The data used in this study is obtained from the SOSE reanalysis dataset (Mazloff et al., 2010). SOSE dataset assimilates a large volume of in-situ measurements including the Argo floats, ship-board CTD, and XCTD profiles, etc. It has a horizontal resolution of (1/6)°×(1/6)° and 42 vertical layers spanning from 2005 to 2010. The model is driven by 6-hourly atmospheric forcing from the National Centers for Environmental Prediction-National Center for Atmospheric Research (NCEP-NCAR) reanalysis, so the results contain the daily variability due to high-frequency forcing. The output provides both monthly and daily resolved data, and thus facilitates evaluating the high-frequency contribution in the entrainment processes.

The calculation of barrier layer thickness (BLT) is based on the method presented in De Boyer Montégut et al. (2007), where the mixed layer depth is calculated using the density threshold:

With the 10 m reference depth and $ \Delta T = 0.2$℃, the isothermal layer depth is defined as the depth at which the temperature decreases by 0.2℃ from the reference depth. The BLT is then calculated as the difference between the mixed layer and isothermal layer depths. This definition has been widely used in the world ocean including the Southern Ocean (e.g., Balaguru et al., 2012b; Bosc et al., 2009; Pan et al., 2018; Qiu et al., 2012).

The mixed layer heat budget equation is expressed as:

where $ {T}_{\mathrm{m}\mathrm{l}} $ is the mixed layer average temperature, $ {Q}_{\mathrm{n}\mathrm{e}\mathrm{t}} $ is the external heat, $ \rho $ is the seawater density, $ {C}_{\mathrm{p}} $ is the specific heat capacity, $ h $ is the mixed layer depth, $ \Delta T $ is the temperature difference between $ {T}_{\mathrm{m}\mathrm{l}} $ and the temperature at the mixed layer base, and the entrainment velocity $ {w}_{\mathrm{e}} $ is defined as:

where the subscript $ -{h} $ indicates the mixed layer base. The positive entrainment velocity ($ {w}_{\mathrm{e}} > 0 $) indicates an entrainment process while it is set to be zero for otherwise (in this study we consider the entrainment processes only). The entrainment velocity can be decomposed into three terms: the local change of mixed layer depth, the lateral induction, and the vertical advection (Eq. (3)). As discussed in Kim et al. (2006), contributions from the last two advective terms can be unambiguously calculated from the model outputs, but the entrainment heat flux due to local mixed layer depth change ($ \Delta T\partial h/\partial t $) is subjected to errors if using ad hoc schemes. To this end, Kim et al. (2006) proposed a new calculation for this term from a heat balance perspective (hereafter K05).

For an entrainment process, the mixed layer depth changes from h⁽ⁿ⁾ to h⁽ⁿ⁺¹⁾ and the average temperature from T⁽ⁿ⁾ to T⁽ⁿ⁺¹⁾ from the time t⁽ⁿ⁾ to t⁽ⁿ⁺¹⁾. Regardless of other processes, the heat balance during the consecutive time step can be written as:

After some algebra, Eq. (4) becomes

where $ {T}_{\mathrm{b}}^{\left(n\right)} $ is the mixed layer base temperature.

There are two differences between the new calculation and the traditional one. One is that the mixed layer depth in the entrainment heat flux term is using the next step value $ {h}^{\left(n+1\right)} $. The other is that the mixed layer base temperature is taken as the average over the mixed layer depth difference ($ {h}^{\left(n+1\right)}-h^{\left(n\right)} $) at the current time step, i.e.,

Figure 1 shows the seasonal BL patterns in the southeastern Indian Ocean deduced from the SOSE dataset. The data are illustrated only from May to October since the BL is very weak in the other half year. The BL phenomenon peaks in June and July with a maximum around 300 m. Thick BLs exhibit a roughly zonal distribution aligned between 40°–55°S, 100°–160°E. Generally speaking, the seasonal variations and patterns well agree the characteristics of the Argo-deduced BL (Pan et al., 2018). The BL formation in the southeastern Indian Ocean stems from the subduction of the warm SAMW, and therefore the BL in this region is characterized by an inverse temperature profile, which can lead to a unique warming effect on the mixed layer from the bottom. In the following, we will concentrate on a strong BL subregion for further analysis (Fig. 1a).

As mentioned in Section 1, various methods have been proposed for the EHF calculation. Figure 2 compares the seasonal variation of the EHF between the K05 and the traditional threshold methods in the BL months. All the methods produce the same temporal variation of the EHF, though the magnitudes are subject to large differences. The mixed layer warming induced by the BL starts in May, reaches the peak in July, and finally dies out in September. The results from the fixed depth difference methods are closer to the K05 with very limited deviations, while the EHFs from fixed temperature difference methods are very sensitive to the definition and thus have relatively large biases. Among all threshold values, the fixed depth difference of 5 m below the mixed layer depth is suggested for this area if we want to continue using the threshold methods. It is noteworthy that this definition may be not the best one for other regions. In fact, the performance of a fixed threshold method varies significantly over the entire Southern Ocean and thus no single threshold value is omniscient. The K05 method should be preferred for a large area or the basin-wide oceans.

To examine the spatial difference between methods, we choose a moderately-performed definition of 20 m depth difference. As shown in Fig. 2, this threshold tends to overestimate the entrainment warming approximately by 0.05℃/month. The EHF distribution is roughly consistent with the BL pattern each month. By comparison, the overestimates are nearly homogeneous over the entire barrier layer region (Fig. 3). The difference is only slightly larger during the strong warming months. It should be noted that there are also negative EHFs at the flanks of the BL region, which results from the normal BL structure, i.e., the isothermal layer is deeper than the density-based mixed layer and no inverse temperature occurs. Overall, the warming effect still plays a dominant role in the BL regions.

The entrainment is a process induced by small-scale turbulence with the occurrence over a much longer time scale. It is thus expected that the cumulative effects of the high-frequency entrainment variability may have a significant difference from the monthly average. Figure 4 demonstrates the EHF calculated using the monthly and daily SOSE data. The EHF difference between them emerges from June to August. The daily EHF warming has a much larger fluctuation and the magnitude can reach as high as 0.25℃/month, nearly double the monthly warming. Even at the lower end of the fluctuation, the daily EHF is roughly the same as the monthly EHF during the strong BL months. In winter, the surface net heat loss is the only dominant term in the mixed layer heat budget of the Southern Ocean, while the other effects are secondary (Dong et al., 2007). However, when considering the high-frequency variability of the mixed layer, we found that the EHF warming in the BL regions is comparable to the surface air-sea heat fluxes, particularly in August when the warming can even surpass the surface cooling (Fig. 4b). This means that the heating from the underlying BL can compensate the mixed layer heat loss from the surface almost over the whole winter. This unique warming effect could not be discerned if neither the BL nor the high-frequency variability is accounted for. In spring, i.e., a transitional season, both the EHF and surface net heat fluxes are reduced to very small values. Their magnitudes are still comparable with ups and downs on both effects.

Equation (5) indicates that the EHF is composed of the relative variation of the mixed layer depth $ \left(1/h\right)\partial h/\partial t $ and the temperature difference between the mixed layer average and the mixed layer base $ \Delta T $. Figure 5 examines the seasonal contributions from these two terms using daily and monthly data respectively. The variation of the temperature difference is mainly due to the decrease of the average mixed layer temperature. Thus, it is of no surprise that the monthly temperature difference is greater than the daily difference, simply because the monthly data have a longer time interval (Fig. 5a). If we assume that the mixed layer temperature decreases linearly within a month, then the daily temperature change would be much smaller than the temperature difference from the daily data. Moreover, the daily-resolved mixed layer variability is also much larger than the monthly one (Fig. 5b), which again highlights the importance of the high-frequency component of the mixed layer variability. The BL region in the southeastern Indian Ocean is located on the course of the strong Antarctic Circumpolar Current (ACC). The high-frequency variation may stem from the rich meso- and submesoscale processes associated with the ACC within the mixed layer. Previous studies revealed the significant vertical heat advection induced by submesoscale in the southern Indian Ocean (Su et al., 2018). Here we emphasize that the small-scale turbulent entrainment processes can also make great contributions to the mixed layer heat content.

On the interannual scale, the EHF in the BL period has more complicated variability. The daily EHF has a relatively larger fluctuation than the monthly EHF, and there is no significant correlation between them (Fig. 6). The EHF anomalies are not negligible compared with the seasonal warming in Fig. 4, indicating that the EHF is subject to large interannual variability. Sallée et al. (2010) pointed out that the Southern Ocean mixed layer depth has a zonally asymmetric response to the Southern Annular Mode (SAM) and the SAM index is positively related to the mixed layer depth anomalies in the southeastern Indian Ocean. The forcing mechanism is attributed to the anomalous northward cold wind associated with the SAM that leads to a buoyancy loss over the southeastern Indian Ocean. This implies a possible linkage between the EHF and SAM via mixed layer deepening. During positive SAM phase, the anomalous negative heat fluxes decrease the mixed layer temperature. It is possible to amplify the temperature difference $ \Delta T $ with the underlying warm water, or to increase the mixed layer depth by convective mixing. Either effect or both together may enhance the EHF. Owing to the limited length of the SOSE data, it is difficult to evaluate the correlation between the EHF and SAM variability. However, we indeed found that the annual EHF deduced from the daily but not monthly data has a very similar trend and variation to the SAM index during this six-year period (Fig. 7). As the EHF is related to the small-scale mixing, the rapid change of mixed layer depth may not be well captured by the monthly data. This is also the reason why the contribution of the entrainment mixing is often taken as secondary, as most available datasets for the Southern Ocean are monthly resolved including the observation-based gridded data and global reanalysis (e.g., Dong et al., 2007; Screen et al., 2010). It should be noted again that the link should be further examined with a longer daily-resolved dataset. This cross-scale interaction between the climate signal and turbulent mixing is an interesting question that worths investigating in the future study.