L Mouysset

Agricultural public policy: green or sustainable?

Keywords: Agriculture, Biodiversity, Public policy instrument, Poverty, Bird, Land-use, Cost-effectiveness

The future of agriculture constitutes a major challenge to the achievement of sustainable development. There are new perspectives on greening(focusing on ecological objectives) and sustainability (combining both ecological and social goals). Academic papers rather study the ecological efficiency of agricultural public policies, while real public policies, such as in the European Common Agricultural Policy, examine both ecological and social considerations. The objective of this paper is to consider economic, social and ecological objectives within the design of agricultural public policies. Using a bio-economic model applied to France, we compare different optimal public strategies. We show that, when the biodiversity objectives are either very limited or very demanding, grassland subsidies are the best instruments from both green and sustainable points of view. However for medium objectives, reducing crops subsidies is the cheapest way to green the CAP, while subsidies on grasslands are the only strategy from a sustainability perspective. Our work highlights new trade-offs related to policy implementation, such as social acceptance or technical difficulties, and the spatial equity of performance among regions.

Introduction

Developing sustainable agriculture will be a major challenge in the coming decades, and particularly with regard to the impact of agriculture on the environment [START_REF] Tilman | Agricultural sustainability and intensive production practices[END_REF]. Changes in agricultural practices, including intensive of cropping, landscape homogenization, loss of semi-natural elements, mechanization and intensive use of inputs, have had several consequences such as increased water pollution [START_REF] Carpenter | Nonpoint pollution of surface waters with phosphorus and nitrogen[END_REF][START_REF] Volk | Towards the implementation of the European Water Framework Directive?[END_REF][START_REF] Tong | Modeling the relationship between land use and surface water quality[END_REF] and the loss of biodiversity [START_REF] Foley | Global consequences of land use[END_REF][START_REF] Tscharntke | Landscape perspectives on agricultural intensification and biodiversity -ecosystem service management[END_REF][START_REF] Tilman | Forecasting agriculturally driven global environmental change[END_REF][START_REF] Chamberlain | Changes in the abundance of farmland birds in relation to the timing of agricultural intensification in England and Wales[END_REF]. In particular, consequences for biodiversity are widespread since many taxa are affected: see [START_REF] Flowerdew | Mammal biodiversity in agricultural habitats[END_REF] for mammals, [START_REF] Sotherton | Changes in plant and arthropod biodiversity on lowland farmland: An overview[END_REF] for plants and [START_REF] Donald | Agricultural intensification and the collapse of Europe's farmland bird populations[END_REF] for birds. These negative effects are due mainly to degradation of habitat quality altering nesting success and survival [START_REF] Benton | Farmland biodiversity: Is habitat heterogeneity the key?[END_REF]. As a result, ecological considerations are increasingly being integrated into agricultural policy. For example, in Europe, such considerations have been introduced since the 90's through the European Common Agricultural Policy (CAP). Notably, agri-environmental schemes were developed to promote protection of biodiversity. However 15 years after their implementation, their effectiveness remains controversial [START_REF] Whittingham | Will agri-environment schemes deliver substantial biodiversity gain, and if not why not[END_REF][START_REF] Kleijn | Mixed biodiversity benefits of agri-environment schemes in five European countries[END_REF][START_REF] Vickery | The role of agri-environment schemes and farm management practices in reversing the decline of farmland birds in England[END_REF]). The management of biodiversity in farmlands is still an open question, especially with ongoing debates about ways to improve the use of the dedicated budget into the CAP .

In addition to these environmental considerations, the socio-economic consequences of the CAP are not ubiquitously positive, even though the agricultural incentives originally were developed for economic reasons. By subsidizing production, the goal was to ensure minimum levels of income for farmers. However, this strategy has encouraged a process of intensification, generating a two-speed agriculture [START_REF] Strijker | Marginal lands in Europe -causes of decline[END_REF]). On the one hand, in the regions capable of intensification, the CAP process has stimulated enlargement of intensive farms and increased yields, generating high incomes and profits. On the other hand, in regions where intensification is impossible, incomes have remained low and a process of land abandonment has started [START_REF] Mottet | Agricultural land-use change and its drivers in mountain landscapes: A case study in the Pyrenees[END_REF][START_REF] Macdonald | Agricultural abandonment in mountain areas of Europe: Environmental consequences and policy response[END_REF], which is deepening the gap between rich and poor farmers. More nuanced social considerations, concerning poverty for example, have now been integrated in to the CAP.

Ecological and social criteria are currently being integrated with the historical economic objectives of the CAP. More precisely, they are part of a structure referred to as the "second pillar," the first pillar being the historical support for production and incomes [START_REF] Lowe | Setting the next agenda ? British and French approaches to the second pillar of the Common Agricultural Policy[END_REF]. Although the public policy instruments that are part of this second pillar seem to be socially acceptable [START_REF] Prager | Stakeholder involvement in agri-environmental policy making-learning from a local-and a state-level approach in Germany[END_REF], they have been strongly criticized for their ecological inefficiency [START_REF] Stoate | Ecological impacts of early 21st century agricultural change in Europe-a review[END_REF][START_REF] Kleijn | Agri-environment schemes do not effectively protect biodiversity in Dutch agricultural landscapes[END_REF]. The gap between the real objectives of the CAP and those studied in the literature opens interesting questions about the objectives of future CAP: do the objectives have to focus on ecological considerations or do they are in line with sustainability perspectives by considering ecological and social criteria? In other words, is the objective to green the CAP or to make it more sustainable? While the second pillar of the CAP is now defined from sustainability point of view [START_REF] Sutherland | Restoring a sustainable countryside[END_REF], the question of greening CAP remains crucial in current debates [START_REF] Scherr | Biodiversity conservation and agricultural sustainability: towards a new paradigm of 'ecoagriculture' landscapes. Philosophical transactions of the Royal Society of London[END_REF]. The originality of this paper lies in its contribution to these two viewpoints. We compare green public policies on the one hands and sustainability policies on the other.

There are various strategies to design public policies. More specifically agricultural public policies frequently are aimed at two different agricultural systems. The first concerns crops, which are characterized by high average incomes, and are associated with strong average negative impacts on biodiversity [START_REF] Chamberlain | Changes in the abundance of farmland birds in relation to the timing of agricultural intensification in England and Wales[END_REF][START_REF] Donald | Agricultural intensification and the collapse of Europe's farmland bird populations[END_REF][START_REF] Stoate | Ecological impacts of arable intensification in Europe[END_REF]. The second concerns grasslands, which are associated with lower average incomes but have a more posi-tively effect on biodiversity by preserving the natural habitats essential of many species [START_REF] Laiolo | Spatial and seasonal patterns of bird communities on italian agroecosystems[END_REF]. However, the impact of land-use and land-use changes on biodiversity is more complex since it has been shown that their ratio is a key element for biodiversity (Robinson et al. 2002). To impact these land-uses, price instruments (taxes or subsidies) are today considered by decision-makers, since they are applied to crops and grasslands respectively within the first and the second pillars of the current CAP.

In this paper, we compare green and sustainability public policies based on economic tools applied to crops or grasslands. Public policy scenarios are designed following an optimal under constraint approach in which the decisionmaker maximizes a welfare criterion under social or ecological constraints. The optimal instruments are analysed and the consequences on different criteria (welfare, social, biodiversity) due to the introduction of these new constraints are compared at different scales.

The paper is organized as follows. Section 2 presents the bio-economic model and the scenarios. Section 3 describes the case study. Sections 4 and 5 are devoted to the results and the discussion.

The bio-economic modelling

The bio-economic model in this paper extends the model developed in [START_REF] Mouysset | How does economic risk aversion affect biodiveristy ? Ecological Applications[END_REF], which links the economic decisions of representative farmers to environmental quality and biodiversity at local scale. In the present paper, a third component related to national scale public policy at has been added. This multi-scale model is depicted in figure 1. A decision-maker chooses public policy scenarios, that impact on the choices of the farmers and thus, in an indirect way, affect habitat quality and biodiversity. Although the ecological equations remain similar1 , the introduction of public policy levers modifies the equations of the economic model.

The economic model of the farmer

As in [START_REF] Mouysset | How does economic risk aversion affect biodiveristy ? Ecological Applications[END_REF], each region r is assumed to be managed by a representative farmer who determines the areas A r,k (t) of each land-uses k to maximize his expected utility depending on the mean and dispersion of incomes. However the income function now includes public policy incentives in addition to agricultural rents:

Inc r (t) = k A r,k (t).gm r,k .(1 + τ k ) (1)
where A r,k (t) denotes the areas dedicated to the land uses k in the region r, gm r,k represents the associated expected gross margin, and τ k the economic incentives applied to the land-use k (τ > 0 for subsidies, τ < 0 for taxes).

Similarly to [START_REF] Mouysset | How does economic risk aversion affect biodiveristy ? Ecological Applications[END_REF], a quadratic form is used for the utility function to characterize the representative agent per small agricultural region:

U r (t) = E[Inc r (t)] -a.Var[Inc r (t)] (2)
(3)

based on the expected income E[Inc r (t)], its risky part Var[Inc r (t)] (based on the covariances between gross margins of land-uses k and k ′ in region r), and the risk aversion level of farmers a (see details in [START_REF] Mouysset | How does economic risk aversion affect biodiveristy ? Ecological Applications[END_REF]).

The maximizing program of farmer's utility in an uncertain context is defined as follows: max

Ar,1;...;A r,K U r (t) (4)
Furthermore, when maximizing the utility, the standard agent must comply with three constraints at each point in time:

|A r,k (t) -A r,k (t -1)| ≤ ε.A r,k (t -1) (5) k A r,k (t) = A r (6)
The first constraint (eq. 5 ) corresponds to a technical constraint where the coefficient ε stands for the rigidity in changes. The second one (eq. 6) is a stability constraint ensuring the total agricultural surface A r per region constant.

The bio-economic indicators for the stakeholder

The performance of the ecological-economic model can be analysed using different indicators. These indicators are used for the design of public policies. The decisions of stakeholder are based on the classical welfare criterion. This welfare corresponds to the total wealth of society on a national scale. It includes the economic states of both private agents (farmers) and public agent (the budget available to subsidize farmers, providing from the general taxes levied society). In other words, welfare corresponds to the evolution in net income generated by the farms, excluding transfers (taxes and subsidies).

The private richness of farmers is analysed through national income Inc(t) (eq. 7) computed as the sum of the product of micro incomes Inc r (t) (as defined in eq. 1):

Inc(t) = r Inc r (t) = r k A r,k (t).gm r,k .(1 + τ k ) (7)
The public wealth corresponds to the non-spent public budget defined as below, where Env represents the initial available envelope:

Budg(t) = Env - r k τ k .gm r,k .A r,k (t) (8)
The total richness is computed as the sum of private and public wealth (eq. 9). Because of the public policies modifying rents or constraints on land-use, the A r,k (t) chosen by the farmers differ according to the scenario, leading to some variations in total wealth.

Rich(t) = Inc(t) + Budg(t) = r k gm r,k .A r,k (t) + Env (9)
Finally we define inter-temporal total richness among time Rich. This corresponds to the sum of total richnesses discounted at the rate ρ from the first year of the projection t 1 to the final time horizon T :

Rich = T t=t1 ρ t-t1 . Rich(t) (10)
Social and environmental criteria are considered as constraints by the decisionmaker. First in a social perspective, the objective is to defence the poorest farmers. Following a maximin perspective [START_REF] Solow | Intergenerational equity and exhaustible resources[END_REF][START_REF] Heal | Valuing the future, economic theory and sustainability[END_REF]), the criterion is then the poorest regional income among the regions:

P oor(t) = min r (Inc r (t)) (11)
The associated intertemporal criteriaon P oor is computed as follows:

P oor = T t=t1 ρ t-t1 . P oor(t) (12)
Second, in an ecological perspective, we use a criterion that ensures a minimum state of biodiversity. Large scale ecological performance is assessed using biodiversity indicators, denoted Biod, but not specified at this stage. This formulation includes the usual biodiversity indices such as species richness, Simpson or trophic indices. They are based on the abundance N s,r (t) of the species s = 1, .., S and the regions r = 1, .., R involved:

Biod(t) = f N 1,r (t), . . . , N S,R (t) (13)

The public policy scenarios

The Laissez-Faire scenario To analyse the impact of public policies, we first define a Laissez Faire (LF) scenario without public intervention (eq. 14), which corresponds to the situation described in [START_REF] Mouysset | How does economic risk aversion affect biodiveristy ? Ecological Applications[END_REF]. Farmers optimize their land-use according to the economic model, win which the only drivers are the expected gross margins and the level of risk aversion. We note the ecological and economic performances of this scenario Rich LF , Biod LF and P oor LF .Then we define public policy scenarios based on the vector of economic incentives τ = [τ k=1 . . . τ k=K ]. Typically the LF scenario is defined as follow:

τ LF k = 0 f or all k (14)
The Green scenario In public policy scenarios, the decision-maker optimally determines the levels of τ using an inter-temporal maximization programme under constraints. The stakeholder makes its choice by maximizing the inter-temporal richness Rich(τ ):

max τ Rich(τ ) (15)
under the budgetary (eq. 16) and the environmental (eq. 17) constraints: 16)

Budg(t) ≥ 0 (
Biod(T ) ≥ (1 + µ) * Biod LF (T ) (17)
The budgetary constraint (eq. 16) ensures that the decision maker does not spend more than the envelope (i.e. the initial envelope and the payments derived from taxes). The ecological constraint (eq. 17) is based on a conservation limit for the biodiversity goal imposed at the temporal horizon T and the performance obtained with the LF scenario. Different values of the percentages µ can be tested between the lowest value µ = 0 and the maximal feasible biodiversity2 corresponding to b = Biod * (T )/Biod LF (T ). These scenarios with the ecological constraint are described as green (GRE) scenarios.

The Sustainable scenario

We also tested sustainable (SUS) scenarios similarly taking account of both social and ecological constraints. In these scenarios, the decision-maker optimally determines the levels of τ by maximizing Rich(τ ) (eq. 15), under the budgetary constraint (eq. 16), the environmental constraint (eq. 17) and a social constraint defined as follows:

P oor ≥ (1 + ν) * P oor LF (19)
This social constraint (eq. 19) is computed with inter-temporal lowest incomes obtained in the LF scenario. The value ν = 0 means that the criterion P oor has to be at least equal to the performance obtained with the LF situation. With the value ν = 10%, the criterion increases by at least 10%. Different values of ν are tested and associated optimal public instruments designed. In this study, SUS scenarios are defined with similar account of ecological and social constraints, i.e. ν = µ. The effect of the social constraint alone will also be presented in the Result section.

Biod * (T ) = max τ Budg ≥ 0 Biod(T ) (18)
3. Case study

Context

As in [START_REF] Mouysset | How does economic risk aversion affect biodiveristy ? Ecological Applications[END_REF], the model is calibrated and applied to metropolitan France, at the Small Agricultural Region (SAR) scale using common birds as the biodiversity metrics. Metropolitan France is split into 620 Small Agricultural Regions (SAR) which exhibits agro-ecological homogeneity. The biodiversity and economic models at local scales described below are thus built and calibrated for each SAR. We then use the model to make projections from t 1 =2009 to T =2050.

Biodiversity data

To measure biodiversity, we focus on common bird populations [START_REF] Ormerod | Editors' introduction: Birds and agriculture[END_REF][START_REF] Sekercioglu | Ecosystem consequences of bird declines[END_REF]). The data are from the French Bird Breeding Survey (FBBS) time series (see the Vigie-Nature website3 ). For details of the monitoring method and sample design see in [START_REF] Jiguet | Method-learning caused first-time observer effect in a newlystarted breeding bird survey[END_REF]. The data provides information about bird abundances across the whole country. Abundance values for each species were available for the period 2002-2008. Among the species included in the survey, we focus on the 20 farmland specialist species which are classified according to their habitat requirements at European level (European Bird Census Council 2007). Table 1 lists the 20 farmland specialist species used as a reference for the European Farmland Bird Index FBI [START_REF] Gregory | The state of play of farmland birds: Population trends and conservation status of lowland farmland birds in the United Kingdom[END_REF]). Consistent with previous analyses which show the relevance of the national FBI to reflect the response of farmland biodiversity to agricultural intensification [START_REF] Doxa | Low-intensity agriculture increases farmland bird abundances in France[END_REF][START_REF] Mouysset | Different policy scenarios to promote various targets of biodiversity[END_REF], we use it as our Biod indicator for ecological constraint (eq. 17) in the design of the public policies. As described in equation 13, we compute a national FBI of 20 farmland specialist species for each SAR:

FBI(t) = s ∈ Specialist N s,nat (t) N s,nat (2008) 1/20 (20)
where N s,nat (t) = 620 r=1 N s,r (t) stands for the total abundance of species s across the 620 small agricultural regions r.

Agro-economic data

Our agro-economic data are from the French agro-economic classification of agricultural land-uses developed by the French Farm Accounting Data Network (FADN) 4 and the Observatory of Rural Development (ODR) 5 . This organization classifies land uses in 14 classes of agricultural systems (see tab. 2). Each SAR is a specific combination of these 14 systems. The areas dedicated to each of the 14 agricultural systems and the associated gross margins relying on tax returns, for the years 2002 to 2008 are available at the ODR website. The budgetary constraint is calibrated with the current French CAP budget. The rigidity and aversion parameters a and ǫ have been calibrated according to a least-square minimization (see [START_REF] Mouysset | How does economic risk aversion affect biodiveristy ? Ecological Applications[END_REF]). As mentioned in the introduction, we focus here on two specific levers for public policy: croplands cop (class 1 in tab 2) and the non-intensive grasslands grass (classes 4-7 in tab 2).

Results

Welfare with green and sustainable optimal scenarios

Figure 2 presents evolutions of the welfare criteria Rich under optimal green GRE (black diamond) and sustainable SUS (triangle) public policies. The xaxis represents the level of constraints: in the case of the sustainability objective µ=ν the strength of both ecological and social constraints, and only µ in the case of the green target. This figure shows first that it is possible to design public policies that can increase either ecological criteria or both ecological and social objectives, but they imply a loss of richness. This decrease is due to the fact that welfare depends solely on land-use distributions and gross margins (excluding subsidies and taxes). The context of market stability (i.e. without any evolution in gross margins) implies that its evolution is due only to a modification in landuse distributions compared to the LF situation. Because the LF is the optimal land use distribution relative to current gross margins, the new distribution can only be either identical to the LF distribution or sub-optimal. In other words, the introduction of incentives due to new constraints and modifications to land use distributions necessarily implies a decrease in welfare. Finally we observe that green and sustainable objectives coincide when the objectives are rather small (between 0% and 5%) or rather high (between 18% and 30%). For medium targets, the two objectives do not imply the same optimal strategy.

Green optimal public policies

Figure 3 shows the optimal GR tools required to maximize the welfare criteria Rich under the budgetary and ecological constraints (tab. 3). The x-axis stands for µ, the strength of the ecological constraint. The red colour represents the levels of ecological constraints achievable with subsidies taking no account of the budgetary constraint. Incentives for grasslands or crops can be used to increase the FBI. The analysis of the instruments highlights that the cost-effective way to satisfy the FBI constraint by playing on crops is to impose a tax on them (i.d. τ * cop < 0). For grasslands, the cost-effective strategies are subsidy based (i.d. τ * grass > 0), yielding an increase in grassland area. Regarding maximum biodiversity performance, we observe that a tax on crops leads at best to medium objectives (up to 17 %) while subsidies on grasslands are capable of achieving the highest performances (30 % or more if the budgetary constraint is relaxed). Finally, figure 3 exhibits some structural limitations for instruments based on croplands: the impact on the FBI remains constant beyond τ cop = -0.47.

Socially optimal public policies

Similarly to figure 3, figure 4 depicts the optimal public policies satisfying budgetary and social constraints (tab. 4). The x-axis stands for ν, the percentage of improvement compared to the Laissez-Faire situation. We observe that, in contrast to the ecological constraint, only subsidies for grasslands are able to satisfy the social constraint. This is explained by the fact that, by enhancing the financial portfolio of farmers, subsidies imply earnings. Since the poorest regions are specialized in grasslands, the only instrument capable of increasing the social objective is a grassland subsidy. Finally, we observe use of this instrument makes it is possible to strongly increase the level of the indicator P oor, up to 84%. The red colour in figure 4 indicates that it is possible to design subsidies that enable an increase beyond ν = 84%. This limit is imposed by the budgetary constraint which limits the design of stronger subsidies.

Welfare under green public policies

Figure 5 compares welfare obtained under optimal instruments based on grassland (solid lines) and cropland (dashed lines) able to satisfy the ecological constraint. The x-axis stands for µ, the strength of the ecological constraint. We observe a switch between instruments able to optimally achieve ecological objectives. While subsidies on grasslands are less costly in terms of welfare for small targets (µ ≤ 5%), taxes on croplands become more efficient for µ ≥ 6%. However, they are not able more than µ ≥ 17%. Then for µ ≥ 18%, the only available strategy is to subsidize grasslands.

Regional distributions

Figure 6 shows regional welfares for tools capable of satisfying the biodiversity constraint with µ = 10%. Since a subsidy on grasslands can also increase the social criteria, figure 6(b) is also related to social perspectives. The intensity of the blue shading represents the loss of welfare. We observe that the land use targeted by public policy is a strong driver of patterns of regional welfare. More precisely, a crop tool is cheaper than a grassland instrument. Because more French regions are specialized in grasslands, the regional costs of grassland instruments are more generalized.

Figure 7 shows the pattern of ecological performances at the regional scale for two optimal tools which satisfy the biodiversity constraint with µ = 10%. Again, figure 7(b) depicting subsidies on grasslands is interesting from a social point of view. The intensity of blue shading represents the decrease in FBI performance and the intensity of red shading indicates the intensity of the improvement. We observe different patterns capable of achieving the same national scores. Figure 7 shows that the crop tool (Fig. 6(a)) slightly positively affects only the half of the regions, with no consequence for the half. This contrasts with the instrument based on grassland (Fig. 6(b)) which affects all of the regions with contrasts among regions that show decreased, stable or strong increase in FBI.

Discussion

Designing agricultural public policies towards sustainable agriculture

The model built here offers a framework for both researchers' and decision makers' thinking about more sustainable management of agriculture in France. Due to the stylization used in the modelling, the results can not be interpreted as exact predictions of future bio-economic states. However, because the model integrates multi-scale and dynamic processes which are calibrated with historical data, it provides relevant ranges of future trends function of different public policy drivers. Its objective is thus to stimulate debates on what strategies to develop for various policy targets, and especially in the perspective of the next reform of the Common Agricultural Policy (CAP). More specifically, we have shown that it is possible to integrate and improve ecological and social criteria through public economic policies.

The first contribution of this study is that it complements knowledge on the economic drivers available to public policy. We highlighted land-use targeted by public policy as an essential driver, determinant of national performances and structuring regional patterns. The lack of attention of this target in the literature can probably be explained by the fact that most models focus only on these two land-uses such as in [START_REF] Barraquand | Biological conservation in dynamic agricultural landscapes: Effectiveness of public policies and trade-offs with agricultural production[END_REF]. This simplification is justified because they constitute crucial levers for designing new socially [START_REF] Prager | Stakeholder involvement in agri-environmental policy making-learning from a local-and a state-level approach in Germany[END_REF][START_REF] Mottet | Agricultural land-use change and its drivers in mountain landscapes: A case study in the Pyrenees[END_REF]) and ecologically [START_REF] Laiolo | Spatial and seasonal patterns of bird communities on italian agroecosystems[END_REF][START_REF] Robinson | Post-war changes in arable farming and biodiversity in Great Britain[END_REF][START_REF] Stoate | Ecological impacts of arable intensification in Europe[END_REF]) improved public policy. However, the symmetry between a negative effect on croplands or a positive effect on grasslands in the two-land use models, raises questions in the real landscapes with multiple land uses. In the present study, we stress the importance of the target since there is no symmetry between them. The two levers do not affect the same regions, according to their specialization. Also, for regions affected by both cropland and grassland policies, the consequences differ because actions related to one land-use impact has an impact, but not necessarily in favour of or at the expense of the other. Hence, the distributions of regional ecological and economic performances depends on the public instruments.

The second contribution is more aligned to ecological perspectives. The optimal crop-based instruments have positive consequences for FBI within almost all regions, whatever the range of land-uses used to replace crops. Consistent with the literature [START_REF] Chamberlain | Changes in the abundance of farmland birds in relation to the timing of agricultural intensification in England and Wales[END_REF][START_REF] Donald | Agricultural intensification and the collapse of Europe's farmland bird populations[END_REF][START_REF] Robinson | Post-war changes in arable farming and biodiversity in Great Britain[END_REF], we confirmed the broadly negative effect of crops on biodiversity. We also confirmed the average positive effect of grassland on biodiversity [START_REF] Laiolo | Spatial and seasonal patterns of bird communities on italian agroecosystems[END_REF]) but we moderated it since it is characterized by contrasting regional patterns. In some regions, the development of grasslands leads to local decrease in the FBI, suggesting that land-uses replaced by grassland have a positive impact on biodiversity. Although on the whole bird communities react positively to grasslands, this land use is not optimal for all single species. If some communities are historically strongly biased towards these species, increasing grassland areas may not have such a positive effect at the local scale.

Implications for the current CAP

These previous theoretical contributions can be interpreted in the context of the current CAP. They give some insights into the design of the best public instruments and understanding the compatibility between green and sustainability objectives. We highlighted in this paper that this compatibility depends on the stringency of the objectives. Our study shows that if the ecological objectives are limited (lower than 5%) or are high (between 18% and 30%), subsidizing grassland is the best strategy for the decision-maker from both green and sustainability points of view: this strategy simultaneously satisfies the ecological and social objectives and minimizes loss of richness. These conclusions unambiguously question the current structure of the second pillar of the CAP based on subsidies for grassland. However, most of the measures in favour of the environment in the current CAP are based on medium objectives. It is because the conclusions in these situations are not trivial that the debate between green and sustainability criteria persists.

First, if the objective is only to green the CAP (i.e. cost-effectively manage biodiversity), playing on the first pillar dedicated to crops is the cheapest strategy. This conclusion questions the social acceptability of taxes. However, due to the structure of our economic dataset which includes the current public incentives, taxes are more understandable as a reduction in current crop subsidies rather than absolute taxes. Then, in contrast to what is commonly understood, reducing first pillar subsidies would be the most relevant strategy to cost-effectively manage biodiversity and greening the CAP with medium ecological objectives. This confirms the interest of the starting debate about the inconsistency of this first pillar with current environmental considerations: some of its subsidies are now subject to environmental conditions. Second, if the objective is sustainability, then the aims are broader and consist of combining ecological and socio-economic (related to poverty for instance) considerations, and subsidies for grassland consistent with the current structure of the second pillar is the only strategy. However, we should keep in mind that this strategy has a cost regarding the biodiversity criterion since loss in welfare is larger than with taxes on crops even if the difference remains small.

In other words, in the case of medium objectives, decision makers need to choose between managing biodiversity or managing it in association with improvements to social criteria and higher welfare losses. The conclusions about the best public policy instrument will depend on this political position. However, it should be noted also that, in any case, these medium objectives are not capable of achieving the classical objective of halting erosion of biodiversity in the short term. We frequently heard about goals related to halting biodiversity loss by 2010, but now the horizon has shifted to 2020. Although our study adopts a more distant and, thus easier, objective (horizon 2050), it still appears that it will be difficult to stop loss of biodiversity (Biod(2050) ≥ Biod(2009) will happen for µ = 43%). This raises questions about the practicality and achievability of such political objectives.

Finally, our study points to the importance of international cooperation since the introduction of ecological and social consideration constraints on a national scale necessarily results in a decrease in welfare. One way to avoid or reduce this decrease would be to re-evaluate prices on a larger scale (here world scale), independent of local public incentives (here national scale).

Perspectives

Our study highlights some interesting trade-offs which are beyond the scope of this paper, but which constitute relevant perspectives for the development of sustainable agriculture public policies.

The first perspective is related to the barriers limiting the implementation of public policy. They differ according to the instrument. Playing on crops to limit negative land uses does not exhibit any social limitations (smallest loss of welfare, equity in ecological performances)s but presents some structural limitations: it is impossible to achieve the highest biodiversity scores since there is no pressure on the land use substituting crop. In contrast, affecting grasslands to develop positive land uses does not exhibit any structural limitations (highest social and ecological scores achievable), but in associated with heavy social limitations (large loss of welfare, heterogeneity of ecological performances, public costs related to the required budget). Thus, we need to debate over choosing the more socially acceptable tool for the short-term (i.e. medium objectives) although it will be impossible later to re-assess the objectives, or choosing the tool that allows the possibility of future re-assessment of objectives although in relation to moderate objectives, it is not the most efficient instrument.

The second perspective is related to the question of spatial equity of performance. We found contrasted patterns for both welfare and biodiversity scores. For biodiversity, the consequences are tow folds: there are ecological impacts (functioning of the ecosystem, ability for dispersion among regions) and social impacts (social impacts of living in a good or bad ecological environment). In this context, taking account of equity criteria in addition to national scores for welfare and biodiversity appears an interesting perspective. It should affect the conclusions, since in our study the instruments capable of achieving the highest national scores are based on heterogeneous regional patterns, clearly highlighting the trade-off between maximizing national score versus regional equity. In this context, it would be relevant to try other schemes to see whether this result (highest national score associated with heterogeneous pattern) is reproduced using other instruments or whether it is inherent in the problematic (at least in our modelling context). Among other schemes, subsidies to diversify land uses could be tested, since the literature identifies the positive impact on biodiversity of such schemes [START_REF] Schläpfer | Returns from hay cultivation in fertilized low diversity and non-fertilized high diversity grassland[END_REF][START_REF] Di Falco | Crop genetic diversity, productivity and stability of agroecosystems. A theoretical and empirical investigation[END_REF][START_REF] Quaas | Uncertainty and sustainability in the management of rangelands[END_REF][START_REF] Mouysset | How does economic risk aversion affect biodiveristy ? Ecological Applications[END_REF]. In this perspective, refining the economic behaviours of farmers in the model might be an interesting improvement.

Finally, green and sustainability strategies raise questions about the productive requirements and open debates at global scale. In the current context of demographic growth, some limitations of production in Europe will imply compensations somewhere else. In this sense, opportunities could arise with the introduction of Ukraine in Europe, where agriculture has dropped in the recent decades.

Figure 1 :

 1 Figure 1: Bio-economic model coupling. The decision maker determines a public policy scenario according to a bio-economic optimization. The farmers choose their agricultural systems by maximizing their utility function under technical constraints. These choices affect the habitats and the bird communities.

Figure 2 :

 2 Figure2: Comparison of the welfare criteria Rich under optimal green (black diamond) and sustainable (triangle) public policies with different levels of constraints.The x-axis stands for the level of constraints: in the case of sustainable objective µ=ν the strength of both ecological and social constraints, and only µ in the case of green target.

Figure 3 :

 3 Figure3: Optimal public policies instruments able to satisfy the budgetary and ecological constraints, based on grasslands (solid lines) or croplands (dashed lines). Red stands for out of the budgetary constraint.

Figure 4 :Figure 5 :

 45 Figure 4: Optimal grasslands incentives τgrass able to satisfy the budgetary and the social constraint. Red stands for out of the budgetary constraint.

  Figure 5: Evolutions of the richness Rich (represented as percentages of the LF performances) with the optimal instruments satisfying the budgetary constraint and the ecological constraint µ based on Biod. Solid lines: incentives on grasslands, dashed lines: incentives on croplands. Red stands for out of the budgetary constraint. Figure 5(b) corresponds to a zoom of figure 5(a) for the small levels of constraint.

Figure 6 :

 6 Figure6: Loss of regional richness obtained with the optimal tools able to satisfy the ecological constraint with µ=10%.

Table 1 :

 1 List of the 20 farmland bird species s

	20 farmland bird species
	(1) Buzzard Buteo buteo
	(2) Cirl Bunting Emberiza cirlus
	(3) Corn Bunting Emberiza calandra
	(4) Grey Partridge Perdix perdix
	(5) Hoopoe Upupa epops
	(6) Kestrel Falco tinnunculus
	(7) Lapwing Vanellus vanellus
	(8) Linnet Carduelis cannabina
	(9) Meadow Pipit Anthus pratensis
	(10) Quail Coturnix coturnix
	(11) Red-backed Shrike Lanius collurio
	(12) Red-legged Partridge Alectoris rufa
	(13) Rook Corvus frugilegus
	(14) Skylark Alauda arvensis
	(15) Stonechat Saxicola torquatus
	(16) Whinchat Saxicola rubetra
	(17) Whitethroat Sylvia communis
	(18) Wood Lark Lullula arborea
	(19) Yellowhammer Emberiza citrinella
	(20) Yellow Wagtail Motacilla flava

Table 2 :

 2 List of the 14 farming systems

See the appendix and[START_REF] Mouysset | How does economic risk aversion affect biodiveristy ? Ecological Applications[END_REF] for the details of the ecological modelling

This maximum Biod * is with the policy determined by the biodiversity maximisation under the budgetary constraint:

http://www2.mnhn.fr/vigie-nature/

http://ec.europa.eu/agriculture/rica/

https://esrcarto.supagro.inra.fr/intranet/

(a) Based on τcop (b) Based on τgrass Figure 7: Evolutions of regional ecological performances obtained with the optimal tools able to satisfy the ecological constraint with µ=10%.

Acknowldgements

We are very grateful to Maia David and Toby Gardner for interesting remarks and suggestions. This work was carried out with financial support from the FRB -Fondation pour la Recherche en Biodiversité -with the MOBILIS programme "Modelling for scenarios and sustainable management of biodiversity, farming and forestry facing climate change" and from the RURAGRI-ERANET TRUSTEE project "Towards RUral Synergies and Trade-offs between Economic development and Ecosystem services" under Grant Agreement 235175.

APPENDIX

A dynamic framework has been chosen. The Beverton-Holt model accounts for the intra-specific competition and the density dependence: N s,r (t + 1) = N s,r (t). 1 + R s,r

where N s,r (t) stands for the bird abundance of species s in a PRA r at year t. The R s,r coefficient corresponds to the intrinsic growth rate specific to each species s in a region r. This parameter takes into account the characteristics of each species such as clutch size, mean reproductive success, number of clutches per year. The variable M s,r captures the ability of the habitat to host the species and the product M s,r (t) * R s represents the carrying capacity of the habitat r.

The habitat variable M s,r (t) is assumed to depend linearly on land-uses as follows:

where A r,k (t) represents the share of the PRA r dedicated to farming systems k at time t. The α s,r,k and β s,r coefficients, specific to each species, quantify how the species s responds to the various farming systems k in a given region r. The β s,r coefficient can be interpreted as the mean habitat coefficient for a species s in a PRA r.


Mohamed Ichchou

Olivier Bareille

come

Energy predictions of turbulent boundary layer induced mid-high frequency structural vibrations

With increasing flight speed and the generalisation of air transportation has appeared a necessity to reduce structural fatigue for aircraft and acoustic/vibration levels for their payload and passengers.

Passengers' comfort, during the past years, has become one of the principal concerns of various actors in the aeronautic industry, since it is actually one of the essential demands of users. Among the factors that are considered to affect the wellbeing of passengers, acoustics and vibrations are ranked at the top of the list. Interior noise related to the external flow has become a significant noise contributor under certain conditions even for ground transportation vehicles.

Turbulent external flow related loads are principally of two types, involving two areas of research.

First, turbulent fluctuations behave as acoustic sources. The acoustic field resulting from turbulence loads the panels which, in turn, transmit noise inside the vehicle. These phenomena are qualified as aero-acoustic. At low Mach numbers, they become important only in the presence of a protruding accessory that induces strong turbulent stresses in a flow region that consequently behaves as an acoustic source.

Fig. 1: Structural vibration due to Turbulent Boundary Layer

Second, the interaction between turbulent flow and external panels causes wall pressure fluctuations that generate panel vibrations, which radiate sound into the passenger or payload compartment (see Fig. 1). In all cases, fluid inertia forces will become greater than viscous forces and a turbulent boundary layer (TBL) develops if velocity is high enough and if the structure is large.

On the one hand, the numerical prediction of such phenomena is costly when processed through conventional methods such as Finite-Element based ones. This is especially true for distributed random excitation, such as TBL excitation, for which the forced response problem requires exponentially more computational effort than the modal analysis of the structure. Some alternative methods, still displacement based, like the dynamic stiffness method and the spectral FEM, (Birgersson et al 2003) succeeded in reducing the number of degrees of freedom and increase accuracy. The elements are formulated and assembled as in the standard FEM while the basis functions are exact solutions to the equations of motion. Normally, the spectral FEM, and the dynamic stiffness method (DSM), consider excitation at the element ends only. Though the above method is efficient and accurate, the relevant information regarding the response of the structure rely on an averaged value. This true to energy based method, to which cross-spectral densities and cross-correlation functions can be related in a straight forward manner. The weakness of energy methods is in the models used for injected power. Provided the TBL excitation can be well described, one can expect some good results. The most common TBL model is attributed to Corcos (Corcos 1967), and has been validated experimentally. Yet, it overestimates the wall-pressure crossspectral density at wave-numbers below the convective peak. Some improved models were therefore developed to give a better representation of the sound source associated to the TBL interacting with surface panels (Zheng 2003). It should be pointed out that the weak coupling assumption (Graham 1997) is accepted in the current study. This assumption corresponds to the assumption, implicit in acoustic analogy analyses, that the basic turbulence structure is essentially unaffected by the acoustic motions. Feedback from the vibrating panel can affect the properties of turbulent flow and its pressure loading. However, when trying to predict the body panel responses, the complexity is such that most studies keep a weak interaction hypothesis, i.e., they neglect panel feedback on flow. This approach, when valid, i.e., without aeroelastic instabilities, allows computation of the panel response from the properties of the wall pressure field experimentally determined on a hard surface. The predictions obtained from the weak coupling approximation have been shown to work well in a wide range of cases (Graham 1997), except for the differences between the theoretical results and measured data mentioned in (Blake 1986) in supersonic flow cases.

On the other hand, TBL excitation is a typical wide frequency band excitation. It should thus be noted that well established and detailed numerical methods like finite element or boundary equation based formulations run into important limitations as frequency increases. Specific medium and high frequency range methods are then needed to deal with the vibrational behaviour in this particular frequency range. Among possible alternatives, two different philosophies can be found in the specialised literature. The first class of analysis methods are based on displacement variables. These methods tend to reach the medium and high frequency domain using both phase and amplitude behaviour. Among these methods we note the work done by C. Soize (Soize 1993) who introduced the structural-acoustic fuzzy theory. Ultimately, the complex ray theory developed by Ladeveze (Ladeveze 1996) can also be considered as displacement approach to medium and high frequencies.

The second class of analysis methods are based on energy variables. The Statistical Energy complex built-up structures. Statistical Energy Analysis offers an alternative form of vibrational and acoustical state. The statistical aspects included are mainly related to the high frequency features.

Modelling uncertainties made deterministic approaches often irrelevant at high frequency. SEA also provides a way of energy transfer analysis. Stored energies, dissipated energies, exchanged powers as well as injected powers, are the main variables. SEA tends to characterise the amplitude of vibrational signals, without any information about phase. This is a major simplification. It offers reduced computational costs, mainly at high frequencies. However SEA suffers from its axiomatic formulation and still requires improvements, as indicated in (Fahy 1994) where an interesting survey and a number of critical comments are given. Beyond these studies, a number of works attempt to enhance the Statistical Energy Analysis robustness and predictivity. Among these attempts we may cite the earlier work of Nefske and Sung (Nefske and Sung 1987) who proposed the use of an energy diffusion model to predict the space spread of energy density within subsystems. Basically, this way of thinking can be viewed as a local energy formalism whereas the SEA formalism is based on global energies of finite subsystems. This model has been improved by

Bernhard and his colleagues (Wohlever andBernhard 1992, Han et al. 1999). Among other contributors to this subject, we may mention (Langley 1995, Carcatera andSestieri, 1995) where interesting discussions are given. Developments at the Ecole Centrale de Lyon in the context of energy models started with the so-called general energy method (Ichchou 1996). The main goal of this method was to reformulate the classical displacement models using four energy variables: the total energy as well as the Lagrangian energy density, the active and the reactive energy flow.

Eliminating the Lagrangian energy density and the reactive energy flow, the general energy formulation leads to an interesting energy model well suited for medium and high frequency dynamics. In fact, from a propagative approach, it has been shown that the Lagrangian energy density and the reactive energy flow is mainly linked to wave interferences or singularities. So that, taking into account only the incoherent contribution of waves to the energy variables leads to the formulation of the local energy approach discussed here and called Simplified Energy Method (SEM).

Two main energy formulations have been formulated. The first one is similar to the heat transfer equation and can be solved through finite element energy based techniques (among others). The second formulation is similar to radiative equations and can be solved through boundary element energy based approaches. The main goal of this paper is to synthesise these formulations in order to show the interest of this local energy approach in the context of predicting TBL induced vibration.

This paper deals with such predictions for the vibroacoustical behaviour of a plate-like structure by means of energy based approaches. These latter are first introduced in section 2, as an alternative to conventional SEA (Statistical Energy Analysis). In section 3, an overview of the existing TBL (Turbulent Boundary Layer) models is given. The choice of one of these is then justified. Section 5 presents the numerical compared results obtained by means of the two energy approaches in an example fully described in section 4.

LOCAL ENERGY APPROACH FORMULATIONS

The main goal of this section is to discuss local energy approach formulations. Dynamical equations of motion of structures will then be expressed in terms of energy. This leads to practical efficient methods especially for high and mid frequency vibrations. We start with a reminder of basic equations of energy conservation in general elastoacoustics. We then formulate the problem in terms of energy flow balance. This is the first step in the local energy as well as global SEA formulation. Then, using a wave approach very similar to the one given in (Ichchou and Jezequel 1996) in the steady state case, smooth energy behaviour is studied for damped structures. This wave approach is combined with statistical representation, in order to reach averaged energy levels for complex dynamical situations. Specifically, the analysis starts with elementary wave energy behaviour. In this context, the discussion starts with a plane wave assumption, leading to a differential energy equation of the associated energy parameters. After that, spherical waves are considered and related energy equations are given. The transition, from elementary waves to a complex waves composed dynamics is then discussed. A fundamental set of required assumptions are thus introduced and commented. The main assumption is related to the non-coherence situation between waves, which leads to a local superposition principle. Finally, a foundation of the local energy approach is given.

PRELIMINARIES

In view of describing the energy transfer inside the medium, two continuous energy fields are introduced. The first energy quantity is merely the total energy density t) , W( x  defined as the sum of the potential energy density and the kinetic energy density.

inj diss p p )) t , ( ( div ) t , ( t W      x x    I (1)
where div is the divergence operator, diss p is the dissipated power density. inj p is connected to exterior forces that will inject power in the system. The damping model adopted here is the same as in SEA (power density being dissipated is proportional to the energy density). Hence:

) t , ( W ) t , ( p diss x x     ( 2
)
where  is the damping loss factor and  the circular frequency. The validity of this relationship has been discussed in the literature (Lyon 1975) about SEA. In order to derive the energy density equations, a wave description of vibrational-acoustical behaviour is adopted. In the subsequent presentation, planes and symmetrical propagating disturbances in an elastic and dissipative medium are considered. Each propagating wave involves partial energy quantities defined as the energy variables associated to those waves. In order to establish a relationship between the partial energies , an additional assumption is introduced. In fact, a superposition principle is assumed and applied here for all quadratic variables:

    i i (3) i , 
being any quadratic variable, representing either a partial energy or a global energy associated with a particular wave field. This assumption is often used in statistical phenomena in physics. In fact, we can find such an assumption in room acoustics, electromagnetism, etc. This assumption is typical in the high frequency literature (see (Lyon 1975) and (Sestieri and Carcaterra 2003) for further details). In the following developments, an intrinsic energy law will be used. This equation is often introduced in order to define the wave velocity i c . Where:

i i i i ) t , ( W c ) t , ( n x x      I (4)
This expression was shown to be valid for conservative wave fields. It will be used in the following as a first step in the energy density equation process for lightly damped structures.

ELEMENTARY WAVE ANALYSIS: PLANE and SPHERICAL WAVES

In (Ichchou andJezequel 1996, Ichchou et al. 2001), it was shown that, considering a plane wave in an isolated medium, as a particular solution of the motion equation, it verifies the energy equation:

0 ) ( 2 2 2 2 2           i i i i i W t W t W W c   (5)
Under continuous forced harmonic excitation (angular frequency ), through time averaging, the following equation is obtained:

0 2 2      i i i W ) ( W c (6)
Or equivalently through linearity in a forced medium

inj i i i p W ) ( W c       2 2 ( 7 )
This equation is equivalent to the equation of heat diffusion with a term of convection. This equation governs the energy density changes. It provides a space average of the predicted energy density. This equation has been commented and discussed in-depth by many authors (see for instance (Langley 1995) and(Ichchou 1996) for further details). Let us consider now symmetrical wave fields. Specifically, we will consider cylindrical waves in dimension 2, and spherical waves in dimension 3. In this case, due to the revolution symmetry, the energy variables are ) , ( t r W i and

) t , r ( i I 
, where r is the radial coordinate. Performing the same demonstration as for plane waves, the following equation is readily obtained:

0 2 1 2 2 2 2 1 2 1 2                i i i i n n i W ) ( t W t W r ) W r ( r c (8)
We can also study the harmonic stationary case, whose equation is:

0 1 2 2 1 2 1 2         i i n n i W ) ( r ) W r ( r c (9)
We notice that this equation is distinct from the equation of heat diffusion. It correctly gives the energy behaviour of infinite systems.

ENERGY FIELDS RECONSTRUCTION: HEAT CONDUCTION EQUATION

From the local energy equations of elementary fields, we will be able to derive the local energy equations for more complex wave fields. Hence, for one dimension, the knowledge of the wave content will lead to the formulation of the problem in terms of the global field. The situation is more complicated for two or three dimension where the wave contents and nature can be of various kinds.

New assumptions are then needed to deal with multi-dimensional cases, like for instance, vibrating Kirchhoff-Love plates. are defined as the superposition of a direct field and a reverberant field. In both techniques, the direct field is calculated assuming a symmetrical wave field as described above. The difference lies in the reverberant field treatment. In Fig. 2 the reverberant field is assumed to be a set of plane waves. This approach leads to a simple differential operator on the reverberant energy density. The second solution considers the reverberant field as the result of secondary sources located at boundaries as shown in Fig. 4. So, only symmetrical wave fields are considered. This solution was first proposed by Kuttruff (Kuttruff 1997). It is also known as the radiosity method. Further details can be found in (Schmitt 2004). Both formulations are explained below. Some indications about their numerical resolution are also briefly given.

ENERGY FIELDS RECONSTRUCTION: HEAT CONDUCTION EQUATION

Let us consider a medium in which only plane waves exist. The medium is assumed to be homogeneous and isotropic, so that all the waves have the same energy velocity, noted for convenience as, c g and which is a physical time and space constant. It was shown that the energy variables, associated to the considered field obeys:

0 2 2 2 2 2            W ) ( t W t W ) W ( c g  (10)
In the case where the considered medium is defined by a distribution of exterior forces, this will be represented by a right-hand side inj p . Under harmonic stationary excitation, this equation becomes simply:

    inj g p t , W ) ( ) t , W ( c      x x   2 2  (11)
It is a heat conduction type equation with a term of convection. Numerical resolution of this equation requires three main parameters. The first one is the energy group velocity. This parameter is known for most conventional structural elements (see reference (Ichchou and Jezequel 1996) for details). The second parameter is the damping loss factor which is assumed to be given and finally the power injected density which is directly related to the source. Calculation of this quantity in the context of TBL excitation is explained below. It should be also pointed out, that Eq 11 needs boundary conditions to be given in terms of energy variables. Detailed discussions of these boundary conditions can be found in many references (see for instance (Ichchou 1996)). In the case considered in this paper, the plate is assumed to be boundary conservative (non absorbing). This condition is found in many classic mechanical situations (clamped, free, pinned,…). So that the non absorbing boundary condition can be simply expressed in terms of energy flow such as:

0    ext W n  ( 12
)
ext n  being the external normal vector to the considered boundary. Eq 11 and 12 can finally be solved within a finite element energy based context, once all the relevant quantities are provided.

ENERGY FIELDS RECONSTRUCTION: RADIATIVE TRANSFER EQUATION

Let us assume that only spherical waves can be propagated in the studied medium, with an energy velocity c g . In this case the energy equation derived above also stands for the global energy and so that for stationary dynamics:

0 1 2 2 1 2 1 2         W ) ( r ) W r (
The destructive interferences assumption allows one to build the solution by superposition of energies associated respectively to direct and reverberated fields (see Fig. 3

                   (15)
f is a scalar function, which depends on the angle between the direction of active intensity and the boundary external normal. M is the point of the considered domain, the primary source density is  S  over  , and the primary sources are proportional to the injected power density inj p . Here, f follows a Lambert directivity law, so that the total energy density and the active intensity somewhere in the considered media can be written simply as:

dP ) M , P ( ) )( P ( d ) M , S ( ) S ( ) M ( dP ) M , P ( G ) )( P ( d ) M , S ( G ) S ( ) M ( W P MP P MP                       H n n H I n n        (16)
The injected power density in the system is equal to

g c  2
because the energy density and the active intensity vector verify the following equation:

g c W div     2 I  (17)
To complete this description, a boundary condition is considered in terms of energy. Indeed, if conservative boundaries are assumed then no energy is flowing from the boundary (the scalar product of the active intensity and of the exterior normal is null). Applying this result to the previous expression of active intensity, a Freedholm integral type equation of the second kind is shown as:

(18)

The resolution of this integral equation is obtained by discretising the boundary in n segments S k .

P k is the middle of the segment S k and Q is a variable point on the segment. The source density is assumed to be constant on each segment and takes the value  k . k n  is the exterior normal vector to the domain at P k . The equation becomes:

                 n k l l S k k Q Q P l k k k g l k dQ ) P , Q ( ) ( d ) P , S ( ) S ( c 1 2 n H n n n H       (19)
The n values of  for a type of wave are obtained in solving the preceding system. After their evaluation, one can obtain the fields of energy density and active intensity in the domain :

                       n l S Q MQ l n l S Q MQ l l l dQ ) M , Q ( ) ( d ) M , S ( ) S ( ) M ( dQ ) M , Q ( G ) ( d ) M , S ( G ) S ( ) M ( W 1 1 H n n H I n n        (20)
In the multidimensional domain, Kuttruff proposed (Kuttruff 1997) an integral equivalent to the one provided here. This method is known as the "radiosity method". It was principally applied in the domain of room acoustics. Further discussions of this formulation can be found in (Schmitt 2004).

  Short running headline: Energy modelling of TBL-induced vibrationKeywords: Turbulent Boundary Layer, Bending vibration, Energy flow methods, Mid-High frequencies.

Fig. 2 :

 2 Fig.2: Energy field reconstruction with plane waves in complex 2 and 3 dimensional media: a reverberant field is considered

Fig. 2

 2 Fig. 2 and 3 show two distinct ways of energy reconstruction. In both techniques, energy variables