Akshay Valsaraj

Ithihas Madala

Nikhil Garg
email: nikhil.garg@usherbrooke.ca

Veeky Baths

Alzheimer's Dementia Detection using Acoustic & Linguistic features and Pre-Trained BERT

Keywords: Speech Recognition, Human-Computer-Interaction, Computational Para-linguistics

Alzheimer's disease is a fatal progressive brain disorder that worsens with time. It is high time we have inexpensive and quick clinical diagnostic techniques for early detection and care. In previous studies, various Machine Learning techniques and Pre-trained Deep Learning models have been used in conjunction with the extraction of various acoustic and linguistic features. Our study focuses on three models for the classification task in the ADReSS (The Alzheimer's Dementia Recognition through Spontaneous Speech) 2021 Challenge. We use the well-balanced dataset provided by the ADReSS Challenge for training and validating our models. Model 1 uses various acoustic features from the eGeMAPs feature-set, Model 2 uses various linguistic features that we generated from auto-generated transcripts and Model 3 uses the auto-generated transcripts directly to extract features using a Pre-trained BERT and TF-IDF. These models are described in detail in the models section.

I. INTRODUCTION

Alzheimer's Dementia (AD) is a progressive Neurodegenerative disorder that affects major brain functions. The usual manifestations of AD include visible changes in memory and language and cognitive function. It also leads to personality changes along with agitation and losing the train of thought while speaking. Unlike other illnesses, the onset of visible symptoms is not observed in the initial stages. Early detection of this disease can prevent further degeneration, improve life quality and life expectancy.

Hence, AD is a major health concern in the struggle for increased life expectancy in the general population. Initially, it was considered to be a rare disease but soon it became closely associated with aging. There is a lot of stigma attached with this disease but with increased research and new treatments for patients with AD, these misconceptions are slowly fading away. Hence, it is the need of the hour to come up with inexpensive, quick and accurate diagnostic tools and early detection methods to provide utmost care to people suffering from early stages of the disease.

Current methods of diagnosis are human intensive, lab intensive and hence expensive and time consuming. While traditional Brain Imaging techniques are effective in detecting the onset of disease, these are expensive and tedious to repeatedly perform on a patient or a healthy individual. Microphones are present in most laptops and smartphones, and most people possess at least one of the two. Thus, speech data could be easily collected from any healthy individual without requiring any medical intervention. This spontaneous speech data could be used for early detection of AD when setup in the form of an interview taken by a virtual assistant and recording the responses.

The comparison of previously proposed models (ADReSS 2021 Challenge) for the classification task is summarized in Table I. This includes the features extracted, ML/DL models used and Accuracy achieved on the test dataset.

In our study, we discuss three different models based on the input features. Model 1: The Acoustic Model, uses various acoustic features from the eGeMAPs feature-set, Model 2: The Lingustic Model, uses various linguistic features that we generated from auto-generated transcripts and Model 3: Pre-trained BERT model + TF-IDF, uses the auto-generated transcripts directly to extract features using a Pre-trained BERT and TF-IDF.

We also propose the implementation of models based on Vision Transformers (ViT) and sPCEN normalized spectrograms as future work.

II. DATASET

We use the ADReSS Challenge Dataset which consists of 242 audio files. These were split into training and test sets with 70% for training and 30% for testing and validation. A set [START_REF] Syed | Automated screening for Alzheimer's dementia through spontaneous speech[END_REF] of recordings of picture descriptions produced by cognitively normal subjects and patients with an AD diagnosis, who were asked to describe the Cookie Theft picture from the Boston Diagnostic Aphasia Exam. The data set for the task was created from a study involving AD patients. The task involved classifying patients into 'ad' (suffering from Alzheimer's) and 'cn' (control subjects). From the data set provided we removed the voice of the interviewer and kept only the voice of the patient using the utterance segmentation files (diarization) provided with the dataset.

III. TRANSCRIPTS For our linguistic model, we require auto-generated transcripts of each audio file. We used the PyTorch based pretrained enterprise-grade Silero STT (Speech-To-Text) English (en v2) model for its ease of use & integration with our code and it is also comparable to the quality of Google's STT according to the benchmarks.

IV. FEATURE EXTRACTION

We calculated various linguistic features such as Brunet's Index, Honore's Statistic, Standardised Entropy, Root Type-Token Ratio and Hyper Geometric Distribution Diversity for our linguistic model. We extracted the eGeMAPS acoustic feature-set for our Acoustic Model. These features have shown to be very effective in previous studies related to Alzheimer's Dementia.

A. Linguistic features

Using the auto-generated transcripts, we decided to extract the following linguistic features based on extensive literature review.

1) Brunet's Index: Its a measure of lexical diversity that has been used in analysis of text and its often considered to be not dependent on text length. Its calculated by using the formula:

W = N V -a (1)
where N is the text length, V is the number of different words and a is a scaling constant which is usually set at 0.172

The richer the vocabulary the lower the values of W and the values usually range between 10 and 20. Patients suffering from dementia tend to repeat their words which leads to a lower lexical diversity, hence an important feature to consider.

2) Honore's Statistic: This feature has been used in the stylometric analyses of texts. It is based on the fact that texts which have a higher proportion of words tend to have a richer vocabulary. Its calculated by using the formula:

R = 100 log (N ) 1 - v 1 V (2)
where v 1 is the number of words used only one, V is the number of different words and N is the text length. Higher values of R correlate to a richer vocabulary and lower values correlate to poorer vocabulary. Patients suffering from dementia tend to have depleting vocabulary, hence an important feature to consider.

3) Standardised Entropy: This measure helps us to check the degradation in the variety of words and word combinations that a patient uses. It is calculated by dividing the word entropy by the log of the total word count. Formula:

-P (xi) × log 2 P (xi) log (N ) (3)
where P (xi) is the probability of the word in the text and N is the total number of words. 4) Root Type-Token Ratio: It has been noticed that in patients who suffer from Alzheimer's, there is a lot of repetition of words in their conversation as they tend to forget what they said and tend to repeat it again. Hence, to measure this we use Type-Token Ratio and to reduce its variance on text length Root Type-Token ratio is used. Its calculated by using the formula:

RT T R = t √ n ( 4
)
where t is the number of unique words and n is the total number of words.

5) Mean Segment Type-Token Ratio:

In this process the text to be analysed is divided into equal segments in terms of the number of words, in this case we used 16 words per segment.After this the TTR is calculated for each segment and the MSTTR is obtained after we take an arithmetic mean of all the TTR.

6) Measure of Textual Lexical Diversity (MTLD):

This measure is similar to MSTTR but in this case the text is varaible and the number of words in each segment varies and the segment ends when the TTR reaches a threshold of 0.72.After that the following formula is obtained to obtain MLTD.

M LT D = L n ( 5
)
where L is text length in number of words and n is the number of segments.

7) Hyper Geometric Distribution Diversity:

This index calculates the probability of finding any token for for each lexical type in a given text, in a random sample of 42 words taken from the same text. The sum of all the probabilities for all lexical types gives the final HDD score.

8) POS-based features:

We used this marker as tagging the text with POS tags can indicate frequent adjectives, verbs and adverbs which indicate more descriptive and meaningful sentences, while frequent adverbs can indicate the ability to relate different utterances to each other. Hence, it can be a useful identifier of patients with Alzheimer's disease. The various POS tags we used are -

• Verb frequency: It is computed by dividing the total number of words with that verb tag by the total number of words spoken by the patient in the recording.

• Noun frequency: It is computed by dividing the total number of words with that verb tag by the total number of words spoken by the patient in the recording.

• Pronoun frequency: It is computed by dividing the total number of words with that Pronoun tag by the total number of words spoken by the patient in the recording.

• Adverb frequency: It is computed by dividing the total number of words with that Adverb tag by the total number of words spoken by the patient in the recording.

• Adjective frequency: It is computed by dividing the total number of words with that Adjective tag by the total number of words spoken by the patient in the recording.

B. Acoustic Features

For the Acoustic Model, we extracted the eGeMAPS feature set on the entire audio file using the opensmile python package. The eGeMAPS feature set is a collection of 88 acoustic features that provide a baseline for evaluation in researches to avoid differences of feature set and implementations. We used a Recursive Feature Elimination (RFE) method to reduce the features which uses an external estimator that assigns weights to features and the RFE is used to select features by recursively considering smaller and smaller sets of feature. By using this technique We found that only 51 features contribute more towards the prediction, hence we decided to proceed with them for training our model.

V. CLASSIFICATION

We used the following ML algorithms as final layers of our models -

A. Support Vector Machine(SVM)

Support vector machines are one of the most prominent and widely used machine learning algorithms used for password authentication and accuracy prediction [START_REF] Nieves | Automatic person authentication using fewer channel EEG motor imagery[END_REF]- [START_REF] Abdel-Hadi | Classification of motor imagery tasks with LS-SVM in EEG-based selfpaced BCI[END_REF]. SVMs are based on the idea that the classes can be separated linearly using a hyper-plane to maximize the spaces between the most classes. In non-linear classification, a kernel function is used to map the non-linear classes to a linear problem by raising their dimension. The most widely used kernel functions are linear, polynomial, radial-basis function (RBF), and sigmoidal. In this study, the Polynomial kernel (degree=4) was found to have better performance than the other kernels.

B. Logistic Regression

It is a supervised Machine Learning algorithm that often uses a Sigmoid function to model a binary dependent variable. Instead of fitting a straight line or hyper-plane in the case of other model,s the logistic regression model uses a logistic function to predict the output of a linear equation between 0 and 1.

C. Random forests

Random Forest is a supervised Machine Learning algorithm [START_REF] Breiman | Random Forests[END_REF] used for classification and regression tasks. It creates multiple decision trees by taking various combinations of classes and samples from the training set. Decision trees are similar to flow charts in terms of structure. A node represents a test on an attribute, each branch of the node holds an outcome of the test, and each leaf node is a class label. The root node is at the topmost position. The impurity in the decision trees constructed is reduced by calculating the Gini index and using it in performing the appropriate splitting measures. The Gini index is calculated using the formula:

Gini = 1 - n i=1 (P i ) 2 (6)
where P i is the probability of a node being classified into a particular class.

VI. MODELS

We have three models -Model 1: The Acoustic Model, uses various acoustic features from the eGeMAPs featureset, Model 2: The Lingustic Model, uses various linguistic features that we generated from auto-generated transcripts and Model 3: Pre-trained BERT model + TF-IDF, uses the autogenerated transcripts directly to extract features using a Pretrained BERT and TF-IDF.

A. Model 1: The Acoustic model

In this model we used the eGeMAPS feature-set and obtained the final 51 features after feature reduction using Recursive Feature Elimination. We then classified them using various machine learning models such as Support Vector Machines (SVM), Linear Regression (LR) and Random Forest (RF).

B. Model 2: The Linguistic model

In this model we used the 13 features obtained from the various linguistic features mentioned in the previous section and classified them using the same machine learning models.

C. Model 3: Pre-trained BERT model + TF-IDF

Attention-based models [START_REF] Vaswani | Attention is All you Need[END_REF] belong to a class of AI models, commonly referred to as sequence-to-sequence models. The goal of these models, as the name suggests, is to generate an output sequence from an input sequence that is generally of different lengths. The main technical innovation of bidirectional transformers (BERT) [START_REF] Devlin | BERT: Pre-training of Deep Bidirectional Transformers for Language Understanding[END_REF] is to apply the bidirectional training of a transformer, a popular attention model, to language modelling. Using this approach, we fine-tune the pretrained BERT model by using the auto-generated transcripts of the audio files. We used the BERT model to obtain the pre-trained word embeddings for each sentence and obtained a total of 768 features and then we concatenated the BERT features with the TF-IDF features obtained from the text. TF-IDF is an information retrieval technique that weighs a term's frequency (TF) in the text and its inverse document frequency (IDF). A TF and an IDF value is obtained from each word and then they are multiplied. The formula to calculate TF-IDF for each word:

tf idf t,d = tf t,d • log N df t (7)
where tf t,d is the number of times the term t occurs in the text d and df t is the total number of documents containing the term t and N is the total number of documents. After obtaining all the features we classified them using the same machine learning models -SVM, LR, RF.

VII. RESULTS

The 5-Fold Cross Validation results of Models 1, 2 and 3 is summarized in Table 2 below. Both LR and SVM performed similarly considering the CV Accuracy achieved. But, considering the Precision, Recall, Specificity and F1 Score, LR came out on top with more consistency and hence we went with LR for all our models for the classification.

VIII. FUTURE WORK

A. Deep Acoustic Model: Vision Transformer and sPCEN

Vision Transformer (ViT) [START_REF] Dosovitskiy | An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale[END_REF] is now the state-of-the-art architecture for image classification tasks. The first layer of the Vision Transformer proposed in the paper involves segmentation of images into small patches (16x16). These patches are then linearly embedded, followed by the addition of position embeddings and then fed into a standard Transformer Encoder. The final classification token was also tweaked according the number of classes required for the task. We worked on implementing ViT for AD classification by constructing melspectrograms and fine-tuning pre-trained ViT, but we were not able to achieve respectable results.

We propose to tweak the first layer such that we take a segment of dimension 128x16 which keeps the full height of the spectrogram (assuming the height of the spectrogram is 128 pixels and width is a multiple of 16) and train the model from scratch on the ADReSS dataset. This way of segmentation may improve the learnability of the model specifically for spectrograms, due to the way spectrograms are constructed. Previous studies have shown that breaking the audio files into 3-second clips and then constructing spectrograms separately for each clip improves accuracy.

We also propose to normalize spectrograms with sPCEN [START_REF] Lostanlen | Per-Channel Energy Normalization: Why and How[END_REF]. A typical spectrogram uses linear frequency scaling so that each frequency range is the same number of Hertz apart, shown to outperform the point-wise logarithm of melfrequency spectrogram as an acoustic front-end. However, in the case of PCEN (Per-Channel Energy Normalization), it improves the robustness of the channel distortion. It has the ability to Gaussianize and whiten the background of acoustic recordings which is due to the operations of temporal integration, adaptive gain control, and dynamic range compression. Hence it is shown to outperform the point wise logarithm of Mel-frequency spectrogram.

B. Ensemble model

We would like to explore further by combining the Deep Acoustic model with our Model 3 (Pre-trained BERT) to find out if it improves the prediction accuracy.

  This work was funded by the Department of Science and technology, Government of India, vide Reference No: SR/CSI/50/2014 (G) through the Cognitive Science Research Initiative (CSRI). We acknowledge financial supports from the EU: ERC-2017-COG project IONOS (# GA 773228).

Table 3 ,

 3 we have summarized the Test Set results. Clearly the Model 3 with the Pre-trained BERT + TF-IDF features is outperforming the Acoustic and Linguistic models. This is inline with the previous studies where Deep Learning based linguistic models outperform basic ML based models and acoustic models.

			TABLE II
	5-FOLD CROSS VALIDATION RESULTS OF MODELS 1, 2 AND 3
	Class	CV Accuracy	Precision	Recall	Specificity	F1 Score
	Model 1
	LR	0.640	0.645	0.741	0.522	0.690
	RF	0.580	0.607	0.630	0.522	0.618
	SVM	0.640	0.629	0.815	0.435	0.710
	Model 2
	LR	0.660	0.867	0.464	0.909	0.605
	RF	0.620	0.800	0.429	0.864	0.558
	SVM	0.620	0.846	0.393	0.909	0.537
	Model 3
	LR	0.700	0.731	0.704	0.696	0.717
	RF	0.618	0.562	0.600	0.632	0.581
	SVM	0.700	0.667	0.636	0.75	0.651
	In


E Parente

M Farano

J.-Ch Robinet

P De Palma

S Cherubini

Continuing invariant solutions towards the turbulent flow

A new mathematical framework is proposed for characterizing the coherent motion of fluctuations around a mean turbulent channel flow. We search for statistically invariant coherent solutions of the unsteady Reynolds-averaged Navier-Stokes equations written in a perturbative form with respect to the turbulent mean flow, using a suitable approximation of the Reynolds stress tensor. This is achieved by setting up a continuation procedure of known solutions of the perturbative Navier-Stokes equations, based on the continuous increase of the turbulent eddy viscosity towards its turbulent value. The recovered solutions, being sustained only in the presence of the Reynolds stress tensor, are representative of the statistically coherent motion of turbulent flows. For small friction Reynolds number and/or domain size, the statistically invariant motion is almost identical to the corresponding invariant solution of the Navier-Stokes equations. Whereas, for sufficiently large friction number and/or domain size, it considerably departs from the starting invariant solution of the Navier-Stokes equations, presenting spatial structures, main wavelengths and scaling very close to those characterizing both large-and small-scale motion of turbulent channel flows.

This article is part of the theme issue 'Mathematical problems in physical fluid dynamics (part 2)'.

Introduction

Understanding the dynamics of wall-bounded turbulent flows is a formidable challenge yet to be fully achieved, since turbulence is a complex phenomenon appearing in a variety of states and patterns which compete with the laminar state [START_REF] Barkley | The rise of fully turbulent flow[END_REF]. In order to derive low-order models of the turbulent dynamics, one should focus on the coherent part of the turbulent motion [START_REF] Panton | Overview of the self-sustaining mechanisms of wall turbulence[END_REF], which is known to contribute much more to the momentum of the flow than the chaotic fluctuations at small scales. The most typical example of coherent motion in turbulent flows are the streaks, which have been first recognized by Kline et al. [START_REF] Kline | The structure of turbulent boundary layer flows[END_REF] as 'surprisingly well-organized spatially and temporally dependent motions'. These elongated structures are continuously generated by the lift-up mechanism [START_REF] Landahl | A note on an algebraic instability of inviscid parallel shear flows[END_REF] within the selfsustained cycle theorized in the nineties [START_REF] Hamilton | Regeneration mechanisms of near-wall turbulence structures[END_REF][START_REF] Waleffe | On a self-sustaining process in shear flows[END_REF]. Although this self-sustained process was initially observed in the inner layer region [START_REF] Hamilton | Regeneration mechanisms of near-wall turbulence structures[END_REF][START_REF] Waleffe | On a self-sustaining process in shear flows[END_REF][START_REF] Jiménez | The autonomous cycle of near-wall turbulence[END_REF], a growing body of evidence has recently indicated that equivalent, mutually independent regeneration cycles are active in the logarithmic and outer regions as well [START_REF] Hwang | Amplification of coherent streaks in the turbulent Couette flow: an input-output analysis at low Reynolds number[END_REF][START_REF] Hwang | Self-sustained process at large scales in turbulent channel flow[END_REF][START_REF] Hwang | Self-sustained processes in the logarithmic layer of turbulent channel flows[END_REF][START_REF] Hwang | Statistical structure of self-sustaining attached eddies in turbulent channel flow[END_REF][START_REF] Hwang | Inner-outer interactions of large-scale structures in turbulent channel flow[END_REF][START_REF] Cossu | Self-sustaining processes at all scales in wall-bounded turbulent shear flows[END_REF], giving rise to large-and very-large-scale motion [START_REF] Kovasznay | Large-scale motion in the intermittent region of a turbulent boundary layer[END_REF][START_REF] Komminaho | Very large structures in plane turbulent Couette flow[END_REF][START_REF] Kim | Very large-scale motion in the outer layer[END_REF][START_REF] Del Alamo | Spectra of the very large anisotropic scales in turbulent channels[END_REF][START_REF] Hutchins | Evidence of very long meandering features in the logarithmic region of turbulent boundary layers[END_REF]. The statistical and dynamical features of these self-sustaining motions associated with streaks and quasi-streamwise vortices at different scales are consistent with Townsend's attached eddy hypothesis [START_REF] Townsend | The structure of turbulent shear flow[END_REF]. Moreover, the same three fundamental mechanisms that compose the regeneration cycle of wall turbulence have been found to sustain several invariant solutions of the Navier-Stokes equations in the form of equilibria, travelling waves or (relative) periodic orbits [START_REF] Nagata | Three-dimensional finite-amplitude solutions in plane Couette flow: bifurcation from infinity[END_REF][START_REF] Waleffe | Three-dimensional states in plane shear flow[END_REF][START_REF] Kawahara | Periodic motion embedded in plane Couette turbulence: regeneration cycle and burst[END_REF][START_REF] Waleffe | Exact coherent structures in channel flow[END_REF][START_REF] Faisst | Travelling waves in pipe flow[END_REF][START_REF] Hof | Experimental observation of nonlinear traveling waves in turbulent pipe flow[END_REF][START_REF] Wedin | Exact coherent structures in pipe flow: travelling wave solutions[END_REF][START_REF] Eckhardt | Turbulence transition of pipe flow[END_REF][START_REF] Duguet | Relative periodic orbits in transitional pipe flow[END_REF][START_REF] Gibson | Equilibrium and traveling-wave solutions of plane Couette flow[END_REF][START_REF] Schneider | Snakes and ladders: localized solutions of plane Couette flow[END_REF][START_REF] Willis | Revealing the state space of turbulent pipe flow by symmetry reduction[END_REF][START_REF] Deguchi | The emergence of localized vortex-wave interaction states in plane Couette flow[END_REF][START_REF] Gibson | Spanwise-localized solutions of planar shear flows[END_REF][START_REF] Zammert | Streamwise and doubly-localised periodic orbits in plane Poiseuille flow[END_REF][START_REF] Park | Exact coherent states and connections to turbulent dynamics in minimal channel flow[END_REF][START_REF] Barnett | Streamwise localization of traveling wave solutions in channel flow[END_REF][START_REF] Budanur | Relative periodic orbits form the backbone of turbulent pipe flow[END_REF], which compose the chaotic saddle sustaining transient turbulence [START_REF] Hopf | A mathematical example displaying features of turbulence[END_REF]. Typical coherent structures populating transitional and turbulent flows, such as streaks and streamwise vortices, have been successfully captured by these fully nonlinear, dynamically unstable solutions of the Navier-Stokes equations which populate state space and support turbulent dynamics with their entangled stable and unstable manifolds. Trajectories in the state space may approach one of these solutions, remain in their neighbourhood for a finite time before being pushed away along one of the unstable directions to approach other solutions through heteroclinic orbits [START_REF] Farano | Computing heteroclinic orbits using adjoint-based methods[END_REF], resulting in a chaotic walk in the state space. Many efforts have been done for deriving low-order models based on these invariant solutions, allowing one to accurately describe the statistical properties of a turbulent flow [START_REF] Cvitanović | Recurrent flows: the clockwork behind turbulence[END_REF]. Recently, Chandler & Kerswell [START_REF] Chandler | Invariant recurrent solutions embedded in a turbulent twodimensional Kolmogorov flow[END_REF] successfully applied the periodic orbit theory to the case of a two-dimensional Kolmogorov flow at a moderate Reynolds number. However, a low-order model of a fully developed three-dimensional turbulent flow is yet to be achieved, and would probably require the discovery of many new invariant solutions at sufficiently high Reynolds number. Unfortunately, the computation of such invariant solutions at large Reynolds number is a hard challenge, due to the multiple bifurcations they typically undergo. In the case of fully turbulent flows at high friction Reynolds number, a valuable approach for computing invariant solutions is resorting to large-eddy simulations (LES), as proposed by Hwang & Cossu [START_REF] Hwang | Self-sustained process at large scales in turbulent channel flow[END_REF][START_REF] Hwang | Self-sustained processes in the logarithmic layer of turbulent channel flows[END_REF]. In particular, choosing filtering widths larger than those typically used for reproducing results of direct numerical simulations (DNS), allows one to filter out a large range of scales that could not be resolved within the chosen numerical grid. This approach, relying on 'overfiltered' LES, with the Smagorinsky constant C s controlling the strength of the filtering, is well suited for investigating the self-sustained nature of coherent large-scale motion, as done at first for the channel [START_REF] Hwang | Self-sustained process at large scales in turbulent channel flow[END_REF][START_REF] Hwang | Self-sustained processes in the logarithmic layer of turbulent channel flows[END_REF] and Couette [START_REF] Rawat | On the self-sustained nature of large-scale motions in turbulent Couette flow[END_REF][START_REF] Rawat | Travelling-wave solutions bifurcating from relative periodic orbits in plane Poiseuille flow[END_REF] flows at relatively low Reynolds numbers and for the channel and asymptotic suction boundary layer flow at large friction Reynolds numbers [START_REF] Hwang | Invariant solutions of minimal large-scale structures in turbulent channel flow for re τ up to 1000[END_REF][START_REF] Azimi | Self-sustained large-scale motions in the asymptotic suction boundary layer[END_REF]. Recent studies [START_REF] Hwang | Self-sustained process at large scales in turbulent channel flow[END_REF][START_REF] Hwang | Self-sustained processes in the logarithmic layer of turbulent channel flows[END_REF][START_REF] Rawat | Travelling-wave solutions bifurcating from relative periodic orbits in plane Poiseuille flow[END_REF] have demonstrated that both large-scale and log-layer coherent motions can be self-sustained, since they survive in both channel and Couette flow, when smaller-scale active motions are artificially quenched and replaced by purely dissipative structures. However, this overfiltered approach does not allow one to investigate the nature of the energy transfer between coherent structures of different scales. Despite the motion at large scales can be sustained even when the wall cycle is quenched, in high-Reynolds-number turbulent flows multiple and non-trivial interactions exist between coherent structures at different scales [START_REF] Cimarelli | Cascades and wall-normal fluxes in turbulent channel flows[END_REF][START_REF] Cho | Scale interactions and spectral energy transfer in channel flow[END_REF]. Very recent works [START_REF] Doohan | Minimal multi-scale dynamics of near-wall turbulence[END_REF] have shown that wall-normal energy is transferred from large to small scales inducing energy production at the wall via the Orr mechanism, while a non-negligible amount of energy is transferred from small to large scales [START_REF] Cho | Scale interactions and spectral energy transfer in channel flow[END_REF], possibly due to small-scale sinuous streak instability [START_REF] Doohan | Minimal multi-scale dynamics of near-wall turbulence[END_REF]. Thus, coherent structures at different scales are intimately connected by direct and inverse cascade mechanisms by which energy is transmitted scale-by-scale among different regions of the flow [START_REF] Cimarelli | Cascades and wall-normal fluxes in turbulent channel flows[END_REF].

A deeper understanding of the energetic bond connecting small-and large-scale structure in turbulent flows can be achieved by the computation of statistically invariant coherent states which characterize the multiple-scale, coherent part of the motion around the turbulent mean flow, without any filtering of small-scale structures. Towards this aim, this work provides a new mathematical framework for the computation of statistically invariant equilibria, travelling waves or (relative) periodic orbits characterizing the motion of turbulent fluctuations around the mean flow. This is achieved by seeking for statistically invariant coherent solutions of the unsteady Reynolds-averaged Navier-Stokes equations written in a perturbative form with respect to the turbulent mean flow, using a suitable approximation of the Reynolds stress tensor. Unlike the classical invariant solutions of the Navier-Stokes equations, these solutions are sustained only in the presence of the Reynolds stress tensor, being representative of the statistically coherent motion of turbulence. This set of equations has been found efficient for characterizing extreme events having an energy spectrum very similar to that of the fully turbulent flow [START_REF] Farano | Optimal bursts in turbulent channel flow[END_REF][START_REF] Farano | Nonlinear optimal large-scale structures in turbulent channel flow[END_REF]. We show in this paper that, continuing in this statistical framework known invariant solutions of the Navier-Stokes equations at high friction Reynolds numbers, statistically invariant motions containing both large-and small-scale coherent structures such as streaks and streamwise vortices are obtained, with main wavelengths corresponding to the typical ones recovered in turbulent flows. The paper is structured as follows. In §2, the problem formulation is provided. In §3, the continuation procedure allowing the computation of the statistically invariant solutions is described. Relevant results are discussed in §4, and conclusions are drawn in §5.

Problem formulation

The incompressible flow in a channel is governed by the Navier-Stokes equations, which describe the dynamics of the instantaneous state variables q = [u, p] T , where u(x, t) is the velocity field and p(x, t) is the pressure.

When studying the flow dynamics in the vicinity of the laminar state, the state variables can be decomposed as the sum of the laminar base flow Q = [U, 0, 0, P] T and a perturbation q = [u , v , w , p ], leading to the Perturbative Navier-Stokes (PNS) equations

∇ • u = 0 and ∂u ∂t = -(u • ∇)u -(u • ∇)U -(U • ∇)u -∇p + 1 Re ∇ 2 u , ⎫ ⎪ ⎬ ⎪ ⎭ (2.1)
where Re = 3U b /2(h/ν) is the Reynolds number, defined on the basis of the bulk velocity U b = 1 -1 U(y) dy (where 3U b /2 corresponds to the value at the centreline of the parabolic laminar flow with mean velocity U b ), the half channel height h and the kinematic viscosity ν. Several invariant solutions of these equations, such as (relative) equilibria or periodic orbits, have been computed in the past decades [START_REF] Nagata | Three-dimensional finite-amplitude solutions in plane Couette flow: bifurcation from infinity[END_REF][START_REF] Waleffe | Three-dimensional states in plane shear flow[END_REF][START_REF] Kawahara | Periodic motion embedded in plane Couette turbulence: regeneration cycle and burst[END_REF][START_REF] Waleffe | Exact coherent structures in channel flow[END_REF][START_REF] Faisst | Travelling waves in pipe flow[END_REF][START_REF] Hof | Experimental observation of nonlinear traveling waves in turbulent pipe flow[END_REF][START_REF] Wedin | Exact coherent structures in pipe flow: travelling wave solutions[END_REF][START_REF] Eckhardt | Turbulence transition of pipe flow[END_REF][START_REF] Duguet | Relative periodic orbits in transitional pipe flow[END_REF][START_REF] Gibson | Equilibrium and traveling-wave solutions of plane Couette flow[END_REF][START_REF] Schneider | Snakes and ladders: localized solutions of plane Couette flow[END_REF][START_REF] Willis | Revealing the state space of turbulent pipe flow by symmetry reduction[END_REF][START_REF] Deguchi | The emergence of localized vortex-wave interaction states in plane Couette flow[END_REF][START_REF] Gibson | Spanwise-localized solutions of planar shear flows[END_REF][START_REF] Zammert | Streamwise and doubly-localised periodic orbits in plane Poiseuille flow[END_REF][START_REF] Park | Exact coherent states and connections to turbulent dynamics in minimal channel flow[END_REF][START_REF] Barnett | Streamwise localization of traveling wave solutions in channel flow[END_REF][START_REF] Budanur | Relative periodic orbits form the backbone of turbulent pipe flow[END_REF].

Conversely, when studying the dynamics of coherent structures characterizing the turbulent flow, it can be appropriate to move the point of view in the vicinity of the turbulent mean flow. This is achieved by using a Reynolds decomposition approach similar to that used by Eitel-Amor et al. [START_REF] Eitel-Amor | Hairpin vortices in turbulent boundary layers[END_REF] and Farano et al. [START_REF] Farano | Optimal bursts in turbulent channel flow[END_REF], where the flow vector is expressed as the sum of a mean flow Q = [U, P] T = [U, 0, 0, P] T (where • denotes long-time and space averaging along the streamwise and spanwise directions) and a fluctuation q = [ ũ, p] T , comprising the coherent and incoherent part of the perturbations of the mean flow. Time-and space-averaging along the wall-parallel directions the Navier-Stokes equations, and subtracting these averaged equations from the Navier-Stokes equations leads to the perturbative Reynolds-averaged Navier-Stokes (PRANS) equations, which describe in a statistical way the nonlinear evolution of fluctuations of the mean turbulent flow as

∇ • ũ = 0 and ∂ ũ ∂t = -( ũ • ∇) ũ -( ũ • ∇)U -(U • ∇) ũ -∇p + 1 Re ∇ 2 ũ + ∇ • ũ ũ, ⎫ ⎪ ⎬ ⎪ ⎭ (2.2)
where the term ũ ũ is the Reynolds stress tensor τ . Note that steady solutions of the PRANS equations, as well as the mean flow itself, are sustained by the Reynolds stress term, which is in turn sustained by the coherent and incoherent part of the fluctuations. The mean velocity profile for channel flow is well approximated by the analytical expression proposed by Reynolds & Tiedermann [START_REF] Reynolds | Stability of turbulent channel flow, with application to Malkus's theory[END_REF] dU

+ dy = - Re τ y ν + T (y) , (2.3)
Re τ = u τ h/ν being the friction Reynolds number based on the friction velocity u τ = √ τ w /ρ, where τ w is the wall shear stress, and ν + T = ν T /ν is the ratio between the total viscosity ν T = ν + ν t and the kinematic viscosity ν, ν t being the turbulent eddy viscosity. The total eddy viscosity is modelled using Cess analytic approximation [START_REF] Cess | A survey of the literature on heat transfer in turbulent tube flow[END_REF], as assumed in previous works [START_REF] Reynolds | The mechanics of an organized wave in turbulent shear flow. part 3. Theoretical models and comparisons with experiments[END_REF][START_REF] Alamo | Linear energy amplification in turbulent channels[END_REF][START_REF] Hwang | Mesolayer of attached eddies in turbulent channel flow[END_REF]. Since this mean velocity profile is not solution of the Navier-Stokes equations, in order to close the problem, the divergence of the Reynolds stress tensor τ in equation (2.2) needs to be modelled. A common way to write this term is using the Boussinesq's Eddy Viscosity hypothesis τ ij = -ũi ũj = ν t S ij [START_REF] Pope | Turbulent flows[END_REF], where S is the shear stress tensor. Considering a fully developed channel flow whose statistics are averaged in the streamwise and spanwise direction, the divergence of the Reynolds stress tensor has only two non-zero components, i.e. dτ 12 (y)/dy and dτ 22 (y)/dy (note that the latter term cannot be incorporated in the pressure term since the other isotropic components have derivative equal to zero). The former term is approximated using the above mentioned eddy viscosity hypothesis, while the latter is modelled by using the rescaling proposed by Chen et al. [START_REF] Chen | Quantifying wall turbulence via a symmetry approach. Part 2. Reynolds stresses[END_REF], as ũ ṽ = -ν t dU dy and ṽ ṽ+ = ũ ṽ+ l

+ 22 l + 12 2 , (2.4)
l + 12 , l + 22 being the Reynolds stress lengths defined as

l + i2 = c i2 y + (i+2)/2 ⎛ ⎝ 1 + y + y + sub 4 ⎞ ⎠ 1/8 ⎛ ⎝ 1 + y + y + buf 4 ⎞ ⎠ -(1+i)/4 1 -r 4 4(1 -r) 1 + r corei2 r 2 i/4 , (2.5)
where i = 1, 2, r = 1y is the distance from the centreline, y + sub = 9.7 is the sublayer thickness, y + buf = 41 is the buffer layer thickness, r core12 = 0.27 and r core22 = 0.3 are the central core layers, and the parameters c 12 , c 22 are function of these quantities (see [START_REF] Chen | Quantifying wall turbulence via a symmetry approach. Part 2. Reynolds stresses[END_REF] for further details). This analytical formulation has been validated by comparing the mean turbulent flow and the Reynolds stress tensor components with those obtained by DNS at Re = 3300 (Re τ = 180) [START_REF] Kim | Turbulence statistics in fully developed channel flow at low Reynolds number[END_REF] and at Re = 12 450 (Re τ = 590) [START_REF] Moser | Direct numerical simulation of turbulent channel flow up to re τ = 590[END_REF]. The results obtained are computed in this range of Reynolds numbers, for which turbulence is fully developed (not spatially patterned) and both the mean flow and the Reynolds stress are accurately described by the chosen analytical approximation.

Continuation from the PNS to the PRANS framework

Statistically invariant travelling waves solutions of the PRANS equations are sought for by continuation of known invariant solutions of the PNS equations. In particular, as sketched in figure 1, a homotopy procedure is used for continuously passing from equations (2.1) to (2.2), which have an almost identical structure, except for the steady flow used as reference and for the presence of the Reynolds stress tensor. Since these quantities depend directly on the turbulent eddy viscosity, the continuation is performed by continuously increasing this quantity from zero to its characteristic turbulent value expressed by the Cess model [START_REF] Cess | A survey of the literature on heat transfer in turbulent tube flow[END_REF]. Towards this purpose, we define an effective turbulent eddy viscosity, ν t , where is a real positive number in the range [0, 1], and ν t is expressed as with η = η + 1 defined in the domain [0, 2], κ = 0.426 and A = 25.4, as assumed in previous works [START_REF] Alamo | Linear energy amplification in turbulent channels[END_REF][START_REF] Hwang | Mesolayer of attached eddies in turbulent channel flow[END_REF][START_REF] Pujals | A note on optimal transient growth in turbulent channel flows[END_REF]. Continuation from the PNS to the PRANS equations is achieved by increasing the coefficient from 0 to 1, and using the effective turbulent eddy viscosity ν t in the analytical expression of the Reynolds stress tensor components τ 11 , τ 12 in equation (2.4) and in the mean flow profile in equation (2.3), where ν + T = 1 + ν t /ν. The procedure consists in selecting a known travelling wave solution of the Navier-Stokes equations, subtracting the laminar flow solution for defining the corresponding perturbation, which is a travelling wave solution of equations (2.1), and continuing it in using the following equations:

ν t = ν 2 1 + κ 2 Re 2 τ 9 (2 η -η2 ) 2 (3 -4 η + 2 η2 ) 2 1 -exp (| η -1| -1)Re τ A 2 1 2 - 1 2 . ( 3
∇ • ũ = 0 -C ∂ ũ ∂x = -( ũ • ∇) ũ -( ũ • ∇)U -(U • ∇) ũ -∇p + 1 Re ∇ 2 ũ -∇ • ν t S(U) U(y) = 1 - Re 2 τ y Re(1 + (ν t /ν)) dy, 0, 0 T , ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ (3.2)
where C is the phase velocity of the Galilean frame in which the solution is steady. When = 0, Re 2 τ = 2Re and one recovers U = 1y 2 = U, the Reynolds stress tensor components being null. In this limit, equations (3.2) coincide with the PNS equations (2.1) and consequently ũ coincides with u . When = 1, one gets the turbulent expression of ν t given in equation (3.1), so that the turbulent mean velocity profile and the Reynolds stress tensor components in equations (2.4) are obtained. Note that, similarly to a homotopy procedure, the solutions obtained for =]0, 1[ have no physical sense. Moreover, for = 1, ũ can have a non-zero mean, since U represents the mean flow only in turbulent conditions, which are achieved only for = 1. The whole procedure, which is sketched in figure 1, is implemented within the open-source software Channelflow (channelflow.ch) [START_REF] Gibson | In preparation[END_REF].

Continuation from the PNS to the PRANS equations is performed enforcing a constant volume flux and consequently fixing the bulk velocity. Thus, while increases from 0 to 1, the friction Reynolds number grows from the laminar (Re L τ ) towards the turbulent (Re T τ ) value as

Re τ = (1 -)Re L τ + Re T τ = [(1 -)u L τ + u T τ ] h ν , (3.3)
where u L τ = 3U b ν/h is the friction velocity of the laminar flow and u T τ is the friction velocity of the turbulent flow, which, using Dean's approximation for the skin friction in fully turbulent flow [START_REF] Dean | Reynolds number dependence of skin friction and other bulk flow variables in two-dimensional rectangular duct flow[END_REF], can be expressed as

u T τ = 0.073 2 U 2 b 2U b h ν -0.025 . (3.4)

Results

As a first attempt aiming at validating the approach, we take as a starting point for the continuation procedure the travelling wave solution TW2 obtained by Gibson & Brandt [START_REF] Gibson | Spanwise-localized solutions of planar shear flows[END_REF] in a small domain at low Reynolds number. This invariant solution of the NS equations is computed at Re = 2300 in a domain of extension 2π × 2 × π , with 32 × 97 × 64 points in the streamwise (x), wall-normal (y) and spanwise (z) direction, respectively. As shown in figure 2a, the TW2 solution is continued with respect to the Reynolds number up to Re = 3800, which is sufficiently high for displaying featureless (not patterned) turbulence. As shown in figure 3a, the TW2 solution at this value of Re consists of two layers of counter-rotating vortices and slightly modulated streaks along the lower wall, and one layer of counter-rotating vortices along the upper wall (not shown). This solution is continued to the PRANS formulation by increasing the eddy viscosity as explained in section (3). Continuation is performed at fixed Reynolds number Re = 3800, enforcing a constant volume flux. The variation of the streamwise velocity norm during this continuation procedure is shown in figure 2b. After an initial drop, the streamwise velocity norm increases with , reaching for = 1 a value about 60% larger with respect to its initial value. Moreover, when = 1, the friction Reynolds number reaches Re τ = 134.521, and the statistically invariant solution TW2 T , shown in figure 3b, is obtained. Note that this friction Reynolds number is rather low for a fully developed turbulent flow, we thus expect the solution to slightly change when continued towards the PRANS framework. Comparing this statistically steady solution with the starting travelling wave, one can observe that the quasi-streamwise vortices (blue and red isosurfaces) are almost unchanged, while strong differences can be noticed on the velocity streaks, which are less fragmented, more streamwise-aligned and shifted towards the wall. This shift of the streaks towards the wall can presumably be due to the wall-normal variation of the eddy viscosity used in the PRANS equations. As discussed in [START_REF] Hwang | Mesolayer of attached eddies in turbulent channel flow[END_REF], since in the near-wall region and lower part of the logarithmic region, the eddy viscosity grows linearly with y, coherent structures are allowed to reach larger amplitudes, so they protrude towards the wall. Whereas, they weaken in the upper part of the logarithmic region and outer region, due to the large value of ν t . Moreover, the streaks considerably increase their size in the spanwise direction, and show a peculiar triangular shape. This peculiar shape compares very well with the streaky mean flow obtained by optimally forcing a turbulent channel flow, while the vortical structures remember closely the most energetic DMD mode recovered on top of this streaky flow [START_REF] De Giovanetti | Streak instability in turbulent channel flow: the seeding mechanism of large-scale motions[END_REF]. The streaks reach down towards the wall, where they are almost streamwise aligned, while they present some wiggles close to the streamwise vortices. Moreover, their structures becomes almost identical on the two walls, despite in the upper wall much weaker vortices are found (not shown). These unexpected differences with respect to the starting TW2 solution might arise from the fact that, when continuing the PNS equations (2.1) towards the PRANS ones (2.2), invariant perturbations of the laminar base flow transform into statistically coherent fluctuations of the mean flow. Thus, the observed structural change of the travelling wave solution can be simply due to the change of reference from the base to the mean flow, rather than to an intrinsic modification of the coherent motion. An answer to this important point can be found by directly comparing (a) ( b) The difference between these two flow field is found to be of O(10 -4 ), three orders of magnitude smaller than the perturbation maximum amplitude, thus validating our procedure at such a low friction Reynolds, for which the dynamics of fluctuations of the mean flow should not strongly differ to that of perturbations of the base flow after a mere change of reference.

0 0 0 0 1 1 1 1 2 2 2 2 1 1 Z Z X X Y Y Y Y 2 2 3 3
Once our approach has been validated, we attempt to increase the friction Reynolds number for reaching values typical of fully turbulent flows. First, we tried to continue the TW2 T solution further in Re, but the convergence of higher-Re solutions was very slow and time-consuming. Conjecturing that the domain might be too small for capturing statistically steady coherent structures typical of higher-Reynolds number flows, we have continued TW2 T in the streamwise direction up to L x = 9.54, while keeping U b fixed. This solution, provided in figure 4a, is very similar to that previously shown, showing coherent, large-scale streaks with smaller vortices on top of them. One can again note the strong similarity of this solution with the main energetic structures found in a forced DNS of turbulent channel flow [START_REF] De Giovanetti | Streak instability in turbulent channel flow: the seeding mechanism of large-scale motions[END_REF]. However, this solution is again characterized by a rather low Re τ = 134.5, thus the difference between the relative instantaneous flow fields, shown in figure 4b, is again very small. This TW is then continued in Reynolds number up to Re = 5945, corresponding to Re τ = 199.0, which is only slightly higher than that Motivated by these results, we have chosen a new starting point of the continuation procedure, namely, a spanwise-localized travelling wave solution called TW2 -1 [START_REF] Gibson | Spanwise-localized solutions of planar shear flows[END_REF], whose similarity to flow structures characterizing the near-wall cycle has been reported in the literature. This travelling wave solution, which has been obtained by continuation of TW2 after windowing on a larger spanwise domain [START_REF] Gibson | Spanwise-localized solutions of planar shear flows[END_REF], is similar in structure to TW2, although being mostly concentrated towards one wall. We have first obtained the TW2 -1 solution at Re = 3300 in the domain of size 2π × 2 × 6π , with 32 × 97 × 324 points in the streamwise, wall-normal and spanwise direction, respectively. The TW2 -1 solution is continued at first with respect to the bulk velocity, in order to increase its friction Reynolds number. The resulting travelling wave is shown in figure 5a for Re = 3300. This spanwise-localized solution consists of slightly modulated streaks flanked by streamwise-inclined vortices, which are weaker on the upper wall, where only one streaks pair is observed, and stronger on the bottom wall, where two pairs of streaks are recovered.

This TW2 -1 solution is then continued to the PRANS equations varying the parameter from 0 to 1, at fixed Re = 3300 and U b . Figure 5b their size in the spanwise direction, reaching a width which appears to be close to that typical of large-scale motion, λ z ≈ 1.5h [START_REF] Lee | Spatial organization of large-and very-large-scale motions in a turbulent channel flow[END_REF]. For reaching higher values of Re τ , continuation with respect to Re of TW2 -1 is performed, keeping U b fixed. The variation of the norm of the streamwise velocity of TW2 -1 during the Re-continuation is found in figure 6a, showing a continuous increase of this quantity, while the wall-normal velocity considerably decreases with Re (not shown). Several solutions at increasing values of Re have been then continued in towards the PRANS framework, as shown in figure 6b. Figure 7a,c,e provides the TW2 -T solutions at different values of the Reynolds number up to 6500 (corresponding to Re τ = 380.03). The whole structure of the travelling wave remains similar to that recovered at Re = 3300, although the streaks become stronger while the counter-rotating vortices slightly weaken. More importantly, as shown in figure 7b,d,f, the difference between the instantaneous flow fields of the corresponding TW solutions of the PNS and PRANS equations consistently increases, now reaching the same order of magnitude of the perturbation itself. This means that the structural modifications of TW2 -1 T with respect to TW2 -1 at high Re τ , do not depend only on the change of reference from the laminar base flow to the mean turbulent flow, but to an intrinsic difference in the coherent disturbances dynamics. Figure 7b,d,f shows a cross-section of the low-and high-speed streaks of TW2-1 (black and white lines) at different Reynolds numbers, together with the difference between the instantaneous flow fields of TW2 -1 and TW2 -1 T at the same values of Re (shaded contours). One can note that the largest modifications are observed on the lateral low-speed streaks, which increase their spanwise size and move towards the wall, and on the high-speed ones at the wall, which appear to be change their spanwise size too (not shown). However, while the streamwise velocity is modified in a large part of the domain, the counter-rotating vortices change exclusively in a very narrow zone between the low-and high-speed streaks. The result of these modification is a consistent increase of the spanwise size of most of the streaks, which reach a length comparable to the channel half-height, typical of large scale motion [START_REF] Hwang | Statistical structure of self-sustaining attached eddies in turbulent channel flow[END_REF][START_REF] Del Alamo | Spectra of the very large anisotropic scales in turbulent channels[END_REF][START_REF] Lee | Spatial organization of large-and very-large-scale motions in a turbulent channel flow[END_REF]. A smallersize streak is observed as well, having spanwise size O(100) (in inner units), close to that typical of wall streaks. However, this smaller-scale coherent structure does not appear directly linked to the wall cycle, since, as shown in figure 8a, its streamwise-averaged velocity profile extracted at z = 9 and scaled with respect to the inner units, appears not to be independent of Re τ , as one would expect for wall-cycle related structures. In particular, its peak scales approximately with Re 1/2 τ , as one would expect for large-scale structures [START_REF] Hwang | Mesolayer of attached eddies in turbulent channel flow[END_REF]. It is worth to note that the same scaling with respect to Re τ characterizes the lateral large-scale streaks, whose inner-scaled velocity profiles are shown in figure 9a. However, comparing figures 8b and 9b, one can also note that, while the lateral streaks present a robust outer scaling, confirming their large-scale nature, for the smallscale central streak the velocity profiles at different Re τ do not collapse at large values of y. Thus, the small-scale central streak cannot directly be related with the wall cycle, but neither to the large-scale structures, probably being linked to the secondary motion induced by the vortical structures placed at the centre of the domain. These counter rotating vortices placed in the region between the large-scale streaks, reaching much higher distances from the wall, remember the typical vortical structures recovered in large-scale motion (also called bulges).

A quantitative analysis of the main spanwise wavelengths of TW2 -1 T has been carried out by computing the premultiplied energy spectra of the streamwise, wall-normal and spanwise velocities of this solution at Re = 4500 (corresponding to Re τ = 275.47), which are shown in figure 10a,b,c, respectively. Concerning the streamwise velocity, the lowest-wavenumber peak (k + z = 0.00127 or k z = 0.35 in outer units) corresponds to the size of the TW envelope, being close to the spanwise domain size. Two other peaks are recovered for k + z = 0.0065 and k + z = 0.0155, corresponding to wavelengths λ z = 3.5 and λ z = 1.47 in outer units, respectively, lying in the range of the typical spanwise size of large-scale motion (reported to be λ z = 1 -3 in outer units). A higher-frequency, weaker peak is found for k + z = 0.047, corresponding to λ + z = 135, which is rather close to the typical spanwise size of wall-streaks. Whereas, the wall-normal and spanwise spectra are both characterized by one peak only, at k + z = 0.048 (corresponding to λ + z = 130.08) and k + z = 0.026 (λ + z = 235.32), respectively. Notice that these wavelengths are much lower than the dominant ones of the streamwise velocity spectra, being closer to those typical of the wall cycle. Very similar spectra are recovered at Re = 5500 and Re = 6500 (corresponding to Re τ = 328. [START_REF] Park | Exact coherent states and connections to turbulent dynamics in minimal channel flow[END_REF] and Re τ = 380.03, respectively), indicating that the structure of TW2 -1 T remains robust when the Reynolds number increases. Moreover, it is interesting to evaluate the scaling of the relevant structures of these solutions with respect to those of the single self-sustaining attached eddy [START_REF] Hwang | Statistical structure of self-sustaining attached eddies in turbulent channel flow[END_REF], which is composed of a long streaky motion reaching the near-wall region, self-similar along y = 0.1λ z and a shorter vortical structure carrying all the velocity components, self-similar along y = 0.5 -0.7λ z . A similar scaling is found in the TW2 -1 T solution, with the two dominant peaks in the streamwise velocity spectrum having y ≈ 0.1 -0.15λ z while a scaling of y ≈ 0.58λ z , y ≈ λ z is found in the spanwise and wall-normal spectra, respectively. The statistically steady solution presented here, composed of large streaky structures and short vortical ones carrying all velocity components, is thus similar in shape and wavelengths to the self-sustaining structures of the attached eddy theory [START_REF] Hwang | Statistical structure of self-sustaining attached eddies in turbulent channel flow[END_REF]. However, in this solution, large-and small-scale structures are not torn apart, but tied together in a non-trivial way, representing one potential first brick for the development of a low-order model of turbulence dynamics.

Conclusion

In this work, we propose a new mathematical framework for characterizing the coherent motion of turbulent fluctuations around the mean flow in a turbulent channel, using a statistical point of view. In particular, we search for statistically invariant coherent solutions of the unsteady Reynolds-averaged Navier-Stokes equations written in a perturbative form with respect to the turbulent mean flow, using a suitable approximation of the Reynolds stress tensor. For doing so, we set up a continuation procedure of known invariant solutions of the perturbative Navier-Stokes equations, based on the continuous increase of the eddy viscosity towards its turbulent value. The recovered solutions are sustained only in the presence of the Reynolds stress, thus being representative of the coherent motion of turbulent flows. The travelling wave TW2 has been first used as a starting point of the continuation procedure, and continued to the turbulent framework up to friction Reynolds number Re τ ≈ 134.52. Although structural changes are found in the solution, when resorting to instantaneous quantities the statistically invariant motion results to be only marginally different to the corresponding invariant solution of the Navier-Stokes equations. This was expected, since turbulence is not fully sustained at such low values of the friction Reynolds number, so that the dynamics of statistically coherent motion of fully turbulent flows remains close to that of transient turbulence and transition. However, by taking the spanwise-localized solution TW2 -1 as a new starting point, and continuing it to the statistically turbulent framework at sufficiently large friction number (Re τ ≈ 380.03), the statistically invariant motion considerably departs from the starting solution. This solution is characterized by large-scale and small-scale streaks reaching the wall, accompanied by rather small vortical structures further from the wall. These structures, as well as the main wavelengths and scaling of this statistically invariant solution are very close to those typical of the coherent motion in turbulent channel flows. In particular, the dominant wavelengths of the streamwise velocity premultiplied energy spectrum correspond to the typical spanwise size of large-scale structures (1.5 -3.5 times the half channel height) and are characterized by a scaling y ≈ 0.1 -0.15λ z , consistent with the attached eddy hypothesis. Whereas, spanwise lengths typical of the wall cycle and a scaling of y ≈ 0.58λ z , y ≈ λ z are found in the spanwise and wall-normal spectra, respectively. Thus, the statistically steady solution presented here, constituted by large streaky structures and short vortical ones carrying all velocity components, is similar in shape and wavelengths to the self-sustaining structures of the attached eddy theory [START_REF] Hwang | Statistical structure of self-sustaining attached eddies in turbulent channel flow[END_REF], although composed by large-and small-scale structures tied together in a non-trivial way. This statistically invariant solution may potentially represent one brick for the development of a low-order model of turbulence dynamics. It should be remarked that the comparison of the main wavelengths and scalings of this statistically invariant solution with those of the attached eddy has been limited to the spanwise direction, since the present solution has been obtained in a rather small streamwise domain. Future work will aim at continuing this or other solutions towards larger streamwise domains, as well as towards much higher friction Reynolds numbers. Moreover, new statistically invariant solutions might be obtained using as a starting point filtered snapshots of the turbulent flows, instead of continuing known invariant solutions of the Navier-Stokes equations. Finally, statistically periodic solutions can be recovered as well, providing relevant information about the temporal dynamics of the coherent part of the fluctuations in the considered statistical framework.

This might be a considerable step forward towards the development of reduced-order models of flows.

Figure 1 .

 1 Figure 1. Schematic visualization of the continuation procedure from the Perturbative Navier-Stokes equations ( = 0) to the Perturbative Unsteady Reynolds-averaged Navier-Stokes equations ( = 1).

Figure 2 .

 2 Figure 2. Continuation diagram of the streamwise velocity norm of TW2 with respect to Re (a) for = 0, and with respect to for Re = 3800 (b).

Figure 3 .

 3 Figure 3. Invariant TW2 solution for = 0 (a) and = 1 (b) for Re = 3800: isosurfaces of negative streamwise velocity ((a) u = -0.11, (b) ũ = -0.15, light grey in the printed version, yellow online) and Q-criterion (Q = 0.1) coloured by the streamwise vorticity (dark grey in the printed version; red for positive, blue for negative online). (a) = 0, (b) = 1. (Online version in colour.)

Figure 4 .Figure 5 .

 45 Figure 4. (a) Statistically steady travelling wave solution TW2 T , for Re = 3800 and L x = 9.544: isosurfaces of negative streamwise velocity (ũ = -0.15, light grey in the printed version, yellow online) and Q-criterion (Q = 0.08) coloured by the streamwise vorticity (dark grey in the printed version, red for positive, blue for negative online). (b) Difference of the instantaneous flow fields of TW2 and TW2 T (shaded contours) and streamwise velocity perturbation associated with the travelling wave solution TW2 (black line for negative, white line for positive) for Re = 3800 and L x = 9.544.(a) ũ, (b) u TW2 Tu TW2 . (Online version in colour.)

Figure 6 .Figure 7 .Figure 8 .

 678 Figure 6. Continuation diagram for the travelling wave solution TW2 -1 versus Re for U bulk = 1: norm of the streamwise velocity (a) and of the wall-normal velocity (b). (Online version in colour.)

Figure 9 .

 9 Figure 9. Streamwise-averaged velocity profile of the TW2 -1 T extracted at the z location where the lateral large-scale streaks present their maximum value, scaled in inner and outer units. (Online version in colour.)

Figure 10 .

 10 Figure 10. Premultiplied one-dimensional spanwise spectra of the streamwise (a), wall-normal (b) and spanwise velocity (c) of TW2 -1 T at Re = 4500. (a) k z E uu (k z ), (b) k z E vv (k z ), (c) k z E ww (k z ). (Online version in colour.)

Data accessibility. Supporting data are accessible at the following link: https://zenodo.org/record/5190924#.

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Authors' contributions. E.P. carried out the computations and the data analysis under the indications of J.C.R., S.C. and P.D.P. M.F. helped in the development of the numerical code. S.C. conceived and designed the study, and drafted the manuscript. All authors read and approved the manuscript.

All authors gave final approval for publication and agreed to be held accountable for the work performed therein.

Competing interests. The authors declare that they have no competing interests. Funding. This research was funded by MIUR grant no. PRIN 2017X7Z8S3_002 LUBRI-SMOOTH. This work was granted access to the HPC resources of IDRIS under the allocation 2020-A0072A06362 and 2021-A0092A06362 made by GENCI.


V Maurice

R Chutani

C Carlé

S Keshavarzi

S Queste

L Gauthier-Manuel

J.-M Cote

R Vicarini

R Boudot

N Passilly
email: nicolas.passilly@femto-st.fr

Versatile microfabricated alkali vapor cells using local sealing

Keywords: alkali vapor cells, microfabrication, atomic devices, I

We report on a new technique to form and seal microfabricated cells that alleviates some of the limitations imposed by current solutions. The technique consists in integrating single-use sealing structures that can be actuated through laser irradiation, either to isolate cavities from each other or to connect them. Inspired by the well-established glass-blowing techniques, it reproduces on a wafer the "make-seal" and "break-seal" features found in conventional cell filling setups. By allowing cells to be evacuated, filled and sealed after having been fully-formed, this technique opens new venues for atomic devices. Here, we demonstrate both features and present one use case where alkali dispensers are used in association with non-noble gasses despite their gettering ability.

INTRODUCTION

Atomic sensors or clocks are nowadays among the best devices to probe physical quantities or deliver stable frequency signals. The interrogation of hot alkali vapors contained in microfabricated cells, in particular, has provided a path toward miniature devices that can be deployed in a variety of systems and environments [START_REF] Kitching | Chip-scale atomic devices[END_REF]. Microfabricated cells are key in making such devices small and accessible as they allow replacing traditional glass-blown cells by cheap chip-size components. To structure them, the use of silicon and glass wafer stacks assembled by anodic bonding has been widely adopted [START_REF] Liew | Microfabricated alkali atom vapor cells[END_REF], but various solutions have been proposed to fill them with alkali vapor and buffer gas, each with its pros and cons. In many solutions, having pure alkali metal during sealing is avoided to prevent its reaction or evaporation. For this purpose, the alkali metal can be introduced inside the cell under a stable form (alkali chromate [START_REF] Douahi | New Vapor Cell Technology for Chip Scale Atomic Clock[END_REF] or molybdate [START_REF] Maurice | Microfabricated vapor cells filled with a cesium dispensing paste for miniature atomic clocks[END_REF], azide [START_REF] Woetzel | Microfabricated atomic vapor cell arrays for magnetic field measurements[END_REF] or chloride [START_REF] Liew | Microfabricated alkali atom vapor cells[END_REF]), which can eventually be reacted by external irradiation in order to release alkali vapor. However, none of these techniques successfully meet the needs of all atomic devices. For instance, azide, due to its fixed stoichiometry, typically imposes large amounts of buffer gas, whereas alkali molybdate-based dispensers, which contain a Zr/Al getter, prevent gasses such as nitrogen from being used.

In this context, we propose a new technique to lift the limitations associated with current cell microfabrication techniques. It consists in transposing the make-seal and break-seal features commonly found in glass cell fabrication assemblies into wafer-integrated structures. Such structures allow connecting or hermetically isolating inner cavities [START_REF] Maurice | Gas cell for an atomic sensor and method for filling a gas cell[END_REF]. This solution offers many advantages, such as the ability to use a single dispenser to fill tens of cells, to introduce gasses that were not compatible with the getter, or to target a higher vacuum inner atmosphere.

II. METHODS/RESULTS

We developed two types of seals. The first one, the make-seal, consists in a thin glass membrane that can be deflected to seal the port of an underlying microchannel. The deflection is achieved by locally heating and melting the membrane with a CO 2 laser as it undergoes a pressure difference. Hermetic sealing is achieved when the membrane touches and fuses with the glass substrate around the port of the channel, forming a ring-shaped sealing area. This structure can be made by standard micromachining processes such as anodic bonding, deep reactive ion etching and laser ablation.

We designed a wafer layout where a single source of alkali metal feeds an array of cells, each featuring a make-seal structure, through a network of microchannels (Figure 1). We used a cesium pill dispenser to act as a source and we monitored the establishment of the atomic density after activating the pill. After a while, a saturated vapor is obtained in the cells and cesium droplets are observed. The cells are then individually sealed using the make-seals. Finally, the cells are diced, resulting in 4 mm x 6 mm chips with pure Cs vapor and no dispenser.

The second type of seal, the break-seal, consists in a thin silicon wall separating an auxiliary cavity from the main cavity of the cell. The wall initially provides an hermetic seal and can consequently be broken by various means. Here, we use an ablation laser to breach the wall without compromising the overall hermeticity of the cell.

To demonstrate this feature, we designed a layout where several auxiliary cavities surround the main cavity of the cell. Since the auxiliary cavities are blind holes, they can be filled with buffer gas at the first anodic bonding step, while the second bonding step defines the content of the main cavity of the cell, which in our case was evacuated. After introducing cesium vapor, we use the break-seals to release the content of the auxiliary cavities into the cell cavity.

Linear absorption and CPT spectroscopy is performed to assess the hermeticity, on the one hand, of the make-seal after the seal is made and, on the other hand, of the break-seal before the seal is broken on the long term.

III. DISCUSSION/INTERPRETATION

Combining make-seal and break-seal structures offer novel possibilities. We demonstrate here an immediate benefit: the ability to mitigate the major limitations imposed by pill-dispensers. While dispenser pills can be relatively costly, tedious to insert and cumbersome, we show that a single dispenser can be used to fill multiple cells. Besides, a quenching gas such as nitrogen can now be introduced within the cells without being absorbed by the pill [START_REF] Franz | Enhancement of alkali optical pumping by quenching[END_REF].

Other use cases can be envisioned. For instance, for chip-scale atomic clocks based on microwave transitions, various gas mixture ratios can be obtained at the same time, providing cells targeting different operating temperatures [START_REF] Vanier | On hyperfine frequency shifts caused by buffer gases: Application to the optically pumped passive rubidium frequency standard[END_REF].

External sources of atomic or molecular species can be envisioned providing new solutions when dispenser pills or stable reactants are not readily available or when electric discharge through the buffer gas prevents anodic bonding from being used.

IV. CONCLUSIONS

By transposing the techniques used in conventional vapor cell fabrication to microfabricated cells, we have proposed a more versatile way of making microfabricated vapor cells alleviating the shortcomings of current techniques. In the future, we will work on reducing the footprint of the make-seals and improving the yield by experimenting with different dimensions and materials. Next, we will focus on ultra-high vacuum regimes and we will investigate its ability to seal a cell under vacuum while minimizing outgassing, which is relevant for optical frequency references [START_REF] Newman | Architecture for the photonic integration of an optical atomic clock[END_REF] and passively pumped cold-atom devices [START_REF] Boudot | Enhanced observation time of magneto-optical traps using micro-machined non-evaporable getter pumps[END_REF].

Figure 1 :

 1 Figure 1: An array of microfabricated cells featuring make-seals connected to a common source of cesium (before sealing and dicing).