Ethan Pickering

Aaron Towne

Peter Jordan

Tim Colonius

Resolvent-based modeling of turbulent jet noise

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I. INTRODUCTION

The goal of this work is to develop jet-noise models founded upon the physics of turbulent flows that are both low-rank and that provide insights into the mechanisms primarily responsible for noise generation. Resolvent analysis (McKeon and Sharma, 2010), also known as input-output analysis [START_REF] Jovanović | From bypass transition to flow control and data-driven turbulence modeling: An input-output viewpoint[END_REF], provides a useful framework for achieving these goals. The central idea of the resolvent framework is similar to that of an acoustic analogy [START_REF] Goldstein | A generalized acoustic analogy[END_REF][START_REF] Lighthill | On sound generated aerodynamically i. general theory[END_REF], whereby a forcing term, related to the statistics of the hydrodynamic near-field turbulence, gives rise, through a linear operator, to the observed far-field sound. The resolvent framework differs in two important ways. First, the operator is decomposed into its singular components that represent the maximal amplification between the forcing and the output. This permits the resulting acoustic field to be described as low rank, and thus limits the forcing statistics that must be modeled. Secondly, the full linearized Navier-Stokes equations are used as the propagator, and we seek a modal basis that represents both near and far-field coherent structures.

Before recent advances in computational power, the idea of modeling both the hydrodynamic component along with the acoustics would have been seen as both a Postdoctoral Associate; ) epickeri@mit.edu b Assistant Professor of Mechanical Engineering c CNRS Research Director d Frank and Ora Lee Marble Professor of Mechanical Engineering unnecessary and computationally taxing. However, the ability to resolve both components of the flow is in fact a benefit. Starting with the experimental findings of [START_REF] Mollo-Christensen | Jet noise and shear flow instability seen from an experimenter's viewpoint[END_REF] and [START_REF] Crow | Orderly structure in jet turbulence[END_REF], it has become clear that coherent structures in the hydrodynamic near-field are directly responsible for farfield sound [START_REF] Jordan | Wave packets and turbulent jet noise[END_REF]. These structures take the spatio-temporal form of wavepackets and have been found to be the dominant source for aft-angle sound [START_REF] Jordan | Wave packets and turbulent jet noise[END_REF], as well as partial contributors to sideline noise [START_REF] Jeun | Input-output analysis of Mach 0.9 jet noise[END_REF][START_REF] Papamoschou | Wavepacket modeling of the jet noise source[END_REF]. These wavepackets may be linked to the early works of [START_REF] Crighton | Stability of slowly diverging jet flow[END_REF] [START_REF] Michalke | Instability of a compressible circular free jet with consideration of the influence of the jet boundary layer thickness[END_REF]) who hypothesized that coherent structures could be described as linear instability modes of the mean flow via modal analysis. However, it has now become apparent that the correct representation of wavepackets is that of a highly-amplified response to turbulent fluctuations, which is directly found via the resolvent framework.

Resolvent analysis uses the Singular Value Decomposition (SVD) to decompose the linear resolvent operator, identifying sets of orthogonal forcing/input and response/output modes, and ranking them in terms of the corresponding energetic gain between the forcing and response. This is particularly important as it allows our model to self-select the most relevant amplification mechanisms for noise generation. This allows for a natural truncation of the resolvent basis that produces a reducedorder model, or in other words, a reduced-rank acoustic analogy.

Several studies have applied resolvent analysis to develop low-rank jet models [START_REF] Cavalieri | Wavepacket models for jet dynamics and sound radiation[END_REF][START_REF] Jeun | Inputoutput analysis of high-speed axisymmetric isothermal jet noise[END_REF][START_REF] Lesshafft | Resolvent-based modelling of coherent wavepackets in a turbulent jet[END_REF]. The existence of relatively low-rank responses in round, turbulent jets was shown by [START_REF] Schmidt | Spectral analysis of jet turbulence[END_REF], with significant agreement between structures found through spectral proper orthogonal decomposition [START_REF] Towne | Spectral proper orthogonal decomposition and its relationship to dynamic mode decomposition and resolvent analysis[END_REF] (SPOD) of a high-fidelity experimentally-verified large-eddy simulations (LES) of jets (Brès et al., 2017 , 2018). Of particular relevance to this study are "acoustic resolvent modes" induced by performing resolvent analysis with an output domain defined over a region where fluctuations are purely acoustic. Through implementation of an acoustic output domain, resolvent analysis is able to filter out energetic, but acoustically irrelevant structures in the near-field. [START_REF] Jeun | Inputoutput analysis of high-speed axisymmetric isothermal jet noise[END_REF] performed such an analysis and found that for a Mach 1.5 jet, at Strouhal number St = 0.33 and azimuthal wavenumber m = 0, that the first resolvent mode reconstructs 57% of the acoustic energy, but through inclusion of the next 23 resolvent modes the reconstruction improved to 70% of the acoustic energy. This study looks to perform a similar analysis, in that we compute many acoustic resolvent modes and assess how well they reconstruct the acoustic energy. However, we also look to reduce the rank of the far-field significantly with the use of an eddy-viscosity model [START_REF] Pickering | Optimal eddy viscosity for resolvent-based models of coherent structures in turbulent jets[END_REF] and generalize the performance of the resolvent framework across frequencies St = 0 -1, azimuthal wavenumbers m = [0 -2], and for two turbulent jets at Mach numbers of 0.9 and 1.5.

For a resolvent jet model to fully reconstruct flow statistics, and in this case those of the acoustic field, a resolvent-based model must incorporate sub-optimal modes [START_REF] Schmidt | Spectral analysis of jet turbulence[END_REF] and correctly describe correlations (i.e. covariance) between modes inherent to turbulent flow [START_REF] Towne | Resolventbased estimation of space-time flow statistics[END_REF]. These correlations are analogous to the concept of "jittering", used to describe temporal modulations of acoustic sources, that has been shown to be critical for accurately describing the acoustic field in turbulent jets [START_REF] Cavalieri | Jittering wave-packet models for subsonic jet noise[END_REF]. In our approach, such temporal modulations, or jittering, may be represented through second-order statistics via the statistical representation of the resolvent operator [START_REF] Towne | Spectral proper orthogonal decomposition and its relationship to dynamic mode decomposition and resolvent analysis[END_REF])

S yy = RS f f R * , (1)
where S yy and S f f are the cross-spectral density (CSD) tensors of the response and the forcing respectively and R is the resolvent operator. This equation shows that if the forcing CSD, describing spatial correlations, can be modeled [START_REF] Towne | A statistical jet-noise model based on the resolvent framework[END_REF][START_REF] Zare | Colour of turbulence[END_REF], then the resolvent operator identically reconstructs the flow statistics, S yy . If the forcing were spatially uncorrelated, S f f = Λ, where Λ is a diagonal matrix, then the eigenvectors of S yy , which are the SPOD modes of the outputs, are aligned with the eigenvectors of RR * [START_REF] Towne | Spectral proper orthogonal decomposition and its relationship to dynamic mode decomposition and resolvent analysis[END_REF], or the response modes of the resolvent operator, R. However, the uncorrelated condition is rarely met, resulting in discrepancies between resolvent and SPOD modes that must be resolved through modeling S f f .

One approach for modeling, at least partially, S f f has been through the inclusion of a turbulence model to the resolvent operator. This approach has been implemented via an eddy-viscosity model in several flow configurations, from wall-bounded [START_REF] Hwang | Amplification of coherent streaks in the turbulent couette flow: an input-output analysis at low Reynolds number[END_REF][START_REF] Morra | On the relevance of Reynolds stresses in resolvent analyses of turbulent wall-bounded flows[END_REF] to free shear flows [START_REF] Pickering | Optimal eddy viscosity for resolvent-based models of coherent structures in turbulent jets[END_REF]. The latter study, quantifying the effect on turbulent jet modeling, found that the use of an eddyviscosity model (utilizing only quantities available from RANS models) significantly improved the agreement between SPOD and resolvent modes, thus reducing the effort required to model the effective S f f by diminishing the magnitude of the off-diagonal terms. We utilize the same eddy-viscosity model in the present work to better model the acoustic field.

This paper explores an approach to describe the coupling between resolvent modes that is necessary for reconstructing the acoustic field with a minimal set of resolvent modes. The coupling provides directional and energetic variability in acoustic radiation inherently important for noise prediction [START_REF] Cavalieri | Jittering wave-packet models for subsonic jet noise[END_REF]. Determination of the coupling between modes is performed by leveraging an ensemble of LES realizations which are projected on to a limited (i.e. low-rank) set of acoustic resolvent modes. From these projections we attain a (drastically) reduced-order cross-spectral density between the retained modes-a Hermitian, frequency-dependent matrix of size n × n that accurately represents the acoustic field.

Organization of the manuscript is as follows. We first briefly describe the LES databases used, the main details pertaining to resolvent analysis, and present the statistical description of the resolvent framework for reconstructing the acoustic field and estimating the reduced order covariance matrix in § II. In § III we present resolvent modes and LES reconstructions in the resolvent basis for one frequency-wavenumber pair for the Mach 1.5 jet before generalizing the approach to both jets over St = [0, 1] and m = [0, 2], and to both the near-and far-field acoustic regions. In the near-field section we compare the impact of including a RANS eddy-viscosity model to the resolvent operator and find it presents a significantly more efficient resolvent basis. We then present results for the far-field, along an arc at 100D from the nozzle, and show that reconstructions for both jets may be found using only the optimal resolvent mode. Finally, we conclude with a discussion on how the correct forcing coefficients may be estimated for a predictive jet noise model.

II. METHODS

A. Large Eddy Simulation database

The LES database and resolvent analysis are fully described in [START_REF] Schmidt | Spectral analysis of jet turbulence[END_REF] and [START_REF] Towne | Spectral proper orthogonal decomposition and its relationship to dynamic mode decomposition and resolvent analysis[END_REF]. Transonic (Mach 0.9) and supersonic (Mach 1.5) jets were computed using the flow solver "Charles"; details on numerical methods, meshing, and subgrid-models can be found in Brès et al. (2018) and [START_REF] Brès | Unstructured large-eddy simulations of supersonic jets[END_REF] along with validation cases conducted at PPRIME Institute, Poitiers, France for the Mach 0.9 jet (Brès et al., 2018). The Mach 0.9 and 1.5 jets have Reynolds numbers of Re j = ρ j U j D/µ j = 1.01 × 10 6 and Re j = 1.76 × 10 6 , respectively, where subscript j gives the value at the center of the jet, ρ is density, µ is viscosity, and M j is the Mach number M j = U j /c j , with c j as the speed of sound at the nozzle centerline.

Throughout the manuscript, variables are nondimensionalized by the mean jet velocity U j , jet diameter D, and pressure ρ j U 2 j , with the resulting equation of state p = ρT γM 2 j , with T denoting temperature and γ the ratio of specific heats. Frequencies are reported in Strouhal number, St = f D/U j , where f is the frequency in Hertz. The database consists of 10,000 snapshots separated by ∆tc ∞ /D = 0.2 and 0.1 for the M j = 0.9 and M j = 1.5 jets, respectively, with c ∞ as the ambient speed of sound, and interpolated onto a structured cylindrical grid x, r, θ ∈ [0, 30] × [0, 6] × [0, 2π], where x, r, θ are streamwise, radial, and azimuthal coordinates, respectively. Variables are reported by the vector

q = [ρ, u x , u r , u θ , T ] T , (2)
where u x , u r , u θ are the three cylindrical velocity components.

To generate an ensemble of flow realizations for computing statistical averages, the LES database of 10,000 snapshots is segmented into bins of 256 snapshots, with an overlap of 75%, and under the implementation of a Hamming window, resulting in 153 realizations of the flow. Each realization is then decomposed in the azimuthal direction and in time. The temporal decomposition provides a resolution of St = 0.026 and St = 0.0217 the M j = 1.5 and M j = 0.9 jets, respectively, and the azimuthal decomposition is valid up to m = 68, however, the acoustically relevant azimuthal wavenumbers are much smaller [START_REF] Juve | Filtered azimuthal correlations in the acoustic far field of a subsonic jet[END_REF] and only azimuthal wavenumbers m = [0 -2] are considered in this paper.

Considering the LES database only extends to r/D = 6, we implement a Kirchhoff surface (as described in [START_REF] Freund | Noise sources in a low-Reynolds-number turbulent jet at Mach 0.9[END_REF])), to the azimuthally and temporally transformed realizations of the flow. In doing so, we create an ensemble of far-field realizations located along an arc, with angle φ, of 100D from the nozzle at each frequency and azimuthal wavenumber. As done in [START_REF] Brès | Unstructured large-eddy simulations of supersonic jets[END_REF], and associated experiments (Schlinker et al., 2009 , 2008), we specifically compute the acoustics for the aft-angle sound from φ = 100 -160 and find our acoustic far-field is in close agreement (within 2dB) with the far-field of the LES calculation.

B. Resolvent analysis

For the round, statistically-stationary, turbulent jets considered in this manuscript, the compressible Navier-Stokes, energy, and continuity equations are linearized via a standard Reynolds decomposition and Fourier transformed both in time and azimuthally to the com-pact expression

(iωI -A m )q m,ω = L m,ω q m,ω = f m,ω , (3)
where ω = 2πSt is the frequency, m is the azimuthal wavenumber, I is the identity matrix, A m is the frequency independent linear operator, L m,ω is the total forward linear operator, q m,ω is the response in each variable, and f m,ω constitutes the nonlinear forcing. Meanflow quantities used in the operator are derived from a RANS model, fitted closely to the LES mean flow. Although the mean flows are similar, the computation of a RANS model, using the standard κ -closure equations, also provides an eddy-viscosity field that may be included in the resolvent operator. This is done following results of [START_REF] Pickering | Optimal eddy viscosity for resolvent-based models of coherent structures in turbulent jets[END_REF] that presented substantially improved agreement between SPOD and resolvent modes with the inclusion of an eddy-viscosity model. The eddyviscosity used here are computed as µ T = cC µ k 2 / , where c and C µ are scaling constants (c = 0.2, C µ = 0.0623 for the M j = 0.9 and C µ = 0.0554 for M j = 1.5 jet), k is the turbulent kinetic energy field, and is the turbulent dissipation field. We stress that the constants of C µ are only altered here for a close comparison to the LES and to allow for a demonstration of the ability of strictly RANS quantities to be used for computing accurate resolvent modes. These values should not be considered general or as recommended values of C µ for future studies. Continuing with the derivation of the resolvent/input-output operator, we rewrite equation (3) by moving L m,ω to the right-hand side to give,

q m,ω = L -1 m,ω f m,ω = R m,ω f m,ω , (4)
where R m,ω = L -1 m,ω is the standard resolvent operator. To then specify particular domains for both the response and forcing, we may write the above as

q m,ω = R m,ω Bf m,ω , (5)
and define the output variable

y m,ω = Cq m,ω , (6)
where B and C are input and output matrices. The latter matrix, C, is used to isolate the acoustics in the near-field, or propagate fluctuations to the far-field. Each of these cases are detailed in Appendix A. Inserting equation (5) into equation ( 6) gives the input-output relationship,

y m,ω = CR m,ω Bf m,ω = H m,ω f m,ω , (7)
where H m,ω = CR m,ω B is the resolvent input-output operator from f m,ω to y m,ω . Then by introducing the compressible energy norm of [START_REF] Chu | On the energy transfer to small disturbances in fluid flow (Part I)[END_REF],

q 1 , q 2 E = q * 1 diag T γ ρM 2 , ρ, ρ, ρ, ρ γ(γ -1) T M 2 q 2 rdxdrdθ = q * 1 W q 2 , (8)
(where superscript * denotes the complex conjugate transpose) via the matrix W to the forcing and response (W f = W y = W ) the weighted resolvent input-output operator, Ĥm,ω is obtained:

Ĥm,ω = W 1/2 y H m,ω W -1/2 f . (9)
Resolvent modes may then be found by taking the singular value decomposition of the weighted resolvent inputoutput operator giving

Ĥm,ω = Ûm,ω Σ m,ω V * m,ω , (10)
where the optimal response and forcing modes are contained in the columns of

U m,ω = W -1/2 y Ûm,ω , with U m,ω = [u 1 m,ω , u 2 m,ω , ..., u N m,ω ], V m,ω = W -1/2 f Vm,ω , V m,ω = [v 1 m,ω , v 2 m,ω , ..., v N m,ω ], and Σ m,ω = diag(σ 1 m,ω , σ 2 m,ω , ..., σ N m,ω
) are the optimal gains [START_REF] Towne | Spectral proper orthogonal decomposition and its relationship to dynamic mode decomposition and resolvent analysis[END_REF]. The unweighted resolvent input-output operator may then be recovered as:

H m,ω = U m,ω Σ m,ω V * m,ω W f . (11)
C. Statistics

The statistics we are interested in are contained within the cross-spectral density tensor, which may be found for the desired output space by multiplying the resolvent equation by its complex conjugate transpose and taking the expectation [START_REF] Towne | Spectral proper orthogonal decomposition and its relationship to dynamic mode decomposition and resolvent analysis[END_REF]

y m,ω y * m,ω = H m,ω f m,ω f * m,ω H * m,ω , (12)
giving

S yy,m,ω = H m,ω S f f,m,ω H * m,ω , (13)
where S yy,m,ω and S f f,m,ω are the CSD tensors of the response and the forcing, respectively. For brevity, we drop the subscripts m and ω and note that all CSD tensors and resolvent matrices must be defined for specific m and ω pairs in the remainder of the manuscript.

As mentioned earlier, this representation shows that if the forcing CSD tensor is known, then the resolvent operator reconstructs the response statistics. However, the forcing CSD is generally unknown. There are at least two potential avenues for modeling it. The first is to directly model S f f . To aid in such modeling efforts, S f f may be computed directly from full LES data [START_REF] Towne | A statistical jet-noise model based on the resolvent framework[END_REF], or estimated from limited flow statistics [START_REF] Towne | Resolventbased estimation of space-time flow statistics[END_REF]. A second approach is to modify the resolvent operator by supplementing the governing linearized equations with an appropriately linearized turbulence model. In [START_REF] Pickering | Optimal eddy viscosity for resolvent-based models of coherent structures in turbulent jets[END_REF], an eddy-viscosity model was considered and LES data was used to determine an optimal eddy viscosity field that would align, insofar as possible, the modes of S qq (i.e. the full response statistics) with those of RR * (identical to HH * when C = B = I). They found this to substantially reduce the magnitude of the off-diagonal terms of S f f , at least as the near-field coherent structures were concerned, consequently simplifying the number of terms that must be modeled.

In this study, we combine both modeling approaches. We first utilize the eddy-viscosity approximation of [START_REF] Pickering | Optimal eddy viscosity for resolvent-based models of coherent structures in turbulent jets[END_REF] and then estimate a low-order approximation of the forcing CSD for the acoustic field. To do the latter, we return to equation ( 13) and expand the resolvent input-output operator through its singular value decomposition,

S yy = U ΣV * W f S f f W f V ΣU * (14)
and define a covariance matrix

S ββ = V * W f S f f W f V ,
where β is the projection of the forcing upon the resolvent input modes, β = V * W f f . This gives

S yy = U ΣS ββ ΣU * (15)
which can be rearranged to solve for the covariance matrix,

S ββ = Σ -1 U * S yy U Σ -1 . ( 16
)
In its current state, the covariance matrix is exact, maintaining a full size of the system and permitting approximately 10 11 degrees of freedom (i.e. S ββ ∈ C 5NxNr×5NxNr ). To obtain a low-rank model of S ββ from the LES data, we compute S ββ with a truncated set of n resolvent modes, Ũ ∈ C 5NxNr×n , as,

Sββ = Σ-1 Ũ * S yy Ũ Σ-1 . (17)
This reduces the size of the covariance matrix to n×n, in that the degrees of freedom are now drastically reduced to O(10 0 -10 1 ).

With Sββ , we may ask several questions: How well does Sββ reconstruct S yy in the truncated resolvent basis, where the reconstructed CSD is computed as

Syy = Ũy Σ Sββ Σ Ũ * y . (18)
May Sββ be further reduced (e.g. neglect off-diagonal terms) and can Sββ be modeled? For the latter two questions, we reserve discussion to § III C, where we both propose a forcing model and neglect the off-diagonal terms.

III. RESULTS

A. Near acoustic field

We begin by providing detailed results for a single frequency and azimuthal wavenumber pair of the M j = 1.5 jet using the RANS eddy-viscosity resolvent operator and a near-field acoustic C output matrix (r/D = [5,20] in the fluctuating pressure field, details provided in A 1). Figure 1 presents the first three resolvent modes computed with the restricted acoustic output domain and then recast in the full domain by setting C = I and calculating U q = L -1 V y for M j = 1.5, St = 0.26, and m = 0. The associated gain of these modes, normalized by the first resolvent gain, are [1, 0.17 decreasing with higher modes), indicating the first resolvent mode has at least six times the amplification to its associated forcing as the following resolvent modes.

The resolvent response modes show a particular pattern of acoustic beams. For the first mode there is a single, energetic acoustic beam, propagating at a shallow angle to the jet axis. The first suboptimal mode consists of two acoustic beams, similar to what was found by [START_REF] Jeun | Inputoutput analysis of high-speed axisymmetric isothermal jet noise[END_REF]. This pattern is shown by the next suboptimal mode, with three beams located at the perimeter of the first suboptimal. Although not shown, this behavior continues for further suboptimal modes.

Figure 2 compares three specific realizations of the m = 0, St = 0.26 field from the LES, q, to the threemode reconstructions of these fields found by projection. The reconstructions are found by q = Ũq α, where α = Ũ + * z W z z and z is an acoustic subset of the LES domain (r/D = [5, 6] of the pressure field) and Ũ + * z is the psuedoinverse projecting the LES domain z and resolvent output domain y. Further details using the psuedoinverse to project resolvent modes on other spaces is provided in Appendix B. From figure 2 we see that the three resolvent modes are able to accurately reconstruct the different radiation patterns evident in the LES realizations. Clearly there is constructive and destructive reinforcement amongst the three resolvent modes in order to produce the LES realizations.

For a more quantitative assessment of the ability of the resolvent modes to reconstruct the acoustic field, we compute and compare the power spectral density (PSD) of the acoustic field, which is located in the diagonal terms of S yy , by dB, ∆dB = 10log diag( Syy -S yy ) , at r/D = 6 in figure 3. This is again performed for St = 0.26, m = 0, but is now averaged over all k = 153 realizations. In addition to the three resolvent mode set, results are also shown for 5 and 10 mode sets. With just three modes we see that the peak directivity is well captured, with minor improvements (and diminishing returns) in the off-peak directivity with increasing numbers of modes.

We now extend our comparison to Strouhal numbers ranging from 0 to 1 and azimuthal wavenumbers 0-2 and assess the overall ability of the truncated resolvent basis to reconstruct the acoustic field. Figure 4 compares the PSD from the LES to its n-rank resolvent-basis reconstructions with n = 1, 3, 5, 10, and 20. The rank-1 model results are similar to those of Sinha et al. [START_REF] Sinha | Wavepacket models for supersonic jet noise[END_REF], who used parabolized stability equations and projected onto the first SPOD-mode at each St -m pair. However, we show here that once additional modes are included, the reconstructions are substantially improved: the 20-mode model shows close agreement with the LES for all frequencies and azimuthal modes, while even the 3-mode model is quantitatively accurate for m = 0 and m = 1.

To quantify the error between the reconstructed and LES PSD, as well as succinctly present similar results for the M j = 0.9 case, we propose the error metric,

= St x (PSD LES -PSD Recon ) 2 PSD 2 LES dStdx, (20)
and present them in figure 5. Shown are the errors for both Mach numbers, three azimuthal wavenumbers, modes retained (i.e. n = [1,3,5,10,20]), as well as with (filled symbols) and without (hollow symbols) the use of the RANS eddy-viscosity field. The filled symbols for the M j = 1.5 case provide a reference between the quantitative measure and the qualitative visualization of figure 4. Comparing the two jet regimes, it is apparent that a larger number of modes are required to reconstruct the near acoustic field of the M j = 0.9 jet. For example, about 10 modes are necessary to obtain a similar quantitative match as compared to just three modes at M j = 1.5. This is consistent with multiple past observations where the M j = 0.9 jet possesses non-negligible contributions from suboptimal modes that are correlated, or, as described in the time domain, as being linked via "jittering" [START_REF] Cavalieri | Jittering wave-packet models for subsonic jet noise[END_REF], thus requiring many modes to reconstruct the acoustic field [START_REF] Freund | Turbulence and soundfield POD analysis of a turbulent jet[END_REF][START_REF] Towne | Stochastic and nonlinear forcing of wavepackets in a Mach 0.9 jet[END_REF]. Figure 5 also shows significant improvements in reconstructing the near-acoustic field when including the RANS eddy-viscosity field when compared to results using a constant turbulent Reynolds number of Re T = 3 × 10 4 . We note that previous results [START_REF] Pickering | Optimal eddy viscosity for resolvent-based models of coherent structures in turbulent jets[END_REF] only considered RANS eddy-viscosity resolvent models with respect to the dominant near-field hydrodynamic SPOD modes. While the rank-1 models for the M j = 1.5 jet are similar with and without the eddy viscosity, the remaining reconstructions show a strong and clear advantage to the adopted eddy-viscosity approach. Particularly as sub-optimal modes are added to the basis, the eddy-viscosity model converges rapidly toward the LES whereas the turbulent-Reynolds-number model shows little improvement. This result is consistent with our previous findings [START_REF] Pickering | Optimal eddy viscosity for resolvent-based models of coherent structures in turbulent jets[END_REF], which showed a more profound effect of the eddy viscosity on sub-optimal modes associated with the Orr-mechanism than on modes associated with the Kelvin-Helmholtz mechanism, where the latter are dominant over most of the frequency-wavenumber space being considered here.

B. Far-Field Results

We now extend the eddy-viscosity enhanced resolvent basis to the far-field, and aim to find the modes that are optimal on an arc 100D from the nozzle and a range of polar angles from φ = 100 • to φ = 160 • (where φ = 180 • lies on the downstream axis). The domain is depicted in figure 6.

Figure 7 presents the magnitude of the first three resolvent modes along the arc for both jets at St = 0.26 and m = 0. The same three-beam structure apparent in figure 1 is evident here, with the dominant one-beam mode peaking at φ ≈ 150 • . This progression in beam number and location continues in the higher mode numbers not visualized here. Also plotted in figure 7 are the magnitude of modes found via spectral proper orthogonal decomposition (SPOD) of the LES data. These modes, which optimally reconstruct the CSD of the far-field arc, are useful to compare to the resolvent modes since a close correspondence between resolvent and SPOD modes indicates that the resolvent mode forcings are mutually uncorrelated [START_REF] Towne | Spectral proper orthogonal decomposition and its relationship to dynamic mode decomposition and resolvent analysis[END_REF]. Indeed, we see a reasonable agreement between the far-field SPOD and resolvent modes. The amplitudes and exact locations vary slightly, but such close agreement suggests that an uncorrelated model may suffice.

Figure 8 shows the near-field signatures of the dominant three far-field modes plotted in figure 7 for the M j = 1.5 jet. This plot should be directly compared to figure 1, which showed the dominant three near-field modes. Outside the jet, the modes are nearly indistinguishable. Within the jet (along the x-axis), there are differences that can be associated with the larger hydrodynamic wavepacket imprint left in the near-field modes and missing in the far-field ones.

We now assess how well the computed resolvent modes reconstruct the PSD of the far-field region across St ∈ [0.1, 1] and m = [0, 1, 2] in figure 9 using the same PSD error metric as figure 5. For both jets at m = 0, the rank-1 resolvent reconstruction provides substantial agreement between the LES, at 30% and 40% for M j = 0.9 and 1.5 respectively. The nonzero azimuthal wavenumbers, with the exception of M j = 1.5 and m = 1, require many additional modes to achieve error levels comparable to the rank-1 m = 0 error. This higherrank behavior is similar to what was observed when reconstructing with near-acoustic-field modes for non-zero azimuthal wavenumbers in the previous section.

C. A simple fit/model

Considering we may reconstruct (i.e. to 30-40% error) the far-field acoustics at low-rank, we now ask whether we can define a simple forcing model. One approach would be to propose a form of the forcing crossspectral density tensor, S f f , and project this form onto the resolvent input modes to produce a reduced-order matrix Sββ . Despite some clear trends for the dependence of S f f on mean flow quantities [START_REF] Towne | A statistical jet-noise model based on the resolvent framework[END_REF], there does not yet exist a general form for estimating S f f . We investigate here an alternative approach of directly estimating Sββ . That is, we focus on the expansion coefficients rather than the forcing itself.

The estimated covariance matrix Sββ presents the least square reconstruction of the observed data and contains both the amplitudes and correlations necessary to force each resolvent mode. Where the forcings are uncorrelated, the estimated Sββ matrix becomes diagonal and only n coefficients (albeit at each azimuthal wavenumber and frequency) require modeling. However, even if the forcing is uncorrelated, minor errors or discrepancies in the data, data-processing, computation of resolvent mode, etc., result in a full Sββ matrix. Further, as rank increases, the statistical uncertainty in the terms becomes greater, reducing our hope for successful modeling. Thus, we explore whether neglecting off-diagonal terms is sufficient for a model, but note that this approach provides no guarantees for success; precisely stated, the approximation is not guaranteed to converge as the number of retained modes is increased [START_REF] Towne | Spectral proper orthogonal decomposition and its relationship to dynamic mode decomposition and resolvent analysis[END_REF].

To limit uncertainty and prevent over-fitting, we assume that the forcing is uncorrelated (i.e. diagonal) and that the projection of the data with the first resolvent mode,

Sββ = Σ -1 1 U * 1 S yy U 1 Σ -1 1 = λ β , (21)
possesses the lowest uncertainty. These values for the two jets and three azimuthal wavenumbers are shown in figure 10. For the M j = 1.5 jet, we see that the forcing amplitudes for m = 0 and m = 1 fall upon lines of constant slope for the most acoustically significant frequency ranges, St = 0.1 -0.8. The m = 2 data similarly collapse to a line of constant slope, however, the trend is not as clear. Similar observations also hold for the M j = 0.9 jet. We also stress that these curves depend on both the data and the resolvent gains, Σ. Including the gains is crucial to collapsing the observed trends. We now look to fit the data with simple curves of the form, λm,ω = a m St bm .

(

) 22
For the nonzero azimuthal wavenumbers, the data represent the sum of both the clockwise (m) and counterclockwise (-m) directions about the round jet, such that a m represents a +m +a -m , or 2a +m , since a +m ≈ a -m , as either rotation about the jet is of equal probability. The exponent, b m is unaffected by this symmetry. Figure 10 provides the lines of best fit, where the fits are computed over the region of St = 0.13 -0.7 for the M j = 1.5 and St = 0.22 -1 for the M j = 0.9. This region is admittedly chosen by observation from the data, however, necessity for clipping the low-frequency ranges is a result of a finite domain where modes (as well as the LES data itself) begin to exit the domain. This results in resolvent modes that, although possessing an inner product of unity, are not fully contained in the domain (introducing errors in the computed gain values) and give the observed plateau in forcing energy. The domain issue is further supported considering we find the frequency cutoff may be related by adjusting the Strouhal number by Mach number, St 0.9 = St 1.5 × 1.5/0.9, meaning each range is associated with the same range of acoustic Strouhal number, St c∞ . The upper bound for the M j = 0.9 case extends to St = 1.17, however, we cap the upper bound to St = 1 as done throughout this manuscript.

Table I provides the fit coefficients for each jet and azimuthal wavenumber. At present, we do not have any physical interpretations of these fits, other than the obvious fact that the power law gives an expected decrease in energy as frequency increases (and thus the length scales of the structures decrease). We suspect we may find similar curves via projection of the resolvent forcing modes with the turbulent kinetic energy or other meanflow quantities, but leave this for future work.

To determine how well such curves predict the data, we use the fitted curves to compute,

Syy (φ) = λm,ω Ũ Σ 2 Ũ * , (
where St min = 0.22 and 0.13 for M j = 0.9 and M j = 1.5, respectively. Figure 9 provides the reconstruction error for the rank-1 model for both jets and three wavenumbers. For the M j = 1.5 jet, the rank-1 model yields a close approximation of the reconstructions at 40% for both m = 0 and m = 1. For M j = 0.9, the m = 0 error is about 10% worse than the projections, but still at 40% for a rank-1 model. The m = 1 case here and both m = 2 cases perform similarly to the perfect reconstructions, but each of these is relatively poor at reducing the error.

With the rank-1 prediction in hand, we conclude by computing the overall sound pressure level (26) Figure 11 presents calculations of the OASPL from both jets using the 100D Kirchhoff surface values from the LES and the rank-1 resolvent model considering m = [0, 2] and St = 0.13 -1. For the M j = 1.5 jet, the values are only close at downstream angles and where the jet is the loudest. From φ = 140 -155 the model is within 1-3dB. A striking difference the resolvent model and data, is that the former peaks at an angle of about 5 degrees higher than the LES data. Interestingly, a similar experiment and simulation in [START_REF] Brès | Unstructured large-eddy simulations of supersonic jets[END_REF] disagreed by the same angle. Although 100D data from the experiment is not available, projecting the resolvent modes onto this data would likely result in better alignment, as shifting the LES data by 5 degrees results in a significantly improved estimate (within 0.5dB from φ = 135 -150). However, as the ultimate source of the discrepancy is unknown, we avoid making any corrections to the model based on these observations. We see similar behavior between the KS surface and the resolvent model for the transonic case. The rank-1 m = [0, 2] resolvent model presents agreement of the peak OASPL to within 2dB at peak noise angles. We stress that this result for the M j = 0.9 jet is rather surprising as many previous studies, although computed in the near-field, found the acoustic field required many modes to agree within 2dB [START_REF] Freund | Turbulence and soundfield POD analysis of a turbulent jet[END_REF][START_REF] Towne | Stochastic and nonlinear forcing of wavepackets in a Mach 0.9 jet[END_REF]. This shows that the application of both the KS surface to 100D and eddy-viscosity model included in our resolvent analysis significantly reduces the rank of the acoustic jet problem. Further, we note that this transonic jet has been extensively verified by experimental data in the near-field and at ρ = 50D, and, although we extend the results to 100D, the peak angles of the KS and the resolvent model are closely aligned when compared to the M j = 1.5 case.

IV. CONCLUSIONS

We formulated resolvent analysis to serve as an acoustic analogy by relating the near-field resolvent forcing to both the near-and far-field acoustic regions. Leveraging the availability of an LES database, we examined resolvent-based reconstructions of the acoustic PSD for turbulent M j = 0.9 and M j = 1.5 jets. We represented the forcing cross-spectral density matrix with a truncated set of resolvent modes and approximated the amplitudes of the modes with best-fit expansion coefficients of realizations from the LES acoustic field. We found that models comprising of just a single resolvent mode can accurately reconstruct the acoustic field for the first two azimuthal modes for a M j = 1.5 jet and the m = 0 azimuthal mode for the M j = 0.9 jet. To reconstruct higher azimuthal modes, the resolvent basis must be increased to at least 5 modes (i.e. m = 2 and m = 1, 2 for M j = 1.5 and M j = 0.9, respectively). In both jets, the use of an eddy-viscosity model in the resolvent formulation led to clearly superior results compared to a fixed turbulent Reynolds number.

Based on the ability of the rank-1 reconstructions to describe the PSD, we investigated a simple model to collapse the forcing coefficients to one scaling function per azimuthal wavenumber (and Mach number). We found that a power law representation, with only a scaling and an exponent, suffices to model the coefficient of the optimal resolvent mode. Fortunately, the first resolvent mode contains much of the acoustic energy, and reductions of the gain for this specific mode (related to the KH mechanism) are likely to provide the greatest reductions in the peak noise of the acoustic field.

The rank-1 m = [0, 2] resolvent models estimate the peak noise to within 2dB for both the M j = 1.5 and M j = 0.9 jets at peak noise angles. Further, the ability of the resolvent basis to describe much of the acoustic field with only a handful of modes across multiple Mach numbers, a large range of frequencies, and the acoustically dominant azimuthal wavenumbers is promising. This shows that the resolvent framework already contains the appropriate acoustic functions to describe jet noise. In future work, we will seek a fully predictive model by estimating the forcing coefficients from mean flow quantities available from RANS.

  FIG. 1. The first three resolvent modes of fluctuating pressure, q p . Red and blue contours vary from ± 20% of the maximum fluctuating pressure of each mode, ±0.2||q p ||∞. Mj = 1.5, St = 0.26, m = 0.

FIG. 3 .

 3 FIG. 3. Comparison of pressure PSD values by dB at r/D = 6 for the LES ensemble and reconstructions in the resolvent basis using 3, 5, and 10 resolvent modes.

  FIG. 5. Error between the LES PSD and the reconstructed PSD by number of resolvent modes retained (i.e. n = [1, 3, 5, 10, 20]). Filled symbols indicate modes using the RANS eddy-viscosity model, while hollow symbols give those using a turbulent Reynolds number, ReT = 3 × 10 4 .

FIG. 6 .

 6 FIG. 6. Schematic of the far-field arc at 100D from the nozzle exit. The angle along the arc is defined as φ, with 0 • on the upstream axis and 180 • on the downstream axis. The red portion of the arc denotes the region of interest, φ = 100 • -160 • and the acoustic beam presented is the first resolvent mode for Mj = 1.5, St = 0.26, m = 0, found for the far-field region.

FIG. 7 .

 7 FIG. 7. Magnitude of the first three resolvent (left) and SPOD (right) modes computed on the far-field arc for the Mj = 0.9 (top) and Mj = 1.5 (bottom) jet at St = 0.26 and m = 0.

  FIG. 8. The near-field of three resolvent modes of fluctuating pressure, q p , computed considering the 100D arc from φ = 100 • -160 • . Red and blue contours vary from ± 20% of the maximum fluctuating pressure of each mode, ±0.2||q p ||∞. Mj = 1.5, St = 0.26, m = 0.

  23) where Ũ represents the truncated resolvent basis to rankn. Additionally, as we cannot expect our methods to have accurately captured such large structures in the finite domain used, we use the piece-wise function λm,ω = a m St bm for St > St min (24) λm,ω = a m St bm min for St ≤ St min ,

  yy (φ, m, St)) .

  FIG.10. Values of the reconstruction projection coefficient, λ β , of the first resolvent mode for the azimuthal wavenumbers m = 0 -2 and their associated fits, the parameters of which are provided in table I.

  FIG. 11. OASPL of the Mj = 0.9 (left) and Mj = 1.5 (right) turbulent jets at 100D from the nozzle over the arc φ. The solid black line denotes the OASPL from the Kirchhoff surface values found from propagating the near-field LES pressure field, while the solid line gives the resolvent model estimation. In each case, the OASPL only considers the acoustically relevant m = [0, 2] and St = 0.13 -1 contributions.

Submitted to J. Acoust. Soc. Am. / 25 August 2021

Submitted to J. Acoust. Soc. Am. / 25 August 2021

Submitted to J. Acoust. Soc. Am. / 25 August 2021

Submitted to J. Acoust. Soc. Am. / 25 August 2021

Submitted to J. Acoust. Soc. Am. / 25 August 2021

Submitted to J. Acoust. Soc. Am. / 25 August 2021

ACKNOWLEDGMENTS

The authors would like to thank André Cavalieri, Oliver Schmidt, and Georgios Rigas for many productive discussions on topics related to this paper. This research was supported by a grant from the Office of Naval Research (grants No. N00014-16-1-2445 and N00014-20-1-2311) with Dr. Steven Martens as program manager. E.P. was supported by the Department of Defense (DoD) through the National Defense Science & Engineering Graduate Fellowship (NDSEG) Program. P.J. acknowledges funding from the Clean Sky 2 Joint Undertaking (JU) under the European Union's Horizon 2020 research and innovation programme under grant agreement No 785303. Results reflect only the authors' views and the JU is not responsible for any use that may be made of the information it contains. The LES study was performed at Cascade Technologies, with support from ONR and NAVAIR SBIR project, under the supervision of Dr. John T. Spyropoulos. The main LES calculations were carried out on DoD HPC systems in ERDC DSRC.

 [5,20]so that the LES could be directly projected onto the resolvent basis, however, the LES database (i.e. the saved data from the LES) only extends to r/D = 6. Although one could define an output matrix C that only includes that surface at r/D = 6, the resolvent modes may still contain hydrodynamic behavior (unless allowed to propagate further from the jet), thus we use the larger domain to ensure the modes are entirely acoustic. Using the larger domain presents a clear loss of orthogonality in the space represented by the LES domain, which is alleviated by truncating the modes to r/D = [5, 6] (after computing the resolvent SVD) and implementing a Moore-Penrose inverse such that a least-squares fit of the LES in the resolvent basis can be performed. While previous studies have suggested the use of a filter based on the turbulent kinetic energy of the jet within the input matrix B [START_REF] Towne | A statistical jet-noise model based on the resolvent framework[END_REF], we take B to be identity for both the near-and far-field analyses for the sake of generality.

Far-field acoustic output matrix

To define an input-output relationship from the nearfield forcing to the far-field acoustics, we introduce a Kirchhoff surface and apply it as a linear operator. We define three radii: R as the radial coordinate of the nearfield cylindrical surface, r as the coordinate pertaining to the far-field cylindrical surface, and ρ representing the distance from the nozzle in spherical coordinates (e.g. ρ/D = 100 for this study). As described in § II B, the input-output problem is defined as

where the output matrix C R,ρ is the total Kirchhoff operator that maps the near-field cylindrical surface, R, to the far-field spherical surface, ρ. This operator is linearly composed of many Kirchhoff surfaces, C R,r , detailed next.

The cylindrical Kirchhoff operator is comprised of several linear operations to ensure accurate results and is defined as,

where C R is a surface selection matrix (∈ R N surface ×5NrNx ), N is an interpolation matrix from a non-uniform grid to a uniform grid with ∆x/D = 0.025 (∈ R N uniform ×N surface ), T is a Tukey windowing matrix (using a taper value of 0.75) that extends over the Kirchhoff surface to reduce spectral leakage (∈ R N uniform ×N uniform ), P is a padding matrix extending the uniform grid with a total of 2 n points (n is set to 15) for computing the upstream and downstream wave propagation, as well as ensuring sufficient accuracy in the transform of the initial surface (∈ R 2 n ×N uniform ), D is the discrete Fourier transform (DFT) matrix (∈ R 2 n ×2 n ), and H contains the derived Hankel functions of the Kirchhoff surface from [START_REF] Freund | Noise sources in a low-Reynolds-number turbulent jet at Mach 0.9[END_REF], with entries along the diagonal for each azimuthal wavenumber, for a specified radial distance, r, from the surface at R (∈ R 2 n ×2 n ). However, the above operator only supports one specified radial distance from the cylindrical surface at R, a linear combination of C R,r and a proper selection of streamwise points is required to construct a spherical arc. Thus, the linear expression to construct the total Kirchhoff operator is then

where x i represents the streamwise location in the 100D arc and r i represents the radial extent to which the Kirchhoff surface must propagate from surface R to the farfield arc ρ for the respective streamwise location. Points are defined along the arc from φ = 100 • -160 • with a resolution of ∆φ = 0.5 • .

APPENDIX B: NON-ORTHOGONAL PROJECTIONS OF RESOLVENT MODES

The statistical relations presented in § II B are valid when U and V are orthogonal bases in the same space as y and f , respectively. However, in the case of the nearfield calculations, U is defined over a larger space than y and a pseudo inverse must be constructed to find the least-square solution to the above projections. First, we truncate the output modes U to the output space x/D = [0, 30] and r/D = [5,6] in the pressure field and define the associated output matrix as C z where z denotes the new restricted space. Applying C z to both the LES data and resolvent modes gives the ensemble of realizations z and resolvent modes U z . In addition to reducing the domain space, we also truncate the resolvent response basis to a limited set of n modes, as discussed above, represented as Ũz . There are now two important consequences of reducing the resolvent domain from C y to C z . The first is a correction to the gain to the domain C z . Since both output domains share identical input modes we have,

and the gain of the new domain is

where, by definition, u * i,y W y u i,y = 1. The second is a loss of orthogonality. Fortunately, we may still determine a least squares fit of the data by computing the Moore-Penrose inverse of W 1/2 z Ũz , (W

z , and projecting it onto the CSD of z to estimate Sββ

This approach is similar to the one taken by [START_REF] Towne | Resolventbased estimation of space-time flow statistics[END_REF] for assimilating partially observed flow statistics.


T Winkel

S Rambal

STOMATAL CONDUCTANCE OF SOME GRAPEVINES GROWING IN THE FIELD UNDER A MEDITERRANEAN ENVIRONMENT

A semi-empirical model of stomatal conductance was used to analyse porometer measurements on different grapevine cultivars under field conditions. Stomatal conductance was expressed as a function of quantum flux density, water vapour deficit, air temperature and leaf water potential.

Where possible, the parameters describing the partial functions were estimated from field data and provide physiological insights into the transpiration process. They indicated differential stomatal sensitivity to air humidity in the various cultivars, a characteristic which could be related to their geographical origins.

INTRODUCTION

The most generally successful model of evaporation from closed canopies is the well-known Penman-Monteith equation [START_REF] Monteith | Evaporation and environment[END_REF]. This model has been applied both to single leaves and to canopies. In the latter case, it has been chiefly used where fluxes might be expected to be predominantly onedimensional. In orchards or vineyards, horizontal heterogeneity requires a description in more than one dimension. For such row crops, the vegetation, by means of the stomata, plays a determining role in the control of bulk evaporation flux [START_REF] Van Bavel | Climate and estimated water use by an orange orchard[END_REF][START_REF] Hicks | Eddy fluxes over a vineyard[END_REF]. Discontinuous canopies are aerodynamically rough and wind is only weakly attenuated [START_REF] Riou | Variation de la vitesse du vent ~ l'int~rieur et audessus d'une vigne[END_REF] so that aerodynamic conductance of water vapour, which is proportional to eddy velocity and roughness length, is likely to be very high (Thorn, 1975 ). In such cases, transpiration from the vegetation is strongly coupled with the air stream above the canopy and, because the transpiration rate is then closely proportional to the product of stomatal conductance and water vapour pres-sure deficit of the air, changes in stomatal conductance become of paramount importance [START_REF] Thom | Momentum, mass and heat exchange of plant communities[END_REF][START_REF] Jarvis | Stomatal control of transpiration: scaling up from leaf to region[END_REF].

Modelling mass transport and evaporation from this type of canopy should therefore involve the most likely realistic identification of the variables and laws governing stomatal functioning under field conditions. Among environmental factors that affect stomatal opening, solar radiation, soil water availability, atmospheric vapour pressure deficit and temperature are known to be important [START_REF] Sheriff | Stomatal aperture and the sensing of the environment by guard cells[END_REF][START_REF] Schulze | Carbon dioxide and water vapor exchange in response to drought in the atmosphere and in the soil[END_REF]Turner, 1986a). Age, position in the canopy, internal CO2 concentration and hormonal equilibrium of the leaves, and previous growing conditions, also influence stomatal behaviour [START_REF] Thomas | Stomatal response to leaf water potential as affected by preconditioning water stress in the field[END_REF][START_REF] Blackman | Cytokinins, abscissic acid and the control of plant water balance[END_REF][START_REF] Field | Leaf-age effects on stomatal conductance[END_REF]. Whereas these intrinsic factors have been rarely investigated, several models have been proposed that relate stomatal aperture to simultaneous variations of weather factors, soil water deficit and plant water potential [START_REF] Jarvis | The interpretation of the variations in leaf water potential and stomatal conductance found in canopies in the field[END_REF][START_REF] Jones | Plants and microclimate. A quantitative approach to environmental plant physiology[END_REF][START_REF] Avissar | A model to simulate response of plant stomata to environmental conditions[END_REF].

When based on physical parameters, these models allow investigation of the regulation of water exchanges under natural conditions with relatively simple input data [START_REF] Whitehead | Stomatal response to environmental variables in two tropical forest species during the dry season in Nigeria[END_REF]. Vegetation types can also be compared to improve understanding of the differential sensitivity to drought and generate hypotheses concerning the underlying mechanisms of water stress adaptation. In the case of cultivated vegetation, this knowledge will help to assess and compare the water requirements of various crops growing under given climatic conditions. In this paper, we examine the effects of weather variables and leaf water potential on the stomatal response of three grapevine cultivars grown in the field under Mediterranean climatic conditions.

MATERIALS AND METHODS

Site description and data collection

Field plots were located in the Aude valley, southern France (43°13'N, 2 ° 50'E), at two sites 3.8 km apart with similar climatic conditions. The area has a Mediterranean-type climate with a potential evaporation rate of ~ 1100 mm year -1 and a mean annual rainfall (1961)(1962)(1963)(1964)(1965)(1966)(1967)(1968)(1969)(1970)(1971)(1972)(1973)(1974)(1975)(1976)(1977)(1978)(1979)(1980) of ~ 600 mm, of which < 14% occurs during the summer [START_REF] Canet | El6ments climatologiques du Languedoc-Roussillon[END_REF]. Different cultivars of grape were studied at the two sites, which differed in their soil type and soil water storage capacity. Available soil water has been used to indicate the difference between both sites; it was calculated from field capacity and minimum water storage, measured in 1986-1987 with a neutron moisture gauge. Soil was assumed to be at field capacity after a few days of drainage following a rainy period.

The first site consisted of two plots on a loamy soil in a flood plain, planted with grape cultivars carignane and merlot. Available soil water was 180 mm for the 0-130-cm layer. The second site consisted of two plots on a stony terrace with a thick calcareous hard pan at 80-120-cm depth; plots were planted with carignane and shiraz vines. The available soil water, measured as described above, was only 120 ram.

Vines were between 6 and 15 years old, and were therefore assumed to have fully developed root systems; soil water profiles showed root water uptake beyond 2-m depth in the plots on the terrace. The carignane vines were goblet pruned, while merlot and shiraz were cordon trained.

A steady-state porometer (LI-1600, LI-COR Ltd., Lincoln, NE, U.S.A. ) was used for simultaneous measurements of stomatal conductance, quantum flux density, water vapour deficit and air temperature (these latter two were measured within the porometer cup). These measurements were replicated six times. Concurrent measurements of leaf water potential, replicated three times, were obtained with a standard Scholander-type pressure chamber (PMS 1000, PMS Inst., Corvallis, OR, U.S.A. ). Data were collected on mature leaves in the sunny part of the canopy from dawn to ~ 16:00 h (LST) on 8 days spaced throughout most of the growth cycle (13 June, 27 June, 9 July and 15 August, 1986;24 May, 2 June, 9 July and 11 September, 1987).

Modelling procedure

The method used was detailed by [START_REF] Jarvis | The interpretation of the variations in leaf water potential and stomatal conductance found in canopies in the field[END_REF] and [START_REF] Jones | Plants and microclimate. A quantitative approach to environmental plant physiology[END_REF]. The model was based upon known relationships between stomatal conductance (gs, mmol m-2 s-1 ) and quantum flux density (Q, #tool m-2 s-1 ), water vapour pressure deficit (D, kPa), air temperature ( T,, o C ) and leaf water potential (~, MPa). Its general form is g~ =gsm "g(Q).g (D).g(T~).g (v])

(1)
where gsm is maximum conductance of a given vine cultivar and each g is the partial function for the indicated independent variable (0 < g < 1 ) (Fig. 1 ).

The parameters that describe stomatal opening in response to the four independent variables were estimated from field measurements by non-linear least squares regression using Marquardt's method [START_REF] Draper | Applied Regression Analysis[END_REF] or trial and error estimations. This approach enabled us to take into account the functional relationships controlling stomatal behaviour and to interpret parameters in a physical sense [START_REF] Reed | An analytical model for field measurement of photosynthesis[END_REF]. As stated by [START_REF] Jarvis | The interpretation of the variations in leaf water potential and stomatal conductance found in canopies in the field[END_REF], such a model is not mechanistic because it does not link environmental factors with stomatal functioning at the cellular level. Nevertheless, its semi-empirical character makes it useful for the interpretation of field observations and prediction of stomatal conductance, even for environmental conditions not directly tested. Two limitations of this approach must be outlined. First, probable interactions between variables are ignored. Response functions given in the literature were obtained in controlled environments in which only one or two factors were varied at a time. In reality, all the factors act simultaneously in interconnected processes and interactions are expected. However, to our knowledge, such complex combinations between factors have not yet been formalized. Secondly, the data sets collected in the field usually show strong correlations between variables, resulting in possible bias in estimation of the parameters [START_REF] Jarvis | The interpretation of the variations in leaf water potential and stomatal conductance found in canopies in the field[END_REF]. The narrow range of certain variables could also be a handicap.

Variables and partial functions

Solar radiation

The stomatal response to sunlight is influenced by several internal factors, including leaf age and plant water status [START_REF] Squire | Stomatal behaviour in the field[END_REF][START_REF] Field | Leaf-age effects on stomatal conductance[END_REF]. This leads to quite variable relationships between conductance and sunlight [START_REF] Cowan | Stomatal behaviour and environment[END_REF]. However for well-watered plants it is usually considered that stomatal conductance shows a hyperbolic response to sunlight. This relationship can be described by the equation

g( Q) = l-exp(-Q/K~) (2)
where Q stands for the quantum flux density or photosynthetically active radiation. The parameter, K1, is derived from the minimum Q value required for a nearly maximum stomatal aperture. For a number of species, this Q value is ~ 400/~mol m -2 s-1 [START_REF] Squire | Stomatal behaviour in the field[END_REF][START_REF] Jones | Plants and microclimate. A quantitative approach to environmental plant physiology[END_REF], in accord with the values of 200 W m-2 of global radiation given by [START_REF] Kriedemann | Effects of irradiance, temperature and leaf water potential on photosynthesis of vine leaves[END_REF] and of 300/~mol m -2 s -1 found by [START_REF] Liu | Soil-plant water relations in a New York vineyard: resistances to water movement[END_REF] for grapevine cultivars.

Assuming 95 % relative stomatal conductance at a quantum flux density of 400 /tmol m -2 s -1, we calculate a K1 value of 133/tmol m -2 s -1. This value is applied for all the studied grapevine cultivars, making the hypothesis that light acts on stomatal aperture by stimulating metabolic processes at the cellular level, which are independent of varietal type.

Water vapour pressure deficit

Many species close their stomata in response to increased water vapour pressure deficit [START_REF] Sheriff | Stomatal aperture and the sensing of the environment by guard cells[END_REF][START_REF] Schulze | Carbon dioxide and water vapor exchange in response to drought in the atmosphere and in the soil[END_REF]. The stomatal response to humidity could be linear or curvilinear depending on the control system involved; a direct feedforward response results in a linear relationship [START_REF] Camacho-B | Efficiency and regulation of water transport in some woody and herbaceous species[END_REF][START_REF] Farquhar | Responses to humidity by stomata of Ni cotiana glauca L. and Corylus avellana L. are consistent with the optimisation of carbon dioxide uptake with respect to water loss[END_REF], whereas a feedback response via leaf water status leads to a non-linear relationship (Johnson and Ferrel, 1983). We compared both types of relationships

g(D)=I-K2.D ifD<l/K2 (3) g(D)=O ifD> l/K2 and g(D) =exp( -Ke "D) (3a)
where D is the vapour pressure deficit of the air near the leaf and K2 is a parameter estimated from the data set.

Air temperature

The effect of air temperature on stomatal conductance is difficult to separate from that of humidity because vapour pressure deficit and air temperature are correlated. However it is widely accepted [START_REF] St~lfelt | The effect of temperature on opening of the stomatal cells[END_REF]) that stomatal conductance reaches a maximum at 30-35 ° C. Such a response curve may be written [START_REF] Jones | Plants and microclimate. A quantitative approach to environmental plant physiology[END_REF]

g(T)=I-K3.(T~-To) z if (Ta-To)2<I/K3

(4)

g(T) =0 if (Ta -To) 2> 1/K3
where Ta is the air temperature near the leaf and To is the optimum temperature for stomatal opening (g (To) = 1 ); To and K3 were estimated from the field measurements.

Leaf water potential

Although bulk leaf water potential is the biological variable most often considered in studies of stomatal function, it is not considered to directly control plant water loss. Instead, turgor pressure in guard cells and hormonal regulators are reported to play the dominant roles [START_REF] Zeiger | The biology of stomatal guard cells[END_REF][START_REF] Schulze | Carbon dioxide and water vapor exchange in response to drought in the atmosphere and in the soil[END_REF]. Moreover, Turner (1986b) pointed out how different rates of soil drying and air vapour deficit could induce a lack of correlation between leaf conductance and leaf water potential. Nevertheless, partly because of ease of measurement in the field, leaf water potential remains a widely used state variable for functional rather than mechanistic models such as we are seeking. The response of conductance to leaf water potential can be modelled with a two-parameter relationship [START_REF] Rambal | Mod~lisation de l'utilisation de l'eau et de la production v~g~tale d'une steppe Rhantherium suaveolens Desf. de la zone aride sud tunisienne[END_REF][START_REF] Jones | Plants and microclimate. A quantitative approach to environmental plant physiology[END_REF][START_REF] Campbell | Soil Physics with Basic[END_REF] g(~,) =1/(1+ (~,/~]~)K4)

(5)

where ~ is the critical leaf water potential (i.e. the value required to reduce maximum conductance by half) and K4 is an empirically derived parameter that describes the rate of decrease ofgs with ~.

APPLICATION OF THE MODEL

Determination of parameters

The model requires seven parameters; gsm, K1, K2, To, K3, ~ and K4. For each vine plot, field measurements of each independent variable were randomly assigned to one of two data sets, one for estimation of the parameters (Table 1 ) and the other for validation of the model.

Maximum stomatal conductance of each of the three cultivars was directly estimated from the field measurements by taking the highest value observed in 1986-1987 (mean of six replicated measurements). Assuming this parameter to be only under genotypic dependence, the same value was adopted for carignane on both sites. Statistical differences appear between the carignane and shiraz cultivars (P= 0.01 ), while merlot, with an intermediate value, differs from the other two only at the 10% probability level. Values were similar Numbers in parentheses show the standard error of the mean.

to those given in the literature for field-grown cultivars [START_REF] Ksrner | Maximum leaf diffusive conductance in vascular plants[END_REF][START_REF] Katerji | Etude in situ du fonctionnement hydrique et photosynth6tique d'une vigne conduite en lyre[END_REF]). The parameter K1, which describes the stomatal response to quantum flux density, was set at a value of 133/lmol m -2 s-1 for the four plots, as previously explained.

Estimation of the other parameters from field observations was less straightforward. Plots ofgs against each independent variable generally show considerable scatter and the upper boundary of points -theoretically representative of the stomatal response to one variable when the others are not limitingusually appears discontinuous and irregular [START_REF] Jarvis | The interpretation of the variations in leaf water potential and stomatal conductance found in canopies in the field[END_REF]. In addition, the functional relationship between gs and each driving variable was difficult to fully characterize, partly because the range of variables was restricted (Table 2). These field measurements however are representative of the "non-ideal data of the kind frequently gathered" [START_REF] Reed | An analytical model for field measurement of photosynthesis[END_REF]. In view of these difficulties, we initialized the model by setting the values of the parameters To, ~u~ and/<4. This was done by trial and error, keeping values realistic with respect to the limits of leaf functioning and leading to a percentage of explained variance as high as possible. K2 and/<3 were obtained by non-linear regression.

From our field measurements, a temperature of 30 ° C was found to be optimal for the stomatal conductance of the three cultivars and K4 was set to 2, representing the progressive decrease ofg~ with a decrease in leaf water potential. The estimates of ~,½ adopted to run the model were -1.4 MPa for carignane, -1.6 MPa for merlot and -1.8 MPa for shiraz. These values correspond fairly well with those derived by stratifying the data and using a boundary-line analysis [START_REF] Webb | Use of the boundary line in the analysis of biological data[END_REF]) (Fig. 2). The estimated values of K2 and K:~ for different cultivars showed a positive correlation, reflecting the interdependence between air temperature and humidity. They revealed significant differences between vine cultivars, especially regarding their reaction to water vapour deficit. As for/<3, its estimates may be relevant only to the range of observed air temperature as they imply complete stomatal closure at ~ 15 ° C, which seems an unrealistic value. The model accounts for 80-88% of the variation in the four data sets (Table 1 ). Because the alternative assumptions of the linear and exponential response of conductance to air humidity explained similar proportions of variation, we adopted the simpler one (eq. 3 ).

Validation of the model

The model was tested by comparing the observations of the second data set with the stomatal conductances estimated from the input variables of this set, with the parameters derived from the first set of measurements. Although Fig. 3 shows a large spread of points, simulated and measured values did not differ significantly, except in the case of merlot (Table 3). Likewise, diurnal varia- tions of simulated conductances were similar to those measured in the field, the model being particularly able to represent the closure of the stomata at midday (Fig. 4).

The most noticeable divergence between predicted and observed values occurred early in the day, when rapid changes in light level and wind speed may outpace the capacity for stomatal response {Williams, 1983). Systematic differences also occurred on certain dates. Estimated conductance was relatively accurate in the four plots on 9 July, but was generally underestimated on 15 August and overestimated on 13 June. This may represent the effect of season on plant development. In particular, the maximum stomatal conductance is known to increase during the development cycle of the plant [START_REF] Jones | Plants and microclimate. A quantitative approach to environmental plant physiology[END_REF][START_REF] Chaves | Photosynthesis and water relations of grapevines growing in Portugal. Response to environmental factors[END_REF] and the adoption of a constant value for gsm would therefore tend to overestimate conductance at the beginning of the vegetation period, when the leaves are not yet fully expanded. Underestimation of conductance in mid-season (15 August) is less easy to explain, but may have been linked to the growth history. Pre-conditioning by repeated soil water depletion can lead to a progressive decrease in the threshold leaf water potential for stomatal closure [START_REF] Thomas | Stomatal response to leaf water potential as affected by preconditioning water stress in the field[END_REF][START_REF] Ackerson | Stomatal response of cotton to water stress and abscissic acid as affected by water stress history[END_REF]. For the grapevine, although they did not concern studies with repeated water stress cycles, the results of [START_REF] Dtiring | Diurnal changes in water relations and abscissic acid in field grown Vitis vinifera cultivars. I. Leaf water potential components and leaf conductance under humid temperate and semiarid conditions[END_REF] and [START_REF] Chaves | Photosynthesis and water relations of grapevines growing in Portugal. Response to environmental factors[END_REF] gave an explanation for this phenomenon. They showed that in stressed vines (i.e. with low bulk water potential) the turgor pressure necessary for the stomatal aperture is maintained by a decrease in osmotic potential. This decrease results from an active accumulation of solutes in the leaf tissues when water stress develops.

DISCUSSION

Our results, with data from the literature, permit some hypotheses to be put forward on the mechanisms that regulate water transport at the leaf level. The similarity of parameters estimated for the two different plots of carignane, and the divergence between cultivars, suggest that genetic differences in stomatal response to the environment were more important than site differences.

A degree of differentiation among cultivars was first apparent in values of gsm; maximum stomatal conductances of the merlot and shiraz cultivars were respectively, 20 and 35% below that of carignane, the difference being significant between carignane and shiraz only (P= 0.01). Although a genetic component seems likely to be involved, this would need to be confirmed by a systematic comparison of grapevine cultivars. Moreover, the maximum stomatal conductance of shiraz was likely to have been limited by previous conditions of cultivation, particularly lack of irrigation since planting time, which would have been a harsh constraint for this cultivar, known for its drought sensitivity. Analysis of stomatal responses to vapour pressure deficit and leaf water potential provides some insight into the development of drought resistance by grapevines.

The low rate at which the stomata react to a decrease in leaf water potential (parameter K4 ) is typical of plants grown in the field, where soil water depletion usually occurs gradually and allows plants to respond by osmotic adjustment or control via phytohormones, and continuous root growth (Turner, 1986a;[START_REF] Rambal | From daily transpiration to seasonal water balance: an optimal use of water?[END_REF]. Osmotic adjustment in particular has been demonstrated for grapevines growing under semi-arid conditions (Diiring and Loveys, 1982). This process provides an explanation for the decrease in the critical potential for stomatal closure in plants subjected to progressive water stress. It also explains the differences between field and controlled environment studies. In our experiment, ~ reached a value of -1.8 MPa for shiraz, whereas [START_REF] Kriedemann | Effects of irradiance, temperature and leaf water potential on photosynthesis of vine leaves[END_REF] found a critical potential of -1.3 MPa for the same cultivar grown in containers without irrigation. The lower critical potential of shiraz, as compared with merlot or carignane, could be linked to vegetative characteristics. [START_REF] Ludlow | Adaptative significance of stomatal responses to stress[END_REF] stated that plants exhibiting stomatal adjustment cannot survive without some tolerance to dehydration, since stomata that adjust osmotically remain partially open and water loss continues. He also added that this tolerance to dehydration is usually associated with slow rates of growth and development. On the other hand, Turner (1986b) outlined the advantage presented by this type of functioning in the maintenance of water uptake; it allows continued root growth at low water potential and enables the plant to explore a greater volume of soil. We observed both features in shiraz; its rooting depth exceeded 2 m and its vegetative development appeared significantly reduced compared with the neighbouring carignane plants (450 and 590 g, respectively, of dry mass of shoot per plant in 1987).

Plants that cannot endure dehydration avoid low water potential by sensitive control of water loss from the leaves. Direct stomatal response to air humidity plays a determining role here. Evidence for this response in grapevines is contradictory, particularly regarding the sensitivity of unstressed plants [START_REF] Diiring | Untersuchungen zur Umweltabhiingigkeit der stomat~iren Transpiration bei Reben. I. Beleuchtungst~rke und Luftfeuchtigkeit[END_REF](Diiring, , 1987;;[START_REF] Lange | Mittaglischer Stomataschluss bei Aprikose (Prunus armeniaca ) und Wein ( Vitis v inifera) im Freiland trozt guter Bodenwasser-Versorgung[END_REF][START_REF] Kliewer | Effect of irrigation, crop level and potassium fertilization on Carignane vines. I. Degree of water stress and effect on growth and yield[END_REF]. Considering the processes underlying this humidity sensing, the distinction between feedforward (eq. 3) and feedback (eq. 3a) control is of importance as it involves quite different behaviours regarding economy in use of water. [START_REF] Cowan | Stomatal behaviour and environment[END_REF] demonstrated theoretically how a feedforward response confers a distinct advantage in dry environments. Our data do not allow us to distinguish between the two possibilities, although they clearly indicate a differential sensitivity to air vapour deficit for vine cultivars grown in the field.

The more pronounced humidity response of the carignane and merlot vines, which also show the higher threshold potentials, is in accord with Ludlow's statement that humidity response is of special importance for plants lacking tolerance to dehydration [START_REF] Ludlow | Adaptative significance of stomatal responses to stress[END_REF]. Via this mechanism, transpiration loss is restricted when atmospheric demand becomes too high, helping to increase the daily water use efficiency. The very high productivity of these two cultivars [START_REF] Winkler | General Viticulture[END_REF] supports this contention. Loveys and Kriede-mann (1973) and then [START_REF] Diiring | Untersuchungen zur Umweltabhiingigkeit der stomat~iren Transpiration bei Reben. I. Beleuchtungst~rke und Luftfeuchtigkeit[END_REF] found greater stomatal responses to humidity in stressed vines; while Schulze and Kiippers (1979) found that longterm stress did not alter the slope of the conductance/humidity relationship, but rather decreased the range of stomatal response. In our study, one or other of these processes should have induced differences among the two carignane vines according to their contrasting soil water resources. This was not observed, which suggests stress avoidance by the carignane grown on the terrace, probably by means of its deep root system.

Further, the similarity of K2 for the two different plots of carignane suggests that this parameter is a species characteristic. [START_REF] Kriedemann | Vineleaf photosynthesis. In: Oenological and Viticultural Research In-stitute[END_REF] points out that even though inference of genetic influences from physiological observations is highly tentative, geographic origins of grapevine taxa can provide useful information about their physiological characteristics. This also seems to be true for grapevine cultivars traditionally linked to a given region, in which physiological characters may have been influenced by their evolution in specific environments [START_REF] Rives | Les origines de la vigne[END_REF]. The carignane, merlot and shiraz cultivars we studied originated, respectively, from Spain (Aragon), the Bordeaux region and the RhSne valley. Aragon has a strong semi-arid climate [START_REF] Font Tullat | Climatolog~a de Espafia y Portugal[END_REF], while the other two regions show summer rain deficits partly limited by periods of mist [START_REF] Rey | Le d~terminisme ~cologique de la rgpartition des plantes m~diterran~ennes en Aquitaine[END_REF]. The contrasting responses of carignane and shiraz (P=0.01) therefore appear consistent with their geographical origin; carignane, well known for its hardiness, could have adapted to climatic drought by developing sensitivity to air vapour deficit, a capacity that shiraz, native to a more mesic region, would not have developed to the same extent. While the contrast between carignane and shiraz could be analysed through their stomatal reaction to only one environmental factor (viz. air humidity), understanding the intermediate position of merlot (high stomatal sensitivity, mesic origin) needs to consider the plant as a whole. Plant response to water stress involves, besides regulation at the stomatal level, several mechanisms acting at larger time scales [START_REF] Rambal | From daily transpiration to seasonal water balance: an optimal use of water?[END_REF]; hydraulic resistance to water flow between roots and leaves, and root/shoot ratio, could also provide an explanation for the differences in water relations between cultivars.

Fig. 1 .

 1 Fig. 1. Partial functions of the stomatal conductance response to quantum flux density (Q), water vapour deficit (D), air temperature (T) and leaf water potential (~) (redrawn after Jones, 1983 ).

Fig. 2 .Fig. 3 .

 23 Fig. 2. Scatter diagram of stomatal conductance (g,) against leaf water potential (~) measured on the two carignane plots, with the curve derived from eq. 5 with ~/~ = -1.4 MPa and K4 = 2.

Fig. 4 .

 4 Fig. 4. Diurnal variations of observed (solid line) and simulated (dotted line) stomatal conductance (the vertical bars show the standard deviation of the observed values).

TABLE 1 Parameter values of the model derived from the first data set (L and E refer to linear and exponen- tial humidity responses, respectively, r 2 is the coefficient of determination, n is the number of observations Flood plain Terrace

 1

		Merlot		Carignane	Carignane	Shiraz
	gsm (mmo1 m -2 s -1)	440		560		560		360
		(41)		(41)		(41)		(14)
	K1 (flmolm -2 s -l)				133
	To (°C)				30
	K,				2
	~u~ (MPa)	1.6		1.4		1.4		1.8
	K2 (kPa-' >< 10 -3)	88 L	114 E	95 L	116 E	96 L	125 E	24 L	26 E
		(13)	(18)	(13)	(18)	(12)	(17)	(12)	(14)
	K3 (~C 2X10 4)	46 L	45 E	51L	50 E	47 L	50 E	40 L	40 E
		(4)	(4)	(4)	(4)	(4)	(4)	(4)	(4)
	r 2 (%)	80 L	82 E	85 L	85 E	83 L	84 E	88 L	88 E
	n=114

TABLE 2

 2 Range of environmental variables used for estimation of the parameters

		Flood plain		Terrace
		Merlot	Carignane	Carignane	Shiraz
	Q (/lmol m -2 S -1)	7-2080	27-2060	34-2050	6-2140
	Ta ( °C )	15.6-38.4	15.6-37.6	17.2-38.0	16-38.6
	D (kPa)	0.42-4.27	0.44-4.45	0.72-5.0	0.48-5.22
	q/(MPa)	0.25-1.57	0.28-1.40	0.32-1.53	0.45-1.63

TABLE 3 Comparison of stomatal conductances observed and simulated for the four

 3 experimental plots ( r is the coefficient of correlation, n is the

number of observations) Flood plain

												Terrace
		Merlot			Carignane					Carignane	Shiraz
	Mean residual	-0.07*				-0.01 "8					-0.03""	-0.03"
	r	0.67				0.65					0.77	0.74
	n	45					42					45	52
	"~ not significant.
	* significantly different at P= 0.05.
	400 /	CARIGNANE t3/06/t986 ....	i	I	SHIRAZ 13/06/1986 I T q
	I-
	-S 8o~ ,'
	0 /	o I	I	I	I	I	[d	I	I		I	I	I
	5	7	9	11 15 15	17 5	7	9 11 13 15 17
	400|	,	09/07/1986 ....					09/07/1986
	I
	-S 8o~
	01	I	L	J	i	I			I	I	I	I
	5	7	9 11 15 15 17	7	9 1t	13 15 17
	480 /			15/08/1986					15/08/1986
	~" 400~-
	32o~-24ot-16o~ 8o r 0 i 5	I 7	I 9	I 11 13 15 J I	t, 17 5	7	I 9	1 11 13 15 17 I I
		TIME (LST)					TIME (LST}

ACKNOWLEDGEMENTS

We are most grateful to M. Grandjanny and M. Maistre for help with the field work. Special thanks are extended to J.C. Jacquinet and his staff of agricultural advisers of the Chambre d'Agriculture de l'Aude. We also thank V. Ghersi-Egea and P. Alpert for improving the quality of the text. This study was conducted with the support of the Programme Interdisciplinaire de Recherches sur l'Environnement (PIREN) of the C.N.R.S. .