Daniel Edgington-Mitchell

Tianye Wang

Petronio Nogueira

Oliver Schmidt

Vincent Jaunet

Daniel Duke

Peter Jordan

Aaron Towne

Waves in screeching jets

The interaction between various wavelike structures in screeching jets is considered via both experimental measurements and linear stability theory. Velocity snapshots of screeching jets are used to produce a reduced order model of the screech cycle via proper orthogonal decomposition. Streamwise Fourier filtering is then applied to isolate the negative and positive wavenumber components, which for the waves of interest in this jet correspond to upstream and downstream-travelling waves. A global stability analysis on an experimentally derived base flow is conducted, demonstrating a close match to the results obtained via experiment, indicating that the mechanisms considered here are well represented in a linear framework. In both the global stability analysis and the experimental decomposition, three distinct wavelike structures are evident; these waves are also solutions to the cylindrical vortex sheet dispersion relation. One of the waves is the well-known downstream-travelling Kelvin-Helmholtz mode. Another is the upstreamtravelling guided jet mode that has been a topic of recent discussion by a number of authors. The third component, with positive phase velocity, has not previously been identified in screeching jets. Via a local stability analysis (LSA), we provide evidence that this downstream-travelling wave is a duct-like mode similar to that recently identified in high-subsonic jets. We further demonstrate that both of the latter two waves are generated by the interaction between the Kelvin-Helmholtz wavepacket and the shock cells in the flow. Finally, we consider the periodic spatial modulation of the coherent velocity fluctuation evident in screeching jets, and show that this modulation can be at least partially explained by the superposition of the three wavelike structures, in addition to any possible modulation of the Kelvin-Helmholtz wavepacket by the shocks themselves.

Introduction

Shock-containing free shear flows frequently exhibit some form of aeroacoustic resonance, the best-known of which are those that produce screech or impingement tones [START_REF] Edgington-Mitchell | Aeroacoustic resonance and self-excitation in screeching and impinging supersonic jets-a review[END_REF]. These resonance mechanisms can be divided into four discrete processes: a downstream-travelling wave [START_REF] Tam | Theoretical model of discrete tone generation by impinging jets[END_REF][START_REF] Gudmundsson | Instability wave models for the near-field fluctuations of turbulent jets[END_REF][START_REF] Sinha | Wavepacket models for supersonic jet noise[END_REF], a downstream-reflection mechanism [START_REF] Manning | A numerical investigation of sound generation in supersonic jet screech[END_REF][START_REF] Suzuki | Shock leakage through an unsteady vortex-laden mixing layer: application to jet screech[END_REF][START_REF] Shariff | A ray tracing study of shock leakage in a model supersonic jet[END_REF][START_REF] Berland | Numerical study of screech generation in a planar supersonic jet[END_REF][START_REF] Edgington-Mitchell | The generation of screech tones by shock leakage[END_REF], an upstream-travelling wave (Tam & Hu 1989a;[START_REF] Shen | Three-dimensional numerical simulation of the jet screech phenomenon[END_REF][START_REF] Bogey | Feedback loop and upwind-propagating waves in ideally expanded supersonic impinging round jets[END_REF][START_REF] Gojon | Oscillation modes in screeching jets[END_REF][START_REF] Edgington-Mitchell | Upstream-travelling acoustic jet modes as a closure mechanism for screech[END_REF][START_REF] Jordan | Jet-flap interaction tones[END_REF], and an upstream-reflection or receptivity mechanism in the nozzle plane [START_REF] Barone | Receptivity of the compressible mixing layer[END_REF][START_REF] Mitchell | The visualization of the acoustic feedback loop in impinging underexpanded supersonic jet flows using ultra-high frame rate schlieren[END_REF][START_REF] Weightman | Nozzle external geometry as a boundary condition for the azimuthal mode selection in an impinging underexpanded jet[END_REF][START_REF] Karami | Receptivity characteristics of under-expanded supersonic impinging jets[END_REF]. Of these, the downstream-travelling wave is generally thought to be the only process where energy is provided to the resonance loop [START_REF] Tam | Theoretical model of discrete tone generation by impinging jets[END_REF]; the growth of the instability wave is driven by the extraction of energy from the mean flow. Amplitude prediction models for aeroacoustic resonance have remained elusive, and while there are several models capable of frequency prediction [START_REF] Powell | On the mechanism of choked jet noise[END_REF][START_REF] Tam | Proposed relationship between broadband shock associated noise and screech tones[END_REF]), these models often struggle to fully explain the staging behaviour typical of jet screech (Mancinelli et al. 2019a). In this context, staging behaviour refers to the tendency of resonant systems to experience discontinuous changes in tone frequency with small changes in operating conditions [START_REF] Davies | Tones from a choked axisymmetric jet[END_REF][START_REF] Powell | Observations of the oscillation modes of choked circular jets[END_REF]Edgington-Mitchell et al. 2015b;[START_REF] Li | Acoustic feedback loops for screech tones of underexpanded free round jets at different modes[END_REF]. In screeching axisymmetric jets, these stages are typically classified into A1 & A2 (m = 0), B & D (flapping), and C (m = 1) helical modes.

Upstream-travelling waves in jet resonance

Screech, like other resonant processes in jets, involves an energy exchange between upstream-and downstream-travelling waves. It is generally accepted that the downstream-travelling wave relevant to resonance in jet screech is the Kelvin-Helmholtz (KH) wavepacket. The nature of the upstream-travelling wave is less clear. [START_REF] Powell | On the mechanism of choked jet noise[END_REF] originally conceived the upstream-travelling wave as a freestream acoustic wave, a view that went unchallenged for many decades. The supporting evidence for this theory was quite strong: a sharp tone is evident in the farfield acoustics, and resonance models based on an upstream-travelling wave with sonic phase speed generally predicted frequency well. It was only in the work of [START_REF] Shen | Three-dimensional numerical simulation of the jet screech phenomenon[END_REF] that an alternative was proposed: that the upstream-travelling wave is not a freestream acoustic wave, but rather a guided jet mode. Evidence for the role of this wave was first provided for subsonic impinging jets [START_REF] Tam | Theoretical model of discrete tone generation by impinging jets[END_REF], then supersonic impinging jets [START_REF] Bogey | Feedback loop and upwind-propagating waves in ideally expanded supersonic impinging round jets[END_REF][START_REF] Jaunet | Dynamics of round jet impingement[END_REF], and finally jet screech [START_REF] Gojon | Oscillation modes in screeching jets[END_REF][START_REF] Edgington-Mitchell | Upstream-travelling acoustic jet modes as a closure mechanism for screech[END_REF]. Frequency prediction models based on the upstream-travelling guided jet mode outperform those that assume a freestream acoustic wave, at least for the m = 0 screech modes (Mancinelli et al. 2019a,b). It should be noted, however, that at this point there is still evidence that some resonant processes are indeed closed by freestream acoustic waves; [START_REF] Weightman | Nozzle external geometry as a boundary condition for the azimuthal mode selection in an impinging underexpanded jet[END_REF] provide evidence for such closures in various supersonic jet impingement cases.

Other waves present in supersonic jets

The guided jet mode is one of three families of waves that can be supported by a supersonic jet. These waves were first visualized by [START_REF] Oertel | Mach wave radiation of hot supersonic jets investigated by means of a shock tube and new optical techniques[END_REF]. Tam & Hu (1989a) then demonstrated that an inviscid high-speed jet, when modelled as a cylindrical vortex sheet, can support three families of waves: the Kelvin-Helmholtz wave, subsonic instability waves, and supersonic instability waves. The supersonic instability waves are only present for very high jet Mach numbers, whereas the subsonic waves are present across all Mach numbers. For supersonic jets, these subsonic instability waves, which can propagate both upstream and downstream, are organised hierarchically according to their azimuthal and radial order. With the exception of the axisymmetric mode of radial order 1, the upstream-travelling waves are confined to a narrow frequency band, whereas the downstream-travelling wave can be supported across a wide range of frequencies. The upstream-traveling wave identified in the vortex-sheet dispersion relation is the guided jet mode that has been demonstrated to play a significant role in resonance, including the global instability of hot jets and wakes [START_REF] Martini | Acoustic modes in jet and wake stability[END_REF]. The downstream-travelling waves have not previously been discussed in the context of supersonic resonance, but they have been discussed extensively for high Mach-number subsonic jets in the works of [START_REF] Towne | Acoustic resonance in the potential core of subsonic jets[END_REF], [START_REF] Schmidt | Wavepackets and trapped acoustic modes in a turbulent jet: coherent structure eduction and global stability[END_REF][START_REF] Jordan | Jet-flap interaction tones[END_REF]. More recently, experimental data presented in [START_REF] Zaman | Pressure fluctuations due to trapped waves in the initial region of high-speed jets[END_REF] has demonstrated that A1 and A2 screech frequencies are closely related to the frequencies of trapped-wave resonance demonstrated in [START_REF] Towne | Acoustic resonance in the potential core of subsonic jets[END_REF]. Though both the upstream-and downstream-travelling waves are associated with the same fundamental mechanism, they have distinct radial structures. In supersonic jets, the upstream-travelling wave has support outside the shear layer of the jet, a requirement for an upstream-travelling wave in a flow that is travelling downstream at supersonic velocity. The downstream-travelling wave by contrast remains essentially trapped within the core of the jet; the work of [START_REF] Towne | Acoustic resonance in the potential core of subsonic jets[END_REF] demonstrated that this mode also obeys the dispersion relation for a soft-walled duct, essentially treating the shear layer of the jet as a pressure-release boundary. While these downstream-traveling duct-like modes have strictly negative phase velocities in subsonic jets, they can have either negative or positive phase velocity in supersonic jets [START_REF] Towne | An investigation of the Mach number dependence of trapped acoustic waves in turbulent jets[END_REF].

Wave interactions in screeching jets

It is generally accepted that screech tones are produced by some interaction between the KH waves and the shock/expansion structure in the jet core. The first model for the prediction of screech frequency was that of [START_REF] Powell | On the mechanism of choked jet noise[END_REF]. Powell assumed that interactions between the downstream-travelling wave and the shocks could be modelled as emission from a phased-array of equispaced monopoles located at the shock reflection points. Observing that screech radiates most strongly in the upstream direction, Powell further assumed maximum upstream directivity as a requirement for screech (i.e. that waves from all three monopole sources would arrive at the nozzle simultaneously and thus provide constructive reinforcement), and on this basis produced his predictive equation for screech,

f = U c s(1 + M c ) . (1.1)
Here f is the frequency of the screech tone, s is the spacing of the shock cells, and U c and M c are the convection velocity and convective Mach number respectively. In a later paper [START_REF] Powell | Observations of the oscillation modes of choked circular jets[END_REF], Powell reconsidered this model and stated that there was no reason to assume perfect reinforcement at the nozzle lip was a requirement for screech. Nonetheless this model does an admirable job of predicting screech across a range of operating conditions, but is incapable of accounting for the mode-staging behaviour typical of aeroacoustic resonance.

An alternative model was proposed by [START_REF] Tam | Proposed relationship between broadband shock associated noise and screech tones[END_REF], an extension of work begun in [START_REF] Tam | Shock associated noise of supersonic jets from convergentdivergent nozzles[END_REF]. In the earlier work, Tam and Tanna construct a model for broadband shock noise, on the assumption of weak interaction between travelling KH waves and stationary shock waves. Key points of the model are recapitulated here, though the nomenclature used is slightly different than in the original paper. The downstreamtravelling KH waves can be modelled as a wavepacket of the form u kh = a(x)ψ(r)e i(k kh x-ωt) .

(1.2)

Here u kh represents a velocity perturbation associated with the KH wavepacket, a(x) is a spatial amplitude distribution, ψ(r) is the radial eigenfunction of the KH wavepacket, k kh is the wavenumber and ω the frequency.

The spatial modulation of velocity (u s ) by the quasi-stationary shock cell structures within the flow was modelled in the work of [START_REF] Tam | Shock associated noise of supersonic jets from convergentdivergent nozzles[END_REF] using the vortex-sheet approach of [START_REF] Prandtl | Über die stationären Wellen in einem Gasstrahl[END_REF] and [START_REF] Pack | A note on prandtl's formula for the wave-length of a supersonic gas jet[END_REF], which can be expressed in simplified form as

u s = ∞ n=1 A n (e iks n x + e -iks n x ) (1.3)
Here A n defines the amplitude of each shock cell mode, while k sn defines the wavenumber, for n = 1, 2, 3 . . . . As stated in [START_REF] Tam | Shock associated noise of supersonic jets from convergentdivergent nozzles[END_REF] and shown more explicitly in [START_REF] Ray | Sound generated by instability wave/shock-cell interaction in supersonic jets[END_REF], the interaction between the KH wavepacket and the stationary shocks can be represented by the product of the two wave expressions. Ignoring amplitude terms, for the first shock cell mode n = 1 this can be written as

u kh u s ∝ e (i(k kh +ks)x-iωt) + e (i(k kh -ks)x-iωt) .
(1.4)

Alternatively, this relation can be directly obtained from the analysis a convective term of the Navier-Stokes equations. Considering only the first shock-cell mode, the mean flow in the streamwise direction U can be written as

U (x, r) = U sm (x, r) + U sh (x, r) 1 2 e iksx + e -iksx , (1.5)
where U sm is the shock-less mean flow (which can be obtained using a low-pass filter, for example), U sh is the slow-varying part of the shock-cell structure (or the envelope), which includes the amplitude term A 1 , and k s is the wavenumber associated with its oscillatory part. The subscripts sm and sh are used to denote the smooth and shock-related parts of the mean. Expanding the velocity as ũ(x, r) = U (x, r) + u(x, r, θ, t), all convective terms of the streamwise momentum equations will have a dependency on both U sm and U sh .

For instance, the linearised form of ũ∂ x ũ -U ∂ x U is given by

ũ∂ x ũ -U ∂ x U ≈ [U sm + U sh 1 2 e iksx + e -iksx ]∂ x u + u∂ x [U sm + U sh 1 2 e iksx + e -iksx ] (1.6) = U sm ∂ x u + U sh 1 2 e iksx + e -iksx ∂ x u + u∂ x U sm + u(∂ x U sh ) 1 2 e iksx + e -iksx + uU sh 1 2 ik s e iksx -ik s e -iksx .
(1.7)

Considering that U sm and U sh are slow varying, the streamwise derivatives of these quantities will be disregarded here, as in locally parallel analyses. Using the normal modes ansatz, disturbances can be written as u(x, r, θ, t) = u (r)e -iωt+ikxx+imθ , where ω is the frequency, k x is the streamwise wavenumber and m is the azimuthal wavenumber. Thus, (1.7) can be rewritten as

ũ∂ x ũ -U ∂ x U ≈ ik x U u e -iωt+ikxx+imθ + +iu U sh 1 2
k s e -iωt+i(kx+ks)x+imθ -k s e -iωt+i(kx-ks)x+imθ .

(1.8)

A clear connection between the most amplified structures in the flow and the generation of waves at other wavenumbers is evident in (1.8). If the wavepacket wavenumber is considered (k x = k kh ), (1.8) becomes equivalent to the well-known expression originally presented by [START_REF] Tam | Shock associated noise of supersonic jets from convergentdivergent nozzles[END_REF], demonstrating that new wavenumbers are energised by the interaction between shocks and the wavepacket.

It is clear from (1.4) and (1.8) that the interaction of the KH wave with the stationary shocks produces two wave-like disturbances for a given frequency. These wavelike disturbances have wavenumbers dictated by the sum and difference of wavenumbers associated with the KH wavepacket (k kh ) and the shock cells (k s ). For cases where the KH wavenumber is smaller than the shock spacing (as is typically observed in screeching jets), the difference wavenumber is negative, and the wave has negative phase velocity. The sum term will always represent a wave that has positive phase velocity. In this work we do not explicitly use the model of [START_REF] Tam | Shock associated noise of supersonic jets from convergentdivergent nozzles[END_REF], however (1.4) will be used to explain phenomena observed in Section 3.

The model of [START_REF] Tam | Shock associated noise of supersonic jets from convergentdivergent nozzles[END_REF] was originally developed to explain broadband shock-associated noise (BBSAN), but [START_REF] Tam | Proposed relationship between broadband shock associated noise and screech tones[END_REF] suggested that screech could simply be considered a special case of BBSAN, citing data from [START_REF] Norum | Measurements of mean static pressure and far field acoustics of shock containing supersonic jets[END_REF]. On this basis the authors developed a predictive model for screech frequency, taking the limit of the BBSAN model of [START_REF] Tam | Shock associated noise of supersonic jets from convergentdivergent nozzles[END_REF] as the observer angle approaches the upstream axis,

f s = U c k s 2π(1 + M c )
.

(1.9)

While this relation provides identical predictions to the model of [START_REF] Powell | On the mechanism of choked jet noise[END_REF], its provenance is rather different. The central thesis of [START_REF] Tam | Proposed relationship between broadband shock associated noise and screech tones[END_REF] is that the screech frequency is selected by the weak interaction of the KH wavepackets with the shock structures; this interaction only produces radiation back to the nozzle lip in a narrow band of frequencies.

Wave modulation

Most existing models for screech assume that the screech tone is produced by some form of interaction between the KH wave and the shock structures within the jet. It is also typically assumed that the KH wavepacket is essentially unaffected by this interaction. Part of the motivation of this paper is the evaluation of the validity of this assumption. It is well recognized that turbulence undergoes significant changes during passage through a shock wave. Both the shock and the turbulence are influenced by this interaction; the shock becomes locally distorted, while the turbulence sees an amplification of intensity and Reynolds stress [START_REF] Ducros | Large-eddy simulation of the shock/turbulence interaction[END_REF]. This amplification has been shown to depend on the scale of the turbulence, with finer scales amplified more than large scales. The interaction of a vortex with a shock strongly depends on the strength of both the shock and the vortex. Passage through a strong shock has been shown to significantly deform the shape of an isolated vortex [START_REF] Grasso | Shock-wave-vortex interactions: shock and vortex deformations, and sound production[END_REF]. The situation in a screeching jet is more complicated: the train of vortices that comprise the wavepacket typically span the sonic line of the jet, meaning some portion of the vortices may pass through the shock cell, whereas components further from the centreline do not. It is thus presently unclear whether the structure or growth of the KH wavepacket is altered or modulated via interaction with the shock cells of the jet.

It is well established that the turbulence fluctuations in the nearfield of the jet are strongly modulated, due to the presence of the standing wave in the acoustic nearfield of the jet [START_REF] Westley | The near field sound pressures of a choked jet when oscillating in the spinning mode[END_REF][START_REF] Panda | An experimental investigation of screech noise generation[END_REF]. Formed by the interaction of the downstream-travelling hydrodynamic waves and the upstream-travelling waves, this standing wave is clearly evident in measures of both fluctuating pressure and velocity. Thus measurements of velocity in screeching jets show strong modulation in the axial direction, but the presence of the standing wave makes it difficult to determine whether this is simply the signature of the standing wave, or the shocks modulating the growth of the KH wavepacket.

Linear models for the screech problem

In general, the initial growth of the KH wavepacket can be well predicted by the careful application of linear stability theory [START_REF] Michalke | Survey on jet instability theory[END_REF][START_REF] Morris | The instability of high speed jets[END_REF], even for highly turbulent jets [START_REF] Cavalieri | Wavepackets in the velocity field of turbulent jets[END_REF]. One outstanding question is how well such linear models perform in shock-containing flows, and whether or not they can provide a description of the non-linear KH-shock interaction when the shock structure is included in the mean flow. There have been limited attempts to apply linear stability theory to shock containing jets. In a global stability analysis of a shock-free jet, [START_REF] Nichols | Global modes and transient response of a cold supersonic jet[END_REF] observed the subsonic modes of Tam & Hu (1989a), and suggested that the upstreamtravelling subsonic modes could underpin resonance in shock-containing flows. [START_REF] Beneddine | Global stability analysis of underexpanded screeching jets[END_REF] conducted a global analysis on a laminar, two-dimensional shock-containing jet, and demonstrated that it exhibits a global instability that matches many of the characteristics of jet screech. [START_REF] Beneddine | Global stability analysis of underexpanded screeching jets[END_REF] were also able to demonstrate the sensitivity of this instability to the thickness of the nozzle lip, which has been observed experimentally for both screeching [START_REF] Raman | Cessation of screech in underexpanded jets[END_REF]) and impinging [START_REF] Weightman | Nozzle external geometry as a boundary condition for the azimuthal mode selection in an impinging underexpanded jet[END_REF] jets. Nonetheless, the validity of linear models for the screech process remains unclear; there are many processes involved in jet screech that are non-linear.

Summary

There remain a broad range of open questions regarding jet screech. What mechanism selects the screech frequency? Which of the processes underpinning screech can be modelled using linear theory? Does the interaction between the KH wavepacket and the shocks affect the growth of the wavepacket? In an attempt to answer these questions, we pair an extensive experimental database with a range of stability analyses. Experimentally, we consider three jets undergoing screech, two jets characterized by an m = 0 instability mode, but with shocks of different strengths, and one jet whose screech is characterized by an m = 1 helical mode, with much stronger shocks. The velocity fields recorded for these jets are decomposed on both an energy and spatial wavenumber basis. A global stability analysis is performed on the experimentally-determined baseflow for the case with the weakest shocks. Local stability analysis is used to interpret some of the results from both the experiment and the global analysis.

Database and Methodology

Experimental Database

The experimental database used here has been well documented in prior literature. All cases considered are from similar experimental facilities (and the same nozzle); however, the details of the velocimetry differ somewhat. Two cases are presented for the m = 0 mode, the A1 and A2 modes of jet screech at pressure ratios of NPR = 2.10 & 2.25 respectively, previously studied in Edgington- Mitchell et al. (2018). Note a slight correction to the Strouhal numbers associated with this data presented in the prior paper. An additional case at a nozzle pressure ratio of NPR = 3.40 is presented, where the flow is characterized by an m = 1 mode associated with the helical C screech mode (Edgington- Mitchell et al. 2014b;[START_REF] Tan | Novel method for investigating broadband velocity fluctuations in axisymmetric screeching jets[END_REF][START_REF] Li | Shock motion and flow structure of an underexpanded jet in the helical mode[END_REF]. A summary of the relevant parameters for the jets is presented in table 1; N P R refers to ratio of stagnation to ambient pressure, M j , U j and D j refer to the ideally-expanded Mach number, velocity and diameter, respectively, and the frequency is non-dimensionalized such that St = f D j /U j . All jets considered here issue from a purely converging nozzle of diameter D = 15 mm, with a radius of curvature of 67.15 mm, ending with a parallel section at the nozzle exit, and an external lip thickness of 5 mm. The nozzles are connected to a large plenum chamber; the area ratio between the nozzle and plenum is approximately 100:1. As a consequence of this high contraction ratio, it is expected that the boundary layer at the nozzle exit will be laminar and extremely thin (below the measurement resolution of the PIV system).

For the NPR = 2.10 & 2.25 datasets, particle images were obtained using a 12-bit Imperx B4820 camera, with a CCD array of 4872 × 3248 px, at an acquisition frequency of 2 Hz. Illumination was provided by a Nd:YAG laser, producing a pair of 6 ns pulses of approximately 160 mJ, separated by ∆t = 1 µs. For the NPR = 3.40 dataset, particle images were obtained using a pair of PCO 4000 cameras mounted orthogonal to the jet, each with a CCD array of 4008 × 2760 px. The resultant velocity fields from the two cameras were stitched together using a convolution with an adaptive Gaussian window [START_REF] Agüí | On the performance of particle tracking[END_REF] with an overlap of 7.5%. Illumination was provided by a Nd:YAG laser, producing a pair of 6 ns pulses of approximately 120 mJ, separated by ∆t = 0.8 µs.

Both jets were seeded with smoke particles, whose diameter was estimated at 600 nm based on observed relaxation times across a normal shock [START_REF] Mitchell | Near-field structure of underexpanded elliptic jets[END_REF]. The pertinent PIV parameters are summarized in table 2. The images were analyzed using a multi-grid cross-correlation algorithm [START_REF] Soria | An investigation of the near wake of a circular cylinder using a video-based digital cross-correlation particle image velocimetry technique[END_REF], where IW 0 and IW 1 refer to initial and final interrogation windows, respectively.

The mean velocity (U, V ) fields are presented in figure 1 for the m = 0 cases and figure 2 for the m = 1 case. For the NPR = 2.10 case, the transverse velocity due to the shocks does not exceed 3% of the jet exit velocity U e , while for the NPR = 3.4 case the transverse velocity is in excess of 20% of the jet exit velocity.

Decomposition of Experimental Database

The turbulent wavepackets that comprise the downstream-travelling component of aeroacoustic resonance typically only represent a small percentage of the total turbulent kinetic energy in a flow (Jordan & Colonius 2013;[START_REF] Jaunet | Two-point coherence of wave packets in turbulent jets[END_REF][START_REF] Towne | Spectral proper orthogonal decomposition and its relationship to dynamic mode decomposition and resolvent analysis[END_REF][START_REF] Schmidt | Spectral analysis of jet turbulence[END_REF]. Eduction of the signatures of these wavepackets from experimental data thus requires some form of modal decomposition, such as those reviewed in [START_REF] Taira | Modal analysis of fluid flows: An overview[END_REF].

Proper orthogonal decomposition (POD) is arguably the most widely-used decomposition method in fluid mechanics broadly [START_REF] Berkooz | The proper orthogonal decomposition in the analysis of turbulent flows[END_REF][START_REF] Sirovich | Turbulence and the dynamics of coherent structures. i. coherent structures[END_REF], and flow resonance in particular [START_REF] Moreno | Low-dimensional model of a supersonic rectangular jet[END_REF]Edgington-Mitchell et al. 2014a, 2015a;[START_REF] Weightman | An explanation for the phase lag in supersonic jet impingement[END_REF]. The highly-periodic nature of resonant flows makes them particularly amenable to POD, as it is typically possible to reconstruct the entire resonance cycle from only the leading POD modes [START_REF] Oberleithner | Three-dimensional coherent structures in a swirling jet undergoing vortex breakdown: stability analysis and empirical mode construction[END_REF]); a travelling wave structure will be defined by a pair of POD modes, with a 90 • phase offset between them [START_REF] Deane | Low-dimensional models for complex geometry flows: application to grooved channels and circular cylinders[END_REF][START_REF] Noack | A hierarchy of low-dimensional models for the transient and post-transient cylinder wake[END_REF]. In the present database the velocity data is not time resolved, and thus decomposition such as spectral POD [START_REF] Towne | Spectral proper orthogonal decomposition and its relationship to dynamic mode decomposition and resolvent analysis[END_REF]) cannot be implemented.

The ability of POD to isolate the structures associated with the resonant process enables a triple decomposition on the basis that the velocity may be represented as the sum of a mean (U), a coherent u c and a stochastic u component after [START_REF] Hussain | The mechanics of an organized wave in turbulent shear flow[END_REF],

u(x, t) = U(x) + u c (x, t) + u (x, t).
(2.1)

To educe the coherent component via POD, an autocovariance matrix (R) is constructed from the velocity snapshots (V) such that R = V T V. The solution of the eigenvalue problem Rv = λv yields the eigenvalues λ and eigenvectors v from which the spatial POD modes (φ) are constructed as

φ n (x, y) = Vv n (t) ||Vv n (t)|| , (2.2)
and the coefficients at each time t for each mode n can be expressed as

a n (t) = v n (t)||Vv n (t)||. (2.3)
Assuming that the leading pair of POD modes will identify fluctuations occurring at the screech frequency ω s , we define [START_REF] Jaunet | Pod-galerkin advection model for convective flow: application to a flapping rectangular supersonic jet[END_REF]): a = a 1 -ia 2 = âe -iωst and ψ = φ 1 + iφ 2 . To ensure that the two leading modes are indeed the modal pair representing screech, the decomposition is conducted only on the transverse component of velocity; in this way the energy associated with the shear-thickness mode identified in the work of [START_REF] Weightman | Signatures of shear-layer unsteadiness in proper orthogonal decomposition[END_REF] is minimized. As a consequence the modes are ranked only on the energy associated with the transverse velocity component. On this basis the temporal Fourier coefficients can be constructed directly from the POD modes, and the coherent fluctuations can be represented as u c (x, y, t) = âe -iωst ψ(x, y).

(2.4)

To identify wavelike structures in the flow, it is advantageous to consider a further decomposition in the streamwise direction,

u c (x, y, t) = âe -iωst k ûc k (y)e ikx .
(2.5)

With the start and end of the domain defined as x 1 and x 2 respectively, the spatial Fourier coefficients are defined as

ûc k (y) = x2 x1 ψ(x, y)e -ikx .
(2.6)

Global stability analysis

We conduct a global linear stability analysis to explore the characteristics and behavior of the waves involved in the screech process. Applying a Reynolds decomposition q(x, r, θ, t) = q(x, r) + q (x, r, θ, t)

(2.7)

to the compressible Navier-Stokes equation and neglecting nonlinear terms yields the linearized Navier-Stokes equation, ∂q ∂t -A(q) q = 0.

(2.8)

Here, q(x, r, θ, t) is a state vector containing velocities and thermodynamic variables; for the round jets considered in this paper we use cylindrical coordinates and velocities and choose density and pressure as the thermodynamic variables. Applying the normal mode ansatz q (x, r, θ, t) = q(x, r) exp (imθ -iωt) , (2.9) to (2.8) yields the eigenvalue problem

(-iωI -A m ) q = 0, (2.10)
where A m is obtained by replacing all azimuthal derivatives in A with im. The azimuthal wavenumber m is an integer due to the periodicity of the mean jet. The global modes of the jet correspond to ω, q pairs that satisfy (2.10) for a given choice of m, i.e., the eigenvalues and eigenvectors of A m . Significant uncertainty remains regarding the effect of shocks and shock-like discontinuities on the validity of a linear stability analysis; we seek to minimize any such effects in two ways. Firstly, we consider only the NPR = 2.10 case, where the shocks are weak, and the transverse velocities are minimal. Much of the compression in a jet operating at this condition is achieved in a continuous fashion through near-isentropic compression waves. Additionally, the inability of tracer particles to faithfully reproduce step changes in velocity is actually of benefit here; the discontinuities around shocks are inherently smoothed by our measurement technique.

The mean flow q(x, r) is obtained from the experimental measurements. To provide a sufficiently large domain for the linear analysis, the experimental domain is extrapolated to a field covering -1 x/D 19 and 0 r/D 4. Density information is not directly available from PIV; however, by assuming that entropy generation by the weak shocks is negligible, an assumption of constant stagnation density along streamlines allows for a calculation of density from the velocity data. This is checked using a tomographic Background-Oriented-Schlieren measurement, with the axisymmetric field extracted from the path-integrated data using an Abel inversion [START_REF] Tan | Measurement of density in axisymmetric jets using a novel background-oriented schlieren (bos) technique[END_REF]; errors due to the isentropic assumption appear to be less than 5%.

The linearized equations are discretized using fourth-order finite differences with summation-by-parts closure at the boundaries. To ensure adequate resolution in the shear layer of the jet and to apply non-reflecting boundary conditions, the computational domain is divided into several zones as shown in figure 3. The core domain covers 0

x/D 8 and 0 r/D 0.3, the shear layer zone covers 0.3 r/D 0.7, and the far-field extends to r/D = 3. Sponge layers are placed upstream (x/D 0) and downstream (x/D 15) of the jet, as well as outside the far-field region. There are N r = 300 points in the radial direction: 100 in the core zone, 100 in the shear layer, and the rest shared between the far-field and the sponge layer. The streamwise direction is discretized using N x = 750 points, with a mapping used to ensure that 60% of these points are placed in the core region. Additional details of the global stability code and on the underlying formulation of the Navier-Stokes equations can be found in [START_REF] Schmidt | Wavepackets and trapped acoustic modes in a turbulent jet: coherent structure eduction and global stability[END_REF] and [START_REF] Babucke | Direct numerical simulation of noise-generation mechanisms in the mixing layer of a jet[END_REF], respectively.

The analysis was performed at a lower Reynolds number as compared to the experiments (Re = ρ j U j D/µ = 10 4 , where ρ j and U j are the ideally expanded density and velocity, and µ is the dynamic viscosity). This choice was motivated by recent work demonstrating that the use of an eddy-viscosity or turbulent Reynolds number to account for the impact of Reynolds stresses improves the agreement between linear analyses and experimental and simulation data for turbulent jets [START_REF] Pickering | Optimal eddy viscosity for resolvent-based models of coherent structures in turbulent jets[END_REF]; no eddyviscosity model was used in the present analysis. The lower Reynolds number has the added advantage of reducing the cost of solving the eigenvalue problem. A consideration of the sensitivity of the result to Reynolds number is presented in Appendix A.

The eigenvalue spectrum obtained from the global eigenvalue problem is presented in figure 4. The spectrum is composed of continuous branches of modes plus a few discrete modes, which are distinct from the main branches and arise at particular frequencies. The least-stable mode is associated with a frequency of St = 0.667, less than a 1% discrepancy from the experimentally observed screech tone at frequency St = 0.67 for this jet. The remainder of the analysis focuses on this least-stable mode.

Local stability analysis

We also conduct a local stability analysis of the same mean flow to aid in identifying the waves observed in the experimental POD and global stability modes. In this case, the appropriate normal mode ansatz is

ξ(x, r, θ, t) = ξ(r) exp (ikx + imθ -iωt) , (2.11)
where k is the streamwise wavenumber for the mode. Setting the mean flow within the operator A to its local value, ξ(x = x 0 , r), at a particular streamwise position x 0 and applying the ansatz (2.11) to (2.8) leads to an eigenvalue problem of the form (-iωI + ikA x + A m,0 ) ξ = 0.

(2.12)

The operator A x is the portion of A associated with x-derivatives while A m,0 contains all remaining terms. For consistency with the local linear stability model of [START_REF] Towne | Acoustic resonance in the potential core of subsonic jets[END_REF] as well as the vortex sheet model used by Tam & Hu (1989b) and others, all viscous terms are neglected. This leads to an eigenvalue problem that is linear in k. Both A x and A m,0 depend on the position x 0 at which the mean flow has been frozen. For a specified m, the local eigenmodes of the jet are given by pairs (k, ω) and vectors ξ satisfying (2.12).

The spatial stability analysis used in this paper is carried out by choosing real values for the frequency ω and solving the eigenvalue problem for the (potentially complex-valued) wavenumber k. Details on the operators and numerics used for the eigenvalue problem are reported elsewhere (Towne 2016).

Results

Wave generation in screeching jets

In figure 5 we compare the spatial structure of the least-stable mode in the global analysis to one of the leading POD modes extracted from the experimental data. In this work the radial co-ordinate r from the stability analysis is represented as the transverse co-ordinate y for consistency with the PIV measurements. A remarkable similarity between the experimental POD mode and the global mode is apparent. While the streamwise velocities are not identical, given that the global mode was calculated from a linearization of an extrapolated base flow, the agreement is surprisingly good. In figure 6 the pressure field associated with the same global mode is presented. While spatially resolved pressure data are not available for the measurement, the pressure field captures many of the recognized features of a screeching supersonic jet, including the strong upstream-propagating acoustic waves, and the downstream-propagating Mach wave radiation.

The normalized coherent fluctuations from the experimental data |ψ| for the m = 0 jets is presented in figure 7, and for the m = 1 jet in figure 8. The coherent fluctuations are quite similar for the two m = 0 modes, with spatial modulation apparent at both the edges of the shear layer and along the jet centreline. For the m = 1 jet, modulation of the axial velocity component is evident both along the lipline and outside the shear layer, while for the transverse velocity component the strongest modulation occurs along the centreline. A comparison of the fluctuations determined from the least-stable global mode with those associated with the leading POD mode pair is provided in figure 9. Qualitatively, the two results are very similar. The streamwise velocity experiences modulation in the jet core and outside the shear layer for both the global analysis and the experimental data. The transverse velocity is even more closely matched. There are however also some key differences: the magnitude of fluctuations in the jet core is higher for the experimental data, and their modulation stronger.

While the fluctuations presented in the preceding figures are all associated with fluctuations at a given frequency, these fluctuations can be associated with a broad range of wavenumbers. Consequently, the wavenumber spectra presented in figure 10 are produced by taking the amplitude of (2.6). Phase velocity is defined as u p = ω/k x ; as the POD produces modes correlated to the screech phenomenon, here ω is fixed at the screech frequency: ω = ω s . Thus the sign of k x determines the sign of the phase velocity; here positive values of k x are associated with a phase velocity in the downstream direction. In this work, we use the sign of the phase velocity as a proxy for the sign of the group velocity. The group velocity determines the direction of energy propagation; waves with negative and positive group velocity may be considered upstream-or downstreamtravelling, respectively. All of the waves in question have phase and group velocities of the same sign in supersonic jets [START_REF] Towne | Acoustic resonance in the potential core of subsonic jets[END_REF], justifying the assumption made in this analysis. The dashed vertical white lines in figure 10 indicate wavenumbers associated with the ambient speed of sound in the upstream and downstream directions, while the dashed vertical red line indicates the wavenumber associated with the average spacing of the shocks in the flow. All three jets have the majority of the energy concentrated at a wavenumber associated with a phase velocity of u p ≈ 0.7U j , with radial structures typical of the classical KH wavepacket, hereafter referred to as the k + kh wave. All three jets also have a component with an upstream phase velocity approximately equal to the speed of sound; this is the signature of the guided jet mode previously documented in screeching jets [START_REF] Gojon | Oscillation modes in screeching jets[END_REF][START_REF] Edgington-Mitchell | Upstream-travelling acoustic jet modes as a closure mechanism for screech[END_REF], hereafter the k - th wave. There is evidence of a third wave at much higher positive wavenumber, observed for all jets, though with different radial structure for the m = 1 case compared to the m = 0 screech modes; this wave will be referred to as the k + t wave. The spectrum of the global mode analysis is cleaner than that of the experimental data as shown in figure 11, but the same three structures visible in the experimental data are likewise visible: an upstream-travelling guided jet mode, a downstream-travelling KH wave, and the high-wavenumber mode. The relative amplitudes (normalized against the k + kh wave) for both the k - th and the k + t mode are significantly stronger in the experimental data, but still clearly visible in the global mode. In the global analysis, the wavenumbers for all three waves are approximately ∆k x D = 0.5 higher than the corresponding wave in the experimental data.

We return now to the model of [START_REF] Tam | Shock associated noise of supersonic jets from convergentdivergent nozzles[END_REF], where it was suggested that the interaction of the KH wavepacket with the stationary shock structures should produce two additional travelling waves in the jet. Due to streamwise variation in the mean flow, the fluctuations associated with the KH wavepacket are spread across a small range of wavenumbers. Likewise, the shock spacing within the jet varies slightly as a function of axial position. Estimates of the wavenumbers associated with both the KH wavepacket and the shock cells are presented in table 3, with a rough estimate of the variation in wavenumber included. We then evaluate (1.4) to produce the two expected wavenumbers (sum and difference) resulting from the interaction between the KH wavepacket and the shock cells. These expected wavenumbers are presented in table 4, along with wavenumbers extracted from the the data presented in figure 10. The observed wavenumbers for all the experimental cases and the global analysis fall within the range of expected wavenumbers produced by (1.4). Thus the model of [START_REF] Tam | Shock associated noise of supersonic jets from convergentdivergent nozzles[END_REF] provides an explanation for the three wave structures observed in the decomposed experimental data and in the global analysis: one structure is the KH wavepacket, while the other two are waves produced by the interaction between the KH wavepacket and the quasi-stationary shock cells. While the model of [START_REF] Tam | Shock associated noise of supersonic jets from convergentdivergent nozzles[END_REF] predicts the wavenumbers of these waves, and provides an explanation for their mechanism of generation, it makes no statement regarding the character of these waves. A cursory examination of figure 10 reveals that each of the three modes has a distinct radial structure. In the following section, we consider the nature of these three waves.

The nature of waves in screeching jets

To better characterize the three wavelike structures in the jet, we consider their spatial features: first their axial variation in amplitude and then their radial structure. The spatial amplitude variation associated with the waves is shown qualitatively via the application of a cosine-tapered bandpass filter in the wavenumber domain. The filter has a half-width of 1.9D, a taper ratio of 0.5, and is centred on the maximum amplitude for each wave. The results are sensitive to the size and type of the filter; these results should thus only be taken as broadly indicative of peak location and general trends. The amplitude of the fluctuations resulting from this filtering is presented for NPR = 2.10 in figure 12 and for NPR = 3.40 in figure 13. The spatial amplitude distribution of the k + kh wavepacket closely resembles the distribution in figures 7 & 8, but without the spatial modulation. This result is unsurprising; the majority of the downstream-travelling energy is associated with the KH wave, and modulation cannot be represented with a single wave. The k - th wave peaks at approximately the fourth shock cell for both jets (often observed as the peak sound source location in screeching jets [START_REF] Mercier | Experimental characterisation of the screech feedback loop in underexpanded round jets[END_REF])), and as previously discussed in Edgington- Mitchell et al. (2018) has support outside the shear layer and in the core of the jet. The k + t downstream wave reaches maximum amplitude slightly upstream of the peak amplitude location for the upstream wave in the NPR = 2.10 jet, but somewhat further upstream for the NPR = 3.40 jet. The fluctuations associated with this wave are almost entirely bound within the core of the jet. At NPR = 2.10 the amplitude of the axial fluctuations are significantly larger than the transverse; at NPR = 3.40 both transverse and axial fluctuations are observed for the k + t wave. A comparison of the radial profiles of streamwise velocity for the three identified structures is presented in figure 14, educed from both experiment and global analysis for the NPR = 2.10 jet. All curves have been lightly smoothed with a moving average filter to reduce noise; no major features have been removed. The radial structure of the k - th wave exhibits a close match between theory and experiment, though the radial decay outside the jet is slower for the global mode. This mismatch in the acoustic field may be a consequence of the PIV measurement being unable to capture weak acoustic perturbations further from the jet. For the k + kh wave, the radial structure for both experiment and LSA exhibits the characteristic double peak of a KH wavepacket, with minima at the same radial location, but with a closer spacing between the peaks for the global mode. The radial structure of the k + t mode is very similar for the two analyses, though there is a small peak at r/D ≈ 0.5 for the global mode that is larger than that observed in the experiment. As noted earlier, the wavenumber for each of the modes identified in the global analysis is about ∆k x D = 0.5 higher than the corresponding wave in the experimental data. Since the global analysis is performed on the experimental mean flow, the spacing of the shock cells is the same for both cases. Given this fact, and the slightly higher wavenumber for the KH wave in the global analysis, the increase in wavenumber for the two interaction waves is consistent with the predictions of (1.4).

The global analysis has done a remarkable job of capturing the same key structures observed in experiment. Critically, it has demonstrated that it is indeed possible to describe non-linear KH-shock interactions using a linear model, provided the shock-cell structure is included in the mean flow: the global mode has been shown to correctly capture the non-linear mechanisms underpinning the production of both upstream-and downstream-travelling waves when coherent turbulent structures are convected through a network of shock cells. It must be noted however that the shocks at N P R = 2.10 are very weak, with much of the compression and expansion happening isentropically. How well such linear models will perform as the shocks increase in strength remains an open question. The analysis has also provided some clarification of the nature of the waves that result from this non-linear interaction. The upstream-travelling wave is the guided jet mode first identified by Tam & Hu (1989a) as a subsonic instability wave. The downstream wave has not previously been discussed in the context of jet screech, but as will be demonstrated it is simply the downstream-travelling form of the same subsonic wave identified by Tam and Hu, which has recently been the subject of extensive investigation by [START_REF] Towne | Acoustic resonance in the potential core of subsonic jets[END_REF] and [START_REF] Schmidt | Wavepackets and trapped acoustic modes in a turbulent jet: coherent structure eduction and global stability[END_REF]. We turn to a local, rather than global, analysis to elucidate the nature of this wave. Figure 15 presents radial eigenfunctions of the downstream-propagating subsonic instability wave as a function of axial position; these eigenfunctions have been produced via a local analysis on the experimentally-derived mean flow at each axial location for the screech frequency St = 0.67. In the region of the flow where the shocks are strongest, both the eigenfunction and the wavenumber (omitted for brevity) of this duct-like wave are significantly modulated by the shocks. Despite this modulation, the wave retains its duct-like character: bounded by the shear layer of the jet. In between the shocks, and further downstream, the wave has the typical structure associated with a k + t mode of the first radial order, as shown in detail in [START_REF] Towne | Acoustic resonance in the potential core of subsonic jets[END_REF]. Immediately downstream of the shocks however, the wavenumber of the wave significantly increases, and the peak amplitude shifts from the centreline to almost the edge of the shear layer. Considering the overall characteristics of the shock-cell structure, it is possible that high pressure regions oppose the radial support of these waves in the core, such that the peak is moved to regions of moderate pressure. The eigenfunctions at two selected axial locations are compared with the profile extracted from the experimentally derived wavenumber spectrum. At x/D = 1.3, the flow is immediately downstream of a shock, and the structure is markedly different to that expected for a k + t wave. At x/D = 2.2, which is close to where the amplitude of this wave peaks according to figure 12, the experiment and stability analysis match well up to r/D ≈ 0.5. The wave is not perfectly trapped by the shear layer in the experimental data, and decays more slowly for r/D > 0.5. With the nature of the three waves now established, we can make an overall statement regarding their behaviours. A non-linear interaction of the KH wavepacket with the shock cells of the jets produces both upstream-and downstream-travelling waves, as predicted by the model of [START_REF] Tam | Shock associated noise of supersonic jets from convergentdivergent nozzles[END_REF]; this is the first direct experimental validation of the theory, and likewise the first demonstration that a linear global analysis can capture this non-linear interaction once the shock-cell structure is included in the mean flow. While the model of [START_REF] Tam | Shock associated noise of supersonic jets from convergentdivergent nozzles[END_REF] predicted that these waves would exist, it is the mean flow that dictates their nature. In essence, the KH-shock interaction can be thought of as a forcing term, with the response to that forcing dictated by the structure of the mean flow. The mean flow supports upstream-and downstream-travelling waves with distinct radial structures (Tam & Hu 1989a): the k - th wave has support both inside and outside the jet, while the k + t wave is (in a vortex-sheet analysis at least) entirely confined to the core of the jet. Thus we now have a full explanation for the results presented here: a "forcing" via the mechanism of [START_REF] Tam | Shock associated noise of supersonic jets from convergentdivergent nozzles[END_REF], and a response in the form of waves predicted by Tam & Hu (1989a). In the following section of the paper, we consider the further interaction between these three waves.

Wave superposition in screeching jets

As alluded to in the introduction of the paper, and demonstrated in figure 7, velocity fluctuations in screeching jets are strongly spatially modulated. In this section we consider possible sources of this modulation. It is already well recognized that the standing wave set up between the downstream-travelling k + kh waves and the upstream-travelling k - th waves results in a periodic modulation of the velocity fluctuations. The effect of this modulation can be removed via the application of a high-pass wavenumber filter with a cut-off at k x = 0; all remaining fluctuations are associated with downstream-travelling waves. The results of this filtering are shown in figures 16 and 17; all periodic modulation of velocity fluctuations outside of the jet lipline have been removed, but all three jets still exhibit periodic modulation within the jet core. For the m = 0 screech cases, there is a periodic oscillation of axial velocity fluctuation amplitude in the jet core, but relatively little modulation of the transverse velocity component. For the helical screech mode, there remains significant modulation of velocity in both the axial velocity (primarily on the high-speed side of the shear layer) and the transverse velocity (at the centreline).

Given the restriction of this modulation to the core of the jet, one candidate is a superposition of the k + kh wave with the trapped k + t waves. To examine superposition between all three waves, dual-peak bandpass filters are applied to each of the three possible wavewave combinations, with both the filtered-wavenumber spectra and resultant amplitude plots presented in figures 18 and 19. Again, it must be emphasized that information is lost in this filtering process, and the results should only be considered a qualitative indication. As such, amplitudes have been normalized by the maximum amplitude in each case for clarity, though the fluctuations resulting from the superposition of the k - th and k + t waves is an order of magnitude weaker than all others. As expected, the superposition of the k - th and k + kh produces the familiar standing-wave pattern in the nearfield of the jet, but is also responsible for significant modulations within the core of the jet. The k - th and k + t superposition is highly periodic and confined to the core of the jet, but is very weak and contributes minimally to the overall fluctuations. The k + t and k + kh wave superposition, while weaker than that of the k - th and k + kh , still results in a significant spatial modulation of velocity fluctuation. As the k + t is confined within the lipline of the jet, so too is the spatial modulation of velocity; outside the jet it is only the superposition of the k - th and k + kh waves that produces spatial modulation. In the flows we consider here, both the k - th and k + t waves result from a non-linear interaction between the k + kh wavepacket and the shocks. The superposition of the k + kh wave with these two new waves thus produces a spatial modulation whose wavelength closely matches the shock spacing in the flow.

Plots of centreline fluctuation amplitude are presented in figure 20. If it is assumed that the bandpass filters only admit fluctuations associated with these waves, then even on the centreline, the majority of the modulation is evidently derived from the superposition of the k - th and k + kh waves, with the superposition of the k + t and k + kh providing a small but non-negligible contribution. When the superposition of all three waves is considered, the modulation in the range 2 x/D 4 is reproduced almost perfectly; outside this range the growth and decay requires the inclusion of lower wavenumbers than those admitted by the filter. There is however a serious limitation with this analysis; even in the absence of either the k - th or the k + t waves, any modulation of the k + kh waves would be contained in wavenumbers k + kh -k s and k + kh +k s . These are of course the same wavenumbers occupied by the k - th and k + t waves. While outside of the shear layer, it seems clear that the majority of observed modulation arises from the previously identified "standing wave" formed by the k + kh and k - th , within the jet core it is difficult to decouple modulation of the overall fluctuations due to wave superposition from modulation of the k + kh by the shock. Preliminary local stability calculations presented in Edgington- Mitchell et al. (2019) suggested that the influence of the shocks on the growth rate of the k + kh wave were minimal, but a deeper analysis using stability tools is required to provide a categorical answer to the question of wave modulation in the core of shock-containing jets.

Conclusion

Tam [START_REF] Tam | Shock associated noise of supersonic jets from convergentdivergent nozzles[END_REF] first suggested that the interaction of the KH wavepacket with the shocks in a supersonic jet would produce waves with both positive and negative phase velocity. Here we have provided the first evidence for these waves in the velocity field of screeching supersonic jets across a range of operating conditions. Further, we have demonstrated that the radial structure of these waves is dictated by the base flow, in accordance with the waves predicted by Tam & Hu (1989a) via a vortex-sheet model. The upstream-travelling wave has support both inside and outside the jet, while the downstream-travelling wave is confined within the core of the jet. This downstream wave is the "trapped" wave described in [START_REF] Towne | Acoustic resonance in the potential core of subsonic jets[END_REF], that treats the jet like a softwalled duct. The presence of these waves has been demonstrated for three different stages of jet screech, with shocks ranging from relatively weak to quite strong. Thus we suggest that as long as the mean flow is able to support these waves, they should be expected in all stages of jet screech, irrespective of shock strength.

We have also demonstrated that the non-linear wave interaction can be captured by a linear global analysis performed on the experimentally-derived mean flow, thanks to the presence of the shock-cell structure in the mean flow. Despite the many non-linear mechanisms active in a screeching jet, the linear analysis predicts the correct screech tone to within ≈ 1%, and the same three wave structures observed in experiment are likewise evident in the global mode. This suggests that, aside from the non-linear wave interaction that drives the upstream-and downstream-travelling waves, the remaining frequency-selection mechanisms (propagation characteristics of the k + kh and k - th waves, receptivity in the nozzle plane) are linear. As a cautionary note however, it must be emphasized that this result has only been demonstrated for the case with the weakest shocks.

Finally, the superposition of these three wave structures and the spatially-modulated velocity fluctuations that result were considered. Outside of the jet, no modulation is evidence once upstream-travelling waves are excluded, consistent with prior work suggesting this modulation is due to interaction between the downstream-propagating upstream-propagating waves. In the core of the jet the result is less clear; some amount of modulation will be due to the two waves generated from the interaction, but the present analysis is unable to decouple this from any direct influence that the shocks have on the Kelvin-Helmholtz wavepacket.

Figure 1 .

 1 Figure 1. Mean velocity fields for the two jets characterized by an m = 0 screech tone, operating at NPR = 2.10 and NPR = 2.25.

Figure 2 .

 2 Figure 2. Mean velocity fields for the jet characterized by an m = 1 screech tone, operating at NPR = 3.40.

Figure 3 .

 3 Figure 3. Calculation domain and mesh structure for the global analysis.

Figure 4 .

 4 Figure 4. Eigenspectrum from global stability analysis for the NPR = 2.10 jet. Red square indicates least-stable mode at St = 0.667.

Figure 5 .

 5 Figure 5. Streamwise velocity (top) associated with the global mode identified at St = 0.667, presented with experimentally-determined streamwise velocity mode associated with the second POD mode (bottom). Both modes are normalized against their maximum value. Results are for the NPR = 2.10 jet.

Figure 6 .

 6 Figure 6. Pressure field associated with the least-stable global mode for N P R = 2.10. The dashed white lines indicate the beginning of the sponge zones; outside these bounds the contours of pressure are only included as a qualitative visualization. The red line indicates Mach wave radiation; the purple line indicates the upstream-propagating acoustic waves.

Figure 7 .

 7 Figure 7. Magnitude of coherent fluctuations for the two jets where the screech mode is an m = 0 azimuthal mode. Upper) Axial velocity fluctuations. Lower) Transverse velocity fluctuations.

Figure 8 .

 8 Figure 8. Magnitude of coherent fluctuations for jet where the screech mode is an m = 1 azimuthal mode. Upper) Axial velocity fluctuations. Lower) Transverse velocity fluctuations.

Figure 9 .

 9 Figure 9. Magnitude of fluctuations from both experiment and global analysis. Upper) Axial velocity fluctuations. Lower) Transverse velocity fluctuations. All results for NPR = 2.10 jet.

Figure 10 .

 10 Figure 10. Wavenumber spectra for all three jets. Left) Axial velocity fluctuations. Right) Transverse velocity fluctuations. Note the different x-axis scale for NPR = 3.4.

Figure 11 .

 11 Figure 11. Wavenumber spectra from both global mode and experiment. Upper) Axial velocity fluctuations. Lower) Transverse velocity fluctuations. All results are for the NPR = 2.10 jet. Note that a logarithmic contour scale is used in this image.

Figure 12 .

 12 Figure 12. Qualitative amplitude distributions associated with the three wavelike structures for the NPR = 2.10 jet (experimental). Top) k - th . Middle) k + kh . Bottom) k + t . Left) Axial velocity fluctuations. Right) Transverse velocity fluctuations.

Figure 13 .

 13 Figure 13. Qualitative amplitude distributions associated with the three wavelike structures for the NPR = 3.40 jet (experimental). Top) k - th . Middle) k + kh . Bottom) k + t . Left) Axial velocity fluctuations. Right) Transverse velocity fluctuations.

Figure 14 .

 14 Figure 14. Radial profiles of streamwise velocity for the three waves discussed in this paper.All waves are normalized to their own maximum value. Results are for the NPR = 2.10 jet.

Figure 15 .

 15 Figure 15. Left) Eigenfunctions of the k + t wave as a function of streamwise position, obtained via local linear stability analysis. Right) Comparison of eigenfunctions at two selected axial locations with profile extracted from figure 12. Results for NPR = 2.10 jet.

Figure 16 .

 16 Figure 16. Magnitude of coherent fluctuations with positive phase velocity for the two jets where the screech mode is an m = 0 azimuthal mode. Upper) Axial velocity fluctuations. Lower) Transverse velocity fluctuations.

Figure 17 .

 17 Figure 17. Magnitude of coherent fluctuations with positive phase velocity for jet where the screech mode is an m = 1 azimuthal mode. Upper) Axial velocity fluctuations. Lower) Transverse velocity fluctuations.

Figure 18 .

 18 Figure 18. Interaction between pairs of waves for NPR = 2.10 jet, represented through streamwise velocity fluctuations. Top) Wavenumber spectra. Bottom) Velocity fluctuation amplitude (normalized).

Figure 19 .

 19 Figure 19. Interaction between pairs of waves for NPR = 3.40 jet, represented through streamwise velocity fluctuations. Top) Wavenumber spectra. Bottom) Velocity fluctuation amplitude (normalized).

Figure 20 .

 20 Figure 20. Centreline streamwise velocity fluctuation profiles for filtered data subsets of NPR = 2.10 jet

Figure 21 .

 21 Figure 21. Strouhal number of the screech mode (left) and the respective growth rate (right) as function of Reynolds number, from global stability analysis.

Figure 22 .

 22 Figure 22. Shapes of the modes for different values of Reynolds number from global stability analysis. Only the real part of the streamwise velocity is shown.

Table 1 .

 1 Jet Conditions

	NPR Mj	Re	St Mode
	2.10 1.09 4.4 × 10 5 0.67 A1
	2.25 1.14 4.7 × 10 5 0.65 A2
	3.40 1.45 8.6 × 10 5 0.26	C

Table 2 .

 2 Non-dimensional PIV Parameters

	Parameter	Value for NPR = 2.1 or 2.25 Value for NPR = 3.4
	IW0	0.12D	0.10D
	IW1	0.030D	0.026D
	Grid Spacing ∆x	0.01D	0.013D
	Depth of Field	0.04D	0.17D
	Light Sheet Thickness	0.1D	0.1D
	Field of View	5.7D x 3.8D	10D x 2.2 D
	Velocity snapshots	9,000	8,000

Table 3 .

 3 Wavenumbers of wave structures

	NPR	k kh D	ksD
	2.10 (Exp) 5.8 ± 0.5	11 ± 0.5
	2.10 (LSA) 6.2 ± 0.5	11 ± 0.5
	2.25	5.5 ± 0.5	9 ± 0.5
	3.4	2.6 ± 0.3	4.6 ± 0.2

Table 4 .

 4 Wavenumber sums and differences

	NPR	k kh D -ksD kxD(k -th )	k kh D + ksD kxD(k + t )
	2.10 (Exp)	-5.2	-5.5	16.8	16.7
	2.10 (LSA)	-4.8	-4.9	17.2	17.8
	2.25	-4.0	-3.5	14.5	15.3
	3.4	-2.0	-2.3	7.2	7.2

† Email address for correspondence: daniel.mitchell@monash.edu

Acknowledgements

Daniel Edgington-Mitchell and Petronio Nogueira's contributions to this work were funded by the Australian Research Council under the Discovery Project scheme via DP190102220.

Appendix A. Sensitivity of Reynolds number in global stability analysis

Considering that the Reynolds number chosen for the global stability analysis differs from the experimental one, and that no turbulent viscosity model is used, it is useful to analyse the influence of this parameter on the prediction of the screech frequency and on the shapes of the modes. Here, the Reynolds number was varied in the interval [2500 30000], and the frequencies of the screech mode (analysed in section 3.1) and the respective growth rates are shown in figure 21. These results show the robustness of the prediction regarding this parameter: the variation of the Strouhal number is only of the order 10 -3 over this range of Reynolds numbers. Also, the growth rate of the screech mode increases as this parameter is increased, as expected in this sort of stability analysis (see [START_REF] Coenen | Global instability of lowdensity jets[END_REF], for instance). Interestingly, the mode does not cross the real axis, and remains stable for all Reynolds numbers analysed. This may be due to the low shock-cell strength for the present case, or even due to the absence of the nozzle in the computational domain.

The shapes of the modes (represented by the real part of the streamwise velocity) for the different Reynolds numbers are shown in figure 22. Overall, all mode shapes are qualitatively similar to the leading POD mode shown in figure 5. For low Reynolds numbers, the increased viscosity leads to modes that are spatially smoother, especially around the shear layer. The streamwise and radial variation are also slower, which leads to more spatially spread mode shapes. As the Reynolds number is increased, the modes


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1IL and 3MLCT excited states modulated by H+ : structure and photophysical properties of [(2-bromo-5-(1H-pyrazol-1-yl)pyrazine)Re(CO)3Br]

Mononuclear rhenium(I) diimine tricarbonyl complexes have been recently subject of many investigations because of their interesting characteristics. This include well-behaved and general synthetic procedures, and, very importantly, remarkable stability. 1,[START_REF] Kumar | Photophysics of Organometallics[END_REF] Uses and potential applications include light capture, [START_REF] Kiefer | [END_REF][4][5][6] anion sensing, 7,8 bio-labeling and therapy, [9][10][11][12][13] and carbon dioxide photo-reduction 4,[14][15][16][17][18][19][20][21] among others. Simple aromatic chelating di-imines, like phenanthroline or bipyridyl are appealing candidates to be explored; owing to their planarity and limited conformational flexibility; since they diminish non-radiative deactivation paths. 22,23 Comparatively less studied, pyrazolyl-pyrazine derivatives are also attractive candidates having conjugated systems and structural rigidity. They are also relatively easy to prepare and to modify upon as request.

Pyrazolyl-pyrazine ligands also offer, when coordinated, an additional nitrogen atom to act as an acid-base center, which would influence the photophysical properties of their complexes, 24,25 adding an extra-feature to be considered for potential applications.

The literature shows pyrazolyl-pyridazine complexes of Fe II , Co II and Ru II . [26][27][28][29][30][31] Transition metal complexes of 2,6-bis(pyrazol-1-yl)pyrazine derivatives have been described as tridentate and meridional ligands, 31 while complexes of mono-substituted pyrazine derivatives are scarce. 29,[32][33][34] Scheme 1. Structural diagram of [(2-bromo-5-(1H-pyrazol-1yl)pyrazine)Re(CO)3Br] (I) and its protonated form IH + .

To the best of our knowledge no previous example of a Re I (d 6 ) complex has been reported with this kind of chelator. Then, expanding our research based on pyrazolyl-pyridazine derivatives, 35 in this work we extend the approach to the use of pyrazolyl-pyrazine as ligands. The synthesis and spectroscopic properties of the new rhenium(I) tricarbonyl complex (I) of the ligand 2-bromo-5-(1H-pyrazol-1-yl)pyrazine (LI), shown in Scheme 1, are detailed. Density Functional Theory (DFT) and Time-Dependent DFT (TD-DFT) calculations have been used to A c c e p t e d m a n u s c r i t p t rationalize these properties in terms of their electronic structures, in comparison with those of the uncoordinated ligand. The effect of acid on the photophysical properties is analyzed.

Experimental

General considerations.

All reagents were used as received from the supplier (Aldrich), with no purification before use. Solvents: dichloromethane (DCM, UvaSol® or SeccoSolv® grade, Merck), chloroform (CHCl3, Analysis grade, EMSURE® Merck), acetonitrile (MeCN, Analysis grade, EMSURE® Merck), ethanol (EtOH, Analysis grade, EMSURE® Merck), methanol (MeOH, Analysis grade, EMSURE® Merck), N,N-dimethylformamide (DMF, Uvasol® grade, Merck), benzene (C6H6, Analysis grade, EMSURE® Merck), toluene (C6H5CH3, Analysis grade, EMSURE® Merck), and trifluoroacetic acid (TFA, for synthesis, Merck) were employed as received. The ligand 2-bromo-5-(1H-pyrazol-1yl)pyrazine (LI) was prepared according to the literature. 35 Standard Schlenk techniques were used for all manipulations.

Synthesis of [(2-bromo-5-(1H-pyrazol-1-yl)pyrazine)Re(CO)3Br] (I).

The complex was prepared following a previously described procedure, 23 by direct reaction of the ligand LI (0.0557 g, 0.262 mmol) with bromotricarbonyl(tetrahydrofuran)-rhenium(I) dimer (Aldrich, 0.121 g, 0.133 mmol) at room temperature in dichloromethane solution (Scheme 2). Upon mixing, an orange solution was instantly obtained. Slow evaporation of the solvent at room temperature lead to yellow crystals of I. Crystalline material yield: 84%. Structural determination. The crystal structure of I was determined by X-ray diffraction at 298 and, to check structural changes and dataset quality improvement, at 100 K, using a Bruker SMART-APEX II CCD diffractometer system. Data reduction was performed with SAINT. 36 Multi-scan absorption corrections were applied using SADABS. 37 The structure solution was by direct methods, completion by Difference Fourier Synthesis and refinement by least-squares using SHELXL. 38 The hydrogen atom positions were calculated after each cycle of refinement with SHELXL using a riding model, with a C-H distance of 0.96 Å. Uiso(H) values were set equal to 1.2Ueq of the parent carbon atom. Table S1 (prepared with Publcif 39 ) shows a summary of structural and refinement details. No significant differences were found between the structures determined at 298 and 100 K. Cyclic Voltammetry. Cyclic voltammograms at room temperature for LI and I were recorded in a CH-INSTRUMENTS 650 E potentiostat in DMF solutions (1.0 mM) using tetrabuthylammonium perchlorate (0.10 M) as supporting electrolyte. Cyclic voltammograms were recorded at 100 mV s -1 between +1.7 and -1.7 V. Before runs, the sample solutions were deoxygenated by bubbling argon gas for 30 min. A vitreous carbon electrode was used as working electrode, a platinum electrode as auxiliary electrode, and an Ag/AgCl electrode as reference electrode. Ferrocene was used as internal standard (E½ = 0.473 V vs Ag/AgCl, Ar); ΔE = 0.069 V (Ar)).

Spectroscopic and photophysical measurements. IR spectrum (4000-400 cm -1 ) of the compound was measured using a Jasco FTIR-4600 spectrophotometer equipped with an ATR PRO ONE. 1 H and 13 C NMR spectra were recorded on a Bruker Avance 400 spectrometer at 298 K, using DMSO as solvent. UV-Vis spectra were recorded on an Agilent 8453 Diode-Array spectrophotometer in the range of 250-700 nm in aerated solvent solutions at room temperature. Emission spectra were measured in a Horiba Jobin-Yvon FluoroMax-4 spectrofluorometer in different solvents at room temperature or in ethanol-methanol glass (4:1, v/v) at 77 K. Luminescence lifetime measurements were carried out with the time correlated single photon counting technique using a PicoQuant FluoTime300 fluorescence lifetime spectrometer. A subnanosecond Pulsed Laser LDH-P-C-405 nm was employed as pulsed light sources (FWHM ~ 54 ps; average power 31 pJ). Time resolved experiments were made in different solutions, either air-equilibrated or argon-saturated. Emission quantum yields (em) were measured using procedures described in literature employing [Ru(bpy)3](PF6)2 in acetonitrile solution as standard. 40 Singlet oxygen, O2( 1 Δg), measurements were carried out in a FluoTime 200 consisting in a multichannel scaler NanoHarp 200. Excitation at 355 nm was achieved with a pulsed laser (FTSS355-Q3, Crystal Laser, Berlin, Germany) working at 1 kHz repetition rate. For the detection of emission at 1270 nm a NIR PMT H10330A (Hamamatsu) was employed. The O2( 1 Δg) quantum yields () were determined by comparing the intensity at zero time of the 1270 nm signals to those of optically-matched solutions of phenalelone as reference ( = 0.95 in DCM). 41,42 The transient difference absorption spectra and transient lifetimes were measured on an Edinburgh LP980 laser flash photolysis spectrometer. The excitation pump source was the Aurora II Integra 30 Nd:YAG/OPO system. The excitation wavelength was 355 nm (laser pulse  10 mJ, 8.2 ns) for all the samples unless otherwise noted. For each measurement, optically dilute solutions were degassed by bubbling argon gas for about 20 min.

Computational Details. All geometry optimizations were performed by DFT calculations at the B3LYP/6-31+G(d,p) [43][44][45][46] level of theory using the Gaussian09 Rev C.01 package of programs (G09), 47 and started from geometry determined by A c c e p t e d m a n u s c r i t p t means of X-rays diffraction. The LANL2DZ basis set was used only for rhenium. The geometry optimization for IH + started at the one of I. The computations for the one electron oxidized I +• or reduced I -• species were done based on the optimized geometry for I. Excitation calculations were performed within the TD-DFT methodology as implemented in G09. Absorption and emission spectra were simulated from the above calculations using the GaussSum 3.0 suite of freely available processing tools. A full width at half-maximum (FWHM) of the Gaussian curves corresponding to 2000 cm -1 was employed to reproduce the spectra. Representations for molecular orbitals were generated using the G09 cubegen tool and have been visualized using VMD and Povray 3.6 programs. 48

Results and discussion

Structural Description. Fig. 1 shows a molecular structure diagram for I. As clear from the figure, the coordination around Re I is a distorted octahedron with facial carbonyl groups. Table 1 shows a summary of the main bond distances and angles. The pyrazolyl-pyrazine bite angle (N1-Re1-N3) is only 74.0(1)°, as required by the ligand. Both rings, pyrazine and pyrazolyl are nearly coplanar, as reflected by the N1-N2-C4-N3 torsion angle, -4.4(5)°. S2, no important differences between the optimized and experimental distances are observed. Fig. 2 depicts the Kohn-Sham orbital diagram of I.

The optical transitions were computed by means of TD-DFT calculations assuming the DFT-optimized ground state geometry. Table 2 summarizes the energy of the major computed transitions, while Fig. 3 shows the simulated spectra of the molecule. The results are consistent with an absorption in the UV-Vis region of metal to ligand charge transfer (MLCT) nature. To study the protonation effect on the electronic structure of I, we have also performed DFT calculations for the protonated complex IH + . Fig. S2. show frontier and near frontier Kohn-Sham orbitals for IH + . For this specie the HOMO-LUMO transition has major d(M+Br) → d*(M+Br+LI) character. As shown in Fig. 2, the HOMO-LUMO GAP for I is computed to be 2.75 eV. The corresponding value for IH + diminishes to 1.60 eV. The same tendency is found for the difference in energy between ground state and the lowest energy triplet state, being 2.16 eV for I and 0.57 eV for IH + . Electrochemistry. To test the electrochemical behaviour and electronic structure of the ligand and the compound, we have carried out cyclic voltammetry experiments for both species in argon-saturated DMF solution. Table 3 shows a summary of the main cyclic voltammetry parameters. Under argon atmosphere, as expected, the positive scan reveals one irreversible oxidation for the complex around 1.1 V vs Fc + /Fc, corresponding to Re I /Re II (Fig. 4, up), whose process is absent in the free ligand LI. Table 3. Summary of cyclic voltammetry results for LI and I. Potentials in V vs Fc + /Fc in DMF with tetrabutylammonium perchlorate (0.1 M) as electrolyte.

Ep,c1 (V) Ep,c2 (V) Ep,c3 (V) Ep,a1 (V) I -1.445 -1.675 -- 1.067 LI -- -- -1.940 --
The negative scan displays a unique irreversible process for the ligand LI, a reduction wave with Ep,c equal to -1.940 V (Fig. 4, below) while two irreversible reduction processes were observed for the complex at -1.445 V and -1.675 V. The first one is attributed to the reduction of the ligand by comparing with Re I diimine ligands reported previously, 49,50 while the second reduction wave at -1.675 V can be assigned to Re I /Re 0 process such as has been observed in similar compounds of Re I when dmf and dichloromethane solutions is used. 51,52 The assignment of redox processes on I is supported by the DFT computed spin density for the one-electron vertically oxidized or reduced species I •+ and I •-(Fig. 5) and is consistent with metal centred oxidation and ligand centred reduction. It is important to note that the reductive behaviour of the rhenium complex is remarkably different from the free ligand, which has been described previously for a series of rhenium complexes with diazine ligands, 53 where it becomes clear that Absorption and emission properties. The absorption spectra for I and LI in DCM and MeCN solutions are shown in Fig. 6. For I, three bands can be observed around 260, 315 and 400 nm. It is noteworthy that the computed and experimental spectra (Fig. 3) are in a satisfying agreement. The bands which are at shorter wavelengths display extinction coefficients in the order of 10 4 M -1 cm -1 (Table 4). These two higher energy bands are also observed for the un-coordinated ligand (LI), being assigned to an intraligand π → π * (IL) transitions. This agrees with the computed excitations, as shown in Table 2. On the other hand, the band centered at 400 nm has an extinction coefficient at least one order of magnitude lower and is consistent with the d(M+Br) → *(LI) charge transfer predicted by TD-DFT (Table 2). The hypsochromic shift of the longest wavelength band observed by changing the solvent from DCM to MeCN is consistent with the MLCT (more precisely MLLCT) assignation. 7a and7b show the emission spectra measured in aerated MeCN and DCM solutions for I upon excitation at 280 nm. Two emission bands around 360 nm and 640 nm can be observed. The broad emission at longer wavelength is assigned to an MLCT excited state, while the higher energy band is similar in energy to the uncoordinated ligand (LI) and is attributed to the πpyrazine→π*pyrazine IL transition. 35 The second emission band of LI (430 nm, shoulder in the case of MeCN) is not observed when coordination to the metal takes place. The lower energy emission of the ligand could be related to a πpyrazol→π*pyrazine IL transition or to the presence of a twisted intramolecular charge transfer (TICT) excited state. 54 These transitions would not be expected following coordination. Emission lifetimes at the higher energy emission bands of I were also coincident with the reported ones for the free ligand (see Table 4, and following sentences). However, when trifluoroacetic acid (TFA) is added to I in airequilibrated DCM solution, an increase of the emission intensity at 470 nm is observed (Fig. 7c), while the intensity at 645 nm decreases. It must be stated that after protonation, the absorption spectrum of complex I remains unchanged (see Fig. S3 in supplementary material), which would be an evidence of an excited-state reaction. Then, just the excited state seems to be modulated when acid is added to the medium. Since the excited state has MLLCT character, it might be protonated even though the ground state it is not (the electron density on the ligand is increased).

The characteristic rigidochromic effect of the emissive MLCT excited state for I is confirmed by the emission spectrum at 77 K in EtOH:MeOH (4:1) shown in Fig. 8. 55 A structureless band centered at 570 nm is observed with higher intensity when compared with emission at 298 K. The weak emission displayed for I at room temperature in the mixture of polar protic solvents is centered around 450 nm, which agrees with a favored emission from the IL excited state for the IH + specie. Time resolved emission experiments allow us to observe a single exponential luminescence decay at 645 nm upon excitation at 405 nm in aerated MeCN and DCM solutions for I (Fig. 9), with lifetimes of 17 and 44 ns, respectively. This emission is associated to the MLCT excited state, while at higher energy (470 nm), a biexponential emission decay is observed, which can be related to the IL singlet excited states with lifetimes similar to the reported ones for the non-coordinated ligand LI. 35 Even though the emission quantum yields (em) and lifetimes () were almost not affected by the presence of oxygen, a triplet character can be attributed to the emissive MLCT excited state due to the ultrafast intersystem crossing that has been reported for this kind of complexes. 56,57 Moreover, a notable difference can be observed between the radiative rate constants (kr) reported in Table 4, for the emission with IL or MLCT character, with the IL emission kr being two orders of magnitude larger. The generation of singlet oxygen, O2( 1 g), was detected following the emission at 1270 nm upon excitation of aerated solutions. The singlet oxygen quantum yields () are included in Table 4. Values were comparable to the reported ones for similar complexes. 58 However, due to the similarity of  with the reported values for the free ligand, 35 it is also possible that singlet oxygen is being sensitized by an IL triplet excited state, then the 3 IL excited state must be close in energy to MLCT. Nevertheless, we have no additional data to prove that the triplet IL plays a significant role in the excited state behaviour.

The MLCT excited state is dynamically quenched after the addition of TFA to the aerated DCM solutions. As shown in Fig. 10, the emission lifetime at 645 nm ( = 44 ns) is reduced after the addition of acid ( = 7 ns at 0.0218 M TFA), with a quenching rate constant kq equal to 5.5 x 10 9 M -1 s -1 (see Stern-Volmer plot in supplementary material, Fig. S4). The protonated excited state or the specie IH + favored the emission from the IL excited state at 470 nm. The time resolved emission spectra (TRES, inset in Fig. 10) shows that the biexponential decay has two components: 1 = 1.72 ns and 2 = 6.89 ns, which are coincident with the reported lifetimes for the transition with IL character of the non-coordinated ligand. 35 Absorption spectral changes must occur if the ground state energy is affected upon protonation, as previously reported for other complexes. The transient absorption spectrum for I in deaerated DCM solution upon excitation at 355 nm is displayed in Fig. 11. The maximum of the transient absorption at 520 nm decays single exponentially with a lifetime  = 50.8 ns, which can be related to the 3 MLCT excited state. Upon TFA addition in absence of oxygen, the maximum in the transient absorption spectrum displayed a hypsochromic shift to 460 nm, being comparable with the transient absorption band previously reported for the pyrazolyl-pyrazine ligand. 35 The trace decay is now biexponential with two lifetimes: the short component,  = 49 ns, and the long component, T2 = 2.50 s. The longer lifetime is coincident with the value reported for the 3 IL excited state of a similar ligand to LI. 35 While the transient absorption for I is still displayed in aerated solutions, it was not possible to observe the transient absorption in aerated solution upon TFA addition. Then, the transition with IL character is more sensible to the presence of oxygen and the long-lived triplet of the protonated species seems to sensitize oxygen readily. The emission bands observed for I at higher energy can be ascribed to a singlet excited state with 1 IL character, while the one at lower energy is assigned to a triplet 3 MLCT excited state. We hypothesize that the trifluoroacetic acid addition to the medium do not produce the IH + specie. In contrast the [START_REF] Kiefer | [END_REF] MLCT excited state may react with the acid to sensitize the IL excited state of the protonated specie ( 3 ILH + ), a long-lived triplet excited state which is very sensitive to the oxygen presence in solution.

Conclusions

The monometallic complex [(2-bromo-5-(1H-pyrazol-1yl)pyrazine)Re(CO)3Br] (I), can be easily synthesized from the reaction of 2-bromo-5-(1H-pyrazol-1-yl)pyrazine (LI) and bromotricarbonyl(tetrahydrofuran)-rhenium(I) dimer. I display a non-regular octahedral geometry around Re I center, a faccorrelation for carbonyls, with pyrazine and pyrazolyl rings highly coplanar. Voltammetry experiments and theoretical DFT calculations support the oxidation is metal-centered processes and the first reduction is ligand centered. Absorption bands were assigned to intraligand π → π* transitions and metal-toligand charge transfer transition. Excitation of I at 280 nm leads to two emission bands, the first one associated to the singlet IL excited state ( 1 IL), while the second one to the triplet MLCT excited state ( 3 MLCT). The 3 MLCT lifetime was sensible to the solvent polarity. The trifluoroacetic acid, when added to the solution, may react with the 3 MLCT excited state yielding the long-lived and oxygen sensitive state 3 ILH + . These findings make I a potential sensor for protons and possibly other cations.

Scheme 2 .

 2 Scheme 2. Synthetic path to I. Elemental Analysis: Flash 2000, Thermo Scientific, Organic elemental analyzer. Calculated (experimental) for C10H5Br2N4O3Re: C, 20.9 (21.3); H, 0.88 (0.95); N, 9.74 (9.13) FTIR-ATR (major peaks cm -1 ): CO: 2021, 1926, 1894; C=N 1481, 1406 C-H: 3146. 1 H NMR (400 MHz (CD3)2SO) δ(ppm) 9.60 (d, J = 0.9 Hz, 1H), 9.31 (d, J = 3.0 Hz, 2H), 9.15 (d, J = 0.9 Hz, 1H), 8.59 (s, 1H), 7.05 (d, J = 2.3 Hz, 1H), 7.04 (d, J = 2.2 Hz, 1H). 13 C NMR (101 MHz, (CD3)2SO) δC(ppm) 195.27, 186.6, 147.25, 146.24, 144.45, 136.60, 132.61, 112.51. ( 13 CNMR and 1 HNMR spectra in Fig. S1).

Fig. 1

 1 Fig. 1 Molecular structure diagram for I as determined at 100 K. Partial atom numbering is included. Displacement ellipsoids drawn at the 50% level of probability. Hydrogen atoms are drawn as spheres of arbitrary radii.

Table 1 .

 1 Selected bond distances (Å) and angles (°) for I as determined by X-ray diffraction at 100 K. 86.77(19) C15-Re1-N3 96.26(16) C14-Re1-C15 89.93(18) N1-Re1-N3 73.98(14) C13-Re1-C15 91.65(18) C14-Re1-Br1 89.06(13) C14-Re1-N1 100.67(16) C13-Re1-Br1 93.33(13) C13-Re1-N1 172.34(16) C15-Re1-Br1 174.84(12) C15-Re1-N1 90.21(15) N1-Re1-Br1 85.01(9) C14-Re1-N3 171.78(15) N3-Re1-Br1 84.30(9) C13-Re1-N3 98.43(16) DFT, TD-DFT and Electronic Structure. To get a deeper insight into the electronic structure of this molecule, we performed DFT calculations by optimizing the gas phase geometry starting from the X-ray structure of I. As clear from Table

Fig. 2 Table 2 .

 22 Fig. 2 Kohn-Sham orbital diagram computed for I, with the following % composition of: Re/Br(coordinated)/CO/LI. Table 2. Major TD-DFT-computed transitions for I, with their oscillator strength and associated charge transfer character. λ / nm f Major Contributions Character 368 0.085 HOMO-4 → LUMO (92 %) d(M+Br) → (LI) 306 0.132 HOMO-6 → LUMO (58 %) (LI) → (LI) 269 0.284 HOMO-6 → LUMO+1 (78 %) (LI) → (LI)

Fig. 3

 3 Fig. 3 TD-DFT computed spectrum for I, including oscillator strength for each transition.

Fig. 4

 4 Fig. 4 Cyclic voltammograms for I (⎯) and LI (⎯) measured in 1.0 mM argon saturated DMF solution. Tetrabutyl-ammonium perchlorate 0.10 M used as electrolyte.

Fig. 5

 5 Fig. 5 DFT computed spin density for the one-electron vertically oxidized and reduced species I •+ (a) and I •-(b).

A

  c c e p t e d m a n u s c r i t p t coordination makes the ligand more electron poor and stabilizes the radical anion.

Fig. 6

 6 Fig. 6 Absorption spectra of I (solid lines) and LI (dashed lines) in aerated DCM and MeCN solutions.

Fig.

  Fig.7a and 7bshow the emission spectra measured in aerated MeCN and DCM solutions for I upon excitation at 280 nm. Two emission bands around 360 nm and 640 nm can be observed. The broad emission at longer wavelength is assigned to an MLCT excited state, while the higher energy band is similar in energy to the uncoordinated ligand (LI) and is attributed to

35

 35

Fig. 7

 7 Fig. 7 Emission spectra of I in: a, MeCN, b, DCM and c, adding TFA to aerated DCM solution (exc 280 nm). LI spectrum was added 35 (dashed lines) for comparison.

  were lower than 10%. a ex = 280 nm. b ex = 280 or 405 nm. c ex = 405 nm. For I, em and τ were evaluated at 645 nm. ex = 355 nm for .

A c c e p t e d m a n u s c r i t p t

59

 59

Fig. 8 .

 8 Fig. 8. Emission spectra of I in EtOH:MeOH 4:1 at 77 K and 298 K (ex = 400 nm).

Fig. 9

 9 Fig. 9 Single exponential emission decays for I in aerated MeCN and DCM solutions (λem = 645 nm and λex = 405 nm).

Fig. 10

 10 Fig. 10 Emission decay for I in aerated DCM solution with and without TFA (λem 485 or 650 nm and λex = 405 nm). Inset: TRES of I upon addition of TFA in aerated DCM solution. For clarity, we just include the higher energy region (420 nm to 550 nm).

  A c c e p t e d m a n u s c r i t p t

Fig. 11 Scheme 3 .

 113 Fig. 11 Transient absorption spectra of I in deaerated DCM solution, in absence (black line) and in presence of TFA (green line), at room temperature and 140 ns after excitation at 355 nm. Inset shows the trace decay at 520 nm upon addition of acid in deaerated DCM solution. Considering the results, we were able to propose the reaction sequence diagram shown in Scheme 3.

A c c e

  p t e d m a n u s c r i t p t

Table 4 .

 4 Summary of photophysical properties measured for LI and I in solution of DCM and MeCN.

	Compound	Solvent	( /	abs / nm

10 4 M -1 cm -1 )

	em/nm	em, air	em,

Ar τair / ns τAr / ns kr, Ar / 10 6 s -1 Δ

		DCM	262 (1.4) 314 (1.0)	360, 440 a	0.0100 a	0.0230 a	1.4 6.7	1.9	12.1	0.09
	LI	MeCN	264 (2.4) 316 (1.7)	372, 432 a	0.0250 a	0.0480 a	1.8 8.4	1.2 5.5

Acknowledgements

The authors gratefully acknowledge partial financial support of Comisión Nacional Científica y Tecnológica, grants FONDECYT 1160546, 1160749, FONDEQUIP EQM 160099, ACT-1404 (IPMaG). AV acknowledges Financiamiento Basal para Centros Científicos y Tecnológicos de Excelencia FB0807. This research has been performed as part of the Chilean-French International Associated Laboratory for "Multifunctional Molecules and Materials" (LIA M3 -CNRS N°1207).

Conflicts of interest

Authors have no interest conflicts to declare.


Anna Lisa Serra

Tobias Vogel

Gauthier Lefevre

Stefan Petzold

Nico Kaiser

Guillaume Bourgeois

Marie-Claire Cyrille

Lambert Alff

Christina Trautmann

Christophe Vallée

Heavy Ions Radiation Effects on 4kb Phase-Change Memory

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche

The crystallization temperature (T c ) can be tuned by engineering the material composition (Ge or N enrichment)

Based on Phase-Change Materials: Ex: Ge 2 Sb 2 Te 5 (GST), GeTe, etc.

• Amorphous phase (logic state 0, RESET)

• Crystalline phase (logic state 1, SET)

Memory programming relies on temperature profile control and not on charge storage

Fig. 1 Fig. 2

 12 Fig.1 Resistivity as a function of temperature of a Phase-Change Material [1]

Fig. 4 Fig. 6

 46 Fig. 4 Schematic of a phase change memory device hit by a heavy ion along the word line [4].

Fig. 7

 7 Fig. 7 TEM (left) and nano-diffraction patterns (right) analyses of as-deposited amorphous GST after irradiation at a fluence of 10 13 ions/cm 2 .

Fig. 5

 5 Fig. 5 Median and ±σ for SET and RESET distributions in GST 4kb arrays before and after irradiations at increasing fluences.

Fig 10

 10 Fig 10 XRD patterns of GGST crystalline layers before (REF) and after irradiations at increasing fluence.

Fig. 9 Fig 11 •

 911 Fig. 9 Median and ±σ for SET and RESET distributions in GGST 4kb arrays before and after irradiations at increasing fluences.

Fig. 14

 14 Fig. 14 Median and ±σ for SET and RESET distributions in GST and IL-PCM 4kb arrays before and after irradiations at increasing fluences.

Fig. 15

 15 Fig. 15 Schematic of the zx-plane of an IL-PCM showing in red the programmed region, demonstrating the contact area reduction.