Jean-Louis Goosse

Dufresne

Olivier Arzel
email: o.arzel@unsw.edu.au

Thierry Fichefet

Hugues Goosse

Jeanlouis Dufresne

Causes and impacts of changes in the Arctic freshwater budget during the 20 th and 21 st centuries in an AOGCM

published or not. The documents may come

Causes and impacts of changes in the Arctic freshwater budget during the 20th and 21st centuries in an AOGCM

Introduction

The freshwater supply to the Arctic Ocean plays a key role in controlling the circulation and water mass properties of this ocean, and the export of this freshwater toward the convection sites located in the Labrador and GreenlandIcelandNorwegian (GIN) Seas significantly affects deep water formation (e.g., [START_REF] Manabe | Simulation of abrupt climate change induced by freshwater input into the North Atlantic Ocean[END_REF][START_REF] Hakkinen | A Simulation of thermohaline effects of a great salinity anomaly[END_REF], Jungclaus et al., 2005) and ultimately the Atlantic meridional overturning circulation (AMOC). Excessive freshwater input to the Arctic Ocean and Nordic Seas could thus influence the climate of the North Atlantic area by decreasing the northward ocean heat transport associated with the largescale AMOC (e.g., [START_REF] Vellinga | Global climatic impacts of a collapse of the Atlantic thermohaline circulation[END_REF][START_REF] Fichefet | Implications of changes in freshwater flux from the Greenland ice sheet for the climate of the 21 st century[END_REF]. It is therefore crucial to analyse the mechanisms of freshwater release from the Arctic Ocean towards subArctic regions on decadal and longer timescales. In particular, this study aims to increase our understanding of the interrelation between changes in freshwater transport toward the convection sites and deep mixing in response to enhanced greenhouse gas conditions.

During the last few decades, the physical Arctic environment has undergone large changes, as reviewed in recent literature (e.g., [START_REF] Serreze | Observational evidence of recent change in the northern high latitude environment[END_REF][START_REF] Moritz | Dynamics of recent climate change in the Arctic[END_REF][START_REF] Overland | Integrated analysis of physical and biological panArctic change[END_REF].

For example, since the mid60s, the annual mean Arctic surface air temperature has raised by about 2°C. Since the late 70s, satellite records show a decreasing linear trend in Arctic sea ice extent at an annual mean rate of about 0.3 10 6 km 2 per decade [START_REF] Cavalieri | 30year satellite record reveals contrasting Arctic and Antarctic decadal sea ice variability[END_REF]. Submarinebased observations indicate that the sea ice thickness in deepwater areas of the Arctic Ocean has decreased by about 1.3 m in average between 19581976 and 19931997 [START_REF] Rothrock | Thinning of the Arctic sea ice cover[END_REF].

River monitoring data show that the average annual discharge of freshwater from the six largest Eurasian rivers to the Arctic Ocean has increased by about 7% from 1936 to 1999, suggesting an increase of precipitation over land in high latitudes of the Northern Hemisphere during that period [START_REF] Peterson | Increasing river discharge to the Arctic Ocean[END_REF]. [START_REF] Hansen | Decreasing overflow from the Nordic seas into the Atlantic Ocean through the Faroe Bank channel since 1950[END_REF] reported that the cold, dense overflows across the GreenlandScotland ridge that feeds the southward branch of the AMOC, has decreased by at least 20 percent during 19502000. [START_REF] Dickson | Rapid freshening of the deep North Atlantic Ocean over the past four decades[END_REF] presented evidence of a rapid and sustained freshening of the system of overflows and entrainment that ventilate the deep Atlantic Ocean over 19652000. [START_REF] Curry | A change in the freshwater balance of the Atlantic Ocean over the past four decades[END_REF] found systematic freshening over much of the water column at both poleward ends of the Atlantic Ocean accompanied by large increases of salinity in the upper water column at low latitudes between the 1950s and the 1990s. Recently, Curry et al. (2005) estimated that the Nordic Seas and the subpolar basins were diluted by an extra 19,000±5,000 km 3 of freshwater input between 1965 and 1995, with about half of this freshwater being added in the late 60s at a rate of 2,000 km 3 per year during the Great Salinity Anomaly (GSA) event [START_REF] Dickson | The "Great Salinity Anomaly" in the northern North Atlantic[END_REF].

Whether the highlatitude freshening observed during the past four decades and the reduced overflows are a manifestation of a weakening of the AMOC remains very controversial. Indeed, the relationship between the largescale AMOC and the northern North Atlantic freshening is difficult to infer from current observations because of the lack of data and the too short period of measurements compared to timescales of internal variability. Nevertheless, [START_REF] Bryden | Slowing of the Atlantic meridional overturning circulation at 25N[END_REF] suggested recently that the AMOC has slowed down by about 30 percent between 1957 and 2004 using five hydrographic transAtlantic sections at 25°N. However, these sections have been performed at different seasons during the year, and we can therefore not exclude that the downward trend of the AMOC reported by these authors be due to a large intraseasonal variability that overwhelms the longterm signal. Using the HadCM3 climate model running over the 20 th century, [START_REF] Wu | Does the recent freshening trend in the North Atlantic indicate a weakening of the thermohaline circulation ?[END_REF] reported a deep freshening trend of the Labrador Sea similar to that seen in observations by [START_REF] Dickson | Rapid freshening of the deep North Atlantic Ocean over the past four decades[END_REF], and obtained yet an enhancement of the AMOC instead of a weakening. The situation is however very different over the 21 st century for which the radiative forcing is much larger than the one experienced over the second half of the past century. In this case models show unequivocally, albeit with various magnitudes, a weakening of the AMOC in response to increasing anthropogenic greenhouse gas forcing (e.g., [START_REF] Houghton | The scientific basis, Contribution of Working Group 1 to the third assessment report of Intergovernmental Panel on Climate Change. 881[END_REF][START_REF] Fichefet | Implications of changes in freshwater flux from the Greenland ice sheet for the climate of the 21 st century[END_REF][START_REF] Gregory | A model of intercomparison of changes in the Atlantic thermohaline circulation in response to increasing atmospheric CO 2 concentration[END_REF]. It is worth noting that in all models used in the Coupled Model Intercomparison Project (CMIP), the AMOC weakening is caused more by changes in surface heat flux than by changes in surface freshwater flux [START_REF] Gregory | A model of intercomparison of changes in the Atlantic thermohaline circulation in response to increasing atmospheric CO 2 concentration[END_REF]. However, increase in both sea ice melting and river flows are generally found in these climatechange scenarios, leading to an increase in the freshwater storage of the Arctic/subArctic regions. As a result, the highlatitude ocean stratification increases and the water mass transformation becomes much less efficient, thus contributing to the AMOC weakening.

Although models agree that the northern North Atlantic freshens in response to global warming, the mechanisms driving the changes in freshwater release from the Arctic Ocean to the Nordic Seas and the subpolar basin of the North Atlantic are still not well understood. Furthermore, the interrelation between changes in deep convection and the freshwater supply to the northern North Atlantic in a warming climate is still not established. This paper arises then from the desire to gain insights into the mechanisms underlying the freshening of the highlatitudes of the Northern Hemisphere, and its link with changes in deep mixing in response to enhanced greenhouse gas conditions. The analysis is based on outputs of simulations performed with an atmosphereocean general circulation model (AOGCM) for the fourth assessment report of the Intergovernmental Panel on Climate Change (IPCC AR4). In this paper, we discussed results from the experiment covering the 20 th century with anthropogenic forcings and the A1B scenario for the 21 st century, which corresponds to an increase of C0 2 until a level of 720 ppm by 2100 as defined by the Special Report on Emission Scenario (SRES). It should be stressed that the analysis devoted to the impact of changes in freshwater flux on deep mixing is based on detrended variables so that the results do not depend on the longterm climate change experienced over the 21 st century, such as the longterm freshening of the high latitudes for instance.

After a brief description of the coupled climate model and its climatology in Section 2, the processes responsible for the longterm freshening of the Arctic Ocean are examined in Section 3, while the link between interannual changes in freshwater fluxes and interannual convection depth anomalies over the 20 th and 21 st centuries is addressed in Section 4. Section 5 deals with the mechanisms influencing the longterm changes in the Fram Strait outflow. Conclusions are finally given in Section 6.

Model description and climatology

Model description and experimental setup

We use the fourth version of the AOGCM developed at the "Institut Pierre Simon Laplace", Paris (IPSLCM4, [START_REF] Marti | The new IPSL climate system model: IPSLCM4, Note du pôle de modélisation[END_REF]. This model is made up of the atmospheric general circulation model LMDz [START_REF] Hourdin | The LMDZ4 general circulation model: climate performance and sensitivity to parameterized physics with emphasis on tropical convection[END_REF], the oceanic general circulation model ORCA (Madec et al., 1998), the thermodynamicdynamic sea ice model LIM [START_REF] Fichefet | Sensitivity of a global sea ice model to the treatment of ice thermodynamics and dynamics[END_REF] and the land surface scheme ORCHIDEE [START_REF] Krinner | A dynamic global vegetation model for studies of the coupled atmosphere biosphere system[END_REF]. Time synchronisation and spatial interpolation of surface fluxes and variables between the atmospheric model and the three other components are ensured through the OASIS coupler [START_REF] Valcke | Proceedings of the 6th International Meeting: High performance computing for computational science[END_REF]. There is no local flux correction in this model. The fresh water associated with ice sheet melting is drained toward the ocean and does not affect the ice sheet geometry (no dynamics). The atmospheric model has a horizontal resolution of 3.75° in longitude and 2.5° in latitude, with 19 levels along the vertical. The oceanic model has a coarse horizontal resolution of 2° in longitude and latitude, with vertical grid spacing increasing from 10 m in the top 150 m to 500 m at the bottom (31 levels).

The outputs from the experiment covering the 20 th century, which includes only the anthropogenic forcing (observed CO 2 and sulphate aerosol concentrations) are first discussed. For the 21 st century, we use outputs from the A1B climate change scenario as defined by the Special Report on Emission Scenarios, which corresponds to a continuous increase of CO 2 concentration until a level of about 720 ppm by 2100 (see [START_REF] Dufresne | Contrasts in the effects on climate of anthropogenic sulfate aerosols between the 20th and the 21st century[END_REF] for a more complete description of those simulations).

Climatology of the Arctic Ocean

A detailed description of the model climatology can be found in [START_REF] Swingedouw | Sensitivity of the Atlantic thermohaline circulation to global freshwater forcing[END_REF].

A summary is given here for the reader's convenience, with a particular focus on the Arctic Ocean.

As most AOGCMs, IPSLCM4 has several biases in its mean climate compared to observations. The AMOC index, defined as the maximum value of the annual mean meridional overturning streamfunction between 500 m and 5000 m depth in the Atlantic basin, is 11.8±0.7 Sv over the 20 th century. This is somewhat weaker than the observational estimates of 15±2 Sv reported by [START_REF] Ganachaud | Improved estimates of global ocean circulation, heat transport and mixing from hydrographic data[END_REF]. [START_REF] Swingedouw | Sensitivity of the Atlantic thermohaline circulation to global freshwater forcing[END_REF] mostly attributed this error to an excessive precipitation less evaporation forcing between 45° and 50°N, which leads to too low sea surface salinities (SSS) in the northern North Atlantic. In the Labrador Sea, this results in the formation of a strong halocline that prevents winter deep oceanic convection. Deep convection sites in the model rather occurs in the GIN Seas in agreement with observations [START_REF] Dickson | Longterm coordinated changes in the convective activity of the North Atlantic[END_REF] and south of Iceland, east of the southern tip of Greenland, with a maximum average ventilation depth of about 1200 m (Fig. 1). The good agreement between the simulated outflow rate of water denser than 27.8 kg m 3 over the GreenlandIcelandScotland ridge (5.4 Sv) and observations (5.6 Sv; [START_REF] Dickson | The production of North Atlantic deep water -sources, rates and pathways[END_REF] suggests that the AMOC bias is rather related to the absence of convection in the Labrador Sea [START_REF] Swingedouw | Sensitivity of the Atlantic thermohaline circulation to global freshwater forcing[END_REF]. This latter feature is accompanied by too large a sea ice extent in the Labrador Sea during winter (Fig. 2). In summer, however, the sea ice edge location agrees reasonably well with observations (Fig. 2). It should be noted that the too weak AMOC simulated by the model is also associated with a cold bias of about 5°C in the northern North Atlantic (45°50°N).

The mean sea ice velocities computed by the model (not shown) are in reasonable agreement with those derived from the 85.5 GHz Special Sensor Microwave/Imager (SSM/I) data by [START_REF] Emery | Satellitederived maps of Arctic and Antarctic sea ice motion: 1988 to 1994[END_REF]. The sea ice drift features the Beaufort Gyre, the Transpolar Drift, the divergent motion on the Siberian shelf and the sea ice export through Fram Strait. The simulated largescale ocean surface circulation (not shown) displays roughly the same features as the mean sea ice drift pattern in the Arctic Ocean, namely the Beaufort Gyre, the Transpolar Drift stream and the East Greenland Current. Both the Bering Strait inflow in the central Arctic (1 Sv) and the Canadian Archipelago throughflow (0.7 Sv) are close to observational data. In the GIN Seas, a cyclonic gyre is simulated, but the inflow of Atlantic waters into the Barents Sea appears too small (0.17 Sv) compared to observational estimates of Blindheim et al. (1989), which rather give about 3 Sv. This deficiency is mostly due to the too weak AMOC simulated by the model. The Fram Strait outflow (0.5 Sv) is therefore weaker than the one observed (3 Sv) in order to balance the mass budget of the central Arctic. Consequently, the modeled Arctic Ocean is somewhat isolated from the northern North Atlantic.

Simulated Arctic freshwater budget over 19512000

In the following, we describe the longterm mean of the volumeaveraged freshwater balance of the Arctic Ocean over 19512000. This 50year period has been chosen in order to avoid potential biases related to the large Arctic climate decadal variability in the model. The freshwater content in a given volume is defined as ∫ (S ref S)dv / S ref dv where S is the salinity and dv the volume increment. The reference salinity S ref is fixed to 34.8 psu and corresponds to the volumeintegrated salinity simulated by the model in the Arctic Ocean which we define as the area delineated by the Bering Strait and the 60 th parallel in the northern North Atlantic. The major contributions to the freshwater budget of the central Arctic Ocean, GIN Seas, and the region located south of Iceland are provided in Tables 1, 2 and 3 respectively while sea ice freshwater fluxes are given in Table 4. The geometry of those three regions is illustrated in Figure 3. It should be stressed that the link between freshwater fluxes and salinity is not straightforward. For example, a positive freshwater flux directed toward a region of mean salinity smaller than the reference salinity could either increase or decrease the mean salinity depending of the magnitude of the freshwater flux, while when it is directed toward a region more saline than the reference salinity, then it always decreases the mean salinity of the considered region.

The largest source of freshwater in the central Arctic Ocean (Table 1) simulated by the model is the river runoff (R hereafter, 3456 km 3 /yr) followed by the Bering Strait water inflow (2831 km 3 /yr) and the atmospheric freshwater flux (precipitation less evaporation; 1136 km 3 /yr; PE hereafter). The magnitudes of these sources are close to estimates of [START_REF] Aagaard | The role of sea ice and other fresh water in the Arctic circulation[END_REF] (AC89 hereafter) for R and PE and Woodgate and Aagaard (2005) for the Bering Strait water inflow. The model reproduces a sea ice export of 196 km 3 /yr through Bering Strait whereas the observations rather show an inflow of 400 km 3 /yr (Woodgate and Aagaard, 2005). Nevertheless, the total (i.e solid plus liquid) freshwater release (2635 km 3 /yr) remains close to the estimation of Woodgate and Aagaard (2005) which is 2500 km 3 /yr.

The largest sink of freshwater in the model central Arctic is the Fram Strait outflow (5034 km 3 /yr) and is roughly equally partitioned between the solid and liquid contributions (Tables 1,4).

The annual mean ice export through Fram Strait is 93,400 m 3 /s over 19512000, a value that compares quite well with data collected by [START_REF] Vinje | Monitoring ice thickness in Fram Strait[END_REF] and [START_REF] Kwok | Fram Strait sea ice outflow[END_REF]. This yields a solid freshwater flux of 2581 km 3 /yr, that matches the estimates of AC89. Although the simulated volume flux at Fram Strait (0.51±0.44 Sv) is roughly onesixth of that observed, the liquid freshwater export there (2453 km 3 /yr) exceeds the estimates of AC89 by about 1600 km 3 /yr. This bias is mainly related to the too large liquid freshwater content (57,000 km 3 ), or the too low salinity, in the central Arctic Ocean in the model which overestimates the one estimated from the climatology of [START_REF] Levitus | Salinity. NOAA Atlas NESDIS[END_REF] by about 19,000 km 3 .

The simulated inflow of salty waters from the North Atlantic toward the GIN Seas (1543 km 3 /yr) is underestimated by almost 50 percent compared to estimates of AC89 (2880 km 3 /yr).

Furthermore, both the simulated sea ice and liquid freshwater exports at Denmark Strait are overestimated by about 600 km 3 /yr and 1200 km 3 /yr respectively compared to observations of AC89, although the volume flux (5 Sv) be fairly realistic. Note that most of the biases underlined in this freshwater budget are likely related to the too weak AMOC simulated by the model and the subsequent too low salinity and temperature values in the northern North Atlantic.

In the GIN Seas (Table 2), the simulated river runoff (424 km 3 /yr) and PE (532 km 3 /yr) are in reasonable agreement with observational estimates of AC89 (420 km 3 /yr and 790 km 3 /yr respectively). The sink of liquid freshwater in those seas is dominated by the freshwater export at Denmark Strait (2116 km 3 /yr) and the inflow of salty waters from the North Atlantic (1543 km 3 /yr), while the source of liquid freshwater is mainly due to the freshwater import through Fram Strait (2453 km 3 /yr) and to the the sea ice to ocean freshwater flux (1826 km 3 /yr).

South of Iceland, the sink of liquid freshwater (Table 3) is mainly caused by the inflow of salty waters originating from the North Atlantic (7212 km 3 /yr) and with a lower magnitude, by the export of liquid freshwater toward the Labrador Sea (1603 km 3 /yr). The source of freshwater south of Iceland is dominated by the inflow of freshwater through Denmark Strait (2116 km 3 /yr) and the export of salty waters toward the GIN Seas (1543 km 3 /yr). Note that PE+R represents the third largest input of freshwater into the GIN Seas (956 km 3 /yr) and into the region located south of Iceland (1406 km 3 /yr) in the model during 19502000. Finally, the freshening of the deep Labrador Sea (2,3003,300 m) identified over 19702000 by [START_REF] Dickson | Rapid freshening of the deep North Atlantic Ocean over the past four decades[END_REF] appears also in the model but with a weaker magnitude (not shown).

Simulated changes in the Arctic freshwater budget over the 21 st century

We briefly describe herein the changes experienced by the freshwater budget at high northern latitudes. In particular, we focus on the regions where deepwater formation takes place in the model, namely the GIN Seas and the region located south of Iceland and on the central Arctic, which is the main source of freshwater for the surrounding seas (Fig. 3). It should be stressed that the Labrador Sea is excluded from our analysis since no deep convection occurs there in the model.

In order to evaluate the changes in freshwater balance, we compare the period 20512100 of the SRES A1B experiment to the period 19512000. Individual contributions to the total liquid freshwater budget of the central Arctic, GIN Seas and region located south of Iceland averaged over a 50year period to filter the decadal variability, are provided in Tables 1, 2 and 3 respectively, while sea ice freshwater fluxes are given in Table 4.

Central Arctic

The freshening of the central Arctic Ocean is dominated by the change in continental runoff, which has increased by about 2035 km 3 /yr (0.065 Sv, 59%) between 19512000 and 20512100 (Table 1). This enhanced river discharge to the Arctic Ocean is an indication of the increase in the integrated highlatitude precipitation over continental areas resulting from an intensified hydrological cycle in response to global warming (Wu et al., 2002). The second most important change concerns the freshwater import through the northern boundary of the Barents Sea which has increased by 2020 km 3 /yr. This is the contribution that experiences the largest relative change since near zero freshwater flux is simulated there over 19512000 (see Table 1). This large shift is due to the freshening of the GIN Seas and also to the strengthening of the northward mass transport through the Barents Sea. The increase in sea ice melting (1325 km 3 /yr change) is the third largest source for the freshening of the central Arctic. Note that the sign of the icetoocean freshwater flux is negative in the central Arctic, indicating that more ice is formed than melted there since it is exported. Finally, the change in the atmospheric freshwater flux (PE) represents only a small contribution (358 km 3 /yr) to the freshening of the central Arctic Ocean. Liquid freshwater export at Fram Strait over 20512100 is more than twice that simulated over 19512000 and appears to dominate the sink of freshwater in the central Arctic (5398 km 3 /yr). The freshwater export through the Canadian Archipelago is enhanced by about 73% compared to 19512000, while the freshwater import through Bering Strait is decreased by about 25%. This latter feature is consistent with the freshening of the central Arctic. This freshening induces a decrease in the mean meridional pressure gradient between the Pacific and the central Arctic, which therefore weakens the Bering Strait throughflow (Coachman and Aagaard, 1966).

GIN Seas

Over 20512100, the linear trend of the annual mean freshwater content in the GIN Seas (315 km 3 /yr) is much smaller than the one of the central Arctic Ocean (Table 2). The implications for deepwater formation are however significant as we will see later on. The largest change is related to the decrease of the northward inflow of salty waters originating from the North Atlantic (3718 km 3 /yr change). Indeed, beyond year 2045 of the simulation, the inflow of Atlantic waters in the GIN Seas tends to decrease the average salinity of the GIN Seas rather than the opposite since the midlatitudes of the North Atlantic become less saline than the Nordic seas. This source of freshening is nearly compensated by the increase of freshwater export at Denmark Strait which undergoes a shift of about 3500 km 3 /yr (0.11 Sv). The increase in freshwater import at Fram Strait and the larger export to the Barents Sea represent the third and fourth largest changes, in absolute value, of the freshwater budget (2945 km 3 /yr and 1866 km 3 /yr respectively). Although sea ice melting occurs further upstream in the East Greenland Current over the second half of the 21 st century compared to the 20 th century, the freshwater flux into the ocean associated with sea ice melting/freezing decreases by 639 km 3 /yr (35% change). This is related to a smaller inflow of ice from the Arctic through Fram Strait. Finally, local changes in PE+R contribute only slightly to the longterm freshening of the GIN Seas (562 km 3 /yr change). In summary, the changes in the freshwater fluxes between the GIN Seas and the surrounding oceans/seas dominate the changes in the freshwater balance of the GIN Seas over the 21 st century.

Region located south of Iceland

Although the linear trend of the annual mean freshwater content in this region is not significant over 20512100 (31 km 3 /yr ), large changes in freshwater fluxes are noticed (Table 3).

The most noticeable modification is a decrease of the northward advection of salty waters originating from the North Atlantic which represents a source of freshening of 4707 km 3 /yr. This is consistent with a reduced salinity in the North Atlantic (2 psu annual mean change between 1951 2000 and 20512100 at midlatitudes) and and the weakening of the largescale AMOC (5 Sv annual mean change by 2100 with respect to 19512000), which induce a reduced salt transport from low to high latitudes. This change is more than compensated by the increase of freshwater export to the Labrador Sea (5385 km 3 /yr). The increase in PE+R is weak and the iceocean freshwater flux has been reduced since less ice is transported through the Denmark Strait.

Links between changes in freshwater fluxes and interannual convection depth anomalies

As a consequence of the longterm freshening and warming of the Arctic Ocean, the model predicts that the convection almost disappears in the GIN Seas and south of Iceland by the end of the 21 st century, with very shallow ventilation depth in March of about 100200 m (not shown).

However, this does not lead to a complete shutdown of the AMOC, as it remains at a value of about 6 Sv by 2100. In the following, we specifically focus on the processes promoting interannual deep convection anomalies in the GIN Seas and South of Iceland over the 20 th and 21 st centuries.

We use detrended timeseries of annual mean freshwater fluxes and March mixedlayer depth which is used as a proxy for convection depth. The analysis is performed separately over 19011980 and 20212080. These peculiar periods have been chosen in order to avoid biases related to the linear detrend applied to the fields. The anomalies represent therefore the difference from the linear trend calculated over these two periods.

GIN Seas

20 th century

Figure 4a shows that the correlation between March mixed layer depth anomalies in the GIN Seas and annual mean salinity anomalies averaged in the upper 100 m and over the East Greenland Current (EGC) region is slightly negative at zero lag. By contrast, a positive mixed layer depth anomaly tends to occur after relatively salty conditions in the eastern part of the GIN Seas. This indicates that the convection tends to be deep when the upper salinity in the EGC is low and when the upper levels of the eastern part of the GIN Seas are anomalously salty. Those features are in agreement with correlations depicted in Figure 5a. These latter show that the maximum of the mixedlayer depth in the GIN Seas occurs about two years after a minimum freshwater import from the North Atlantic. This is perfectly consistent with the idea that water mass formation in the GIN Seas results from the cooling of warm and salty waters originating from the North Atlantic. Note that the variations of upper salinity in the top 100 m in the eastern part of the GIN Seas are not related to the variations in the largescale AMOC but rather to changes in the upper ocean circulation. Indeed, the maximum of the upper salinity there tends to occur one year after the minimum of the AMOC index with a correlation of 0.2. On the other hand, the convection depth reaches its maximum about 2 years after the maximum of the southward sea ice and liquid freshwater fluxes at Fram Strait, with a correlation of 0.4 (Fig. 5a). Indeed, almost all sea ice passing through Fram Strait melts close to Iceland over the 20 th century. The subsequent freshwater anomalies are further advected to the North Atlantic and influence therefore weakly the deep convection that takes place in the other parts of the GIN Seas. This is supported by the regression of winter mean (DecFeb) sea level pressure (SLP) upon the timeseries of annual mean Fram Strait outflow (Fig. 6a) which is characterized by a maximum over Greenland. This pattern does not promote eastward advection of freshwater anomalies transported at Fram Strait toward the eastern part of the GIN Seas where deep convection takes place. As a consequence, the convection depth anomalies are not directly related to the influence of the freshwater transport at Fram Strait but rather to the atmospheric circulation anomalies as discussed below.

The anomalous winter mean atmospheric circulation associated to one standard deviation change in convection depth features stronger than usual northeasterly winds in the northern part of the GIN Seas and westerly winds in the southern part (Fig. 7a). This brings cold air from the Greenland ice sheet and the West Siberian shelf and induces large heat loss in the icefree regions of the GIN Seas, therefore promoting deep convection there. Furthermore, this atmospheric pattern tends to increase the southward sea ice export through the Fram Strait and explains the positive correlation between sea ice export and mixedlayer depth at small negative lags (Fig. 5a).

In addition, when convection is strong, the GIN Seas are characterized by an anomalous cyclonic circulation in agreement with the windstress forcing and the largescale geostrophic adjustment to density increase associated with convection (not shown). This circulation pattern tends to increase simultaneously the inflow of salty Atlantic waters and the southward liquid freshwater flux at Fram Strait, the sea ice export being rather controlled by local windstress forcing [START_REF] Arfeuille | Simulation of the interannual variability of the winddriven Arctic seaice cover 19581988[END_REF][START_REF] Vinje | Fram Strait ice fluxes and atmospheric circulation 19502000[END_REF][START_REF] Koenigk | Variability of Fram Strait sea ice export: causes, impacts and feedbacks in a coupled climate model[END_REF]. This explains the almost phase opposition between correlations depicted in Fig. 5a.

The relationships described here strongly support the idea that interannual convection depth anomalies in the GIN Seas during the 20 th century result from the anomalous atmospheric cooling of surface waters anomalously salty in agreement with observations. The coarse resolution of the model could have an influence on the representation of the processes described above as they do not allow to accurately represent the lateral interactions between the East Greenland Current (EGC) and the remaining interior of the GIN Seas. Nevertheless, it should be mentioned that our results are consistent with those from other coarseresolution coupled atmosphereocean models (e.g., [START_REF] Goosse | A mechanism of decadal variability of the seaice volume in the Northern Hemisphere[END_REF]. They are also coherent with the conclusions of [START_REF] Mauritzen | Influence of sea ice on the thermohaline circulation in the ArcticNorth Atlantic Ocean[END_REF], who showed, using a coupled iceocean model having a higher resolution that the present one, that the sea ice export at Fram Strait does not significantly influence convection in the GIN Seas.

21 st century

In contrast to the 20 th century, the convection depth anomalies in the GIN Seas during the water formation in the GIN Seas tends to occur a few years after an increase in freshwater import from the North Atlantic. However, a noticeable difference from the 20 th century, is that an increase of the deep water formation in the GIN Seas tends to occur a few years after a decrease of the solid and liquid freshwater export at Fram Strait during the 21 st century. This behaviour might be related to the enhanced freshwater exchanges between the EGC and the interior of the GIN Seas. This idea is supported by the fact that the winter mean atmospheric circulation associated with a stronger than usual Fram Strait volume flux (Fig. 6b) displays westerly winds in the northern part of the EGC. This windstress forcing promotes the lateral exchanges of properties between the EGC and the interior of the GIN Seas and induces an increase in the eastward advection of positive sea ice anomalies transported at Fram Strait. The subsequent melting of those sea ice anomalies further weaken the convection.

Additionally, the maximum freshwater export at Fram Strait occurs 4 years after the maximum of convection depth with a strong correlation of 0.8 (Fig 5b). This indicates that modifications of the freshwater export at Fram Strait during the 21 st century might potentially be influenced by changes in convection in the GIN Seas. This contrasts with the 20 th century during which a positive convection depth anomaly tends to occur after an increase in freshwater export at Fram Strait. Indeed, the winter mean atmospheric circulation associated with strong convection during the 21 st century is characterized by southerly winds along the Norwegian coast and in the BarentsKara Seas sector (Fig. 7b). This anomalous circulation first brings warm air from the North Atlantic to the Arctic Ocean. Second, it leads to larger than usual inflow of warm and salty waters in the eastern part of the GIN Seas and in the BarentsKara Seas area. This could explain the higher correlation between convection depth anomalies and changes in northward inflow of salty waters over the 21 st century compared to the 20 th century (Fig. 5b). In addition to the anomalous northward advection of warm air and warm waters associated with strong convection, the temperature anomalies associated with ocean heat release induced by the convection itself are also transported northward in the BarentsKara Seas sector. All these temperature anomalies then yield a decrease in sea ice cover in the Barents Sea that further warm the overlying atmosphere through the reduced insulating effect of sea ice in winter, the temperaturealbedo feedback being very weak during that period of the year. Note that the changes in oceanic heat transport in the Barents Sea does not appear to be influenced by the largescale AMOC, since the correlation between the maximum of the overturning streamfunction in the North Atlantic and the Barents Sea heat transport does not exceed 0.2 for all lags between 10 and +10 yr. This chain of processes is in agreement with the fact that an increase in surface air temperature (SAT) averaged over the Barents Sea occur after an increase in convection depth in the GIN Seas, with a maximum correlation of 0.8 at lag +4yr. It should be mentioned that it has not been possible to establish a strong relationship between changes in convection and changes in the Barents Sea SAT during the 20 th century. This can be inferred from the atmospheric circulation in Fig. 7a which does not promote an increase in the northward advection of warm air or warm waters in the Barents Sea when the convection is strong.

Let us examine the possible consequences of a positive SAT anomaly centered over the Barents Sea during the 21 st century. The regression of the winter mean SLP anomalies upon the timeseries of the annual mean SAT anomalies averaged over the Barents Sea is characterized by a minimum around Spitzbergen with southward winds at Fram Strait and northeastward winds in the eastern part of the GIN Seas and in the Barents Sea (not shown). This anomalous atmospheric circulation induces more cyclonic conditions in the GIN Seas which induce a stronger than usual mass transport and freshwater export through Fram Strait that prevents deep convection, as suggested by the above correlation and regression analyses (Fig. 5b and7b). An increase in freshwater export at Fram Strait tends therefore to occur after an increase in convection depth in the GIN Seas (Fig 5b). This feature is consistent with the correlation between the annual mean Barents Sea SAT and the annual mean liquid freshwater export at Fram Strait which reaches 0.8 at zero lag over 20212080. This negative SLP anomaly is also associated with southward advection of cold air from the central Arctic that tends to weaken the effect of freshwater on deep convection since salinity effects on deep convection overwhelm those associated with temperature. In addition, this anomalous atmospheric circulation tends to increase the oceanic heat transport in the Barents Sea which further amplifies the initial SAT anomaly in the BarentsKara Seas through direct airsea heat exchanges and intermediate sea ice processes (i.e. insulating effect in winter and temperaturealbedo feedback in summer). Finally, advection of warm air from the North Atlantic into the Barents Sea sector by this anomalous atmospheric circulation amplifies also the initial SAT anomaly.

It should be mentioned that we cannot prove from our study if the northward ocean heat transport in the Barents Sea and the subsequent warming of the the atmosphere are able to influence the SLP around Spitzbergen. However, the winter mean SLP pattern associated to one deviation standard change in Barents Sea SAT bears some resemblance with those obtained by [START_REF] Alexander | The atmospheric response to realistic Arctic sea ice anomalies in an AGCM during winter[END_REF] and [START_REF] Magnusdottir | The effects of North Atlantic SST and Sea ice anomalies on the winter circulation in CCM3. Part I: Main features and storm track characteristics of the response[END_REF] who analyzed the response of the atmospheric circulation to realistic perturbations of SST and sea ice extent using standalone atmospheric general circulation models. Finally, the positive feedback found here between the Barents Sea SAT and the Barents Sea ocean heat transport was also recognized by [START_REF] Goosse | Mechanisms of decadal Arctic climate variability in the Community Climate System Model CCSM2[END_REF] in the second version of the Community Climate System Model (CCSM2) under presentday conditions with no changes in anthropogenic forcing.

To summarize, there is, during the 21 st century a positive feedback between the Barents Sea ocean heat transport and the Barents Sea SAT and a negative one between the convection and the freshwater export at Fram Strait. with the upper salinity anomalies averaged in the Irminger Sea (the western part of the box 3) with a correlation of 0.2 at zero lag. This is in consistent with the correlations between changes in annual mean sea ice and liquid freshwater exports through the Denmark Strait and interannual convection depth anomalies (Fig. 5c). The convection depth reaches a maximum about one year after the max imum sea ice and liquid freshwater exports at Denmark Strait indicating that both solid and liquid freshwater fluxes are not the direct cause of the convection anomalies south of Iceland. This con trasts with the studies of [START_REF] Mauritzen | Influence of sea ice on the thermohaline circulation in the ArcticNorth Atlantic Ocean[END_REF], [START_REF] Hakkinen | A Simulation of thermohaline effects of a great salinity anomaly[END_REF] and [START_REF] Holland | The role of iceocean interactions in the variability of the NorthAtlantic thermohaline circulation[END_REF] who argued that convection in the North Atlantic is very sensitive to ice export through the Denmark Strait and its subsequent melting. However, the convection site in those modelling studies is located close to the southern tip of Greenland, a region which is directly affected by sea ice ex port through the Denmark Strait. By contrast, the deepconvection site located south of Iceland in our model is too far eastward of the EGC to be directly influenced by sea ice export through the Denmark Strait.

Region located south of Iceland

20 th century

Close to the Denmark Strait, the annual mean atmospheric circulation associated with strong convection displays northerlies, as it can be deduced from SLP anomalies displayed in Fig. 8. This is consistent with larger than usual southward sea ice and freshwater transports there. In addition, this atmospheric pattern brings cold air from the central Arctic, which further enhances surface ocean heat loss to the atmosphere and thus promotes deep convection south of Iceland. Note that it was not possible to establish a coherent link between the behaviour of the freshwater fluxes at Den mark Strait when the convection is strong (Fig. 5c) and the regression of the winter mean SLP an omalies upon the timeseries of the annual mean convection depth anomalies (Fig. 7c).

In addition, the convection tends to be deep when the upper levels of the eastern part of the box 3 are anomalously salty, with a correlation of 0.5 at zero lag (Fig. 4c). This is coherent with the correlation between the convection depth anomalies and the changes in freshwater import from the North Atlantic which is strongly negative at zero lag (0.8, Fig. 5c). Furthermore, the correlation between the convection anomalies and the annual variations in the northward mass flux in the east ern part of the box 3 peaks at +0.8 at zero lag. These features are in agreement with the regression of the winter mean SLP anomalies upon the timeseries of the annual mean convection depth anom alies (Fig. 7c). The associated windstress forcing induces a northward advection of salty waters ori ginating from the North Atlantic in the convection site area. Finally, the changes in upper salinity in this region are mainly related to changes in the upper ocean circulation rather than variations in the largescale AMOC. Indeed, the maximum upper salinity averaged in the upper 100 m tends to occur 2 years after a minimum of the AMOC index with a correlation of 0.3.

In summary, those results suggest that the inter-annual variability of convection depth south of Iceland during the 20 th century is related to the anomalous atmospheric cooling of waters anomalously salty originating from the North Atlantic.

21 st century

Similarly to the GIN Seas, years characterized by strong convection south of Iceland during the 21 st century tend to be associated with anomalously salty conditions in the upper levels of the Irminger Sea and of the eastern part of the box 3 (Fig. 4d). The correlations are however much higher than during the 20 th century, suggesting than salinity changes could drive the convection depth anomalies during that period. Figure 5d illustrates the correlations between the maximum mixed layer depth and the freshwater exchanges at the Denmark Strait and with the North Atlantic.

Similarly to the 20 th century, the convection is weak two years after the maximum freshwater import from the North Atlantic with a correlation of 0.6 (Fig. 5d), and 2 years after the minimum northward volume flux. This is perfectly consistent with the winter mean atmospheric circulation associated with strong convection (Fig. 7d) which is characterized by northeasterly winds around the convection area, and the low salinity tongue located slightly northward of the souther limb of the subpolar gyre and extending from off Newfoundland eastward at midlatitudes of the North Atlantic. The large increase in freshwater export from the Arctic Ocean to the North Atlantic through the Canadian Archipelago (Table 1) and the Denmark Strait (Table 2) combined with the AMOC weakening and the enhanced hydrological cycle are likely to contribute to the formation of this low salinity tongue of about 30 psu during the 21 st century. These features support thus the idea that the formation of positive convection depth anomalies south of Iceland during the 21 st century results from the weaker than usual northward advection of fresh waters originating from the northwestern part of the North Atlantic.

It should be recalled at this stage that all the variables have been linearly detrended before performing the correlation analyses so that the results do not depend on the longterm climate change experienced over the 21 st century. The different relationships between interannual anomalies of convection and freshwater fluxes between the 20 th and 21 st centuries rather suggest that different mechanisms are involved. Finally, note that the spatial patterns of the correlation maps are similar to those of the regression maps in the GIN Seas and south of Iceland over the 20 th century as well as over the 21 st century (not shown). The correlations do not exceed 0.4 in most of the cases but are statistically significant at the 95% confidence level.

Processes influencing the longterm changes in the Fram Strait outflow

The export of sea ice and liquid freshwater through the Fram Strait strongly affects the deep water formation in the Labrador Sea and the circulation of the North Atlantic Ocean. It is therefore crucial to determine the mechanisms responsible for its longterm changes, and in particular in response to anthropogenic gas forcing. In this section, we specifically focus our analysis on the mechanisms influencing the longterm changes in liquid freshwater export at Fram Strait.

Figure 9 shows the timeseries of the liquid freshwater and volume fluxes at Fram Strait during the 20 th and 21 st centuries. An interesting feature seen in these timeseries is the rapid strengthening of the volume and freshwater fluxes between 1990 and 2010. Such an abrupt shift is not reproduced in a control simulation with preindustrial conditions (Fig. 9). This means that the anthropogenic forcing is responsible for this significant change during that period. Furthermore, the volume and freshwater fluxes appear well correlated with a correlation coefficient up to 0.9 over 19011980 and 20212080 (all fields linearly detrended), suggesting that changes in the Fram Strait volume flux rather than changes in salinity have a significant influence on modifications of the liquid freshwater export at Fram Strait. We therefore investigate herein the mechanisms influencing the Fram Strait outflow rather than on those influencing the freshwater export.

Strengthening of the Fram Strait outflow

The large shift of the Fram Strait outflow simulated between 1990 and 2010 can be thought to be initiated by the reduction of sea ice cover in the Barents Sea which appears as a consequence of the longterm warming of the Arctic climate. The reduction of the annual mean sea ice cover in the Barents Sea over the late 20 th century amplify the warming of the overlying atmosphere in this region mainly in winter through the reduced insulating effect of sea ice during that period. This yields a decrease of the mean SLP in winter around Spitzbergen that causes more cyclonic conditions in the GIN Seas. As a result, the Fram Strait outflow increases as well as the Barents Sea ocean heat transport which ultimately amplify the initial sea ice reduction. The increase in the ocean heat transport reaches 0.02 PW in annual mean between 19011980 and 20212080 (0.025 PW change in winter) and is similar to those obtained in several models poleward of 70°N in response to a doubling of the CO 2 [START_REF] Holland | Polar amplification of climate change in coupled models[END_REF]. In our model, the changes in Arctic surface air temperature in the Barents Sea between 19011980 and 20212080 are the largest over the Barents Sea with values up to 15°C in winter (9°C in annual mean, not shown). This regional amplification of the warming is consistent with the above positive feedback and the enhanced northward ocean heat transport in the Barents Sea. Finally, it should be noted that no similar shifts has been observed during the late 20 th century although the variability of the exchanges between the North Atlantic and the GIN Seas through the Barents Sea has been large during the last decades.

Weakening of the Fram Strait outflow

Around year 2080, the model predicts a second transition threshold beyond which the Fram Strait outflow rapidly weakens (Fig. 9). There is indeed a competition between the positive feedback described above, that is responsible for the stability of state in which the Fram Strait outflow is large, and the geostrophic adjustment to the freshening in the GIN Seas which tends to slow down the cyclonic circulation in the GIN Seas and thus the Fram Strait outflow. In agreement with the decrease of this latter, a strong halocline develops in the central Arctic during the course of the 21 st century with an averaged annual mean SSS anomaly of almost 3 psu between 20802100 and 19011980. The slowing down of the cyclonic circulation in the GIN Seas is also associated with a reduced northward ocean heat transport in the Barents Sea. This contributes to an expansion of sea ice cover in this region as well as to a decrease of the SAT there. Averaged over the Barents Sea area, the surface air cooling reaches 5°C between 20802100 and 20212080, much higher than over the GIN Seas (1°C, not shown). Associated to this strong regional cooling is an increase of the winter mean SLP over the GIN Seas (6 hPa change) that amplifies the initial weakening of the cyclonic circulation in the GIN Seas caused by the longterm freshening.

There is therefore apparently a particular threshold beyond which the influence of the positive feedback between the Barents Sea SAT and the Barents Sea ocean heat transport is not sufficiently strong to sustain the Fram Strait outflow in a somewhat strong state against the freshening of the GIN Seas. Those results more or less suggest that there are two stable oscillatory states for Fram Strait outflow in the model because of those positive and negative feedbacks.

Furthermore, the coarse resolution of our model does not allow to represent all currents, so that we reproduce only strong and weak modes, while in reality, the shape of the transition between these stable states could be more continuous.

Although those changes do not indicate causal relationships, they suggest the possibility that the longterm freshening in the GIN Seas could lead to strong regional cooling in a warming climate, through reorganization of horizontal ocean currents and heat transports in agreement with Russell and Rind (1999) and [START_REF] Schaeffer | Intrinsic limits to predictability of abrupt regional climate change in IPCC SRES scenarios[END_REF][START_REF] Schaeffer | The influence of ocean convection patterns on highlatitude climate projections[END_REF]. [START_REF] Schaeffer | The influence of ocean convection patterns on highlatitude climate projections[END_REF]) obtained a regional cooling south of Spitzbergen between 2060 and 2100 in response to the SRES A1B scenario. The area delineated by the decrease in SAT in their model coincides with the pattern of change in convection depth and expansion of sea ice. In our model however, the pattern of change in SAT is centered over the Barents Sea, and coincides clearly with the area of expansion of sea ice, the convection changes being located southward in the interior of the GIN Seas. This indicates that the regional cooling in our model is rather ascribed to the insulating effect of sea ice rather than to a reduced ocean heat loss induced by a collapse of convection.

Conclusions

Outputs from IPSLCM4 have been used to investigate the processes underlying the long term changes of the freshwater balance of the highlatitude ocean in the Northern Hemisphere and the mechanisms influencing the formation of interannual convection depth anomalies over the 20 th and 21 st centuries. Additionally, the processes influencing the longterm evolution of the Fram Strait outflow have been examined. The analysis has been performed on the basis of simulations performed for the IPCC AR4, namely the experiment covering the 20 th century, which includes only anthropogenic forcings, and the SRES A1B scenario over the 21 st century.

In the central Arctic, the model predicts that changes in continental runoff dominate the longterm freshening over the 21 st century, followed by increase in sea ice melting and increase in precipitation less evaporation. In the convection sites, localized in the GIN Seas and south of Iceland in our model, changes in oceanic freshwater fluxes rather than changes in PE+R are seen to dominate the longterm change of the freshwater balance. We found that the Fram Strait outflow, which represents the main source of freshwater for the Nordic Seas, undergoes a large shift over the late 20 th century. We suggest that this sudden transition is caused by a regional positive feedback in the atmospheresea iceocean system initiated by the retreat of sea ice cover in the Barents Sea as a result of the longterm warming of the Arctic climate. As the climate warms, the sea ice cover reduces in the Barents Sea. This amplifies the warming of the overlying atmosphere there which further increases the SLP over the GIN Seas. As a result the cyclonic ocean circulation in this region strengthens as well as the Fram Strait outflow and the northward ocean heat transport in the Barents Sea that ultimately amplifies the initial sea ice reduction. Over the late 21 st century, the model predicts a second transition threshold beyond which the Fram Strait outflow rapidly weakens. The longterm freshening in the GIN Seas, which affects directly the Fram Strait outflow and indirectly the magnitude of the overall abovementioned positive feedback through its influence on the Barents Sea ocean heat transport, is pointed out as the main contender to explain this large shift.

Additionally, the mechanisms influencing the interannual changes in convection depth have been examined over the 20 th and 21 st centuries on the basis of correlation and regression analyses of detrended variables. This not allows to demonstrate causal effects or to assess the relative contributions of various processes in a particular phenomenon. Additional sensitivity experiments will thus be needed to illustrate the processes addressed herein. However, we were able to show that the mechanisms of interannual changes in deep convection in the GIN Seas and in the region located South to Iceland differ fundamentally between the 20 th and 21 st century. This difference is caused by the strong influence of the interannual changes in oceanic freshwater fluxes on the deep mixing during the 21 st century. In particular, in the GIN Seas, during the 20 th century, positive convection depth anomalies are related to atmospheric cooling of warm and salty waters originating from the North Atlantic in agreement with observations, whereas during the 21 st century, modifications in the southward freshwater export through Fram Strait combined with changes in northward inflow of salty waters from the North Atlantic are seen to play a major role. Conversely, the convection depth anomalies in the GIN Seas during the 21 st century influence the Fram Strait outflow through alterations of the sea level pressure around Spitzbergen. In the other convection site simulated by the model, south of Iceland, interannual convection depth anomalies during the 20 th century are mostly related to atmospheric cooling as in the GIN Seas, whereas during the 21 st century, modifications of the northward advection of fresh waters originating from the northwestern part of the North Atlantic have a significant influence.

It should be stressed that our analysis is based on a particular model using a coarse resolution which, as all climate models, shows several biases for presentday climate conditions.

For instance, the "presentday" oceanic convection pattern simulated by our model is somewhat different from the one observed, with no deepwater production in the Labrador Sea. This bias could thus have an impact on the reliability of the Arctic climate change simulated by the model during the late 21 st century. For example, [START_REF] Schaeffer | The influence of ocean convection patterns on highlatitude climate projections[END_REF] showed that the location of "presentday" convection sites in the ocean is crucial for the probability and strength of a regional cooling signal under global warming. In addition, our model does not include a comprehensive Greenland ice sheet model that has yet been shown to have a significant influence on the climate of the late 21 st century [START_REF] Fichefet | Implications of changes in freshwater flux from the Greenland ice sheet for the climate of the 21 st century[END_REF]. Despite all the abovementioned limitations and biases, several aspects of our results show consistency with previous published results obtained with different coupled models. This study suggests that the importance of oceanic freshwater fluxes in controlling the NADW formation at interannual time scales could significantly increase during the 21 st century in response to increasing greenhouse gases, and that rapid climate transitions in the Arctic system could be trigger by moderate changes in the sea ice cover in the Barents Sea, an issue that must be kept in mind when analysing model results and observations in that region. [START_REF] Rayner | Global analyses of sea surface temperature, sea ice, and night marine air temperature since the late nineteenth century[END_REF] and shown as a thick solid line.

Figures captions

Figure 3. Geometry of the three boxes chosen for the analysis. These include the central Arctic (box 1), the GreenlandIcelandNorwegian Seas (box 2) and the region located to the south of Iceland (box 3). These two latter enclose the region where the convection sites are located over the 20 th century (see Fig. 1). Figure 9. The timeseries of (top) Fram Strait outflow and (bottom) liquid freshwater export at Fram Strait. A 5year running mean has been performed. The thin line denotes the timeseries associated to the preindustrial experiment, while the thick solid line denotes the timeseries associated to the experiment covering the 20 th century with anthropogenic forcings and the SRES A1B scenario over the 21 st century. [START_REF] Aagaard | The role of sea ice and other fresh water in the Arctic circulation[END_REF] and Woodgate and Aagaard (2005).

List of tables

Table 2. Annual mean freshwater budget in the GIN Seas for the ocean averaged over the period 19512000 in the model and observations, and over 20512100 of the SRES A1B experiment (km 3 /yr). Corresponding change between 19512000 and 20512100 is indicated in the right column. Freshwater fluxes are relative to the reference salinity 34.8 psu. Positive values are for sources and negative ones for sinks. Acronym AC89 state for [START_REF] Aagaard | The role of sea ice and other fresh water in the Arctic circulation[END_REF].

Table 3. Annual mean freshwater budget in the region south of Iceland for the ocean (see Figure 3) averaged over the period 19512000 in the model and available observations, and over 20512100 of the SRES A1B experiment (km 3 /yr). Corresponding change between 19512000 and 20512100 is indicated in the right column. Freshwater fluxes are relative to the reference salinity 34.8 psu. Saline waters are referred to salinities greater than 34.8 psu (i.e. the reference salinity). Positive values are for sources and negative ones for sinks. Acronym AC89 state for [START_REF] Aagaard | The role of sea ice and other fresh water in the Arctic circulation[END_REF]. Table 4. Annual mean sea ice (solid) freshwater flux in the Arctic Ocean and in observations, averaged over 19512000 and 20512100 in the model (km 3 /yr). Corresponding change between 19512000 and 20512100 is indicated in the right column. Negative sign indicates a southward sea ice export. AC89 states for Aagaard and Carmarck (1989) and WA05 states for Woodgate and Aagaard (2005). The freshwater flux are computed assuming a mean salinity of ice of 4 psu and a reference salinity of 34.8 psu.

  21 st century are strongly positively correlated (0.8) at lag 1 yr (convection lags) with the upper salinity anomalies averaged over the EGC (Fig 4b), and at zero lag with the upper salinity averaged in the eastern part of the GIN Seas. This indicates that, for this period, salinity anomalies transported in the EGC Seas could potentially induce changes in deepwater mass formation or amplify changes caused by other mechanisms. The correlations between convection depth anomalies and annual mean freshwater flux anomalies at Fram Strait and between Norway and Iceland over 20212080 are illustrated in Figure 5b. Similarly to the 20 th century, a decrease of deep

Fig

  Fig 4c shows that the convection depth anomalies south of Iceland are negatively correlated

Figure 1 .

 1 Figure 1. Simulated mixed layer depth in March averaged over 19512000 (m). Contour interval is 100 m.

Figure 2 .

 2 Figure2. Sea ice concentrations in March (left) and September (right) simulated by IPSLCM4 averaged over 19512000. Contour interval is 0.1. The observed sea ice edge, defined as the 15% concentration for March and September is derived from the HadISST dataset[START_REF] Rayner | Global analyses of sea surface temperature, sea ice, and night marine air temperature since the late nineteenth century[END_REF] and shown as a thick solid line.

Figure 4 .

 4 Figure 4. Correlations between the March mixedlayer depth anomalies and the annual mean salinity anomalies averaged in the upper 100 m in the GIN Seas as a function of lag over 19011980 (a) and 20212080 (b). Solid contours indicate the correlation between mixed layer depth and upper salinity averaged in the eastern part of GIN Seas which correspond to the areas where deep convection takes place. Dashed contours indicate the correlation between mixed layer depth and upper salinity averaged in the East Greenland Current (ab) and in the Irminger Sea (the western part of the box 3, cd). A linear detrend and a 5yr running mean have been performed. Mixed layer depth anomaly lags (leads) the salinity anomaly for negative (positive) lags.

Figure 5 .

 5 Figure 5. (ab) Correlation between March convection depth anomalies in the GIN Seas and annual mean liquid freshwater flux through Fram Strait (solid), annual mean sea ice transport through Fram Strait (dashed) and liquid freshwater import from North Atlantic (solid dotted) as a function of lag over 19011980 (a) and 20212080 (b). (cd) Correlation between March convection depth anomalies South of Iceland and annual mean liquid freshwater flux through Denmark Strait (solid), annual mean sea ice transport through Denmark Strait (dashed) and liquid freshwater import from the North Atlantic (solid dotted) as a function of lag over 19011980 (c) and 20212080 (d). A linear detrend and a 5yr running mean have been performed before the analysis. Mixed layer depth anomaly lags (leads) the anomalous freshwater fluxes for negative (positive) lags.

Figure 6 .

 6 Figure 6. Winter mean (DJF) SLP (hPa) regressed upon the timeseries of the annual mean Fram Strait transport over 19011980 (a) and 20212080 (b) at zero lag. Shown is the change of SLP related to one standard deviation change in annual mean Fram Strait transport. Contour interval is 0.2 hPa. A linear detrend has been perfomed for SLP and Fram Strait transport. Negative values are dashed.

Figure 7 .

 7 Figure 7. Winter mean (DJF) SLP (hPa) regressed upon the timeseries of the March mixedlayer depth in the GIN Seas and in the region South of Iceland over 19011980 and 20212080. Shown is the change of SLP related to one standard deviation change in March mixedlayer depth. Contour interval is 0.2 hPa. A linear detrend has been performed for SLP and mixedlayer depth. Negative values are dashed.

Figure 8 .

 8 Figure 8. Annual mean SLP (hPa) regressed upon the timeseries of the March mixedlayer depth in the region South of Iceland over 19011980. Shown is the change of SLP related to one standard

Figure 1 .

 1 Figure 1. Simulated mixed layer depth in March averaged over 19512000 (m). Contour interval is 100 m.

Figure 2 .

 2 Figure 2. Sea ice concentrations in March (left) and September (right) simulated by IPSLCM4 averaged over 1951 2000. Contour interval is 0.1. The observed sea ice edge, defined as the 15% concentration for March and September is derived from the HadISST dataset[START_REF] Rayner | Global analyses of sea surface temperature, sea ice, and night marine air temperature since the late nineteenth century[END_REF] and shown as a thick solid line.

Figure 3 .Figure 4 .

 34 Figure3. Geometry of the three boxes chosen for the analysis. These include the central Arctic (box 1), the Greenland IcelandNorwegian Seas (box 2) and the region located to the south of Iceland (box 3). These two latter enclose the region where the convection sites are located over the 20 th century (see Fig.1).

Figure 5 .

 5 Figure 5. (ab) Correlation between March convection depth anomalies in the GIN Seas and annual mean liquid freshwater flux through Fram Strait (solid), annual mean sea ice transport through Fram Strait (dashed) and liquid freshwater import from North Atlantic (solid dotted) as a function lag over 19011980 (a) and 20212080 (b). (cd) Correlation between March convection depth anomalies South of Iceland and annual mean liquid freshwater flux through Denmark Strait (solid), annual mean sea ice transport through Denmark Strait (dashed) and liquid freshwater import from the North Atlantic (solid dotted) as a function of lag over 19011980 (c) and 20212080 (d). A linear detrend and a 5yr running mean have been performed before the analysis. Mixed layer depth anomaly lags (leads) the anomalous freshwater fluxes for negative (positive) lags.

Figure 6 .

 6 Figure 6. Winter mean (DJF) SLP (hPa) regressed upon the timeseries of the annual mean Fram Strait transport over 19011980 (a) and 20212080 (b) at zero lag. Shown is the change of SLP related to one standard deviation change in annual mean Fram Strait transport. Contour interval is 0.2 hPa. A linear detrend has been perfomed for SLP and Fram Strait transport. Negative values are dashed.

Figure 7 .

 7 Figure 7. Winter mean (DJF) SLP (hPa) regressed upon the timeseries of the March mixedlayer depth in the GIN Seas and in the region South of Iceland over 19011980 and 20212080. Shown is the change of SLP related to one standard deviation change in March mixedlayer depth. Contour interval is 0.2 hPa. A linear detrend has been performed for SLP and mixedlayer depth. Negative values are dashed.

Figure 8 .

 8 Figure 8. Annual mean SLP (hPa) regressed upon the timeseries of the March mixedlayer depth in the region South of Iceland over 19011980. Shown is the change of SLP related to one standard deviation change in March mixedlayer depth. Contour interval is 0.2 hPa. A linear detrend has been performed for SLP and mixedlayer depth. Negative values are dashed.

Figure 9 .

 9 Figure 9. The timeseries of (top) Fram Strait outflow and (bottom) liquid freshwater export at Fram Strait. A 5year running mean has been performed. The thin line denotes the timeseries associated to the preindustrial experiment, while the thick solid line denotes the timeseries associated to the experiment covering the 20 th century with anthropogenic forcings and the SRES A1B scenario over the 21 st century.

Table 1 .

 1 Annual mean freshwater budget in the central Arctic Ocean averaged over the period 2051 2100 of the SRES A1B experiment (km 3 /yr). Corresponding change between 19512000 and 2051 2100 is indicated in the right column. Freshwater fluxes are relative to the reference salinity 34.8 psu. Positive values are for sources and negative ones for sinks. Acronyms AC89 and WA05 state for

Acknowledgments. H. Goosse is Research Associate with the Belgian National Fund for Scientific Research. This work was conducted within the European project ENSEMBLES (ENSEMBLEbased Predictions of Climate Changes and their Impacts) and the Action Concertée Incitative Changement Climatique "Changement Climatique et Cryosphère" funded by the French Ministry of Research.


M Duval

J.-C Passieux
email: passieux@insa-toulouse.fr

M Salaün

S Guinard

NON-INTRUSIVE MODEL COUPLING: A FLEXIBLE WAY TO HANDLE LOCAL GEOMETRIC AND MECHANICAL DETAILS IN FEA

Keywords: model coupling, multiscale, FEA, parallel computing, crack growth simulation

Computer Aided Engineering (CAE) often involves structural mechanics analysis (most of the time using the finite element method). When dealing with nonlinear complex models on large 3D structures, the computational cost becomes prohibitive. In this paper, we present the recent developments linked to an innovative computing method: nonintrusive coupling. Such a method allows to efficiently taking into account local modifications on an initial existing model in a non-intrusive way: the previously computed analysis is left unchanged. Large scale linear models can thus be easily computed, then localised nonlinear complex models can be used to pinpoint the analysis where required on the structure. After a presentation of the scientific context and a description of non-intrusive coupling methods, we will present its application to crack growth simulation and parallel structure analysis.

1-Context and industrial issues

Every time one wants to create a new mechanical object, a design cycle has to be respected, involving Computer Aided Design (CAD) and Finite Element Analysis (FEA). When dealing with a critical part of a mechanical product, a particular attention should be paid to the FEAand most of the time, it involves complex simulations. This is the point we will focus on in this paper. Indeed, in order to perform a mechanical simulation, one needs both a geometrical and a mechanical model. Some cases require a complex mechanical model, and some other a detailed geometry.

The works which will be presented in this paper try to bring a solution to a recurrent FEA problem: how to compute a complex mechanical problem in a way requiring the least effort? There are two main problems which need to be addressed: geometric complexity and mechanical behavior complexity.

-Geometric modifications in FEA

The most direct way to perform a structural analysis via numerical simulation is to use the CAD model as the geometric model for the simulation. Indeed, the creation of a structure geometric model can be very time consuming. Moreover, for some critical parts, the mesh has to be certified before being used (aeronautical parts for instance). All in all, the fact is that we cannot afford to create a new geometric model or a new mesh each time we need to perform a simulation. Moreover, it appears that in the life cycle of a product, its geometric specifications can change (a crack can appear, some holes can be drilled). One objective of the computing method presented here is to bring a way to reuse an initial geometric model and its finite element mesh to compute structural analysis involving local details.

-Localised complex mechanical behaviour

Another class of complex problems involves mechanical behaviour of the structure we need to analyse. Indeed, in the field of structural analysis, two types of models have to be considered: linear and nonlinear ones. Of course, the time required to complete the analysis will depend on the type of model. The use of certain complex nonlinear models on very large structures fatally leads to considerable cost in terms of computer resources, often beyond what is currently available. The fact is that most of time, a nonlinear model is useful only on a small part of the structure (e.g. localised plasticity), which can be represented elsewhere with a linear model.

Again it is possible to save a lot of computation time by reusing an existing linear model (and the corresponding solution) on a full structure: a nonlinear model will thus be considered only on localised areas (see [AG1]). The objective of non-intrusive coupling is here to make us able to merge several local complex nonlinear models with a global linear one, without actually modifying this last one. In other words, if we need to perform several analysis of a large scale structure, the linear model will be assembled on the full structure only once, whereas the localised analysis will be performed as many times as necessary.

2-Model coupling method and algorithm

As said previously, the non-intrusive coupling algorithm (see [START_REF] Gendre | Nonintrusive and exact global/local techniques for structural problems with local plasticity[END_REF]) aims to perform a structural analysis using two separate models:

-a pre-existing simple model involving the full structure, which will represent the global mechanical behaviour (linear elasticity), -an ad-hoc complex model involving only a small part of the structure, representing the local mechanical behaviour (plasticity for example).

-Coupling algorithm

In this paper, a two scale finite element method is considered. We will denote with the letter the geometric domains and with the letter the mechanical behaviour models. Two coupled overlapping models are considered (see Fig. 1): a global one (involving the full structure , i.e. including a large number of nodes) and a local one (involving only ). Thus the global model will be treated as a coarse linear one, whereas the local one will take into account the localised (potentially nonlinear) behaviour. Basically, the idea is to reach the equilibrium between the global and the local models at the interface by the mean of an iterative algorithm, similar to those used in domain decomposition methods, i.e. solving alternately the Dirichlet (resp. Neumann) problem on the local (resp. global) model until convergence (see Fig. 2). Actually, we seek to replace the global solution from (for example a non-cracked domain) on by the one we would get with (for example a cracked domain).

Let us consider a domain decomposition in the linear case involving and : we then get a monolithic coupling system (see Eq. 1).

(1) Here and stands for the stiffness matrices, for the load vector, and for the displacement fields, and for the interface coupling matrices and for the Lagrange multipliers vector. An iterative algorithm is then set up in order to dissociate the two models when solving the linear system (see Eq. 2).

Here, the non-intrusiveness of the method comes from a fictitious prolongation of the solution from to the full global domain (see Eq. 3 and Eq. 4). We then define so that and .

(3)

Actually, the iterative algorithm tends to replace the global stiffness on by the local one, through an additional right hand side load vector.

(4)

In a few words, the global and the local models are coupled via displacement and effort swap at the interface. The computational cost involved by the algorithm can be significantly smaller than the one involved by a full scale nonlinear complex computation. Still, it may be noted that, as such, the performance of the method is on the stiffness gap between the two models and . Indeed, the more the stiffness gap is important, the more the algorithm will require a large number of iterations to converge. This is very inconvenient for crack propagation simulation: the stiffness gap increases as the crack growths. Fortunately, a Quasi-Newton correction (see [START_REF] Gendre | Nonintrusive and exact global/local techniques for structural problems with local plasticity[END_REF]) allows to get rid of that problem: thanks to the Sherman-Morrison-Woodbury formula, it is possible to modify the tangent stiffness of the global model in a non-intrusive way (see [AG1]).

-Connections with standard methods

For several years, model coupling is used in nearly all the engineering departments. One can cite two main classes of method:

-First, when dealing with large mechanical structures, one often needs a very precise analysis on localised small parts.

As said before, we cannot afford to use a fine mesh and a precise model on the whole structure. Instead, structural zooming is often a solution (see [START_REF] Daridon | Changement d'échelles et zoom structural[END_REF]).

It consists in computing a precise solution on a small area of a structure, using the pre-computed coarse solution on the full structure as boundary conditions. No iteration is applied in such methods.

-Then, one can cite domain decomposition method. These methods allow to efficiently couple models and meshes; nevertheless it requires significant efforts in order to make interconnections between the models (see e.g. [BR1], [START_REF] Brezzi | The three-field formulation for elasticity problems[END_REF] and [START_REF] Farhat | A method of finite element tearing and interconnecting and its parallel solution algorithm[END_REF]). Moreover, no pre-computed solution can be reused, resulting on very large computing resource needs.

In fact, the non-intrusive coupling algorithm provides a generic method which allows coupling several models with the least effort, while preserving the inherent advantages of the methods presented above:

-The global pre-existing model is unmodified.

-Incompatible meshes can be interconnected via a mortarlike method, for example.

-Parallel resolution can reduce the computation time in case of multiple local models.

3-Application to crack growth simulation

The main application we focus on in this paper is crack growth simulation. For a lot of engineers, sustainability in construction (aeronautical, naval) is a priority. Indeed, during the life cycle of a mechanical structure (steel, concrete), cracks can appear, endangering the integrity of the structure. Thus, forecasting the propagation path of such cracks is a major issue for engineers.

Nevertheless cracks locations cannot be known a priori when designing a structure or setting up a finite element mesh for initial structural analysis.

Using the common FEA tools, if one needs to simulate crack propagation, two main solutions are available:

-Set up a crack conforming mesh (see [START_REF] Bittencourt | Quasi-automatic simulation of crack propagation for 2D LEFM problems[END_REF], [START_REF] Chiaruttini | Advanced remeshing techniques for complex 3D crack propagation[END_REF]) at each step of the propagation. When combined to an adapted mesh refinement at crack tip, it leads to accurate results. The main drawback of conforming meshing is the substantial computational resulting cost. For that reason, direct crack meshing is rarely used as such, unless a very fast and efficient remeshing algorithm is set up.

-Use X-FEM method (see [START_REF] Moës | A finite element method for crack growth without remeshing[END_REF]) on an existing mesh. Theoretically, such a method allows the mesh to be not conforming to the crack faces. Nevertheless, most of time, remeshing is necessary at crack tip if one wants to get an accurate enough solution.

All in all, common methods do not allow reuse of existing meshes without (at least local) modifications, resulting in an extra computational cost. Here, the idea is to consider two different models, standing for different scales: a global linear elastic model, representing the full structure (healthy structure) and a local (potentially nonlinear and/or XFEM) model for the cracked domain (see Fig. 3). Using the non-intrusive algorithm for crack propagation simulation (see [START_REF] Gupta | Analysis of three dimensional fracture mechanics problems: A non-intrusive approach using a generalized finite element method[END_REF], [START_REF] Passieux | Local/global non-intrusive crack propagation simulation using a multigrid X-FEM solver[END_REF]) will provide the following advantages: -Remeshing will be necessary only on the local model.

-Nonlinear behaviour will be used only on the local model.

-The global linear model will be assembled once and the stiffness matrix will be factorised only once too. Altogether, the non-intrusive coupling algorithm allows reusing a pre-existing mesh and linear elastic model (i.e. stiffness matrix) in order to perform computationally cheap crack growth simulation.

The results presented in this paper have been computed using Code_Aster, a structural engineering software developed by Électricité de France (see [START_REF]Électricité de France[END_REF]). Both global and local models have been computed as a black box using this software, whereas the coupling (i.e. interface data exchange) has been done using a Python code developed for that purpose.

4-Parallel computing, distributed micro models

The two main features of the non-intrusive coupling are: "non-intrusive" which is the possibility to modify an initial model, and "coupling" which is the possibility to use different models to compute a single structure. This last feature will be developed now more in details. Indeed, it is possible to consider several non-overlapping local models (for example if we want to represent several cracks on the same structure). Each local model will be completely independent from the others, allowing for an efficient parallel solver to be set up, using for instance MPI communications (see Fig. 4). When dealing with nonlinear local models, parallelism is often the only possibility making FEA possible. We want to draw the reader's attention here on an important detail: unlike domain decomposition methods (e.g. FETI method, see [START_REF] Farhat | A method of finite element tearing and interconnecting and its parallel solution algorithm[END_REF]), each local analysis can be carried out independently from the others. An application of this property is assemblies analysis. Indeed, for very large structures, simulation of assemblies is computationally very expensive. Moreover, if the model contains too much contact areas, it becomes difficult to make the analysis possible because of the high complexity of the nonlinear behaviour. Thanks to non-intrusive coupling, it is possible to compute each junction assembly separately: common contact algorithms will be able to perform the computation easily.

We present here another application of parallel computing applied to non-intrusive coupling: a multi-cracked plate (see Fig. 5). We consider here, as an academic test case, three disjoint cracks. From a global coarse mesh, we generate three local refined patches on which an X-FEM model is applied. For a given iteration of the coupling algorithm, the three local models are computed in parallel. The important point here is that there is no direct communication between the local patches. Every interface data exchange (displacement and effort) takes place between the global and one local model, through a coupling engine. It may also be noted that, in that example, no load is applied to the global model; we constrained only three degrees of freedom in order to disable rigid body motion. The only load applied is a hydrostatic pressure on the crack lips. Then, the local loading spreads to the global model only through the additional global right hand side load vector. Moreover, the stiffness gap between the global and the local models is very important here, as we considered three cracks (two of which being emerging). Still, the Quasi-Newton method allows for an important speed-up (see Fig. 6), the number of iteration dropping from about three hundreds to twelve.

5-Integration of research codes into commercial softwares

Finally, the last interest of non-intrusive coupling we will develop is the possibility to easily merge research codes and commercial softwares. Indeed, as presented previously, the only data exchange occurring between the global and the local models is interface displacements and forces. Thus it is possible to compute the global solution from a commercial software using the existing models solvers, and compute the local solution using ad-hoc model developed with any code. Using MPI communication between the two codes (i.e. between the two models, global and local) makes the communication straightforward: we only have to focus about the data we need to exchange, as MPI will provide his own standard for language compatibility (see Fig. 7).

Figure 7: Commercial software -Research code integration

For instance we can cite [START_REF] Gendre | Nonintrusive and exact global/local techniques for structural problems with local plasticity[END_REF] who compute an aeronautical structure analysis with localised plasticity within a linear model from Abaqus/Standard. If we direct our interest on crack growth simulation, we can cite [START_REF] Chahine | Spider-xfem, an extended finite element variant for partially unknown crack-tip displacement[END_REF] and [START_REF] Passieux | Local/global non-intrusive crack propagation simulation using a multigrid X-FEM solver[END_REF] who propose a special treatment for crack tip displacement singular field (analytical solution, adapted radiating mesh) within an elastic linear model.

6-Patch definition and extend: caution

At this point, one could legitimately ask oneself on the manner the patches are defined. In fact, there is no constrain nor generic rule about the way to define the patch extend. In the crack propagation example, we simply select a given number of stitch layers from the global mesh around the crack location. Then those stitches are duplicated and saved as local mesh (see Fig. 3). Any refinement of the freshly generated local mesh is possible, particularly at crack tip. For instance, let us consider a cracked bending plate. From a practical point of view, one can extend the patch as far as desired (see Fig. 8). Nevertheless, the patch extend is not without consequence on the algorithm convergence properties. Indeed, as said previously, the convergence speed depends on the stiffness gap between the local and the global model. In the treated example, the global model stands for a non-cracked plate whereas the local model stands for the cracked one. According to the Saint-Venant principle, the crack influence will decrease when getting far from the perturbation (the crack). Thus, the more the patch extends far from the crack, the less the stiffness gap between the global and the local model will be important, allowing for faster convergence. We plotted on Fig. 9 the number of iteration required to reach the fixed tolerance ( ) as a function of the patch thickness (i.e. the number of global stitches from which the patch has been generated). As predicted, the more the patch extends, the faster the algorithm converges. Without any acceleration, one can nearly divide the required number of iteration by ten, thanks to the patch extend. When using the Quasi-Newton acceleration, the same result stands for true (the number of iterations is divided only by two extending the patch from one to ten layers). Still, the more the patch extends, the more it will be computationally expensive to as it will involve a larger number degrees of freedom. Then engineers' skills must prevail in order to determine the best choice of parameters in such situations.

7-Conclusion and perspectives

In this paper, a non-intrusive coupling approach has been presented. This method allows to take into account local features in existing FEA models without actually modifying it. The main purpose of the method is to make FEA easier, as finite element models preparation can sometimes be more time consuming than the computation itself.

Thanks to this algorithm, we have been able to set up a two and three dimensional crack growth simulation, using Code_Aster for the mechanical computations and MPI based communications for the interface coupling. All of the process is wrapped into a Python API. The distributed implementation of the algorithm we proposed here allows for high performance multi-patch parallel computations. In a near future, we seek to make the method even more flexible by extending it to non-coincident patches (see [START_REF] Lozinski | Numerical Zoom for localized multiscales[END_REF]).

It may be noted that the algorithm can also be used to couple different mechanical representations (e.g. 2D/3D coupling, see [START_REF] Guguin | Techniques de raccord 2D-3D pour l'analyse non-intrusive de structures composites stratifiées[END_REF]) or to couple different analysis methods (deterministic/stochastic models, see [START_REF] Chevreuil | A multiscale method with patch for the solution of stochastic partial differential equations with localized uncertainties[END_REF]) in a non-intrusive way.

This work is supported by the French National Research Agency (Grant ANR-12-MONU-0002 ICARE).

8-References

Figure 1 :

 1 Figure 1: Non-intrusive coupling -Situation overview.

Figure 2 :

 2 Figure 2: Iterative algorithm -Interface data exchange.

Figure 3 :

 3 Figure 3: Non-intrusive FEM/XFEM coupling.

Figure 4 :

 4 Figure 4: Non-intrusive parallel solver.

Figure 5 :

 5 Figure 5: Multi-crack distributed patches.

Figure 6 :

 6 Figure 6: Convergence of the algorithm.

Figure 8 :

 8 Figure 8: Cracked bending plate -Patch extend (1, 5 and 10 layers)

Figure 9 :

 9 Figure 9: Patch extend -Convergence properties

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  AG1] M. A. Akgün, J. H. Garcelon, R. T. Haftka. Fast exact linear and non-linear structural reanalysis and the Sherman-Morrison-Woodbury formulas. International Journal for Numerical Methods in Engineering, 50:1587-1606, 2001. [BR1] H. Ben Dhia, G. Rateau. The Arlequin method as a flexible engineering design tool. International Journal for Numerical Methods in Engineering, 62(11):1442-1462, 2005.