Jean-Philippe Groby

Matthieu Malléjac

Aurélien Merkel

Vicente Romero-García

Vincent Tournat

Daniel Torrent

Jensen Li

Analytical modeling of one-dimensional resonant asymmetric and reciprocal acoustic structures as Willis materials

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Since the seminal work of Willis in the 80's [START_REF] Willis | Variational principles for dynamic problems for inhomogeneous elastic media[END_REF], the eponymous materials have received an increasing attention, because of their analogy with bi-isotropic electromagnetic metamaterials [START_REF] Lindell | Electromagnetic waves in chiral and bi-isotropic media[END_REF]. The Willis coupling parameters couple the potential and kinetic energy in the acoustic conservation relations, therefore enhancing the ability to control waves in metamaterials compared to other materials that do not exhibit such coupling. These parameters have thus been employed to design PT symmetric [START_REF] Merkel | Unidirectional zero sonic reflection in passive PT -symmetric willis media[END_REF], wave front shaping [START_REF] Koo | Acoustic omni meta-atom for decoupled access to all octants of a wave parameter space[END_REF][START_REF] Li | Systematic design and experimental demonstration of bianisotropic metasurfaces for scattering-free manipulation of acoustic wavefronts[END_REF], or non-reciprocal [START_REF] Quan | Nonreciprocal willis coupling in zero-index moving media[END_REF] systems. Willis coupling arises from chiral inhomogeneities [START_REF] Frenzel | Ultrasound experiments on acoustical activity in chiral mechanical metamaterials[END_REF], asymmetric inhomogeneities, nonlocal effects, and nonreciprocal biases [START_REF] Zhai | Active willis metamaterials for ultracompact nonreciprocal linear acoustic devices[END_REF]. Although most of the works to date have focused on the experimental evidence [START_REF] Koo | Acoustic omni meta-atom for decoupled access to all octants of a wave parameter space[END_REF][START_REF] Muhlestein | Experimental evidence of willis coupling in a one-dimensional effective material element[END_REF][START_REF] Liu | Willis metamaterial on a structured beam[END_REF], physical origins [START_REF] Sieck | Origins of willis coupling and acoustic bianisotropy in acoustic metamaterials through source-driven homogenization[END_REF], calculation [START_REF] Ponge | Dynamic homogenization theory for nonlocal acoustic metamaterials[END_REF][START_REF] Shuvalov | Effective willis constitutive equations for periodically stratified anisotropic elastic media[END_REF][START_REF] Nemat-Nasser | Homogenization of periodic elastic composites and locally resonant sonic materials[END_REF], and enhancement [START_REF] Melnikov | Acoustic meta-atom with experimentally verified maximum willis coupling[END_REF] of Willis coupling, only a few have focused on deriving a closed form of these parameters. The present article aims at filling this gap and therefore at easing its physical understanding and engineering use. Effectively, it turns out that various systems rely on asymmetric meta-atoms for perfect absorption in transmission problem [START_REF] Jiménez | Rainbow-trapping absorbers: Broadband, perfect and asymmetric sound absorption by subwavelength panels with transmission[END_REF], non-Hermicity of the acoustic waves [START_REF] Shen | Synthetic exceptional points and unidirectional zero reflection in non-hermitian acoustic systems[END_REF], PT symmetry [START_REF] Aurégan | PT -symmetric scattering in flow duct acoustics[END_REF], or more generally most of the double negative one-dimensional devices [START_REF] Seo | Acoustic metamaterial exhibiting four different sign combinations of density and modulus[END_REF][START_REF] Lee | Composite acoustic medium with simultaneously negative density and modulus[END_REF][START_REF] Sieck | Origins of willis coupling and acoustic bianisotropy in acoustic metamaterials through source-driven homogenization[END_REF].

We focus on three resonant, asymmetric, and reciprocal one-dimensional unit cells and derive closed forms of the corresponding Willis coupling parameters, exhibiting different forms depending on the nature of the asymmetry. By analyzing these Willis parameters, we show that Willis coupling in resonant systems arises from first order Taylor expansion originating differently from multilayer systems [START_REF] Zangeneh-Nejad | Active times for acoustic metamaterials[END_REF][START_REF] Shuvalov | Effective willis constitutive equations for periodically stratified anisotropic elastic media[END_REF][START_REF] Nemat-Nasser | Overall dynamic constitutive relations of layered elastic composites[END_REF] and exhibit the dipolar feature of the coupling via arm terms. In addition, the reciprocity condition directly provides even coupling and vanishing odd coupling [START_REF] Sieck | Origins of willis coupling and acoustic bianisotropy in acoustic metamaterials through source-driven homogenization[END_REF], thus inducing doubt on the existence of odd coupling in one-dimensional reciprocal acoustic systems. While even Willis coupling parameters appear with opposite sign in the propagation matrix, odd Willis coupling appear with identical sign in the propagation matrix for reciprocal structures.

The article is organized as follows. The general procedure for the derivation of the effective parameters is detailed in Section 2. It relies on the Padé's approximation of the total transfer matrix, which links the state vectors at both sides of a unit cell. The procedure is applied in Section 3 to three different resonant asymmetric and reciprocal one-dimensional unit cells. While the first two unit cells are composed of detuned resonators either in parallel or in series of a duct, the third one combines resonators in parallel and in series in the duct, see Fig. 1. Two of these unit cells have already been studied as Willis materials [START_REF] Sieck | Origins of willis coupling and acoustic bianisotropy in acoustic metamaterials through source-driven homogenization[END_REF][START_REF] Muhlestein | Experimental evidence of willis coupling in a one-dimensional effective material element[END_REF], but the closed forms of the coupling parameters have not been provided. The derived forms clearly unveil dipolar feature of Willis coupling, as well as the differences between resonant and non-resonant asymmetries and nature of the resonant asymmetry. The effective parameters are validated against experimental and numerical (derived from [START_REF] Shuvalov | Effective willis constitutive equations for periodically stratified anisotropic elastic media[END_REF]) results for each unit cell in Section 4. The order of Taylor expansion required to correctly model an asymmetric resonant unit cell is questionned. Finally, concluding remarks and perspectives are provided. Additional information are also given in appendices.

Derivation of the Willis coupling parameters

We consider a one-dimensional asymmetric and reciprocal material composed of the periodic repetition of a unit cell of length d. The pressure and particle velocity (alternatively the flow in a duct) form a state vector W T =< p, V >, where T is the transpose operator. This state vector satisfies the following matrix equation which directly arises from the mass conservation and constitutive law:

d dx W = AW. (1)
Assuming an implicit time dependence e -iωt , the matrix A reads as

A = iω χ ρ 1/K -χ , (2)
for Willis type materials, where ρ, K and χ are respectively the density, bulk modulus and Willis coupling parameter. Please note that a usual isotropic and symmetric effective fluid implies χ = 0. The state vectors at both sides of the unit cell, W(d) and W(0), are thus linked by

W(d) = exp (Ad) W(0) = TW(0), (3)
where the term exp (Ad) is the matrix exponential of Ad also known as the transfer matrix T. Among the different ways of evaluating the matrix exponential [START_REF] Moler | Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later[END_REF] is the Padé's approximation. The transfer matrix T is thus approximated in the long wavelength regime, i.e., when the wavelength λ is much larger than the period d, by

T = exp (Ad) ∼ (I -Ad/2) -1 (I + Ad/2) , (4)
where I is the identity matrix. Assuming the transfer matrix T between the state vectors at both sides of the unit cell being known, Eq. ( 4) can be inverted and the expression of the constitutive matrix A becomes

A ∼ d/2 (T + I) -1 (T -I) . (5)
The Padé's approximation is of particular interest compared to other approximations such as the Taylor's expansion (see Appendix A), because it allows us to account for the reciprocity of the material [START_REF] Feng | Effective parameters retrieval for complex metamaterials with low symmetries[END_REF]. This material property imposes det (T) = 1, i.e., t 11 t 22 -t 12 t 21 = 1, where t ij are the elements of the matrix T. Accounting for the reciprocity, Eq. ( 5) can be rewritten in the form

A ∼ 2 d (2 + t 11 + t 22 ) t 11 -t 22 2t 12 2t 21 t 22 -t 11 , (6)
from which we immediately see that the diagonal terms are of opposite signs and we can identify

χ = -2i (t 11 -t 22 ) ωd (2 + t 11 + t 22 ) , ρ = -4it 12 ωd (2 + t 11 + t 22 )
,

and 1 K = -4it 21 ωd (2 + t 11 + t 22 ) , (7)
by comparison with Eq. ( 2). Density, bulk modulus and Willis coupling parameter can then be directly calculated via the elements of the transfer matrix that links the state vectors at both sides of the unit cell. Please note that the odd Willis coupling introduced in Ref. [START_REF] Sieck | Origins of willis coupling and acoustic bianisotropy in acoustic metamaterials through source-driven homogenization[END_REF] is completely canceled from Eq. ( 6) only because of the reciprocal condition and that only even coupling remains. The effective parameters that are derived following this procedure are indicated by a subscript e in the following.

Explicit effective parameters of Willis materials composed of resonant elements

The total transfer matrix T is evaluated thanks to the knowledge of different elementary transfer matrices that are presented in Appendix B. A circular duct of radius r and length d much smaller than the wavelength is considered. Only plane waves propagate in this duct, i.e., the frequencies are lower than that of the first cut-off of the duct and possible evanescent coupling between the unit cell elements is neglected. The reduced density and bulk modulus (or alternatively the wavenumber and reduced impedance) in the duct are ρ and K (or alternatively k = k and Z). Three asymmetric and reciprocal unit cells are considered and are represented in Fig. 1: i) when the duct is loaded by two detuned Helmholtz resonators (HR) located at different positions (Fig. 1(a)) of impedances Z(j) HR , j = 1, 2, ii) when two detuned plates are clamped in the duct (Fig. 1(b)) of impedances Z(j) p , j = 1, 2, and iii) when the duct is loaded by a HR and a plate is clamped in it at a different position (Fig. 1(c)). Each resonant element possesses its own local dynamics which is assumed different from that of the duct, i.e., co-dynamic regime is assumed [START_REF] Auriault | Dynamique des composites élastiques périodiques[END_REF]. This is an important difference with respect to a laminated twomaterial unit cell for example, for which it is clear that a second order Taylor expansion of the total matrix elements in Eq. ( 7) is required to exhibit Willis coupling terms as demonstrated in Appendix C. The order of the resonant element impedances becomes unclear notably around the resonance as explained in Appendix B. Nevertheless, 1/ ZHR (Lorentzian function) and Zp (inverse of a Lorentzian function) are assumed to vary like O ((kd) 2 ) to ensure reciprocity of the configuration in the considered frequency range, see for example the discussion of Appendix D in the presence of a single HR. Note that this relies more on a frequency analysis rather than on a purely kd analysis.

Unit cell composed of a duct loaded by two detuned Helmholtz resonators

We first consider a unit cell of length d composed of a straight circular duct loaded by two detuned HR of respective reduced impedances Z(1) HR and Z(2) HR , assumed to be point-like resonators, and located at l (1) and l (1) + l (2) such that d = l (1) + l (2) + l (3) , as represented Fig. 1(a). Both inverse impedances 1 Z(1) HR and 1

Z(2)

HR are assumed to be O ((kd) 2 ) terms. This configuration is formally that considered in Ref. [START_REF] Merkel | Unidirectional zero sonic reflection in passive PT -symmetric willis media[END_REF], where the loading quarter-wavelength resonators are replaced by HR. The total transfer matrix reads as

W (d) = TW (0) = T l (3) T HR (2) T l (2) T HR (1) T l (1) W (0) , (8)
where the transfer matrices accounting for the propagation along each length and for the loading HR are given in Eqs. (B.1) and (B.2). Assuming kd << λ, in such a way that kl (1) = ζ (1) kd, kl (2) = ζ (2) kd, and kl (3) = ζ (3) kd, with ζ (j) = l (j) /d, j = 1, 2, 3, are also much smaller than the wavelength, and making use of Eq. ( 7) leads to

χ e = ρ 2d
l (3) + l (2) -l (1) Z (1) HR + l (3) -l (1) -l (2)

Z

(2) HR

+ O (kd) 3 ,

ρ e = ρ + O (kd) 3 , 1 K e = 1 K + 1 d - i ω Z (1)
HR - i ω Z(2) HR + O (kd) 3 . (9)
To get a grip on these equations, we compare them to those in the presence of a single HR presented in Appendix D. The Willis coupling term appears as the sum of the two Willis coupling terms arising from each HR. Each of them clearly exhibits a momentum introduced by the resonator as testified by the moment arm term l (3) + l (2) -l (1) for the first HR and the moment arm term l (3) -l (1) -l (2) for the second HR. The Willis coupling parameter vanishes when the symmetry is introduced, for example, when the two HR are identical and when l (3) = l (1) . It appears in the form of an effective density divided by the impedance of the HR. The density term, second line of Eq.( 9), is not the subject of any specific remark and simply reads as the effective density in the duct. The bulk modulus is the sum of the contribution of the effective properties of the duct in the absence of the loading resonators, i.e., 1/ K, and of each HR, i.e., -i/dω Z(j) HR , j = 1, 2. The presence of the HR causes the effective bulk modulus to become negative for frequencies around their resonances [START_REF] Liu | Locally resonant sonic materials[END_REF][START_REF] Fang | Ultrasonic metamaterials with negative modulus[END_REF]. All in all, the effective parameters appear as the sum of those of each elements, i.e., those of each segments and those of the HR, without any particular coupling. When a subperiodicity can be introduced, i.e., when the two HR are identical and when l 2 = d/2, the effective parameters reduce to those of a unit cell of length d/2 consisting in a single, possibly uncentered, HR with Willis parameter exhibting a moment arm term equal to l (3) -l (1) .

Unit cell composed of a duct with detuned clamped plates

We now consider a unit cell of length d, as depicted in Fig. 1(b), composed of a straight circular duct in which two detuned plates, assumed to be point-like resonators, of respective reduced impedances Z(1) p and Z [START_REF] Lindell | Electromagnetic waves in chiral and bi-isotropic media[END_REF] p are clamped at l (1) and l (1) + l (2) such that d = l (1) + l (2) + l (3) . The clamped plates (CP) simply replace the HR when compared to the previous configuration, but this time the impedances are in series instead of being in parallel. This configuration is formally that considered in Ref. [START_REF] Muhlestein | Experimental evidence of willis coupling in a one-dimensional effective material element[END_REF]. The total transfer matrix reads as

W (d) = TW (0) = T l (3) T p (2) T l (2) T p (1) T l (1) W (0) , (10)
where the transfer matrices accounting for the propagation along each length and for the CP are given in Eqs. (B.1) and (B.4). Proceeding similarly as in the previous section, Eq. ( 7) together with the second order Taylor expansion of the total transfer matrix elements provides

χ e = 1 2d K Z p (1) l (1) -l (2) -l (3) + Z p (2) l (1) + l (2) -l (3) + O (kd) 3 , ρ e = ρ + 1 d - iZ p (1) ω - iZ p (2) ω + O (kd) 3 , 1 K e = 1 K + O (kd) 3 . (11)
The Willis coupling term appears as the sum of the Willis terms associated with each plate (see Appendix D). The momentum seems opposite to that imposed by the HR with an arm term l (1) -l (2) -l (3) for the first CP and an arm term l (1) + l (2) -l (3) for the second CP. It appears in the form of an effective compressibility multiplied by the impedance of the plate. Again, this term vanishes when the symmetry is introduced as for example, when the two CP are identical and l (3) = l (1) . The density is the sum of the contribution of the effective density of the duct in the absence of the CP, i.e., ρ, and of each CP, i.e., -iZ p (j) /dω, j = 1, 2. The presence of the clamped plates causes the density to be negative for frequencies lower than their resonances [START_REF] Yang | Membrane-type acoustic metamaterial with negative dynamic mass[END_REF]. Finally, the effective bulk modulus is that in the absence of the clamped plate and is thus not the subject of any specific remark. The effective parameters again appear as the sum of those of each elements, i.e., those of each segments and those of the CP, without any particular coupling. When a plane of symmetry can be introduced, i.e., when the two CP are identical and when l 2 = d/2, the effective parameters reduce to those of a unit cell of length d/2 consisting in a single, possibly uncentered, CP with Willis parameter exhibting a moment arm term equal to l (1) -l (3) .

Unit cell composed of a duct with a clamped plate and loaded by a detuned Helmholtz resonator

We finally consider a unit cell of length d composed of a straight circular duct in which a plate, assumed to be a point-like resonator, of reduced impedance Zp is clamped at l (1) . The duct is in addition loaded by a HR, also assumed to be a point-like resonator, of reduced impedance ZHR and located at l (1) + l (2) such that d = l (1) + l (2) + l (3) . A picture of the configuration is shown in Fig. 1(c). In other words, the first HR is replaced by a CP compared to the configuration studied in Section 3.1 or the second CP is replaced by a HR compared to the configuration studied in Section 3.2. Beyond the expected double negative effective property [START_REF] Li | Double-negative acoustic metamaterial[END_REF], this asymmetric configuration combines impedances in series and in parallel and is similar to that studied in Ref. [START_REF] Sieck | Origins of willis coupling and acoustic bianisotropy in acoustic metamaterials through source-driven homogenization[END_REF]. The total transfer matrix reads as

W (d) = TW (0) = T l (3) T HR T l (2) T p T l (1) W (0) , (12)
where the transfer matrices accounting for the propagation along each length, for the CP, and for the HR are given in Eqs. (B.1), (B.4), and (B.2) respectively. Again, Eq. ( 7) together with the second order Taylor expansion of the total transfer matrix elements provides

χ e = 1 2d ρ Z HR l (3) -l (1) -l (2) + Z p K l (1) -l (2) -l (3) + iZ p ωZ HR + O (kd) 3 , ρ e = ρ - iZ p dω + O (kd) 3 , 1 K e = 1 K - i dωZ HR + O (kd) 3 . ( 13
)
Contrary to the configurations studied in both previous sections, the Willis coupling term is not only the sum of the Willis terms associated with the presence of the CP and of the HR (see Appendix D), but also exhibits a coupling between the CP and the HR thanks to the term iZ p /2dωZ HR . Note that the moment arms are introduced by the first two terms, that related to the presence of the CP and that related to the presence HR, while the coupling term does not present moment arm term. In addition, the Willis coupling parameter never seems to vanish, because the configuration is structurally asymmetric. The density is the sum of the contribution of the effective density of the duct in the absence of the CP and of the HR, and of the CP, i.e., -iZ p /dω. In a similar way, the bulk modulus is the sum of the contribution of the effective bulk modulus of the duct in the absence of the CP and of the HR, and of the HR, i.e., -i/dωZ HR . Note that obviously no symmetry plane can be introduced for this configuration.

Experimental validation of the effective parameters and discussion

All experiments are conducted in a duct of radius r = 2.5 cm, see Figs. 1(a-c). The experimental set-up consists of a 4 microphone measurement system with a pair of microphones upstream and a second pair of microphones downstream of the sample. The microphones that compose each pair are separated by a distance of 2.5 cm. A stept signal from 100 Hz to 1000 Hz with a step of 1 Hz is delivered by a loudspeaker at one end of the duct and an anechoic termination is mounted at the opposite end. Temperature, humidity and athmospheric pressure are recorded for each experiment.

The transfer function between the loudspeaker and each microphone is recorded by a NI USB-4431 acquisition card driven by the INTAC software. Each sample is measured in both direct and reverse orientations in order to form an overdetermined system based on the scattering matrix [START_REF] Niskanen | Deterministic and statistical characterization of rigid frame porous materials from impedance tube measurements[END_REF][START_REF] Merkel | Unidirectional zero sonic reflection in passive PT -symmetric willis media[END_REF] as explained in Appendix E to solve for T , R + , and R -, i.e., the transmission, the direct orientation reflection, and the reverse orientation reflection coefficients. These coefficients are also calculated by the Transfer Matrix Method (TMM) using the total transfer matrix relying on the elementary matrices and the Pade's approximation of transfer matrix relying on the evaluated effective parameters (I -A e d/2) -1 (I + A e d/2), Eq. ( 4),

R + =
where Z is the reduced impedance of the surrounding medium. From the measured direct and inverse orientation reflection and transmission coefficients, the effective parameters are reconstructed following the procedure described in Appendix F. The effective properties can also be directly evaluated from the total transfer matrix as explained in Appendix G, which appears as a numerical version of the procedure described in Ref. [START_REF] Shuvalov | Effective willis constitutive equations for periodically stratified anisotropic elastic media[END_REF].

Figures 2(a-e-i) depict respectively the absolute values of the two reflection and transmission coefficients, calculated by the TMM with the total transfer matrix and by the Pade's approximation using the effective properties derived in Section 3 for the three configurations considered in the present article. A single unit cell is measured for each configuration. This measurement is entirely sufficient, because one-dimensional structures are assumed. The first configuration (see Fig. 1(a)) consists in two detuned HR, the dimensions of the cavities and necks of which are l HR ≈ 230 Hz. The two remaining dimensions l (1) and l (3) are chosen identical, i.e., l (1) = l (3) = 1 cm, such that d = 7 cm. The second configuration (see Fig.

p = 40+380i/ √ ω kg.m -1 , ν (2)
p = 0.1, and E

p = 470-0.007iω kPa already used in Ref. [START_REF] Romero-García | Perfect and broadband acoustic absorption by critically coupled sub-wavelength resonators[END_REF]. The resonant frequencies of both CP are thus f

p ≈ 255 Hz and the two remaining dimensions l (1) and l (3) are chosen almost identical, i.e., l (1) = 5 mm + h

p + h (2)
p . The last configuration (see Fig. 1(c)) is composed of the poroelastic plate and a loading HR, the cavity and neck dimensions of which are l c = 5 cm and r c = 2.15 cm, and l n = 2 cm and r n = 2 mm, separated by a distance l (2) = 5 cm + h p /2. The resonance frequency of the CP is still f p = 255 Hz and that of the HR is f HR = 140 Hz and the two remaining dimensions l (1) and l (3) are again chosen almost identical, i.e., l (1) = 1 cm + h p /2 and l (3) = 1 cm, such that d = 7 cm+h p . The lengths of the unit cells are thus much smaller than the wavelengths over the frequency range considered in each configuration. The necks were manufactured by Fused Deposition Modeling to fit in an initial radius of 1 cm. The necks also present some rugosity inherent to this rapid manufacturing technique, which can influence the viscothermal losses in these resonator narrow elements [START_REF] Huang | 3d-printed sound absorbing metafluid inspired by cereal straws[END_REF]. The configurations were designed to present a reflection coefficient that vanishes at a specific frequency while the reflection coefficient of the reverse sample is different from zero at this frequency. The calculated coefficients are found in very good agrement with the experiments, which therefore valid the derived effective properties and prove the asymmetry of the acoustic response of each configuration. Some discrepancies are visible, notably for the third configuration at very low frequencies. These discrepancies are attributed to measurement complexity at low frequencies and to possible evanescent coupling between the resonant elements that is not accounted for in our modeling. At high frequency, the calculated transmission coefficient based on the effective parameters starts to deviate from its value for the second configuration because the scale separation is not ensured anymore and the Pade's approximation is not valid anymore, see Fig. 2(e). In addition, slight shifts between the coefficients calculated with the total transfer matrix and with the effective parameters can also be notice at low frequencies and around the resonances, see Fig. 2(e) and (j). These discrepencies are attributed to the assumption we made on the dependency of the resonant element impedances, that are supposed to vary like kd. Around the resonances, this assumption is not fully valid and the effective properties would require additional terms to be Taylor expanded to the second order.

The normalized effective properties, i.e., χ e S, ρe /ρ, and K e S/γP 0 , are respectively depicted in Figs. 2(b-f-j), (c-g-k), and(d-h-l) for each configuration. The numerical and experimental effective properties match those evaluated in Section 3. Generally speaking, the real parts of the effective densities (CP) and bulk moduli (HR) are negative in stop bands and follow regular trends [START_REF] Liu | Locally resonant sonic materials[END_REF][START_REF] Fang | Ultrasonic metamaterials with negative modulus[END_REF][START_REF] Yang | Membrane-type acoustic metamaterial with negative dynamic mass[END_REF][START_REF] Li | Double-negative acoustic metamaterial[END_REF]. Some discrepancies are visible for the third configuration again due to the fact that the phenomenon are encountered at very low frequencies, but also to the assumption made on the order of the resonant element impedances. The amplitude of the Willis parameter is usually much smaller than the other ones, partially due to the fact that this parameter is only normalized by the cross-sectional area of the main duct S, in the absence of a normalization value for this parameter. Note that χ e S vanishes away from the resonances in case of detuned HR, while it tends to infinity at low frequency when a plate is involved, see Figs. 2(f,j). A similar remark is made concerning the effective density ρe /ρ. This is due to the presence of the term Zp in both χ e and ρ e , Eqs. ( 11) and [START_REF] Shuvalov | Effective willis constitutive equations for periodically stratified anisotropic elastic media[END_REF], which varies like 1/ω at low frequencies. This also blurs the required order of Taylor expansion to derive these effective parameters even away from the resonances in case of unit cells involving plates as it can be seen in Figs. 2(e) and 2(i). This divergence is nevertheless physical, because a plate should be a rigid wall at low frequency. Note that to reach the request accuracy too near to resonance, non-local effective medium are probably requested, which are outside the scope of the current paper. The dispersion relation is also plotted for the configuration comprising a straight duct with a CP and loaded by a detuned HR in Figs. 2(m-n). The grey areas highlight the frequency range where the real parts of both the effective density and bulk modulus are negative. The viscothermal losses and Willis coupling induce a slight shift in the negative index part. The dispersion relation for the other two configurations (not shown here) are found in good agreement.

Conclusion

We derive closed forms of the effective properties of Willis materials thanks to a Pade's approximation of the total transfer matrix linking the state vectors at both sides of an asymmetric and reciprocal one-dimensional unit cell. We primarily show that the reciprocal condition leads to the unique existence of the even Willis coupling. Similarly to the case of laminated structures, which are usually in nonresonant acoustic structures, second order Taylor expansion of the transfer matrix elements is sufficient to exhibit Willis coupling parameters. Nevertheless, this result relies on a strong assumption on the resonant element behavior and its veracity is questionned around the resonance. Higher order Taylor expansion of the full transfer matrix terms involving the resonant elements may therefore be more suitable. The dipolar feature of the Willis coupling is clearly evidenced, because the coupling parameter presents moment arm terms. This also links Willis material to higher order strain gradient theories [START_REF] Maugin | Non-Classical Continuum Mechanics -A Dictionary[END_REF]. We also evidenced different types of coupling terms due to the asymmetry, either absent when the unit cell consists in detuned identical type resonators or due to a physical asymmetry when the unit cell involves different types of resonators. Beyond the derivation of these closed-forms, we also show that various asymmetric structures can be modeled as Willis materials. Each effective parameter (effective density, effective bulk modulus, and effective Willis coupling parameter) is validated against experimental and numerical results, showing the robustness of the derivation method. This work paves the way for the engineering use of Willis materials, from perfect absorption devices in transmission problems, to double negative structures, but also to metaporous materials and liquid foams [START_REF] Lagarrigue | Absorption of sound by porous layers with embedded periodic arrays of resonant inclusions[END_REF][START_REF] Schwan | Nonlocal boundary conditions for corrugated acoustic metasurface with strong near-field interactions[END_REF][START_REF] Pierre | Resonant acoustic propagation and negative density in liquid foams[END_REF].

where k n and Zn are respectively the wavenumber and reduced impedance of the neck, k c and Zc are respectively the wavenumber and reduced impedance of the cavity, and δ = 0.82(1-1.35r n /r c +0.31(r n /r c ) 3 )r n +0.82(1-0.235r n /r-1.32(r n /r) 2 +1.54(r n /r) 3 -0.86(r n /r) 4 )r n is the correction length, with r n and r c the radius of the neck and of the cavity respectively.

In the absence of losses and correction length, the latter impedance reduces to ZHR = -iZ (1 -k 2 l n S c l c /S n ) /k (S n l n + S c l c ) at low frequencies, from which it follows that 1/ ZHR varies like ω when ω → 0, but like a Lorentzian around the resonance frequency.

dependent density and bulk modulus read as [START_REF] Stinson | The propagation of plane sound waves in narrow and wide circular tubes, and generalization to uniform tubes of arbitrary crosssectional shape[END_REF]

ρ = ρ 0   1 - 2 r iωρ 0 /η J 1 r iωρ 0 /η J 0 r iωρ 0 /η   -1 , K = γP 0   1 + 2 (γ -1)
r iPrωρ 0 /η J 1 r iPrωρ 0 /η J 0 r iPrωρ 0 /η   -1 , (B.6)
where ρ 0 , γ, η, and Pr are respectively the density, specific heat ratio, dynamic density and Prandtl number of the saturating fluid, and P 0 the atmospheric pressure. The reduced density and bulk modulus can then be straightforwardly evaluated by ρ = ρ/S and K = K/S, with S = πr 2 .

Appendix C. Derivation of the Willis parameters in the case of a laminated two-material unit cell

We assume a unit cell of length d composed of a material M (1) of density ρ (1) and bulk modulus K (1) and of length l (1) and a material M (2) of density ρ (2) and bulk modulus K (2) and of length l (2) = d -l (1) . The state vectors at both sides of the unit cell are thus related by the total transfer matrix composed of the multiplication of the transfer matrix modeling the propagation in the material M (1) over the length l (1) and that in the material M (2) over the length l (2)

W (d) = TW (0) = T l (2) T l (1) W (0) , (C.1)
where the expression of T l (2) and of T l (1) are given Eq. (B.1). Assuming kd << λ, in such a way that k (1) l (1) = ζ (1) kd and k (2) l (2) = ζ (2) kd, with ζ (j) = l (j) /d, j = 1, 2, are also much smaller than the wavelength and making use of Eq. ( 7) leads to

χ e = 0 + O (kd) 2 , ρ e = 1 d l (2) ρ (2) + l (1) ρ (1) + O (kd) 2 , and 1 K e = 1 d l (2) K (2) + l (1) K (1) + O (kd) 2 , (C.2)
if we only Taylor expand the elements of the transfer matrix to the first order. In the opposite, it leads to

χ e = iωl (1) l (2) 2d ρ (2) K (1) - ρ (1) K (2) + O (kd) 3 , ρ e = 1 d l (2) ρ (2) + l (1) ρ (1) + O (kd) 3 , 1 K e = 1 d l (2) K (2) + l (1) K (1) + O (kd) 3 , (C.3)
if we Taylor expand the elements of the transfer matrix to the second order. Of particular interest is the fact that the eigenvalues of A e , i.e., the wavevectors, are identical for both orders of expansion (assuming respectively first and second order expansion) and equal to k e = ± iω √ Ke/ρe

, while the eigenvectors exhibit Z ± e = K e (k e ± χ e ) at the second order against Z ± e = K e k e at the first order. In the lossless case, the Willis coupling parameter is purely imaginary.

Appendix D. Derivation of the Willis parameters in case of a unit cell presenting a single resonator This section is motivated by nonlocal aspects of the Willis coupling and the fact that structures are often bounded in practice. Moreover, it helps in a better understanding of coupling terms that arise in case of more complex asymmetric unit cells.

Appendix D.1. Derivation of the Willis parameters in case of a unit cell presenting a single Helmholtz resonator

We consider a unit cell composed of a straight duct of length d and radius r loaded by a HR of reduced impedance ZHR and located at l (1) such that d = l (1) + l (2) . The total transfer matrix reads as

W (d) = TW (0) = T l (2) T HR T l (1) W (0) , (D.1)
where the transfer matrix accounting for the propagation along each length and the HR are provided in Eqs. (B.1) and (B.2). Assuming kd << λ, in such a way that kl (1) = ζ (1) kd and kl (2) = ζ (2) kd, with ζ (j) = l (j) /d, j = 1, 2, are also much smaller than the wavelength, but also that 1/ ZHR is a O ((kd) 2 ) term (because it varies like ω at low frequencies), and making use of Eq. ( 7) leads to

χ e = 1 2d ρ(2) l (2) -ρ(1) l (1) Z HR + iωl (2) l (1) ρ(2) K(1) - ρ(1) K(2) + O (kd) 3 = ρ 2d l (2) -l (1) Z HR + O (kd) 3 , ρ e = 1 d ρ(1) l (1) + ρ(2) l (2) + O (kd) 3 , = ρ + O (kd) 3 , 1 K e = 1 d l (1) K(1) + l (2) K(2) - i ω ZHR + O (kd) 3 . = 1 K - i ω ZHR d + O (kd) 3 . (D.2)
At first glance, the effective parameters appear as the sum of those of a laminated twomaterial unit cell, Eqs. (C.3) and additional terms related to the presence of the HR. The main tube is identical along the segments 1 and 2, thus K(1) = K(2) and ρ(1) = ρ(2) making the Willis parameter associated with the two materials to collapse. The effective bulk modulus and density then read as those proposed in [START_REF] Liu | Locally resonant sonic materials[END_REF][START_REF] Fang | Ultrasonic metamaterials with negative modulus[END_REF]. The presence of the HR affects the bulk modulus, which can become negative around the HR resonance. The Willis parameter only accounts the possible phase shift, when a material is bounded by bounds that do not coincide with the natural unit cell bounds, i.e., when the HR cannot be centered in the unit cell. Please note that the Willis parameter simply vanishes when l (1) = l (2) = d/2. Nevertheless, the term l (2) -l (1) translates a momentum within the unit cell. While ρ e and 1/K e are also O ((kd) 2 ) terms, χ e is a O ((kd) 3 ) term, when 1/ ZHR is assumed to be a O ((kd) 2 ) term.

To prove that 1/ ZHR should be a O ((kd) 2 ) term (at least in the low frequency), we adopt a reductio ad absurdum. Let us assume a centered HR (i.e., the Willis parameter vanishes) and that 1/ ZHR is a O (kd) term for example. The Eqs. (D.2) are thus derived from a first order Taylor expansion of the elements of the total transfer matrix. Under these assumptions, det (T) ≈ 1 -iω ρd/ ZHR + O ((kd) 2 ) = 1 does not satisfy the reciprocity condition. Thus, 1/ ZHR is assumed to be a O ((kd) 2 ) term. Note that this relies more on a frequency analysis rather than on a kd analysis. We will therefore use the power of ω instead of the power of kd to express the order of Taylor expansion in the following.

Appendix D.2. Derivation of the Willis parameters in the case of a unit cell presenting a single clamped plate

We consider a unit cell composed of a straight duct of length d and radius r in which a plate of reduced impedance Zp is clamped at l (1) such that d = l (1) + l (2) . The total transfer matrix reads as

W (d) = TW (0) = T l (2) T p T l (1) W (0) , (D.3)
where the transfer matrix accounting for the propagation along each length and the CP are provided in Eqs. (B.1) and (B.4). Assuming kd << λ, in such a way that kl (1) = ζ (1) kd and kl (2) = ζ (2) kd, again with ζ (j) = l (j) /d, j = 1, 2, are also much smaller than the wavelength, but also that Zp is a O ((kd) 2 ) term (because it varies like ω at low frequencies), and making use of Eq. (D.4)

The effective bulk modulus and density read as those proposed in [START_REF] Yang | Membrane-type acoustic metamaterial with negative dynamic mass[END_REF]. The presence of the CP affects the density, which can become negative around the resonance. The Willis parameter only accounts the possible phase shift, when a material is bounded by bounds that do not coincide with the natural unit cell bounds, i.e., when the CP cannot be centered in the unit cell. Please note that the Willis parameter simply vanishes when l (1) = l (2) = d/2. Nevertheless, the term l (1) -l (2) translates a momentum within the unit cell. Note that the fact that the main tube is identical along the 1 and 2 segments has already been accounted for and that an analysis similar to that performed in Appendix D.1 can be carried out to prove that Zp should be assumed to be a O ((kd) 2 ) term. Nevertheless, this assumption is even more tedious than that made in the case of the Helmholtz resonator, because Zp physically diverges at very low frequency.

Introducing r ± = (Z + Z ± ) / (Z -Z ± ), where Z ± = K (σ ± χ), the reflection coefficients at the interface between a semi-infinite background medium (of impedance Z) and the Willis material in the direct and reverse orientations, these equations can be inverted to yield

r + = -(R -R + + 1 -T 2 ) ± (R -R + + 1 -T 2 ) 2 -4R -R + 2R + , r - = -(R -R + + 1 -T 2 ) ± (R -R + + 1 -T 2 ) 2 -4R -R + 2R - , e -iωσd = T R + r -+ 1 = T R -r + + 1 = R + + r + T r + = R -+ r - T r -.
(F.4)

The sign of the first two equations are chosen to satisfy the passivity condition and χ e , ρ e , and K e are subsequently evaluated. These equations are slightly different from those given in [START_REF] Muhlestein | Experimental evidence of willis coupling in a one-dimensional effective material element[END_REF] and extend the method proposed in [START_REF] Nicolson | Measurement of the intrinsic properties of materials by timedomain techniques[END_REF][START_REF] Weir | Automatic measurement of complex dielectric constant and permeability at microwave frequencies[END_REF], which was already used in [START_REF] Fokin | Method for retrieving effective properties of locally resonant acoustic metamaterials[END_REF].

Figure 1 .

 1 Figure 1. (Color online) Pictures of the three studied Willis unit cells: (a) a straight duct loaded by two detuned HR located at different positions, (b) a straight duct with two detuned plates clampled in it, and (c) a straight duct loaded by a HR and with a plate clamped in it at a different position. A sketch of the configuration is provided below each picture.

  mm, separated by a distance l (2) = 5 cm. The resonant frequencies of both HR are thus f

  1(b)) consists in two CP separated by a distance l (2) = 5 cm + h The first CP is a plastic shim plate of thickness h (1) p = 254 µm and material properties ρ (1) p = 1400 kg.m -3 , ν (1) p = 0.41, and E (1) p = 4.6 (1 -0.03i) GPa already used in Ref. [31]. The second CP is a poroelastic plate of thickness h (2) p = 3.1 mm and material properties ρ (2)

Figure 2 .

 2 Figure 2. (Color online) (a-e-i) |R + | (blue curve), |R -| (magenta curve), and |T |(red curve) as calculated with TMM with the total transfer matrix (dashed line), with the effective parameters (solid line), and as measured experimentally (markers), for a unit cell that is composed of a straight duct loaded by two detuned HR -with two detuned plates clamped in it -with a plate clamped in it and loaded by a detuned HR.Subfigures (b-f-j), (c-g-k) and (d-h-l) depict respectively the real (blue curve) and imaginary (red curve) of the corresponding Willis coupling parameter, normalized density, and normalized bulk modulus. Subfigures (m-n) depict both the real and imaginary parts of the normalized dispersion relation for a unit cell composed of a straight duct with a plate clamped in it and loaded by a detuned HR. The grey regions highlight the frequency band where the real parts of both the effective density and bulk modulus are negative.

( 1 )

 1 p /2 and l (3) = 5 mm + h (2) p /2, such that d = 6 cm + h (1)

1 d

 1 Appendix G. Direct numerical calculation of the effective properties from the total transfer matrixOnce the total transfer matrix T is calculated, it is directly assimilated to exp A num e d. From Eq. (F.1), it is clear that the eigvenvectors of T and A num e are identical and that the exponential of the eigenvalues of A num e d are the eigenvalues of T. Following the idea of[START_REF] Shuvalov | Effective willis constitutive equations for periodically stratified anisotropic elastic media[END_REF], we immediately end up withA num e = Vdiag log Λ ± V -1 , (G.1)where Λ ± are the eigenvalues of T and V the associated eigenvector matrix. The three quantities χ num e , ρ num e , and K num e are subsequently evaluated.

  t 11 -t 12 / Z + Zt 21 -t 22 t 11 -t 12 / Z -Zt 21 + t 22 , R -= -t 11 -t 12 / Z + Zt 21 + t 22 t 11 -t 12 / Z -Zt 21 + t 22 , T = 2 t 11 -t 12 / Z -Zt 21 + t 22 ,

Acknowledgments

J.-P.G., M.M., V.R.-G., and V.T. would like to thank the support of the ANR-RGC METARoom project (ANR-18-CE08-0021). J.L. would like to thank the support of the Research Grants Council in Hong Kong (Grant No. 16302218). D.T. acknowledges financial support through the "Ramón y Cajal" fellowship under grant number RYC-2016-21188.

Appendix A. Padé's approximation versus Taylor's expansion

In the long-wavelength regime, the first order Taylor's expansion of the transfer matrix reads as T = exp (Ad) ∼ I + Ad, (A. [START_REF] Willis | Variational principles for dynamic problems for inhomogeneous elastic media[END_REF] which can be inverted to provide

This expression does not ensure the reciprocity of the material and the diagonal terms are not opposite each other. Taylor's expansion, at least of first order, is not a sufficiently robust tool to derive effective parameters, when compared to Padé's approximation.

Appendix B. Elementary transfer matrices

This section details the different elementary transfer matrices that are used to evaluate the total transfer matrix T.

Appendix B.1. Propagation in a duct of length l

The elementary matrix T l that connects the state vectors at two locations separated from a length l in a straight duct of section S reads as

where k = ω/c = ω/ K/ρ is the wavenumber, K = K/S, ρ = ρ/S and Z = K ρ are respectively the reduced bulk modulus, density and impedance.

Appendix B.2. Impedance in parallel

A flow split at a position s associated with the continuity of pressure is modeled by an impedance Z HR in parallel leading to an elementary matrix T HR

In particular, the impedance of a side branch HR to a duct of radius r, assumed to be a point-like resonator, is [START_REF] Merkel | Control of acoustic absorption in one-dimensional scattering by resonant scatterers[END_REF]

. Impedance in series

A pressure drop at a position s associated with the continuity of the normal velocity (or flow) is modeled by an impedance Z p in series leading to an elementary matrix T p

In particular, the impedance of a h p -thick plate clamped in a circular duct of radius r and section S, assumed to be a point-like resonator, is [START_REF] Bongard | Acoustic transmission line metamaterial with negative/zero/positive refractive index[END_REF] Zp = -iωρ p h p S

where I n and J n are respectively the modified and regular Bessel functions of first kind and order n, and ρ p and k p are respectively the density and wavenumber of the plate. The wavenumber is k 4 p = ω 2 ρ p h p /D p , where

is the bending stiffness, with E p and ν p the Young's modulus and Poisson's ratio of the plate material respectively. Note that poroelastic and viscoelastic plates can be modeled with a complex and frequency dependent Young's modulus and/or density.

The plate impedance reduces to Zp = -iω192ρ p h p 1/ (k p r) 4 -5/384 /S at low frequencies, from which it follows that Zp varies like 1/ω when ω → 0 because a plate tends to a rigid wall ( Zp → ∞) at low frequency, like the inverse of a Lorentzian around the resonance frequency, and like ω otherwise.

Appendix B.4. Viscothermal losses

Circular ducts are considered all along this article. The boundaries give rise to viscothermal losses from viscous and thermal skin depths. Assuming that only plane waves propagate in a cicular duct of radius r, the effective complex and frequency

Appendix E. Measurement of the direct and reverse orientation reflection and transmission coefficients

Let us label the upstream and downstream 1/4 in Grass microphones 1, 2, 3, and 4 from the loudspeaker to the anechoic end. The pressure p j is thus recorded at location x j , the first interface of the structure is located at x s and the structure is of length d. A method based on the scattering matrix is prefered, because anechoic termination is never fully anechoic and thus back propagating waves are always present due to reflection at the end of duct [START_REF] Niskanen | Deterministic and statistical characterization of rigid frame porous materials from impedance tube measurements[END_REF]. The ingoing i and outgoing o waves at the interfaces of the samples (vanishing phase is imposed at these interfaces) can thus be reconstructed to form the first set of equations

Turning the sample over and repeating the procedure (second set of measurements are mark with '), gives another set of equations

These two sets of equations form an overdetermined system to solve for R + , R -, and T .

Appendix F. Recovery procedure of the effective parameters from the measured reflection and transmission coefficients

The two state vectors at both sides of the sample are related by

where diag is the diagonal matrix, Σ ± are the eigenvalues of A and V the corresponding eigenvector matrix. For a Willis material, the constitutive matrix of which is given in Eq. ( 2), Σ ± = ±iω χ 2 + ρ/K = ±iωσ,

, and

1/Kσ (σ -χ) /σ -1/Kσ (σ + χ) /σ . (F.3)


Venkata Suresh Vajrala

Valentin Saunier

Lionel Nowak

Emmanuel Flahaut

Christian Bergaud

Ali Maziz

Carbon Nanofiber/PEDOT Based Macro-porous Composite For High Performance Multifunctional Neural Microelectrode

come   L'archive ouverte pluridisciplinaire

Brain study methods mainly rely on neurochemical detection and electrophysiology, respectively to study the molecular dynamics [1] and electrical activity [2] in the brain cellular environment. Since the brain functioning relies on careful orchestration of both neurotransmitter and neuroelectric activities, coupled monitoring of them holds great promise in elucidating the brain functioning mechanisms and its related diseases e.g. Parkinson's, epilepsy, etc [3]. Most of the conventional implantable neural interfacing technologies are comprised of noble metals on silicon/ polymer substrates. They suffer from inferior electrical properties, mechanical tissue-electrode mismatch, incompatible surface area, and low signal-to-noise ratio [4]. To address the need, researchers around the world are developing innovative soft materials with multimodal functionalities, especially the organic flexible materials such as conductive polymers and carbon allotropes, to fuel the next-generation multimodal neural electrode devices [5].

Here in this work, we present a new macro-porous, conductive, and stable nanocomposite, made up of carbon nanofibers (CNFs) and PEDOT polymer, coated on a flexible neural microelectrode array (Fig. 1A-B). A fully controlled one-shot electrodeposition strategy was developed using chronopotentiometry technique, to coat flexible Au-microelectrode surfaces with macro-porous CNF/PEDOT nanocomposite. The oxidized CNFs were used as a dopant and infused within the conductive PEDOT, electrochemically, in such a way that their excellent electronic, mechanical, chemical properties, and resulting combined performances can be exploited to yield better quality neural recordings, electrical stimulation and neurotransmitter detection. At first, CNFs were oxidized to be dispersible in water (unlike raw CNFs) by vortex and sonication. Later, CNF/PEDOT composite was deposited on Au-electrodes galvanostatically, resulted in a 3D porous and fibrous network (Fig. 1B-D). Then, CNF/PEDOT deposits were characterized through electrochemical impedance spectroscopy (EIS) and pulse current injection. The composite microelectrodes showed low impedance, 16 ± 2 kΩ for a typical electrode area of 1250 µm 2 , at 1kHz (value corresponding to the neuron firing frequency) (Fig. 1E). Voltage transient responses were measured, at continuous 1ms-biphasic current pulses (cathodic first) in biologically relevant aCSF media, resulted in a charge-injection limit value of 7.6 ± 1.3 mC/cm² (Fig. F). Among all reported organic nanostructured composite materials that were deposited on flexible metallic substrates, CNF/PEDOT displayed superior electrical properties and charge injection capabilities [5]. Next, the capability of CNF/PEDOT composite electrode for in-vitro Dopamine (DA) sensing was evaluated by chronoamperometry, at 130mV vs SCE (oxidation potential of DA). The current responses submitted to DA injections resulted in linear regression (Fig. 1G,R²>0.99), with a high sensitivity of 13.4pA/nM.µm², compared to similar electrochemical sensors [6]. Overall, these electrodes displayed excellent electrical properties: low impedance (16 ± 2 MΩ.µm2 at 1 kHz) and high charge injection capabilities (7.5 ± 1.3 mC/cm²) making them promising candidates for bidirectional electrophysiology, while being capable to detect neurotransmitters like dopamine. Finally, integrated upon flexible neural probes, these composite microelectrodes were tested in-vivo on mouse brain slices as recording and stimulating electrodes (Fig. 1H). They displayed excellent performances, being capable to record spontaneous activities of neurons (sharp wave-ripples and single unit action potentials) and allowed to stimulate electrically the brain slice tissues, to obtain safe and stable evoked potentials. The obtained results show a great prospect for CNF/PEDOT composites for developing next-generation microelectrodes for applications in neural therapies.

Figure 1 .

 1 Figure 1. (A) Optical micrograph of flexible neural probe. (B) CNF/PEDOT deposition on 4 Au-microelectrodes of the implant. SEM images, (C) top view and (D) crosssectional view of the composite electrode. (E) EIS measurement on modified

Acknowledgments

Agence Nationale de la Recherche (ANR-15-CE19-0006 and ANR-19-CE19-0002-01) and French RENATECH network.


Grégoire Pigeon

Pierre Durand

Valéry Masson

Evaluating parameterization of anthropogenic heat release in urban land surface scheme from field measurements and energy consumption inventory over Toulouse during CAPITOUL

niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

The anthropogenic heat releases (Q F ) are additional sources of energy in urban areas that must be taken into account in surface schemes dedicated to these areas. The Q F term has been studied with different methodology. The first one is based on energy consumption inventory [START_REF] Sailor | A top-down methodology for developing diurnal and seasonal anthropogenic heating profiles for urban areas[END_REF]. This methodology gives access to statistical values of Q F over large urban areas. It is very difficult to assess realtime spatio-temporal variability of Q F with this methodology. The second methodology is based on measurements of components of the surface energy balance (SEB) [START_REF] Masson | The Canopy and Aerosol Particles Interaction in TOulouse Urban Layer (CAPITOUL) experiment: first results[END_REF][START_REF] Offerle | Heat storage and anthropogenic heat flux in relation to the energy balance of a central European city centre[END_REF]. This methodology can give access to the diurnal temporal variability but is limited for evaluation of spatial variability of Q F . It is based on estimation of residual and has the inconvenient to accumulate all errors of measurement of the other terms of the SEB.

In the surface schemes dedicated to urban areas, the different sources of Q F are differently integrated. The traffic and industrial releases are an additional source of sensible heat flux (Q H ) and latent heat flux (Q LE ). The domestic heating which varies with season can be parameterized. The objective of this paper is to evaluate this parameterization in the TEB scheme [START_REF] References Masson | A physically-based scheme for the urban energy balance in atmospheric models[END_REF] against the CAPITOUL dataset. The CAPITOUL project [START_REF] Masson | The Canopy and Aerosol Particles Interaction in TOulouse Urban Layer (CAPITOUL) experiment: first results[END_REF]) is focused on the study of the SEB of the centre of the city of Toulouse (France) over a whole year period (field campaign from February 2004 to February 2005) and its consequences on the urban boundary layer. Both methodologies of estimation of Q F were applied in the CAPITOUL project.

ESTIMATION OF ANTHROPOGENIC HEAT FROM MEASUREMENTS

In the old core of Toulouse, a 30 m tower was set up on a terrace roof and instrumented with a 3D sonic anemometer, a Licor 7500 (water vapour and carbon dioxide) and a CNR1 (four components of the radiation budget). Turbulent fluxes of sensible heat (Q H ) and latent heat (Q LE ) are computed with the eddy-correlation technique. The residual of measurements is computed as: R=Q*-(Q H +Q LE ). Considering the surface energy balance drawn by [START_REF] Oke | The urban energy balance[END_REF] and neglecting the advection effects, R=∆Q S -Q F where ∆Q S is the storage inside the urban canopy. This term has been computed for daily period over the whole year of the campaign (Figure 1). The daily residual presents a seasonal evolution. It has negative values during end of winter 2004 and winter 2005. During summer the residual can be negative or positive but with lower absolute values. The negative values observed during winter time indicates that the mean residual over one day period is mainly driven by opposite of Q F . On the contrary, the strongest positive values indicate that R can be driven by positive storage during warmer period. It can be the case during a few days when the climatic conditions evolves from a overcast period to a period with strong solar forcing.

ESTIMATION OF ANTHROPOGENIC HEAT FROM INVENTORY

A database has been constituted to evaluate Q F over the whole agglomeration. The original data are:

annual mean of total daily traffic counting for all street sections with more than 2000 vehicles per day available in a Geographical Information System (GIS) and provided by the city administration, consumption factors of vehicles typical of urban areas provided by the COPERT program, electricity consumption around the measurement area and over 14 areas covering the whole agglomeration with a sampling rate of 10 minutes provided by the French national electricity company, annual mean of consumption of other source of energy (gas, fuel, wood) for the different types of housing provided by the regional observatory of energy, number of building of each type of housing in each district of the agglomeration (spatial resolution variable between a few hundreds of meters in the city centre to a few kilometres in suburbs), provided by the French national statistics administration.

The original data have been integrated in a GIS. The annual Q F terms for an area of 500 m around the measurement site have been reported in Table 1 for the different sources (traffic, electricity, other sources) and compared with the mean annual residual of flux measurements (see section 1). On an annual basis, the observations are very comparable with the inventory data. On an annual basis, the mean residual of observations will not be affected by ∆Q S which should be close to 0, otherwise the ground temperature would increase abnormally. In the TEB model, three sources of anthropogenic heat releases are considered: the traffic, the industries and the domestic heating.

To account for traffic and industries, 2 additional sources are added to Q H (and Q LE ). These terms are prescribed by the user when preparing a simulation. Domestic heating is simulated by limiting the evolution of the internal temperature inside the building with a minimum value of 19°C. This internal temperature is used to compute the conduction trough the internal layer of walls or roof [START_REF] Masson | Evaluation of the Town Energy Balance (TEB) scheme with direct measurements from dry districts in two cities[END_REF]. In winter time the internal temperature generally remains steady and is warmer than inner walls or roof temperature. As a consequence the flux of heat between the internal building and the first layer of walls or roof is directed to the outside of the building and represents the domestic heating.

Surface characteristics

The city administration has let at our disposal a GIS describing the agglomeration. It is composed of maps of buildings (2D and 3D), streets, green parks. It also contains orthorectified aerial photography of the agglomeration at a spatial resolution of 0.25 m. This data set has been processed in order to retrieve the information useful for preparation of simulation In order to perform a simulation of TEB comparable with observations from the tower, the characteristics of an area of 500 m around the tower has been prescribed to the model (Table 2). . The roofs are constituted of 3 different layers: two 0.025 m layers of red tiles also made with cooked soil with the same characteristics than the red bricks, a 0.03 m layer of wood whose heat capacity is 2.201 10 6 J m -3 K -1 and thermal conductivity is 0.2 W m -1 K -1

.

Simulation of a cold winter period

A 9 days period from 20 to 28 February 2005 has been simulated with the a surface model using TEB to compute evolution of the urban system (roofs, walls, roads) and the ISBA model (Noihlan and Planton, 1989) for all other covers (here vegetation). The model is forced by the observations at the tower of incoming solar radiation, incoming infra-red radiation, air temperature, air pressure, specific humidity, wind, and precipitation. It computes then all other fluxes of the SEB. The selected period is characterized by an evolution from an overcast situation on the first day to more sunny days (Figure 3). The air temperature is low on average and no trend can be observed for the whole period. There are a few small rain events during the period. Daily cycle of SEB terms computed by the model and averaged over the period are presented on Figure 4 and compared with the observations from the tower. The first graph of Figure 4 represents the net radiation. . The third graph presents the latent heat flux comparison. Q LE is very low during the period. However, small fluxes are present in both observations and simulation and correspond to rain events during the period. Finally the bottom graph of Figure 4 represents the residual for both observations and simulation and also Q F (all source of energy) that has been computed from the simulation. The residual is globally well computed by the simulation. The Q F term (blue triangle) presents a very smooth evolution which results from the cycle of storage of the solar energy by walls or roofs.

Residual from obs. (W m

-2 ) Residual from model (W m -2 ) Q F from model (W m -2 ) -95
-92 87 The mean values of residual (observations and simulation) and Q F over the period have been computed and are reported in Table 3. We can see that the model and the observations are close. The Q F term computed from the model is slightly lower in absolute value but of the same order than the other.

CONCLUSION

The surface energy balance measured over the centre of a European city over a whole year indicates that the anthropogenic heat flux is a quantitative source of energy. This term can reach up to 150 W m -2 during winter period. It has been compared to inventory of energy consumption (traffic, electricity, etc…) and the results are very coherent. A simulation of cold winter period has been performed with the TEB-ISBA model and compared with observations. The model presents a good ability to simulate winter conditions. The Q F term computed from the model is also coherent with observations.

Figure 1 :

 1 Figure 1: Mean daily residual (Q*-(Q H +Q LE )) for one year period. This term represents opposite of daily anthropogenic heat releasesThe residual is very well correlated with the air temperature as can be seen on Figure2with a correlation of 0.77 over the whole year period.

Figure 2 :

 2 Figure 2: Variation of mean daily residual in function of mean air temperature. The correlation between the 2 signals is 0.77.

  Mean annual value of the different sources of anthropogenic heat and comparison with estimation with mean residual of observations

2

  with schemes like TEB. A grid covering the whole agglomeration with a resolution of 100 m has been constituted and contains: fraction occupied by buildings, fraction occupied by roads, fraction occupied by trees, fraction occupied by grass, height of buildings.

  surface characteristics over 500 m radius area around measurement station Walls and roofs are represented by three layers in TEB. Characteristics of these layer have been estimated from observations. Walls are constituted from red bricks of cooked soil of a depth of 0.3 m (3 layers of 0.1 m in TEB). Heat capacity of red brick is 1

Figure 3 :

 3 Figure 3: Incoming solar radiation and air temperature during the simulation period.

Figure 4 :

 4 Figure 4: Comparison between simulation and observation for the different components of the energy balance. Mean diurnal cycle on the period 20-28 February 2005.Observations and the simulation are very comparable for that term. The model has notably a good ability to compute the upward infra-red

Table 3 :

 3 Mean value of residual from observations, residual from model and Q F from model during the period of simulation (20-28 February 2005). Residual is defined as R=∆Q S -Q F .


Sedina Tsikata

Jörg Hermann

Sylvain Iséni

Stéphane Pellerin

Stéphane Pellerin Electrons

Electrons: probing their properties and collective dynamics

Introduction

The complexity of the plasma state requires the use of experimental tools which can provide information on particle properties and dynamics. This is essential to understand processes such as:

• energy exchange • ionization dynamics • development of plasma instabilities and particle transport, among other features. Information on these aspects is key to the development of theoretical models and numerical codes for all plasmas. Regardless of the plasma state and type (magnetized or unmagnetized, hot or cold, high or low pressure) electrons play a key role in determining the discharge features. Diagnosing their properties and dynamics is therefore critical.

Non-invasive versus invasive techniques

Thomson scattering is electromagnetic wave scattering on free charges. Two regimes -incoherent and coherent -provide different and complementary insights into particle properties and dynamics.

Schematic view of an early CTS setup used for the investigation of ion acoustic turbulence in a plasma column using infrared scattering [11] Components of a full ITS implementation on a hollow cathode plasma [7], with the TGS replaced with a volume Bragg grating (VBG) [START_REF] Glebov | Proc. SPIE[END_REF] and single monochromator to filter stray light. This considerably reduces photon collection losses and allows a detection limit as low as 10 16 /m 3 to be achieved. Recent 2D PIC simulation results confirming the coexistence of two small-scale modes in an ExB discharge configuration: the electron cyclotron drift instability along the ExB drift direction and the ionion two stream instability along the electric field direction [START_REF] Hara | Electrons: probing their properties and collective dynamics Diagnostics optiques[END_REF]

  Example of standard triple grating (TGS) spectrometer assembly used for redistribution of scattered light[5] InvasiveNon-invasive example: Langmuir probes Pros • simplicity of implementation • low cost • high spatial and temporal resolution may be achievable Cons • assumptions necessary (Maxwellian EVDF) • unsuited to magnetized plasma regions • perturbative: physical size, sputtering • non-direct interpretation generally necessary example: incoherent Thomson scattering Pros • direct information often accessible • suitable for investigations in magnetized plasma regions • high spatial and temporal resolution, depending on implementation Cons • complexity of implementation • high cost • optimization of implementation required to maximize signal-noise ratio and minimize stray light Coherent Thomson scattering has been used for decades in the investigation of turbulent fluctuations in tokamaks and pinch devices. In recent decades, it has also proven key to the identification and characterisation of instabilities in lowtemperature plasmas. Study of electron density fluctuations in different tokamak operating modes: e.g. reversed shear [1] Coherent Thomson scattering implementations in hot plasmas NSTX implementation for study of identification of modes driving anomalous electron transport (e.g. electron temperature gradients) [2] examples from hot plasmas probing wave at 280 GHz Identification of key scaling laws on Tore Supra [3] and other plasmas Intensity of density fluctuations n e S(k, ω) measured within a collisionless shock wave (Mach number 2.5, deuterium plasma) in a theta pinch [4] Incoherent Thomson scattering (ITS) Coherent Thomson scattering (CTS) • electron density • electron drift velocity • correlated fluctuations in presence of wave activity Incoherent Thomson scattering in low-temperature plasmas Coherent Thomson scattering in low-temperature plasmas Example of Thomson scattered spectrum observed using a highperformance ITS system on the linear plasma machine Pilot-PSI [6] for the study of plasma-surface interactions. Calibrated electron density values were measured in the range 10 20 -10 21 /m 3 . Simultaneous observation of Thomson and Raman spectra in investigations of atmospheric pressure plasma jets [9] Electron density and temperature variations [10] (left) during pulsing of a planar magnetron Ar plasma at 7.5 mTorr (right) ITS provides access to detailed measurements on electron properties, and in recent years, its application to low-temperature plasmas has expanded considerably. Implementations of CTS have provided new information on microturbulence across a range of magnetized plasmas. Indirect information on plasma large-scale structuring can also be obtained from certain implementations. (left) CTS spectrum showing identification of the MHz-frequency electron cyclotron drift instability in an ExB annular plasma configuration [12] and (right) experiment setup (left) Dispersion relation and scaling law identification for ion acoustic turbulence in a hollow cathode made using CTS