Agnès Rivière
email: agnes.riviere@mines-paristech.fr

Nicolas Flipo

Patrick Ansart

Aurélien Baudin

Loic Marlot

PIREN-Seine phase VII -rapport 2018 -Revue des données de température du bassin des Avenelles 1 Revue des données de température du bassin des Avenelles

Une revue des données de température du bassin des Avenelles est présentée en détail. Une évaluation de l'évolution des variations à différentes échelles de temps pour chaque compartiment de l'hydrosystème (air, rivière, zone hyporhéique, berges, et aquifère) est faite.

Une augmentation de 0.05°C/an de la température de l'air et une augmentation du nombre de jours supérieurs à 20 °C sont observés. Dans les aquifères l'augmentation de température est plus marquée, atteignant 0.1 °C/an. Tous les compartiments sont assujettis à des variations saisonnières. Le régime thermique annuel de la rivière est similaire à celui de l'air, avec des températures élevées en période estivale et faibles en période hivernale. La température de l'aquifère est plus faible en période estivale et plus élevée en période hivernale. La température des berges a un cycle entre ces deux régimes.

Les températures de l'air et de la rivière sont soumises à deux cycles liés à la variation du rayonnement solaire incident. Le premier cycle est lié aux saisons. Le second cycle correspond à la rotation de la terre sur elle-même et l'alternance entre le jour et la nuit. Ce second cycle dépend du premier, de la durée du jour et de la nuit évoluant selon la saison. Ceci est modulé par l'ombrage, l'humidité et le vent. Les variations diurnes des compartiments aquifères et berges sont nulles. Celles-ci semblent être non nulles dans les forages.

Points clefs

 Tendance de l'augmentation de la température des différents compartiments de l'hydrosystème de l'ordre 0.1°C/an  Deux tendances sinusoïdales se superposent à l'augmentation de température : saisonnière pour tous les compartiments et diurne pour la rivière et l'air  Régime thermique annuel

Introduction

D'après le rapport du Groupe d'experts intergouvernemental sur l'évolution du climat (GIEC, 2018), le changement climatique à long terme fait référence à une modification des paramètres climatiques dans le temps et dans l'espace (local, régional et mondial) et dont l'impact peut inclure l'élévation du niveau de la mer, la désertification, la diminution des ressources, de la biodiversité et d'eau douce, et l'accentuation des phénomènes hydrologiques et thermiques extrêmes tels que les inondations, les sécheresses et les canicules. La variation de la température devient une variable essentielle de quantification des impacts des changements climatiques afin d'être capable de comprendre l'évolution de la qualité de l'eau dans les différents compartiments mais aussi de prédire leur état écologique futur.

Ce rapport présente l'évolution de la température dans les différents compartiments de l'hydrosystème des Avenelles c'est-à-dire l'air, la rivière, les berges, la zone hyporhéique (zone de mélange entre les eaux souterraines et les eaux de surface) et les aquifères de 2011 à aujourd'hui. Il met en lumière de manière qualitative les mécanismes d'amortissement et de diffusion de la température à différentes échelles de temps depuis l'air et son influence sur la rivière jusqu'aux aquifères de plateau en passant par l'interface napperivière (Figure 1).

Variations de la température de l'air

Les stations de mesures de la température de l'air sont indiquées dans la Figure 6. Cinq stations mesurent la température proche de la surface du sol (5-10 cm) et deux stations mesurent la température à 150 cm. Le Tableau 1 indique le nombre de données par station, le pas d'échantillonnage ainsi que les périodes de mesure.

Figure

Variations annuelles

Les températures de l'air en moyennes annuelles sont tracées dans la ; Nombre de jours chauds par année : c) Melarchez_Old (1962( -1982( ) et d) Boissy (1994( -2018) ) ;

a) b) c) d)

Variations journalières annuelles

Cette partie rend compte de la variation journalière de température de l'air au cours de l'année. Les températures journalières moyennes calculées sur toute la chronique de mesure sont présentées sur la La température des berges est plus faible au mois de mars et plus élevée au cours des mois de septembre à octobre (Figure 16). (iii)

La température de l'aquifère est maximale au cours des mois de décembre et janvier et minimale pendant les mois de juillet à Septembre (Figure 17). Les mesures de la température de l'aquifère sont perturbées par les prélèvements réalisés dans les piézomètres qui ont introduit des altitudes de mesure différentes pour le pzps1. Les données futures permettront de confirmer ou d'invalider le comportement différent enregistré. La température enregistrée au piézomètre pzps6 comporte des variations abruptes. La nappe est très proche de l'altitude de la surface du sol en période de hautes eaux, le transfert de chaleur s'intensifie pendant ces périodes. Aux différentes stations de suivi, les températures varient de manière similaire à celle de la température de l'air. Les stations amont de rivière (Loge, Bergeresse, Melarchez) enregistrent des températures annuelles plus élevées que les stations situées au centre du bassin (Petit Paris, Bertin et Derrier). Les deux stations aval enregistrent des diminutions des températures annuelles alors que toutes les autres stations enregistrent une augmentation. La Figure 21 montre le même constat en été les températures amont et aval sont supérieures à celles enregistrées au centre du bassin. Cependant, en hiver, les températures montrent la même dynamique.

Les stations centrales enregistrent des tendances similaires hormis pendant les périodes estivales (Figure 22). Les cours d'eau en amont du bassin sont situés dans des champs de cultures, leur ripysylve est peu dense contrairement à celle du centre du bassin. La diffusion de la chaleur de surface entre l'atmosphère et le sol n'est ainsi pas atténuée par la végétation. De plus, le cours d'eau est de faible largeur et le débit est faible, ce qui induit une diffusion rapide de la température de l'air dans la rivière. Les températures moyennes mensuelles sont, de ce fait, plus importantes à Bergeresse (Figure 22). La station de Petit Paris enregistre une température moyenne mensuelle plus importante au mois de juin. Ceci est lié au pompage plus intense pendant les mois de juillet et d'aout. Ceci s'explique par l'évolution de la végétation, celle-ci va réduire la radiation solaire en été contrairement en période hivernale, où la végétation est dépourvue de feuilles. Les températures enregistrées à la station de Petit Paris sont inférieures aux températures enregistrées aux autres stations (Figure 18). Cette station est influencée par un pompage en rive gauche. Une hypothèse pour expliquer ce phénomène serait que les débits d'eau pompés induiraient une diminution de la température en rivière en favorisant des échanges nappe-rivière.

La température annuelle de piézomètres de berge est supérieure à celle de la rivière pour les stations Bergeresse et Petit Paris contrairement aux autres stations (Figure 18 et Figure 19). La température de la berge de Bergeresse est plus importante en hiver que celle de la rivière de 3°C. Ceci est lié à la diffusion de la température dans la berge.

Les maxima des températures moyennes mensuelles dans les aquifères ne sont pas tous synchrones. Ils interviennent à la fin de l'autonome pour Granges et Chantemerle, et pzps6 et à la fin de l'hiver pour pzps1, pzps4 et Derrier (Figure 23). Ceci est lié à différents processus : diffusion de la chaleur est modulée par la profondeur du point de mesure, par l'altitude du battement de la nappe, et par le phénomène d'advection.

Conclusion et perspectives

Cette étude montre qu'au cours des dernières années, une augmentation de 0.05°C/ an est observée dans l'air. De plus, le nombre de jours pour lesquels la température est supérieure à 20 °C augmente. Ces deux facteurs se traduisent, pour les eaux souterraines, par une augmentation des températures des berges et des aquifères.

Tous les compartiments sont assujettis à des variations saisonnières. Le régime thermique annuel de la rivière est similaire à celui de l'air, avec des températures élevées en période estivale et faibles en période hivernale. La température de l'aquifère est plus faible en période estivale et plus élevée en période hivernale. La température des berges correspond au mélange de ces deux régimes.

La température de l'air et de la rivière sont soumises à deux cycles liés à la variation du rayonnement solaire incident. Le premier cycle est lié aux saisons. Le second cycle correspond à la rotation de la terre sur ellemême et à l'alternance entre le jour et la nuit. L'influence de ces cycles sur la température des eaux de surface est essentiellement modulée par l'ombrage généré par la végétation. Les variations diurnes des compartiments aquifères et berge sont nulles. Celles-ci semblent être non nulles dans les forages.

Ce rapport montre l'intérêt de poursuivre le maintien de ce réseau de mesure afin d'obtenir suffisamment de données pour rendre compte de l'évolution de conditions thermiques des différents compartiments mais aussi afin de la conceptualisée en vue de prédire l'évolution de l'état écologique du bassin dans un contexte de changements globaux.

Bibliographie

Mouhri , A., Flipo, N., Rejiba, F., de Fouquet, C., Bodet, L., Kurtulus, B., Tallec, G., Durand, V., Jost, A., Ansart, P., Goblet, P., 2013. Designing a multi-scale sampling system of stream-aquifer interfaces in a sedimentary basin. J. Hydrol. 504, 194-206. Rivière, A., Mouhri, A., Bodet, L., Ansart, P., Baudin, A., Berrhouma, A., Carrillo, N., Chauris, H., Cocher, E., Cucch, K., Cuisinier, E., Durand, V., de Fouquet, C., Goblet, P., Hovhannissian, G., Jost, A., Labarthe, B., Martineau, N., Pasquet, S., Rejiba, F., Rubin, Y., Tallec, G., Flipo, N., 2016. Quantifier les échanges nappe-rivière depuis l'échelle locale jusqu'à l'échelle du petit bassin-versant. Rapport PIREN Seine, Dans "Interfaces Nappe-Rivière", Rapport de Synthèse 2011-2015

Tallec, G., Ansart, P., Guérin, A., Delaigue, O., Blanchouin, A., 2015. Observatoire Oracle; Irstea.

Thomas, W., 2018. Température des axes fluviaux du bassin de la Seine au fil de l'eau, au fil du temps. DRIEE, police de l'eau.

Figure 1 .

 1 Figure 1. Evolution temporelle de la température dans les trois compartiments de l'hydrosystème : rivière, berge et aquifère. Les couleurs de fond marquent les saisons : printemps en vert, été en jaune, automne en rose, et hiver en violet

Figure 2 .

 2 Figure 2. Instrumentation du bassin

Figure 4 .

 4 Figure 4. Chronique des précipitations totales journalière et débit à la station des Avenelles

Figure 5 .

 5 Figure5. Précipitation mensuelle

Figure 7 .

 7 Il est difficile de statuer sur un gradient entre les différentes zones du bassin, ce qui s'explique par le fait que le bassin est de petite taille et très plat. Les températures de Melarchez_old et de Boissy sont mesurées à 150 cm du sol et les autres températures sont mesurées à la surface du sol. Les données récentes semblent indiquer que la température est plus faible sous la ripisylve à l'aval du bassin. Les chroniques supérieures à 20 ans indiquent une tendance générale à l'augmentation de 0.05°C/an. Le nombre de jours inférieurs à 5°C diminue significativement entre les deux périodes 1962-1982 et 1994-2018 (Figure 7 a et b) ; celui-ci est en moyenne de 75 jours avant 1982 et seulement de 40 jours depuis 1994. Le même constat a été fait par Thomas (2018) sur le bassin de la Seine. Le nombre de jours avec une température moyenne supérieure à 20 °C a quant à lui augmenté entre les deux séries ; il était de 13 jours entre 1962 et 1983.

Figure 8

 8 Figure 8 c et d) et de 20 jours entre 1994 et 2018.Sur la période d'étude de 2010 à 2017, le nombre de jours froids par année oscille régulièrement avec une période de 3 ans environ. Par contre, le nombre de jours chauds (T>20°C) augmente sur les trois dernières années(2015, 2016 et 2017). Nous allons par la suite regarder comment les différents compartiments de l'hydrosystème transforment ce signal (surface du sol, rivière, berge et aquifère)

Figure 9 Figure 9 .Figure 10 .Figure 11 .

 991011 Figure 9. Régime des variations des températures journalières : a) Mélarchez (1962-1983) et b) Mélarchez (2010-2018), c) Loge

Figure 12 .

 12 Figure 12. Amplitude diurnes de l'air en fonction du temps. River Lab correspond à la station Avenelles

Figure 13 .

 13 Figure 13. Position des stations de jaugeage. River Lab correspond à la station Avenelles

  Figure et Figure 15). La hauteur d'eau, la largeur de la rivière et la densité de végétation est plus faible à l'amont du bassin. Le centre du bassin est caractérisé par une ripysylve dense et une fréquence de mouilles et seuils plus importante que sur le reste du bassin. Ceci induit des variations de hauteurs d'eau conséquentes le long du cours d'eau. La ripylsylve de l'aval du bassin est sporadique, la rivière est plus large et la profondeur des mouilles est moins importante qu'au centre du bassin. Le nombre de valeurs extrêmes et les valeurs de ces extrêmes diminuent d'amont en aval du bassin. Les variations de températures sont plus importantes à l'aval qu'au centre du bassin, la ripysylve étant moins dense à l'aval et à l'amont. Ceci est principalement lié à la hauteur d'eau, la largeur de la rivière, et la ripysylve qui amortissent la diffusion de la température de l'air dans la rivière. La température de la berge de Bergeresse est maximale au mois d'août, elle atteint sa valeur minimale au mois de mars comme les autres berges. Ceci présage une conduction importante de la température de rivière pendant la période estivale. Cette hypothèse est appuyée par les variations brutales des températures observables sur la Figure 16 a. La station de Petit Paris est influencée par le pompage d'où des variations brutales des températures (Figure 16 b) au moment des pompages non réguliers.

Figure 15 .Figure 15 .Figure 167 .Figure 17 .

 151516717 Figure 15. Températures journalières de rivière : a) Loge, b) Bergeresse, c) Mélarchez, d) Petit Paris, e) Bertin et f) Derrier

Figure 18 .Figure 24 .Figure 25 .

 182425 Figure 18. Température de la rivière annuelle

Station Variable Début Fin Nb de données Pas de temps

	Tableau 1. Caractéristiques des stations de mesures de la température de l'air
	Avenelles et AvAv	T° air	2011	-	56296	horaire
	Boissy	T° air (150 cm)	1994	-	184282	horaire
	Boissy_0	T° air (10 cm)	2007		61881	horaire
	Loge	T° air	2013	-	35311	horaire
	Melarchez	T° air	2015	-	24476	horaire
	Melarchez_old	T° air	1963	1982	7305	journalière
	Theil	T° air	2013	-	43274	horaire
	6. Position des mesures de la température de l'air. River Lab correspond à la
		station Avenelles

  Sept stations de jaugeage sont présentes sur le bassin et cinq stations LOMOS (Figure13et Tableau 2). Quatre stations de jaugeage comportent une mesure de température en rivière (La loge, Mélarchez, Avenelles et Theil). Chacune des stations LOMOS possède une mesure de température en rivière. La Figure14représente la position de tous les piézomètres sur le bassin. Les piézomètres donnant des réponses intermédiaires entre la température de berge et la température aquifère ne sont pas présentés dans ce rapport.

	Tableau 2. Caractéristiques des stations de mesures de la température des trois compartiments rivière,
			berge, aquifère
	Station	Variable	Début Fin Nb de données Pas de temps
	Loge	T° rivière	2012	-	33424	horaire
	Melarchez	T° rivière	2014	-	218433	horaire
	Bergeresse	T° rivière	2012	-	209544	15 minutes
	Petit_Paris	T° rivière	2012	-	164705	15 minutes
	Bertin	T° rivière	2012	-	210953	15 minutes
	Derrier	T° rivière	2012	-	164705	15 minutes
	Avenelles	T° rivière	2011	-	54891	horaire
	AvAv	T° rivière	2013	-	180534	15 minutes
	Theil	T° rivière	2012	-	48559	horaire
	Bergeresse	T° berge	2012	-	198721	15 minutes
	Petit_Paris	T° berge	2012	-	209718	15 minutes
	Bertin	T° berge	2012	-	216197	15 minutes
	Derrier	T° berge	2012	-	184838	15 minutes
	AvAv	T° berge	2013	-	166780	15 minutes
	Pzps1	T° aq	2012	-	190140	15 minutes
	Pzps4	T° aq	2012	-	188136	15 minutes
	Pzps6	T° aq	2012		190725	15 minutes
	Chantemerle	T° aq	2011	-	107861	horaire
	Granges	T° aq	2009	-	77682	horaire
	Derrier	T° aq	2010	-	65065	horaire

PIREN-Seine phase VII -rapport 2018 -Revue des données de température du bassin des Avenelles


Hansjörg Mixdorff
email: hmixdorff@bht-berlin.de

Navneet Nayan
email: nnayan@ec.iitr.ac.in

Albert Rilliard
email: albert.rilliard@limsi.fr

Preeti Rao
email: prao@ee.iitb.ac.in

Debashis Ghosh

DEVELOPING A CORPUS OF AUDIO-VISUAL ATTITUDINAL EXPRESSIONS IN HINDI

Keywords: Attitudes, Hindi, audio-visual perception

, and forms part of an intercultural endeavour aimed at studying prosodic and facial expressions of social affect across languages. Our corpus includes a total of 16 attitudes, such as arrogance, surprise, politeness etc. portrayed by 19 speakers (10f, 9m) of which three experts selected the best four males and females and the better of two turns for subsequent analysis. A followup experiment with more participants employed a total of 512 stimuli, 256 hereof full audio-visual stimuli, 128 audio-only and 128 silent video stimuli. Results indicate higher ratings for emotionally loaded attitudes such as irritation and doubt as compared to, for instance, irony or seductiveness. Reduced modality stimuli were rated more poorly.

INTRODUCTION

Human communication always has a social goal. Information about e.g. the mental state, emotions, mood or attitudes of the speaker and listener is conveyed during the dialog. The affective state is influenced, for instance, by the situation or role of the dialog partners. Mutual understanding of the social intention between communication partners should not be difficult as long as they grow up in the same or at least a similar culture. Interaction between partners from different cultures sometimes leads to wrong interpretations of the social expression. It has been shown that the verbal and non-verbal expressions depend on the culture in which we grow up. A study by Shochi et al. investigated twelve social attitudes e.g. surprise, irritation, command-authority for prosodic effects in British English, French and Japanese [START_REF] Shochi | Intercultural perception of English, French and Japanese social affective prosody[END_REF]. Intercultural comparison of linguistic and paralinguistic effects has enjoyed growing attention as the knowledge about how verbal and non-verbal social affects are expressed in different languages is paramount for mutual understanding between different cultures. The primary goal of the work reported in the current paper was the recording of a Hindi speech corpus of social attitudes for an intercultural comparison. It follows the experimental design developed by Rilliard et al. [START_REF] Rilliard | Social face to face communication -American English attitudinal prosody[END_REF], discussed in Section 2. In earlier works we collected corpora of attitudinal expressions in German [START_REF] Hönemann | Social Attitudes -Recordings and Evaluation of an audio-visual Corpus in German[END_REF] and Cantonese [START_REF] Lee | Free Labeling of Audio-visual Attitudinal Expressions in Cantonese[END_REF] following the same experimental design. These corpora were judged by Hindi speaking subjects [START_REF] Mixdorff | Perception of Audio-visual Expressions in German and Cantonese by Native Speakers of Hindi[END_REF]. Now the corresponding Hindi attitudinal data were gathered. This paper presents the process of data collection and a perception study exploring the credibility of the 16 social attitudes (see Table 1 for complete listing), e.g. doubt, surprise and politeness of 19, later 8 Hindi speaking subjects. Section 2 presents the experimental design and the social and linguistic criteria of situations employed to elicit the attitudes from our subjects. In the same section the technical setting for the recordings, as well as the process of the recordings with dialog examples are discussed. The perceptual evaluation of the social attitudes by native Hindi raters is described in section 3. Section 4 presents the results of the evaluated expressions which includes analyses with regards to speaker and attitude. Section 5 concludes this paper with discussion and conclusions.

EXPERIMENT SETUP

Situational attitudes

16 Attitudes are performed by 19 speakers (10f, 9m) which are elicited through a short dialog between each speaker and the experimenter. The dialogs led to two target utterances: "mairee naach rahee thee" (मै री नाच रही थी) (engl. Mary was dancing) and "ek kela" (एक क े ला) (engl. a banana). In the remainder of this article we will refer to the English translations to identify the respective phrase.

For each expression of attitude, a test dialog was executed in order to prepare the speakers. This dialog was designed according to different social situations differing in the following social and linguistic aspects (A => speaker, B => experimenter):

• Type of speech act: propositional / social attitude • Hierarchical distance between speaker A and speaker B: A > = < B 1 • Social distance between speaker A and speaker B: 1-friend, 2-known, 3-unknown

Surprise (SURP):

A and B are friends and the same age. A did not know that B can sing well. One day, B has A listen to his beautiful voice. The scene is at a friend's home.

Technical setting and recording process

We used a Nikon D850 HD camera set up 1m away from the speaker. Additionally, the acoustic signal was recorded through a dynamic AKG D5 microphone placed under the chin of speaker. A laptop positioned in front of the speaker displays the description of each social situation, the related test dialog and target dialog permitting the speakers to prepare for the ensuing task. The experimenter performed each situation with the subject in order to immerse him/her in the context for the attitude. After a short break the target dialog containing the target phrase followed immediately. The complete sequence of dialogs was recorded twice. An example test and target dialog for the attitude "irritation" (IRRI) could run as follows: A: Sorry, but don't smoke please. B: Okay, okay… A: Don't smoke, please! B: What did Mary do last night? A: Mary was dancing. I already told you three times (are you deaf?) In the test dialog A is angry because B starts to smoke and A wants to stop him. The sentence in bold 1 Notions of distance, cf. Spencer-Oatey (1996) is the target utterances which A should speak with an irritated expression. The sentences in brackets are only for familiarizing the speaker with the task, and are not uttered The main dialog ends with the target utterance "Mary was dancing". The target phrases were cut out of the session video. We recorded a total of 19 speakers x 16 attitudes x 2 phrases x 2 trials=1216 target phrases performed by 19 speakers. Participants were students of IIT Roorkee and compensated for their time.

EVALUATION

Before we can subject the data to further analysis or employ them in perceptual tests we need to ensure the quality of presentation. To this end we first had the complete corpus of 1216 files judged by three experts who ranked each video clip on a scale from 9 (convincing) to 1 (implausible), given the intended attitude. The experts were familiar with the elicitation paradigm and supposed to scrutinize the data accordingly. In order to facilitate the tiring task we created four batches of 240 stimuli and one of 256. In all experiments the stimuli were presented via a survey on the PsyToolkit server [START_REF] Lee | Free Labeling of Audio-visual Attitudinal Expressions in Cantonese[END_REF] [START_REF] Mixdorff | Perception of Audio-visual Expressions in German and Cantonese by Native Speakers of Hindi[END_REF]. After an introduction to the experiment design, listing all 16 attitudes in English as well as Hindi, the stimuli were presented in a randomized order, first displaying for 900 ms a screen with the intended attitude in English and Hindi and a fixation cross in the location where the video was going to appear, then the video for its respective duration was shown. Subsequently the viewers were able to input their judgment via a graphical interface by clicking on a line between "very well" corresponding to a score of 9 and "not at all" corresponding to 1. A progress bar was shown for orientation. The experts viewed the full audio-visual (A/V) stimuli and helped us identify the four best male and female speakers, as well as the better turn of the two we had collected. Hence we yielded 8 x 32=256 utterances for the ensuing main experiment with a larger number of participants. In addition to the 256 full A/V stimuli, from these we created 128 silent video (V) and 128 audio-only (A) stimuli, selecting the better one of "Mary was dancing" or "a banana". Hence the experiment employed a total of 512 stimuli, the quality of which was judged by 32 Hindi speaking raters (23m, 9f). Due to the large number of utterances to be evaluated, we created a PsyToolkit survey with four sets of 128 stimuli each, while ensuring that the reduced A and V stimuli were not in the same set as the corresponding A/V stimuli. This means that each set was rated by eight judges. After hearing/watching the stimulus only once raters had to judge the plausibility of each attitude during a time window of ten seconds. We collected 4096 scores, 32 of which (0.8%) were invalid due to timeouts. We hence included in total 2034 valid judgments for A/V stimuli 1016 for A, and 1014 for V type stimuli. The experiment took on average 14 minutes to complete.

RESULTS

The first analysis presented in this section concerns the judgments on a scale from one to nine assigned by the three expert judges. Statistical tests were performed using the SPSS statistical toolkit [START_REF] Shochi | Intercultural perception of English, French and Japanese social affective prosody[END_REF]. In the second part we look at the judgments of the larger participant group on the reduced set.

Expert Ratings

The raters generally judged the female speakers (score average/s.d. 7.32/1.95) better than the male speakers (7.06/2.10), or in other words, of the eight best rated speakers six were females. Interestingly, the distance "between the genders" is greater for certain attitudes such as ADMI or WOEG than others like DOUB or IRRI. In Table 1, columns three and four list means and s.d. for all 16 attitudes. The judgments of the raters indicated a slightly better performance of the speakers at the second trial (average of 7.23) than the first one (7.16). The reason could be that the speakers knew the task after the first trial and on that account, it was easier to prepare the expressions. For the ensuing larger study, we selected for each attitude/target phrase the repetition that was judged better. We did not find any significant difference between the target phrase "a banana" and "Mary was dancing".

As witnessed in earlier studies, complex attitudes like IRON and SEDU were judged more poorly than coping strategies with strong emotions such as IRRI, SURP, CONT and DOUB.

In contrast, POLI which is actually a socially conditioned register also ranges among the highest scorers. In our German data this was the opposite. Obviously politeness exhibits special features in Indian society which are not as strong in the West or just represents a "normal way of speaking". Interestingly, also the neutral question (QUES) received rather low ratings. We assume that the facial expression together with the interrogative mode often counteracted the intended "neutrality" and rather conveyed doubt or suspicion.

There is a strong negative correlation between the mean score for each stimulus (Pearson's R of -.765, p < .001) and its standard deviation, indicating that the raters agreed more on highly rated stimuli. There is also a weak correlation between the reaction time and the score (Pearson's R of -.147, p < .001).

Ratings on subcorpus of best speakers

As explained above we selected a subcorpus of the best four male and best four female speakers and had them judged by a larger group of participants. Figure 1 presents snapshots of examples of different attitudes of these speakers and their mean scores and s.d. In general, results among the "non-experts" follow similar tendencies as those of the experts. If we correlate their mean stimulus-wise judgments with those of the experts on the 256 full AV stimuli that both experiments had in common, we yield a Pearson's R of .385 (p < .001). Noticeably, the non-experts extend their range of scores much more towards the lower end. Hence their mean score across the whole data set is almost one point below that of the experts (7.32 vs. 8.12, for AV stimuli) Since we look at many more judges their s.d.s are much higher at 1.89 vs. 0.68 of the experts.

Attitude

The most important difference between the two experiments besides the smaller number of stimuli presented is the presence of full A/V as well as V and A stimuli. On average full A/V stimuli were judged better (mean/s.d. of 7.32/1.89) than V (6.86/2.10) and A (6.88/2.10). This is confirmed by independent samples Kruskal-Wallis test between AV and A or V, respectively. Table 2 lists mean scores and standard deviations for each attitude depending on the modality. Interestingly, for the so-called "neutral" attitudes DECL and QUES the audio-only mode yields better ratings than the full AV, indicating the contribution of the acoustic channel, but also DOUB and SURP seem to be conveyed well acoustically. On the video-only data, IRRI and ARRO, two strongly negative attitudes, surpass audio-only by far and show the importance of the visual channel.

DISCUSSION AND CONCLUSIONS

This study presented a recently developed Hindi corpus of attitudinal expressions and first results of its analysis. Starting from an initial set of 19 speakers and 1216 video clips, we first selected the best four male and female speakers for subsequent analysis. We found that the females were the better performers, especially at expressing ADMI.

On the remaining data set we performed a rating test of full AV as well as reduced A and V versions. Our results are in line with those of earlier studies on different languages [3][4]. Reduced stimuli are generally judged more poorly than the full ones, with a few exceptions. This result shows that both the acoustic signal and the visual cues are vital in the portrayal of attitudes. Future work will entail perception tests with viewers from different cultures. Work is in progress on a free labeling task with German participants. We also plan analysis of acoustic data as well as visual cues. Although we already observed many kinds of head movements, the famous Indian "head wobble" or "wag" occurs very seldom. Of the 256 stimuli only three exhibit this head movement. They all pertain to the UNCE condition.

Acknowledgements

This work was supported by DFG grant Mi625-32 to Mixdorff. Thanks to Priya G L for video editing.

  1

  A and B are almost the same age and know each other. A is sitting next to B. Suddenly B starts smoking and A is very angry; he wants him/her to stop, expressing his irritation toward speaker B. The scene is in a public place.

	• Valence of speech act: positive / negative
	• A dominates B: yes / no
	The social situations conceived are adapted from
	Rilliard et al. [2]. As examples detailed descriptions
	of three of the social situations are presented in the
	following:
	Admiration (ADMI): A and B are almost the same
	age and know each other well. Both love Samosas,
	and talk about the delicious ones they had the day
	before at Prakash's restaurant. The scene is at a
	coffee shop.
	Irritation (IRRI):

Table 1 :

 1 List of attitudes, abbreviations used in the paper and mean score and s.d. found in the expert study for 1216 stimuli. Examples from the four best male and female speakers performing various attitudes, mean and s.d. of scores.

		abbrev-iation	mean	standard deviation
	admiration	ADMI	7.35	2.18
	arrogance	ARRO	7.42	1.81
	authority	AUTH	7.38	1.62
	contempt	CONT	7.25	2.00
	neutral statement	DECL	7.30	1.78
	doubt	DOUB	7.62	2.05
	irony	IRON	6.60	2.03
	irritation	IRRI	7.63	2.07
	obviousness	OBVI	7.10	1.96
	politeness	POLI	7.46	1.57
	neutral question	QUES	6.50	2.37
	seductiveness SEDU	6.78	2.29
	sincerity	SINC	6.95	2.02
	surprise	SURP	7.42	2.11
	uncertainty	UNCE	7.30	1.93
	walking-on-eggs	WOEG	7.06	2.11

Table 2 :

 2 List of attitudes. mean score and standard deviation as a function of the attitude and modality.


Florian Faucher

come

Numerical study of quantitative inverse wave problems in visco-acoustic and visco-elastic media

Numerical study of quantitative inverse wave problems in visco-acoustic and visco-elastic media.

Florian Faucher1 ,

We consider the propagation of waves in media with attenuation. Attenuation is a frequencydependent phenomenon and several constitutive laws exist to consider it, each leading to different models of wave equations. In the inverse problem for the reconstruction of the physical properties from the measurements of waves, considering a medium with attenuation leads to additional unknown coefficients to recover (those that represent the attenuation), while the precise attenuation law characterizing the medium is also unknown a priori. In this work, we consider the inverse wave problem for visco-acoustic and visco-elastic media. We use time-harmonic formulations which allow us to unify the models of attenuation using complex-valued parameters and illustrate the wave propagation depending on the attenuation model. We carry out quantitative reconstruction with attenuation model uncertainty using an iterative minimization procedure. Furthermore, we also consider the case of reflecting boundaries surrounding the sample, which lead to multiple reflections within the domain and complicate the reconstruction.

Project-Team Makutu, Inria Bordeaux, University of Pau and Pays de l'Adour, France.


Adrien Bella

Noé Lahaye

Gilles Tissot

Internal tides energy transfers and interactions with the mesoscale circulation in two contrasted areas of the North Atlantic

Keywords: Internal tide, mesoscale flow, energy budget, vertical modal decomposition, high resolution numerical simulation

The energy budget of the internal tide and its life cycle is investigated with a high resolution numerical simulation and a vertical normal mode decomposition. Two areas of interest are considered: the Azores Islands over the mid Atlantic ridge and the Gulf Stream offshore the North of the US East coast shelf break. Low mode (1 and 2) internal tides are found to propagate from 100 km (mode 2) to 1000 km (mode 1) away from their generation sites. Waves loose a significant portion of their energy as they propagate through the Gulf Stream, in contrast to the Azores domain. In the Gulf Stream domain, the mesoscale circulation is responsible for energy transfers from low to high modes internal tides, while the topographic scattering is dominant in the Azores area. This transfer of energy toward high modes favours energy dissipation. The mesoscale is significant in the energy budget of modes higher than mode 1 for both domains, and for all baroclinic modes in the Gulf Stream area. The internal tide is found to extract or loss energy toward the mesoscale circulation, but this accounts for less than 14%, of the energy scattered from low internal tide modes to higher ones once summed over all contributions of the modal energy budget.

Introduction

Internal tides are a category of oceanic internal inertia-gravity waves that are generated when the astronomical tide interacts with topographic features such as shelf breaks, ridges or seamounts. They are encountered in many areas in the ocean [START_REF] Zaron | Baroclinic tidal energetics inferred from satellite altimetry[END_REF], Zhao et al., 2016, among others] and are of crucial importance for the mixing of deep water and the closure of the Meridional Overturning circulation [START_REF] Munk | Abyssal recipes ii: energetics of tidal and wind mixing[END_REF]]. The role of mesoscale currents for the energy pathway starting from the barotropic tide input and leading to this mixing is not yet fully understood. As a different thread of motivation, internal tides exhibit the same spatial scale as (sub-)mesoscale features, which renders challenging the estimation of the geostrophic velocities and the disentanglement of the two type of motions from satellite altimetry ( [START_REF] Ponte | Incoherent signature of internal tides on sea level in idealized numerical simulations[END_REF]). This problem is exacerbated by the ability of the new SWOT mission to obtain a resolution of 20 km to 30 km, compared to the approximately 100 km of current altimeter products [START_REF] Ballarotta | On the resolutions of ocean altimetry maps[END_REF]). It is therefore needed to better understand the dynamics of the internal tide if we hope to disentangle them from the geostrophic field via data assimilation for instance.

Since a decade or so [e.g. [START_REF] Arbic | Concurrent simulation of the eddying general circulation and tides in a global ocean model[END_REF], global and regional realistic simulations have reached a sufficient resolution to be able to represent both internal tides and mesoscale features, opening the way to studying their interactions. Most studies focusing on these interactions use the vertical modal decomposition framework [START_REF] Gill | Atmosphere-Ocean dynamics[END_REF], which allows to get a clear separation of the barotropic and baroclinic tides as well as alleviating the computational cost associated with processing 3D simulation outputs.

The importance of the mesoscale circulation and buoyancy field have been shown by [START_REF] Kelly | Internal-tide interactions with the gulf stream and middle atlantic bight shelfbreak front[END_REF] in a realistic configuration over the Gulf Stream area for the energy budget of the first baroclinic mode (denoted mode 1). They quantified the energy transfers between the mode 1 and the mesoscale circulation as well as its relative importance compared to topographic effects, showing that the mesoscale explains first order interactions pattern visible on the mode 1 energy flux divergence. Similarly, [START_REF] Pan | Interactions of internal tides with a heterogeneous and rotational ocean[END_REF] have analysed the energy budget of the mode 1 in a realistic setup including the mesoscale circulation and heterogeneous buoyancy field in the Gulf Stream area and the Mallau Island, leading to the same conclusion concerning the importance of mesoscale -internal tides interactions. Both studies have also considered realistic simulations with baroclinic currents, and have used a modal decomposition approach to define their energy budget. The deviation of internal tide energy flux have been analysed by [START_REF] Duda | Internal tidal modal ray refraction and energy ducting in baroclinic gulf stream currents[END_REF], indicating a refracting behaviour of a Gulf Stream like current on beams of internal tides energy fluxes. [START_REF] Dunphy | Focusing and vertical mode scattering of the first mode internal tide by mesoscale eddy interaction[END_REF] have shown in an idealised setup that the interactions of the first baroclinic internal tide mode with the barotropic eddy and the first baroclinic eddy mode lead to a modification of the energy flux propagation and a scattering of energy toward higher modes, respectively.

In the present paper, our focus will be on the energy budget yielded by a modal analysis and we will in particular extend the work of other studies to the 10 first baroclinic modes, as well as studying the couplings between modes.

The contribution of the mesoscale flow and buoyancy field will be taken into account and the relative importance of the coupling between modes as well as their spatial pattern will be quantified in an area where topographic feature are prominent, and another area with a strong mesoscale circulation.

The document is organised as follows: section 2 will explain the theoretical framework with the modal decomposition and the governing dynamical equations. Section 3 will detail the dataset used and computational techniques. Section 4 will present results. Section 5 closes this article with a conclusion.

Governing equations and energy budget

To investigate the modal internal tide energy budget, we rely on linearized equations of motions projected on vertical normal modes. The derivation follows previous studies (e.g. [START_REF] Kelly | Internal-tide interactions with the gulf stream and middle atlantic bight shelfbreak front[END_REF]) and is only briefly depicted here. It starts from the hydrostatic primitive equations under the Boussinesq approximation with a free surface [e.g. [START_REF] Vallis | Atmospheric and Oceanic Fluid Dynamics: Fundamentals and Large-Scale Circulation[END_REF]:

∂tu h + u • ∇u h + f ⃗ ez ∧ u h = -∇ h p -∇Π tide , (1a)
∂zp -b = 0, (1b)

∂tb + u • ∇b = 0, (1c)
∇ • u = 0, ( 1d
)
where u is the velocity, b the buoyancy, p the reduced pressure (divided by ρ0, the reference density), and the index h denotes horizontal component of a 3D vector. Other variable names follow standard notations. Here, the forcing and dissipation terms have been omitted on purpose, except for the tidal potential term Π tide .

Next, the flow is decomposed into low-frequency mesoscale flow and a high-frequency component that includes the internal tide: i.e., for the velocity, we split u = U + ũ. We will further decompose the slow variables of the buoyancy into a long-time mean and a slow variable part: B = B + B ′ (• denotes long time average), and introduce the Brunt-Väisälä frequency: N 2 = ∂zB and N 2 = ∂z B . Subtracting the low frequency equations (1) to the initial system and further assuming that nonlinearities amongst the high-frequency flow are negligible, one obtains the following linear system of equations:

∂t ũ + U • ∇ũ + ũ • ∇U + f ⃗ ez ∧ ũ = -∇p -∇Π tide , (2a) ∂z p -b = 0, (2b) ∂t b + U • ∇ b + ũh • ∇ h ( B + B ′ ) + w( N 2 + N ′2 ) = 0, (2c)
∇ • ũ = 0. ( 2d
)
Note that, since the above system of equations is linear in the high-frequency variables, and assuming that most of the mesoscale variability is of timescale longer or equal to the cutoff period used in the internal tides complex demodulation (3 days), band-pass filtering of these equations leaves the system unchanged. Hence, in our study, we will focus on the semi diurnal (including M2 and S2 frequencies) internal tide with a period centred on 12.2 h, relying on the above set of equations where the highfrequency variables describe this frequency band. These equations are completed by the usual linearized boundary conditions at the mean surface η and the bottom at z = -H:

p(η) = g η, (3)
w(η) = ∂t(η) + U h (η) • ∇ h (η), (4)
w = -ũ h (-H) • ∇H. (5)
We thus have the internal tide part of the primitives equations that are now ready to be projected on a set of verticals modes. The latter are given locally by the standard Sturm-Liouville problem for internal waves in a rotating stratified ocean [e.g. [START_REF] Gill | Atmosphere-Ocean dynamics[END_REF][START_REF] Vallis | Atmospheric and Oceanic Fluid Dynamics: Fundamentals and Large-Scale Circulation[END_REF], assuming a flat ocean, a horizontally-homogeneous stratification profile and no background flow:

Φ ′ n N 2 ′ + Φn c 2 n = 0, with Φ ′ n = 0 at z = -H, and gΦ ′ n + N 2 Φn = 0 at z = η, (6)
where prime denotes vertical derivative. The obtained set of vertical modes Φn is complemented by another set of vertical functions which obey:

φ ′ n = Φn, Φn ′ = - N 2 c 2 n φn, (7)
and these modes follow the orthogonality conditions:

η -H ΦmΦn dz = η -H N 2 c 2 n φmφn dz + g c 2 n , φm(η)φn(η) = Hδmn. (8)
Physical fields can then be expanded over these bases through:

[u h , p] = n [un, pn]Φn, (9)
[w, b] = n [wn, N 2 bn]φn, (10)
and the modal projection coefficient are obtained as follows:

[un, pn] = ⟨[u h , p], Φn⟩ , (11)
wn = 1 c 2 n φn, w N 2 + g H w(η)φn(η) , (12) bn
= 1 c 2 n ⟨φn, b⟩ + g H b(η) N 2 (η) φn(η) , (13)
with the inner product defined as such: ⟨f, g⟩ = 1 H η -H f (z)g(z) dz. Note that the two bases have a parametric dependency on the horizontal coordinates because of the varying topography H(x, y) and stratification profile N 2 (z; x, y). In this paper, we are going to consider the modes 0 to 10, with the mode 0 being the barotropic tide, and all others being the baroclinic internal tide.

The dynamical equations are then obtained by projecting the tidal momentum equations and continuity on a mode Φm, and buoyancy equations on the mode φm, using the relation exposed above plus integration by part and Leibniz formula, and substituting the buoyancy modal amplitude by the pressure anomaly (after projection of the hydrostatic balance on φm) [START_REF] Kelly | Internal-tide interactions with the gulf stream and middle atlantic bight shelfbreak front[END_REF][START_REF] Kelly | The vertical mode decomposition of surface and internal tides in the presence of a free surface and arbitrary topography[END_REF][START_REF] Pan | Interactions of internal tides with a heterogeneous and rotational ocean[END_REF]. Omitting the • notation from now on, one obtains -after a tedious but straightforward derivation:

∂tum + ∇ h pm + f ⃗ ez ∧ um = - n Umn • ∇un - n unU Φ mn - n un • U ∇ mn - n wnU z mn - n pnTmn - 1 H ∇Π tide (φm(η) -φm(-H)), (14a) ∂tpm -c 2 m wm = - g H φm(η)U h (η) • ∇η - n U p mn • ∇pn - n pn φm, U h • ∇ N 2 c 2 n φn + n un • (Bmn -Bmn) + n wn φm, φnN 2 , ( 14b
)
∇ h • (Hum) + Hwm = H n unTmn, (14c)
with:

Tmn = ⟨Φm, ∇ h Φn⟩ , Umn = ⟨Φm, U h Φn⟩ , U Φ mn = ⟨Φm, U h • ∇ h Φn⟩ , U ∇ mn = ⟨Φm, Φn∇U h ⟩ , U z mn = ⟨Φm, φn∂zU h ⟩ , U p mn = φm, Utot N 2 c 2 n φn , Bmn = ⟨φm, Φn∇B⟩ , Bmn = φm, Φn∇ h B .
We then take the scalar product of the projected momentum equation ( 14a) with un and multiply the pressure equation ( 14b) with pm, substitute wn using the continuity equation ( 14c) to obtain the energy budget of the internal tide mode m in presence of a slowly varying mesoscale circulation and buoyancy field:

temporal variation of energy ∂ ∂t u 2 m 2 + p 2 m 2c 2 m + divergence of energy flux 1 H ∇ h • (Humpm) =
pm c 2 m n pn φm, U h • ∇ h N 2 c 2 n φn - Tidal potential work um • ∇Π tide (φm(η) -φm(-H)) . ( 16
)
By anticipation of the results in figure 3 in section 4.2, the term Rm gathers all contributions that are not of primary importance (and will not be discussed in this paper). This equation describes the different energy exchanges (coupling) between modes as well as energy sources and sinks. Most coupling terms can be represented with coupling matrices -noted Kmn here -that will be shown below in section 4.2 ( Figure 2). Generally, this coupling matrix contains an anti-symmetric part, describing net exchanges of energy between modes, and a symmetric part, with the diagonal included, that describes gain/loss of energy for the internal tide.

The first three terms in the right-hand side (RHS) arise from the presence of a mesoscale flow, while the next two ones include contributions from associated buoyancy perturbations as well as seasonal variations. The term labelled as slowly variable stratification, arises during the derivation of the CSW buoyancy equation 14 because of our choice to construct the modal bases by using a stationary profile of stratification in the problem 6. In case of a definition of the vertical modal bases with the instantaneous stratification profile, this term would vanish. Indeed, for a linear propagation of an internal wave without topography, mesoscale and dissipation, the vertical modes diagonalise the operator, thus decouples the components from each other. By time-averaging the stratification profile, the associated vertical mode basis becomes inadequate to perform this decoupling. The slowly variable stratification term then arises, taking into account in the energy equation coupling between modes associated with this linear propagation. This term can be interpreted as energy exchanges between modes during a linear propagation of an internal wave without topography, mesoscale and dissipation. Since vertical modes are eigenfunctions of this associated idealised operator, we associate the variable stratification term to the inadequacy of the basis to represent internal wave dynamics (through the time-averaged stratification profile). We do not expect, however, that using a basis defined with a slowly time-varying stratification would change significantly the magnitude of the dominant interaction terms discussed in this paper (see e.g. section 4.2), and in particular the mesoscale and topographic contributions.

Finally, Cmn = -Cnm is the topographic scattering, which includes the generation of baroclinic tide from the barotropic tide (Cm0). Note that this term has no symmetric part -it can only redistribute energy amongst the vertical modes.

Once averaged in time over a sufficiently long period (typically, one month), the time-variation of the energy vanishes and the above modal energy equation reduces to the following, simplified form:

∇ h Fm = - n Kmn + Rm, ( 17
)
where Fm denotes the modal energy flux, Kmn gathers the energy transfers, sources and sinks that are not negligible and Rm gathers all the neglected terms. These include a contribution from the free surface and the tidal potential, which would both vanish under a rigid-lid assumption for the baroclinic modes -in agreement with the fact that we find they have negligible magnitude in our diagnostics (cf. Sect. 4). The remaining term in Rm involves horizontal variations of the modes and associated background stratification, which is small since a year average is used.

3 Data and method

eNATL60 simulation

The high-resolution realistic simulation of the North Atlantic Ocean eNATL60 [START_REF] Brodeau | Ocean-next/eNATL60: Material describing the set-up and the assessment of NEMO-eNATL60 simulations[END_REF] is used to diagnose the energy budget of the internal tide following eq. ( 15). It uses the Nucleus for European Modelling of the Ocean (NEMO) model, which solves the primitive equations under the Boussinesq and hydrostatic approximations with an Arakawa C grid and z-coordinate with partial step [Madec and the NEMO team, 2008]. The eNATL60 run includes astronomical tidal forcing with the M2, S2, N2, O1 and K1 frequencies. In addition, surface forcing from the 3-hourly ERA-interim (ECMWF) reanalysis is used, which enables the simulation to develop a realistic mesoscale field. The horizontal resolution of the simulation is 1/60 • (about 1.5 km in the mid latitudes) and features 300 vertical levels with a thickness starting from less than 1 m at the top of the ocean to 100 m at the bottom. We processed the hourly outputs of the horizontal velocity u, sea level, temperature and salinity fields (from which we computed the pressure and stratification. We will use the 2009 October month over two subdomains representative of different dynamical regimes of the ocean. The first one is centred over the Azores Islands and the North Mid-Atlantic Ridge. The second one is located offshore the North of the US East coast and includes a portion of the Gulf Stream. The Azores domain is characterised by the predominance of topographic features such as the mid Atlantic ridge exhibiting low amplitudes topographic variation over 10 km, and a group of seamount with strong slope and a typical scale of 100 km. Comparatively to the Gulf Stream area, the mesoscale flow is weak. The Gulf Stream domain is marked by a strong mesoscale activity and a large continental slope leading to a flat abyssal plain with a few surrounding seamounts (see Fig. 1 discussed below).

Filtering and computing methods

In order to obtain the vertical normal modes bases, the mean stratification N 2 is computed from the time average of temperature and salinity using the nonlinear equation of state from TEOS-10. The Sturm-Liouville problem ( 6) is then discretised and solved on the staggered vertical grid in each horizontal cell, giving the two modal bases on the T grid, at the centre (in the horizontal) of each cell. From there, the modal amplitude un, vn and pn are obtained by projecting the corresponding fields u and p, following (11)-( 13). Since the horizontal grid is staggered, the horizontal velocity is located at the edge of each cell. We thus interpolate the base Φn before projecting the velocities. Unfortunately, this step induces a loss of orthogonality for the newly interpolated basis. The coefficients determined by innerproduct between the horizontal velocity fields and the interpolated basis are corrected a posteriori by inverting the cross-correlation matrix between the interpolated basis and the local basis issued from the Sturm-Liouville problem (6) on the u, v-grid. This allows us to obtain the projection coefficients un and vn defined on the u, v-points. Vertical velocity is then computed using the continuity equation formulated with the discrete scheme employed in the NEMO code [Madec and the NEMO team, 2008].

The next step consists in separating the mesoscale contribution from the semi diurnal internal tide. The latter is extracted by means of a complex demodulation at a central semi-diurnal frequency ω = 1.415×10 -4 rad s -1 and a low-pass cutoff cutting period of 3 days. The mesoscale flow is directly lowpass filtered with the same cutting period. The mesoscale slowly-variable buoyancy fields is computed from the daily-averaged temperature and salinity, and the corresponding stratification is then estimated from this buoyancy field and a mean sound speed profile [using standard formula, see e.g. [START_REF] Vallis | Atmospheric and Oceanic Fluid Dynamics: Fundamentals and Large-Scale Circulation[END_REF].

Since our modal decomposition is inaccurate in areas of shallow water, a mask is applied to remove all locations shallower than 300 m in the Azores domain, and 250 m in the Gulf Stream domain. The mask is slightly shallower in the Gulf Stream domain in order to retain the topographic generation occurring at the shelf break.

Since high modes are expected to propagate over shorter distances from their generation site compared to low modes, we usually group the modes 4-10 together, and interpret them as part of dissipative processes in the energy budget. Finally, all terms of equation ( 15) display small scale patterns, particularly pronounced when the mesoscale and internal tides interact. These patterns are not of particular physical significance for the diagnostics discussed in this paper. Therefore, we apply a Gaussian spatial filter before displaying any term in (15), with a kernel width equal to 25 grid points (a little less than 40 kilometres). This procedure is only applied for map plotting.

Results

The energy budget of internal tides in both domains are examined in this section, first by considering the geographical distribution of their generation, propagation and sinks, and secondly by quantifying the spatially integrated and temporally averaged energy budget.

Life cycle of the internal tide

Both the Azores Islands and Gulf Stream domains are sites of powerful internal tides generation, exposed to contrasted background conditions. The divergence of the energy flux averaged in time over one month (for the period of October 2009) is shown in Figure 1, along with the mesoscale currents at 52 m depth and the topography. In the Azores area, mode 1 energy flux originates from the seamounts and propagates far away (at least 1000 km), indicating a moderate energy loss during the propagation (Fig. 1, top left panel). Modes 2 (not shown) and 3 propagate over shorter distances. All dominant sources and sinks are located around the topography (visible in Figure 1). For the third mode, we find a local loss of energy just next to the generation at the seamount, of the same order of magnitude but smaller value. This property holds for higher modes as well (not shown). In the Gulf Stream domain, in contrast, the input of energy for mode 1 is located at the shelf break, producing a strong offshore beam. This beam manifests strong energy loss as it propagates as can be seen in the negative patch displayed in the mode 1 energy flux divergence, Fig. 1, bottom left panel. A first hot spot of sinks for the mode 1 is directly next to the shelf-break, another being at the encounter with the strong north-eastward current. This beam is also refracted by the Gulf Stream, a behaviour previously reported in [START_REF] Duda | Internal tidal modal ray refraction and energy ducting in baroclinic gulf stream currents[END_REF]. As we will show in the next section (cf. Figures 3 and2), this loss of energy is mainly caused by the advection terms in equation ( 15) and is largely converted in mode 2 energy. Mode 3 (Fig. 1, bottom right panel) displays a more complex repartition of sources and sinks, with an overall gain of energy at and near the shelfbreak and an overall loss of energy further away from the land, where it encounters the current. The sources and sinks of energy from modes 1 and 3 are therefore much less localised around the prominent topographic feature than in the Azores domain, and clearly exhibit strong interactions with the mesoscale currents.

Importance of the different contributions in the energy transfers

We now focus on quantifying the causes of the multiples interactions of the internal tide with its surrounding which results in the pattern in the energy flux divergence previously described (Fig. 1).

Detailed view of coupling terms.

To this aim, the modal energy budget ( 15) is integrated over the two domains considered, averaged over the October month and the different contributions are displayed under their matrix form in figure 3. The matrix are decomposed into their anti-symmetric part (upper triangle) denoting a transfer of energy from mode to mode, referred afterwards as scattering, and symmetric part (diagonal plus lower triangle) showing sources or sinks for the internal tide. We have only plotted terms of ( 15) that are of first order for at least one mode in one domain.

Amongst these terms, the topographic contribution is at least one order of magnitude higher than all others for the barotropic mode in both domains and for the first baroclinic mode in the Azores domain. However, for the Gulf Stream area, the advection coupling becomes of similar magnitude for the first mode and exceed the topographic scattering for modes 2 and 3. In the Azores domain, this is non negligible for modes 2 and 3 but never exceeds the topographic contribution. In this region, the second largest contribution is given by the variable stratification coupling for the baroclinic modes. In The Gulf Stream area, the mesoscale vertical shear production and the horizontal buoyancy gradient loss are also of first order. However, as already found in [START_REF] Kelly | Internal-tide interactions with the gulf stream and middle atlantic bight shelfbreak front[END_REF], these two terms seem to compensate each other leaving a near zero energy gain for mode 1 -which the authors explained as resulting from the thermal wind balance holding for the background flow. The same phenomenon approximately occurs for higher modes, cf. Figure 2: the lower triangles of buoyancy and shear matrix cancel out.

These dominant processes are dominated by their anti-symmetric part. The topographic contribution is purely anti-symmetric by construction, while the sum of the symmetric part for a given mode for the advection coupling is always smaller than 10% of the sum of the anti-symmetric part when this term is significant in regard to the topographic scattering. This quantity never exceeds 20% for the variable stratification coupling in the Azores domain. For the Gulf Stream domain however, it is close to one for the first baroclinic mode, indicating that symmetric and antisymmetric parts have comparable magnitudes. The background flow shear and buoyancy gradient couplings have a dominant symmetric part, but these two terms are cancelling each others. Last, if we sum all the contributions of the modal energy budget (15), the symmetric transfer of energy is found to be worth less or equal to 14% for modes 1 , 3 in both domains and 2 in the Azores one. Therefore, the interaction between the mesoscale circulation and the internal tides mainly contribute to a scattering of modes equivalent to the one produced by the topography without exchanges between the internal tides and the mesoscale circulation and buoyancy field.

As a last comment, Figure 2 shows that topographic scattering is the only process that significantly transfers energy between non-neighbouring modes. For the other dominant processes, the energy transfer toward non-neighbouring modes is one order of magnitude lower than the one toward a neighbouring mode, except for the variable stratification coupling in the Gulf Stream area. For the mesoscale contributions, this implies that only the third mode is able to transfer energy toward what we consider as part of dissipative processes.

Modal energy budget

We are now looking at a more aggregated view of (15) thanks to figure 3 which shows the modal energy budget of mode 1, 2 and 3 for both domains. Here we have separated the energy transfers with the following decomposition: We also have separated the symmetric part (*) of the antisymmetric one, and have furthermore divided the latter into an energy exchange coming from lower modes, and an energy exchange going from higher modes (**). The flux coming from lower modes is again divided into an energy transfer coming from the barotropic mode (****), and transfers coming from the baroclinic ones (***). A positive value means a transfer of energy toward the mode of interest.

Topographic Scattering : Cnm = pmun • Tnm -pnum • Tmn Advection of the interal tide : Anm = um • (Umn • ∇un) -um • unU Φ mn + pm c 2 m U p mn • ∇pn Shear of background flow : ∇Unm = un • U ∇ mn • um + um • U z
If we consider i the index of the mode of interest and Anm the energy transfer matrix already separated into symmetric and antisymmetric part as in figure 2, then we have :

**** = A0i *** = i-1 n=1 Ani ** = -4 n=i+1 Ain * = i n=0 Ain -4 n=i+1 Ani
Comparing the integrated value of the energy divergence flux, the Gulf-Stream domain is overall less energetic than the Azores domain: mode 1 has a divergence around 4 time larger in the last domain than in the first and modes 2 and 3 are also more energetic in the Azores area. In the results displayed in figure 3, the mode 1 in the Gulf-Stream domain is the only one to have an overall negative energy flux divergence.

The barotropic mode has an importance in the energy budget of mode 1 to 3 only through the topographic scattering, and is dominant in the energy budget of mode 1 for both areas. However, it becomes less important for modes 2 and 3, with a contribution to the topographic scattering toward mode 3 inferior to the one caused by modes 1 and 2, but still significant.

Among the first order energy transfers exposed in all panels of figure 3, none transfers energy from small scale to large scale. Mesoscale and topographic couplings therefore create a forward energy cascade.

Last, figure 3 enables identifying the terms of the modal energy budget (15) that are of negligible importance. The advection of ssh and stationary stratification are negligible, which justifies gathering these terms in equation ( 15) in a small residual term. Similarly, the astronomical tidal potential (not included in Figure 3) only projects weakly on the baroclinic modes. The mean of temporal variations of potential and kinetic energy are also small: one month is enough to flatten the two weeks periodic variation introduced by the spring neap cycle.

Conclusion

The semidiurnal internal tide energy budget was characterised based on a vertical mode decomposition, allowing a detailed investigation of the energy transfers amongst different vertical scales -and associated horizontal scales -of the internal tide field. We showed that the Azores area, a region of weak mesoscale activity and strong topographic features, is essentially dominated by the topographic induced energy transfer from low to higher modes. While this effect is dominant for low modes, variable stratification and -to a lesser extent -advection of the internal tide by the mesoscale flow become of similar importance for high modes (equal or larger than 2). Some questions remain concerning the precise role of the variable stratification, in relation with the definition of the modal basis expansion and how this definition would change the energy exchanges between modes. In the Gulf Stream area, advection of the internal tide by the background flow is as significant as the topographic scattering -the transfer of energy caused by the advection becoming more important than the one caused by topographic scattering starting from mode 2 -, while the variable stratification term is less important. However, the net impact of the mesoscale circulation on the energetic of internal tide is mainly limited to a scattering of energy from the low to high-order modes. In comparison, exchange of energy between the internal tide and the mesoscale fields are negligible: the energy exchanged by modes 1 to 3 with the background is always less than 14% of the total energy transfer for the respective mode.

The type of modal analysis conducted in the present paper on a high resolution primitive equation simulation is able to give some insights on the energetic impact of mesoscale circulation and its associated buoyancy field on the internal tide and the transfer of energy from the astronomical tide toward dissipation. This work could be extended by investigating the deep water mixing caused by these interactions. We have in particular overlooked for now the internal tide dissipation, but it is worth investigating in the near future.

Finally, the temporal variability of mesoscale -internal tides interactions will also be studied in the near future. 15). The divergence of the modal flux is plotted in red, while others contributions are separated into the symmetric part (green) and the anti-symmetric part where interaction with lower modes and with higher modes (orange) are separated. The anti-symmetric part coming from low modes is furthermore separated into fluxes coming from the barotropic mode (blue) and fluxes coming from the baroclinic modes (fuchsia). A positive value means a gain of energy for the mode.

Processes taken into account are Topo : C nm , Adv : A nm , shear : ∇U nm , ssh + Ns : S nm , Buo : ∇B nm , Strat : N nm .

  background flow um • (Umn • ∇un) -um • unU Φ mn + stratification by the background flow

Figure 1 :

 1 Figure 1: One-month average of the horizontal energy flux divergence for mode 1 (left column) and mode 3 (right column), for the Azores domain (top) and the Gulf Stream domain (bottom). A positive value indicates a gain of energy for the mode. Black arrows show the square root of the corresponding modal horizontal energy flux. A spatial Gaussian filter with a half modal wavelength kernel has been applied on these fields in order to remove the local alternate of positive and negative values and make the plot cleaner. Green streamlines indicate, qualitatively, the mesoscale horizontal currents at 52 m depth, and isobaths at 1000 m, 2000 m and 3000 m are superimposed in grey contours.

  mn wn Horizontal gradient of buoyancy field : ∇Bnm = pm Hc 2 m un • ( Bmn + Bmn) Surface contribution and advection of the mean stratification profile : Snm = -pm Hc 2 m gφm(η)Uh(η) • ∇η -pm c 2 m n pn φm, U h • ∇ h

Figure 2 :

 2 Figure 2: Matrices K nm summarising the mean energy transfer integrated over the whole subdomains between a mode n (row) and a mode m (column). The upper-right triangle is the anti-symmetric part, and the lower-left triangle (diagonal included) the symmetric part. The sign convention follows eq. (15): a positive value indicates a gain of energy for the reference -"receiving" -mode m from n mode and conversely. Modes 4 to 10 are grouped together. From top to bottom are the coupling caused by the topography, the internal tide advection by the background flow, the terms implying the horizontal and vertical shear of the background flow and the term implying the gradient of the buoyancy field (both variable and annual mean). Notice the different range of values in the colorbar for the topographic coupling terms compared to the other ones. 9

Figure 3 :

 3 Figure3: Mean modal budget for modes 1 (top), 2 (middle row) and 3 (bottom) for the Azores domain (left) and the Gulf Stream (right) domains, as described by equation (15). The divergence of the modal flux is plotted in red, while others contributions are separated into the symmetric part (green) and the anti-symmetric part where interaction with lower modes and with higher modes (orange) are separated. The anti-symmetric part coming from low modes is furthermore separated into fluxes coming from the barotropic mode (blue) and fluxes coming from the baroclinic modes (fuchsia). A positive value means a gain of energy for the mode. Processes taken into account are Topo : C nm , Adv : A nm , shear : ∇U nm , ssh + Ns : S nm , Buo : ∇B nm , Strat : N nm .