domain: phys.cond.cm-ms spi phys chim

text:
Guy Molénat

Benjamin Galy

Michael Musi

Louïse Toualbi

Marc Thomas

Helmut Clemens

Jean-Philippe Monchoux

Alain Couret

Plasticity and brittleness of the ordered o phase in a TNM-TiAl alloy

Keywords: A. titanium aluminide alloys, B. brittleness and ductility, dislocation structure, plastic deformation mechanisms, C. powder metallurgy, F. transmission electron microscopy

In order to identify the reasons for the brittleness of a TNM-TiAl alloy (Ti51.05Al43.9Nb4Mo0.95B0.1 in at.%), the plastic deformation behaviour of the ordered o phase is investigated. The corresponding powder was densified by Spark Plasma Sintering to produce a near lamellar microstructure made of γ/α2 lamellar colonies surrounded by  and o grains.

Then, the room temperature tensile behaviour is studied by carrying out tensile tests and additional in-situ straining experiments in a transmission electron microscope (TEM). The TNM alloy exhibits a limited ductility, with the fracture surface of test specimens showing cleavage facets along lamellar interfaces. In addition, the o grains contain nano-precipitates of the o phase and deform plastically by <111> superdislocations dissociated into two superpartial dislocations separated by an antiphase boundary. These dislocations glide in {011} planes and are observed to be localized into pile-ups, which is related to the o strengthening precipitation within the o grains. This localised hardening behaviour leads to stress concentration in grain boundaries which is assumed to be responsible for delamination along lamellar interfaces in neighbouring lamellar colonies.

Introduction

The new generation of intermetallic -TiAl-based alloys suitable for high-temperature structural applications usually contains refractory elements in order to improve mechanical properties at elevated temperatures via the reduction of dislocation mobility. These elements, e.g. V, Nb, Cr, Mo, Re or W, also promote the formation of the body-centred cubic (bcc) phase as disordered  phase (A2 structure) at high temperatures and ordered o phase (B2 structure) at lower temperatures. As mentioned in the exhaustive review of Appel et al. [START_REF] Appel | Gamma Titanium Aluminides: Science and Technology[END_REF] on TiAl alloys, the role of this phase on mechanical properties is still unclear: "A possible concern for the solidifying alloys is the reported reduction in mechanical properties such as room temperature ductility, high-temperature strength and creep resistance associated with the presence of the /B2 phases". A review of the current literature does confirm that the ductility of these β/o phases remains an open question. Indeed, on one hand, the o phase is considered to present very limited plastic deformation. In a Nb and Mo containing TiAl alloy, Hoppe et al. [START_REF] Hoppe | Deformation-induced internal stresses in multiphase titanium aluminide alloys[END_REF] showed a dislocation-free o grain located between two deformed lamellar colonies and assumed that an elastically transmitted deformation occurs between colonies promoted by a high internal stress. In-situ high-energy X-ray diffraction tensile experiments were performed at room temperature (RT) on sheets of the so-called TNM-TiAl alloy with a nominal composition of Ti-43.5Al-4Nb-1Mo-0.5B (in at.%) [START_REF] Erdely | Lattice and phase strain evolution during tensile loading of an intermetallic, multi-phase -TiAl based, alloy[END_REF]. In this study, a linear variation of the lattice strain for the o-reflections was found as a function of the applied stress, which was interpreted as indicativeof no plastic deformation in this phase. Sun et al. [START_REF] Sun | Alloying Mechanism of Beta Stabilizers in a TiAl Alloy[END_REF] measured the tensile behaviour of TiAl alloys with different amounts of Fe, Cr, V and Nb and found a decrease in plastic elongation with the increase in the content of alloying elements, which is associated with an increase in the proportion of o phase. Based on the observation of cleavage fracture facets in o grains and despite the presence of dislocations, these authors concluded to the brittleness of this phase.

On the other hand, Morris et al. [START_REF] Morris | Deformation mechanisms and slip transfer in a Ti-44Al-2Mo alloy[END_REF][START_REF] Morris | Deformation mechanisms within the B2, L10 and DO19 structures of a Ti-44Al-2Mo alloy, Gamma Titanium Aluminides[END_REF] investigated the RT mechanical properties of a Ti-44Al-2Mo (at%) alloy which shows that plastic elongation (0.5%) was higher in the as-HIPed condition with a higher o phase volume fraction (15%), compared to the other process conditions. Besides, these authors observed two types of dislocations in the o grains: <100> dislocations and dissociated <111> superlattice dislocations, which formed pile-ups nucleated at grain boundaries under the effect of deformation in adjacent grains. Th limited ductility of this alloy was attributed to both a low mobility of dislocations in the o phase and the lack of slip activity in the 2 phase. Finally, Leitner et al. [START_REF] Leitner | Fracture and Rcurve behavior of an intermetallic -stabilized TiAl alloy with different nearly lamellar microstructures[END_REF], who listed all the microstructural features playing a role on the fracture toughness of  containing TNM-TiAl alloys, did propose a negative effect of this phase as it forms a continuous seam at lamellar colony boundaries.

Based on these discrepancies in the literature about the role of the o phase on plasticity and fracture properties of γ-TiAl based alloys, the present work aims at clarifying this open question by combining an experimental approach of post-mortem and in-situ Transmission Electron Microscope (TEM) observations. For this purpose, a pre-alloyed powder with the chemical composition of the TNM-TiAl alloy, Ti51.05Al43.9Nb4Mo0.95B0.1, in at.%, was densified by Spark Plasma Sintering (SPS). The metallurgical features and mechanical properties of this alloy family are summarised elsewhere [START_REF] Clemens | Design of Novel -Solidifying TiAl Alloys with Adjustable /B2-Phase Fraction and Excellent Hot-Workability[END_REF][START_REF] Mayer | Intermetallic β-solidifying γ-TiAl based alloys: from fundamental research to application[END_REF]. Consistently with the Nb and Mo contents, the o phase is a constituent of this alloy associated with the  and 2 phases.

In TiAl alloys, the -stabilizing element Nb limits the formation of the γ-TiAl phase (L10 structure) and shifts the  and o phase field region to higher Al concentrations. The o phase is known to transform into -type phases at temperatures ranging between 500 °C and 1100 °C, depending on alloy composition and thermal treatments [START_REF] Bendersky | The formation of the ordered -related phases in alloys of composition Ti4Al3Nb[END_REF][START_REF] Wallgram | Technology and mechanical properties of advanced -TiAl based alloys[END_REF]. This transformation of -type hexagonal phases in bcc phases is similar to that commonly observed in commercial  or near- Ti base alloys [START_REF] Silcock | An X-ray examination of the  phase in TiV, TiMo and TiCr alloys[END_REF][START_REF] De Fontaine | The omega phase transformation in Titanium alloys as an example of displacement controlled reactions[END_REF][START_REF] Prima | Evidence of a-nanophase heterogeneous nucleation from w particles in a b-metastable Ti-based alloy by high-resolution electron microscopy[END_REF][START_REF] Devaraj | Experimental evidence of concurrent compositional and structural instabilities leading to x precipitation in titanium-molybdenum alloys[END_REF][START_REF] Smilauerova | On the completeness of the  →  transformation in metastable  titanium alloys[END_REF][START_REF] Ballor | A review of the metastable omega phase in beta titanium alloys: the phase transformation mechanisms and its effect on mechanical properties[END_REF], or more generally in alloys of group IV elements such as Ti, Zr and Hf [START_REF] Ballor | A review of the metastable omega phase in beta titanium alloys: the phase transformation mechanisms and its effect on mechanical properties[END_REF][START_REF] Banerjee | Phase transformations: examples from titanium and zirconium alloys[END_REF][START_REF] Banerjee | Omega phase transformationmorphologies and mechanisms[END_REF]. The transformation occurs through local chemical rearrangements associated with the collapse of {111} bcc planes, leading to the formation of double layers, the latter alternating with single layers. Because there are four {111}o planes in a single grain, four rotational orientation variants can be generated.

In the case of intermetallic TiAl alloys, extensive research has been done to determine the completeness of the bcc/hexagonal transformation, as well as the related chemical rearrangements [START_REF] Stark | In Situ Observation of Various Phase Transformation Paths in Nb-Rich TiAl Alloys during Quenching with Different Rates[END_REF][START_REF] Schloffer | Evolution of the o phase in a -stabilized multi-phase TiAl alloy and its effect on hardness[END_REF][START_REF] Wang | Atomic-scale observations of B2→ ω-related phases transition in high-Nb containing TiAl alloy[END_REF][START_REF] Klein | Impact of Mo on the 0 phase in -solidifying TiAl alloys: An experimental and computational approach[END_REF]. In particular, this transformation was investigated in a cast TNM alloy with a chemical composition similar to that of the alloy studied here [START_REF] Schloffer | Evolution of the o phase in a -stabilized multi-phase TiAl alloy and its effect on hardness[END_REF][START_REF] Klein | Impact of Mo on the 0 phase in -solidifying TiAl alloys: An experimental and computational approach[END_REF]. In-situ high-energy X-ray diffraction heating experiments allowed Schloffer et al. [START_REF] Schloffer | Evolution of the o phase in a -stabilized multi-phase TiAl alloy and its effect on hardness[END_REF] to measure the solvus temperature of o phases to be 825 °C. Klein et al. [START_REF] Klein | Impact of Mo on the 0 phase in -solidifying TiAl alloys: An experimental and computational approach[END_REF] proposed that the growth sequence was dominated by the redistribution of Mo into the  matrix, i.e. by the Mo diffusivity.

Materials and methods

The pre-alloyed TNM powder produced by argon gas atomization with the Electrode Induction Melting Gas Atomization (EIGA) technique [START_REF] Gerling | Power metallurgical processing of intermetallic gamma titanium aluminides[END_REF] was densified by SPS using the thermal and pressure cycle described elsewhere [START_REF] Voisin | Temperature control during Spark Plasma Sintering and application to up-scaling and complex shaping[END_REF]. To achieve a lamellar microstructure containing o grains, the dwell temperature ranged between 1300 °C and 1360 °C [START_REF] Voisin | Microstructures and mechanical properties of a multi-phase -solidifying TiAl alloy densified by spark plasma sintering[END_REF].

Throughout the paper, the indicated temperatures correspond to the actual temperatures of the powder and material (for detailed information regarding the SPS process the reader is referred to Ref. [START_REF] Voisin | Temperature control during Spark Plasma Sintering and application to up-scaling and complex shaping[END_REF]). Cylindrical billets with a 36 mm diameter and an 8 mm thickness were sintered under four different conditions: three different dwell temperatures, one of them (1360 °C) repeated with a 60 min instead of the usual 2 min (see Table 1). No heat treatment was applied to the material after sintering. Three specimens suitable for RT tensile tests (10 -4 s -1 strain rate) were machined from each billet.

A JEOL 6490 Scanning Electron Microscope (SEM) was used to characterize the microstructures in back-scattered electron (BSE) mode, and to analyse the fracture surfaces in secondary electron (SE) mode. Electron dispersive spectroscopy (EDS) analyses were performed with a FEI Helios 600i SEM equipped with an Oxford detector. The volume fractions of the o phase in the different materials were measured by X-ray diffraction (XRD) analysis on an AXS D8 Advanced diffractometer from Bruker. A JEOL 2020 TEM operating at 200 kV was used to analyse both the dislocations microstructures generated by the tensile tests (postmortem microscopy) and the controlling dislocations mechanisms (in-situ RT straining experiments) with a Gatan straining holder). Every TEM samples for both post-mortem and insitu experiments were taken from the AM504 billet (Table 1).

Microstructure of the as-SPSed TNM alloy

Regarding the four billets under examination (Table 1), it appears that billet AK639 with the lowest dwell temperature exhibits a lower volume fraction of the o phase compared to the three other billets. Billet AM504 with a specific long duration at dwell temperature shows larger lamellar colonies. The common microstructural feature for the four materials corresponds to large lamellar colonies with o grains along boundaries (see Fig. 1). As described in section 4, these similar microstructures result in similar RT mechanical behaviour (Fig. 3).

Regarding the lamellar microstructure, electron backscatter diffraction (EBSD) fails to delimit the colony size since this technique analyses each lamellae of a given colony as a different grain (indeed, in a colony the lamellae can take six different orientations, which manifest in ordered domains, twins or pseudo-twin relationships [START_REF] Zghal | A quantitative tem analysis of the lamellar microstructure in TiAl based alloys[END_REF]). Therefore, the colony size has been measured using a manual procedure applied on SEM micrographs. For each measurement, one hundred adjacent colonies were isolated and their size measured using the free ImageJ software. The grain size is defined as the diameter of a circle with the same area as the respective colony. The average values as well as the corresponding standard deviations (σ) are summarized in Table 1 for the four specimens. The relatively high value of σ (in the order of half the average colony size) reflects a large scattering of the colony sizes that typically range from a few microns to nearly one hundred µm., The volume fraction of the o phase as measured by XRD is also reported in Table 1, does increase with dwell temperature up to 1325 °C and then remains more or less constant. Therefore, the microstructure presented in Fig. 1 (billet AK746 for SEM analysis, Figs.

1a-e, and billet AM 504 for TEM image of Fig. 1f) is representative for the different sintered materials. The SEM micrograph of Fig. 1a shows a near lamellar microstructure made of γ/α2 lamellar colonies surrounded by borders with  and o grains exhibiting black and white colours, respectively. The o phase is distributed in different ways as evidenced by the corresponding enlargements (Figs. 1b-e) of Fig. 1a: -Extended shapes, situated at junctions between colony boundaries (enlargement framed in yellow, Fig. 1b), -Elongated features, generally located along a boundary between a border and a lamellar colony (enlargement framed in red, Fig. 1c), -Series of parallel, elongated and thin noodle-shaped phase immerged in a  grain, generally oriented perpendicularly to the boundaries (enlargement framed in blue, Fig. 1d),

-Small precipitates, lath-or noodle-shaped (enlargement framed in orange, Fig. 1e) and, -Bright zones located inside lamellar colonies (bright arrows in Fig. 1a).

Figure 1f shows a TEM micrograph of an area with the o phase distributed at the junction between several colony boundaries according to the situation illustrated in Fig. 1b.

darkest parts marked o() exhibit the so-called tweed contrast typical for the o-phase containing -type precipitates. Figure 2a depicts a diffraction pattern comparable to that previously reported for the bcc phases of conventional disordered Ti alloys [START_REF] De Fontaine | The omega phase transformation in Titanium alloys as an example of displacement controlled reactions[END_REF][START_REF] Devaraj | Experimental evidence of concurrent compositional and structural instabilities leading to x precipitation in titanium-molybdenum alloys[END_REF][START_REF] Ballor | A review of the metastable omega phase in beta titanium alloys: the phase transformation mechanisms and its effect on mechanical properties[END_REF][START_REF] Banerjee | Omega phase transformationmorphologies and mechanisms[END_REF] or intermetallic TiAl alloys. [START_REF] Bendersky | The formation of the ordered -related phases in alloys of composition Ti4Al3Nb[END_REF][START_REF] Wallgram | Technology and mechanical properties of advanced -TiAl based alloys[END_REF]. The o phase is oriented with a <110>o direction of the zone axis parallel to the electron beam in this diagram, and the additional spots are due to the presence of two 0 variants. Looking at the (222 ̅̅̅̅̅ )o direction in particular, several spots are believed to originate from the variant for which the c axis is parallel to the (222 ̅̅̅̅̅ )o direction: one fundamental reflexion and two superstructure spots, resulting from the long range ordered structure. However, it may be argued that similar patterns have been seen in Ti and Zr alloys exhibiting a disordered  structure [START_REF] De Fontaine | The omega phase transformation in Titanium alloys as an example of displacement controlled reactions[END_REF][START_REF] Devaraj | Experimental evidence of concurrent compositional and structural instabilities leading to x precipitation in titanium-molybdenum alloys[END_REF][START_REF] Ballor | A review of the metastable omega phase in beta titanium alloys: the phase transformation mechanisms and its effect on mechanical properties[END_REF][START_REF] Banerjee | Omega phase transformationmorphologies and mechanisms[END_REF].

On the basis of image simulations, Sabeena [START_REF] Sabeena | Microstructural characterization of transformation products of bcc  in Ti-15 Mo alloy[END_REF] proposed that double diffraction explains the presence of these spots in a Ti-15Mo alloy.

Figure 2b depicts a dark field micrograph of a o area taken with g=(0002)o superlattice diffraction vector, where the o precipitates seem to have circular morphology. By counting the number of precipitates and determining the precipitates volume fraction, the  precipitates are found to occupy a few volume percent (0.4 to 3.6 %) of the material and the order of magnitude of the distance between precipitates is about 27 nm (390 precipitates counted for one variant in a 182x394x434 nm 3 volume). Using the methods described in Ref. [START_REF] Delmas | Quantitative TEM study of the precipitation microstructure in aluminium alloy Al(MgSiCu) 6056 T6[END_REF], a precise determination of the volume fraction, the precipitates size and spacing will be carried out in a forthcoming investigation. In addition, work is underway to determine the precise crystallographic structure of these o precipitates, with a particular attention to their ordering state. Former studies have shown that the precipitation of  phase requires either an ageing at intermediate temperatures, e.g. 700°C [START_REF] Bendersky | The formation of the ordered -related phases in alloys of composition Ti4Al3Nb[END_REF] or 850°C [START_REF] Schloffer | Evolution of the o phase in a -stabilized multi-phase TiAl alloy and its effect on hardness[END_REF], or a slow cooling rate of 10 K/min [START_REF] Stark | In Situ Observation of Various Phase Transformation Paths in Nb-Rich TiAl Alloys during Quenching with Different Rates[END_REF]. This is consistent with our SPS experiments that exhibit cooling down from the dwell temperature (1300 °C or above, see Table 1) at a cooling rate of 100K/min inside the graphite mold maintained in the SPS setup [START_REF] Voisin | Temperature control during Spark Plasma Sintering and application to up-scaling and complex shaping[END_REF].

Mechanical tests and fracture surfaces

The RT true stress vs. true strain curves (10 -4 s -1 strain rate) are shown in Figure 3a for the four SPS billets.

Yield stress (YS) is the stress at which plastic deformation starts and is defined as the stress at 10 -3 % plastic elongation, R0.2 denotes the tensile stress at 0.2 % plastic elongation, UTS the ultimate tensile stress, and the term A represents the plastic elongation at rupture. As can be seen in Table 2, the four billets exhibit similar mechanical properties. Indeed, as already emphasized in as-SPSed B-containing TiAl alloys with the so-called GE composition (Ti-48Al-2Cr-2Nb, in at.%) [START_REF] Luo | Refinement of lamellar microstructures by boron incorporation in GE-TiAl alloys processed by Spark Plasma Sintering[END_REF], as soon as the dwell temperature is situated in the single  or + phase field, microstructures and mechanical properties are highly reproducible. The alloy densified at 1325 °C appears to have slightly improved properties resulting from a higher plastic elongation, which leads to a higher ultimate tensile strength (UTS). This trend is not easily explained from the measurements of the average grain size and of the volume fraction of o phase. Figure 3b shows the curves obtained for four specimens extracted from two billets both densified at 1325°C (billets AK746 and AM797). These four samples exhibit very similar properties with reduced deviations on the different measured stresses (YS, R0.2, UTS) and on the plastic elongation at failure.

The plastic elongation at rupture for all hot-worked billets tested is between 0.5 and 1.0 %. As noted by Appel et al. [START_REF] Appel | Gamma Titanium Aluminides: Science and Technology[END_REF], this limited RT ductility is typical for -solidifying alloys, which possess a fairly high RT strength. However, using SPS as processing route appears to result in a slight increase in RT plastic elongation by generating uniform and non-textured microstructures. 2 for details). 45c-d, Table 2). Figure 4a shows representative cleavage facets, the size of which matches that of lamellar colonies. The cleavage is likely to occur along lamellar interfaces. Between these cleavage facets, transgranular crack propagation is observed, as emphasized in Figure 4b showing at higher magnification some steps between the cleaved planes, which might be in the  lamellae. The micrographs in Fig. 4c and Fig. 4d depict an area both in backscattered mode and by a Mo EDX SEM map, respectively. The o phase is indicated by the bright yellow regions. This demonstrates that the o areas do not show any particular brittle behaviour.

Study of the deformation processes for o grains

Characterization of the deformation processes operating in o grains was carried out by both TEM post-mortem analyses of thin foils extracted from RT deformed samples until rupture (specimen AM 504, Table 2) and TEM in-situ straining experiments in samples cut from the same AM 504 billet.

Dislocation microstructure in deformed samples

About fifteen grains were examined by TEM and the presence of dislocations was observed in most of them (Figures 5 and6). On the general view (Fig. 5a), several groups of dislocations are observed. As shown in the two inserts (Figs. 5b andc), dislocations are out of contrast with the (01 ̅ 1) and (1 ̅ 01) reflexions demonstrating that their Burgers vector is parallel to the [111] direction, as this is the case for every dislocations in this area (circled detail labelled F serves as a mark in Figs. 5a-c). Dislocations are generally organized in pile-ups, where traces on the thin foil surfaces permit to determine their glide plane. For the pile-up of three dislocations (marked A, Fig. 5a), the trace is parallel to the (1 ̅ 01) plane. A tilt experiment (not reproduced here), for which this pile-up was observed at different inclinations tilted around this trace has clearly confirmed that it is located in the mentioned (1 ̅ 01) plane, which is a glide plane for these Figure 6, which is taken under higher magnification, gives two examples of the structure of these <111> dislocations. In Fig. 6a, dislocations are localized in a pile-up, whereas Fig. 6b shows an isolated long dislocation, the rectilinear segments of which are parallel to the screw direction of the dislocation. Dislocations are often pinned at small obstacles, as indicated by small white arrows in Fig. 6a. Every dislocations clearly appear as paired dislocations in every cases but with notable differences in spacing. In particular in the pile-up, various spacing is likely to occur from the presence of pinning points (arrowed). Without pinning points, the spacing of paired dislocations is approximately constant all along the line with an apparent value (da) ranging from 5 to 10 nm. The long dislocation also exhibits macrokinks, which are reminiscent of the dislocation structures in L12 intermetallic compounds such as Ni3Al and Ni3Ga [START_REF] Molénat | Dislocation mechanisms in Ni,Al at room temperature. In situ straining experiments in TEM[END_REF][START_REF] Bontemps | The geometrical configuration of kinks on screw superdislocations in Ni,Al deformed at room temperature[END_REF]. It is even tempting to underline an alignment of the leading and trailing partials bordering a macrokink as evidenced in Ni3Al and Ni3Ga alloys deformed at RT [START_REF] Molénat | Dislocation mechanisms in Ni,Al at room temperature. In situ straining experiments in TEM[END_REF][START_REF] Couret | Glide sequences of deformation in the (111) plane of Ni3Ga single crystals in the yield stress anomaly[END_REF] as the result of jump amplitudes equal to the width of the antiphase boundary separating the two superpartial dislocations of the a<011> dislocations involved in these face-centered cubic type intermetallic alloys. The obtained experimental results can be interpreted as follows:

• a<111> dislocations are dissociated into two identical dislocations bordering an antiphase boundary (APB), consistently with the following dissociation scheme: a<111> → a/2<111> + APB + a/2<111> (as the lengths of the Burgers vectors of these a/2<111> partial dislocations and a<111> dislocations result from the long range ordered structure, they are generally called superpartial dislocations and superdislocations).

• Dislocations are often confined in pile-ups, gliding in {110} type planes.

• Assuming that the APB ribbon lays in the gliding plane, a dissociation width ranging from 10 to 20 nm can be deduced from the apparent dissociation distances measured on the rectilinear dissociated dislocation segments.

• Extrinsic obstacles pin superpartial dislocations, leading to a wide range of dissociation distances of the superdislocations.

In-situ observation of the movement of a<111> dislocations in βo grains

The movement under stress of these a<111> dislocations was studied by performing in-situ RT straining experiments in a TEM (note that Figs. 7-10 were made from the video sequences available for each figure; it is worth noting that these sequences are at real speed). The main tendency of these dislocations is to move in pile-ups as illustrated by Fig. 7 (video here), where a small defect, labelled F, is used as a reference point. In Fig. 7a on by a fourth one (Fig. 8c). It can be seen that the spacing between the two superpartials of a given superdislocation is increased with the distance of the considered superdislocation from the pile-up tip (in Fig. 8a, dislocation d4 is more widely dissociated than dislocation d2, for example), as that was observed in the bulk (Fig. 6a). The spacing between the two superpartial dislocations of leading dislocation d1 is even difficult to observe due to the resolution limit of this micrograph. Between Fig. 8a and Fig. 8b, however, all three dislocations have moved on.

Examination of the video shows that these dissociated superdislocations move in a jerky manner, mainly by independent jumps of the superpartials. Figs. 8d-f (video here) follow the movement of another a<111> superdislocation in a different grain, which is not strictly speaking part of a pile-up even though several other dislocations are simultaneously active in the same area. In this case, imaging conditions did not allow the right resolution to be achieved for visible dislocation dissociations. Dislocations are elongated along their screw orientation and anchored at many pinning points (some of them are indicated by small triangles on two dislocations visible in the frame corresponding to Fig. 8e). A dislocation marked D is followed between frame 1 (D1 in Fig. 8d) and frame 3 (D3 in Fig. 8f, 1 minute later). The whole dislocation jumps between positions D1 and D2 between frames Fig. 8d and Fig. 8e (dashed circled small defect is a fixed spatial reference). In position D2, the dislocation is divided into two segments separated by a pinning point (marked PP). Subsequently, probably due to the higher strength of this pinning point, dislocation D splits into two segments that move independently by the same jerky motion (Fig. 8e), leading to the formation and expansion of a loop.

From these images, it is possible to estimate the distance between the pinning points anchoring these moving a<111> screw dislocations: the apparent distance DAP is about 30 nm (distance DAP plotted on frame Fig. 8e), which corresponds to a true distance of 35 nm considering the inclination of the screw dislocation line. This order of magnitude is compatible with the apparent distance between  precipitates (see Fig. 2b), thereby confirming that dislocation anchoring occurs on these  precipitates.

Investigation of the cracking process by in-situ straining experiment

In the o grain, where the sequence shown in Fig. 7 was taken, four pile-ups were in fact observed in almost parallel planes. Figs. 9a-d (video here) present the evolution of one of them labelled PU in Fig. 9a. The dislocations moved upwards in the observed area and Fig. 9a is a first micrograph of the area when PU already reached the grain boundary labelled GB located at the upper part of the micrograph. The rest of the dynamic sequence was recorded inside the o grain ('below' GB). Between Figs. 9b andc, the apparent width of PU decreases with the accumulation of new dislocations in the pile-up. This effect is due to the relative slip of the respective part of the grain located on both sides of pile-up PU. Suddenly, the grain cracked at the level of the pile-up, see Fig. 9d. Regarding Fig. 9e, it shows a o grain characterized by post-mortem TEM in a thin foil extracted from a RT tensile specimen. The pile-up labelled PU, evidenced in Fig. 9e, demonstrates that localization of deformation in the form of large pileups (in addition to the small pile-ups already mentioned in Figs. 5 and6) is also possible in the bulk and not limited to in-situ tests in thin specimens.

Discussion

The present TEM studies have shown that the o phase can be plastically deformed at RT by dislocations, consistently with the work of Morris et al. [START_REF] Morris | Deformation mechanisms and slip transfer in a Ti-44Al-2Mo alloy[END_REF][START_REF] Morris | Deformation mechanisms within the B2, L10 and DO19 structures of a Ti-44Al-2Mo alloy, Gamma Titanium Aluminides[END_REF]. It is recalled that these authors investigated a Ti-44Al-2Mo (at%) alloy and observed a<111> and a<001> dislocations presumably nucleated in o grains by the local stress generated on the grain boundaries by mechanical twins emitted in neighbouring  grains. In the present work, deformation is shown to occur by a<111> superlattice dislocations, which are often situated in pile-ups. Some evidences of deformation in o areas originated by twins in  grains have also been observed (Fig. 5). Not shown here, other families of dislocations have also been observed and are currently under investigation. Many of the features of these a<111> superlattice dislocations are reminiscent of observations previously obtained for deformation systems active in other intermetallic compounds (ordinary dislocations in the  phase of TiAl, superlattice dislocations in L12 (Ni3Al and Ni3Ga) and D019 structures (Ti3Al)):

• The dissociation scheme determined in section 5.1 is the one expected from the literature concerning intermetallic alloys with B2 structure [START_REF] Paidar | Three types of dislocation core structure in B2 alloys[END_REF], even though in NiAl B2-type intermetallic alloys the core of these dislocations has been found to be very compact by TEM [START_REF] Veyssière | Weak-beam study of (111) superlattice dislocations in NiAl[END_REF][START_REF] Sun | Characteristics of (111) slip in a NiAl [00l] single crystal[END_REF]. These superlattice dislocations are dissociated into two identical superpartials bordering an antiphase boundary ribbon as in Ni3Al [START_REF] Molénat | Dislocation mechanisms in Ni,Al at room temperature. In situ straining experiments in TEM[END_REF][START_REF] Bontemps | The geometrical configuration of kinks on screw superdislocations in Ni,Al deformed at room temperature[END_REF] and Ni3Ga [START_REF] Couret | Glide sequences of deformation in the (111) plane of Ni3Ga single crystals in the yield stress anomaly[END_REF], both with the L12 fcc-type ordered structure, and also in α2-Ti3Al [START_REF] Legros | Prismatic and basal slip in Ti3Al : I. Frictional forces on dislocations[END_REF].

Superpartials exhibit superkinks, each of them resulting of a dislocation elementary jump.

• The dislocations are elongated along their screw directions, with sometimes some rectilinear parts for the superpartials, as observed in Ni3Al and Ni3Ga, as well as during the prismatic glide in α2-Ti3Al. This is generally explained by a superpartial core spreading out of the glide plane, which leads to a lattice friction stress (Peierls stress), thus hindering dislocation glide.

• Dislocations glide is made of quick jumps (in less than the time between two successive video frames i.e. 1/50 s) between two elongated screw configurations. This jerky motion is also the result of the lattice friction stress [START_REF] Couret | Dissociations and friction forces in metals and alloys[END_REF]. Indeed, dislocations transit from a stable sessile configuration due to the core spreading, to a non-stable glissile configuration allowed by the thermally activated recombination of the core under the effect of the local stress, before again falling in the stable configuration.

• Dislocations are anchored at pinning points, which are o precipitates. This situation is analogous to that of ordinary dislocations in the  TiAl phase [START_REF] Viguier | Modeling the flow stress anomaly in gamma TiAl. 1. Experimental observations of dislocation mechanisms[END_REF][START_REF] Sriram | The geometry and nature of pinning points of 1/2 (110] unit dislocations in binary TiAl alloys[END_REF][START_REF] Couret | An in situ study of ordinary dislocations glide in TiAl alloys[END_REF], in which the pinning was attributed to small chemical clusters [START_REF] Zghal | Pinning points anchoring ordinary and Shockley dislocations in TiAl alloys[END_REF], but also to the case of precipitation hardened Al base alloys [START_REF] Delmas | Straining mechanisms in aluminium alloy 6056. In-situ investigation by transmission electron microscopy[END_REF], in which dislocations anchor on precipitates made of crystallographic phases different from that of the matrix.

• Loops formation leading to dislocation multiplication processes are often observed.

These mechanisms are frequently evidenced when extrinsically pinned dislocations are subjected to friction forces: their screw character induced by the Peierls stress allows intense cross-slip events, in a similar manner as ordinary dislocations in TiAl [START_REF] Viguier | Modeling the flow stress anomaly in gamma TiAl. 1. Experimental observations of dislocation mechanisms[END_REF][START_REF] Sriram | The geometry and nature of pinning points of 1/2 (110] unit dislocations in binary TiAl alloys[END_REF][START_REF] Couret | An in situ study of ordinary dislocations glide in TiAl alloys[END_REF].

Regarding the o phase present in the investigated TNM-TiAl alloy, a<111> superdislocations experience a friction force and must overcome the  precipitates in order to continue gliding, precisely in the {110} planes, the densest planes of the B2 structure. The tendency for these dislocations to align along their screw direction is less marked than in other intermetallic alloys, especially by considering whole dislocations (Figs. 8 d-f). Sometimes, this alignment seems to be more pronounced for partial dislocations (Figs. 8 a-c). On the other hand, the anchoring points are very close with a separation distance of the order of 35 nm smaller than that measured for ordinary dislocations in the TiAl -phase (150 nm [START_REF] Couret | An in situ study of ordinary dislocations glide in TiAl alloys[END_REF][START_REF] Zghal | Pinning points anchoring ordinary and Shockley dislocations in TiAl alloys[END_REF]). On the reverse, this distance is close to that separating the precipitates of Al alloys (30 nm) [START_REF] Delmas | Straining mechanisms in aluminium alloy 6056. In-situ investigation by transmission electron microscopy[END_REF], in which the structural hardening due to these precipitates is known to control deformation. This suggests that  precipitates have a dominant role here on the dislocations mobility.

Furthermore, the trend of having dislocations moving in pile-ups with a tendency for decorrelation of the superpartials, provides a strong evidence ev that superdislocations are far from the pile-up tip, as been observed during in-situ straining experiments. It is tempting to conclude that this pile-up effect results from the structure of the  precipitates which are crossed by dislocations. An analogous situation is found in the short-range order effect detected in the -phase of Ni base superalloys [START_REF] Pettinari | Local order in industrial and model γ phases of superalloys[END_REF], where the formation of pile-ups results from the greater facility of dislocations to move in the wake of the head dislocations once the ordered structure has been sheared. To check the key role of  precipitates on the formation of dislocations pile-ups , a specific TEM in-situ RT experiment was carried out in  grains of a Ti-44Al-3Mo alloy (at. %) characterized by the absence of  precipitates due to the absence of Nb. As shown in Fig. 10 (video here), dislocations observed in the starting situation (Fig. 10a) have been progressively emitted in the  grain from a grain boundary situated out of the frame on the right side of the picture. Dislocations are spread all over the area. The extremity of a dislocation marked by the letter F which will not move anymore is used as a reference point in the next three images. An indexation of these dislocations allowed us to determine that their Burgers vector is parallel to the <111> crystallographic direction and to deduce that these dislocations have a marked screw character. A TEM analysis under weak-beam conditions, which is not reproduced here, showed that these dislocations are dissociated into two partials separated by an APB ribbon of the order of 7 nm on screw segments. Regarding the dynamics of these dislocations, Figs. 10b-d In summary, the movement of a<111> dislocations and the formation of corresponding pileups appear to result from the presence of  nanoprecipitates that seem to play a key role in the dynamics of the dislocations involved. It should be noted that further information is needed to fully understand the glide mechanism of these dislocations. Work is currently in progress to obtain quantitative results on that particular topic. This experimental study did not reveal any particularly brittle behaviour of the  phase since we did observe deformation dislocations and no brittle rupture surface. On the other hand, we evidenced a strong tendency to form pile-ups of large numbers of dislocations. Even though pile-ups are enhanced by in-situ experiments, however, it is worthwhile pointing out that a pileup of the same type has been found in a post-mortem sample and pile-ups made of few dislocations, probably embryos of large pile-ups, were also evidenced. As failure is linked to a localised event that does not necessarily need to occur many times in the material, it can be assumed that the occurrence of only a few pile-ups in the bulk material (or even just one if it is well oriented with respect to the neighbouring lamellar colony) is sufficient to induce failure of the TiAl alloy. These pile-ups contain many dislocations, involving high stress concentrations that can lead to the cleavage along a lamellar interface if the lamellar colony is well oriented.

Consistently with the limited slip activity of the 2 phase [START_REF] Morris | Deformation mechanisms and slip transfer in a Ti-44Al-2Mo alloy[END_REF][START_REF] Morris | Deformation mechanisms within the B2, L10 and DO19 structures of a Ti-44Al-2Mo alloy, Gamma Titanium Aluminides[END_REF] and with the observation of intensive cleavage in lamellar interfaces, the low ductility of the present alloy is thus attributed to a stress mediated cleavage process along lamellar interfaces. As Tables 1 and2 show, beyond a few microns (around 20 m) the size of the lamellar colonies has no additional detrimental effect on fracture (A= 0.5 %) because the surface of the cleaved interface certainly exceeds the critical size for an irreversible failure. Similarly, the proportion of  phase has little influence since the fracture results from a localised process in the material and not from the average deformation of the  phase. However, only a certain volume fraction of  phase is required to make the stress concentration possible in some sites.

Conclusion

The deformation microstructure and the fracture surface have been studied in a TNM-TiAl alloy (Ti51.05Al43.9Nb4Mo0.95B0.1) densified by Spark Plasma Sintering. RT tensile tests were coupled with TEM in-situ RT straining experiments. A cleavage mechanism was observed in the lamellar interfaces of the γ/α2 colonies but not in the o grains. The latter were found to be plastically deformable by a<111> superlattice dislocations, which are dissociated into two superpartials separated by an antiphase boundary. These dislocations mainly glide in pile-ups lying in {110} planes and involving many dislocations. In addition, the o phase contains o nanoprecipitates, which anchor the dislocations, thus limiting their propagation and enhancing pile-ups configurations.

In summary, this study shows that fracture of -containing TiAl alloys occurs by cleavage along the interface plane of the lamellar γ/α2 colonies, in a similar manner as in lamellar TiAl alloys with no  phase. However, this cleavage mechanism is premature since  precipitates that are present in the  phase, as a result of Nb addition in the TNM alloy, promote stress concentrations by the formation of dislocation pile-ups in this phase. Therefore, such dislocation pile-ups are believed to trigger cleavage in the neigbouring lamellar colonies.

Fig. 1 .

 1 Fig. 1. Microstructure of TNM-TiAl: (a) SEM micrograph of specimen AK746 taken in BSE

Fig. 2 .

 2 Fig. 2. Characterization of the o precipitates in o grains. a) Diffraction pattern with

Fig. 3 .

 3 Fig. 3. True RT stress-strain curves for the four investigated SPS treatments. (a) one curve for

Fig. 4 .

 4 Fig. 4. SEM characterization of the fracture surface of RT deformed samples until rupture (AK

[

  111] dislocations. This is also the case for the pair of dislocations marked B. The trace of the main pile-up marked C is not straight but curved (see dashed line) indicating changes in the glide plane. The analysis of the surrounding  grains showed that these pile-ups can be associated with the locking of a twin on the grain boundary, as in case D, or not, as in the case of pile-up C. Hence, Fig.5dconfirms the strong influence of the deformation occurring in the  phase on that proceeding in the o phase. In this area, three parallel noodle-shaped o grains are situated within one  grain deformed by twins. Clearly, some dislocations are seen in the noodle-shaped constituents in planes following those of the twin habit planes, thus indicating that the emission of dislocations in the o phase results from the transmission of deformation from surrounding  phase.

Fig. 5 .

 5 Fig. 5. Characteristic occurrence of dislocations observed by TEM (weak-beam conditions) in

Fig. 6 .

 6 Fig. 6. Two examples of the structure of a<111> superdislocations observed by post-mortem

Fig. 7 .

 7 figure, the pile-up tip is visible in the observed area and a dislocation loop (arrowed) was formed

Figure 8

 8 Figure8details the elementary mechanisms of propagation of these dislocations. Figures8a-c

(

  Figs. 8d-f show the situation in another sample. Figs. 8a-c show a pile-up tip, where dislocations

Fig. 8 .

 8 Fig. 8. Propagation of a<111> superdislocations in βo grains during in-situ RT straining

Fig. 9 .

 9 Fig. 9. Pile-up growing and crack formation in a βo grain; (a-d) dynamics of a growing pile-

  allow to follow the path of a dislocation indicated by an arrow: this dislocation occurs in the observed area (Fig.10b) and moves individually to position indicated in Fig.10.c, until reaching the position indicated in Fig.10.d. The trace left on the surface corresponds to a (101) gliding plane. Dislocations activated in the o phase of this material with no  precipitates are therefore similar to the dislocations investigated in the o phase of the TNM alloys with  precipitates, but keeping in mind that they move individually without forming pile-ups. It is worth noting that even if it would be interesting to carry out mechanical tests to determine the mechanical impact of the absence of  precipitates, we did not do so: a fair comparison with the TNM alloys would indeed require the manufacturing of a material with a similar microstructure. This is really a long and difficult task including atomization of the right pre-alloyed powder, SPS densification etc, that is out of the scope of this work.

Fig. 10 .

 10 Fig. 10. Propagation of a<111> superdislocations in a βo grain with no  precipitates during

Table 1

 1 Microstructural characterization of the obtained as a function of the SPS dwell temperature.

	Dwell temperature (°C)	1300	1325	1360	1360 -60 min -
	Specimen designation	AK639	AN849	AL824	AM504
	Other SPS billets used in the study		AK746
			AM797
	Average colony size (µm)	17.27	17.38	17.11	28.04
	Standard deviation (µm)	11.27	9.07	8.95	13.15
	Vol.% of the βo phase	5.6	8.2	8.4	9

Table 2

 2 RT tensile tests results (10 -4 s -1 strain rate).

	Specimen name Dwell temperature		YS (MPa) R0.2 (MPa) UTS (MPa) A (%)
	AK 639	1300 °C	Spec 1	559	771	923	0.73
			Spec 2	554	759	910	0.75
	AK 746	1325 °C	Spec 1	591	814	995	0.92
			Spec 2	583	792	961	0.81
	AM797		Spec 1	559	755	914	0.93
			Spec 2	545	748	905	1
	AL 824	1360 °C	Spec 1	538	770	901	0.54
	AM 504	1360 °C -60 min -	Spec 1	554	779	914	0.52
			Spec 2	528	763	880	0.44

A SEM study of the fracture surfaces is presented in Figure

4

for two RT deformed samples (specimens AK 639 -Figs. 4a-b and AM504 -Figs.

Acknowledgements

This study has been conducted in the framework of the cooperative Austrian-French project "Hi-TiAl -18-CE91-0008-01" co-supported by the French Agence Nationale de la Recherche (ANR) and the Fonds zur Förderung der wissenschaftlichen Forschung (FWF). This project has received funding from the European Union's Horizon 2020 research and innovation programme under grant agreement No 823717 -ESTEEM3.


domain: sdv sde

text:
J P Lhomme

C Montes

F Jacob

L Prévot

Evaporation from heterogeneous and sparse canopies: on the formulations issued from some multi-source representations

Keywords: evaporation, heterogeneous and sparse canopy, multi-source modelling

Evaporation from heterogeneous and sparse canopies is often represented by multisource models which take the form of electrical analogues based upon resistance networks.

. Agric For Meteorol 84: 187-205] as a basis, it is shown that the stomatal characteristics of the foliage (amphistomatous or hypostomatous) generate different formulations. New generic and more concise equations, valid in both configurations, are derived. The differences between the patch and layer approaches are outlined and the consequences they have on the composition and formulation of component fluxes are specified. Then, the issue of calculating the effective resistances of the single-layer model from multi-source representations is addressed. Finally, a sensitivity analysis is carried out to illustrate the significance of the new formulations.

Introduction

The energy transfers within heterogeneous and sparse canopies are often represented by multi-source models. The first models of this kind were designed for sparse canopies to separately account for vegetation and substrate (soil) contributions [START_REF] Shuttleworth | Evaporation from sparse crops-an energy combination theory[END_REF]; [START_REF] Choudhury | A four-layer model for the heat budget of homogeneous land surfaces[END_REF][START_REF] Shuttleworth | The theoretical relationship between foliage temperature and canopy resistance in sparse crops[END_REF]. Subsequently, more complex representations were proposed to account for heterogeneous canopies with various components (main vegetation, herbaceous substrate, bare soil) [START_REF] Dolman | A multiple-source land surface energy balance model for use in general circulation models[END_REF][START_REF] Huntingford | An intercomparison of single and dual-source vegetation-atmosphere transfer models applied to transpiration from Sahelian savannah[END_REF], but always keeping the Shuttleworth-Wallace model as a basis. In the "clumped" model of [START_REF] Brenner | The effect of clumping and stomatal response on evaporation from sparsely vegetated shrub-lands[END_REF], specially adapted to semi-arid shrub-lands, the substrate layer is divided into two sub-layers: one corresponds to the soil under the foliage and the other to the bare soil. The ERIN model of [START_REF] Wallace | Evaporation and radiation interception by neighbouring plants[END_REF] addresses the issue of evaporation in multi-species canopies through an extension of the Shuttleworth-Wallace model. From principles similar to those of [START_REF] Brenner | The effect of clumping and stomatal response on evaporation from sparsely vegetated shrub-lands[END_REF], [START_REF] Verhoef | A SVAT scheme describing energy and CO2 fluxes for multicomponent vegetation: calibration and test for a Sahelian savannah[END_REF] developed a multi-source model, where the substrate is divided into three sub-layers: grasses, herbs and bare soil.

The multi-source models are generally represented by resistance networks, which combine surface and air resistances within the canopy. They simulate fairly well evaporation provided appropriate parameterizations for component resistances are available. However, it appears that the writing of the total flux of evaporation from the component fluxes and the combination of elementary resistances differs depending on the authors and can lead to apparent inconsistencies and even inexact formulations. For instance, whereas in the first models (e.g. [START_REF] Shuttleworth | Evaporation from sparse crops-an energy combination theory[END_REF] the component fluxes of the multi-source representation are simply added, in the subsequent ones (e.g. [START_REF] Dolman | A multiple-source land surface energy balance model for use in general circulation models[END_REF]) they are weighted by the relative area of each source without clear justification. In line with the pioneering work of [START_REF] Wallace | Interactions in mixed-plant communities: light, water and carbon dioxide[END_REF] on the modelling of interactions in mixed-plant communities, the main goal of the current study is to carefully examine the formulation of these multisource models in order to clearly establish their common ground, the differences between them and the correct way to formulate them. This examination is made through the most emblematic ones [START_REF] Shuttleworth | Evaporation from sparse crops-an energy combination theory[END_REF][START_REF] Brenner | The effect of clumping and stomatal response on evaporation from sparsely vegetated shrub-lands[END_REF] and three particular issues will be addressed in relation to the main objective.

The first issue is linked to the stomatal characteristics of the canopy (amphistomatous versus hypostomatous leaves) and the impact they have on the basic equations of the multilayer representation. Although the Verhoef-Allen model accounts for differences in stomatal characteristics, the introduced modifications are not systematically applied in many other studies based on multi-source models. In the case of vineyard for example, several papers apply the Shuttleworth-Wallace model (Ortega-Farias et al. 2007;[START_REF] Zhang | An evaporation model for sparsely vegetated canopies under partial root-zone irrigation[END_REF][START_REF] Ortega-Farias | Parameterization of a two-layer model for estimating vineyard evapotranspiration using meteorological measurements[END_REF] or the clumped model [START_REF] Zhang | Comparison of three evapotranspiration models to Bowen ratio-energy balance method for a vineyard in an arid desert region of northwest China[END_REF][START_REF] Poblete-Echeverria | Estimation of actual evapotranspiration for a drip-irrigated Merlot vineyard using a three-source model[END_REF] without correction for the hypostomatous characteristics. We propose here new generic formulations which are valid in both configurations (amphi-and hypo-stomatous).

The second issue deals with the distinction between the patch (or uncoupled) approach and the layer (or coupled) approach. As already explained by [START_REF] Lhomme | Comments on dual-source vegetation-atmosphere transfer models[END_REF] and [START_REF] Daamen | Modeling energy fluxes from sparse canopies and understorys[END_REF], the choice of a coupled or uncoupled model has significant consequences in the formulation of the basic equations. When a coupled (also called interactive or layer) model is used, component fluxes are strictly additive, whereas in an uncoupled (patch) model, component fluxes should be weighted by the relative area of each patch. This point will be addressed and discussed in relation to the clumped model of [START_REF] Brenner | The effect of clumping and stomatal response on evaporation from sparsely vegetated shrub-lands[END_REF].

The third issue addressed here occurs when the relative complexity of a multi-source model should be represented by a single-layer model, i.e., the "big leaf" model of Penman-Monteith. How should one combine the component resistances of the multi-source approach to calculate the effective resistances of the single-source? Since the aggregation procedures proposed in the literature are diverse and raise questions [START_REF] Were | Analysis of effective resistance calculation methods and their effect on modelling evapotranspiration in two different patches of vegetation in semi-arid SE Spain[END_REF], the problem will be discussed and solutions will be proposed.

The paper is divided into three main sections which examine successively the twolayer representation [START_REF] Shuttleworth | Evaporation from sparse crops-an energy combination theory[END_REF], the "clumped" or three-source model [START_REF] Brenner | The effect of clumping and stomatal response on evaporation from sparsely vegetated shrub-lands[END_REF] and the calculation of single-source effective resistances.

Numerical simulations and sensitivity analyses are presented in the last section to illustrate the interest of the new formulations.

A new and generic formulation of the two-layer model

The two-layer model of [START_REF] Shuttleworth | Evaporation from sparse crops-an energy combination theory[END_REF] represents the evaporation from a stand of vegetation composed of two main sources: a substrate (which can be bare soil or grass) and an upper canopy (main foliage). The corresponding resistance network is shown in Figure 1. The resulting formulation for total evaporation slightly differs depending on the distribution of stomata on the leaves. The amphistomatous case (stomata distributed on both sides of the leaves) is the case implicitly considered in the original formulation [START_REF] Shuttleworth | Evaporation from sparse crops-an energy combination theory[END_REF][START_REF] Shuttleworth | The theoretical relationship between foliage temperature and canopy resistance in sparse crops[END_REF]. More general equations are established here: they take into account the two possible patterns of stomatal characteristics (amphistomatous and hypostomatous).

In the two-layer model, total evaporation (λE) from the canopy is expressed as the simple sum of two components: λEf (foliage) and λEs (substrate): (2)

s f E E E    + = . ( 1
The foliage boundary-layer resistance for water vapour is written similarly

) 2 /( , , , LAI r n r l a v f a = .
(3) However, since each leaf side is a heat source, the foliage boundary-layer resistance for sensible heat remains the same in both cases (amphistomatous or hypostomatous) and is defined as

n r LAI r r v f a l a h f a / ) 2 /( , , , , , = = . ( 4
)
The hyperstomatous case (stomata only on the upper side of the leaves) is similar to the hypostomatous case and the corresponding expressions of bulk resistances are identical.

The combination equation (Penman-Monteith type) for the foliage takes the following form derived in Appendix A:

) / ( / , , , , , h f a f s h f a m p f f r r n r D c A E + +  +  =    . ( 5
)
For the substrate, resistances are always expressed per unit area of land surface and the boundary-layer resistance is assumed to be the same for sensible heat and water vapour.

Consequently, the corresponding combination equation is simply written as

) / 1 ( / , , , s a s s s a m p s s r r r D c A E + +  +  =    , ( 6
)
where rs,s is the substrate resistance to evaporation and ra,s the aerodynamic resistance between the substrate and the source height. Expressing Dm as a function of Da (vapour pressure deficit at the reference height zr) leads to [START_REF] Shuttleworth | Evaporation from sparse crops-an energy combination theory[END_REF], Eq. ( 8))

  ) /( ) ( p a a m c r E A D D    +  -  + = . ( 7
)
The general expression of total evaporation λE is obtained by introducing Eq. ( 7) into Eqs. ( 5) and (6). At this stage, two formulations of λE are possible: the one which follows the strict formalism of Shuttleworth and Wallace's original equations and an alternative one, considered as more synthetic, which is proposed hereafter.

a. Original formulation

If we respect the original formalism, the resultant equation is written as

s s f f PM C PM C E + =  , (8)
with

    ) /( 1 ) /( ) ( , , , , s a a s s s a a s s a a p s r r r r r A A r D c A PM + + +  + -  - +  =   , (9)     ) /( ) ( ) /( ) ( , , , , , , , , , h f a a f s h f a a h f a a f h f a a p f r r r r n r r r A A r D c A PM + + + +  + -  - +  =   . ( 10
)
In these equations A represents the available energy for the whole stand (foliage and substrate): A = Af + As. The steps of the calculation are identical to those given in the original article of [START_REF] Shuttleworth | Evaporation from sparse crops-an energy combination theory[END_REF]. The coefficients Cf and Cs are simple combinations of the basic air and surface resistances. They are expressed as ( )

a f a s s f a f s f R R R R R R R R R C + + + = , (11)
( )

a f a s s f a s f s R R R R R R R R R C + + + = , (12) with (
)

f s h f a f r r n R , , ,   + +  = , ( 13
) ( ) s s s a s r r R , ,   + +  = , (14)
( )

a a r R  +  = . ( 15
)
The two-layer representation of sparse canopies does not have the same mathematical form for amphistomatous and hypostomatous canopies. Some adjustments should be made when passing from one type of canopy to another: the PMf and Rf terms undergo a change and it is easy to verify that when n = 1 (amphistomatous case), the original equations of [START_REF] Shuttleworth | Evaporation from sparse crops-an energy combination theory[END_REF] are retrieved. Additionally, we have to note that the partition of the original formulation into two "Penman-Monteith" type components (foliage and substrate) is not really well-designed insofar as each part has a relatively complex form and does not represent the respective component evaporation (foliage and substrate). This can be even confusing and misleading. An alternative and simpler formulation is proposed hereafter.

b. Alternative formulation

By differently collecting the terms in the basic equations (the details of the calculation are given in Appendix B) it can be shown that the total flux of evaporation can be written as

a s a s s h f a f f p s f r r A P r A P E P P E / ) ( ) ( , , , +  + + +  =      . ( 16
)
In Eq. ( 16) λEp represents the potential evaporation from the sparse canopy expressed as

   +  +  = a a p p r D c A E / (17)
The other terms are defined as follows:

' ' ' ' ' ' ' s a f a s f f a s R R R R R R R r P + + = (18) ' ' ' ' ' ' ' s a f a s f s a f R R R R R R R r P + + = (19) h f a f s f f r n r R R , , , ' ) ( /    + + = = (20) s a s s s s r r R R , , ' ) 1 ( /    + + = = (21) ) 1 ( / '    + = = a a a r R R (22)
This new formulation of the two-layer model has three main advantages: (i) first it is more concise than the traditional one; (ii) it avoids the confusion generated by separate evaporation terms; (iii) it involves the "climatic demand" λEp, which can be convenient and useful in many applied studies [START_REF] Lhomme | Towards a rational definition of potential evaporation[END_REF]. We also note that Eq. ( 16) has correct asymptotic limits. If there is no substrate evaporation (As is zero and rs,s is infinite), Eq. ( 16) reduces to the conventional Penman-Monteith equation for a closed canopy with n = 1:

  ) /( 1 ) /( , , , , , h f a a f s h f a a a p r r r r r D c A E + + +  + +  =    (23)

A reformulated clumped model

Layer or patch approach?

As already mentioned, the choice between the two representations is critical in the sense that it leads to different manner of aggregating the elementary fluxes (simple sum or area weighted addition). The choice should be dictated by the way the aerodynamic resistance above the canopy (between zm and zr) is defined. If the patches of vegetation or bare soil are large enough to allow different aerodynamic resistances to be defined for each patch, an uncoupled representation should be chosen (see Fig. 2). On the other hand, if the different sources are close to each other and do not allow the definition of separate aerodynamic resistances, a layer approach should be preferred (see Fig. 1). [START_REF] Daamen | Modeling energy fluxes from sparse canopies and understorys[END_REF] explained: "the patch model is fully justified at the scale where a boundary layer is fully developed over each patch and edge effects between patches are insignificant, but as the size of the patches decreases this model may be less valid". From a turbulent transfer perspective, McNaughton and van den Hurk (1995) also showed that the coupled (interactive) model is a simplification of more complex and realistic Lagrangian models and consequently more widely applicable than the patch model.

The clumped model of [START_REF] Brenner | The effect of clumping and stomatal response on evaporation from sparsely vegetated shrub-lands[END_REF], which is similar to the multi-species canopy representation described by [START_REF] Wallace | Evaporation and radiation interception by neighbouring plants[END_REF], constitutes in fact a modified two-layer model analogous to the Shuttleworth-Wallace model: the layer representing the soil surface is divided into two sub-layers (soil under the foliage and bare soil) and the component fluxes mix together at canopy source height before experiencing the same aerodynamic resistance above the canopy (Fig. 3). It was used and reworked by [START_REF] Domingo | Evapotranspiration model for semi-arid shrub-lands tested against data from SE Spain[END_REF] and [START_REF] Were | Analysis of effective resistance calculation methods and their effect on modelling evapotranspiration in two different patches of vegetation in semi-arid SE Spain[END_REF][START_REF] Were | Aggregating spatial heterogeneity in a bush vegetation patch in semi-arid SE Spain: a multi-layer model versus a single-layer model[END_REF] for sparse bush vegetation in semi-arid Spain and by [START_REF] Zhang | Comparison of three evapotranspiration models to Bowen ratio-energy balance method for a vineyard in an arid desert region of northwest China[END_REF][START_REF] Zhang | An evaporation model for sparsely vegetated canopies under partial root-zone irrigation[END_REF] for vineyard in an arid region of China. Following the logic of the layer model, the total flux of evaporation emanating from the canopy should be written in a simple additive form

bs vs f E E E E     + + = (24)
with subscript i = f for the foliage, i = vs for the soil below the vegetation and i = bs for the bare soil. This form of the conservation equation differs from the one used in the original model of [START_REF] Brenner | The effect of clumping and stomatal response on evaporation from sparsely vegetated shrub-lands[END_REF] where the total flux of evaporation is written as an areaweighted form of the component fluxes

bs vs f E F E E F E     ) 1 ( ) ( - + + = , ( 25
)
F representing the fractional cover of the foliage. Although a formulation similar to Eq. ( 25)

has been used by [START_REF] Dolman | A multiple-source land surface energy balance model for use in general circulation models[END_REF] and [START_REF] Verhoef | A SVAT scheme describing energy and CO2 fluxes for multicomponent vegetation: calibration and test for a Sahelian savannah[END_REF], we have to stress that this form of the equation is not concordant with the resistance network representing the model (Fig. 3). Given that all the component fluxes mix together at canopy source height and that a sole aerodynamic resistance is defined above the canopy, a layer approach should be preferred. We develop below the new equations of the clumped model when (i) the simple additive form of the conservation equation is used instead of the area weighted form and (ii)

when the stomatal characteristics of the foliage are taken into account.

New formulations

The component resistances should be expressed in a way similar to the two-layer

) 1 /( 1 , , F r r bs s bs s - = ( (27)
) ) 1 /( 1 , , F r r bs a bs a - = . ( 28
) 29
In fact, given the basic formalism (Ohm's law type) of component fluxes, dividing the resistances by the fractional areas is equivalent to multiplying the corresponding fluxes by the same quantity. In the original clumped model of [START_REF] Brenner | The effect of clumping and stomatal response on evaporation from sparsely vegetated shrub-lands[END_REF], where a mix of patch and layer approach is used, the surface and aerodynamic resistances of the substrates are logically expressed per unit area of land surface and consequently they are not divided by their relative area (F or 1-F). This is consistent with the patch formulation of Eq. ( 25), but as explained above, it is the approach itself which is questionable.

Each evaporation term can be expressed in the form of a Penman-Monteith equation (Eqs. ( 5) and ( 6)), n having the same significance as in the Shuttleworth-Wallace model (section 2). Replacing Dm by its expression as a function of Da (Eq. ( 7)) leads to the following

formulation bs bs vs vs f f PM C PM C PM C E ' ' ' + + =  (30) with     bs or vs i r r r r r A A r D c A PM i a a i s i a a i i a a p i = + + +  + -  - +  = ) /( 1 ) /( ) ( , , , ,   (31)     ) /( ) ( ) /( ) ( , , , , , , , , , h f a a f s h f a a h f a a f h f a a p f r r r nr r r r A A r D c A PM + + + +  + -  - +  =   . ( 32
)
The three coefficients C' are simple combinations of the component air and surface resistances and are detailed in Appendix C. We note that the modifications needed to take into account the stomatal characteristics of the canopy (through the parameter n) are exactly similar to the ones made to the Shuttleworth-Wallace model (Eq. ( 32) versus ( 10) and ( 57) versus ( 13)).

As for the Shuttleworth-Wallace model, it is possible to obtain an alternative formalism and to write more concisely the resultant expression of λE by collecting differently the terms of the equation (derivation similar to that described in Appendix B). With the coefficients P' defined in Appendix D, the equation equivalent to Eq. ( 16) in the two layer model is written in the clumped model as

+ +  + + + +  =      , ( 33
)
where λEp represents the potential evaporation defined by Eq. ( 17). It can be verified that the asymptotic limits of this formulation are correct. When the vegetated part of the substrate tends to zero, rs,vs tends to infinite, ' vs P and Avs tends to zero and Eq. ( 33) transforms into Eq. ( 16).

Big leaf model: aggregation of component resistances

The problem to solve

In the modelling of surface fluxes at different scales, the canopy spatial heterogeneities are often averaged by using simpler representations. One way to proceed is by using a bulk-transfer approach based on a single-layer representation and the concept of kB -1 introduced by [START_REF] Owen | Heat transfer across rough surfaces[END_REF] and largely discussed since then [START_REF] Garratt | Momentum, heat and water vapour transfer to and from natural and artificial surfaces[END_REF]. However, this concept, which is defined as the logarithm of the ratio between momentum and heat roughness length, is questionable and considered as not perfectly sound from a physical standpoint [START_REF] Verhoef | Some practical notes on the parameter kB -1 for sparse canopies[END_REF][START_REF] Lhomme | Sensible heat fluxradiometric surface temperature relationship over sparse vegetation: parameterizing B -1[END_REF]. Another way to proceed is by calculating effective parameters. Presently, the problem to solve is how to correctly represent a heterogeneous canopy by a simple combination equation (Penman-Monteith model) in which the effective surface and air resistances are expressed from the component resistances of the multi-source model. The basic model [START_REF] Monteith | Evaporation and the environment[END_REF] is commonly written as

) / 1 ( / a e s a a p r r r D c A E + +  +  =    , ( 34
)
where e s r is the effective surface resistance and ra is the aerodynamic resistance calculated between the mean source height of the canopy zm and the reference height zr (Fig. 4a). The original Penman-Monteith equation is strictly valid for a full covering canopy (big leaf model) and the surface resistance of the foliage is reduced to its stomatal component (expressed from leaf stomatal resistance as n rs,l /2LAI). The canopy boundary-layer resistance for sensible heat and water vapour is neglected or assumed to be incorporated in the aerodynamic resistance above the canopy (ra).

When a two-layer or clumped approach is considered, it is clear that the effective surface resistance ( e s r ) should be put in series with the aerodynamic resistance (ra) above the canopy. It is less clear, however, how the component surface and air resistances within the canopy should be combined into the effective surface resistance. [START_REF] Were | Analysis of effective resistance calculation methods and their effect on modelling evapotranspiration in two different patches of vegetation in semi-arid SE Spain[END_REF][START_REF] Were | Aggregating spatial heterogeneity in a bush vegetation patch in semi-arid SE Spain: a multi-layer model versus a single-layer model[END_REF], for instance, have tested with observed data of evaporation several aggregation procedures previously discussed by [START_REF] Blyth | Effective resistance to sensible and latent heat flux in heterogeneous terrain[END_REF]: in parallel or in series, weighted by the relative area F or not. The specific objective of this section is to examine this issue from a theoretical standpoint and to identify the correct way of physically combining the component resistances of multi-source models into the effective resistances of a single-layer model. The two-layer case will be thoroughly examined. Then, we will show how the equations can be extended to the clumped model.

General expressions for the two-layer model

The Penman-Monteith model results from the combination of two basic Ohm's law type formulations, one for sensible heat and the other for latent heat (Fig. 4b). Written with effective resistances they read

) /( ) ( , e h a a a m p r r T T c H + - =  , ( 35
) ) /( ) ) ( * )( / ( e v a a m p r r e T e c E + - =    . ( 36
)
Tm is air temperature at canopy source height, e h a r , is the effective resistance for sensible heat (which includes only air resistances within the canopy) and e v r is the one for water vapour (which includes air and surface resistances). Both resistances should be logically added to the aerodynamic resistance above the canopy (ra). Combining Eqs. ( 35) and ( 36

+ = + = , (38)
( 2 1 1 1 + + + = + + + = . ( , , , , , 1 )
) 39
This set of equations constitutes the unique rationale to aggregate the elementary resistances within the canopy in order to calculate the effective resistances of a big-leaf model from a two-layer representation. Two remarks should be made, however. First, the effective resistance for water vapour e v r is a complex arrangement of air and surface resistances and does not allow air and surface components to be separated into two bulk resistances in series.

Second, the resultant combination equation does not have the common formalism of Eq. ( 34),

where the simple ratio of a surface resistance to an aerodynamic resistance appears in the denominator of the equation, allowing the effects of the air to be separated from those of the surface. Nevertheless, this simple ratio can be obtained if the air resistances within the canopy are neglected, as is the case in the original Penman-Monteith equation. This approximation can be justified since it is well known that evaporation depends much more on stomatal resistance and LAI than on internal air resistances. With this approximation the effective resistance for sensible heat e h a r , is equal to 0. This means that Eq. ( 37) transforms into Eq. ( 34) and the effective resistance for water vapour e v r becomes e s r defined as

s s l s s s f s e s r r n LAI r r r , , , , 1 2 1 1 1 + = + = , ( 40
)

Decoupling surface and air resistances (2-layer model)

In order to obtain the familiar formalism of the big leaf model it is convenient to split the effective resistance for water vapour transfer e v r into two resistances put in series: one including only surface components and the other only air components:

e v a e s e v r r r , + = . ( 41
)
This assumption is physically valid to a first approximation if we consider that it roughly respects the path followed by water vapour in its transfer from the plant to the open air. In this case, the effective surface resistance e s r is logically defined as the sum of parallel resistances and obtained from Eq. ( 40). On the other hand, the effective aerodynamic resistance e v a r , can take two forms depending on the stomatal characteristics of the foliage.

In amphistomatous canopies, the same expression for the effective aerodynamic resistance (calculated as the sum of parallel resistances) can be applied simultaneously to sensible heat and water vapour: given by Eq. ( 38). Putting Eq. ( 41) into Eq. ( 37

E + + +  + +  =    . ( 42
)
Although this formulation is not perfectly sound from a physical standpoint, it is certainly more realistic than the one obtained by neglecting the air resistances within the canopy.

The hypostomatous case is more complicated since the effective aerodynamic resistance for water vapour e v a r , differs from the one for sensible heat e h a r , . For the two-layer model these effective aerodynamic resistances are respectively calculated as

s a l a s a h f a e h a r r LAI r r r , , , , , , 1 2 1 1 1 + = + = , ( 43
) s a l a s a v f a e v a r r LAI r r r , , , , , , 1 1 1 1 + = + = . ( 44
)
Following steps similar to those developed in Appendix A, the combination equation in the hypostomatous case writes as

  ) /( ) ( ) /( , , , e h a a e s e v a a e h a a a p r r r r r r r D c A E + + + +  + +  =    . ( 45
)
It is not possible to obtain the simple ratio of a surface resistance divided by an aerodynamic resistance. The decoupling of surface and air resistances does not allow obtaining a "strict" Penman-Monteith formalism.

Extension to the clumped model

The equations developed for the two layer model can be easily extended to the clumped model by splitting the substrate component into its two sub-components of area F (vegetated soil) and 1-F (bare soil) (see section 3.1). The resistances expressed per unit area of substrate being denoted by the upper-script 1, Eqs. ( 38) and ( 39) should be rewritten with

1 , 1 , , 1 1 bs a vs a s a r F r F r - + = , ( 46
) 1 , 1 , 1 , 1 , , , 1 1 bs a bs s vs a vs s s a s s r r F r r F r r + - + + = + . ( 47
)
When the air resistances within the canopy are disregarded, 1/rs,s in Eq. ( 40) should be replaced by Eq. ( 47) in which air resistances ( 1,vs a r and 1 ,bs a r ) are put to zero. When surface and air resistances are decoupled, Eqs. ( 43) and ( 44) should be similarly modified by expressing 1/ra,s as in Eq. ( 46). The formulations developed in section 4 are based upon a strict physical background with different levels of approximation and should be preferred to the empirical combinations tested by [START_REF] Were | Analysis of effective resistance calculation methods and their effect on modelling evapotranspiration in two different patches of vegetation in semi-arid SE Spain[END_REF][START_REF] Were | Aggregating spatial heterogeneity in a bush vegetation patch in semi-arid SE Spain: a multi-layer model versus a single-layer model[END_REF].

Sensitivity analysis

Numerical simulations were undertaken to assess the sensitivity of evaporation to some incorrectness or approximations in the formulation of multi-source representation. For the sake of convenience the analysis was carried out on the basis of the two-layer model. The parameterizations used to formulate available energy and component air resistances are given in Appendix E. Calculations are made for a canopy with a height (zh) equal to 1 m under the following meteorological conditions at a reference height of 2 m above the canopy: Rn = 400 W m -2 , Ta = 25 °C, Da = 10 hPa, ua = 2 m s -1 . These values were already used by [START_REF] Shuttleworth | Evaporation from sparse crops-an energy combination theory[END_REF] in the sensitivity analysis of their model.

Amphistomatous versus hypostomatous leaves

In section 2 it was shown that the formulation of the two-layer model differs depending on whether the canopy is amphistomatous or hypostomatous. The practical problem arises when the common formulation (strictly valid for an amphistomatous foliage) is applied to a hypostomatous one. Our simulation consisted in comparing for a hypostomatous canopy the "true" evaporation rate λE0, calculated using the right formulae (Eqs. ( 8) or ( 16) with n = 2), with the "erroneous" evaporation rate λE', calculated with the equations valid for an amphistomatous one (putting n = 1), except for the bulk stomatal resistance (Eq. ( 2)), which logically should be calculated as hypostomatous (n = 2). The error is generated in fact by an under-estimation of foliage boundary-layer resistance for water vapour ra,f,v (Eq. ( 3)): its hypostomatous value being twice its amphistomatous value (for instance, for LAI = 3, 166 s m -1 against 83 s m -1 ). Fig. 5 illustrates the corresponding error made on the evaporation rate as a function of LAI for different values of leaf stomatal resistance (rs,l) and substrate resistance (rs,s). Under the standard conditions specified above, the relative error, calculated as δλE = (λE'-λE0)/λE0, is around 10 % for LAI greater than 1, which is not negligible and justify the use of different formulations.

Effective resistances

The aim of the exercise is to compare the evaporation rate calculated using the big leaf model and different approximations for the effective resistances with the "true" evaporation rate (λE0) calculated using the two-layer model. The simulations are made here for an amphistomatous canopy. Three levels of approximation are considered: (1) λE1 is calculated by the general expressions (Eqs. ( 37), ( 38), ( 39)), which are supposed to be the most accurate ones; (2) λE2 is obtained by the expressions derived by neglecting the air resistances within the canopy (Eqs. ( 34), ( 40)); (3) λE3 when surface and air resistances are decoupled (Eqs.

(42), ( 38), ( 40)). In each case, a relative error (δλEi) is calculated as in section 5.1 under the weather conditions specified above. Fig. 6 illustrates the results obtained as a function of LAI for different values of leaf stomatal resistance (rs,l) and substrate resistance (rs,s). It is clear that λE1 yields the most accurate results with a relative error (δλE1) generally less than 5 %. λE3 provides a relatively good approximation mainly for high LAI (> 2) and high substrate resistance with a relative error generally lower than 5 %. λE2 yields the less accurate estimates, mainly when the substrate resistance is low or nil; nevertheless, the approximation can be acceptable for large LAI, provided substrate resistance be large enough.

Conclusion

In several papers of the scientific literature, the multi-source representation of heterogeneous and sparse canopies raises some questions in relation to the correct formulation of the physical processes. When multi-source models are used to represent evaporation, some basic principles should be respected in writing the equations. The resistance network chosen to represent the physical processes necessarily dictates the correct formalism for combining the component fluxes and resistances. In relation to that, three particular points have been addressed. The first point stresses and details the fact that the basic formulations and the resulting equations slightly differ when the foliage is amphistomatous and hypostomatous. Generic equations, valid in both situations, have been established and it has been found that not taking into account the differences can lead to errors on the evaporation rate of about 10 ra: aerodynamic resistance between the source height and the reference height (s m -1 ) ra,f,h: bulk boundary-layer resistance of the foliage for sensible heat (s m -1 ) ra,f,v: bulk boundary-layer resistance of the foliage for water vapour (s m -1 ) ra,s: aerodynamic resistance between the substrate and the source height (s m -1 ) ra,vs: aerodynamic resistance between the vegetated soil and the source height (s m -1 ) ra,bs: aerodynamic resistance between the bare soil and the source height (s m -1 ) rs,f: bulk stomatal resistance of the foliage (s m -1 ) rs,s: substrate resistance to evaporation (s m -1 ) rs,bs: bare soil resistance to evaporation (s m -1 ) rs,vs: vegetated soil resistance to evaporation (s m -1 ) rs,l: leaf stomatal resistance (one side) (s m -1 ) ra,l: leaf boundary-layer resistance for sensible heat and water vapour (one side) (s m -1 ) e s r : effective surface resistance to water vapour within the canopy (s m -1 ) e v r : effective resistance to water vapour transfer within the canopy (s m -1 ) e h a r , : effective air resistance to sensible heat transfer within the canopy (s m -1 ) e v a r , : effective air resistance to water vapour transfer within the canopy (s m -1 ) Appendix A: Derivation of Eq. ( 5)

The steps of the derivation are the same as those used for deriving the Penman-Monteith equation. Sensible and latent heat fluxes emanating from the foliage are written respectively (Fig. 1)

h f a m f p f r T T c H , , / ) ( - =  , ( 48
) ) /( ) ) ( * )( / ( , , , v f a f s m f p f r r e T e c E + - =    . ( 49
)
Tf is foliage temperature and e*(Tf) is the saturated vapour pressure at temperature Tf.

Linearizing the difference of saturated vapour pressure between the foliage and the canopy source height and combining Eqs. ( 48) and ( 49) with the energy balance (

f f f E H A  + = ) leads to   h f a v f a f s h f a m p f f r r r r D c A E , , , , , , , / ) ( /
+ +  +  =    . ( 50
)
Given that ra,f,v = n ra,f,h (Eq. ( 4)), we obtain (Monteith and Unsworth 1990, p.188)

  h f a f s h f a m p f f r r n r D c A E , , , , , / / + +  +  =    (51)
with n =1 and 2 respectively for amphistomatous and hypostomatous leaves.

Appendix B: Derivation of Eq. (16)

Introducing Eq. ( 7) into Eqs. ( 5) and ( 6) and adding the two equations gives

        ) / 1 ( / ) ( ) / ( / ) ( , , , , , , , , s a s s s a a a p s h f a f s h f a a a p f r r r r E A D c A r r n r r E A D c A E + +  +  -  + +  + + +  +  -  + +  =          . ( 52
)
Introducing the terms Rf ' and Rs' defined by Eqs. ( 20) and ( 21) yields

        ' , ' , , ) ( ) ( s a a p s a s f a a p h f a f R r E A D c r A R r E A D c r A E        +  -  + +  + +  -  + +  = . ( 53
)
By collecting the terms in λE we obtain

  ) ( ) ( ) ( ) )( ( ' , ' , , ' ' ' ' ' ' ' ' f s a s s h f a f s f a p a s f a s f s f R r A R r A R R D c r R R A r R R R R E +  + + + +  = + +  +     . ( 54
)
Introducing the term Ra ' defined by Eq. ( 22) gives

    ) ( / ) ( ) ( ' , ' , , ' ' ' ' ' ' ' f s a s s h f a f a a p a s f s f a s f R r A R r A r D c A r R R R R R R R E +  + +  + = + +    . ( 55
)
Eq. (B4) can be transformed into

        + + +  +         +  +  + + +         +  = ) ( / ) ( ) ( ' ' ' ' ' , ' , , ' ' ' ' ' ' ' ' s f a s f s a s f h f a f s a a p s f a s f a s f R R R R R r A R r A R r D c A R R R R R r R R E       . ( 56
)
Taking into account the definitions of Ps and Pf given by Eq. ( 18) and ( 19) leads to Eq. ( 16).

Appendix C: Coefficients of the modified clumped model (conventional formulation)

We define:

( ) f s h f a f r r n R , , ,   + +  = , ( 57
) ( ) vs s vs a vs r r R , ,   + +  = , (58) ( ) bs s bs a bs r r R , ,   + +  = , (59)
( )

a a r R  +  = . ( 60
)
The coefficients of Eq. ( 30) are written as

( ) DE R R R R C a f bs vs f / ' + = , (61)
( ) DE R R R R C a vs bs f vs / ' + = , ( 62
) ( ) DE R R R R C a bs f vs bs / ' + = , (63)
DE being defined as

a bs vs a vs f a bs f bs f vs R R R R R R R R R R R R DE + + + = . ( 64

)

  Each evaporation component is calculated separately from equations of the Penman-Monteith type involving the corresponding available energy (Af or As) and the vapour pressure deficit Dm at the mean canopy source height zm, assumed to be located at the apparent sink for momentum (zero plane displacement height d + roughness length z0). The component resistances are expressed per unit area of land surface. So, leaf stomatal resistance rs,l (one side) and the corresponding leaf boundary-layer resistance for latent heat ra,l (one side) should be divided by the transpiring surface expressed per unit area of land surface: 2LAI for amphistomatous leaves and LAI for hypostomatous leaves. For convenience we introduce the parameter n (n = 1 for amphistomatous leaves and n = 2 for hypostomatous leaves), which allows the bulk stomatal resistance to be written as

  model of Shuttleworth-Wallace (i.e. per unit area of land surface) since all the component fluxes mix together at canopy source height. Foliage resistances (stomatal and boundarylayer) have the same expressions as those of the two-layer model and are given by Eqs. (2), (3) and (4). Substrate resistances, however, should have different expressions since the exchange surfaces are not the same. If the resistances expressed per unit area of vegetated soil or bare soil are written with the upper-script 1, the resistances should be divided by the relative area of the corresponding surface (vegetated or bare soil) to obtain the component resistances of the model (per unit area of land surface):

  soil to be isothermal (at effective temperature Tm), the effective resistances should be calculated as the parallel sum of the component resistances expressed per unit area of land surface. This means that for the two-layer model we have

  ) leads to the common formalism of the Penman-Monteith model, where the ratio of a surface resistance divided by an aerodynamic resistance appears in the denominator

%

  . The second point concerns the conservation equation. If all the component fluxes mix together at canopy source height and experience the same aerodynamic resistance above the canopy, they should be simply added and not weighted by the relative area of each component. The weighing by the relative area should be applied not to the fluxes but to the component resistances. The third point deals with the effective resistances which allow a multi-source model to be reduced to a single-layer model. It is shown that the way of combining the elementary resistances within the canopy to formulate the effective air and surface resistances is physically determined as the parallel sum of component resistances: here also the hypostomatous case slightly differs from the amphistomatous one. More simple and operational formulations of the effective resistances can be obtained by means of some legitimate assumptions with different degrees of confidence illustrated by numerical simulations. Nomenclature A: available energy of the whole crop (W m -2 ) Af: available energy of the foliage (W m -2 ) As: available energy of the substrate (W m -2 ) Avs: available energy of the vegetated soil (W m -2 ) Abs: available energy of the bare soil (W m -2 ) Rn: net radiation of the whole crop (W m -2 ) G: soil heat flux (W m -2 ) H: sensible heat flux from the complete canopy (W m -2 ) λE: latent heat flux from the complete canopy (W m -2 ) Hi: component heat flux (i = f, s, vs, bs) (W m -2 ) λEi: component latent heat flux (i = f, s, vs, bs) (W m -2 ) Da: vapour pressure deficit at reference height (Pa) Dm: vapour pressure deficit at canopy source height (Pa) Ta: air temperature at reference height (°C) Tm: air temperature at canopy source height (°C) Ti: surface temperature of component i (i = f, s, vs, bs) (°C) ua: wind speed at reference height (m s -1 ) ea: vapour pressure at reference height (Pa) em: vapour pressure at canopy source height (Pa) e*(T): saturated vapour pressure at temperature T (Pa) cp: specific heat of air at constant pressure (J kg -1 °C-1 ) ρ: air density (kg m -3 ) γ: psychrometric constant (Pa °C-1 ) Δ: slope of the saturated vapour pressure curve (Pa °C-1 ) Canopy structural characteristics: d: canopy displacement height (m) F: fractional cover of foliage (dimensionless) LAI: leaf area index (m 2 m -2 ) n: parameter with value of 1 for amphistomatous and 2 for hypostomatous foliage zr: reference height (m) zh: mean canopy height (m) zm: mean canopy source height (= d+z0) (m) z0: canopy roughness length (m) Component resistances:

Fig. 1 .

 1 Figures captions

Fig. 2 .

 2 Fig. 2. Patch representation of evaporation from a sparse crop: schematic diagram showing the resistance network and potentials: subscript 1 is for foliage component and subscript 2 for substrate component.

Fig. 3 .

 3 Fig. 3. Resistance networks and potentials for a clumped representation (Brenner-Incoll model) of convective fluxes (sensible and latent heat) from a heterogeneous canopy (see the list of symbols).

Fig. 4 .

 4 Fig. 4. Resistance network and potentials for a one-layer representation (Penman-Monteith model) of convective fluxes (sensible and latent heat) from a full covering canopy: a) basic Penman-Monteith model; b) Penman-Monteith model with effective parameters accounting for air resistances within the canopy.

Fig. 5 .

 5 Fig. 5. Variation as a function of canopy LAI of the relative error δλE (expressed in percentage) made on the evaporation rate of a hypostomatous canopy when calculated with the two-layer equations valid for an amphistomatous one: a) for different values of leaf stomatal resistance rs,l (with rs,s = 500 s m -1 ); b) for different values of substrate resistance rs,s (with rs,l = 400 s m -1 ).

Fig. 6 .

 6 Fig. 6. Variation as a function of canopy LAI of the relative error δλE (expressed in percentage) made on the evaporation rate when calculated with a big leaf model and effective resistances estimated with different levels of approximation: a) with the physically soundest formulation (δλE1); b) when air resistances within the canopy are neglected (δλE2); c) when surface and air resistances are decoupled (δλE3).

Figure 1 .

 1 Figure 1. Resistance networks and potentials for a two-layer representation (Shuttleworth-Wallace model) of convective fluxes (sensible and latent heat) from a sparse canopy (see the list of symbols).

Figure 3 .

 3 Figure 3. Resistance networks and potentials for a clumped representation (Brenner-Incoll model) of convective fluxes (sensible and latent heat) from a heterogeneous canopy (see the list of symbols).

Figure 4 .

 4 Figure 4. Resistance network and potentials for a one-layer representation (Penman-Monteith model) of convective fluxes (sensible and latent heat) from a full covering canopy: a) basic Penman-Monteith model; b) Penman-Monteith model with effective parameters accounting for air resistances within the canopy.

Figure 5 .

 5 Figure 5. Variation of the relative error δλE (expressed in percentage) made on the evaporation rate of a hypostomatous canopy when calculated with the two-layer equations valid for an amphistomatous one. The variation is plotted as a function of canopy LAI: (a) for different values of leaf stomatal resistance rs,l (with rs,s = 500 s m -1 ); (b) for different values of substrate resistance rs,s (with rs,l = 400 s m -1 ).

Fig. 6 .

 6 Fig. 6. Variation of the relative error δλE (expressed in percentage) made on the evaporation rate when calculated with a big leaf model and effective resistances estimated with different levels of approximation: (a) and (b), with the physically soundest formulation (δλE1); (c) and (d), when air resistances within the canopy are neglected (δλE2); (e) and (f), when surface and air resistances are decoupled (δλE3). The variation is plotted as a function of canopy LAI.

The coefficients of the resultant equation are written as

Appendix E: Parameterizations used in the simulation process

The parameterization used to simulate the component air resistances and the distribution of available energy within the canopy are taken and adapted from Shuttleworth and [START_REF] Shuttleworth | Evaporation from sparse crops-an energy combination theory[END_REF], [START_REF] Choudhury | A four-layer model for the heat budget of homogeneous land surfaces[END_REF] and [START_REF] Shuttleworth | The theoretical relationship between foliage temperature and canopy resistance in sparse crops[END_REF].

The net radiation reaching the substrate Rns is calculated from the net radiation above the canopy Rn following Beer's law

Soil heat flux is calculated as a given fraction of Rns (G = 0.2 Rns). Consequently, available energies are obtained as: A = Rn-G, Af = Rn-Rns and As = Rns-G. For the sake of convenience, the aerodynamic resistance above the canopy (ra) is calculated in neutral conditions. It is expressed as a function of wind speed ua at reference height zr

where d = 0.63 zh, z0 = 0.13 zh and k is von Karman's constant. The aerodynamic resistance between the substrate (with a roughness length z0s = 0.01 m) and the canopy source height (d + z0) is calculated as