Impact of the Indonesian Throughflow on Agulhas leakage

Using ocean models of different complexity we show that opening the Indonesian Passage between the Pacific and the Indian oceans increases the input of Indian Ocean water into the South Atlantic via the Agulhas leakage. In a strongly eddying global ocean model this response results from an increased Agulhas Current transport and a constant proportion of Agulhas retroflection south of Africa. The leakage increases through an increased frequency of ring shedding events. In an idealized two-layer and flat-bottom eddy resolving model, the proportion of the Agulhas Current transport that retroflects is (for a wide range of wind stress forcing) not affected by an opening of the Indonesian Passage. Using a comparison with a linear model and previous work on the retroflection problem, the result is explained as a balance between two mechanisms: decrease retroflection due to large-scale momentum balance and increase due to local barotropic/baroclinic instabilities.


Introduction
The Indonesian Passage forms an equatorial gateway between two major ocean basins which is a unique feature in the world ocean.Its existence allows for an input of Pacific water, the Indonesian Throughflow (ITF), that crosses the Indian Ocean and feeds its western boundary current, the Agulhas Current (AC), independently of Indian Ocean wind stress forcing.The ITF takes part in the so-called "warm water route" (Gordon, 1986) that brings thermocline water from the Pacific and Indian Oceans to the Atlantic Ocean through Agulhas leakage.This water also takes part on the Southern Hemisphere "supergyre" (Speich et al., 2007), feeds the Atlantic Meridional Overturning Circulation and may have an impact on its variability and strength (Biastoch et al., 2008a;Beal et al., 2011).One of the main issues of current research, and the topic of this paper, is the impact of the strength of the ITF on the Indian Ocean circulation and the Agulhas leakage.
The most direct model experiment to address this issue is to compare the global ocean circulation in the present-day world with that having a continental configuration in which the Indonesian Passage is (artificially) closed.Such model experiments have been carried out with both ocean-only models (Hirst and Godfrey, 1993;Lee et al., 2002) and global climate models (Schneider, 1998;Wajsowicz and Schneider, 2001;Song et al., 2007) using decadal-long simulations.In both classes of models, the presence of the ITF strengthens the South Equatorial Current and the AC, and the volume transport of the East Australian Current decreases.For example, this type of response of the Indo-Pacific ocean circulation was found both in the GFDL CM2.1 climate model (Song et al., 2007) and in the MITgcm ocean-only component (with a similar horizontal resolution of about 1 • ) as used in Lee et al. (2002).
In these model studies, not much attention was given to the impact of the ITF on the Agulhas leakage and the South Atlantic Ocean circulation.A crucial component of the Indo-Atlantic connection is the retroflection of the AC.In ocean models with a horizontal resolution of only 1 • , this retroflection is controlled by the wrong dynamical (i.e.viscous) processes.A much higher horizontal resolution is required to reach an inertial (or turbulent) retroflection regime (Dijkstra and de Ruijter, 2001a; Le Bars et al., 2012) which is thought to be more realistic regime.In addition to a more adequate representation of the boundary layer flow, the representation of mesoscale flows is also crucial to model a correct Indo-Atlantic exchange (Biastoch et al., 2008b).
Published by Copernicus Publications on behalf of the European Geosciences Union.

D. Le Bars et al.: Indonesian Throughflow and Agulhas leakage
From a theoretical point of view, two categories of processes are believed to play a role in determining the volume of Agulhas leakage.The large-scale forcing or "far field" (De Ruijter et al., 1999) sets the strength of the AC and the position of the Agulhas Return Current.A leakage of Agulhas water to the South Atlantic is possible because of the gap between the southern tip of the African continent (37 • S) and the southern limit of the subtropical gyre, materialized by the latitude of maximum westerlies, where the wind stress curl vanishes (around 48 • S).Secondly, local dynamics also plays an important role in the Agulhas retroflection.The inertia of the AC was shown to increase the proportion of retroflection (Ou and De Ruijter, 1986;Dijkstra and de Ruijter, 2001a;Le Bars et al., 2012).Although its role is not yet fully understood, the rectification of instabilities is also crucial for the mean leakage.Barotropic instabilities were shown to decrease the degree of retroflection (Dijkstra and De Ruijter, 2001b) but mixed barotropic/baroclinic instabilities may lead to a significant increase of the retroflection (Le Bars et al., 2012).
The Indonesian Throughflow is also part of the large-scale forcing of the Agulhas System.The inflow of water increases the strength of the AC but simultaneously water has to return south of Tasmania.In this case Hirst and Godfrey (1993) showed that potential vorticity constraints direct the flow zonally in the South Atlantic from Africa to South America.Through the input of negative vorticity at the coast there can be a flow southward which next is able to return eastward at the latitude of south Tasmania.In this case we would expect a decreased degree of retroflection with the increase of the ITF.However the situation is more complex because the Indonesian Throughflow -by strengthening the AC -also strengthens nonlinear processes south of Africa that are expected to increase the degree of retroflection.How these processes combine and impact the volume of Agulhas leakage will be analyzed in detail here.
The possible effect of the ITF on the Agulhas leakage will be analyzed using two 75 yr-long simulations with the strongly eddying version (with a 0.1 • horizontal resolution) of the Parallel Ocean Program (POP).In both simulations (closed vs open Indonesian Passage), the wind stress is the same and there is no restoring of the surface salinities.These model simulations therefore provide new insights on the dynamics of the Agulhas retroflection and the effect of the presence of the ITF on the Agulhas leakage in an inertial to turbulent retroflection regime.Results from simulations with a two-layer shallow-water model in an idealized basin geometry, allowing for more detailed sensitivity studies, are subsequently used to interpret the POP results.To understand better the physics of the previous two models, a linear model is also used to investigate the importance of nonlinear processes in the connection between the ITF and the Agulhas leakage.Its results are derived and discussed in Appendix A.
In Sect. 2 we describe the POP model simulations and provide an overview of the idealized shallow-water model used.In Sect. 3 we assess the strengths and weaknesses of POP for the simulation of the Agulhas retroflection and Agulhas leakage by comparing the model simulated (with realistic bathymetry) sea surface height field to satellite altimetry data.Main results on the impact of the ITF on the western Pacific, on the Indian Ocean and on the Agulhas retroflection are presented in Sect. 4 and further analyzed using the idealized shallow-water model in Sect. 5. A summary and discussion of the results follows in Sect.6.

Model simulations
Two numerical models are used in this paper.A global ocean model (POP) and a regional, idealized shallow-water model (HIM).

POP configuration
The global ocean simulations analyzed here were performed using the Parallel Ocean Program (POP, Dukowicz and Smith, 1994) developed at Los Alamos National Laboratory.The configuration is based on that used by Maltrud et al. (2010), with an average 0.1 • horizontal resolution and 42 vertical levels, allowing for a maximum depth of 6000 m.The atmospheric state used to force the model is based on the repeat annual cycle (normal-year) Coordinated Ocean Reference Experiment (CORE1 ) forcing dataset (Large and Yeager, 2004), with 6-hourly forcing averaged to monthly.Wind stress is computed offline using the Hurrell sea surface temperature (SST) climatology (Hurrell et al., 2008) and standard bulk formulae; evaporation and sensible heat flux were calculated online also using bulk formulae and the model predicted SST.Precipitation was also taken from the CORE forcing dataset.Sea-ice cover was prescribed based on the −1.8 • C isoline of the SST climatology, with both temperature and salinity restored on a timescale of 30 days under diagnosed climatological sea-ice.
As initial conditions we used the final state of a 75 yr spin-up simulation described in Maltrud et al. (2010) using restoring conditions for salinity.The freshwater flux was diagnosed during the last five years of the spin-up simulation and the simulations we present in this paper use this diagnosed freshwater flux.This makes the ocean circulation free to adapt to changes in continental geometry, i.e. here the closing of the Indonesian Passage.Using this model we did a control simulation (POP CTRL) with realistic bathymetry and another one (POP NoITF) on which the Indonesian Passage was closed by small land bridges.Both POP simulations were continued under the same surface forcing for 75 yr, up to model year 150.

HIM configuration
To test dynamical mechanisms and help the interpretation of the POP simulations, we use an idealized configuration of the Hallberg Isopycnal Model (HIM, Hallberg (1997)).It uses a time-splitting scheme to solve the hydrostatic primitive equations in spherical coordinates on an Arakawa Cgrid.The model domain (Fig. 1) and the boundary conditions are similar to that used in Le Bars et al. (2012).The domain is centered on the Indian Ocean with the African continent and Madagascar in the western part and the Australian continent in the eastern part.There are 3 basins in this setup and we will call them Atlantic Ocean, Indian Ocean and Pacific Ocean even though the sizes of the Atlantic and Pacific basins are much smaller than in reality.This setup allows for water to go from the Pacific to the Indian Ocean.
We use zonal periodic boundary conditions between 60 • S and 55 • S to allow an Antarctic Circumpolar Current to develop.To reduce its intensity we have added a ridge at 130 • E; the longitudinal extent of this ridge is 1 • and its height is fixed to 2500 m.The rest of the domain has a flat bottom.We use a horizontal resolution of 0.1 • and 2 vertical layers.Lateral friction is parameterized with a Laplacian horizontal eddy viscosity with a constant background value of 500 m 2 s −1 and a Smagorinsky coefficient of 0.15.We apply no-slip lateral boundary conditions and the model is forced by a steady zonal wind stress (Fig. 1) of the form: where θ is the latitude and with θ 0 = 32 • S and α = π /32.The wind-stress pattern is fixed but its amplitude is taken as control parameter and ranges from 0.05 N m −2 to 0.3 N m −2 .
For each amplitude of the wind stress we run the model to reach a statistical equilibrium and for analysis we take an average of five years of each experiment.A set of 6 experiments under different wind stresses is performed for a control experiment (HIM CTRL) and we compare the results with a set of simulations in which the Indonesian Passage is closed (HIM NoITF); see Table 1 for a summary of the parameters for the two cases.

POP model -data comparison
To assess the skills of the POP model in simulating the Agulhas retroflection, we compare the sea surface height (SSH) simulation results for the case POP CTRL with satellite altimeter data produced by Ssalto/Duacs and distributed by AVISO, with support from CNES (http://www.aviso.oceanobs.com/duacs/).The dataset of the dynamic topography for the altimeter results is provided on a 1 3

Volume transports and Agulhas leakage
The time mean dynamic topography deduced from the satellite observation and from POP CTRL for the Indian Ocean and South Atlantic Ocean is plotted in Fig. 2. The average transport of the main currents and strait transports expected to be impacted by the ITF are computed over a 40 yr period for both POP CTRL and POP NoITF and are given in Table 2.In POP CTRL the transport of the ITF is 15 Sv which is close to the estimation based on observations by Sprintall et al. (2009).The Mozambique Channel transport is 17 Sv also close to the 16.7 Sv observed by Ridderinkhof et al. (2010).The transport of the AC at 32 • S is 92 Sv, much larger than the 69.7 Sv ± 4.3 Sv observed by Bryden et al. (2005).However, Bryden et al. (2005) only registered the current strength over a period of 8 months, which is not enough given the important annual and inter-annual variability; the yearly averaged transport in POP CTRL varies between 74 Sv and 104 Sv.Another possible explanation for the discrepancy in AC transport values is the relatively strong recirculation in the southwest Indian Ocean (see Fig. 3a-b) that can locally break the Sverdrup balance and increase the transport.
www.ocean-sci.net/9/773/2013/Ocean Sci., 9, 773-785, 2013 τ 0 −τ 0 0 Fig. 1.Domain of the idealized model HIM with continents (black), ridge (orange), profile of wind stress forcing and important sections (ITF: Indonesian Throughflow, MC: Mozambique Channel, AC: Agulhas Current, L: Leakage).Also indicated is the regions where tracer concentration is fixed to 1 (red hatching) and 0 (blue hatching).The passive tracer is used to compute the leakage, it is initialized to zero in the white regions but is free to evolve in time.
Computing Agulhas leakage is difficult because of the highly turbulent nature of the flow south of Africa.Lagrangian tracers (Biastoch et al., 2008b;Van Sebille et al., 2009a) or passive Eulerian tracer (Le Bars et al., 2012) are required to compute an accurate time series.Unfortunately in these methods the tracers need either to be advected online or offline but using high frequency temporal output (at most a few days), which was not possible with the POP simulations.Another method is to use thresholds of temperature and salinity to separate Indian Ocean waters from its surroundings (Rouault et al., 2009;van Sebille et al., 2010).However, if the temperature and salinity of the South Indian Ocean change, which is the case for an opening of the Indonesian Passage, it introduces a bias in the computation and hence this method is not suited for the purpose of our study.
As we will see later (Fig. 7), the path of the Agulhas eddies does not change when the Indonesian Passage opens.For this reason, we choose here to compute the surface Eulerian volume flux across a fixed section that includes the direct eddy path and the Benguela Current (see Fig. 2).The value thus computed is 21 Sv (Table 2).This is higher than the estimation of Richardson (2007) who found 15 Sv using observations from subsurface floats and surface drifters.The higher value that we obtain here is probably due to the method we use, some of the flow that we measure is Atlantic water and not Agulhas leakage.However our main concern here is not to have a precise measure of the leakage in the control experiment but to measure the difference with the case without ITF.Therefore this method is based on the reasonable assumption that the ITF impact on the volume transport of Atlantic water at the section we choose is small compared to the ITF impact on the Agulhas leakage.

Agulhas retroflection
To assess the performance of POP CTRL in modeling the AC retroflection we compare the average and standard deviation of the SSH (using the last 20 yr of the model simulation) with satellite altimetry (Fig. 3a-b).The model adequately simulates a strong AC, its retroflection/recirculation south of Africa and the meandering Agulhas Return Current (ARC).The regions of high SSH variability in the Mozambique Channel, the area southeast of Madagascar and at the AC retroflection are also captured by the model.However, as already noticed by Maltrud and McClean (2005) in a similar POP simulation, the area of retroflection is broader and the variability is higher than observations in this region.In addition, the path of the eddies is very regular, leading to a unique well defined eddy corridor while in the satellite observations three dominant paths are found (Dencausse et al., 2010a).
To further analyze these differences between model and observations, we computed the retroflection position from satellite dynamic topography and from the model sea surface height (Fig. 4a-b).The method used is the same as Dencausse et al. (2010b): we choose a SSH contour representative of the AC and ARC cores and look for its western most point for each output file.Note that the output frequency is weekly for satellite and monthly for model data.The average retroflection position deduced from satellite data is localized at 18 • E, after the current has left the coast and forms a free jet.In the model, the highest probability of retroflection is around 26 • E, just north of the Agulhas Plateau (Fig. 4b).After this point, the AC often breaks up into eddies instead of being a narrow jet following the coast like it is seen in the altimetry.This results in an average retroflection position at 23 • E in the model, which is 5 • too far east.The same behavior of the AC was seen in a 0.1 • horizontal resolution regional configuration of the Hybrid Coordinate Ocean Model (HYCOM) and was greatly improved by replacing the 2nd order momentum advection scheme by a 4th order scheme (Backeberg et al., 2009).The 2nd order momentum advection scheme of POP could then be responsible for the early retroflection of the AC.
A too easterly retroflection position might be one of the causes for the regular path of the Agulhas eddies.In fact, most of the time in the model the eddies enter the Atlantic at the southern tip of the continental margin of South Africa at 37 • S, whereas the observations show that eddies can form as far south as 42 • S and that the path they follow depends on the formation location (Dencausse et al., 2010a).Another possible cause for the very regular path of the eddies could be related to the seasonal cycle atmospheric forcing used in this simulation.Coupling POP with an atmospheric model showed an increased dispersion of the eddy paths (McClean et al., 2011).It is interesting to notice that the combination of a retroflection too far east and a regular path of the Agulhas eddies is common in state-of-the-art ocean models.It can be seen in global high-resolution models like OFES (Masumoto et al., 2004) and even in ECCO2 (Menemenlis et al., 2008) in which data assimilation is used.However, a regional configuration of ROMS (Penven et al., 2006) and a regional high-resolution nest based on NEMO-ORCA (Biastoch et al., 2008b) seem to perform better in this region.Analyzing the processes causing the differences between these model results would be useful for the ocean modeling community but is outside of the scope of this paper.
In summary, the results from POP compare more favorably with observed sea surface height field properties on aspects of Agulhas retroflection and Agulhas leakage than all models previously used in Indonesian Passage closure studies (as discussed in the introduction).Remaining discrepancies arise probably due to (still too) high values of mixing coefficients and the fact that the model is forced by mean seasonal fluxes.

Effects of closure of the Indonesian passage
As mentioned in Sect.2, the POP CTRL and POP NoITF simulations were run from model year 75 to year 150.The AC transport and the top 1000 m average potential density in the AC retroflection region show that after year 110 a statistical equilibrium of the surface flow seems to be reached (Fig. 5).We therefore choose to focus on the period year 110 to year 150.During this period the AC transport has an important inter-annual variability, the 5 yr running mean shows a range between 88 and 97 Sv but the average density is more stable.

Mean transports
The spatial pattern of the difference of the 40 yr average horizontal velocity fields (POP CTRL minus POP NoITF) is shown in Fig. 6.Changes in volume transports across important sections due to the closure of the Indonesian Passage are given in Table 2.As expected by Sverdrup theory (Hirst and Godfrey, 1993), opening the Indonesian Passage results in a decrease of the East Australian Current (EAC) transport between 20 and 35 • S. At 32 • S the decrease is 13 Sv which is almost equal to the ITF transport.This leads to a weakening of both the Tasman Front transport and the EAC extension transport.The Tasman Leakage is hard to compute south of Tasmania because it is very close to the eastward Antarctic Circumpolar Current (ACC).For this reason we have chosen a meridional section at 120 • E south of the Australian continent (indicated in Fig. 6) where the westward flow coming from the Tasman Leakage is far from the ACC and we can determine it more accurately.The input of water from the Pacific to the Indian Ocean through the Tasman Leakage decreases by about 2 Sv, so we can conclude that an opening of the Indonesian Passage leads to increase of the net flow of water into the Indian Ocean of about 13 Sv.
The opening of the Indonesian Passage also leads to an increase of the South Equatorial Current transport and strengthens both the Mozambique Channel flow (+15 Sv) and the South East Madagascar Current (SEMC) flow (+3 Sv).The Mozambique Channel transport would be reduced to almost nothing (2 Sv) if there was no ITF (Table 2).Assuming that the Tasman Leakage inflow does not reach as far north as the SEMC, we see that the 15 Sv increase of ITF leads to a 18 Sv total increase of the Mozambique Channel and SEMC.This could be due to an increase inertial recirculation in the SEMC (Fig. 2) or a small eastward migration of the South Indian Ocean Counter Current (Siedler et al., 2006;Palastanga et al., 2007).
The increased transport of the Mozambique Channel and the SEMC is only compensated by a small decrease of the Tasman Leakage transport leading to an increase of the AC transport of 10 Sv.It is interesting to note that given the increase and decrease of the sources of the AC one would expect an increased AC transport of 16 Sv.The reason why it is not seen here is likely to be related to an increase of the "Leeuwin Current system", defined by Hirst and Godfrey (1993) as the combination of southeastward geostrophic flow around 25 • S, southward flow at the west coast of Australia, coastal downwelling and a northwestward return flow at depth.The surface part of this system is seen to be strengthened by the ITF (see Fig. 2b,c) as found previously by Hirst and Godfrey (1993) and Valsala and Ikeda (2007).

Agulhas Leakage
The increase in AC transport due to the opening of the Indonesian Passage leads to both an increase of Agulhas leakage to the Atlantic (+3 Sv) and an increased transport of the Agulhas Return Current.In the South Atlantic the additional leakage mostly stays in the subtropical gyre, increasing the  Brazil Current by (+3 Sv).A little part seems to escape further north to strengthen the North Brazil Current (+1 Sv).
The increased AL originates from an increase of the number of Agulhas eddies (see Table 2) which leads to increased variability of the SSH in the eddy corridor from 21 to 27 cm (Fig. 7).The variability of the longitudinal position of the retroflection also increases (not shown) but the average stays at the same position (Fig. 4b, c).This means that we find an important strengthening of the AC transport that has no effect on the mean retroflection position.This is not what would be expected from the non-linear theory developed by Ou and De Ruijter (1986) in which the separation location was shown to be very sensitive to the AC transport.
There are different hypotheses to explain the small sensitivity of the retroflection position.The control by bottom topography, in this case the Agulhas plateau, might be strong as suggested by Matano (1996) and could cause a "locking" of the retroflection in one position (De Ruijter et al., 1999).However even though in the models the retroflection is often located north of the Agulhas Plateau (Fig. 4) the mean position is 3 • further west due to repeated intrusion of the AC west of the Agulhas Plateau.Time series of the low frequency variability of the longitudinal position show an important variability (Fig. 8a) suggesting that the retroflection position is not locked by topographic features.
If we look at the relation between the low frequency variations of the AC and the longitudinal position of the retroflection, we see that in this model there is no simple linear relation (Fig. 8b).For most of the values of the AC transport the average retroflection is between 23 and 23.5 • E. The only remarkable event is a far east retroflection associated with a very weak AC and this could be due to the fact that a weaker current would be more sensitive to the effect of bottom topography.We see that even though we do not find a general relation between AC and retroflection position there are events (of five years duration) during which the AC transport is stronger combined with an eastward shift of the retroflection.Such a result was used by Van Sebille et al. (2009b) to illustrate the importance of the inertial overshoot mechanism (using another high resolution model) but in POP also the opposite response is found frequently.

Analysis using HIM
With the idealized model setup, we can control the retroflection regime by changing the amplitude of the wind stress (Le Bars et al., 2012).Hence, performing sensitivity studies with the wind stress strength for both an open (simulations referred to as HIM CTRL) and closed Indonesian Passage (simulations indicated as HIM NoITF ) can help interpret the POP results.In HIM CTRL, the ITF varies from 1 to 16 Sv over the wind stress range used (0.05-0.3 N m −2 ).
An important indicator of the Agulhas retroflection is the retroflection index R (Dijkstra and de Ruijter, 2001a; Le Bars et al., 2012), defined by where L and AC are the Agulhas leakage and the AC transport, respectively.Hence, R is a non-dimensional number that represents the proportion of AC transport that is section, R is very similar for POP CTRL (R = 0.78) and POP NoITF (R = 0.79).This result is surprising, as it means that opening the ITF does not affect the proportion of water that is retroflected.
To assess the generality of this result we compare Rvalues computed for HIM CTRL and HIM NoITF for different wind stress amplitudes.As can be seen Fig. 9a, the retroflection index R is insensitive to the presence of the ITF for all wind stress amplitudes.As the AC transport increases with the ITF, an opening of the Indonesian Passage will always strengthen the leakage (Fig. 9b).The fact that R is insensitive to the ITF strength is intriguing as for an increase in AC transport due to a stronger wind, R increases (Le Bars et al., 2012).This suggests that the partitioning of the transports in the leakage and return current is dominated by the wind field.
where the superscripts O and C refer to the HIM CTRL and HIM NoITF results, respectively.From Fig. 9a, we deduce that for all wind stress amplitudes τ , we have approximately A comparison of the time mean of the left hand side and right hand side of (Eq.4) is shown in Fig. 10.Although the leakage has a large variability, there is a reasonable balance of both time mean terms.The balance (Eq.3) does not strictly hold for POP, but for a wind stress amplitude between 0.25 and 0.3 N m −2 , corresponding to an ITF transport of 14 to 16 Sv in HIM CTRL, the result (Eq.4) provides an increased leakage of around 4.5 Sv, which is at least consistent with what was found in POP.
The fact that POP and HIM give similar results concerning the retroflection index even with different continent geometry, bottom topography, vertical coordinates and surface forcing highlights the generality of our results.

Summary and discussion
In this work we have extended the results of previous modeling studies on the role of the ITF in the global climate system, by focussing on the impact of the ITF on the volume of Agulhas leakage (AL) in a strongly eddying ocean model (POP at 0.1 • horizontal resolution).In this version of POP, boundary layer flows as well as meso-scale eddies are better resolved that in the previous modeling efforts related to the climate effects of the ITF (Schneider, 1998;Lee et al., 2002;Wajsowicz and Schneider, 2001;Song et al., 2007).
In POP the AL increases due to the opening of the Indonesian Passage.This response was shown to be due to the combined effect of an increase AC transport due to the additional input of water from the Pacific and a constant proportion of Agulhas Current retroflection (constant retroflection index).The increase of AL is associated with an increased number of Agulhas eddies.The issue of the effect of ITF on the heat and salt transport from the Indian to the South Atlantic and its impact on the Atlantic Meridional Overturning is outside the scope of this paper because it involves a complex competition between changes of the so-called "warm route" through the AL and the "cold route" through the Pacific and the Drake Passage (Gordon, 1986).
A two-layer model was used to investigate the change of the retroflection index with opening/closing the Indonesian Passage.We show that for a wide range of wind stress forcing the retroflection index is insensitive to the strength of the ITF.For this model we are also able to deduce a semi-empirical formula that relates the ITF transport and the leakage.
The fact that the retroflection index stays constant is intriguing.As mentioned in the previous section, one would expect an increase retroflection index due to mixed barotropic/baroclinic instabilities related to an increase Agulhas Current forced by Indian Ocean winds (Le Bars et al., 2012).The experiment realized here is however different because the input of water comes from the Pacific Ocean and needs to return south of Australia.To isolate this additional large-scale condition we solve the linear momentum and mass conservation equations on the same domain.The derivation is presented in Appendix A and we discuss here the main results.In a linear model the retroflection index is independent of the wind stress.This can be understood because it lacks all the mechanisms put forward by Dijkstra and de Ruijter (2001a) and Le Bars et al. (2012) to explain the change of the retroflection index, i.e. viscosity, inertia or turbulence, depending on the dynamical regime.In this model, the retroflection index decreases when the Indonesian Passage is opened which is consistent with the potential vorticity constraints on the flow and the fact that the water has to return at a latitude south of Australia.
The fact that the retroflection index does not change when the Indonesian Passage opens is then the result of two mechanisms: (i) increase barotropic/baroclinic instabilities that are expected to increase the retroflection and (ii) large-scale linear momentum and mass balances that lead to decrease the retroflection.Understanding why these two effects nearly cancel each other for each amplitude of the wind forcing is a challenge left for future research.

Linear model
To further understand the physics of the previous results we use a steady linear one-layer quasi-geostrophic model.We want to investigate whether the fact that the retroflection index stays constant when the ITF is open is the result of steady linear mass and vorticity balances.Using the same geometry as used for the HIM model, we define seven sections (see Fig. A1) and assuming a steady state we have the following mass balances: AC + 1 = ITF , (A1b) We now write the steady quasi-geostrophic equations in the form F = 0, with where H is the thickness of the layer having a constant density ρ.Furthermore, f is the Coriolis parameter and A H the lateral friction coefficient.We now integrate F over three closed contours C 1 , C 2 and C 3 as indicated in the Fig. A1; i.e.F.ds = 0. Inertia is neglected, friction is only considered in the Agulhas Current and the contour integrated pressure gradient cancels for closed contours.On C 1 this gives where Using the vorticity balance at the WBC then, under no-slip boundary conditions, it is possible to express the horizontal friction as a function of the wind stress (Eq.2.11.6 from Pedlosky, 1996):

Fig. 2 .
Fig. 2. Contour of mean dynamic topography and standard deviation (colors, in cm) from (a) altimetry data, (b POP CTRL and (c) POP noITF.Sections used forTable 2 are also shown here (EAC: East Australian Current at 30 • S, TL: Tasman Leakage, NBC: North Brazil Current at 6 • S, SEMC: South East Madagascar Current, BC: Brazil Current at 32 • S, AL: Agulhas leakage, AC: Agulhas Current at 32 • S, MC: Mozambique Channel).

Fig. 3 .
Fig. 3. Contour of mean dynamic topography and standard deviation (colors, in cm) from (a) satellite altimetry, (b) POP CTRL and (c) POP noITF near the tip of South Africa.

Fig. 4 .
Fig. 4. Probability density of the retroflection position from (a) satellite altimetry, (b) POP CTRL and (c) POP NoITF simulation.The black crosses indicate the average retroflection position from the all time series.Bathymetric contours are at depth of 1000 m, 2000 m and 3000 m.

Fig. 5 .
Fig. 5. Time series of the five-year running average (a) AC transport at 32 • S and (b) average potential density of the upper 1000 m in the retroflection area (40 • S to 37 • S and 20 • E to 25 • E).Black lines for POP CTRL and red for POP NoITF.We see the important interannual variability of the AC and the spin-up time of approximately 35 yr of the surface density.

Fig. 6 .
Fig. 6.Velocity difference (arrows, with the length of the arrow corresponding to 12 cm s −1 indicated) between POP CTRL and POP NoITF.The difference of the magnitude in cm s −1 is shown in color.Velocities are vertically averaged from the surface to the depth of 1500 m and temporally over the last 40 yr of the simulations.Sections used forTable 2 are also shown here (EAC: East Australian Current at 30 • S, TL: Tasman Leakage, NBC: North Brazil Current at 6 • S, SEMC: South East Madagascar Current, BC: Brazil Current at 32 • S, AL: Agulhas leakage, AC: Agulhas Current at 32 • S, MC: Mozambique Channel).

Fig. 7 .
Fig. 7. Contour of mean dynamic topography and standard deviation (colors, in cm) from (a) POP CTRL and (b) POP ITF.This zoom on the Agulhas eddy corridor shows the increase variability due to an opening of the Indonesian Passage.The black line shows the section used to compute the Agulhas leakage.
Fig. 8. (a) Five-year running mean of the longitude of the retroflection in POP CTRL (black) and POP NoITF (red).(b) Longitude of the retroflection as a function of AC transport for POP CTRL (black) and POP NoITF (red) also filtered with a five-year running mean.Bigger crosses represent the average AL corresponding to a certain range of 1 Sv of AC.
which reflects the independence of R on the ITF amplitude.In addition, in this model the difference between the AC transport in the HIM CTRL and HIM NoITF is equal to the ITF in the model, i.e.C AC (τ ) ≈ O AC (τ ) − O ITF (τ ).It follows directly from (3) that the leakage difference is given by O

Fig. 9 .
Fig. 9. Retroflection index (a) and Agulhas leakage (b) as a function of wind stress for HIM CTRL (black) and HIM NoITF (red).Each cross represent a statistical steady state and vertical bars represent the 95 % confidence interval for the computation of the mean Agulhas leakage.

Fig. 10 .
Fig. 10.Comparison of the right hand side and the left hand side of Eq. (4) for the equilibrium solutions of HIM CTRL and HIM NoITF.
Fig. A1.Domain of computation with the main sections along which the transport is computed and the contours of integration C 1 , C 2 and C 3 .

Table 1 .
Relevant parameters of the idealized experiments used for both open ITF (HIM CTRL) and closed ITF (HIM NoITF) cases.

Table 2 .
Mean transport (in Sv) across different sections of POP CTRL and POP NoITF rounded to integer values.The standard deviation is given between brackets.Note that the numbers are rounded to 1 Sv.
Table 2 are also shown here (EAC: East Australian Current at 30 • S, TL: Tasman Leakage, NBC: North Brazil Current at 6 • S, SEMC: South East Madagascar Current, BC: Brazil Current at 32 • S, AL: Agulhas leakage, AC: Agulhas Current at 32 • S, MC: Mozambique Channel).