cannot cross the equator. Figure 13 shows the transmission ratio of $E_{\textrm{Makassar}}$, $E_{\textrm{Molucca}}$ and $E_{\textrm{Halmahera}}$ to the incoming wave energy, propagating eastward off the southern coast of Java Island ($E_{\textrm{IO}}$). The transmission rate of the Makassar Strait increases as the wave period increases in the shorter periods and is almost constant in the longer periods. The Kelvin waves approaching the Makassar Strait are not expected to pass smoothly, because the width of the strait at the narrowest part of the Makassar Strait is narrower than 1/5 of the deformation radius. Figure 13 also shows the theoretical energy transmission rate through the Makassar Strait calculated based on the Kelvin wave transmission theory through the strait narrower than the deformation radius (Durland and Qiu 2003). The energy transmission rate becomes smaller for shorter period Kelvin waves because the phase of the incoming Kelvin waves changes before the adjustment in the strait is completed. Comparing the theory (black dashed line in Fig. 13) and our results (blue line in Fig. 13), the energy transmission rate is almost constant for sufficiently long period in both cases. However, the constant values are very different: the value is almost 1 in the theory while it is about 0.3 in the model results. This discrepancy may be due to the inability to accurately estimate the incoming energy into the Makassar Strait for the model result and to the lack of the viscous effect in the theory of Durland and Qiu (2003) as pointed out by Johnson and Garrett (2006). Nevertheless, their dependencies on the incoming wave period show the similar tendency. Thus, it is reasonable to consider that the smaller energy transmission rate for the shorter period Kelvin waves in the model is due to incomplete adjustment in the strait under the rapid phase change of the incoming Kelvin waves, as discussed in Durland and Qiu (2003).

Unlike the Makassar Strait, the transmission rates of the Molucca Sea and Halmahera Sea do not increase with increasing the period of the incoming wave and have a maximum and a negative minimum in the periods shorter than 3 months. This peculiar behavior may be related to the complicated wave propagation around the Halmahera Island associated with the additional eastern pathway of the wave energy within the Indonesian archipelago. Figure 13 also shows that the transmission rates of the Molucca Sea and Halmahera Sea are much smaller than that of the Makassar Strait in all the periods, suggesting the main route of the wave energy through the Makassar Strait. It is interesting to note that the sum of the transmission rate for the three passages in Fig. 13 is far less than 1, which seems to be caused by the effect of energy dissipation.

Horizontal distributions of the energy dissipation rate indicates that the strong energy dissipation appears along the major route of the energy flux from the Lombok Strait to the Makassar Strait (not shown). The dissipation rate is larger in the west at each latitude within the archipelago, suggesting the importance of the western boundary layer as in the results of the Pacific experiments. Since there are no significant meridional energy fluxes across 12$^\circ$ S between 110$^\circ$ E and the Australian coast in the Indian Ocean and 10$^\circ$ N between the Philippines coast and 140$^\circ$ E in the Pacific Ocean (see Fig. 10a, b), the value $1-E_{\textrm{PA}}/E_{\textrm{IO}}$ can be considered as the dissipation rate within the Indonesian Seas, which does not show clear dependence on the forcing period as in the cases of Pacific experiments (Fig. 7). This discrepancy between the two sets of experiment is attributed to the different causes of the energy dissipation. The amount of the energy dissipation in the Pacific experiments is determined by the shape of the western boundary near New Guinea and wavelength of the incoming waves (i.e., forcing period) as described in Sect. 3.2.2. For the case of incoming Kelvin waves from the Indian Ocean, on the other hand, horizontal velocity shear in the narrow straits provide strong energy dissipation. Therefore, the energy dissipation rate depends on the wave energy pathway in the Indonesian archipelago. The pathway is almost identical for the forcing period longer than two months, causing little changes in the dissipation rate. However, the appearance of the secondary eastern route of the energy pathway in the very short period modulates the dissipation rate within the Indonesian Seas (Fig. 12).

It is noted that the experiment with the shortest period forcing shows very small amount of the wave energy across 4$^\circ$ N/4$^\circ$ S (Fig. 10c) and enters the Indonesian Seas (Fig. 12). This small wave energy is due partly to the less energy input in the forcing region because a period of 10 days is not sufficiently long for the generated Kelvin waves to exit the forcing region before the direction of the wind forcing is reversed. The energy dissipation along the west coast of Sumatra Island, which is found only in the high-frequency cases, may also affect the small magnitude in the shortest period case.

5 Summary and discussion

The detailed pathways of the equatorial wave energy through the Indonesian archipelago and the processes responsible for the formation of the pathways are investigated using a 1.5-layer reduced gravity model, for the incoming waves both from the Pacific Ocean and from the Indian Ocean. The energy transmission rates between the two basins are also quantitatively explored for a wide range of the forcing period. To evaluate the wave energy flux in the equatorial region, the formulation proposed by Aiki et al. (2017) is utilized. This energy flux analysis scheme enables us to show directly how the energy of incoming waves from the Pacific Ocean reaches the Indian Ocean and vice versa.

For the case of incoming Rossby waves from the Pacific Ocean, most of the wave energy propagates southward through the Halmahera Sea and reaches the Indian Ocean via the Banda and Timor Seas. It turns out that the wave energy propagating around the island chain in the Banda Sea cancels the southward energy flux along the easternmost route and has the main pathway shifted to the western side of the island chain. Another pathway to the Indian Ocean via the Makassar Strait also shows significant magnitude of energy flux, but the energy entering the Indonesian Seas through the Makassar Strait is about a quarter of the one through the Halmahera Sea. This wave energy flux distribution is different from the transport distribution of the ITF mean flow which enters the Indonesian archipelago mainly through the Makassar Strait (Gordon and Fine 1996; Gordon 2005). Therefore, it is suggested that not only the western pathway via the Makassar Strait but also the eastern pathway via the Banda Sea should be considered to investigate the impacts of the variabilities in the tropical Pacific Ocean on the Indonesian archipelago.

The energy budget analysis indicates that both the transmitted and reflected wave energy decreases significantly for the wave period shorter than 1.5 years, which is mainly due to the increase in energy dissipation along the northern coast of New Guinea. The different characteristics of the energy propagation for the shorter period waves are related to the geometry of the western boundary of the equatorial Pacific Ocean. The zonal wavelength of the first meridional mode equatorial Rossby wave at the 1.5-year period is about 4000 km, which is equivalent to about two times the zonal width of New Guinea. For this reason, the meridional wall approximation adopted by Clarke (1991) is appropriate when the period of incoming Rossby wave is longer than 1.5 years.

The inclination of the New Guinea coast also affects mass flux along the coast. When the western boundary is inclined, the total incoming mass flux normal to the coastline decreases because the phase difference along the boundary induces incoming and outgoing mass fluxes simultaneously (Cane and Gent 1984; McCalpin 1987). This decrease of incoming mass flux becomes more significant as the wavelength of the incident wave becomes shorter. Thus, the decrease of reflection rate with decreasing wave period shown in this study (see Fig. 7) is partly due to the change in the total mass flux balance along the western boundary. However, the reduction of reflection rate in the present study is more rapid than that explained by the mass flux balance only, and this difference may be explained by the energy dissipation in the boundary layer.

It is worth mentioning that additional experiments with advective terms show similar results with about 2% decrease in the transmission rate and about 2% increase in the reflection rate (not shown), therefore, nonlinear effects may have some influences on the Rossby wave reflection on the western boundary as suggested by Yuan and Han (2006) and Yuan et al. (2019) as well as the wave intrusion into the Indonesian archipelago as suggested by Hu et al. (2022). Despite the decrease of wave energy entering the Indonesian archipelago, the behaviors of waves in the archipelago are similar to those in the linear experiments. It is also noted that the nonlinear experiments in the present study do not include ITF like mean flow, thus no wave-mean flow interaction is taken into account. The assumptions of linear wave and no mean flow in the formulation of Aiki et al. (2017) make it difficult to discuss further on these issues in terms of the wave energy flux. These would be the problems to be investigated in the future.

In the case of the Kelvin waves propagating from the equatorial Indian Ocean, the incident wave energy onto the eastern boundary is divided almost equally between the northern and southern hemispheres for the period longer than one month. The energy in the southern hemisphere continues to propagate eastward along the Sumatra and Java Islands as the coastal Kelvin waves. When the incoming wave period is shorter than one month, most of the wave energy is distributed to the southern hemisphere. The eastward propagating coastal Kelvin waves enter the Indonesian Seas through the Lombok and Ombai Straits and then propagate northward through the Makassar Strait to reach the western Pacific. The wave energy flux also indicates another pathway to the western Pacific via the Banda Sea (see Fig. 11a), which has not been discussed in the previous studies. This additional pathway appear clearly only when the incoming wave period is shorter than 2 months. The energy budget analysis indicates that about 15% of the incoming wave energy reaches the western Pacific for the incoming wave period longer than 1 year. The transmission rate has a peak at 20-day period with a value of about 27% due to the appearance of the additional pathway.

This Banda Sea route of the wave energy flux may appear because the wave energy transmission coefficient through the narrow channel depends also on the ratio of channel width to the deformation radius. The wave energy transmission coefficient quickly decrease as the ratio becomes small when the channel is narrower than the local deformation radius (Durland and Qiu 2003). When the incoming Kelvin waves with a period of about one month enter the Indonesian archipelago, much energy can propagate through the Lombok Strait, where the width of the strait is about one third of the deformation radius at the latitude of the Lombok Strait. However, while the narrowest width of the Makassar Strait is almost the same as that of the Lombok Strait, the deformation radius at the latitude of Makassar Strait is much larger than that for the Lombok Strait. Therefore, the ratio becomes smaller ($\sim$ 0.1) and the wave energy cannot penetrate fully into the Makassar Strait, where the strong energy dissipation occurs instead. The pathway needs to detour to a wider region, generating the new pathway in the Banda Sea. Because observed transport variability at the ITF outflow passages suggests the existence of Kelvin waves forced by periodic winds with the period of 28–46 days (Drushka et al. 2010), these intraseasonal waves penetrating into the Indonesian Seas may play important role in the inter-basin wave energy exchange and variability within the Indonesian archipelago.

Most of the incident wave energy enters the Indonesian archipelago through the Lombok Strait in the present study. The wave signals from the equatorial Indian Ocean, however, have also been observed in the Ombai Strait (Molcard et al. 2001; Potemra et al. 2002; Sprintall et al. 2009; Drushka et al. 2010). Drushka et al. (2010) suggest that the Kelvin wave signals in the Ombai Strait is due to the downward propagation that prevent the incoming Kelvin wave from passing through the shallow Lombok Strait. The 1.5-layer reduced gravity model does not include such vertical process, thus, may not be sufficient for the realistic representations of the Kelvin wave penetration into the archipelago. However, the high transmission rates and the additional eastern energy pathway for intraseasonal waves are still apparent in sensitivity experiments without the Lombok Strait (not shown) since the wave energy turns around the Lesser Sunda Islands and coming back along the northern coast to the longitude of Lombok Strait.

Although only 10% of the incoming wave energy from the Pacific Ocean reaches the southeastern Indian Ocean, the wave signals from the Pacific Ocean certainly affect the interannual variability in the Indian Ocean, such as the Leeuwin current (Feng et al. 2003) and the Ningaloo Niño (Kataoka et al. 2014). On the other hand, about 10% of the incoming wave energy from the Indian Ocean reaches the Pacific Ocean in the present study. The ocean waves from the Indian Ocean, however, have not been observed in the western Pacific Ocean. This difference in the impact of ocean waves in reality may be due to the difference in period of the dominant variations or intensities of the forcing between the Pacific and Indian Oceans, which would be the important questions for future works.

Appendices

Appendix 1: Formulation of energy flux

A new formulation of the energy flux proposed by Aiki et al. (2017) (hereafter AGC17) is briefly introduced here. Following AGC17, we assume linear waves in the absence of a mean flow on an equatorial $\beta$-plane. Then, the linear shallow water equations are nondimensionalized, with the time scale of $1/\sqrt{c\beta }$ and length scale of $\sqrt{c/\beta }$, as:

$$\begin{aligned}&\frac{\partial u}{\partial t} - yv+\frac{\partial p}{\partial x}=0,&\nonumber \\&\frac{\partial v}{\partial t} + yu+\frac{\partial p}{\partial y}=0,&\nonumber \\&\frac{\partial p}{\partial t} + \frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0,&\end{aligned}$$

(1)

where p indicates pressure. Manipulation of the equations 1 yields the wave energy equation

$$\begin{aligned} \frac{\partial }{\partial t}(u^2+v^2+p^2)/2+\nabla \cdot \langle up,vp \rangle =0, \end{aligned}$$

(2)

where $\langle \cdot \rangle$ means a horizontal vector. According to this energy Eq. 2, the divergence of pressure flux, $\nabla \cdot \langle up,vp \rangle$, accurately represent the time rate of energy change at a particular location. However, the pressure flux itself does not always point in the direction of the group velocity vector, i.e.,

$$\begin{aligned} \langle up,vp \rangle \ne \left\langle \frac{\partial \omega }{\partial k},\frac{\partial \omega }{\partial l}\right\rangle (u^2+v^2+p^2)/2, \end{aligned}$$

(3)

where $\omega$ is wave frequency, k and l are zonal and meridional wavenumber, respectively. In mid-latitudes, this problem can be avoided by taking into account the pressure flux associated with geostrophic flows (Orlanski and Sheldon 1993). In contrast, in equatorial regions, we cannot consider geostrophic flows. Therefore, another diagnostic quantity that represents the difference between the two sides of Eq. 3 is required to evaluate the energy flux.

Matsuno (1966) has derived a solution to Eq. 1 on the equatorial $\beta$-plane, which is shown as

$$\begin{aligned}&v=A\cos {\theta }\exp (-y^2/2)H^{(n)},&\nonumber \\&u=(\omega y v_{\theta }-kv_{y\theta })/(\omega ^2-k^2),&\nonumber \\&p=(kyv_\theta -\omega v_{y\theta })/(\omega ^2-k^2),&\end{aligned}$$

(4)

where A is wave amplitude, $H^{(n)}$ is the Hermite polynominal with n being the meridional mode number, $\theta$ is wave phase ($kx-\omega t$). The subscript represents partial differentiation. Using the solutions 4, phase averaged zonal pressure flux can be written as

$$\begin{aligned} \overline{up}=\overline{vv}(2\omega k+1)/[2(\omega ^2-k^2)]+[\overline{v_yv}(2\omega k)-y\overline{vv}(\omega ^2+k^2)]_y/[2(\omega ^2-k^2)^2], \end{aligned}$$

where overbar denotes the phase average. In the same way, the wave energy can be decomposed into two parts,

$$\begin{aligned} \left( \overline{u^2+v^2+p^2}\right) /2= \overline{vv}(2\omega ^2+k/\omega )/[2(\omega ^2-k^2)]\\+[\overline{v_y v}(\omega ^2+k^2)-y\overline{vv}(2k\omega )]_y/[2(\omega ^2-k^2)^2]. \end{aligned}$$

Then, we can obtain an analytical expression for difference between the right and left sides of 3 to yield

$$\begin{aligned} (\partial \omega /\partial k)(\overline{u^2+v^2+p^2})/2-\overline{up}=\frac{-(\overline{pv_\theta })_y-(2\overline{u_{tt}v_\theta })_y}{2k(1+2\omega ^3/k)}. \end{aligned}$$

Finally, the right-hand side of the Eq. 3, i.e., zonal component of the group velocity times wave energy can be rewritten in phase averaged form:

$$\begin{aligned}{} & {} (\partial \omega /\partial k)(\overline{u^2+v^2+p^2})/2=\overline{up}+(\overline{p\varphi }/2+\overline{u_{tt}\varphi })_y, \nonumber \\{} & {} \quad \varphi \equiv -v_\theta /(k+2\omega ^3). \end{aligned}$$

(5)

Meridional component of the group velocity times wave energy can also be derived in the same way. Furthermore, the definition of $\varphi$, the Eq. 5, can be written as

$$\begin{aligned} \nabla ^2 \varphi -y^2\varphi -3\varphi _{tt}=-v_\theta /\omega =q, \end{aligned}$$

where q is the Ertel’s potential vorticity. Therefore the scalar quantity $\varphi$ can be estimated without using Fourier analysis, and then we can diagnostically estimate energy flux vector even with coastal boundaries. Moreover the scalar quantity $\varphi$ is also applicable to mid-latitude waves. Thus, we can trace wave propagations at all latitude with coastal boundary using the AGC17 scheme.

In the present study, we use simplified energy flux, called as level-2 flux in AGC17, to reduce the computational cost. Note that the level-2 flux provides an approximate expression for energy flux based on the group velocity of both low- and high-frequency equatorial waves. To calculate the level-2 energy flux vector of AGC17 scheme, the following approximated inverse problem in dimensionalized form is solved numerically, using results obtained during the last forcing period of our simulations:

$$\begin{aligned} \nabla ^2 \varphi ^{\text {app}} - \left( f/c\right) ^2 \varphi ^{\text {app}} =q, \end{aligned}$$

(6)

where $q = \frac{\partial v}{\partial x} - \frac{\partial u}{\partial y} -(f/c^2)p$ is the linearized Ertel’s potential vorticity and p indicates pressure. Then, using $\varphi ^{\text {app}}$ obtained by solving the Eq. 6, we calculate the level-2 energy flux vector in the dimensionalized form:

$$\begin{aligned} \overline{\mathbf{\textrm{V}}p}+\nabla \times (\overline{p\varphi ^{\text {app}}})/2, \end{aligned}$$

where $\textbf{V}$ is horizontal velocity vector and overbar denotes the phase average.

Appendix 2: Response of the western boundary layer to the incoming equatorial Rossby waves

In the Sect. 3.2.2, we have discussed how the shape of the western boundary affects the incoming equatorial Rossby waves. This appendix details the formation of the meridional currents across the equator in case of the incoming Rossby waves from the Pacific Ocean. Since the shape of the western boundary of the equatorial Pacific Ocean is not purely meridional, mass distribution in the western boundary layer depends on the zonal wavelength of the incoming Rossby waves (Cane and Gent 1984; McCalpin 1987). For sufficiently low-frequency case, mass flux entering the western boundary layer is equatorially symmetric and has maxima away from the equator (upper panel of Fig. 14). To emit the equatorial Kelvin wave, whose zonal mass flux is concentrated around the equator, equatorward boundary flows are formed on both the northern and southern sides of the equator. On the other hand, the high-frequency Rossby waves form the equatorially antisymmetric mass distribution in the western boundary layer, resulting in the meridional flow across the equator (bottom panel of Fig. 14). This strong cross-equatorial flow can be seen west of 138$^\circ$ E in Fig. 9b and propagates westward at the phase speed of the incoming Rossby waves.

Fig. 14

Schematic diagram of the inclined western boundary and incoming Rossby waves. Positive (negative) sea level anomalies of the equatorial Rossby waves are represented as red (blue) color shades. Black arrows indicate the directions and magnitudes of the western boundary flow (color figure online)

Figure 9b also shows strong cross-equatorial current east of 138$^\circ$ E, which cannot be attributed to the mass distribution in the western boundary layer because this region is located outside of the diffusive western boundary layer. Moreover, this meridional flow east of 138$^\circ$ E shows a clear westward phase propagation with eastward propagation of the wave packet, which implies eastward group velocity. It is suggested, therefore, that short Rossby waves or Yanai waves with eastward energy propagation are emitted from the western boundary layer in the high-frequency case.