Previous studies indicated that NBSWs are able to enhance the turbulent mixing in the upper ocean (Babanin and Haus, 2009; Dai et al., 2010; Huang and Qiao, 2010; Qiao et al., 2016). The ability to simulate the SST and MLD can obviously be improved via the incorporation of the related NBSW-induced turbulent mixing schemes into OGCMs (Lin et al., 2006; Xia et al., 2006; Song et al., 2007; Aijaz et al., 2017; Wang et al., 2019). According to Yuan et al. (2011, 2013), Zhuang et al. (2020) expressed the vertical viscosity, Bus, and diffusivity, BTs, generated by the non-breaking surface waves (NBSW) as follows: 1Bus=7412⋅hsw2⋅∑i=13∑j=13∂ui∂xjsw2/∑i=13∑j=13∂ui∂xjx3=0swBTs=1σ7412⋅hsw2⋅∑i=13∑j=13∂ui∂xjsw2/∑i=13∑j=13∂ui∂xjx3=0sw, where hsw is the SWH, and ∑i=13∑j=13∂ui∂xjsw is the averaged velocity shear module of the sea surface waves and can be calculated as 2∑i=13∑j=13∂ui∂xjsw2=2∫∫kΦswKsw2ωsw2⋅cosh⁡2Ksw(x3-H)sinh⁡2Ksw⋅(-H)dk1dk2, where ωsw is the surface wave frequency in a typical frequency range: ωsw>N. N denotes the Brunt–Väisälä frequency, Ksw is the wavenumber and H is the water depth. Φsw=ηsw⋅ηsw∗ is the wavenumber spectrum of hsw, ηsw is the Fourier kernel function of hsw, i.e., hsw=∫∫ηswexp⁡ik1x+k2y-ωtdk1dk2 (here the superscript * means the conjugate value), and k1 and k2 are the horizontal components of the wavenumber in the x and y directions.

Apart from NBSWs, the residue of the wave transport flux is also able to contribute to inducing mixing in the ocean circulation through the Reynolds average upon characteristic wavelength scale (Yang et al., 2009, 2019). Yang et al. (2009) proposed a mixing scheme for the wave transport flux residue (WTFR), which has been adopted in OGCMs (Shi et al., 2016; Yu et al., 2020). The results show that the simulated SST and MLD are remarkably improved, especially in summer and in the strong current regions. In tropical cyclone conditions, the performance of the model to simulate ocean response could also be greatly improved if the wave transport flux residue mixing scheme is introduced. The coefficients of the wave transport flux residue mixing are expressed as follows: 3BSM1=∫∫kωk1Ekcosh⁡2K(x3-H)sinh⁡2K⋅(-H)dk1dk2BSM2=∫∫kωk2Ekcosh⁡2K(x3-H)sinh⁡2K⋅(-H)dk1dk2, where Ek represents the wavenumber spectrum, which can be calculated from the wave spectrum model; other variables are the same as in Eq. (2).

In the stratified ocean interior, ITs are able to provide about half of the mechanical power required for the ocean interior turbulent mixing (Wunsch and Ferrari, 2004; Zhao, 2018; Vic et al., 2019; Whalen et al., 2020). However, current field observations are insufficient for constructing the whole internal-tide map in the IO. Satellite altimetry is able to provide sea surface height (SSH) measurements to observe the global ITs (Ray and Mitchum, 1996). Zhao et al. (2016) presented a method to extract the M2 ITs by fitting plane waves to satellite altimeter data in individual windows with a size of 160 km × 160 km. In this technique, the least square fitting algorithm is adopted to determine the amplitude and phase of one plane wave. This procedure can be repeated three times to extract the three most dominant M2 internal waves, superposition of which gives the final internal tidal solution. In this study, the turbulent mixing generated by M2 semi-diurnal ITs will be derived from the SSH amplitude (Zhao et al., 2016). Other principal tidal constituents will be studied in the future. For simplicity, the mode-1 M2 ITs, which mainly originate from regions with steep topographic gradients, are considered because the depth-integrated energy and SSH amplitudes of the mode-2 M2 ITs are much smaller than mode-1 ones (Zhao, 2018).

Yuan et al. (2013) presented a second-order turbulence closure model to estimate the turbulence kinetic energy and dissipation in terms of the velocity shear module of non-breaking waves. The subsurface displacements of ITs, pressure anomalies and currents can be derived from the SSH amplitudes following vertical models (Zhao and Alford, 2009; Wunsch, 2013; Zhao, 2014). The detailed derivation process about the velocity shear module of the internal tide can be found in Appendix A. For the mode-1 M2 IT, vertical viscosity, Bui, and diffusivity, BTi, generated by the velocity shear can be written as follows: 4Bui=7412⋅hiw2⋅∑i=13∑j=13∂ui∂xjiw2∑i=13∑j=13∂ui∂xjx3=0iwBTi=1σ7412⋅hiw2⋅∑i=13∑j=13∂ui∂xjiw2∑i=13∑j=13∂ui∂xjx3=0iw, where hiw is the SSH amplitude of the mode-1 M2 IT. ∑i=13∑j=13∂ui∂xjiw is the averaged velocity shear module of the internal tides in a simple monochromatic form and can be calculated based on the unified linear theory under general ocean conditions (Yuan et al., 2011). The expression can be written as 5∑i=13∑j=13∂ui∂xjiw2=2hiwKiw2ωiw2⋅ϕ2ωiw-12sin⁡2∫-Hx3ϕωiwKiwdx3+2ϕ2ωiwsin⁡2∫-H0ϕωiwKiwdx3, where 6ϕωiw=N2-ωiw2ωiw2-f2, where ωiw is the M2 tidal frequency, and Kiw is the wavenumber. Under the influence of the Earth's rotation, the dispersion relation can be written as 7ωiw2=Kiw2cn2+f2, where f is the inertial frequency, and cn is the eigenvalue speed, which is the phase speed in a non-rotating fluid. The expression can be written as 8cn2=∫-H0Nx3dx3nπ2, where n is the mode number that is set to be 1 here.

The effects of the new vertical mixing schemes are introduced into OGCMs. The modified equations can be written as 9∂Ui∂t+Uj∂Ui∂xj=F+ΠU∂C∂t+Uj∂C∂xj=G+ΠC, where xj (j=1, 2, 3) represents the x, y and z axes of the Cartesian coordinates, and Ui (i=1, 2, 3) and C denote the mean velocity current components and one of the two tracers including the potential temperature and salinity, respectively. The second terms on the left-hand side of Eq. (9) are the advection ones. F represents the sum of the terms on the right-hand side of the momentum equations including the Coriolis force, pressure gradient force, horizontal diffusion, molecular viscous force and external forcing terms. G represents the sum of the terms on the right-hand side of the tracer equations including horizontal diffusion, molecular diffusivity force, and heat and freshwater flux terms. ΠU and ΠC denote the modified vertical diffusive terms and can be expressed as 10ΠU=∂∂x3Km∂Ui∂x3+∂∂x3Bus∂Ui∂x3+-BSM1∂Ui∂x1-BSM2∂Ui∂x2+∂∂x3Bui∂Ui∂x3ΠC=∂∂x3Kh∂C∂x3+∂∂x3BTs∂C∂x3+-BSM1∂C∂x1-BSM2∂C∂x2+∂∂x3BTi∂C∂x3. The new vertical diffusive terms ΠU and ΠC can be divided into four parts as shown in Eq. (10). The first term on the right side denotes the original diffusive term, where Km and Kh are vertical eddy viscosity and diffusivity calculated by the classic Mellor–Yamada 2.5 (M-Y 2.5) scheme (Mellor and Yamada, 1982). The M-Y 2.5 scheme is a level-2.5 turbulence model based on the modification of the material derivative and diffusive terms. The mixing coefficients Km and Kh can be calculated as the turbulence characteristics ql multiplied by a stability function associated with the Richardson number, where q represents the turbulent fluctuation velocity, q2/2 is the turbulence kinetic energy and l is the mixing length scale. The turbulence kinetic energy q2/2 can be estimated from the local shear production, buoyancy and dissipation based on the q2-q2l closure equations in the atmospheric boundary. Actually, the M-Y 2.5 scheme was proposed based on the assumption of a rigid surface and did not consider the effects of surface and internal waves (Qiao et al., 2004; Huang and Qiao, 2010; Huang et al., 2011), which is regarded as one of the major reasons for the insufficient mixing in the upper-ocean simulation. The remaining terms represent the new diffusive terms generated by surface waves and internal tides, which are described in Sect. 2.1–2.3.

The three-dimensional MASNUM ocean circulation model (Han and Yuan, 2014; Zhuang et al., 2018) is used to evaluate the effects of NBSW- and IT-generated turbulent mixing and WTFR-induced mixing. The two-level single-step Eulerian forward–backward time-differencing scheme and the σ–Z–σ hybrid vertical coordinate are adopted in the MASNUM ocean model. The forward–backward scheme with a spatial smoothing method should be superior to the leapfrog scheme because of more stability and more computational efficiency (Han, 2014). Han (2014) and Han and Yuan (2014) have tested the modeling ability of the MASNUM model compared with the POM. The results showed that the MASNUM model could produce quite identical simulation results as the existing models with only half the computer cost.

The model domain is in an area of 50∘ S–30∘ N, 0–135∘ E (Fig. 1) with a horizontal resolution of 1∘ / 6 by 1∘ / 6. 5 surface σ layers and 31 intermediate Z layers. Three bottom σ layers are used in the vertical direction in order to obtain the vertical grid spacing with a high resolution in the upper ocean. The topography of the model is downsampled from the global General Bathymetric Chart of the Oceans 2008 (GEBCO_08) with a resolution of 1 by 1. The minimum depth is set to 5 m. The maximum depth is set to 5000 m, avoiding artificial influences at deep-water depths. The topography has been smoothed using the dual-step five-point-involved spatial smoothing method (Han, 2014) to make the calculation more stable. The topographic gradients were not considered to be a key factor in this study because the climatological experiments in this study are inappropriate to directly simulate the ITs.

The initial temperature and salinity are interpolated based on annual mean Levitus94 data (Levitus and Boyer, 1994; Levitus et al., 1994) with the horizontal resolution of 1∘ by 1∘ and 33 vertical layers. The initial velocities are set to 0. The gravity-wave radiation conditions (Chapman, 1985) were used as the lateral boundary conditions, which are very important for basin-scale modeling in this study. The simulated variables, including velocities, temperature, salinity and SSH, on the lateral boundary grids are calculated in an explicit numerical form. In the explicit form, the values of the related variables obtained from the daily global climatologic model results by the MASNUM model with the horizontal resolution of 1∘ / 2 by 1∘ / 2 are also used. The lateral boundary conditions are time-dependent with an updating period of 1 d. The surface forcing including the momentum, heat and wind stress fluxes is calculated from the monthly mean surface fields of the National Centers for Environmental Prediction/National Center for Atmospheric Research (NCEP/NCAR) reanalysis dataset with the horizontal resolution of 1∘ / 4 by 1∘ / 4. We calculated the multiyear monthly mean surface forcing results based on the time series of the NCEP/NCAR data from 1948 to 2021. Then the model is driven by the monthly mean surface forcing results, which repeats in every climatologic year. The time step size of the barotropic mode is set to 30 s, while that of the baroclinic mode is 900 s. The model is integrated from the quiescent state for 10 climatological years. The simulated results in the last 1 year are compared with the monthly World Ocean Atlas 2013 (WOA13) and the Ocean Surface Current Analyses Real-time (OSCAR) climatologic data, which can be regarded as the true solution of the climatological numerical experiments.

It is worth noting that the time interval of 10 years should be appropriate for ocean simulation from the quiescent state to a relatively stable circulation background. The average kinetic energy, which can be regarded as a model stability index, fluctuated obviously in the first 2 years, then became stable gradually in the third year and was completely steady from the fourth to the 10th year. The conclusion is similar to many previous studies (e.g., Xia et al., 2006; Qiao et al., 2010; Han, 2014; Yu et al., 2020).

The MASNUM wave spectrum model (Yuan et al., 1991, 1992; Yang et al., 2019) is used to simulate the parameters of surface waves in the IO. The energy-balanced equations are solved in the model based on the wavenumber spectrum space. The characteristic inlaid scheme is adopted for the wave energy propagation to improve the original wave model (Yuan et al., 1992). The wave model has been validated by observations (Yu et al., 1997) and widely accepted in ocean engineering and numerical simulation (e.g., Qiao et al., 1999; Xia et al., 2006; Qiao et al., 2010; Shi et al., 2016; Yang et al., 2019; Yu et al., 2020; Sun et al., 2021). The results showed that the simulated SWH and mean wave period are consistent with satellite observations.

The model domain, resolution, topography and surface wind stress flux data are consistent with those in the MASNUM ocean circulation model. The boundary conditions are from the JONSWAP spectrum (Hasselmann et al., 1973). The wave model is integrated from the quiescent state for 10 climatological years with the same period as the ocean circulation model. Actually, the configuration of the wave model is simpler than the OGCM, and the model design in this study is almost the same as that in Xia et al. (2006) and Qiao et al. (2010). Therefore, we believe that the experiment using the MASNUM wave model is able to characterize the spatial pattern and variation of surface waves in the IO.

The wave spectrum Ek is calculated from the MASNUM wave model, and then ωsw, Ksw and hsw can be estimated. Thus, the new mixing coefficients including Bus, BTs, BSM1, BSM2, Bui and BTi are calculated directly from Eqs. (1)–(4).

To assess the effects of the NBSW, WTFR and IT on the vertical mixing and simulated thermal structure in the upper ocean, five experiments (Table 1) are denoted as Exp 1–5 and designed as follows.

Exp 1 (benchmark experiment). The model is integrated with the classic M-Y 2.5 turbulence closure model (Mellor and Yamada, 1982), which is broadly used in OGCMs.

Exp 2. Same as Exp 1, except with the classic M-Y 2.5 scheme and the NBSW-generated turbulent mixing scheme. This experiment is designed to evaluate the effect of NBSWs.

Exp 3. Same as Exp 1, except with the classic M-Y 2.5 scheme and the NBSW- and IT-generated turbulent mixing schemes. This experiment is designed to evaluate the effects of NBSWs and ITs. The experiment with the M-Y 2.5 scheme and the IT-generated turbulent mixing scheme (Exp 3.5) is omitted in this study because the deviation of the temperature between Exp 1 and Exp 3.5 is too small. The possible reason is that ITs are often considered to enhance vertical mixing in the ocean interior from the thermocline to abyssal regions (Munk and Wunsch, 1998; Wunsch and Ferrari, 2004; Kunze et al., 2006); therefore, it could be insufficient for the incorporation of only ITs into the M-Y 2.5 scheme to draw warmer water from the surface into the interior. This implies that only the IT is unable to improve the upper-ocean simulation.

Exp 4. Same as Exp 1, except with the classic M-Y 2.5 scheme and the WTFR-induced mixing scheme. Comparisons between Exp 2 and Exp 4 are implemented to evaluate the two mechanisms through which the surface waves affect the upper-ocean vertical mixing.

Exp 5. Same as Exp 1, except with the classic M-Y 2.5 scheme, NBSW- and IT-generated turbulent mixing, and the WTFR-induced mixing scheme. This experiment is designed to evaluate the effects of NBSWs, ITs and WTFR.

It is worth noting that the climatological experiments, which should be regarded as the multiyear mean simulation, are designed in this study, so it is inappropriate for the simulated results to be compared with the Argo data because there should be a considerable difference between the climatologic data and real-time in situ observations. The WOA13 data, which represent the multiyear (1955–2012) mean results, and the multiyear (1993–2021) mean OSCAR data will be a good choice to evaluate the ocean climatological modeling.

As a typical example, the vertical distribution of the monthly mean vertical temperature diffusive terms in logarithmic scale along the zonal transect of 10.5∘ S in January and July is shown in Figs. 2 and 3. As expressed in Eq. (10), the following are presented: the calculated vertical diffusive term based on the M-Y 2.5 scheme, which is written as ∂∂x3Kh∂T∂x3 (KHT for short) and calculated from Exp 1; the NBSW-generated turbulent mixing scheme, which is written as ∂∂x3BTs∂T∂x3 (BTST) and calculated from Exp 2; the IT-generated turbulent mixing scheme, which is written as ∂∂x3BTi∂T∂x3 (BTIT) and calculated from Exp 3; and the WTFR-induced mixing scheme, which is written as -BSM1∂T∂x1-BSM2∂T∂x2 (BSMT) and calculated from Exp 4. The BTIT can be calculated independently as part of the diffusive terms in Exp 3. Figures 2 and 3 show the comparisons among these diffusive terms along 10.5∘ S, which is a typical transect to show the difference, in January and July. There are two reasons for the choice of the 10.5∘ S transect. Firstly, the Madagascar–Mascarene regions (0–25∘ S in the western Indian Ocean) are considered to be a hot spot for the generation of semi-diurnal ITs (Zhao et al., 2016). Both NBSWs and ITs should grow fully and become large enough for the comparison of the diffusive terms. Secondly, the 10.5∘ S transect is typical to show the spatial pattern among the diffusion terms because of the stronger surface waves and ITs (Kumar et al., 2013; Zhao et al., 2016; Ansong et al., 2017).

One can see that all of the terms decay with the depth below the sea surface. In January, BTST is >10-2 ∘C s-1 in the upper 30 m layer of most regions, the values of which are too high to show in Figs. 2b and 3b, and obviously greater than other terms, implying that the NBSW-generated turbulent mixing is dominant in layers with depths less than 30 m. Similar to BTST, BSMT (Fig. 2d) is also induced by NBSW and directly generated by the surface wave orbital velocity, but the values are about 4 to 6 orders smaller than those of BTST. However, in July, BSMT may affect greater depths than BTST and KHT, especially in some regions with large topographic relief. In the ocean interior with depths from 40 to 130 m, BTIT (Fig. 2c) is about 10-6 ∘C s-1 and significantly higher than the other three terms in some regions such as the eastern Atlantic (5–10∘ E), the western Indian Ocean (50–70∘ E) and the western Pacific (122–125∘ E) because of the effects of the IT. It is worth noting that the vertical distribution of the eddy diffusivity (Kh, BTs and BTi) is very similar to the diffusive terms. Especially in January, BTs is the largest in the upper 30 m layers and BTi is generally larger in the ocean interior with depth deeper than about 40 m. Kh and BTs decay with depth below the sea surface, and the delay rate of BTs is obviously slower than Kh, so BTs is larger than Kh in the ocean interior. The high-value layers (>10-5 m2 s-1) of Kh are as thin as about 20 m in January and up to about 80 m partially in July, while the high-value layers of the BTs are generally about 70–100 m in both January and July. When the depth is larger than 40 m, the value of BTi appears to be about 10-5–10-3 m2 s-1.

The climatologic experiments are designed in this study because of the NCEP monthly climatological sea surface flux forcing fields and the daily global climatological lateral boundary conditions in the simulation, so the WOA13 monthly climatology data can be used in comparisons as a reference.

Figures 4–7 show the comparisons of the upper-ocean temperature vertical structure between the WOA13 data and the model results of the five experiments along transects of 30.5∘ S and 7.5∘ N, corresponding to SIO and north of the equatorial Indian Ocean (EIO), in January and July. In the Southern Hemisphere, the 30.5∘ S transect is typical to show the effects of the three schemes on the temperature modeling. The temperature structure along 10.5∘ S, which is used to show the comparison of the diffusive terms in Sect. 4.1, is omitted here because there is a non-ignorable difference between the WOA13 data and the simulation results, especially in the eastern Indian Ocean, and the effects of the NBSW in the tropical area are relatively non-obvious, which is regarded as a long-standing issue (Qiao et al., 2010; Zhuang et al., 2020). In the Northern Hemisphere, the temperature structure along the 7.5∘ N transect, which is located in the south of the Arabian Sea and Bay of Bengal and regarded to be representative for modeling evaluation, is discussed. One can see that the difference by subtracting the monthly mean results of Exp 1 from the monthly WOA13 data is the largest among the five experiments along the two transects in January and July. In the ocean interior, the temperature of Exp 1 is extremely lower than the WOA13 data, which implies that less surface water is transferred into the layers with depths from 30 to 100 m in Exp 1 because of the insufficient vertical mixing process simulated by the classic M-Y 2.5 scheme. Compared with Exp 1, the difference for Exp 2 decreases remarkably because the NBSW strengthens the vertical mixing and improves the upper-ocean simulation, which has been proved many times by previous studies (Lin et al., 2006; Huang et al., 2011; Qiao et al., 2016; Zhuang et al., 2020).

The difference for Exp 3 is much smaller than that of Exp 1 and Exp 2 because of the incorporation of the IT-generated turbulent mixing, especially in layers with depths between 20 and 50 m. This implies that the IT strengthens the vertical mixing of the ocean interior and improves the simulation further. It is worth noting that the experiment with the classic M-Y 2.5 scheme and the IT-generated turbulent mixing scheme is omitted; the reason is that the results have not been improved if only the IT-generated turbulent mixing is incorporated because the simulated surface mixing is insufficient and even deteriorated in some regions because colder water will be drawn from the lower layers with depths deeper than 100 m into the upper ocean.

However, the simulation is slightly improved in Exp 4 compared with Exp 1 because the BSMT, which is induced by the WTFR, is remarkably smaller than the BTW, so the WTFR-induced mixing is too insufficient to significantly improve simulating the upper-ocean temperature structure. Similarly, there is less difference between Exp 3 and Exp 5, implying that the effects of the WTFR on enhancing vertical mixing are much weaker than the NBSW in the surface layers and the IT in the ocean interior.

In Exp 1, the simulated temperature along 30.5∘ S is cooler than the WOA13 data, while the temperature bias becomes reversed with similar magnitude in Exp 2 and Exp 3. The reason should be that the multiyear monthly mean surface forcing fields, on one hand, were smaller than the actual values, which leads to insufficient heat transfer from the atmosphere to the ocean. After 10 climatologic years of modeling, the temperature in the ocean interior became obviously cooler than the WOA13 data. On the other hand, NBSWs and ITs enhanced the vertical mixing as well as the heat transfer, so more heat entered the ocean interior and the SST became cooler. Additionally, the Haney equation (Haney, 1971), which improves the large-scale thermal coupling of the ocean and atmosphere, is used to modify the surface heat flux. However, a disadvantage of the Haney modifying method is the destruction of the heat balance, so solar radiation will continuously increase in the ocean surface. Therefore, the accumulation of heat during the 10-year modeling will make the temperature bias in Exp 2 and Exp 3 reversed, with similar magnitude in Exp 1. Furthermore, the temperature from the annual mean Levitus94 data, which are used as the initial fields, is cooler by about 3∘ than that from the WOA13 data in the Antarctic Circumpolar Current (ACC) region from the surface to 200 m depth and warmer by about 0.5∘ in the NIO and tropics when the depth is deeper than 200 m. This should make the simulated temperature cooler than the WOA13 data in the SIO and become warmer in the NIO and the tropics. The improvement of NBSWs and ITs in the temperature simulation is obvious because of the smaller errors in Exp 2 and Exp 3, although the difference of the temperature bias between Exp 1 and Exp 2–3 is substantial.

Figure 8 shows the monthly variability of the root mean square errors (RMSEs) of the temperature in the upper 100 m layers between the WOA13 data and the model results. Actually, the RMSEs are calculated based on the simulated temperature only in the whole IO (the regions outside the IO have been removed) as the following expression: 11RMSE=∑i=1im∑j=1jm∑k=1kstmi,j,k-twi,j,k2im×jm×ks, where tm and tw represent the model results and the WOA13 data for the monthly mean temperature, and im, jm and ks mean the number of grids in the whole IO in the horizontal and vertical directions. The RMSE can be regarded as a spatial average deviation of the three-dimensional temperature fields. Therefore, the RMSE should be statistically robust because the calculated result is unique if the spatial range of the temperature field is determined.

The study area is divided into three zones (Zones 1–3 marked in Fig. 1). The zone partition of the IO in this study is designed based on previous studies and the dynamic patterns of the IO. On one hand, previous studies (Talley et al., 2011; Kumar et al., 2013, 2018) showed different zone partitioning criteria, which often included the NIO, SIO and tropical regions. On the other hand, the principal upper-ocean flow regimes of the IO are the subtropical gyre of the SIO and the monsoonally forced circulation of the tropics and NIO. All effects of the Indonesian throughflow (ITF) should also be considered. Taking the above factors into account, the whole IO was divided into three parts. Zone 1 represents the NIO including the Arabian Sea and the Bay of Bengal. Zone 2 represents the tropics and subtropical regions in the SIO with all effects of the subtropical gyre and the ITF. Zone 3 represents the region in the south of Zone 2 in the SIO. In Zone 2, there is a complete cyclonic circulation system between the Equator and 20∘ S, consisting of the westward South Equatorial Current on the south side, the eastward South Equatorial Countercurrent on the north side, the northward East African Coastal Current and the ITF. The effects of the M2 internal tides, which are generated in the northern regions around Madagascar, are produced throughout the whole west region in Zone 2.

In Zone 1, the RMSEs for Exp 2 are smaller than for Exp 1 in all of the months, indicating the improvement of the NBSW in the upper-ocean simulation in the NIO. Compared with Exp 2, the RMSEs for Exp 3 are smaller in most of the months except November, December and January. This implies that the IT enhances vertical mixing and improves the simulation further. The possible reason for few effects of the IT from November to January is that, on one hand, the mixed layer depths in the NIO are relatively shallower in boreal winter so that the averaged velocity shear module of the internal tides is smaller and the IT-induced mixing is weaker; on the other hand, the strength of the surface waves is more intensive, so the NBSW-induced mixing is relatively sufficient. The largest declines occurred in May, when the RMSE decreased 14.0 % from 1.72∘ (Exp 1) to 1.49∘ (Exp 2) and 19.1 % from 1.72∘ (Exp 1) to 1.40∘ (Exp 3).

In Zone 2, the NBSW is ineffective because the RMSEs for Exp 2 are almost equal to, or even larger than, those for Exp 1. This is a long-standing issue about the trivial effects of the NBSW in the tropical area (Qiao et al., 2010; Zhuang et al., 2020), implying that only the NBSW should not be enough to improve the tropical simulation. To solve this issue, coupled atmosphere–wave–ocean–ice modeling is one solution (Song et al., 2012; Wang et al., 2019). Another way is incorporation of the additional mechanism into OGCMs, such as the IT-generated turbulent mixing added in Exp 3 and Exp 5. The RMSEs for Exp 3 are obviously smaller than for Exp 1 and Exp 2 in the whole climatologic year except March, implying that the combination of the NBSW and the IT is able to improve the simulation of the temperature structure in the tropical area. Additionally, the RMSEs in Zone 2 are smaller than in Zones 1 and 3 on the whole, and the RMSEs in Zone 2 for Exp 3 and Exp 5 are even less than 0.9∘ in half of the climatologic year, indicating much accurate simulation in the tropical area.

In Zone 3, the results are similar to those in Zone 1. The RMSEs for Exp 2 are smaller than for Exp 1 in most months, and the RMSEs for Exp 3 are the smallest ones among the first three experiments. The largest declines occurred in January, when the RMSE decreases 20.8 % from 1.50∘ (Exp 1) to 1.18∘ (Exp 2) and 25.7 % from 1.50∘ (Exp 1) to 1.11∘ (Exp 3). The situation also indicates significant improvements from the combination of the NBSW and IT in simulating the upper-ocean temperature structure.

Furthermore, in Zones 1–3, the effects of WTFR are much weaker and similar to those in Figs. 4–7 because the RMSEs for Exp 4 and 5 are almost equal to, and even larger than, those for Exp 1 and 3. The possible reason is that the values of the WTFR-induced diffusion terms are about 4 to 6 orders smaller than NBSW, which is too low to enhance vertical mixing, especially in the surface layers.

The thermal structure in the regions with depths from 100 to 300 m are also compared with the WOA13 data. The simulated temperature is generally cooler than the WOA13 data along 30.5∘ S at depths between 100 and 300 m, while it is dramatically warmer along 7.5∘ N. The distribution pattern of the simulated temperature in the ocean interior (from 100 to 300 m or deeper) seems cooler in SIO and warmer in NIO and the tropics. One of the reasons is that the Haney equation (Haney, 1971) is used to modify the climatologic surface heat flux and brings excessive heat into the ocean interior in the simulation. Some more accurate surface forcing data with higher resolution will be used in future simulations.

Furthermore, the intermediate and deep-water masses in the IO are often effected by the Southern Ocean, including Antarctic Intermediate Water, Circumpolar Deep Water and North Atlantic Deep Water. These cooler water masses are carried by the meridional overturning circulation into the IO throughout the south of the South Equatorial Current in the subtropical Indian Ocean (Talley et al., 2011), but the situation did not appear in the simulated current fields. Therefore, another important reason should be that it is hard to accurately simulate the meridional overturning circulation in the present experiments, especially the meridional transport of heat. This makes the simulated temperature cooler or warmer than the WOA13 data along 30.5∘ S and 7.5∘ N when the depth is deeper than about 100 m.

In addition, it is worth noting that the initialization design is also important for ocean modeling. The comparison between the annual mean temperature between the Levitus94 and WOA13 data shows that the temperature from the Levitus94 data is obviously cooler than that from the WOA13 data in the ACC regions (45–75∘ E, 35–50∘ S), while it is generally warmer in the whole IO at depths from 200 to 500 m. The WOA13 data contain more mesoscale information than the Levitus94 data. Therefore, the inaccurate initial field could also be one of the reasons why the simulated temperature in the ocean interior is different from the WOA13 data. A series of high-resolution real-time numerical experiments for the circulation in the IO will be carried out to examine the influence of different initial fields, parameterization schemes, surface fluxes and open boundary conditions in the future. It is worth noting that the detailed analysis of the deep ocean is omitted here because the vertical mixing in the upper ocean (0–100 m) is the main focus of this study.

Lozovatsky et al. (2022) demonstrated that internal wave instabilities appear to be a dominant mechanism for generating energetic mixing based on an analysis of in situ observations of the turbulent kinetic energy dissipation rate and buoyancy frequency profiles. Actually, designing a universal and flexible IT-induced mixing scheme for ocean modeling based on in situ observations still needs a lot of work. The three schemes introduced in this study are just preliminary research on the contribution of upper-ocean vertical mixing.

The thermocline structure, which is normally defined as the depth of the 20∘ isothermal (Z20), can affect SST variability via vertical water exchange and thereby modulate air–sea interaction events (Talley et al., 2011). Vertical displacements of the thermocline depth at the equatorial region are regarded as one of the distinctive features in the IO. Because of the weak westerly at the Equator, the IO shows a deeper and reversed slope of Z20 compared to its counterpart in the Pacific Ocean. Analysis of the mean state thermocline structure is very important because the thermocline variability is related to the Indian Ocean Dipole, ITF and El Niño–Southern Oscillation at the interannual timescale according to previous studies (e.g., Chambers et al., 1999; Gordon et al., 2003; Liu et al., 2017). Figure 9 shows the variation of the Z20 depths along the Equator in January and July. As another indicator of the thermal structure in the upper 100 m layers, the depths of the 26∘ isothermal (Z26) are also plotted in Fig. 9a and b. The RMSEs of the Z20 and Z26 depths are calculated and plotted in Fig. 9c and d.

From Fig. 9 one can see that the Z20 and Z26 depths are both shallow in the west and deep in the east. The simulations of the thermal structure in the five experiments depict this pattern successfully, but there is still an obvious difference between the WOA13 data and the results, especially in the east regions in January for Z26 depths and in July for Z20 depths. One of the main reasons should be that the ITF may be simulated inaccurately because of inaccurate topography in the Indonesian regions and the open boundary conditions. The accurate simulation of the ITF should be a difficult issue because of the complicated topography and ocean–atmosphere interaction in the Indonesian Archipelago. Many OGCMs are incapable of reproducing the patterns of the ITF (Nagai et al., 2017; Santoso et al., 2022). Another reason is that this area is full of eddies produced by horizontal velocity shear, but our ocean model still lacks an accurate and reasonable parameterization of eddy-induced mixing, which needs more future work. For Z26 depths, the RMSEs for Exp 1 and Exp 4 are the largest in almost all of the months; this implies that the WTFR-induced mixing has little effect on the modeling, which is consistent with the comparison results above (Figs. 4–8). The NBSW- and IT-generated turbulence mixing can improve the simulated thermal structure as two of the key factors because of the smallest RMSEs (from February to July for Exp 2 and from August to January for Exp 3 and Exp 5). For Z20 depths, the NBSW and the IT have negative effects on the modeling because the RMSEs for Exp 2–Exp 5 are obviously larger than those for Exp 1. The reason is that the Z20 isothermal simulated in Exp 1 is generally deeper than the WOA13 data because the simulated temperature in the regions with depths from 130 to 200 m is warmer than the WOA13 data, and the enhanced vertical mixing induced by the NBSW and the IT will make the Z20 isothermal deepen further and deviate more from the WOA13 data (solid curves in Fig. 9a and b). Therefore, more optimization and improvement of the experimental design will be implemented in future work to make the simulated results more accurate.

In order to evaluate the modeling results further, the existing Argo-derived gridded products, which are named Barnes objective analysis-Argo (BOA-Argo) datasets (Li et al., 2017), are also chosen. The climatologic monthly mean BOA-Argo data (multiyear mean from 2004 to 2014) are used and can be downloaded directly from ftp://data.argo.org.cn/pub/ARGO/BOA_Argo/ (last access: 5 September 2022). The BOA-Argo data with 49 vertical levels from the surface to 1950 m depth are produced based on refined Barnes successive corrections by adopting flexible response functions. A series of error analyses is adopted to minimize errors induced by the nonuniform spatial distribution of Argo observations. These response functions allow BOA-Argo to capture a greater portion of mesoscale and large-scale signals while compressing small-sale and high-frequency noise. The performance of the BOA-Argo dataset demonstrates both accuracy and retainment of mesoscale features. Generally, BOA-Argo seems to compare well with other global gridded datasets (Li et al., 2017).

Figures 10 and 11 show the comparison of the temperature structure between the monthly BOA-Argo data and the model results in January. The vertical distributions are similar to those from the WOA13 data (see panel a in Figs. 4, 6, 10 and 11). The difference between the BOA-Argo data and the model results along 30.5∘ S is also similar to the WOA13 data. Compared with Exp 1, the difference for Exp 2 often decreases remarkably, and the difference for Exp 3 is much smaller than that of Exp 1 and Exp 2 because of the incorporation of the IT-generated turbulent mixing, especially in the layers with depths between 20 and 50 m. In addition, the improvement of the NBSW and IT along 7.5∘ N is not obvious; this conclusion is also similar to that for the WOA13 data. This implies that the three mixing schemes introduced in this study may not be appropriate in the marginal sea simulation that is full of small-scale and mesoscale processes. In order to solve the issues about the accuracy, we attempt to design high-resolution real-time numerical modeling experiments in the NIO (or the Arabian Sea and the Bay of Bengal only), as well as finer simulation of surface waves and more accurate estimation of ITs.

The mixed layer (ML), which is characterized by quasi-uniform temperature and salinity, is crucial in understanding the physical processes in the upper ocean. The MLD variability is influenced by many processes including wind-induced turbulence, surface warming or cooling, air–sea heat exchange and turbulence–wave interaction (Chen et al., 1994; Kara et al., 2003; de Boyer Montégut et al., 2004; Abdulla et al., 2019). There are different methods to define the MLD (Kara et al., 2003). The threshold criterion, which is a widely favored and simple method for finding the MLD (Kara et al., 2003; de Boyer Montégut et al., 2004), is used in this study. In the threshold criterion, the MLD is defined as the depth at which the temperature or density profiles change by a predefined amount relative to a surface reference value. Various temperature threshold criteria were used to determine the MLD globally, such as 0.2∘ in de Boyer Montégut et al. (2004), 0.5∘ in Monterey and Levitus (1997), 0.8∘ in Kara et al. (2003) and 1.0∘ in Qiao et al. (2010). Therefore, considering the vertical temperature distribution pattern in the IO, we choose one of the typical threshold criteria (1.0∘) to define the MLD and attempt to make the effects of NBSWs and ITs on the simulated MLD more obvious. In this study, the MLD is defined as the depth at which the temperature is lower than the SST by 1.0∘ and is used to show the upper-ocean thermal structure.

Figure 12 shows the comparisons of the MLDs between the WOA13 data and the model results in January. The MLDs for Exp 1 are generally shallower than WOA13 in the whole IO and the Southern Ocean because of insufficient simulated mixing processes, which leads to underestimation of the vertical mixing in the upper ocean, especially during summer. The conclusion is similar to the global and regional simulations in previous studies (Kantha and Clayson, 1994; Qiao et al., 2010; Wang et al., 2019; Zhuang et al., 2020). The accumulation of weak vertical mixing during the 10-year climatologic modeling will make more heat staying in the surface layer, which will lead to warmer SST and shallower MLDs. In fact, from Fig. 12 one can see that the obviously shallower MLDs are generally in the ACC regions where the simulated vertical mixing from the original experiment is dramatically weak. In addition to the ACC regions, the obviously shallower MLDs also appear in the east regions of the Arabian Sea because of the weak vertical mixing. Furthermore, the simulated MLDs in most of the tropical and southern regions of the IO are partially shallower than the WOA13 data. Adopting the threshold criterion of 1.0∘, the simulated MLDs were shallower than the WOA13 data because of the warmer SST and cooler temperature in the ocean interior. Comparisons among the MLDs for Exp 1–Exp 3 show that the NBSW and IT may enhance upper-ocean mixing and make the simulated MLDs closer to the WOA13 data. The MLs for Exp 2 and Exp 3 are extremely deepened, especially in the tropical IO and the Southern Ocean. The RMSEs of the MLDs between the WOA13 data and the model results decrease 13.2 % and 14.6 % from 21.9 m (Exp 1) to 19.0 m (Exp 2) and to 18.7 m (Exp 3), respectively. However, the effects of the WTFR seem to be trivial in the whole area because there is almost no improvement from Exp 4–5 to Exp 1 and 3. The RMSEs for Exp 4–5 are larger than for Exp 1 and 3, and the larger deviations in Exp 4–5 mostly occur in the Southern Ocean, implying that the WTFR does not work well in the Southern Ocean in austral winter.

In this subsection, the simulated horizontal velocities in the surface layer are analyzed to evaluate the effects of NBSWs and ITs on ocean currents. Previous studies indicated that NBSWs and ITs have complicated impacts on simulated currents for OGCMs (e.g., Huang and Qiao, 2010; Wu et al., 2019). Only the results simulated in Exp 1–Exp 3 are discussed in detail, and the results in Exp 4 and Exp 5 are omitted here because the effects of the WTFR-induced mixing are relatively small. This situation is similar to the simulated temperature structure and MLDs in Sect. 4.2 and 4.3.

Figure 13 shows the comparisons of the surface velocities between the monthly mean OSCAR data (Bonjean and Lagerloef, 2002) and the model results of Exp 1–Exp 3 in January and July. The simulated surface velocities are chosen as those at the depth of 2 m interpolated by the model results. The OSCAR surface current products with a horizontal resolution of 1∘ / 3 × 1∘ / 3 and a time resolution of 5 d are constructed from the altimeter SSH, scatterometer winds, and both radiometer and in situ SST. The velocities are calculated based on a combined formulation including geostrophic balance, Ekman–Stommel dynamics and a complementary term for SST gradients (Bonjean and Lagerloef, 2002; Dohan, 2017). Johnson et al. (2007) demonstrated that OSCAR products are able to provide accurate estimates of the surface time mean circulation. A 29-year (1993–2021) time series of the OSCAR surface current data is collected and used to calculate the monthly mean climatologic currents, which are regarded as the reference in the comparison. In Fig. 13a and b, the spatial distribution of the surface current fields is presented as the anticyclonic subtropical gyre in the SIO and the monsoonally forced circulation of the tropics and the NIO (clockwise in boreal summer and anticlockwise in boreal winter). The eastern boundary current (Leeuwin Current) is not obvious because of the too small magnitude of the velocities in the climatologic data. Figure 13c–h present the difference by subtracting the monthly OSCAR data from the mean results of the three experiments. Generally, the simulated results contain most features of the surface currents in the IO. However, the simulated velocities are relatively too large to the southwest of the Indian Peninsula in January and too small in the Somali Current and the ACC regions in both January and July. The reason should be that the spatial and temporal resolution as well as the accuracy of the surface forcing data are insufficient. One can see that there is only a little difference in the simulated currents among the three experiments. The relatively smallest RMSEs for Exp 3 indicate that the NBSW and IT are able to improve the simulated surface currents.

Furthermore, we calculated the three-dimensional vertical vorticity and eddy kinetic energy (EKE) in Exp 1–Exp 5 to evaluate the effects of mixing induced by NBSWs and ITs on the mesoscale eddy activity. However, the difference of the vertical vorticity and the EKE among the five experiments was too complicated to summarize some dynamic processes and physical mechanisms. The reason should be that the climatological modeling in this study, on one hand, may be inappropriate to analyze mesoscale or small-scale processes because of the relatively coarse resolution, smoothed surface forcing, open boundary conditions and topography data; on the other hand, the induced vertical mixing may not be a key mechanism for eddy activity, as previous studies indicated that surface waves affect eddies through the interaction among the turbulence, circulation and Langmuir circulation when the turbulent Langmuir number is small (Jayne and Marotzke, 2002; Romero et al., 2021); subharmonic instability may transfer the energy from the internal tides to the shear-induced turbulent diapycnal mixing (MacKinnon and Gregg, 2005; Pinkel and Sun, 2013). Especially in the east region of the tropical Indian Ocean, the effects of the ITF on mesoscale or small-scale processes have not yet been simulated exactly in existing OGCMs (e.g., Nagai et al., 2017; Santoso et al., 022). Additional improvements of the mixing schemes and the ocean modeling will be studied further in the future.