Tichng Zhng Wiqing Wng Qing Xi Kxiu Liu Dongxio Wng g Xinrong Wu Xioshung Zhng Kng Xu Wny Yun

a Key Laboratory of State Oceanic Administration for Marine Environmental Information Technology, National Marine Data and Information Service, Ministry of Natural Resources, Tianjin, China

b State Key Laboratory of Tropical Oceanography, South China Sea Institute of Oceanology, Chinese Academy of Sciences, Guangzhou, China

c Southern Marine Science and Engineering Guangdong Laboratory (Guangzhou), Guangzhou, China

d Innovation Academy of South China Sea Ecology and Environmental Engineering, Chinese Academy of Sciences, Guangzhou, China

e Institute of Deep-sea Science and Engineering, Chinese Academy of Sciences, Sanya, China

f Center for Ocean Mega-Science, Chinese Academy of Sciences, Qingdao, China

g School of Marine Sciences, Sun Yat ‐Sen University, Zhuhai, China

Keywords:Indian Ocean Shallow overturning circulation Meridional overturning streamfunction Vertical overturning streamfunction

A B S T R A C T The calculation of the meridional overturning streamfunction in the southern Indian Ocean is biased by the Indonesian Throughflow. Therefore, this study applies the vertical overturning streamfunction to diagnose the shallow overturning circulation in the Indian Ocean. Using the Ocean General Circulation Model for the Earth simulator output, improvements with the vertical overturning streamfunction compared with the meridional overturning streamfunction are explored. The results show that the vertical overturning streamfunction smoothly connects the shallow overturning circulations of the northern Indian Ocean and the southern Indian Ocean with the whole cycle of the subtropical cell and the cross-equatorial cell. The vertical overturning streamfunction shows a much cleaner shallow overturning circulation, which is underestimated by the meridional overturning streamfunction. It shows that the shallow overturning circulation has a magnitude of ～13 Sv (1 Sv ≡10 6 m 3 s ? 1 ),of which the subtropical cell accounts for ～8 Sv. In addition, the vertical overturning streamfunction captures a clockwise overturning cell in the upper 600 m layer between 30°S and 34°S. This cell has a magnitude of about ? 5 Sv and probably corresponds to the wind-forced subtropical gyre. Therefore, the vertical overturning streamfunction provides a new approach for estimating the shallow overturning circulation in the Indian Ocean.

1. Introduction

The Indian Ocean is one of the most critical junctions in the global oceanic circulation. The Indian Ocean overturning, in particular, receives tropical waters from the Pacific Ocean ( Miyama et al., 2003 ;Talley, 2013 ; Zhang et al., 2019 ) and the Indian Ocean accounts for more than 70% of the global ocean heat gain in the upper 700 m during the warming hiatus ( Lee et al., 2015 ). The diagnosis of the Indian Ocean overturning is fundamental to our understanding of the global climate and its variability ( Wang et al., 2014 ; Zheng et al., 2018 ; Wang, 2019 ).

Streamfunctions have been widely used to diagnose overturning circulation through simplifying complex three-dimensional and timevarying overturning circulation into two-dimensional time-averaged circulation ( Groeskamp et al., 2014 ). The streamfunction has been defined using different combinations of coordinates, including geographic and thermohaline coordinates ( Zika et al., 2012 ). A classic example is the meridional overturning streamfunction, which represents the zonally averaged ocean circulation in meridional and vertical coordinates. It should be noted that only a velocity field that is non-divergent in two coordinates can be represented by a streamfunction ( D??s et al., 2012 )and hence the meridional overturning streamfunction is valid in the global ocean and the Atlantic Ocean. However, in the Indian Ocean, the meridional overturning streamfunction should be used carefully as the non-divergent condition is challenged by the Indonesian Throughflow(ITF).

To implement the non-divergent constraint, the meridional overturning streamfunction in the Indian Ocean is hard to define, especially in the southern Indian Ocean and for the shallow overturning. In the northern Indian Ocean (north of 8°S), the velocity field is non-divergent and the meridional overturning streamfunction is valid ( Miyama et al.,2003 ; Hu and Godfrey, 2007 ). However, in the southern Indian Ocean(between 8°S and 34°S), the ITF induces a net poleward transport and the magnitude of the net transport is comparable to the magnitude of the shallow overturning cells ( Schott et al., 2002 ; Zhang et al., 2019 ).By assuming that the ITF waters cross the Indian Ocean in the upper layers, the meridional overturning streamfunction is seemingly valid for diagnosis of the deep overturning circulation in the Indian Ocean( Drijfhout and Garabato, 2008 ; Wang et al., 2012 ). Using the tracing method, Zhang et al. (2019) tried to separately examine the Lagrangian partial meridional overturning streamfunction of the ITF in the Indian Ocean. However, a significant gap exists at 8°S in the Lagrangian partial meridional overturning streamfunction and how to connect the meridional overturning streamfunction in the southern Indian Ocean and in the northern Indian Ocean remains unaddressed. Using the Helmholtz decomposition, Han and Huang (2020) separated the total meridional overturning streamfunction in the Indian Ocean into divergent and nondivergent parts. Their results showed that the pattern of the deep cells in the Indian Ocean might still be influenced by the ITF, which conflicts with earlier results that the ITF’s influence on the meridional overturning streamfunction is limited to within the upper layers ( Wang et al.,2012 ; Zhang et al., 2019 ). It is worth noting, however, that in the study of Han and Huang (2020) , the open boundary at the ITF was modified from the zonal 8°S section to the meridional 115°E section. This modification causes that ITF’s influence on the meridional overturning streamfunction might be contaminated by other currents across the 115°E section (for example, the East Gyral Current) ( Liu et al., 2005 ).

Nurser and Lee (2004) introduced the vertical transport streamfunction, which has cast new light on the diagnosis of overturning circulation. Following their work, we define a zonally averaged vertical overturning streamfunction, which is similar to the meridional overturning streamfunction, to provide a new representation of the shallow overturning circulation in the Indian Ocean. Our study also illustrates the shallow overturning circulation of the Indian Ocean from the perspective of vertical overturning.

The rest of the paper is organized as follows. In Section 2 , the streamfunction and model data are briefly introduced. In Section 3 , the shallow overturning circulation is diagnosed from the meridional and vertical overturning perspective. In Section 4 , a comparison with the existing method is discussed. Finally, the results are summarized in Section 5.

2. The streamfunction and model data

In this section, the meridional overturning streamfunction is firstly discussed. Then, the definition of the vertical overturning streamfunction is described. And lastly, the model data used to construct the overturning streamfunction are briefly introduced.

2.1. Meridional overturning streamfunction

The meridional overturning streamfunction is defined as the meridional volume transport across the zonal—vertical plane at latitude

y

below depth

z

. In the Eulerian space, the meridional overturning streamfunction can be calculated as follows:

where

x

and

x

are the eastern and western boundaries, respectively;

v

is the meridional velocity. In this study, the open boundary of the Indian Ocean at the ITF is set to the zonal 8°S section rather than the meridional 115°E section used in Han and Huang (2020) , to minimize contamination from the eastern open boundary. The southern open boundary of the Indian Ocean is set to the 34°S section as in Wang et al. (2012) , to respect the non-divergent condition.In the non-divergent case,

ψ

(

y,z

) is equal to the upwelling through the horizontal surface at depth

z

north of latitude

y

because of mass conservation ( Fig. 1 ) ( Zika et al., 2012 ; D??s et al., 2013 ). Hence, the meridional overturning streamfunction could also be used to implicitly diagnose the mass volume transport in the vertical direction. Sometimes,to diagnose the exchanges of waters of different densities or temperatures, the density coordinate or the temperature coordinate are also used to construct the meridional overturning streamfunction instead of the depth coordinate ( Wang et al., 2012 ; Zhang et al., 2019 ).

Fig. 1. Schematic showing the equivalent relationship of the meridional overturning streamfunction ψ and vertical overturning streamfunction φin the nondivergent case. For non-divergent ocean that is enclosed by land masses on the west, east, and north sides, the vertical transport φin the upper layer needs to be balanced by equivalent meridional transport ψ in the lower layer.

In the divergent case, the meridional overturning streamfunction is non-existent. In this case, Eq. (1) gives a meridional transport function,representing the accumulated meridional volume transport below the depth

z

at latitude

y

. The corresponding vertical volume transport cannot be derived directly by comparison of the meridional transport function at different latitudes due to zonal sources or sinks.

2.2. Vertical overturning streamfunction

The vertical overturning streamfunction was first suggested as the upward transport of fluid colder than a given temperature across a constant height surface ( Nurser and Lee, 2004 ). After being transformed into Eulerian space, it represents the vertical volume transport of fluid north of a given latitude across a constant height surface:

where

w

is the vertical velocity.The streamfunction

φ

(

y,z

) can be used as a complement to

ψ

(

y,z

)because it indicates the vertical transport explicitly and directly. In the non-divergent case,

φ

(

y,z

) is equal to

ψ

(

y,z

) because of mass conservation ( Fig. 1 ), and hence indicates the meridional flows. In the divergent case, the vertical overturning streamfunction is non-existent and Eq. (2) gives a vertical transport function.

2.3. Model data

The Ocean General Circulation Model for the Earth simulator (OFES)output ( Masumoto et al., 2004 ) is used in this study. The model is quasiglobal ( ± 75°latitude), with a horizontal resolution of 0.1°and 54 levels in the vertical direction ( Masumoto et al., 2004 ). The annual mean climatological meridional velocity field and vertical velocity field are used in this analysis to construct the overturning streamfunctions.

There are two major reasons that the OFES data are chosen in present study. One is that they keep a constant horizontal resolution and fixed vertical

z

coordinates as the same as the original model grid. The overturning streamfunctions can be carried out on this invariable grid without remapping of the original velocity fields. The other is that both the meridional velocity and the vertical velocity can be directly obtained from these OFES data and then the corresponding overturning streamfunctions can be directly given.

3. The shallow overturning circulation

The shallow overturning circulation of the Indian Ocean mainly consists of the subtropical cell in the southern Indian Ocean, the crossequatorial cell connecting the two hemispheres, and the equatorial roll( Miyama et al., 2003 ; Lee, 2004 ; Schott et al., 2009 ). In the climatological mean state, the subtropical cell and the cross-equatorial cell are anticlockwise, characterized by northward transport mainly within the western boundary current and southward returning mainly within Ekman transport ( Schott et al., 2002 ; Nagura and McPhaden, 2018 ).The upwelling of the subtropical cell mainly occurs near the Seychelles Dome. The upwelling of the cross-equatorial cell is mainly along the coasts of Somalia and Oman. The subduction of both cells mainly occurs in the southeast Indian Ocean ( Schott et al., 2002 ). The equatorial roll is clockwise, almost confined to the surface-mixed layer, and driven by southerly cross-equatorial winds ( Miyama et al., 2003 ).

In this section, the shallow overturning circulation is briefly shown from the meridional overturning perspective as a reference, and that from the vertical overturning perspective is then determined in detail.

3.1. Meridional overturning perspective

Many studies have diagnosed the shallow overturning circulation of the Indian Ocean from the meridional overturning perspective. Lee and Marotzke (1997) demonstrated the existence of an anticlockwise shallow overturning cell (14 Sv, 1 Sv ≡10ms1 ) with a general circulation model. By calculating the meridional overturning streamfunction in the northern Indian Ocean (north of 8°S), Miyama et al. (2003) estimated the magnitude of the cross-equatorial cell and the equatorial roll to be about 6 Sv and ? 6 Sv (the negative value represents clockwise overturning), respectively.

Fig. 2 shows the meridional overturning streamfunction calculated by Eq. (1) with the OFES annual climatology. We can see that the anticlockwise subtropical cell, the anticlockwise cross-equatorial cell, and the clockwise equatorial roll are clearly identified. The transport of the anticlockwise cross-equatorial cell and the clockwise equatorial roll is about 5 Sv and ? 4 Sv respectively, which is comparable to that estimated by Schott et al. (2002) and Miyama et al. (2003) . This justifies that the OFES data are suitable for diagnosis of the overturning circulation in the present study.

A remarkable characteristic of the meridional overturning streamfunction is that the streamline is discontinuous at 8°S ( Fig. 2 ). In the domain north of 8°S, the anticlockwise overturning circulation has a magnitude of about 10 Sv. In the domain south of 8°S, the net poleward transport induced by the ITF reduces the northward transport in the subsurface, resulting in the anticlockwise overturning cell having a magnitude of only ～8 Sv. In other words, the estimation based on the meridional overturning streamfunction possibly underestimates the transport of the overturning circulation in the southern Indian Ocean.

3.2. Vertical overturning perspective

Fig. 3 shows the vertical overturning streamfunction diagnosed from the OFES annual climatology following Eq. (2) . Compared with Fig. 2 ,the streamline is continuous near 8°S and shows closed cycles of the subtropical cell and the cross-equatorial cell. As a whole, the overturning circulation has a magnitude of 13 Sv, of which the subtropical cell and the cross-equatorial cell account for 8 Sv and 5 Sv, respectively.

In the domain north of 8°S, the vertical overturning streamfunction is almost equal to the meridional overturning streamfunction, which is consistent with the analysis in Fig. 1 . A point P1 (3°S, 75 m) is chosen to further examine the equivalence.

ψ

(P1) represents the 9.81 Sv meridional transport across the zonal 3°S section deeper than 75 m, and

φ

(P1) represents the 9.47 Sv vertical transport across the horizontal 75 m plane north of 3°S. The small discrepancy (～0.34 Sv) might be due to numerical errors as the vertical velocity is usually much smaller than the meridional velocity.In the domain south of 8°S, the vertical overturning streamfunction presents a similar pattern to the meridional overturning streamfunction but with a different magnitude. A point P2 (13°S, 75 m) is chosen to further explain the difference.

ψ

(P2) represents the 8.31 Sv meridional transport across the zonal 13°S section deeper than 75 m, and

φ

(P2)represents the 13.10 Sv vertical transport across horizontal 75 m plane north of 13°S. The large discrepancy (～4.79 Sv) is closely related to the ITF. About 2 Sv of the discrepancy can be explained by the upwelling of the ITF waters between 8°S and 13°S ( Zhang et al., 2019 ). The remaining discrepancy (～2.79 Sv) might due to the contamination induced by the net poleward transport of the ITF.

In the open mouth of the Indian Ocean between 30°S and 34°S, there is a clockwise overturning cell (～5 Sv) located in the upper 600 m layer( Fig. 3 ). The figure shows that water enters the Indian Ocean from the south in the upper 200 m layer, subducts to 200—600 m and then exits the Indian Ocean. From a barotropic perspective, this cell possibly involves Southern Ocean waters crossing 34°S into the Indian Ocean, turning anticlockwise within the wind-forced subtropical gyre, and shrinking toward the western boundary ( Talley, 2011 ; Zika et al., 2012 ). The subduction of this cell can also be inferred from the meridional overturning streamfunction, although it implies that the subduction sources are from the ITF rather than from the Southern Ocean. The ITF waters weakly subduct between 30°S and 34°S according to the tracing pathways of the ITF waters in the Indian Ocean ( Durgadoo et al., 2017 ; Zhang et al.,2019 ). Although the ITF waters cross 34°S in the Indian Ocean, the transport is mainly confined to the upper 200 m layer ( Zhang et al., 2019 ). In contrast, the Southern Ocean waters of more than 60 Sv cross 34°S into the Indian Ocean within Ekman transport in the interior, turn anticlockwise, and strongly subduct within the subtropical gyre ( Karstensen and Quadfasel, 2002 ; Wang et al., 2012 ; Talley, 2011 ; Ma and Lan, 2017 ).The dominant subduction of this clockwise overturning cell more likely comes from the Southern Ocean rather than from the ITF. In addition,one likely reason that this cell is not captured by the meridional overturning streamfunction is that the southward transport induced by the ITF waters overwhelms the northward transport of this cell. Hence, we suggest that this clockwise overturning cell mainly originates from the Southern Ocean rather than from the ITF, as illustrated in Fig. 3 .

Fig. 2. The meridional overturning streamfunction diagnosed from the OFES annual climatology. Positive (negative) streamlines correspond to anticlockwise (clockwise) circulation.

Fig. 3. The vertical overturning streamfunction diagnosed from the OFES annual climatology. Positive (negative) streamlines correspond to anticlockwise (clockwise)circulation. The chosen points P1 (3°S, 75 m) and P2 (13°S, 75 m) are marked by the yellow stars.

Fig. 4. Conceptual illustration of the shallow overturning circulation in the Indian Ocean. Note that the ITF marked here represents the net poleward transport and the so-called subtropical gyre cell represents the clockwise overturning cell between 30°S and 34°S.

4. Discussion

In this section, we discuss the relation between the vertical overturning streamfunction and the streamfunction proposed by Han and Huang (2020) . For an ocean model based on volumetric conservation,the continuity equation is

where (

u,

v,

w

) denote the velocity in the three coordinates: longitude

x

, latitude

y

, and depth

z

, respectively. Zonally integrating Eq. (3) for the Indian Ocean leads to

where

u

and

u

denote the zonal velocities at the western and eastern boundaries of the model domain, respectively. We denote the zonally integrated flow components as

and Eq. (4) can be rewritten as

The western boundary is simply set at land, to give

u

≡0 . The eastern boundary is set at the meridional 143°E section where

u

≡0 is approximately satisfied. The open boundary of the ITF is set at the zonal 8°S section between 115°E and 143°E. The other sources, such as the transport of the Malacca Strait, are negligible compared with the ITF( Durgadoo et al., 2017 ; Han and Huang, 2020 ). In this sense, Eq. (6) can be rewritten as

Eq. (7) shows that the non-divergent condition in terms of(

V

(

y,

z

)

,

W

(

y,

z

)) is satisfied except at the 8°S section. Then, we decompose it into a source part and a non-divergent part as

where the first term (

V

(

y,

z

)

,

W

(

y,

z

)) on the right-hand side is a function of the

V

(

z

) , and the second term (

V

(

y,

z

)

,

W

(

y,

z

)) accounts for the remaining part and maintains the non-divergent condition. A special solution of Eq. (9) is listed as follows:

The non-divergent condition of this special solution can be easily demonstrated and then the corresponding streamfunction can be induced. From this point, the vertical overturning streamfunction in this study is similar to the non-divergent meridional overturning streamfunction defined in Han and Huang (2020) . However, the remaining part is different. In our decomposition, the remaining part is assumed to be a horizontal flow. However, Han and Huang (2020) assume that the remaining part is an irrotational flow.

To further compare the performance of these two methods, we take Fig. 11b in Han and Huang (2020) as an example. Han and Huang (2020) show a shallow overturning circulation with a magnitude of 16 Sv. We calculate the meridional overturning circulation using the same data with our method, and find that the result also shows a shallow meridional overturning cell of 16 Sv (not shown). In the deep ocean,however, there is a discrepancy. Han and Huang (2020) show a deep anticlockwise cell with a magnitude of 14 Sv, whereas our result shows a magnitude of only 4 Sv. The discrepancy might be caused by the different assumptions for the remaining part. We assume that the remaining part is a horizontal flow, whereas Han and Huang (2020) assume that it is an irrotational flow.

The vertical overturning streamfunction has two main limitations.One limitation is that it implicitly assumes that the ITF waters are horizontally transported in the Indian Ocean, which is not the general situation (e.g., Jensen, 2003 ; Song et al., 2004 ; Valsala and Ikeda, 2007 ;Zhang et al., 2019 ). The other is that it is non-unique and the present solution is a special one. Deeper research will be conducted in a future study.

5. Summary

Given the existence of the ITF waters from the Pacific Ocean into the Indian Ocean, this study applies the vertical overturning streamfunction to diagnose the shallow overturning circulation in the Indian Ocean. Using OFES data, the vertical overturning streamfunction is constructed to give a new representation of the shallow overturning circulation in the Indian Ocean. Fig. 4 conceptually summarizes the major overturning cells, including the subtropical cell, the cross-equatorial cell, the equatorial roll, and the clockwise overturning cell between 30°S and 34°S.The first three components are well known and the last one is novel.

Compared with the meridional overturning streamfunction, the vertical overturning streamfunction shows several improvements. First, the vertical overturning streamfunction smoothly connects the overturning circulation between the northern Indian Ocean and the southern Indian Ocean. The streamline of the vertical overturning streamfunction is continuous near 8°S and shows the whole cycles of the subtropical cell and the cross-equatorial cell. In contrast, the streamline of the meridional overturning streamfunction is discontinuous at 8°S due to the ITF waters, resulting in the subduction branch and the upwelling branch of these two cells being cut off.

Second, the vertical overturning streamfunction describes a much cleaner shallow overturning circulation in the Indian Ocean. It shows that the subduction of the shallow overturning circulation mainly occurs between 12°S and 30°S, with a magnitude of ～13 Sv. The upwelling of the subtropical cell mainly occurs between 6°S and 12°S, with a magnitude of ～8 Sv. Both these processes are underestimated by the meridional overturning streamfunction.

Third, the vertical overturning streamfunction captures a clockwise overturning cell in the open mouth of the Indian Ocean between 30°S and 34°S. This cell probably corresponds to the wind-forced subtropical gyre, which involves the Southern Ocean waters crossing 34°S into the Indian Ocean, turning anticlockwise and shrinking toward the western boundary ( Talley, 2011 ). This cell has a magnitude of about ? 5 Sv,which probably represents the inter-basin mass exchange between the Indian Ocean and the Southern Ocean.

The presented vertical overturning streamfunction provides a new approach for estimating the shallow overturning circulation in the Indian Ocean. It partly excludes the interruption from the ITF and exhibits a much cleaner shallow overturning circulation. Future work should focus on further validation and generalization of this method.

Funding

This study was jointly supported by the National Key Research and Development Program of China [grant number 2016YFC1401803],the National Natural Science Foundation of China [grant numbers 41976019 and 42076020], the Strategic Priority Research Program of the Chinese Academy of Sciences [grant number XDA20060502], the open project of the State Key Laboratory of Tropical Oceanography,South China Sea Institute of Oceanology, Chinese Academy of Sciences[grant number LTO1910], the Research Program of the Southern Marine Science and Engineering Guangdong Laboratory (Guangzhou) [grant number GML2019ZD0306], and the Key Research Program of the Chinese Academy of Sciences [grant number ZDRW-XH-2019-2].

Acknowledgments

We are grateful to the OFES group for providing the simulation data.