Loïc Favergeon
email: favergeon@emse.fr

Michèle Pijolat

Influence of water vapor pressure on the induction period during Li 2 SO 4 •H 2 O single crystals dehydration

Keywords: Induction period, nucleation, water vapor pressure, lithium sulfate monohydrate

The dehydration of Li 2 SO 4 •H 2 O single crystals at 80°C has been studied by means of both isothermal thermogravimetry at 2.6, 3.6 and 4.6 hPa of water vapor and environmental scanning electron microscope. Thermogravimetric experiments allowed the determination of induction periods.

Distributions of these isoconversion induction periods for a large number of single crystals showed that increasing the water vapor pressure produced a longer induction period. Moreover the shape of the distributions of the induction periods over a large number of single crystals changed from one mode at 2.6 hPa to two modes at 3.6 and 4.6 hPa. Using an environmental scanning electron microscope, this result could be attributed to differences in nucleation rates at edges and faces of the single crystals.

Introduction

The decomposition reactions of solids, or reactions between a solid and a gas, involve the appearance and growth of nuclei. If the growth process is generally described as the advance of the interface between the reactant and the product, it must be acknowledged that the nucleation process remains really misunderstood. Indeed, it is very difficult to find in the literature some quantitative information about the kinetics of this process. The influence of atmosphere [1][2][3][4][5][6] and of reactant crystallography [7][8] on the nuclei shape has been put in evidence in the case of the dehydration of copper sulfate pentahydrate. Some authors have also shown that nucleation occurs selectively at the edge of single crystals for the thermal dehydration of magnesium oxalate dihydrate [9] and zinc acetate dihydrate [10], but without quantitative data. Moreover at our knowledge there is no detailed work about induction periods in the field of thermal decomposition of solids, even though the induction time reflects the kinetics of the nucleation process.

In order to have a better description of the nucleation process, we have chosen the reaction of dehydration of lithium sulfate monohydrate single crystals. This reaction is known to occur in one single-step:

( ) ( ) ( ) g O H s SO Li s O H SO Li 2 4 2 2 4 2 + = ⋅ (1)
The kinetics of this reaction has been extensively studied on powders [11][12][13][14][15] or single crystals [16][17][18][19][20][21][22][23][24][25][START_REF] Favergeon | Proceedings NATAS conference[END_REF].

These previous works have shown that the isothermal kinetic curves obtained for powders are sigmoids and various interpretations have been proposed for the kinetic model of transformation. Studies on single crystals led to R n or A n laws. For such A n laws however, the nuclei are supposed to appear in the bulk of the solid phase. If this situation is really encountered in precipitation phenomena in materials, this is not the case in the reactions of decompositions or when solids react with gases. Indeed, the nuclei of the new phase must appear at the surface of the solid particles. Hence, when such reactions lead to sigmoids, A n laws must not be used to account for the kinetics.

In a previous paper [25], we have presented a kinetic modeling of experimental data obtained with single crystals, based on Mampel's model assumptions [START_REF] Mampel | [END_REF]28] and Monte-Carlo calculations. The advantage of such a model was to provide a description of the nucleation and growth processes very near to the physical reality by taking into account the real shape and size of the single crystal, the nuclei appearance at random at its surface, and their inward isotropic growth. From the Monte-Carlo numerical simulations, it was possible to get the values of the areic frequency of nucleation noted γ (in number of nuclei.m -2 .s -1 ) and the areic reactivity of growth noted φ (in mol.m -2 .s -1 ) for each single crystal. We observed that, at same temperature and water vapor pressure, the values of the areic frequency of nucleation varied from a single crystal to another, contrarily to the values of the areic reactivity of growth, showing that the nucleation kinetics depends also of the surface state of the single crystals. This outstanding result, as well as the observation of an induction period on the thermogravimetry curves, led us to investigate more precisely the nucleation process.

This article reports experimental data on the induction periods on a large number of lithium sulfate monohydrate single crystals for three conditions of water vapor pressure: 2.6, 3.6 and 4.6 hPa. In addition, the dehydration reaction has been followed by means of environmental scanning electron microscopy at 3.9 and 5.2 hPa. These in situ observations are discussed in order to interpret the data on the distributions of induction periods, as well as the fluctuating values of the areic frequency of nucleation.

Experimental: material and set-up

Lithium sulfate monohydrate single crystals were grown through slow evaporation of a saturated aqueous solution (using powdered materials (Aldrich 99%) and pure water) at room temperature. Single crystals thus obtained have the typical shape of hexagonal plates, those having a mass higher than 1.9 mg were selected for kinetic measurements. A sample shape is represented in Figure 1 which shows the view of a single crystal by means of scanning electron microscope JEOL JSM6400. The kinetic curves were obtained by means of isothermal and isobaric thermogravimetry at 80°C, in a water vapor atmosphere. A symmetrical thermobalance was used in static conditions (Setaram MTB 10 -8 ), the water vapor pressure is fixed using a thermoregulated water bath : the temperature fixed the water vapor pressure. After introduction of the sample in the close chamber at room temperature then evacuation up to a vacuum of 0.001 hPa, a pressure of water vapor equal to 123 hPa was established (this value being higher than 93 hPa which is the dehydration equilibrium pressure at 80°C, in order to prevent the sample from dehydration during the heating up to 80°C). When the temperature was stabilized at 80°C, the water pressure was then rapidly decreased to the chosen pressure for the experiment by a short pumping, and then maintained constant during each experiment due to the thermoregulated bath, the total pressure being that of water vapor. The whole apparatus was located in a box heated at 52°C in order to avoid cold points.

The time necessary to decrease the pressure from 123 hPa to the pressure of experiment was about one minute. The origin of the time scale was arbitrarily chosen to be the moment at which the pressure of experiment was reached. This procedure was perfectly reproducible (as shown by repeating several

experiments on Li 2 SO 4 •H 2 O powders).
From thermogravimetric data, fractionnal conversion is calculted as follows:

( )

0 0 m m m t m f - - = α (2)
where m(t) is the sample mass at time t, m 0 the initial sample mass and m f the final sample mass.

In situ imaging was obtained using an environmental scanning electron microscope (ESEM) FEI/Philips XL30 ESEM FEG. This apparatus allows to examine a single crystal under controlled temperature and controlled water vapor pressure, without metal coating.

Results and discussion

Kinetic curves obtained from thermogravimetry showed that an induction period was systematically observed for each single crystal, whatever the water vapor pressure is. Figure 2 represents typical curves of the fractional conversion versus time α(t) for 2.6, 3.6 and 4.6 hPa at 80°C. Induction periods can be distinctly seen in the magnified parts of Figure 2 for all water vapor pressures. The "theoretical" definition of the induction period is the date of appearance of the first nucleus.

Obviously, the thermobalance was not able to detect the mass loss corresponding to one nucleus, and this "theoretical" induction period is impossible to measure by this way. It was however possible to determine the time necessary for the mass loss to be higher than the noise of the thermobalance (i.e. 0.5 µg) and to get the distribution of these "experimental" induction periods for several tens of single crystals. The drawback of this method was to give "experimental" induction periods which were dependent of the initial mass of the single crystal. Thus we decided to measure the time necessary to reach the fractional conversion α=0.00187, value which was in fact obtained for a mass loss of 0.5 µg for the lightest single crystal (i.e. 1.9 mg). We called this time the "isoconversion" induction period.

Figure 3 illustrates the method to determine this isoconversion induction period in the case of a single crystal decomposed at 80°C and 2.6 hPa (the "experimental" induction period is also represented). In order to be totally independent of the size of the initial solid (i.e. initial volume and initial surface of the single crystal), it would have been more rigorous to measure the time necessary to reach the value For a group of single crystals dehydrated in the same conditions of temperature and water vapor pressure, we obtained different values of isoconversion induction periods. According to our previous study [25], the kinetics of the growth process was characterized by a constant value of the areic reactivity of growth (φ, in mol.m -2 .s -1 ) due to constant temperature and water vapor pressure during the experiment. As a consequence, differences observed in the isoconversion induction periods, for same temperature and pressure conditions, were attributed to differences in the nucleation process, and more precisely in the dates of appearance of the first nucleus.

Of course, this must be related to the dispersion in the values of the areic frequency of nucleation γ which were obtained from our kinetic modeling [25], as previously mentioned in the introduction part.

For example, for six single crystals, the values of γ spread out from 1.37 10 3 to 2.97 10 3 nuclei.m -2 .s -1 (more than a factor 2) whereas those of φ did not vary significantly (from 7.32 10 -5 to 10.08 10 -5 mol.m - 2 .s -1 ) [START_REF] Favergeon | Proceedings NATAS conference[END_REF].

Using a large number of single crystals, we could represent the distributions of the isoconversion induction periods at 80°C. Figure 4 shows these distributions obtained with three distinct water vapor pressures: 2.6, 3.6 and 4.6 hPa. The isoconversion induction period corresponds to the time of reaction at α=0.00187. At this moment, it is obvious that a very large number of nuclei have already appeared and grown. However, the spreading of the distributions over a time scale displayed over 2400s for 2.6 hPa, 6300s for 3.6 hPa and 11700s for 4.6 hPa reflects an important effect of the water vapor pressure on the time required for the formation of the first nucleus.

The most striking feature which comes out from Figure 4a, b andc concerns the change in the shape of the distribution from the lowest to the highest water vapor pressure. The distribution obtained at 2.6 hPa exhibits a maximum for 1200-1500s, and the data are rather well distributed on both sides of this maximum. For 3.6 and 4.6 hPa, the shape of the distribution is bi-modal since in both cases, two maxima can be distinguished: the first maximum is located before 2000s in both cases; the second one is shifted to longer times when the pressure increases. An explanation for the presence of one or two maxima has been possible due to in situ observations using an environmental scanning electron microscope.

Two in situ experiments were carried out at fixed temperature and water vapor pressure. The micrograph views obtained with the environmental scanning electron microscope are presented on Figure 5 (at 70°C under 3.9 hPa) and Figure 6 (at 70°C under 5.2 hPa). Even if temperature and water vapor pressure are different (but in the same range) for thermogravimetric and microscopic experiments, which does not permit a quantitative analysis, the results of both types of experiments allow to qualitatively discuss the influence of water vapor pressure on the nucleation process during the dehydration.

On Figures 5 and6 one can see at once a face and an edge of a single crystal. The cracks that can be observed at various times of observations must be considered as the consequence of the growth of the nuclei that were formed earlier in the same area. Indeed the expansion coefficient (proportional to the ratio of the molar volumes between the final and the initial phases) is equal to 0.78 in the case of the Li 2 SO 4 •H 2 O dehydration, which involves mechanical stress and thus cracks during the transformation. At 70°C and 3.9 hPa (Figure 5), even if the first few cracks appear on the face of the single crystal, the cracks observed are much more numerous on the edge than on the face of the single crystal. At 70°C and 5.2 hPa (Figure 6) all the cracks were found to appear on the surface, the edge being unchanged even after more than three hours of reaction. From these observations it could be deduced that the appearance of nuclei on the edges of the single crystal is favored in the case of the lowest vapor pressure (3.9 hPa).

Nucleation on the face is observed under both water vapor pressures 3.9 and 5.2 hPa.

From these observations, it is possible to propose an explanation for the shape of the induction periods distribution. The presence of two maxima in the distribution reveals that the nuclei appear statistically according to two distinct rates. Under a pressure of 3.9 hPa, nuclei were observed to appear on both edges and faces. This corresponds to the case of Figures 4b and4c, respectively under 3.6 and 4.6 hPa.

Consequently the two maxima on both distributions can be related to the nucleation on edges and faces.

As the pressure increases, the intensity of the first maximum seems to decrease, which is in agreement with the observations of Figure 6: at 5.2 hPa, the majority of nuclei were observed to appear on the face. At a lower pressure, as 2.6 hPa in the case of Figure 4a, the first nuclei are most likely formed on the edges rather than on the faces of the single crystals. Figure 7 represents the various shapes of distributions in a schematic way, from 2.6 hPa up to 5.2 hPa, illustrating both nucleation modes on edges and on faces.

One can see that nucleation on the edges is faster than on the faces: this is not so surprising since defects (at the atomic scale) are expected to be much more numerous on the edges than on the faces, and also because atoms or molecules situated on the edges are known to be energetically more reactive than those on faces.

Nevertheless, our results seem to indicate that the water vapor pressure had an inhibiting effect when the nucleation takes place on the edges. Moreover, nucleation taking place on the faces seems to be delayed when water vapor pressure is increased. These results suggest that kinetics of nucleation on the edges and on the faces of the single crystals are different. Thus it seems that the mechanisms of nucleation should be different, which appears obvious if one considers that the formation of a nucleus may involve the following steps (a nucleation mechanism can be write by analogy with the classical nucleation theory [START_REF] Kaschiev | Nucleation : basic theory with applications[END_REF][START_REF] Markov | Crystal growth for beginners[END_REF]):

(a) appearance of defects on the surface (here: H 2 O vacant sites), (b) surface diffusion of these defects, (c) defect aggregates formation up to a critical size.

In the present study, the defects, that are thought to be water molecules vacancies, are expected to have rates of appearance which take different values if exposed to different water vapor pressures, and if one considers edges or crystallographic planes. A microscopic model involving steps (a) to (c) has been developed in order to attempt to calculate distributions of the dates of appearance of the first nucleus [START_REF] Favergeon | [END_REF]. This work indicates that the frequency of defect formation on the surface is responsible for the position of the maximum of the distribution, which confirms our interpretation.

Conclusion

Isothermal and isobaric thermogravimetry was used to follow the dehydration of a large number of Li 2 SO 4 •H 2 O single crystals. For each of them, an induction period was determined for a conversion α s equal to 0.00187. This isoconversion induction period was found to vary from a single crystal to another so that distributions could be represented for experiments performed at 80°C and under 2.6, 3.6 and 4.6 hPa of water vapor. The shapes of these distributions were analyzed and discussed according to previous results on the values of the areic frequency of nucleation and to in situ observations of cracks appearing at the edges and/or faces of the single crystals.

It could be concluded that two modes of nucleation process have to be considered, according to the appearance of the nuclei on the edges and on the faces of the single crystals. If nucleation on the edges is faster than on the faces, we noted that when the water vapor pressure is increased, nucleation on the edges no longer takes place, and the nucleation on faces is delayed.

Fig. 1 .

 1 Fig. 1. Scanning electron microscope image of a Li 2 SO 4 •H 2 O single crystal.

Fig. 2 .

 2 Fig. 2. Kinetic curves α(t) for Li 2 SO 4 •H 2 O single crystals dehydrated at 80°C under three different water vapor pressures: 2.6, 3.6 and 4.6 hPa.

Fig. 3 .

 3 Fig.3. Experimental induction period and isoconversion induction period determined on a kinetic curve α(t).

  and S 0 are the initial volume and the initial surface of the single crystal. Nevertheless, the use of α s instead of of about 10 % on the determination of the induction period. As the surface of each single crystal has not been determined, we have chosen to determine the time necessary to reach α S .

Fig. 4 .

 4 Fig. 4. Distributions of isoconversion induction periods obtained for dehydration of Li 2 SO 4 •H 2 O single crystals at 80°C and: (a) 2.6hPa, (b) 3.6hPa and (c) 4.6 hPa of water vapor pressure.

Fig. 5 .

 5 Fig. 5. Views of a Li 2 SO 4 •H 2 O single crystal during dehydration with the environmental scanning electron microscope (T=70°C, P(H 2 O)=3.9 hPa).

Fig. 6 .

 6 Fig. 6. Views of a Li 2 SO 4 •H 2 O single crystal during dehydration with the environmental scanning electron microscope (T=70°C, P(H 2 O)=5.2 hPa).

Fig. 7 .

 7 Fig. 7. Scheme of the isoconversion induction periods distributions for different water vapor pressures at 80°C.

Acknowledgments

We wish to thank Françoise Valdivieso for her contribution to this work, Michel Soustelle for helpful discussions and Gilbert Thollet, engineer in the GEMPPM of the INSA of Lyon for his help in environmental scanning electron microscope experiment.


Gharsalli Mazen
email: mezen@imel.org

French firm's financing choices: towards a reconciliation of the static trade off theory and the pecking order theory? By

Keywords: capital structure, trade off theory, pecking order theory I

Introduction

Although the capital structure policy is not a new area of research, it remains one of the most interesting and puzzling topics of research. The theories and explanations that have emerged have resulted in an enormous body of theoretical and empirical research. However, no consensus has been reached.

The financial literature in this field was marked by the contributions of [START_REF] Modigliani | The Cost of Capital, Corporation Finance and the Theory of Investment[END_REF], Miller (1977) and [START_REF] Myers | The Capital Structure Puzzle[END_REF]. The contribution of these authors relates, primarily, to the recognition of two theses. The first relates to the absence of impact of the leverage on the market value of the firm. Indeed, the celebrate article of [START_REF] Modigliani | The Cost of Capital, Corporation Finance and the Theory of Investment[END_REF] constitutes the starting point of the majority of the studies relating to the leverage. These two pioneers, by adopting a certain number of hypotheses, primarily the existence of perfect and frictionless capital markets, prove their irrelevance theorem. According to the irrelevance theorem the firm's financing policy should not affect the firm's value or its cost of capital. The firm's value is solely determined by its investment decisions. This obviously implies that there are no interactions between corporate finance and investment decisions.

This thesis was the subject of several criticisms, which primarily related to the existence of a perfect market. The fact that the researchers neglected the imperfections of the market, in particular, the effect of the taxation; costs of bankruptcy; the costs of agency and the costs of signaling, was regarded as a limitation with this contribution. By introducing market imperfections, two research orientations appeared. The first axis, supported by Kraus and Litzenberg (1973), Scott (1976), Myers (1977) takes into account the imperfections of the market such as the taxation and the costs of bankruptcy and suggests that firms seem to get an optimal, value maximizing debt equity ratio by trading off the advantages of debt against the disadvantages.

The second axis takes into account the imperfections of the market such as the agency problems and asymmetries of information [START_REF] Myers | The Capital Structure Puzzle[END_REF][START_REF] Myers | Corporate Financing and Investment Decisions When Firms Have Information That Investors do not Have[END_REF]. It considers the choice of the structure of the capital as a solution to the inefficiencies which characterize the decisions of investment of the firm related to the asymmetric information between managers and investors. Managers know more about the true value of the firm and the firm's riskiness than less informed outside investors. To avoid the underinvestment problem, managers will seek to finance the new project using a security that is not undervalued by the market, such as internal funds or riskless debt. Therefore, this affects the choice between internal and external financing. Thus, to mitigate information asymmetry, firms follow a financing hierarchy: internal funds are preferred above external financing and if the latter becomes necessary, safe debt is preferred over risky debt and equity issues are the lowest end choice. So, research in the capital structure field is dominated by two theories: the static trade off theory and pecking order theory. The main purpose of this study is to examine the empirical validity of the pecking order and the trade off theory, for a sample of French firms over the period 1998 to 2002.

The rest of the paper is organized as follows. In the next section the theoretical background and hypotheses are presented. In section 3 and 4, we test the static trade off theory and the pecking order theory against the irrelevance theorem of [START_REF] Modigliani | The Cost of Capital, Corporation Finance and the Theory of Investment[END_REF]). The study uses the models proposed by Shyam [START_REF] Sunder | Testing Static Trade Off against Pecking Order Models of Capital Structure[END_REF]. It employs the partial adjustment model and a regressing model of the change in total debt level on the internal funds deficit. Furthermore, to consider the specificity of the French context (the short term debt is not negligible) we have to supplement this approach by introducing short term vision in financial deficit. In section 5, to reconcile the STT and the POT, this study tests the pecking order model and trade off model in the same regression. Finally, the conclusions are presented in section 6.

II.

Theories and related literature:

1. Static Trade Off:

In their second seminal paper, [START_REF] Modigliani | Corporate Income Taxes and the Cost of Capital: A Correction[END_REF] have altered the underlying argument of their classical proposition of capital structure. They incorporate the corporate income tax and contend that the value of the firm, if levered, equals the value of the firm if unlevered plus the value of the generated tax benefit. [START_REF] Modigliani | Corporate Income Taxes and the Cost of Capital: A Correction[END_REF] as [START_REF] Modigliani | The Cost of Capital, Corporation Finance and the Theory of Investment[END_REF] ignore the agency and bankruptcy costs of debt. To certain limits, the presence of agency and bankruptcy costs of debt may outweigh its tax benefit, suggesting the existence of an optimal target financial debt ratio, under which the firm's value is maximized (Kraus andLitzenberg (1973), Scott (1976), Myers (1977)). The static trade off theory (STT) implies that firms have a target debt ratio and try to move towards this target.

According to [START_REF] Myers | Capital Structure[END_REF], the optimal point can be attained when the marginal value of the benefits associated with debt issues exactly offsets the increase in the present value of the costs associated with issuing more debt. The benefits include the tax shield whereas the costs include expected financial distress costs1 .

The static trade off predicts that:

• Firms that possess intangible assets or high growth opportunities (high bankruptcy costs) tend to be less debt than those with mostly tangible assets (bankruptcy costs relatively low). • The risky firms tend to less being involved in debt, anything being equal in addition.

• The debt ratio target should be different for different industries that must depend in each branch of the evolution of the tax advantage and the danger of bankruptcy. • The mature firms, which usually have many tangible assets, are more indebted than firms whose main business growth is largely based on research and development of technology and advertising. • The most profitable firms are more indebted.

• The existence of a negative relationship between earnings variability and firm leverage.

• Firms with high marginal rate of imposition (most profitable and whose variation of earnings is low) are high debt.

• By controlling profitability, the firms with tax benefits unrelated with high debt have small debt ratios.

The pecking order theory:

The pecking order theory (POT) developed by [START_REF] Myers | The Capital Structure Puzzle[END_REF] is an alternative capital structure theory. It predicts that, due to asymmetric information and transactions costs, firms adopt a hierarchical order of financing preferences so that internal financing is preferred over external financing. If external financing is needed, firms first seek debt funding. Equity is only issued as a last resort. This ranking was motivated with reference to the [START_REF] Myers | Corporate Financing and Investment Decisions When Firms Have Information That Investors do not Have[END_REF] adverse selection problem that arises because managers are more knowledgeable than outsiders (investors). [START_REF] Myers | Corporate Financing and Investment Decisions When Firms Have Information That Investors do not Have[END_REF] claim that if the firm finances its new project by issuing new securities, these securities will be under priced. This is because managers cannot credibly convey the quality of their existing assets and available investment opportunities to potential investors. As a result, outsiders may not be able to discriminate between good and bad projects, consequently interpreting the firm's decision to issue new securities as a sign of possible bad news and then pricing new securities accordingly.

The Pecking Order Theory stipulates that:

• Firms prefer internal financing over external financing. However, if in the presence of profitable investment opportunities, they choose external financing.

• They adapt their payout ratio of dividends to their investment opportunities. The policy is rigid in order to have a sufficient cash flow which enables it to finance its investment opportunities.

• A rigid dividend policy and unpredictable fluctuations in profitability and investment opportunities of the firm suggests that internal cash flows may be higher or lower than investments. If they are lower, the firm primarily uses its liquidity or its portfolio of securities. If they are higher, the firm repays his debt or investing in liquid assets or marketable securities. If the surplus continues, it may gradually increase its payout ratio target.

• If external funding is necessary, the firm issues less risky funding, first, the debt, then hybrid securities such as convertible bonds and in the last, it issues new shares. Pecking Order theory does not predict a ratio of debt (a mix of debt and equity) well defined target because there are two types of capital: internal and external. One is at the top of the hierarchy and the other is at the bottom of the hierarchy. The debt ratio represents the accumulated firm's external financing needs.

Empirical evidence :

Although many previous studies have examined the pecking order theory and the static tradeoff theory, there is no consensus on the superiority of one of the theories.

In their recent study, Dufour and [START_REF] Molay | La Structure Financière Des PME Françaises : Une Analyse Sectorielle Sur Données De Panel, Crises et nouvelles problématiques de la Valeur[END_REF] analyzes the capital structure of 1535 French SME observed over a period of 8 years. Two representations of financing behavior are tested: the first one considers that companies implement a debt policy to reach a target debt ratio while the second one relying on a pecking order of financing considers that there is no such a target ratio. Statistical tests validate the first approach. The choice of financing of French SMEs confirms the greatest explanatory power of the target ratio explanation. The industrial sector of the firms does not affect these results.

In similar study, [START_REF] Molay | La structure financière du capital : tests empiriques sur le marché français[END_REF] tests two alternative theories of capital structure: the pecking order theory and the static trade off theory. The empirical tests conducted on a sample of French firms listed on the Paris stock ex change show that their financing choice seems to be more in line with the pecking order theory than with the static trade off theory. The tested firms prefer internal financing to external financing and, when using external financing, debt is preferred over equity.

In more recent study conducted in the context of Central and Eastern European countries, [START_REF] Delcoure | The determinants of capital structure in transitional economies[END_REF] finds a positive relationship between firms leverage ratios on the one hand, and asset tangibility, non debt tax shield, and taxes on the other hand. In addition, a negative relationship between leverage ratios and profitability is found. Furthermore, puzzling findings relating to the relationship between firm size and earnings volatility with the leverage ratios are found, as the significant signs change across countries and among the different dependent variables. Finally, [START_REF] Delcoure | The determinants of capital structure in transitional economies[END_REF] concludes that the pecking order hypothesis, the trade off theory, and the agency theory explain the capital structure puzzle only partially in his sample countries.

Syham - [START_REF] Sunder | Testing Static Trade Off against Pecking Order Models of Capital Structure[END_REF] test the pecking order theory and trade off theory in the US market. For pecking order theory, they regress the firm's net debt issues on its net financing deficit. They find that the estimated coefficient on the deficit variable is close to one. Syham - [START_REF] Sunder | Testing Static Trade Off against Pecking Order Models of Capital Structure[END_REF] interpret this result as evidence supporting pecking order theory because a shortfall in funds is first met by debt. Furthermore, they find that the power of trade off theory in explaining new debts issues is better than pecking order theory because when the pecking order model and trade off model are nested in the same regression, all cases of pecking order model are rejected (they use the net financing deficit as an additional explanatory variable in their trade off theory model).

III. Static Trade Off Theory: Empirical Evidence

To test the static trade off theory (STT) against the irrelevance theorem ( [START_REF] Modigliani | The Cost of Capital, Corporation Finance and the Theory of Investment[END_REF]), we rely firstly on the approach initiated by Shyam [START_REF] Sunder | Testing Static Trade Off against Pecking Order Models of Capital Structure[END_REF], and secondly by adding a few tests to account for possible effects (sector, size, risk and capital intensity) on the behavior debt of the French firm.

The description of data, the presentation of the variables studied and the hypotheses tested are successively assessed in this section concludes with a presentation and interpretation of results.

Data and methodology:

Our sample consists of 122 French firms belonging to different sector (Agroalimentary, chemicals, pharmaceuticals and cosmetics, basic industry, general industry and service). Given the differences in approach to funding policy between financial firms and non financial ones, it seemed appropriate to eliminate the financial firm. The choice of sectors and enterprises has been constrained by data availability.

The data used in the study are mainly accounting data on balance sheets, income statements and cash flow. They are provided by the financial markets authority (AMF). The study period extends from 1998 to 2002 i.e. 610 observations.

To test the static trade off theory for the selected sample of French firms, models (M1) (M2) (M3) (M4) are then estimated on data consisting of 121 firms multiply by 5 observations. To this end, we first calculated the change in long term debt for each firm and each year. Next, we determined the variation in the ratio of long term debt. Finally, the long term debt target was calculated in two ways: with the approach set by Shyam [START_REF] Sunder | Testing Static Trade Off against Pecking Order Models of Capital Structure[END_REF] and on the basis of a debt target sector average.

Dependent Variables:

The explanatory variables are classified into two groups depending on whether one wants to test the model (M1) or model (M2). Indeed the model (M1) uses the change in the amount of long term debt us variable while the model (M2) refers to the change of the ratio of long term debt.

The variation of the long term debt is determined as follows:

∆D it = D it D it 1
Where D it : the long term debt of the i th firm at time t as it appears in the balance of the i th firm.

For each firm considered, it has 5 annual observations. In total and for the 122 firms in the sample used, we have 610 observations of the change of the long term debt.

The variation in the ratio of long term debt is calculated as follows:

1 1 - - - =
Where: D it : the long term debt of i th firm at time t A it: the book value of assets of i th firm at time t. It is equal to the difference between total assets and net current liabilities (Shyam [START_REF] Sunder | Testing Static Trade Off against Pecking Order Models of Capital Structure[END_REF]).

Independent variable:

For describing the explanatory variable of the models (M1) and (M2), it is necessary to determine as a preliminary the long term debt targets such as definite by Shyam Sunder and Myers (D* it : the target ratio of the i th firm) and such as we defined it as a sectoral average (D* jt : the ratio targets of the j th sector). CAPI it : the book value of the credit of i th firm at the period T Thereafter, we calculated the explanatory variables z it and z' it (w it and w jt ) : the distance between the target value and the delayed value (the target ratio and the ratio delayed) for one year of the long term debt ; determined by the following expression:

z it = D* it D it 1 1 1 * - - - = (global) z' it = D* jt D it 1 1 1 * ' - - - = (sectoral)
Where: d* it : the debt ratio targets of i th firm d* jt : the debt ratio targets of j th sector

Regression Model and hypotheses:

The static trade off theory rests on the assumption according to which the managers are in search of an optimal structure of the capital. Random events can move away them from this objective. Nevertheless, the managers must converge there gradually. If the optimal debt ratio is stable, it is then possible to note a behavior of return to the average (Mean reverting behavior). A simple form of the model of adjustment explains the variations of the debt ratio by the deviations of the current ratio from the target ratio.

The first stage of our research consists in checking this result. For that, we carried out the following regressions:

( ) + - + = -1 * (M1) +         - + = - - 1 1 * (M2) ( ) + - + = -1 * (M3) +         - + = - - 1 1 * (M4)
The hypotheses to be tested are written as follows:

   ≠ = 0 : 0 : 1 0
For each firm and each year, variables ∆D it (D* it D it 1 ) and (D* jt D it 1 ) were reported at book value of assets of each firm only as a precaution against heteroscedasticity [START_REF] Sunder | Testing Static Trade Off against Pecking Order Models of Capital Structure[END_REF]). Furthermore, our analysis is limited to the carrying value of debt and the debt ratio is expressed only at book value.

In order to highlight the possible effects of size 2 , risk 3 and capital intensity 4 on the long term debt from the French firms in the sample, we adopted an approach which is to classify firms into two categories around the median of each variable: SMEs and large firms, less risky and risky, capital intensive and low capital intensive and then estimate the Model (M1) (M2) (M3) (M4).

The model estimation model estimation is performed through the software "SPSS".

Regression results

The first step of our research is to estimate the models (M1) and (M2). Estimation of the model (M1) shows the following results ( Table 1). Analysis of the results of the estimation of model (M1) shows that the coefficient associated with the variable (D* it -D it 1 ) is not affected by the expected sign ( 0.414 < 0). Nevertheless, it verifies the assumption that its value is less than the unit ( 0.414 < 1). In addition, to the 5% level, the variable (D* it -D it 1 ) is contributory to the explanation of debt financing in the long term.

At 5% level, the intercept is significant. The coefficient of determination of the model (M1) is 0.27. Moreover, the results of the estimation of model (M2) that the variable is measured by the variation long term debt are as follows:

2 The approximate variable used is the average of the capital engaged for each firm 3 The approximate variable used is the standard deviation of the capacity of self financing divided by the average of this last. 4 The approximate variable used is the average of ratio investment by capital engaged for each firm 5 Values in parentheses indicate the statistical "t" of student 6 ** Indicates that the coefficient is significant at 5% level. The estimation of model (M2) shows a finding that the use of debt as measured by the variation of long term debt is significantly related to the distance between the target ratio and the ratio of delayed debt long term. This result is consistent with expectations of the STT.

In addition, the intercept is not significant. This finding invalidates the hypothesis that change in long term debt partially absorbs the difference between the target and the lagged value of long term debt.

Moreover, the speed of adjustment towards the target value of long term debt is low. Indeed, an adjustment factor of 41.4% reflects the relatively rapid adjustments in respect of long term debt. In addition, the coefficient on the variable z it is significantly less than unity, suggesting the existence of positives adjustment costs.

The estimation of model (M2) shows results consistent with expectations. Indeed, the coefficient on the distance between the target ratio and the ratio of delayed long term debt is positive and significant at 5% (b TA = 0.853 and t statistic = 14,930). An adjustment factor of 85.3% reflects a relatively high speed of adjustment and shows that the variation in the ratio of long term debt absorbs quickly the distance between the target ratio and the ratio of delayed long term debt.

2.1.Regression results of firms according to the restraint characteristic:

The estimation results of models (M1) and (M2) are presented in Table 2.

The estimation of the model (M2) for small firms show that at the 5% level, small firms converge significantly and partially to the target level of long term debt (b TA = 0.760 and Student's t=7.442). In addition, the adjustment factor is less than unity suggesting the existence of positive adjustment costs associated with market imperfections. A speed adjustment of 76% shows that the change in long term debt absorbs quickly the distance between the target and the lagged value of long term debt for small firms. Estimation of the model (M1) for small firms leads to similar results to those obtained in the analysis of model (M2).

The results of the estimation of model (M2) for large firms shows that the ratio of long term debt of large firms in the sample converges quickly and significantly to the target ratio (b TA =0.939 and student's t = 16.654).

Estimation of the model (M1) for the least risky companies shows that at the 5% level, the long term debt of firms in the sample moves away significantly from the target level of long term debt. This finding invalidates the hypothesis that the change in long term debt partially absorbs the difference between the target and the lagged value of long term debt. Moreover, regression results of the model (M2) show that the ratio of long term debt converges significantly and quickly to the target ratio (b TA =0.987 and Student's t =15.758).

The results of the estimation of model (M1) for risky firms show that at the 5% level, the long term debt of French firms is significantly closer to the target (b TA = 0.690 and Student's t =11.986). This result confirms the hypothesis that change in long term debt partially absorbs the difference between the target and the lagged value of long term debt. The estimation of model (M2) shows results very close to those observed in the analysis model (M1).

Estimation of the model (M1) for lowly capital intensity firms highlight a negative and significant adjustment factor at the 5% level (b TA = 0.644 and Student's t = 19.292). Thus, the debt ratio moves away partially from the target value.

The results of the estimation of the model (M2) for highly capital intensive firms are consistent with predictions of the model of capital structure adjustment. Indeed, the adjustment coefficient is positive and significant (b TA = 0.931 and Student's t =10.029). In addition, the adjustment factor is less than unity, suggesting the existence of positive adjustment costs. Estimation of the model (M1) shows the results very close to those observed during the analysis of the model (M2).

In sum, the test of the STT against the irrelevance theorem for our sample of French firms shows that:

Firms tend to converge to an optimal overall capital.

The existence of a sectoral effect is validated. By choosing an optimal structure by sector, the change in long term debt absorbs, with a specific speed to each sector, the difference between the target and the lagged value of long term debt.

The existence of positive adjustment costs that the French firms of the sample does not completely and probably converge at any time to the target level of long term debt. Nevertheless, the costs of deviations from an optimal structure are generally not high enough to prevent French firms from the sample used to make more use of long term debt.

The existence of a size effect. Indeed, large firms in our sample seem to converge more quickly towards the target level of long term debt.

The existence of risk effect: the less risky firms those who's their cash flow is more even, seem to converge more quickly towards the target level of long term debt.

The most capital intensive firms seem to converge more quickly towards the target level of long term debt.

IV. Pecking Order Theory : Empirical Evidence

This part aims to test the Pecking Order Theory which can explain the financial behavior of French firms in the sample against the irrelevance theorem ( [START_REF] Modigliani | The Cost of Capital, Corporation Finance and the Theory of Investment[END_REF]). Firstly, we selected the approach initiated by Shyam [START_REF] Sunder | Testing Static Trade Off against Pecking Order Models of Capital Structure[END_REF] and also extend this approach to be compatible with the French view (the short term debt does is not negligible). The description of data, presentation of the variables studied and the identification of hypotheses tested are successively presented in this part which ends with a presentation and interpretation of results.

Data and methodology :

To test the POT we used the same sample, the same study period and the same data sources used in the empirical study conducted in the second section. Thus, the study was conducted on 122 French firms from different sectors over the period from 1998 to 2002.

An examination of the POT for the sample of French firms selected requires measuring the funds flow deficit in each period and for each firm. Data are collected from balance sheets and financial statements. To this end, we determined the variables needed to calculate the funds flow deficit of firms. Moreover, the explanatory variables as defined by Shyam Sunder and Myers (1999) were calculated.

1.1.Dependent variables:

It should be remembered that our study is based on the same explanatory variables used in the empirical analysis in second section of the present work.

The variation of the long term debt is determined as follows:

∆D it = D it D it 1
Where: D it : the long term debt of i th firm at time t as it appear in his balance sheet

For each firm considered, it has 5 annual observations. In total and for the 122 firms in the sample used, we have 610 observations of the variation of the long term debt.

The variation in the ratio of long term debt is calculated as follows:

1 1 - - - =
Where: D it : the long term debt of i th firm at time t.

A it : the net book of assets of i th firm at time t. It is equal to the difference between total assets and net current liabilities [START_REF] Sunder | Testing Static Trade Off against Pecking Order Models of Capital Structure[END_REF]).

1.2.Independent variable:

For purposes of this study, we consider that a firm faces in each period to fund requirements. The funds flow deficit of the firm in year t are given by:

DEF t = DIV t + X t + ∆W t + Rt C t = ' DEF' t = DIV t + X t + ∆W t + ∆Ch t + ∆DCT t + R t C t ' ' =
Where:

C t = operating cash flows after interest and taxes.

Div t =dividend payments.

X t = capital expenditure.

∆W t = net increase in working capital. R t = current portion of long term debt at start of period.

CH t = liquidity and short term investments.

DCT t = the short term debt.

D t = the long term debt.

A t = net book assets 7 d t = D t / A t : the book debt ratio.

1.3.The estimation technique and testable hypotheses:

In its simplest form, the POT states that when internal cash flows of the firm do not adjust its investment opportunities and real commitments on distribution of dividends, it emits then the debt without risk. She never proceeds with the issuance of new shares except where it can only issue junk bonds and when the bankruptcy costs are high. Based in part on the approach Shyam [START_REF] Sunder | Testing Static Trade Off against Pecking Order Models of Capital Structure[END_REF], the funds flow deficit of the firms are assumed to be met by the change of the ratio of long term debt and the change in long term debt, and secondly 7 equals net total book assets less current liabilities.

we have reformulated the funds flow deficit so that it can account for the short term vision. The purpose of this study is to verify this result. For this model (M5) (M6) (M7) and (M8) are then estimated.

+ + = (M5) + + = (M6) + + = ' (M7) + + = ' (M8)
Where: D it : the long term debt issued by i th firm (or repaid if surplus). DEF i: the financial deficit of the i th firm. def it: the financial deficit of i th firm divided by its total equity. ∆d it : the change in the ratio of long term debt.

DTOT it : net total debt issued (or repaid if surplus) by the firm i.

DEF' i : fixed needs of i th firm.

∆dTOT it : the change in the ratio of total net debt.

def' it : the corrected financial funds flow deficit of the i th firm divided by equity.

The test of significance of the coefficient b PO is written as follows8 :

   ≠ = 0 : 0 : 1 0
In order to test the POT, the two hypotheses that a = 0 and b po = 1 are verified. The model estimation is tested by software "SPSS".

Rgression results:

Analysis of the results of estimating models (M5) shows that the coefficient (b PO ) is different from unity and statistically significant at 5% level. This result suggests that at the 5% level the variable DEF has a significant contribution to explaining the change of long term debt. Moreover, with a correlation coefficient which equals zero results of the estimation of model (M6) are not taken into account.

The results of the estimation model (M7) did not differ from those obtained when estimating the model (M5). Indeed, the coefficient associated with the variable deficit is different from unity and statistically significant at 5% level. Thus, the indicator of deficit plays a role in explaining the variation in total net debt. In addition, the intercept is different of zero and statistically. In contrast, when the observed variable is the debt ratio and not the change of debt, the results do not support the POT.

2.1.Regression results by sector:

The results of estimating models (M5) (M6) (M7) and (M8) are presented in Table (4). The results of the estimation of the model (M5) show that at 5% level the variable deficit is significantly contributory to explaining the variation in long term debt for the Agro alimentary sector. The results of the estimation of model (M6) did not differ from those obtained when estimating the model (M5).

For the chemicals sector, pharmaceuticals and cosmetics, the indicator deficit plays a role in explaining the variation in long term debt (total debt net) in the model (M5), (M6) , (M7) and ( 8).

The regression results of models (M7) and (M8) for basic industry sector and general industry sector are consistent with predictions of the POT. Indeed, the deficit variable contributes significantly to explaining the change in net total debt and net total debt ratio.

For the service sector, the regression results of the model (M5) show that the indicator of the deficit plays a role in explaining change in long term debt.

2.2.Regression results according to the feature selected:

The results of the regression models (M5) and (M6) show that the DEF variable is explanatory of the change of the long term debt (rate of long term debt) for the large firms, the least risky firms and the low capital intensive firms. In addition, regression results of the models (M7) and (M8) are not different from those obtained during the estimate of the models (M5) and (M6).

Ultimately, the test of the theory POT against the irrelevance theorem for the sample of the French firms retained leads to the following results:

The financial deficit contributes only partially to the explanation of the change of the long term debt for our total sample. Consequently, the hypothesis of POT can be accepted.

The existence of a sectoral effect for the short term vision. Indeed, the change of the long term debt is explained significantly by the deficit for the sectors agro alimentary, chemistry, pharmacy, cosmetic and service whereas the change of the clear total debt is explained significantly by the deficit corrected for the basic industry and general industry sectors.

The existence of a size, risk and capital intensive intensity effects. Indeed, the indicator of financial deficit plays a significant role in the explanation of the variation of the long term debt for the large firms, the least risky firms and the low capital intensive firms.

With the obviousness, our empirical results do not make it possible to conclude on a better explanation of the behaviors by a model or the other. From where the idea, already suggested by Syham [START_REF] Sunder | Testing Static Trade Off against Pecking Order Models of Capital Structure[END_REF] to combine the two approaches to try to explain the variety of the behaviors.

V.

Combining Trade off and Pecking order models:

Previous studies have shown that understanding the behavior of firms in terms of choice of financing passes certainly through conciliation and not confrontation of the two analytical frameworks best known.

This paragraph proposes to analyze models combining each of the factors that might explain the financial behavior of French firms in the sample by the static trade off theory or the pecking order theory. We would then know whether the explanatory power and significance of the coefficients of the models (M1), (M2), ( M3), (M4), ( M5) and (M6) will they be improved taking into account both specific factors to the STT and to POT.

Regression Model:

In order to verify the improvement in explanatory power and significance of the coefficients of the models (M1), (M2), ( M3), (M4), ( M5) and (M6), we tested the following regressions.

( )

+ + - + = -1 * (M9) + +         - + = - - 1 1 * (M10) ( ) + + - + = -1 * (M11) + +         - + = - - 1 1 * (M12)

Regression results of the Model combining partial adjustment and the hierarchy of financing models:

The results of the models estimation (M9) and (M10) show that their coefficients of determination have increased considerably, which means that the addition of the indicator of the distance between the target and the lagged value of long term debt improves the explanatory power of models (M5) and (M6).

The combination of the two models of capital structure has improved the significance of the coefficient associated with the variable DEF.

2.1.Regression results by sector:

The estimate of the models (M11) and (M12) led to results similar to those obtained during the estimate of the models (M9) and (M10).

Ultimately, the combination of the partial adjustment model and the hierarchy of financing model improve considerably the statistical capacity of the hierarchical model and confirm the assumptions of the POT in the majority of the sectors selected. How to explain that the combination of two approaches predicting the different behaviors, to see contradictory, makes it possible better to account for the reality of the practices that each model separately?

One can imagine two explanations:

• The managers have a target ratio but they show opportunism, in the way which they take to reach it. Nevertheless, this way is directed by the hierarchical financing.

• There are two non reconcilable types of behavior, but which cohabit in our sample, and probably in the population. The taking into account of two models makes it possible to increase the variance explained without justifying of a real combination of the practices in the same firm.

VI. Discussion and conclusion:

This paper presents the empirical tests on data from a sample of French firms and which are intended to explain their behavior in terms of financing.

In the second section, we empirically tested the static trade off theory against the irrelevance theorem for a sample of 121 French firms from different sectors over a period from 1998 2002. To do this we adopted the approach of Shyam [START_REF] Sunder | Testing Static Trade Off against Pecking Order Models of Capital Structure[END_REF]. Our studies highlight the following results:

A convergence to an global optimal structure of capital from firms in the sample.

The existence of a sectoral effect is validated. By choosing an optimal structure sector, the change in long term debt absorbs with a speed specific to each sector, the difference between the target and the lagged value of long term debt.

The existence of positive adjustment costs that make firms in the sample probably did not converge completely at any time to the target level of long term debt. Nevertheless, the costs of deviations from an optimal structure are generally not high enough to prevent French firms from the sample used to make more use of long term debt.

The existence of size effect. Indeed, large firms in sample chosen seem to converge more quickly towards the target level of long term debt (which is consistent with the assumptions of STT).

The existence of risk effect: the less risky firms seem to converge more quickly towards the target level of long term debt.

The most capital intensive firms seem to converge more quickly towards the target level of long term debt (STT predicts that investment is positively related to debt).

The third section presents an empirical study of the behavior dictated by the POT. We empirically tested the relationship between the change in long term debt and financial deficit, firstly, as defined by Shyam [START_REF] Sunder | Testing Static Trade Off against Pecking Order Models of Capital Structure[END_REF] and then by introducing the specific French context which do not neglect the short term debts, a new financial deficit is calculated.

Empirical tests of the theory of the POT against the irrelevance theorem show that:

The financial deficit contributes only partially to the explanation of the change in long term debt for the full sample of French firms. Therefore, the hypotheses of the POT are accepted.

The existence of a sectoral effect for short term vision. Indeed, the change in long term debt is significantly explained by the deficit for the agro alimentary, chemicals, pharmaceuticals, cosmetics and service sectors while the change in total net debt is significantly explained by the deficit adjusted for industry sectors basic and general industry.

The existence of a size, risk and capital intensity effect. Indeed, the indicator of financial deficit plays a significant role in explaining the variation in long term debt for large firms, less risky and low capital intensive firms.

This paper concludes with an empirical study combining the model of the Trade Off and the hierarchy of financing. The inclusion of two schemes of analysis greatly improves the statistical power of the hierarchical model and confirms the hypotheses of the POT in the majority of the selected sectors.

In sum, this study shows that we can not formally reject either of the two theories explaining financing choices. However, it confirms the importance of considerations provided by the static trade off theory.

However, if this study helps to better understand financial behavior of French firms in the sample, it does not include less serious restrictions. Indeed:

The partial adjustment model that we applied to determine the convergence behavior of French firms in the sample supposes the stability of the target level of long term debt. However, the determinants of debt such as the variability of profits, tangibility, growth opportunities, size, profitability, tax savings which are not linked to debt ... are likely to vary over time for same firm. This captures the dynamic nature of capital structure. We should therefore use a debt target which changes in the time.

Moreover, the adjustment factor used in the adjustment model is assumed constant over time and identical for all firms. Nevertheless, the speed of adjustment towards optimal capital structure depends on several factors specific to the firm such that the distance between the target and the lagged value of long term debt, the firm size, growth opportunities, profitability, general economic conditions, etc. Thus, the adjustment is done at a rate specific to each firm (Heshmati ( 2001)).

In addition, the hierarchical model we used does not necessarily evaluate the pecking order theory. According Chirinko and Singha (2000), the hierarchical model initiated by Shyam [START_REF] Sunder | Testing Static Trade Off against Pecking Order Models of Capital Structure[END_REF] is likely to be rejected even when the firms' behavior is consistent with the assumptions of the POT. The hierarchical model can be also used even when the financial behavior firms is inconsistent with the POT. However, the results demonstrated for the sample of French firms open several avenues for future research. Indeed, alternative tests are necessary to enable beneficiaries to identify the determinants of capital structure and also enable to distinguish between rival hypotheses.

Finally, a better understanding of the relationship between the financing of the firm measured by stock (the levels of debt and equity in relation to the target level) and the financing of the firm measured in flow (which title to be emitted at a given moment) can reconcile the different theories of capital structure (Barclay and Smith (1999)).

  period of the study. It corresponds to five years (T=5).

  The coefficient of determination of the model (M2) is 0.359. The estimation results of models (M1) and (M2) are reported in Table(1).Regression results of model (M1) of the aggregate basis show that at the 5% level, the long term debt of firms in the sample moves away significantly from the target level of the long term debt (b TA = 0.414 < 1 and Student t 12.162). Contrary to the predictions of the adjustment model of capital structure, the coefficient associated with the variable (D * it Dit 1) is negative and significant at 5% level.

Table 1 :

 1 Regression results of model estimates (M1) and (M2)

		M1 (<D it )	M2 (<d it )
		301729	0.028
	Regression constant	(2.634)*	(1.429)
		0.414	0.853
	Independent variable	( 12.162)**	(14.930)**
	R 2	0.270	0.359
	=umber of observation	605	605

Table 2 :

 2 Regression results of models (M1) and (M2) for French firms in the sample according to each characteristic restraint:

	Small Business	M1 (<D it)	M2 (<d it )
		2547	0.58
	Regression constant
		(2.061)*	(1.621)
		0.501	0.760
	Independent variable	(10.319)**	(7.442)**
	R 2	0.358	0.226
		580697	0.002
	Regression constant
		(3.595)*	(0.104)
		0.414	0.939
	Independent variable	( 8.823)**	(16.654)**
	R 2	0.276	0.576
	Less risky firms
		480284	0.00
	Regression constant
		(3.187)*	( 0.020)
		0.465	0.987
	Independent variable	( 10.521)**	(15.758)**
	R 2	0.359	0.558
	Risky firms
		20552	0.059
	Regression constant
		(0.475)	(1.634)

Table 3 :

 3 Regression results of model (M5) (M6), (M7) and (M8)

		(M5)	(M6)	(M7)	(M8)
		44615	0.042	283802	0.062
	Regression constant	(0.619)	(1.709)	(2.037)*	(0.894)
		0.680	0.000002	0.175	0.003
	Independent variable	(19.253)**	( 0.167)	( 4.010)**	( 0.607)
	R 2	0.482	0.000	0.039	0.001

Table 4 :

 4 Regression results models (M5) (M6) (M7) and (M8) for French firms by sector.

		(M5)	(M6)	(M7)	(M8)
	Agroalimentary sector
		1426	0.019	463292	0.148
	Regression Constant	(0.017)	(0.303)	( 1.269)	( 0.407)
		0.692	0.127	1.02	1.061

Table 5 :

 5 Regression results of the models (M5) (M6) (M7) and (M8) for French firms according the feature selected.

		(M5)	(M6)	(M7)	(M8)
	Agroalimentary sector
		1426	0.019	463292	0.148
	Regression Constant	(0.017)	(0.303)	( 1.269)	( 0.407)
		0.692	0.127	1.02	1.061
	Independent variable	(6.805)**	(1.70)	(2.324)	( 5.621)*
	R 2	0.556	0.072	0.127	0.461
	Chemical and pharmaceutical sector
		51627	0.023	134321	0.005
	Regression Constant	(0.856)	( 0.711)	( 0.887)	(0.103)
	Independent variable	0.564	0.362	0.383	0.217
		(10.394)**	(3.685)*	( 5.764)**	( 2.046)*
	R 2	0.675	0.207	0.390	0.074
		243687	0.038	222056	0.033
	Regression Constant	(3.348)*	(1.404)	(1.389)	(0.195)
		0.135	0.084	0.334	0.002
	Independent variable	(2.248)*	(1.265)	( 4.607)**	( 0.368)
	R 2	0.062	0.02	0.216	0.002
		226445	0.002	254689	0.063
	Regression Constant	(2.219)*	(0.048)	(2.696)**	( 0.871)
		0.199	0.333	0.268	0.664
	Independent variable	(1.782)	(3.761)*	( 4.578)**	(8.583)
	R 2	0.025	0.104	0.147	0.377
	Service sector
		248312	0.078	782252	0.018
	Regression Constant	( 1.035)	( 0.949)	(1.620)	( 0.085)
	Independent variable	0.789	0.369	0.153	0.869
		(11.982)**	(2.735)*	( 1.725)	(3.145)
	R 2	0.589	0.07	0.028	0.089

[START_REF] Myers | The Capital Structure Puzzle[END_REF] focuses on the tradeoff between tax benefits and bankruptcy costs when describing the static tradeoff theory. Several other papers incorporate more factors into the static tradeoff theory, like agency costs (see, e.g., De Miguel and Pindado,

[START_REF] Myers | Capital Structure[END_REF] Flannery and Rangan, 2006).

This test is done through the Student t statistic


A S A U R E T

D C É B R O N

C M O R I Z E

D M L E B

Experimental and numerical study of mean zonal flows generated by librations of a rotating spherical cavity

We study both experimentally and numerically the steady zonal flow generated by longitudinal librations of a spherical rotating container. This study follows the recent weakly nonlinear analysis of Busse (2010), developed in the limit of small libration frequency -rotation rate ratio, and large libration frequency -spin-up time product. Using PIV measurements as well as results from axisymmetric numerical simulations, we confirm quantitatively the main features of Busse's analytical solution: the zonal flow takes the form of a retrograde solid body rotation in the fluid interior, which does not depend on the libration frequency nor on the Ekman number, and which varies as the square of the amplitude of excitation. We also report the presence of an unpredicted prograde flow at the equator near the outer wall.

Introduction

Longitudinal librations (reported below as librations) are periodic oscillations of a rotating container about its axis of rotation. Despite the fact that theses oscillations are time-dependent, it has been recently suggested that they can generate non-linearly a steady axisymmetric flow in the liquid interior through the Ekman boundary layer [START_REF] Busse | Mean zonal flows generated by librations of a rotating spherical cavity[END_REF]. A better knowledge of this resulting flow is of great interest in geo-and astrophysics (see for instance [START_REF] Noir | An experimental and numerical study of librationally driven flow in planetary cores and subsurface oceans[END_REF] where libration is driven by gravitational interactions and is used to investigate the interior structure of planets (e.g. [START_REF] Margot | Large amplitude libration of Mercury reveals a molten core[END_REF][START_REF] Van Hoolst | The librations, shape and icy shell of Europa[END_REF].

Despite of the possible applications, flows driven by libration in rotating containers have not been much studied. [START_REF] Aldridge | Axisymmetric inertial oscillations of a fluid in a rotating spherical container[END_REF] have observed experimentally that inertial modes can be excited by libration at particular resonance frequencies, which has been confirmed numerically by [START_REF] Rieutord | Linear theory of rotating fluids using spherical harmonics -II. time-periodic flows[END_REF]. However in the case of [START_REF] Aldridge | Axisymmetric inertial oscillations of a fluid in a rotating spherical container[END_REF], the experimental results are measurements of pressure differences between two points on the axis of rotation, and do not bring information about the resulting flow created in the interior. [START_REF] Tilgner | Driven inertial oscillations in spherical shells[END_REF] has investigated numerically the linear response to the forcing in the case of a spherical shell and has shown that the presence of an inner core only marginally modifies the resonance frequencies. More recently [START_REF] Noir | An experimental and numerical study of librationally driven flow in planetary cores and subsurface oceans[END_REF] have studied experimentally by direct flow visualization the presence of centrifugal instabilities in the form of Taylor-Görtler vortices near the outer boundary, by varying the frequency and the amplitude of libration. The same group has also performed LDV measurements of libration-driven zonal flows in a librating cylinder [START_REF] Noir | Experimental study of libration-driven zonal flows in a straight cylinder[END_REF] in the case of high frequency librations and axisymmetric simulations in a spherical shell [START_REF] Calkins | Axisymmetric simulations of libration-driven fluid dynamics in a spherical shell geometry[END_REF]. Finally, a complete weakly non-linear theory of the zonal flow driven by low frequency librations in a sphere has been recently developed by [START_REF] Busse | Mean zonal flows generated by librations of a rotating spherical cavity[END_REF] in the absence of direct resonant forcing of any inertial wave. To the best of our knowledge, the main features of this analytical solution have not yet been validated quantitatively. This is the aim of the present work, combining experimental and numerical approaches. The paper is organized as follows. §2 gives a brief summary of the governing equations and of the weakly nonlinear analysis of [START_REF] Busse | Mean zonal flows generated by librations of a rotating spherical cavity[END_REF]. In §3 we present the experimental setup and the numerical model used in this study. Then experimental and numerical results are compared to the theory in §4. Discussion and conclusion are given in §5.

Weakly nonlinear theory

Let us consider a spherical cavity of radius R filled with a homogeneous and incompressible fluid of kinematic viscosity ν. In the inertial frame the cavity rotates with an angular velocity

Ω(t) = Ω 0 + ∆Ω 2 cos(ω lib t) k, (2.1)
where Ω 0 is the mean rotation rate, ∆Ω is the amplitude of libration, ω lib the libration frequency and k the unit vector in the direction of the rotation axis. Using respectively R and Ω 0 -1 as lengthscale and timescale, the dimensionless equations of motion written in the frame rotating at the angular velocity Ω 0 and the sidewall boundary conditions are given by

∂u ∂t + u • ∇u + 2 k × u = -∇p + E∇ 2 u (2.2a) ∇ • u = 0 (2.2b) u = ǫ k × r cos(ω t) at |r| = 1, (2.2c)
where r is the spherical radial coordinate, u the velocity measured in the rotating frame, p the modified pressure taking into account centrifugal forces, E = ν/Ω 0 R 2 the Ekman number, ǫ = ∆Ω/2Ω 0 the normalized amplitude of libration and ω = ω lib /Ω 0 the normalized librational frequency. The librational forcing appears in the problem through the boundary condition (2.2c). This problem has been recently solved by [START_REF] Busse | Mean zonal flows generated by librations of a rotating spherical cavity[END_REF] in the limit:

√ E ≪ ω ≪ ǫ ≪ 1.
(2.

3)

The limit of small Ekman number allows splitting of the velocity field into two parts: a component U describing the inviscid flow in the interior and a boundary layer component u. Following the weakly non-linear method previously used in the case of precession [START_REF] Busse | Steady fluid flow in a precessing spheroidal shell[END_REF]) and tidal forcing [START_REF] Suess | Viscous flow in a deformable rotating container[END_REF], [START_REF] Busse | Mean zonal flows generated by librations of a rotating spherical cavity[END_REF] finds an expression for the steady zonal flow in the inviscid interior Ū in the limit of low libration frequencies

Ū = ǫ 2 k × r f (|k × r| 2 ) (2.4a) where f (x 2 ) = 259 x 2 -360 2400(1 -x 2 ) .
(2.4b)

The function f (x 2 ) represents the average difference in angular velocities between the container and the fluid divided by ǫ 2 . For 0 r 1, f (x 2 ) is negative, i.e. the fluid is expected to rotate in the retrograde direction. Moreover this differential rotation is nearly constant up to r ∼ 0.6 with a mean value of -0.154. The zonal flow can thus be assimilated in the bulk to a retrograde solid body rotation superimposed on the mean rotation, whose amplitude is independent of the libration frequency and Ekman number and changes as -0.154 ǫ 2 . In the present paper, we verify experimentally and numerically these main features. Note that (2.4.b) diverges for x = 1, i.e. near the outer boundary at the equator. Here, the analytical approach requires the introduction of a specific scaling due to the singularity of the Ekman boundary layer (Busse 2010).

Methods

Experimental setup

Figure 1 shows a photo and a schematic view of the experimental setup used in this study, which is the same as the one used by [START_REF] Morize | Zonal winds driven by tides[END_REF] to study zonal flows driven by tides. It consists in a hollow sphere, of radius R = 10 cm, which was molded in a transparent silicone gel to allow flow visualization. The sphere is filled with water and seeded with Optimage particles of 100 µm in diameter and of density 1 g.cm -3 ± 2%.

The sphere is set in rotation about its vertical axis (Oz) with a mean angular velocity Ω 0 up to 85 r.p.m with a precision of ±0.3%. Once a solid body rotation is reached (typically in ∼ 10 minutes), a librational motion is set using sinusoidal oscillations of the angular velocity of the sphere of the form ǫ cos(ω lib t) where ω lib can be chosen between 0.6 and 120 r.p.m with a precision of ±0.3%. In terms of dimensionless numbers, we have explored the following ranges: the Ekman number E = ν/Ω 0 R 2 ∈ [10 -5 ; 10 -4 ], the ratio between the libration and spin frequency ω = ω lib /Ω 0 ∈[0.04; 0.1] and the amplitude of libration ǫ ∈[0.02; 0.15] with a precision of ±0.6%.

In order to measure the velocity field in the equatorial plane induced by librational forcing, we used a rotating particle image velocimetry (PIV) system. A miniature wireless camera 1/4 ′ Sharp HighQ CCD 29.4 × 22 mm of resolution 576 × 768 pixels rotates at a constant angular velocity Ω 0 and measurements are made from above through the transparent top surface. The PIV particles are illuminated by a laser sheet, of thickness 3 mm, produced by a continuous laser (4 W) in the equatorial plane. We wait for about 20 oscillations after turning on the libration forcing to ensure that the response of the fluid is well established, then we start acquiring pictures for PIV measurements using a video transmitter-receptor system. Velocity fields are computed using DPIVSoft [START_REF] Meunier | Analysis and minimization of errors due to high gradients in particle image velocimetry[END_REF]) on a 60 × 80 grid with a spatial resolution of 3 mm, close to the laser sheet thickness. We look for the time independent axisymmetric zonal flow induced by the libration whereas the forcing of the sphere is of the form ǫ cos(ωt). Velocity fields are thus time-averaged over several periods of libration in order to eliminate the time dependent term. This also significantly enhances the signal to noise ratio. However, our experimental setup only allows collection of PIV data for a limited time. Hence, we cannot set the libration frequency at too low a value because we would not be able to average out the resulting data over enough periods to have a correct velocity profile. In the experiments we have considered frequencies in the range 0.04 ω 0.1. Higher and lower libration frequencies have been studied using numerical methods, that we describe in the following section.

Numerical approach

In addition to the experiments, we have performed axisymmetric numerical simulations of the flow within a sphere of radius R in rotation with an angular velocity Ω(t) = Ω 0 (1 + ǫ cos(ωt))e z . We use a commercial software, Comsol Multyphisics c , based on the finite elements method to solve this problem. The numerical grid consists of two domains: (i) a boundary layer domain of thickness 0.035 R all along the outer boundary and the axisymmetric axis, which is discretised in the direction normal to the boundary into 25 quadrilateral elements with initial thickness of 5.10 -5 R and a stretching factor of 1.2; (ii) a bulk zone with triangular elements. All elements are of standard Lagrange P 1 -P 2 type (i.e. linear for the pressure field and quadratic for the velocity field). Note that the finite element method does not induce any particular problem around r = 0 and that no stabilization technique has been used in this work. The temporal solver is IDA [START_REF] Hindmarsh | Sundials: Suite of nonlinear and differential/algebraic equation solvers[END_REF], based on backward differencing formulas. At each time step the system is solved with the sparse direct linear solver PARDISO †. The number of degrees of freedom (DoF) used in the simulations is constant and equal to 157 869 DoF. Our numerical model solves the Navier-Stokes equations in the frame rotating at the velocity Ω 0 e z , with no-slip boundary conditions and a fluid initially at rest in this frame (i.e. a solid body rotation at Ω 0 in the inertial frame). At time t = 0, libration of the outer boundary is turned on and computations are pursued until a stationary state is obtained, which is reached typically in less than 10 libration periods. The velocity is then averaged in time over 5 libration periods to obtain the steady zonal flow. Results are non-dimensionalised as in the experiment and in the theory. The numerical model has been validated in reproducing the experimental results of [START_REF] Aldridge | Axisymmetric inertial oscillations of a fluid in a rotating spherical container[END_REF]. In their paper they define a fixed libration Reynolds number Re ω = ω lib R 2 /ν = 6.2 • 10 4 and their applied angular velocity is given by Ω(t) = Ω 0 + ǫ ω lib cos(ω lib t) where ǫ = 8.0 π/180 rad.

(3.1) Pressure measurements from our numerical simulation when systematically changing the libration frequency ω are presented in figure 2.a and show an excellent agreement with the experimental results of [START_REF] Aldridge | Axisymmetric inertial oscillations of a fluid in a rotating spherical container[END_REF] and the numerical results of [START_REF] Rieutord | Linear theory of rotating fluids using spherical harmonics -II. time-periodic flows[END_REF], which validates the numerical model. In figure 2.a, each peak corresponds to the resonant forcing of a given inertial mode of the rotating sphere and is labeled by two integers (n, m)(see [START_REF] Aldridge | Axisymmetric inertial oscillations of a fluid in a rotating spherical container[END_REF]. There is no inertial mode for |ω| 2 and the modes are progressively damped when 1/ω increases due to the reduced coupling between the container's oscillation and the fluid interior as well as to the † www.pardiso-project.org increased viscous damping as the structure of the forced modes becomes more complex. The velocity in the z-direction for the mode (2, 1) in the notation of [START_REF] Aldridge | Axisymmetric inertial oscillations of a fluid in a rotating spherical container[END_REF] is presented in figure 2.b and shows the inertial wave excited by libration forcing as well as the structure of the outer boundary layer. In the next section, following Busse (2010), we investigate the limit ω ≪ 1 where the forcing of inertial modes is negligible, but where a global zonal flow is excited. opposed to the rotation of the sphere. From this velocity field we can plot an averaged azimuthal velocity profile at the equator for given parameters in term of dimensionless quantities as a function of the radial distance (Fig. 3.b). Experimentally, due to optical deformation induced by the planar air-silicone and spheroidal silicone-water interfaces, it is not possible to measure the profile for r > 0.85. We observe the steady zonal flow already visible in figure 3.a. Moreover we can directly compare this dimensionless quantity with the analytical solution given by Busse ( 2010) and we observe an excellent agreement up to r ∼ 0.6 with no adjustment parameter. An example of the velocity field obtained numerically is shown in figure 4, which also exhibits good agreement with the analytical solution in the bulk. For r 0.6 a deviation in the prograde direction with respect to the theoritical profile due to the librating outer boundary is observed and will be discussed below. But for now, we concentrate on the mean zonal flow induced in the bulk.

Results

We have performed series of experiments and numerical calculations to systematically check the effect of the three control parameters E, ǫ and ω on this bulk zonal flow. To do so, we define a reproducible method to synthetise the experimental and numerical data. Since the function f (x 2 ) (2.4.b) may be considered as constant up to r ∼ 0.6, which means that the predicted zonal flow is almost a solid body rotation up to r ∼ 0.6, we take the average value of the measured | Ū |/r between r = 0.1 and r = 0.6, i.e. the non-dimensionnalized mean angular velocity, and compare it with the theoretical value 0.154ǫ 2 . Note that experimental results are represented by bars (see for example figure 5) which represent both the uncertainties of the PIV measurements and the deviation of the measured velocity profile from a pure solid body rotation.

In figure 5 we investigate the influence of the amplitude of libration ǫ on the zonal flow. Experimentally we set E = 2.3 × 10 -5 , ω = 0.07 and we systematically change ǫ between 0.02 and 0.15. To explore a larger range of amplitude we have also performed numerical simulations with E = 4 × 10 -5 , ω = 0.04 and ǫ ∈[0.01; 0.2]. Both experimental and numerical results are quantitatively compatible with the theory with no adjustment parameter. The steady azimuthal velocity scales as ǫ 2 for a large range of ǫ. When ǫ becomes larger than 0.2 the weakly nonlinear hypothesis cannot be used anymore because terms of higher order cannot be neglected. We also notice that even in a range of values where the condition ω ≪ ǫ in (2.3) is not fully satisfied, the zonal wind intensity still scales as ǫ 2 . In fact, rather than the more restrictive condition (2.3), we only require that (i) E ≪ 1 to decouple the bulk and boundary layer flows, (ii) √ E ≪ ω ≪ 1 in order to neglect the excitation of inertial waves [START_REF] Aldridge | Axisymmetric inertial oscillations of a fluid in a rotating spherical container[END_REF] and to ensure that the spin-up effect of the libration is confined inside the outer boundary layer, and (iii) ǫ ≪ 1 to remain in the weakly nonlinear regime.

In figure 6 we report the systematic study of the influence of the Ekman number on the zonal flow. Experimental results are compatible with the no-Ekman dependence predicted by [START_REF] Busse | Mean zonal flows generated by librations of a rotating spherical cavity[END_REF] in a large range of Ekman number with no adjustment parameter. This is confirmed numerically up to E ∼ 10 -3 . For larger values of the Ekman number, the condition (2.3) is not fulfilled and we cannot assume that the effect of spin-up is negligible in the bulk. We have also noticed numerically that further decreasing E for a given ω = 0.1 (which is not so small) leads to a deviation from the theory. Indeed, forced inertial modes are not expected to be negligible anymore and can perturb the zonal flow. In particular, nonlinear self-interaction of these forced mode can drive localised zonal winds (e.g. [START_REF] Morize | Zonal winds driven by tides[END_REF]. These peculiar behavior appears out of the asymptotic limit (2.3) under consideration here and will be the subject of a future study.

Figure 7.a shows that the amplitude of the zonal wind in the bulk does not depend on ω, as suggested by Busse. Nevertheless, as can be seen in figure 4, the flow near the outer wall changes with ω. We expect this prograde flow to be related to the same mechanism of boundary layer ejection near the critical latitude as the prograde jets described in [START_REF] Noir | An experimental and numerical study of librationally driven flow in planetary cores and subsurface oceans[END_REF]. To quantify the distance at which the real flow deviates from the analytical solution, we identify the value r min where the velocity has a minimum. The thickness of the layer where the prograde flow develops is plotted in Fig. 7.b and is found to scales as 1/ √ ω, which is representative of a skin effect. So if ω becomes too small at a fixed E, the layer where the effects of external walls are important is visible in the bulk and perturbs the zonal flow.

Conclusion

In this paper, combining numerical and experimental studies, we report the first quantitative measurements of the steady flow driven by longitudinal librations in a rotating sphere. This approach confirms the main features of the weakly nonlinear theory of Busse (2010): a retrograde differential rotation induced by the libration of the sphere takes place, which may be assimilated to a solid body rotation for r < 0.6. It is also shown that the amplitude of this steady zonal flow is independent of ω and E and scales as ǫ 2 . Note that the same features have been observed experimentally in a librating cylinder [START_REF] Noir | Experimental study of libration-driven zonal flows in a straight cylinder[END_REF] and numerically in a librating spherical shell [START_REF] Calkins | Axisymmetric simulations of libration-driven fluid dynamics in a spherical shell geometry[END_REF] and thus appear to be generic of librating flows.

The main differences between our results and the theoretical profile of Busse (2010) arise close to the outer boundary at the equator. There, we observe a prograde flow in a layer of thickness proportional to 1/ √ ω. The analytical resolution of this peculiar feature would necessitate a special treatment since it appears at the critical latitude of the outer boundary layer [START_REF] Busse | Mean zonal flows generated by librations of a rotating spherical cavity[END_REF]. This, as well as the experimental study of a librating spherical shell, will be the subject of forthcoming studies.

Figure 1 .

 1 Figure 1. (a) Photo and (b) sketch of the experimental setup.

Figure 2 .

 2 Figure 2. (a) Time-averaged crest-to-trough amplitude of pressure difference Cp between the center and the pole of the sphere for various frequency ratios ω = ω lib /Ω0 and for a libration forcing Ω0 + ǫ ω lib cos(ω lib t) with ǫ = 8.0 π/180rad and Reω = ω lib R 2 /ν = 6.2 • 10 4 . The squares stand for our numerical values, the circles for experimental values of Aldridge & Toomre (1969) and the line for the theoretical plot. (b) Velocity in the z-direction (i.e. in the direction of the rotation axis) at time ωt = 3π/2 [2π] for the mode (2, 1) in the notation of Aldridge & Toomre(1969), corresponding to 1/ω = 1.066.

Figure 3 .

 3 Figure 3. (a) Time-averaged velocity field obtained by PIV measurement in the equatorial plane for E = 1.15 × 10 -5 , ǫ = 0.08 and ω = 0.1. The background is colored as the norm of the horizontal velocity. The center of the sphere is at (0, 0). (b) Mean experimental dimensionless azimuthal velocity (symbols) corresponding to the velocity field of Fig. 3.a and comparison with the theoretical results of Busse (2010) (dashed-dotted line).

Figure 3 .

 3 Figure 3.a shows an example of the velocity field obtained by PIV measurement in the equatorial plane. The stationary flow is azimuthal and axisymmetric. Besides, as the system rotation is clockwise, the zonal flow corresponds here to a retrograde circulation

Figure 4 .

 4 Figure 4. Dimensionless time-averaged velocity profiles obtained by numerical simulation in the equatorial plane with E = 5 × 10 -5 , ǫ = 0.2 and ω = 0.03 (dashed line), ω = 0.06 (dotted line) and ω = 0.1 (dashed-dotted line), compared with the inviscid analytical solution of Busse (2010)(continuous line).

Figure 5 .

 5 Figure 5. Dimensionless average amplitude of | Ū |/r between r = 0.1 and r = 0.6 as a function of the amplitude ǫ. Experimental results (bars) for E = 2.3 × 10 -5 , ω = 0.07, and numerical results (squares) for E = 4 × 10 -5 , ω = 0.04 are compared with the theoretical value of Busse (2010) (dashed-dotted line).

Figure 6 .

 6 Figure 6. Dimensionless average amplitude of | Ū |/rǫ 2 between r = 0.1 and r = 0.6 as a function of the Ekman number. Bars are experimental results for ǫ = 0.08 and ω = 0.1 and squares are numerical results for ǫ = 0.2 and ω = 0.06. The dashed-dotted line shows the theoretical result of[START_REF] Busse | Mean zonal flows generated by librations of a rotating spherical cavity[END_REF]. The velocity profiles have been rescaled by ǫ 2 following the results presented in figure5. Experimentally, errorbars increase with the Ekman number mainly due to the fact that our experimental setup does not allow us to average the velocity on a sufficient number of periods when increasing E.

Figure 7 .

 7 Figure 7. (a) Dimensionless average amplitude of | Ū |/rǫ 2 between r = 0.1 and r = 0.6 as a function of the frequency of libration ω; squares are numerical values (E = 5 × 10 -5 , ǫ = 0.2) and bars are experimental data (E = 1.5 × 10 -5 , ǫ = 0.1). The dashed-dotted line shows the theoretical result of Busse (2010). The velocity profiles have been rescaled by ǫ 2 following the results presented in figure 5. (b) Distance of the minimum of the velocity profile from the outer boundary as a function of ω obtained by numerical simulation for E = 5 × 10 -5 , ǫ = 0.2. The dotted line scales as 1/√ ω, representative of a skin effect.

† Email adress for correspondance: sauret@irphe.univ-mrs.fr