S Somot

F Sevault

M Deque

M Crepon

M Déqué

M Crépon

st Century Climate Change Scenario for the Mediterranean using a coupled Atmosphere-Ocean Regional Climate Model

Keywords: Climate change, regional climate modelling, air-sea coupling, Europe, Mediterranean, coupled model 2-meter temperature, DJF CRU

21st century climate change scenario for the Mediterranean using a coupled Atmosphere-Ocean Regional Climate Model Samuel Somot, Florence Sevault, Michel Déqué,

Introduction

According to the Third Assessment Report of the Intergovernmental Panel on Climate Change [START_REF] Ipcc | Climate Change 2001. The scientific basis. Contribution of Working Group I to the third assessment report of the IPCC[END_REF], the climate over the Mediterranean basin may become warmer and drier during the 21 st Century. It has even been described as one of the main climate change "hot-spots" [START_REF] Giorgi | Climate change hot-spots[END_REF]) in a study based on changes in the mean and in the interannual variability of temperature and precipitation. These results of 21 st Century climate change scenarios have been obtained both with global oceanatmosphere coupled climate models [START_REF] Giorgi | Climate change hot-spots[END_REF]Bi 2005, Giorgi 2006) and with atmosphere only regional climate models [START_REF] Gibelin | Anthropogenic climate change over the Mediterranean region simulated by a global variable resolution model[END_REF][START_REF] Déqué | Global high resolution versus Limited Area Model scenarios over Europe: results from the PRUDENCE project[END_REF][START_REF] Gao | Projected changes in mean and extreme precipitation over the Mediterranean region from high resolution double nested RCM simulation[END_REF], Christensen and Christensen 2007). However each of these approaches has its own shortcoming. The global coupled models or AOGCMs (Atmosphere-Ocean General Circulation Models) that take into account oceanatmosphere interactions do not resolve the spatial scales required to correctly represent the Mediterranean climate. This climate is strongly driven by the regionalscale mountains, valleys and local winds, which are not resolved until the spatial resolution reaches approximately 50 km [START_REF] Li | Regional atmospheric, marine processes and climate modelling (chapter 7)[END_REF].

Over Europe and the Mediterranean Basin, the state of the art of regional models or ARCMs (Atmosphere Regional Climate Models) that resolve spatial scales up to 50 or 20 km do not include the ocean feedback. They often use low resolution SST anomalies coming from previously run AOGCMs with no active feedback [START_REF] Christensen | A summary of the PRUDENCE model projections of changes in European climate by the end of this century[END_REF]. The representativeness of these SST anomalies in the local seas such as the Baltic Sea, the Black Sea or the Mediterranean Sea is doubtful. Indeed, these seas are not very well simulated by the AOGCMs, which is rather worrying since the impact of the SST changes of the local seas on the regional climate is relatively important [START_REF] Kjellström | Atmospheric response to different sea surface temperatures in the Baltic Sea: Coupled versus uncoupled regional climate model experiments[END_REF][START_REF] Somot | Modélisation climatique du bassin méditerranéen : variabilité et scénarios de changement climatique[END_REF][START_REF] Li | Atmospheric GCM response to an idealized anomaly of the Mediterranean sea surface temperature[END_REF]).

Considering the current available computer power, it is not possible to run long term high resolution global coupled simulations. Consequently, developing Atmosphere-Ocean Regional Climate Models (AORCM) is the practical way to test the impact of regional coupling on regional climate change. For the European area, such a model has already been set up for the Baltic Sea [START_REF] Döscher | The development of the coupled ocean-atmosphere model RCAO[END_REF][START_REF] Räisänen | European climate in the late twenty-first century: regional simulations with two driving global models and two forcing scenarios[END_REF], Kjellström et al 2005). Significantly different results for climate change scenarios have been obtained by comparing the non-coupled regional model with its coupled version. [START_REF] Kjellström | Atmospheric response to different sea surface temperatures in the Baltic Sea: Coupled versus uncoupled regional climate model experiments[END_REF] claimed for instance that the SST anomaly produced by the AORCM along the 21 st Century is far more realistic that the one deduced from an AOGCM. However, the Mediterranean Sea is more complex to simulate than the Baltic Sea. The long (some decades) memory of the Mediterranean sea state is mainly due to the long overturning time of the Mediterranean ThermoHaline Circulation (MTHC). The MTHC is an anti-estuarine circulation [START_REF] Wüst | On the vertical circulation of the Mediterranean Sea[END_REF], which enables the storage of a large amount of heat because it can transfer surface water anomalies towards the intermediate and deep layers [START_REF] Somot | Transient climate change scenario simulation of the Mediterranean Sea for the 21st century using a high-resolution ocean circulation model[END_REF]. Moreover the MTHC is known to have different stability states (deep or intermediate, [START_REF] Béthoux | Budgets of the Mediterranean Sea. Their dependence on the local climate and on the characteristics of the Atlantic waters[END_REF]Gentili 1999, Myers and[START_REF] Myers | Stability of the Mediterranean's thermohaline circulation under modified surface evaporative fluxes[END_REF] that can affect the SST pattern and, subsequently, the regional climate. Recent modelling studies suggest that the MTHC could be strongly weakened during the 21 st Century under the influence of the global climate change [START_REF] Thorpe | Modelling the sensitivity of the Mediterranean Outflow to anthropogenically forced climate change[END_REF][START_REF] Somot | Modélisation climatique du bassin méditerranéen : variabilité et scénarios de changement climatique[END_REF][START_REF] Bozec | La circulation thermohaline de la Mer Méditerranée sous les climats présent et future[END_REF][START_REF] Somot | Transient climate change scenario simulation of the Mediterranean Sea for the 21st century using a high-resolution ocean circulation model[END_REF]. Such an event could impact the regional climate and should be taken into account when assessing the climate change response over the Euro-Mediterranean area. Therefore, since the simulation of the MTHC requires high resolution atmospheric fluxes [START_REF] Myers | On the importance of the choice of wind stress forcing to the modeling of the Mediterranean Sea circulation[END_REF][START_REF] Somot | Modélisation climatique du bassin méditerranéen : variabilité et scénarios de changement climatique[END_REF] as well as a high resolution ocean model (Madec et al. 1991, Lascaratos and[START_REF] Lascaratos | A high-resolution three-dimensional numerical study of intermediate water formation in the Levantine Sea[END_REF], the use of a high resolution atmosphere-ocean regional climate model (AORCM) seems to be suitable.

The CNRM (Centre National de Recherches Météorologiques) has developed an AORCM coupling a variable resolution version of the global ARPEGE-Climate model and a Mediterranean limited area version of the OGCM OPA called OPAMED.

As far as we know, this new model named S.A.M.M. for Sea-Atmosphere Mediterranean Model is the first AORCM dedicated to the Mediterranean Basin. It is described in Section 2 along with the validating dataset. This paper aims at addressing two different goals. The first one is to validate this new coupled model. This is done by comparing the simulated atmospheric fields with observed data and with the corresponding ARCM in a non-coupled mode (Section 3). The second goal consists in describing the AORCM climate change response and analyzing the differences between the AORCM and the corresponding ARCM in terms of climate change for the Euro-Mediterranean area (Section 4). The representativeness of the obtained differences compared to other uncertainty sources evaluated in the European project PRUDENCE [START_REF] Déqué | An intercomparison of regional climate models for Europe: assessing uncertainties in model projections[END_REF] will be discussed in Section 5 as well as the comparison with the corresponding AOGCM. We conclude in Section 6. This paper is dedicated to the study of the SST and of the atmospheric fields. The oceanic validation of the AORCM has been carried out in [START_REF] Somot | Modélisation climatique du bassin méditerranéen : variabilité et scénarios de changement climatique[END_REF] and the analysis of the ocean response to the climate change will be done in another paper except for the analysis of the sea level change that is studied in Tsimplis et al (2007, this issue).

Data and model setup

Data

The atmospheric data used in this study are the CRU data [START_REF] New | A high-resolution data set of surface climate over global land areas[END_REF]. They are globally gridded on a 0.5° resolution grid over land areas. 2-meter temperature and precipitation are available monthly for the 20 th Century. They were used to validate the mean state and the interannual variability of the SAMM model during the 1961-1990 period. European boxes has been defined following [START_REF] Christensen | A summary of the PRUDENCE model projections of changes in European climate by the end of this century[END_REF] and [START_REF] Jacob | An intercomparison of regional climate models for Europe: Model performance in Present-Day Climate[END_REF] in order to simplify the readability of this validating section. The boxes are defined in Table 1 (or in Figure 4 in [START_REF] Christensen | A summary of the PRUDENCE model projections of changes in European climate by the end of this century[END_REF]. Only land points are considered.

TABLE 1

Concerning the sea, the only parameter validated in this study is the SST, as it is the one known to mainly influence the climate. The ARCM was forced by the so-called RSSTs [START_REF] Smith | Reconstruction of historical sea surface temperatures using empirical orthogonal functions[END_REF], which are also used for the validation. The RSST dataset shows a weak interannual variability for the Mediterranean Basin due to the way the dataset was created. Therefore, we also use the SST analysis of the more recent ERA40 reanalysis [START_REF] Simmons | The ERA-40 Project Plan[END_REF] to validate the SST variability of the model. Note however that the weakness of the interannual variability of the RSST dataset does not seem to influence the results of the current study as shown in section 3.2.

Model description

Regional coupling strategy

Different ways of doing regional climate modelling have been described in the literature for both ocean and atmosphere models. One can either use high resolution global circulation models, limited area regional climate models forced by data (often reanalysis for the atmosphere or temperature and salinity 3D climatology for the ocean data) or a global variable resolution model focused on the area of interest.

Consequently, AORCMs dedicated to a particular area can be based on different types of regional climate and ocean models. However, the general principle is always the same: an air-sea coupling is applied only over a part of the domain and data are used to force the model elsewhere. For example, [START_REF] Döscher | The development of the coupled ocean-atmosphere model RCAO[END_REF] used an ARCM forced by lateral boundary conditions taken from the ERA15 reanalysis and coupled to a regional ocean model of the Baltic Sea forced by climatological data in the North Sea. [START_REF] Aldrian | Modelling Indonesian rainfall with a coupled regional model[END_REF] also used an ARCM forced by the ERA15 and NCEP reanalyses but coupled to a global stretched ocean model forced by additional atmospheric forcing outside the coupling domain.

The Mediterranean Sea is a semi-enclosed basin separated from the Atlantic Ocean by the narrow and shallow strait of Gibraltar. The time scales associated with the Mediterranean Sea temporal variability are ten times smaller than those of the Atlantic THC (10-100 years versus 1000 years). So it is logical to simulate it with a limited area ocean model forced by 3D ocean data in a buffer zone dealing with the Atlantic Ocean as in [START_REF] Döscher | The development of the coupled ocean-atmosphere model RCAO[END_REF]. Moreover, the Mediterranean basin being located in the mid-latitudes, it is influenced by and in turn influences many global climate processes such as the NAO, the African monsoon and the Indian monsoon (see [START_REF] Lionello | Mediterranean Climate Variability[END_REF] for a review). Finally, the complex orography surrounding the basin, the local climate and the Mediterranean cyclogenesis require spatial high resolution [START_REF] Somot | Modélisation climatique du bassin méditerranéen : variabilité et scénarios de changement climatique[END_REF]. We have consequently chosen a global and stretched AGCM to represent the atmospheric part of the coupled model.

The regional atmosphere model

In the present study, we used the variable resolution version (or stretched version) of the ARPEGE-Climate model [START_REF] Déqué | High-Resolution climate simulation over Europe[END_REF]. This stretching ability allows this version of ARPEGE-Climate to be at the same time an AGCM because it is global and an ARCM because of its high resolution over the area of interest.

The model used here is the same as in [START_REF] Gibelin | Anthropogenic climate change over the Mediterranean region simulated by a global variable resolution model[END_REF] and we just recall here its main features. The spectral truncation is T106, with 31 vertical levels located mainly in the troposphere (exactly those of the ERA15 reanalysis, [START_REF] Gibson | ERA description. ECMWF Reanalysis project report series[END_REF].

The time step is 30 min. The pole of stretching is located at the centre of the Mediterranean basin (40°N, 12°E) and the stretching factor is 3. The grid has 120 pseudo-latitudes and 240 pseudo-longitudes (with a reduction near the pseudo-poles to maintain the isotropy of the grid). As a result, the maximum horizontal resolution reaches 0.5°, which corresponds to approximately 50 km over the Mediterranean Basin, and has a minimum of 4.5° in the Pacific. [START_REF] Déqué | High-Resolution climate simulation over Europe[END_REF] and [START_REF] Machenhauer | Validation and analysis of regional present-day climate and climate change simulations over Europe[END_REF] have shown that the variable resolution version of ARPEGE-Climate realistically reproduces the seasonal and geographical variations of the main climatological parameters over Europe. The climate simulation is reasonably close to the CRU climatology [START_REF] New | A high-resolution data set of surface climate over global land areas[END_REF] over Europe and around the Mediterranean Basin as shown by [START_REF] Gibelin | Anthropogenic climate change over the Mediterranean region simulated by a global variable resolution model[END_REF] and [START_REF] Jacob | An intercomparison of regional climate models for Europe: Model performance in Present-Day Climate[END_REF], the largest deficiency being a too rainy winter.

The high resolution Mediterranean Sea model

In this study, we used a Mediterranean Sea limited area version of the primitive equation numerical model Ocean PArallel (OPA, [START_REF] Madec | OPA 8.1, Ocean General Circulation Model, Reference Manual[END_REF]. This model, called OPAMED, is based on the 8.1 version of OPA and is consequently very close to the one developed for the MERCATOR project (http://www.mercator-ocean.fr, [START_REF] Drillet | The MEDDIES in the Mercator North Atlantic and Mediterranean Sea eddy-resolving model[END_REF][START_REF] Béranger | The dynamic of the Sicily Strait: A comprehensive study from observations and models[END_REF][START_REF] Alhammoud | Surface circulation of the Levantine Basin: comparison of model results with observations[END_REF]. It is described in details in [START_REF] Somot | Modélisation climatique du bassin méditerranéen : variabilité et scénarios de changement climatique[END_REF] and [START_REF] Somot | Transient climate change scenario simulation of the Mediterranean Sea for the 21st century using a high-resolution ocean circulation model[END_REF] in which it has been used to study the impact of the climate change on the Mediterranean Sea with atmospheric forcing provided by ARPEGE-Climate. Forcing from the ERA40 reanalysis [START_REF] Somot | Modélisation climatique du bassin méditerranéen : variabilité et scénarios de changement climatique[END_REF] and from the ECMWF analysis [START_REF] Bozec | Mediterranean oceanic response to the interannual variability of a high-resolution atmospheric forcing : a focus on the Aegean Sea[END_REF] have also been tested for present climate studies.

The horizontal resolution of OPAMED is 1/8°x1/8°cos(φ), φ being the latitude. This is equivalent to a range of 9 to 12 km from the north to the south of the domain with square meshes. It has 43 vertical Z-levels with a stretched distribution (from ∆Z = 6 m at the surface to ∆Z = 200 m at the bottom with 25 levels in the first 1000 m). The maximum depth is 4100 m in the Mediterranean Sea (Eastern Basin).

The OPAMED grid is tilted and stretched at the Gibraltar Strait to better follow the SW-NE axis of the real strait. The Gibraltar Strait is represented with a two grid-point wide strait. The Atlantic ocean was modelled as a buffer zone with a restoring to climatology as described in [START_REF] Somot | Transient climate change scenario simulation of the Mediterranean Sea for the 21st century using a high-resolution ocean circulation model[END_REF]. A time step of 20 minutes was applied.

The regional coupled model

The regional coupled model used in this study was based on the coupling of the two models described above: ARPEGE-Climate and OPAMED. They were coupled with the OASIS coupler [START_REF] Valcke | OASIS 2.4 Ocean Atmosphere Sea Ice Soil, User's Guide[END_REF] with a daily time step. Over the coupling zone (ocean model domain), the models exchange SST, heat, water and momentum fluxes.

Technical details about the coupling can be found in [START_REF] Sevault | Couplage ARPEGE-MEDIAS -OPA-MEDITERANEE. Les étapes. Note de Centre[END_REF] and [START_REF] Somot | Modélisation climatique du bassin méditerranéen : variabilité et scénarios de changement climatique[END_REF].

It is worth noting that the use of a Mediterranean Sea limited area model for the ocean leads to neglect a possible global feedback loop. Indeed, we assumed that the influence of the Mediterranean Outflow Waters (flowing into the Atlantic Ocean at a 1000m depth) on the Atlantic SST (through the Atlantic thermohaline circulation) and then on the Euro-Mediterranean climate is weak.

Models setup and simulations

In this paper, we compared two 140-year simulations following the SRES-A2 scenario [START_REF] Ipcc | Climate Change 2001. The scientific basis. Contribution of Working Group I to the third assessment report of the IPCC[END_REF]. One is performed with the ARCM in a standalone mode and the other with the AORCM. The IPCC-A2 scenario is a pessimistic scenario in which the CO 2 concentration is about 815 ppm in 2099.

Simulation with the ARCM

The experimental set-up (SST and radiative forcing) was the same as in [START_REF] Gibelin | Anthropogenic climate change over the Mediterranean region simulated by a global variable resolution model[END_REF]. Up to year 2000, SST was prescribed from interannual observed data [START_REF] Smith | Reconstruction of historical sea surface temperatures using empirical orthogonal functions[END_REF]. GHG gas and aerosol concentrations were imposed from observed values. Then, from 2000 onwards, the simulation was forced by SRES-A2 radiative forcing (GHG and aerosol concentrations). Modified SSTs were also used as surface boundary conditions for the ARCM. They were computed exactly as in [START_REF] Gibelin | Anthropogenic climate change over the Mediterranean region simulated by a global variable resolution model[END_REF] except for the choice of the SRES scenario (A2 instead of B2). This computation used global low resolution AOGCM simulations (A2 scenario and control run) coming from the CNRM [START_REF] Royer | Simulation of climate changes during the 21-st century including stratospheric ozone[END_REF], Douville et al. 2002) to obtain SST anomalies. These anomalies were then simply added to [START_REF] Smith | Reconstruction of historical sea surface temperatures using empirical orthogonal functions[END_REF] climatology in order to force the ARCM.

The year numbering corresponds to the SST chronology used to force the atmospheric model. This does not imply that the daily atmospheric circulation of the model (or even the climate of the year) coincides with the actual one since no other data were assimilated in the model. Only the climate statistics are reproduced by climate models. We performed a 1 year long spin-up (1960) in order to stabilize the soil scheme.

Simulation with the AORCM

Up to year 2000, the same observed data as for the ARCM were applied as boundary conditions except for the Mediterranean SST (coupling zone). The coupled simulation was performed without any relaxation at the sea surface nor in the deep layers but we applied a heat flux correction (see below).

Explicit river runoff discharges were applied to make the water budget more realistic.

A monthly mean climatology covering the 1900-1984 period was extracted from the RivDis UNESCO database [START_REF] Vörösmarty | Global river discharge database[END_REF]. For the Nile River, we only kept the data collected after the Aswan dam was built in 1964. In OPAMED, the Black Sea was considered as a river with a fresh water flux equal to the E-P-R budget of the Black Sea [START_REF] Stanev | Sea level variations and their dependency on meteorological and hydrological forcing: Analysis of altimeter and surface data for the Black Sea[END_REF].

We simulated the Atlantic Ocean by a buffer zone (6°W-11°W, 32°N-40°N) where we applied a 3D relaxation towards the [START_REF] Reynaud | A new analysis of hydrographic data in the Atlantic and its application to an inverse modeling study[END_REF]

Heat flux correction

Because of bias in the air-sea fluxes of ARPEGE-Climate (the heat loss is generally too strong when averaged over the whole basin in winter), a simple heat flux correction was applied in the coupled model over the Mediterranean Sea. It consists in adding a constant term in the equation of the temporal evolution of the sea surface temperature (noted T) as follows :

M C t y x Q t t y x T + = ∂ ∂ ) , , ( ) , , ( (1)
where Q is the heat flux provided by the atmospheric model depending on T and on atmospheric variables. T and Q depend on longitude (x), latitude (y) and time (t)

whereas M C (equivalent to a heat flux) only depends on the month M (12 values) and is spatially averaged over the Mediterranean Sea surface. This correction does not directly act upon the atmosphere model but influences it in removing the SST bias. It is worth noting that the correction does not change the spatial and temporal variability of the coupled model and also helps to stabilize the AORCM when it is run for a long period of time. The correction has been designed from a preliminary ocean simulation forced by ARPEGE-Climate with a SST relaxation towards observed data as well as from simulations with the AORCM. The yearly mean value of the correction is equal to 28 W/m 2 correcting the too strong average heat loss, whereas it amounts to -13 W/m 2 in August and +69 W/m 2 in January.

Validation

The validation was mainly done for the 1961-1990 period, a period long enough to take into account the interannual variability of the climate system. This validation was only done on yearly averages of model output.

SST

The heat loss, spatially averaged over the Mediterranean Sea and temporally over the 30-year period, amounted to -3.9 W/m 2 . This value is consistent with in-situ evaluations of this term which are of -7 ± 3 W/m 2 [START_REF] Béthoux | Budgets of the Mediterranean Sea. Their dependence on the local climate and on the characteristics of the Atlantic waters[END_REF][START_REF] Bunker | A note on the heat balance of the Mediterranean and Red Seas[END_REF].

The SST bias of the uncorrected AORCM was equal to -0.9°C (1961-1990 value) with respect to the RSST data [START_REF] Smith | Reconstruction of historical sea surface temperatures using empirical orthogonal functions[END_REF] [START_REF] Brasseur | Seasonal temperature and salinity fields in the Mediterranean Sea: Climatological analyses of a historical data set[END_REF] and 19.3°C for ERA40 analysis averaged over the 1961-1990 period [START_REF] Simmons | The ERA-40 Project Plan[END_REF].

Both corrected and uncorrected AORCM simulations show an interannual standard deviation of the Mediterranean SST equal to 0.22 K (1961-1990 period, after removing the trend), which is in agreement with the ERA40 SST analysis (σ = 0.25 K). Besides, we found that the heat flux correction does not modify the model interannual variability of the SST. In addition, during the same period, a trend in SST was observed: +0.019°C/year (but only +0.011°C/year for the 1961-1980 period).

This positive trend is caused by a weak imbalance between the yearly mean heat loss by the sea surface (-3.9 W/m 2 ) and the heat flux gain by the Gibraltar Strait transport (+4.4 W/m 2 ). It is concluded that the heat flux correction was efficient in correcting the AORCM biases in SST and heat flux.

The atmospheric fields

Over land, we performed the validation using the different boxes described in Section 2. Over these boxes, we compared the temperature at 2m and the precipitation provided by the AORCM and ARCM simulations to the CRU observed gridded dataset. Tables 2a and2b summarize the results, showing that the regional coupled model is as good as the regional forced atmosphere model at the same resolution and close to the CRU values.

Winter bias

In winter, the AORCM shows a cold and dry bias over the southern Europe boxes improvements were observed in the AORCM compared to the ARCM. The two models have comparable quality and the differences may be attributed to the chaotic behaviour of the atmosphere and to the inter-decadal variability of the models.

As in winter, the AORCM simulates an interannual variability that is not different from the one simulated by the ARCM. Moreover, a validation of the intermediate seasons has also been carried out (not shown) with similar conclusions for the mean and the standard deviation of the temperature and precipitation.

State of the art

Following the model intercomparison of the PRUDENCE project [START_REF] Christensen | PRUDENCE employs new methods to assess European climate change[END_REF], [START_REF] Jacob | An intercomparison of regional climate models for Europe: Model performance in Present-Day Climate[END_REF] analyzed the biases of 11 ARCM runs over Europe at a 50km resolution over the same boxes at the present ones. These values could be seen as the state of the art of the RCM biases for climate simulations over Europe.

Considering the 32 compared values (2m-temperature and precipitation, winter and summer, 8 boxes), one can tell that the AORCM was the best model in 22% of the cases, was in the first half in 69% and was the worst only one time out of 32.

We can thus conclude that the present AORCM is a good model when compared to the state of the art of uncoupled regional models. Moreover, the bias and the interannual variability simulated by the two models are statistically comparable. The coupled and non-coupled ARPEGE-Climate versions are then equivalent for the present-climate.

Climate change response and impact of the coupling

In the following, the response of the models to climate change is studied in terms of the difference between the 2070-2099 and the 1961-1990 mean climatological parameters.

Impact on the SST of the Mediterranean Sea

In the AORCM, the yearly mean SST averaged over the Mediterranean basin (East of the Gibraltar Strait) increased by +2.6°C with a minimum in spring (+2.3°C) and a maximum in summer and autumn (+2.9°C). This is in agreement with non-coupled ocean scenarios [START_REF] Somot | Modélisation climatique du bassin méditerranéen : variabilité et scénarios de changement climatique[END_REF]Sevault 2005, Somot et al. 2006). Figure 1

FIGURE 1 FIGURE 2

The SST anomalies used to force the ARCM in the Mediterranean Sea were provided by a low resolution AOGCM run [START_REF] Royer | Simulation of climate changes during the 21-st century including stratospheric ozone[END_REF], also see Figure 2). The mean SST of this model increases by +2.5°C with a minimum in spring (+2.1°C) and a maximum in summer (+3.1°C). The yearly mean anomalies are similar in the AORCM and in the AOGCM (and consequently in the ARCM). However, the seasonal cycle of the AORCM, as well as the spatial pattern were significantly changed with respect to the ones of the AOGCM (as shown when comparing Figure 1 and Figure 2). Winter, spring and autumn were warmer in the AORCM whereas summer was colder. These differences are the signature of a damping of the SST anomaly seasonal cycle. The maximum warming in summer was weaker and the minimum warming in spring was stronger. In order to estimate the different behaviour of the two models, we have computed the parameter DD which is defined as :

DD = (AORCM-future -AORCM-present) -(ARCM-future -ARCM-present) (2)
A positive value of DD means that the SST warming of the AORCM is stronger than that of the ARCM (and therefore of the AOGCM). Figure 3 shows a map of the DD parameter for SST. The DD value was also calculated for other variables.

FIGURE 3

The DD in SST, noted DD(SST), reached +0.3°C in winter (value significantly different from zero at a 99% level with respect to the interannual variability of the AORCM SST), +0.2°C in spring (significant at a 95% level), -0.2°C in summer (significant at a 90% level) and +0.04°C in autumn (not significant).

In winter, DD(SST) was maximum in the northern-western part of the Mediterranean basin (see Figure 3a). In spring the maximum of DD(SST) was located in the northeastern part of the basin (+0.3°C in the Levantine Basin and the Adriatic Sea). In summer, DD(SST) was minimum in the south-eastern part of the Mediterranean Sea (see Figure 3b). No spatial differences were found in autumn. The spatial differences in the SST anomaly between the models are not easy to explain but they are probably due to a better resolution of the surface currents and the thermohaline circulation of the Mediterranean Sea in the AORCM compared to the AOGCM where the Mediterranean is crudely modeled with a coarse grid (2° in longitude).

Impact on the two meter temperature and the precipitation

For the same 8 boxes as for the validation, Tables 3a and3b summarize the temperature and precipitation anomalies for the AORCM (see also Figure 4 and 5 for winter and summer) as well as the difference of the anomalies (equivalent to DD in temperature and precipitation).

TABLE 3a and TABLE 3b

At the end of the 21 st Century, a strong warming was simulated for all seasons. This warming has the same characteristics as that simulated by the regional models used in the PRUDENCE project. [START_REF] Déqué | Global high resolution versus Limited Area Model scenarios over Europe: results from the PRUDENCE project[END_REF] described this mean behaviour as a warming that is stronger in summer than in winter (1), stronger over land than over sea (2), warmer in the eastern part of Europe in winter (3) and in the southern part in summer (4).

These four characteristics were also found in the present simulations and are presented in Figure 4. As an example, for the France box, the AORCM warming was stronger in summer (+4.6°C) than in winter (+2.8°C), weaker than the East-Europe warming in winter (+4.2°C) and stronger than the British Isles warming in summer (+2.7°C).

FIGURE 4 FIGURE 5

In the AORCM (land points only), the weakest warming, which is nevertheless highly significant, occured in the British Isles box in spring (+1.7°C) and the warmest box was the Mediterranean box in summer (+5.2°C). Smaller areas with warming higher than 6°C can be seen in winter in Russia (Figure 4a) and in the Balkan area in summer (Figure 4b).

For precipitation, Table 3b also shows a pattern consistent with the PRUDENCE results [START_REF] Déqué | Global high resolution versus Limited Area Model scenarios over Europe: results from the PRUDENCE project[END_REF]. A wetter climate in winter (see Figure 5a) for the northern part of Europe (significant for the British Isles, Mid-Europe, East-Europe, Scandinavia) with a drying over the Mediterranean Sea and a drier climate during summer (significant everywhere except for Scandinavia). The summer drying (see Figure 5b) was maximum for the southern part of Europe (Iberian Peninsula, France, Mediterranean, East-Europe). It reached less than one mm/d in Romania. Note that the drying behaviour was also true for the intermediate seasons for the Iberian Peninsula,

France and Mediterranean boxes (figures not shown). [START_REF] Giorgi | Climate change hot-spots[END_REF] proposed to characterize the regional climate change response by using a Regional Climate Change Index (RCCI). This index is based on an ensemble of global air-sea coupled models and on regional mean precipitation change, mean surface air temperature change and change in the interannual variability of precipitation and temperature. In that study, the RCCI permits to identify the Mediterranean (MED) and North-Eastern Europe (NEE) regions as the main climate change "hot-spots" with a RCCI reaching 19. Over the 26 regions defined by [START_REF] Giorgi | Climate change hot-spots[END_REF], the RCCI values ranges from 5 to 19. We computed the RCCI values over the boxes used in our study as well as over the regions defined by [START_REF] Giorgi | Climate change hot-spots[END_REF]. We used a global warming value of +3.25°C in order to compute the Regional Warming Amplification factor (RWAF, see [START_REF] Giorgi | Updated regional precipitation and temperature changes for the 21 st century from ensembles of recent AOGCM simulations[END_REF]. The corresponding average global precipitation change amounted to +4% in the AORCM. These values were computed from the AORCM simulation (2070-2099 period) including ocean areas.

The main difference with the study of [START_REF] Giorgi | Climate change hot-spots[END_REF] is that we computed the RCCI using summer (June-July-August) and winter (December-January-February) values instead of wet and dry season values to be consistent with the frame of our study.

For the AORCM, RCCIs equal to 19 and 20 were obtained over the MED and NEE regions defined by [START_REF] Giorgi | Climate change hot-spots[END_REF]

Impact of the coupling

Now, we would like to describe the differences between the responses of the AORCM and the ARCM by comparing the two corresponding runs. Only statistically significantly differences, underlined in grey in Tables 3a and3b, will be pointed out and discussed.

The warming simulated by the AORCM was stronger than the ARCM one. Such a statement is true over a large part of the domain in winter (Figure 6a) mainly in the northern part of Europe (British Isles, France, Mid-Europe, Alps, East-Europe). The DD value was rather large: it reached +0.9°C over the East-Europe box, that is to say it accounts for 20% of the AORCM warming.

FIGURE 6

In spring, the area in which the warming is stronger in the AORCM than in the ARCM is restricted to the East-Europe box (+0.9°C) whereas it also reached the Mediterranean box in summer (+0.8°C, Figure 6b). In autumn, no significant signal appeared in the box averages. However, the warming was significantly weaker in the AORCM over the Middle East.

Concerning the precipitation in winter (Figure 7a), the AORCM was generally wetter than the ARCM in northern Europe even if only the British Isles box was significantly affected (the AORCM anomaly was twice as high as the ARCM anomaly). The summer drying at the end of the Century was significantly stronger in the AORCM (Figure 7b) over the same regions as for the temperature (Mediterranean, East-Europe) and also in autumn over the Mediterranean box. In these cases as well, the climate change signal of the AORCM was more than twice that of the ARCM.

FIGURE 7

The AORCM-ARCM differences in terms of climate change response seem to be rather large compared to the climate change signal itself. Moreover these changes are spatially and physically consistent (see Figures 6a,6b, 7a, 7b) as well as statistically significant with respect to the interannual variability (see Tables 3a and3b). All these differences are compared to other sources of uncertainty in Section 5.

Understanding of the coupling impact

Winter

In winter, climate change responses can be often explained by changes in the large scale circulation whose signature is well marked on the mean sea level pressure (MSLP). Figure 8 represents a map of DD(MSLP) in winter expressing the difference of the anomalies between the AORCM and the ARCM (see Equation 2 for a definition). A significant negative DD(MSLP) was observed over the north of Europe, which was centred in Scandinavia (DD(MSLP) = -4 hPa in the Scandinavia box, statistically significant with a 95% level).

FIGURE 8

This favours the advection of Atlantic air mass over central Europe, which is responsible for the observed temperature and precipitation anomalies (see Figures 6a and7a). A dipole pattern can also be noted for the cloud cover and the water available for precipitation with higher values over northern Europe and lower values in the south. It is worth noting that the AORCM-ARCM difference in terms of 2m temperature, precipitation, water content and cloud cover looks like a NAO+ pattern.

The link between the DD(SST) (larger warming for the AORCM in the north-western part of the Mediterranean Sea) and the DD(MSLP) is not yet explained and is not consistent with sensitivity studies performed with an atmosphere model forced by SST that has been modified over the Mediterranean Sea [START_REF] Li | Atmospheric GCM response to an idealized anomaly of the Mediterranean sea surface temperature[END_REF].

Spring

Like in winter, the DD(MSLP) showed a negative anomaly but now located over central Europe. This could lead to an enhanced advection of warm and wet air masses coming from the Mediterranean Sea in the AORCM compared to the ARCM. In addition, the DD(SST) is positive in spring and more important in the north-eastern part of the Mediterranean Sea in the AORCM. Both these effects could contribute to warm East-Europe. Finally, a positive feedback of the snow cover seems to be sufficient to create the DD(T 2m ) described for the EE box (+0.9°C, see Table 3a andFigure 9).

FIGURE 9

Summer

For the summer, [START_REF] Rowell | Causes and uncertainty of future summer drying over Europe[END_REF] proposed three "driver" mechanisms and one positive feedback for explaining a higher drying and warming over eastern

Europe and the Mediterranean area in the climate change context. These mechanisms can be used to explain the AORCM-ARCM differences.

The first driver mechanism is linked with a lower soil moisture (SM) in spring. As no significant DD(SM) was obtained in spring, this mechanism is not relevant in our case. The second one is based on large-scale circulation differences. This mechanism does not correspond to the AORCM -ARCM difference either, because no significant DD(MSLP) was found in summer. The last one however seems to explain part of our signal. It is based on a larger land-sea contrast leading to an increase in atmospheric water-holding capacity higher over land than over sea. This implies a strongly reduced relative humidity for continental air masses advected from the maritime areas leading to a decrease in convective precipitation, cloud cover and an increased warming (increase in short-wave radiation, warming by sensible heat flux, decrease in evaporation). This third driver mechanism corresponds to the AORCM situation because the SST warming was lower in the AORCM than in the ARCM in summer, whereas the continental warming was higher, thus increasing the land-sea contrast.

The positive feedback between the soil moisture depletion in summer and the decrease in evaporation and convective activity, which was also mentioned by [START_REF] Rowell | Causes and uncertainty of future summer drying over Europe[END_REF], is also supposed to play an important role in the simulated AORCM-ARCM differences by amplifying the "land-sea contrast" mechanism.

These mechanisms (land-sea contrast and summer soil moisture feedback) are reinforced by the fact that the atmospheric water divergence (E-P) over sea increases less in the scenario in the AORCM than in the ARCM. This leads to negative DDs with respect to the water vapour (see Figure 10), cloud cover (see Figure 11) and soil water content over the Mediterranean countries and East-Europe areas. The summer drying and warming of this part of Europe is consequently enhanced (see Figure 6b and 7b).

FIGURE 10 FIGURE 11

Autumn

In autumn, changes between the ARCM and the AORCM do not concern Europe but the Middle East. In the AORCM, there was a large increase in atmospheric water divergence over the Eastern part of the Mediterranean Sea together with a water convergence over the Middle East with more clouds and more rain explaining the observed temperature and precipitation changes (figures not shown). As the AORCM and the ARCM simulated equivalent SST differences, (same seasonal mean, same spatial pattern), high-frequency coupling processes must explain this significant difference. However, the study of these processes is complex and requires computationally expensive sensitivity tests with the AORCM that are beyond the scope of this paper.

Discussion

5.1 How large are the coupled/non-coupled differences?

From the previous results, we can claim that a source of uncertainty in predicting the climate change in Europe could come from the choice between an ARCM and an AORCM with an interactive Mediterranean Sea. Now we would like to evaluate how large this new source of uncertainty is in comparison to other sources due to model choice and pointed out by the European PRUDENCE project [START_REF] Christensen | A summary of the PRUDENCE model projections of changes in European climate by the end of this century[END_REF]). These other sources in regional climate modelling comprise the choice of the ARCM for a given forcing GCM, the choice of the forcing GCM, the choice of the SRES scenario and the internal model variability [START_REF] Déqué | An intercomparison of regional climate models for Europe: assessing uncertainties in model projections[END_REF]. For a given SRES scenario (A2 for example), [START_REF] Déqué | An intercomparison of regional climate models for Europe: assessing uncertainties in model projections[END_REF] proved that the larger sources of uncertainty lie in the choice of the forcing GCM in all seasons and in the choice of the ARCM only in summer.

Note that the uncertainty linked to the internal model variability of a given RCM or of a given GCM has not been evaluated in this study. For atmosphere models, this internal variability can be linked to different initial states of the atmosphere or to different SST forcings. We assume that this internal model variability is negligible with respect to the other sources of uncertainty what is often true [START_REF] Déqué | An intercomparison of regional climate models for Europe: assessing uncertainties in model projections[END_REF].

For the AORCM, we also assume that the initial state of the Mediterranean Sea model is not an important source of internal model variability. This should be evaluated through ensemble simulations in the future.

Using the four GCMs and the eleven RCMs of the PRUDENCE project, we were able to compute the InterModel Distance (IMD) for temperature and precipitation denoted IMD GCM and IMD RCM respectively. The IMD RCM was computed using the Hadley

Center GCM (HadAM3H) as a common boundary forcing for the various RCMs.

These indexes can be considered as an evaluation of the uncertainty due to the choice of the model that is to say due to the spreading of the model responses around their mean response.

For example, the IMD RCM for temperature was computed as :

∑ < ∆ - ∆ - = j i j i RCM T T N N IMD 2 ) ( ) 1 ( 2 (3)
where i and j are used to number the models from 1 to 11 (for the RCMs); ∆T i is the climate change response of the model i and N(N-1)/2 is the number of model couples.

The larger the IMD index, the larger the difference in the model responses. Note that the IMD is exactly equal to the intermodel standard deviation multiplied by 2 .

The IMD computation has been carried out in the cases where there was a statistically significant difference between the AORCM and the ARCM responses as provided by the so-called DD values (see grey boxes in Tables 3a and3b). A value of DD larger than the IMD implies that the impact of the regional coupling on the regional climate change response is greater than the impact of using different GCMs or RCMs. Results are summarized in Table 3a for temperature and Table 3b for precipitation.

In three seasons (winter, spring and autumn) and for two parameters (temperature and precipitation), DD was always larger than IMD RCM and had, at least, the same order of magnitude as IMD GCM . In summer, the situation is the same as in winter over the Mediterranean box but different over the East-Europe box. Indeed IMD RCM and IMD GCM are very large over this last box in summer (more than 1.3°C and 0.4 mm/d) and they are both larger than DD. The high value of IMD GCM can be explained by the strong warming of the Hadley Center model in this area compared to other GCMs.

For IMD RCM , the high value of the index is explained by the physics of the Eastern-Europe climate in summer: convective weather situations, strong atmosphere-land interaction and important contribution of the soil water content in the determination of the 2m temperature [START_REF] Rowell | Causes and uncertainty of future summer drying over Europe[END_REF]. This specific climate and these threshold effects on the soil water content might explain why the RCMs are so free with respect to their common lateral boundary conditions and then able to diverge.

Our study proves that whatever the summer behaviour over the EE box may be, the uncertainty due to using coupled or non-coupled RCM is large and should not be neglected a priori. Consequently this new source of uncertainty should be taken into account when dealing with European climate change.

Comparison with the AOGCM response

In order to acquire a better understanding of the differences between the AORCM and the ARCM responses to climate change (DD values), we also compared their responses to that of the corresponding AOGCM (same physics, [START_REF] Royer | Simulation of climate changes during the 21-st century including stratospheric ozone[END_REF]. The AOGCM and the ARCM have the same SST response to climate change by construction (see Section 2) whereas the AORCM has a different response in SST (see Section 4). However, over the Mediterranean Sea surface, the AOGCM and the AORCM are coupled and the ARCM is forced. But, the ARCM and the AORCM have the same resolution over the Mediterranean Sea whilst the AOGCM has a lower one. Therefore, comparing the three model responses to climate change could also help to find out which factor among the resolution, the Mediterranean coupling or the Mediterranean SST difference sparks off the observed AORCM-ARCM differences. [START_REF] Gibelin | Anthropogenic climate change over the Mediterranean region simulated by a global variable resolution model[END_REF] have already compared the ARCM and the AOGCM in SRES-B2 simulations in the same configuration (the SST response of the AOGCM was used to force the ARCM). They concluded that the resolution has a weak effect on the climate change impact at the European scale.

The comparison between the three models was carried out over the boxes where the DD parameters were significantly different from zero (see Tables 3a and3b). In winter, for all the significant boxes, we found that the AOGCM response is equal to the ARCM response even if these models have a different resolution and a different coupled/non-coupled mode. This implies that the AORCM-ARCM differences would only be explained by the Mediterranean SST difference given by the two models during this season. This SST difference is mainly due to an explicit simulation of the Mediterranean thermohaline circulation due to the high-resolution ocean part of the AORCM.

The same behaviour was observed in spring for the temperature over the East-Europe box showing that the difference in SST response is the source of the AORCM-ARCM difference, as it is in winter. The ARCM, AORCM, AOGCM comparison did not bring clear conclusion in autumn.

In summer the situation is different. Over the Mediterranean and East-Europe boxes, the AOGCM response was very similar to the AORCM response. The prevailing factor explaining the DD values is consequently the coupling over the Mediterranean Sea because the non-coupled ARCM has a different response. For this season, the interactive Mediterranean Sea enhanced the simulated warming for the areas under Mediterranean influence.

Conclusion

As far as we know, this is the first time that the climate change response over Europe and the Mediterranean basin has been studied using a high resolution coupled Atmosphere-Ocean Regional Climate Model (AORCM). We showed that the spatial and seasonal patterns of the climate change over Europe simulated by the AORCM were comparable with non-coupled regional simulations run with the corresponding ARCM or with other European RCMs. However, the climate change signal was enhanced over Europe in the AORCM simulation. In winter, the warming is significantly larger over the following areas: British Isles, France, Mid-Europe, Alps and Eastern-Europe with values reaching +0.9°C for the AORCM-ARCM difference in the climate change response (up to 25% of the signal).

The north of Europe is also wetter in the AORCM in winter. These differences can be explained by a MSLP anomaly that is probably linked to a higher warming of the Mediterranean SST in the AORCM. In summer, the Mediterranean and the East-Europe areas are significantly warmer (25% of the signal) and drier (50% of the signal) in the AORCM. Three factors could explain the former behaviour: a weaker warming of the AORCM SST for that season, a weaker humidity advection from the Mediterranean Sea to these areas and a positive feedback of the soil water content.

High-frequency coupling interactions could also be part of the explanation as well as differences in the spatial pattern of the Mediterranean SST response. Differences in spring and autumn were also pointed out but were not so well explained.

A better consistency between the SST, the air-sea fluxes and the vertical structure of the atmosphere (air-sea coupling) as well as a better representation of the Mediterranean thermohaline circulation and the SST pattern (high resolution ocean model) are probably the key-processes explaining the differences between the AORCM and the ARCM (with SST coming from a previous AOGCM run). They also give more confidence in the AORCM results over the Euro-Mediterranean area.

However, a detailed study of the ocean part of the AORCM simulation (compared to the forced ORCM simulation or to the ocean part of the AOGCM run) is needed to better understand that point. Moreover, mechanism studies through sensitivity tests would also be necessary to really understand which processes might explain the differences described in winter and summer. Using the simulated SST response of the AORCM (spatially filtered or not) to force the ARCM model could help understanding the roles of the mean SST difference, the SST pattern difference and the coupling effect.

Compared to other sources of uncertainty arising when evaluating the European climate change through regional climate modelling (PRUDENCE project), the choice between non-coupled and coupled RCMs appears to be as important as the choice of the RCM or the choice of the forcing GCM are. This result confirms the results obtained by [START_REF] Kjellström | Atmospheric response to different sea surface temperatures in the Baltic Sea: Coupled versus uncoupled regional climate model experiments[END_REF] for the Baltic Sea. Therefore, we recommend that this new source of uncertainty should not be neglected when evaluating the climate change response over Southern and Eastern Europe, which are the areas where the impact of the regional coupling seems to be the most important. However a multimodel approach is necessary to legitimate the results of our study. For example, ensemble simulations performed with the same AORCM and different initial ocean states are required to estimate the internal variability of these models. This multimodel approach should be achieved through a dedicated international cooperation between different research centres owing to the enormous computing effort required.

It is actually planned in the European CIRCE project where 6 different AORCMs dedicated to the study of the Mediterranean climate change will be used. We hope that this intercomparison framework will also allow to develop non-flux corrected regional coupled models for the Mediterranean Sea in comparing and improving the air-sea flux parameterizations within the RCMs.

Figures Captions

  climatology as described in[START_REF] Somot | Transient climate change scenario simulation of the Mediterranean Sea for the 21st century using a high-resolution ocean circulation model[END_REF]. The relaxation was weaker close to the Gibraltar Strait (100day restoring time scale) than it was by the western limit of the model (3[START_REF] Group | MEDAR/MEDATLAS 2002 Database. Cruise inventory, observed and analyzed data of temperature and bio-chemical parameters[END_REF] for the Mediterranean Sea and from the[START_REF] Reynaud | A new analysis of hydrographic data in the Atlantic and its application to an inverse modeling study[END_REF] climatology for the Atlantic part of the model. A 20-year long spin-up was run before starting to study the simulation. The first 10 years were performed with OPAMED in the standalone mode and a strong 3D relaxation for temperature and salinity, whereas the 10 last years of the spin-up were run with the coupled model.After year 2000, the SST outside the Mediterranean Sea and the radiative forcings evolved as they did in the ARCM run. River discharges and buffer zone T-S characteristics also changed following the A2 scenario using an anomaly mode. The Atlantic T-S anomalies were computed as differences between a given year and the present-climate state from the outputs of the AOGCM already used to compute the SST anomalies for the ARCM simulation. For the river discharges, the anomalies were applied as multiplying factors of the climatology. These factors were computed for each Mediterranean river catchment basin from the ARCM runoff outputs as a ratio between future and present climate. The anomalies were then filtered and evolved every decade from year 2000. Details about the ocean configuration and about the implementation of the external forcing in the scenario are given in[START_REF] Somot | Modélisation climatique du bassin méditerranéen : variabilité et scénarios de changement climatique[END_REF] and[START_REF] Somot | Transient climate change scenario simulation of the Mediterranean Sea for the 21st century using a high-resolution ocean circulation model[END_REF].

  shows the spatial pattern of this response. Minimum warming occurred along the Atlantic Water pathway in winter and summer and in the north-eastern part of the Mediterranean Sea in winter. Maximum warming was encountered in the shallower parts of the sea in winter (North of the Adriatic and Aegean Seas and Gulf of Gabes) and in the northwestern part in summer (Adriatic Sea, Tyrrhenian Sea and Liguro-Provençal basin).

  The SAMM model (Sea-Atmosphere Mediterranean Model) coupling a global variable-resolution atmosphere model (50-km over the Mediterranean area) to a 10-km Mediterranean Sea model has been designed to represent the major atmospheric and oceanic characteristics of the Mediterranean basin which is mainly driven by orography, small scale physical processes and narrow straits. A SRES-A2 scenario has been carried out with SAMM from 1960 to 2099. An anomaly mode was used to simulate the evolution of the external forcing (SST outside the Mediterranean Sea, Atlantic T-S characteristics, river runoff fluxes) in addition to the GHG and aerosol concentrations. A simple heat flux correction was successfully applied over the Mediterranean Sea. It allowed us to remove the SST bias and to ensure the model stability without changing either the SST spatial variability or the interannual variability. Moreover, we verified that the AORCM was as good as the corresponding ARCM (ARPEGE-Climate) for the present climate and belongs to the state-of-the-art of the European regional climate models. The climate change simulated by the AORCM shows an important warming at the end of the 21 st Century for all seasons with maximum values (higher than +6°C) in the eastern part of Europe in winter and in the southern part of Europe in summer. A drying was generally simulated over Europe and the Mediterranean Basin except for northern Europe where the climate mainly becomes wetter. Over southern Europe, the decrease in precipitation between the 1961-1990 period and the 2070-2099 period often reached -30% in summer.

Figure 1 :

 1 Figure 1: AORCM response to the climate change for the 30-year average sea surface

Figure 2 :

 2 Figure 2: Same as Figure 1 but for ARCM SST response.

Figure 3 :

 3 Figure 3: 30-year average sea surface temperature difference in °C (a) in winter and

Figure 4 :

 4 Figure 4: Same as Figure 1 but for 2-meter temperature (a) in winter and (b) in

Figure 5 :

 5 Figure 5: Same as Figure 1 but for precipitation (a) in winter and (b) in summer (in

Figure 6 :

 6 Figure 6: Same as Figure 3 (DD values) but for 2 meter temperature (a) in winter and

Figure 7 :

 7 Figure 7: Same as Figure 3 but for precipitation (a) in winter and (b) in summer (in

Figure 8 :

 8 Figure 8: Same as Figure 3 but for mean sea level pressure (in hPa) in winter.

Figure 9 :

 9 Figure 9: Same as Figure 3 but for 2m temperature (in °C) in spring.

Figure 10 :

 10 Figure 10: Same as Figure 3 but for atmosphere water vapour (in mm) in summer.

Figure 11 :

 11 Figure 11: Same as Figure 3 but for nebulosity (in %) in summer.

  is the mean climate change response of the AORCM between the 1961-1990 period and the 2070-2099 period. ∆(AORCM) -∆(ARCM) is the difference between the AORCM and ARCM climate change responses (denoted DD in the text). A value noted *** or ** or * means that the difference is statistically significant with a 99% or 95% or 90% significance level. The significant values of DD are highlighted by grey boxes. The InterModel Distances (IMD GCM , IMD RCM ) are computed only for significant values of DD (see the text for the definition of IMD) and should be compared to the DD values.

Figure 1 :Figure 2 :

 12 Figure 1: AORCM response to the climate change for the 30-year average sea surface

Figure 4 :Figure 5 :Figure 6 :Figure 7 :

 4567 Figure 4: Same as Figure 1 but for 2-meter temperature (a) in winter and (b) in

Figure 8 :

 8 Figure 8: Same as Figure 3 but for mean sea level pressure (in hPa) in winter.

Figure 9 :

 9 Figure 9: Same as Figure 3 but for 2m temperature (in °C) in spring.

Figure 10 :

 10 Figure 10: Same as Figure 3 but for atmosphere water vapour (in mm) in summer.

Figure 11 :

 11 Figure 11: Same as Figure 3 but for nebulosity (in %) in summer.

TABLE 2a

 2a In summer, the AORCM was too warm compared to the CRU dataset over the East-Europe box and too cold over the Mid-Europe, Scandinavia and Alps boxes.

	and TABLE 2b
	3.2.2 Summer bias
	Temperature biases are nevertheless always less than 1°C. Precipitation biases are
	also weak for this season (less than 0.3 mm/d) although the model is significantly too
	wet for Iberian Peninsula, France, Mid-Europe and Scandinavia. No systematic

  in good agreement with his study. Moreover RCCIs higher than 16 were obtained over all our own boxes included in the MED and NEE

regions: 20 over Iberian Peninsula (IP), 17 over the Mediterranean (MD) and 24 over East-Europe (EE). The good agreement with Giorgi's results allows us to have some confidence in the climate change signal simulated by the AORCM and singles out the Mediterranean Basin and North-Eastern Europe as the climate change "hot-spots" of the AORCM simulation.

Table Captions Table 1 :

 Captions1 Definition of the European sub-areas used in the text expressed in degrees in longitude and latitude.

Table 2a :

 2a Validation table for winter (December-January-February) and summer (June-July-August) for the two-meter temperature (in °C, box averages for the boxes defined in Table1): CRU data (values from[START_REF] Jacob | An intercomparison of regional climate models for Europe: Model performance in Present-Day Climate[END_REF], AORCM bias compared to CRU, and AORCM-ARCM difference. A value noted *** or ** or * means that the difference is statistically significant with a 99% or 95% or 90% significance level. The significant values are highlighted by grey boxes for the AORCM bias and the AORCM-ARCM difference.Tab. 2b: Same as Table2abut for precipitation (in mm/d).

Table 3a :

 3a Climate change impact for the 2-meter temperature for the four seasons and for the boxes defined in Table1. ∆(AORCM)

Table 3b :

 3b Same as Table 3a but for precipitation (in mm/d).

	Area	West East South North
	1 (BI) British Isles	-10	2	50	59
	2 (IP) Iberian Peninsula -10	3	36	44
	3 (FR) France	-5	5	44	50
	4 (ME) Mid-Europe	2	16 48	55
	5 (SC) Scandinavia	5	30 55	70
	6 (AL) Alps	5	15 44	48
	7 (MD) Mediterranean	3	25 36	44
	8 (EE) East-Europe	16	30 44	55

Table 1 :

 1 Definition of the European sub-areas used in the text expressed in degrees in longitude and latitude.

	AORCM bias	-0.3	-0.3	-0.3	-0.7**	-0.9**	-0.8**	0.1	0.7**
	AORCM -ARCM	-0.2	-0.5	-0.5	-0.4	-0.2	-0.4	-0.5	-0.4

Table 2a :

 2a Validation table for winter (December-January-February) and summer (June-July-August) for the two-meter temperature (in °C, box averages for the boxes defined in Table1): CRU data (values from[START_REF] Jacob | An intercomparison of regional climate models for Europe: Model performance in Present-Day Climate[END_REF], AORCM bias compared to CRU, and AORCM-ARCM difference. A value noted *** or ** or * means that the difference is statistically significant with a 99% or 95% or 90% significance level. The significant values are highlighted by grey boxes for the AORCM bias and the AORCM-ARCM difference.Tab. 2b: Same as Table2abut for precipitation (in mm/d).

		BI	IP	FR	ME	SC	AL	MD	EE
					precipitation, DJF
	CRU data	3.4	2.7	2.4	1.7	1.8	3.2	3.1	1.3
	AORCM bias	0.2	-0.9** 0.8**	1.4** 0.7**	-0.1	-1.1**	0.4**
	AORCM -ARCM	-0.2	0.0	-0.1	-0.2	0.0	-0.2	0.0	0.0
					precipitation, JJA
	CRU data	2.5	1.1	1.8	2.3	2.4	4.0	1.5	2.6
	AORCM bias	0.0	0.2**	0.3**	0.3** 0.3**	-0.2	0.2	0.0
	AORCM -ARCM	0.1	0.1	0.2	0.1	0.1	0.2	0.1	0.1

Table 3a :

 3a Climate change impact for the 2-meter temperature for the four seasons and for the boxes defined in Table 1. ∆(AORCM) is the mean climate change response of the AORCM between the 1961-1990 period and the 2070-2099 period. ∆(AORCM) -∆(ARCM) is the difference between the AORCM and ARCM climate change responses (denoted DD in the text). A value noted *** or ** or * means that the difference is statistically significant with a 99% or 95% or 90% significance level. The significant values of DD are highlighted by grey boxes. The InterModel Distances (IMD GCM , IMD RCM ) are computed only for significant values of DD (see the text for the definition of IMD) and should be compared to the DD values.

		BI	IP	FR	ME	SC	AL	MD	EE
					2-meter temperature, DJF
	∆(AORCM)	2.2*** 2.9*** 2.8***	3.2*** 4.2*** 3.3***	3.2***	4.2***
	∆(AORCM)-∆(ARCM)	0.5**	0.3	0.7**	0.8**	-0.1	0.7**	0.4	0.9**
	IMD GCM / IMD RCM	0.6 / 0.3		0.3 / 0.4 0.5 / 0.5		0.8 / 0.6		0.8 / 0.5
					2-meter temperature, MAM
	∆(AORCM)	1.7*** 3.3*** 2.2***	2.5*** 3.6*** 3.0***	3.3***	4.2***
	∆(AORCM)-∆(ARCM)	0.0	0.2	0.2	0.3	0.6	0.3	0.4	0.9**
	IMD GCM / IMD RCM								no data / 0.8
					2-meter temperature, JJA
	∆(AORCM)	2.7*** 4.8*** 4.6***	3.6*** 2.7*** 4.7***	5.2***	4.6***
	∆(AORCM)-∆(ARCM)	0.1	0.1	0.3	0.5	-0.2	0.5	0.8**	0.9**
	IMD GCM / IMD RCM							1.0 / 0.7	2.0 / 1.3
					2-meter temperature, SON
	∆(AORCM)	3.1*** 3.8*** 3.7***	3.5*** 3.2*** 3.5***	3.6***	3.6***
	∆(AORCM)-∆(ARCM)	0.2	0.2	0.5	0.2	-0.3	0.2	-0.1	-0.3

Table 3b :

 3b Same as Table 3a but for precipitation (in mm/d).

		BI	IP	FR	ME	SC	AL	MD	EE
					precipitation, DJF
	∆(AORCM)	0.8***	0.0	0.3	0.4** 0.5***	0.3	-0.2	0.2**
	∆(AORCM)-∆(ARCM)	0.4**	-0.1	0.2	0.3	0.2	0.3	-0.1	0.1
	IMD GCM / IMD RCM	0.3 / 0.3
					Precipitation, MAM

Acknowledgments

This work was supported by the ENSEMBLES European project (contract GOCE-CT-2003-505539) financed by the European Commission under the 6 th Framework Programme and by the CYPRIM project in the program ACI -FNS "Aléas et Changements Globaux" of the Ministère de l'Enseignement et de la Recherche (French Research Department). We also thank Dr. Aurore Voldoire for computing AOGCM values, Dr Sébastien Conil, Dr Anna Pirani, Dr Nellie Elguindi and Jeanne Colin for helpful comments and suggestions.

** 0.2 AORCM -ARCM -


domain: math shs

text:
Tinne Hoff Kjeldsen

Morten Blomhøj
email: blomhoej@ruc.dk

Students' video tutorials as a means for supporting and analysing their reflections on the limit concept

Keywords: Limit concept, calculus, student produced video-tutorials, student-student interactions, theories on the learning of mathematical concepts, learning environment

We discuss how a theory based design for students' production of video-tutorials explaining problem solving activities in a first year calculus course can support the students' conceptual learning. We focus on the limit concept and show how the production of tutorials can facilitate the students' interactions and reflections and at the same time provide a rich source for analyzing their learning difficulties in relation to key concepts. The theory based design of students' productions of tutorials and the following analyses can inform and support the development and implementation of the learning environment in order to facilitate better the students' conceptual learning.

Introduction and research questions

The many papers in the literature of mathematics education devoted to calculus signify the difficulties with the teaching and learning of calculus with the limit concept as one of the fundamental challenges (Rasmussen & Borba, 2014). In this paper we present and discuss a theory based design and its implementation for students' productions of tutorials aimed at supporting the students' learning of the limit concept. The research was related to a teaching experiment in a first year calculus course at Roskilde University, where the students produced tutorials with the app Explain Everything or similar apps1 . Building on theories explaining general learning difficulties with mathematical concepts such as the notion of students' concepts images [START_REF] Vinner | Images and definitions of the concept of function[END_REF]) and [START_REF] Sfard | On the dual nature of mathematical conceptions: Reflections on processes and objects as different sides of the same coin[END_REF] model for formation of mathematical concepts and the importance of communication in the learning of mathematics [START_REF] Sfard | Thinking as Communicating[END_REF] a didactic tutorial activity was designed. The purpose was to get the students to express their images of key mathematical concepts in dialogues with their fellow students and to support their concept formation process in particularly with respect to their use of symbolic representations and their reification of key concepts such as the derivative, integral and limit concepts as mathematical objects. The activity was designed and implemented as an assignment; namely to produce in small groups video tutorials explaining the group's solution to selected exercises and the mathematical basis hereof, see figure 1. In the assignment specific requirements for the workflow of the students' production were given in order to support the didactic purpose. The tutorials were used as a resource by the students at the course and in preparing for the final.

In the design and implementation of the experiment the basic idea was to use the digital tutorial genre to create a theory directed learning environment that encouraged the students to communicate with each other about mathematical theory, concepts and techniques. The tutorial genre mediated student-student interactions and hereby supported students' reflections on key mathematical concepts. The didactical assumption was that by focusing the students' attention on the production of tutorials, aimed at supporting their fellow students' learning, it would be possible to create situations where small groups of collaborating students would activate and express their own understandings and images of the mathematical concepts in focus. In addition, the digital tutorials could be (and was) used by students for their own retention of the problem solving techniques and their mathematical foundationparticular in preparing for the final. Furthermore, re-viewing and discussing the tutorials could support the students' reflections on their conceptual learning or be used as a basis for common reflections.

With respect to the scope and focus of the working group Implementation of Research Findings in Mathematics Education (see the introduction to TWG23), the implementation aspect of this paper and further related work is the design and organization in the classroom of the digital tutorial in creating a learning environment in which research findings on students' concept formation and the importance of communication are brought to use in the teaching of a first year calculus course. We are developing our practice of mathematics teaching through interplay with researchand in this process we are also engaged with research, so in the present paper, implementation of research results and research are intertwined.

We audio recorded the students' reflections during their work with the tutorials. The transcripts of these records, the tutorials, the students' performance at the course and observations during the course form the empirical basis for addressing our three didactical research questions (RQ):

1. In what sense and degree can a learning environment focusing on the students' productions of tutorials and other digital products with subject matter content support the students' formation of key concepts in calculus?

2. Which learning difficulties related to the key concepts in calculus can be revealed and theoretical explained through analyses of such student activities and products?

3. How can the findings from such analyses be used to further develop the learning environment and the way in which the students' products are used in the calculus course?

In this paper we address the three RQs focusing on the limit concept.

The context of the experiment

The experiment was performed at the calculus course in the natural science bachelor program at Roskilde University. There were 32 first year students in the course, which was taught in English using a typical American calculus textbook [START_REF] Adams | Calculus, eight edition[END_REF]. During the previous few years, it had become evident that the students' had problems explaining and applying the concepts and methods of calculus in their subsequent courses. One possible explanation for the increasing difficulties for the students in developing their conceptual learning in the calculus course could be changes in their prerequisites concerning key mathematical methods and concepts from high school, which are fundamental for calculus such as algebra, variable, function and the limit concept. CAS and graphing software for calculators and computers are used intensively in mathematics teaching at high school in Denmark and students learn instrumented techniques to solve standard problems by means of such tools. Similar experiences are reported in other educational systems. [START_REF] Barbé | Didactic restrictions on teachers' practice-The case of limits of functions in Spanish high schools[END_REF] have analyzed the situation in Spanish high schools, in particular the incoherence in the mathematical organizations of the theory of limits at high school level. Similar incoherencies have been found with other key elements of upper secondary level calculus in Denmark and American textbooks for first year calculus courses at the university level, due to the elimination of the topological parts of mathematical analysis (Winsløw 2015, pp. 200-203). In general, the extended use of it-instrumented techniques, in particular CAS-based techniques, poses challenges for the teaching of calculus in high school and at introductory courses at university entrance level. Gücler (2014, p.4) addresses the role of the teacher in students' learning of the limit concept in such a teaching practice and pinpoints the importance of the teacher in challenging the students to consider limits both from a process and an object perspective in different mathematical contexts. In particular, the students need to communicate their conceptual understanding in order to really experience both of these perspectives and typically they will need specific support in order to grasp the idea of the limit of a function as a mathematical object.

Our approach to this challenge is to experiment with the learning environment so as to make the students produce and publish their own mathematical products in the form of tutorials in the digital genre through the use of various app's and video-recordings with tablets. The students' products combined oral and written explanations and presentations of their mathematical task and its content matter with visual and dynamical elements.

Implementation of the experiment: mediatized presentations and assessment

The students' fabrications of the mediatized tutorials were connected to their usual homework of sets of traditional problems and exercises. During the course, the students handed in three sets of homework that each consisted of between 10 to 21 standard problems and exercises from the text book. The students were allowed to discuss the solutions to the exercises with each other, but they handed in an individual presentation of solutions to the exercises. In order to participate in the final, a student had to have all three homework sets approved by the teacher. After the deadline for handing in a homework set, the exercises of the set were distributed between the students who in small groups produced a video tutorial of a solution to one of the exercises using the app Explain Everything or by other means. The video assignments were accompanied by a set of requirements for the product and a specific work flow which the students had to follow. The instructions given to the students can be seen in figure 1.

Each student also participated in a project work in which the students worked in groups with different subjects from the text book that were not covered in class. Each group wrote a technical report of two pages, which was supplemented by various video and/or other visual and oral digital products.

Finally, the assessment criteria were changed. The written test towards the end of the course was replaced by an oral test. The oral examination consisted of two elements: (1) a 12-15 minutes group presentation of the group's project work. Every student was required to participate actively in the presentation of his/hers project work;

(2) an individual examination of 8-10 minutes following the group presentation. After the presentation of the project work, the group left the examination room and hereafter each student was called back in for the individual examination. Each student drew by chance one of the three homework sets of exercises, and made a 5-6 minutes presentation of selected parts of the subject matter covered by the exercises in the set. The presentation by the student was followed by 4-5 minutes of questioning in the remaining parts of the course and/or homework sets. The alignment of the requirements for the students' work during the course and the final was an important aspect of the design of the new learning environment. The oral communication skills which were need for the final were trained through the student-student interactions in the design phase of the video productions as well as in the actual fabrication of the videos of the homework exercises and the mediatized parts of the project work. The mediatized products of the project report were directly aligned with the first part of the final, and the video tutorials were directly aligned with the second part of the final. The bank of video tutorials of the total amount of exercises in the portfolio sets that were produced by the students during the course helped the students prepare for the second part of the oral examination. The students carefully designed tutorials that explained clearly and in depth how to solve the exercises, because they were to be used by their fellow students in their preparation for the final.

Analyzing the students' reflections on the limit concept

The students' tutorials were examined during the course in order to identify those which were rich enough to be analyzed for answering RQ 2 with respect to the limit concept. Among them, a few were selected for discussions with the students in the classroom. Later, the material might be used for analyses that focus on other key concepts. Here, we illustrate with one case how we have analysed the students' work with the tutorials for answering RQ 2 focusing on the limit concept.

Guided by the theories, we looked for indicators of students' image concept, of students revealing a conceptual understanding, and of students using a process oriented conception when we analysed the data. The data is transcripts of the audio recordings of one group of students while they produced their first tutorial of an exercise from the first homework set. The students' video is also included in our analysis.

The students' objective was to explain the limit concept using the exercise of finding the limit:

9 9 6 2 2 3 lim     x x x x
To begin with, the students observed that the limit cannot be found simply by inserting x=3 in the expression, since the denominator is zero for x=3. They mentioned the technique of factorization in order to handle this challenge. This technique is used in several examples in the text book. During their work, the students had software for graphical and CAS analysis at their disposal. However, before implementing this technique they began to reflect on the meaning of the limit concept. The students realized that the video should be aimed at "all the students that are struggling with calculus", as student 2 phrased it. Accordingly, they took on the challenge of explaining the limit concept in their tutorial. Their dialogue went as follows: S1:

Wait a second and now I think I'm lost a little bit. What is this limit thing actually doing?

S2:

We have to watch the limit, so we have to watch the value that our function is approaching, when it's, when the axis approach …[S2 draws a graph of the function on the computer.] S1:

Ah, yeah so we can see here, that's true. Okay trace, so we can see here, so we take x value 3, eeh.. to here, so it's pretty close to 0.! S1: Makes a lot of sense. We can also zoom in! S1: I think you just say that a limit is a number that is (inaudible) by. We have a value that is approaching a number ... right?

S2:

A variable that is approaching a number?

S1:

We have a function. We have variables in a function that is getting approached by a number. .. It's like we're getting infinitely close, we're getting infinitely close to a number in the function without reaching it. That's a limit.

Here we see that there is a close interplay between, on the one hand, the students' understanding of the particular problem in hand and their ideas and images for the limit concept in general, and on the other hand, the challenge of producing an explanatory tutorial for their fellow students. The situation forces the students to activate and verbalize their images of the limit concept. It becomes clear for the students that their understanding is not sufficiently developed for explaining the limit concept clearly in their own words in the tutorial. In the process of writing the manuscript for their tutorial the students consulted the text book of the definition of the limit concept. The dialogue continued as follows:

S1: yeah, okay, but let's first answer the first one. ... What's an introduction to the topic. I think an introduction would just be like "A limit is an. you can, you can say limit of x when x approaches 2, it's (over) the limit. S1: yeah okay, but the limit is numbered, but it never reaches that number, that's not the idea. The limit is just taking it to the limit.

S2: okay!

The students are getting into a deep discussion of how to understand the limit concept. Student 1 is quite persistent in his process-based thinking of a limit, and he has difficulties with accepting that the limit of a function can actually be reached. Student 2, however, switches to a more object-based thinking in their discussion of how they should introduce the limit concept in their tutorial. In some of her utterances she is getting close to formulating the definition of a limit in her own words, and she seems to be aware of the fact that the value of the function can be forced arbitrary close to its limit in a certain point x=c by choosing x to be sufficiently closebut not equalto c.

After a while student 1 begins to switch into a more object-based thinking of limits. He says: S1: a limit is a number that is being. A limit is a value, you have (on there), the y axis, when you're approaching a number on the x axis…. But is it mathematically correct to say that? It's like if you have a function, and you go towards the number.

In their tutorial the students explain in detail doing the algebraic manipulation by hand how to get to the identity:

) 3 )( 3 ( ) 3 ( 9 9 6 2 2 2        x x x x x x
, which for x≠3 is equal to

) 3 ( ) 3 (   x x .
They explain in the tutorial how the limit of the given expression can be evaluated by substituting 3 for x in "the redefined function" and that this yields the result 0. The students also explain that the given function is not continues for x equal to 3: "It has a hole in its graph for x equal to 3, while the redefined function is continues in this point and has the value 0". However, the crucial mathematical argument is not stated or explained explicitly in their tutorial; namely that since the two expressions are equivalent for all real value of x ≠3, then Discussion and some initial conclusions Focusing on the limit concept, our findings in relation to the three RQs are:

Ad (1): The learning environment encourages the students to articulate and communicate their understanding of key concepts such as the limit concept in various situations and forms of representations. As illustrated in the analysis presented, in the process of producing the tutorial, the students engaged in dialogues about the limit concept using their own language. In their tutorial they used oral, written and visual forms of expressions in communicating their conceptual understanding and the techniques they used in solving their problem. The crucial didactical feature of the learning environment is the focus on the students' productions of the tutorials aimed at their fellow students. Together with the prescribed work flow and the requirements specified in figure 1, the focus on the productions of the tutorials, encouraged and enabled the students to express and reflect on their own conceptual understanding. In general, from analyzing the students' dialogues during their production of their tutorial it is evident that only when the students began to work on the video production they became fully aware of what their task really was about, how the technique they used could be explained, and why their results were correct. Ad (2): From the analysis it is clear that the students found it very difficult to really understand and use the formal definition of the concept of a limit of a function from the text. Especially student 1 insisted on understanding the limit concept as a process or actually as two connected processes. He did not focus on the quality of the limit of a function, and he emphasized wrongly that the limit is a number which will not be reached by the function. This phenomenon can be understood by means of the concept of students' concepts images [START_REF] Vinner | Images and definitions of the concept of function[END_REF]. Their research explains and evidences that formal concept definition only become meaningful to students to the extent in which they are unfolded and concretized by personal experiences. The phenomenon can be further analyzed by means of the processobject duality of mathematical concepts [START_REF] Sfard | On the dual nature of mathematical conceptions: Reflections on processes and objects as different sides of the same coin[END_REF].

In the tutorial, the students explain how to use the technique of factorization to find the limit of the given function. They reached a new expression for the function, which is equivalent to the given expression for all real values of the independent variable x except for x=3. Only the new expression is defined for x=3 and can therefore be evaluated in x=3, which yields 0. However, in the tutorial, the students did not really explain why the limit for x → 3 have to be the same for the two expressions since they are equal for all values of x except for x=3. [START_REF] Güçler | Examining the discourse on the limit concept in a beginning-level calculus classroom[END_REF][START_REF] Güçler | The role of symbols in mathematical communication: The case of the limit notation[END_REF] found and analyzed similar difficulties in students' learning of the limit concept. Ad (3): In general, analyses of the students' productions of the tutorials and the tutorials themselves allow pinpointing learning difficulties related to key concepts in calculus such as the limit concept, which are theoretically explainable. The analyses can provide ideas for variation of problems to be dealt with in the tutorials in order to invoke and challenge the students' different concept images of the key concepts in calculus. Such analyses can inform the development of the learning environment in order for the theories to be used in the practice of teaching for identifying and help overcoming students' learning difficulties. Moreover, anchored to the students' experiences with the tutorial, theories can help students develop a sound meta-learning related to the formation of mathematical concepts.

Also, the tutorials can be used in whole class teaching as a point of departure for discussing the relations between the techniques used to solve the different types of problems addressed and the mathematical theory explaining them and hereby contribute to the development of the mathematical organization in the calculus course.

Figure 1 :

 1 Figure 1: Assignment, product requirement and workflow for video production. 'Tim' in step 4 of the work flow refers to the teaching assistant. The students were requested to perform a quality test through his approval before they continued to step 5

  reached, it can be reached. I just. You don't have to say if it's reached or defined or undefined, you just say that it's a number that the function approaches as x approaches. As x gets closer and closer to the c. [Referring to the notation in the textbook]

		"!
	S2:	value that
	S1:	it's actually like…
	S2:	a value or number, maybe a number. It's a number that…
	S1:	yeah it is a number that is getting approached, but is never reached!
	S2: it can be S1: but I mean it doesn't reach the limit, that's the idea.

S2:

it has… it could. It can because you can write, because that's only the...

S1:

it's still the number it's going towards but… S2:

The pedagogical set up was developed in collaboration with Maja Bødtcher-Hansen, University of Copenhagen.Thematic Working Group

Proceedings of CERME10

Proceedings of CERME10