Pablo Ruiz Fabo

Clara Martínez Cantón

Thierry Poibeau

Distant Rhythm: Automatic Enjambment Detection on Four Centuries of Spanish Sonnets

Introduction

Enjambment takes place when a syntactic unit is broken up across two lines of poetry (Domínguez Caparrós, 2000: 103), giving rise to different stylistic effects (e.g. increased emphasis on elements of the broken-up phrase, or contrast between those elements), or creating double interpretations for the enjambed lines (García-Paje, 1991).

In Spanish poetry, the syntactic configurations under which enjambment takes place have been described extensively, and detailed studies on the use of enjambment by individual authors exist (see Martínez Cantón, 2011 for an overview)1 . However, a larger-scale study to identify enjambment across hundreds of authors spanning several centuries, enabling distant reading [START_REF] Moretti | Distant Reading[END_REF], was not previously available.

Given that need, we have developed software, based on Natural Language Processing, that automatically identifies enjambment in Spanish, and applied it to a corpus of approx. 3750 sonnets by ca. 1000 authors, from the 15th to the 19th century. What is the interest of such large-scale automatic analyses of enjambment? First, the literature shows a debate about which specific syntactic units can be considered to trigger enjambment, if split across two lines, and whether lexical and syntactic criteria are sufficient to identify enjambment. Second, the stylistic effects that enjambment permits are also an object of current research [START_REF] Martínez Fernández | La voz entrecortada de los versos[END_REF]. Systematically collecting large amounts of enjambment examples provides helpful evidence to assess scholars' current claims, and may stimulate novel analyses. Finally, our study complements Navarro's (2016) automatic metrical analyses of Spanish Golden Age sonnets, by covering a wider period and focusing on enjambment.

The abstract is structured thus: First we provide the definition of enjambment adopted. Then, our corpus and system are described, followed by an evaluation of the system. Finally, findings on enjambment in our diachronic sonnet corpus are discussed. The project's website provides details omitted here for space reasons.2

Enjambment in Spanish

Syntactic and metrical units often match in poetry. However, this trend has been broken since antiquity for various reasons [START_REF] Parry | The distinctive character of enjambement in Homeric verse[END_REF]) on Homer, or Flores Gómez (1988) on early classical poetry).

In Spanish tradition, enjambment 3 is considered to take place when a pause suggested by poetic form (e.g. at the end of a line or across hemistichs) occurs between strongly connected lexical or syntactic units, triggering an unnatural cut between those units. [START_REF] Quilis | La estructura del encabalgamiento en la métrica española: Contribución a su estudio experimental[END_REF] performed reading experiments, proposing that the following strongly connected elements give rise to enjambment, should a poetic-form pause break them up:

1. Lexical enjambment: 4 Breaking up a word. 2. Phrase-bounded enjambment: Within a phrase, breaking up sequences like "noun + adjective", "verb + adverb", "auxiliary verb + main verb", among others. 5 3. Cross-clause enjambment: Between a noun antecedent and the pronoun heading the relative clause that complements the antecedent.

Besides the enjambment types above, [START_REF] Spang | Ritmo y versificación. Teoría y práctica del análisis métrico[END_REF] noted that if a subject or direct object and their related verbs occur in two different lines of poetry, this can also feel unusual for a reader, even if the effect is less pronounced than in the environments identified by Quilis. To differentiate these cases from enjambment proper, Spang calls these cases "enlace", translated here as "expansion". [START_REF] Quilis | La estructura del encabalgamiento en la métrica española: Contribución a su estudio experimental[END_REF] was the only author so far to gather recitation-based experimental evidence on Spanish enjambment. His typology is still considered current, and was adopted by later authors, although complementary enjambment typologies have been proposed, as Martínez Cantón (2011) reviews. Our system identifies Quilis' types, besides Spang's expansion cases.

3 The term for enjambment in Spanish studies is "encabalgamiento". 4 We translate Quilis' terms thus: "lexical enjambment" stands for "encabalgamiento léxico" or "tmesis". "Phrase-bounded enjambment" stands for "encabalgamiento sirremático", and "cross-clause enjambment" stands for "encabalgamiento oracional". 5 For Quilis' complete list of syntactic environments that can trigger enjambment, and the types identified by our system, see https://sites.google.com/site/spanishenjambment/enjambment-types

Corpus

The corpus is based on two public online collections6 (García González, R. (ed.), 2006a, 2006b). The first one covers 1088 sonnets by 477 authors from the 15th-17th centuries. The second one contains 2673 sonnets by 685 authors from the 19th century. We created scripts to download the poems, remove HTML and extract dates of birth and death for the authors. 7Table 1 shows the distribution of authors and poems by century. The corpus covers canonical as well as minor authors, inspired in distant reading approaches [START_REF] Moretti | Graphs, Maps, Trees. Abstract Models for Literary History[END_REF][START_REF] Moretti | Distant Reading[END_REF].

Period

System description

The system has three components: a preprocessing module to format input poems uniformly, an NLP pipeline, and the enjambment-detection module itself.

The NLP pipeline is IXA Pipes [START_REF] Agerri | IXA Pipeline: Efficient and ready to use multilingual NLP tools[END_REF]. Its results for contemporary Spanish are competitive. Our system uses it to obtain part-of-speech tags, syntactic constituency (e.g. verbphrase, noun-phrase) and syntactic dependencies (e.g. direct object).

The enjambment detection module is rule and dictionary-based, and exploits the information provided by the NLP pipeline. Rules (30 in total) of different characteristics identify enjambed lines, assigning them a type among a list of 12 types, based on the typology in Section 2.8

• Some rules are very shallow and only take parts of speech into account.

• Some rules additionally exploit constituency info.

• Some rules use dependency information, e.g. to detect "subject / object / verb" relations. • For any type of rule, custom dictionaries can restrict rule application to a set of terms. E.g. certain verbs govern arguments introduced by one specific preposition; we itemized these verbs and their prepositions in a dictionary, to complement information provided by the NLP pipeline or correct parsing errors.

Enjambment annotations are output in standoff format. The project's site provides details.9

System evaluation and discussion

Test-corpus

To evaluate the system, we created two reference-sets (SonnetEvol and Cantos20th), manually annotating enjambment in them.

1. SonnetEvol: 100 sonnets (1400 lines) from our diachronic sonnet corpus of ca. 3750 sonnets (Table 1). This test-set contains 260 pairs of enjambed lines.10 2. Cantos20th: 1000 lines of 20 th century poetry [START_REF] Colinas | Noche más allá de la noche[END_REF], showing natural contemporary syntax. We identified 277 pairs of enjambed lines.

The distribution of enjambment types in the test-corpora is balanced (Table 2). The SonnetEvol diachronic test-corpus is balanced across periods (Table 3).11

We annotated the Cantos20th corpus in order to assess the system's performance on contemporary Spanish with natural diction, compared to its behaviour with the SonnetEvol corpus, which includes some archaic constructions and often shows an elevated register.

For the evaluation reported here, each sonnet was annotated by a single annotator. Obtaining multiple annotators' input on the same sonnet to assess inter-annotator agreement [START_REF] Artstein | Inter-coder agreement for computational linguistics[END_REF]) is part of our ongoing work.

Enjambment-detection tasks evaluated

We defined two enjambment-detection tasks:

1. Span-match: the positions of enjambed lines proposed by the system must match the positions in the reference corpus for a correct result to be counted. 2. Typed span-match: for a correct result, both the positions and the enjambment type assigned by the system to those positions must match the reference.

System results and discussion

Precision, recall and F1 were obtained. 13 Table 4 provides overall results for both corpora. On the SonnetEvol corpus. Number of items per type, Precision (P), Recall (R) and F1 on the typed span-match enjambment detection task. * The types are described on our site: http://sites.google.com/site/spanishenjambment/enjambment-types 13 The definitions for Precision (P), Recall (R) and F1 were the usual: 14 E.g. per-type results for the Cantos20th corpus, or breakdowns for SonnetEvol per period (overall and per type), see https://sites.google.com/site/spanishenjambment/evaluation For untyped detection (span-match), F1 reaches 80% in the SonnetEvol corpus, whereas F1 for typed detection is 66.31%. For the contemporary Spanish corpus (Cantos20th), F1 is higher: 80.63% typed detection, 86.51% span-match. This reflects additional difficulties posed by archaic language and historical varieties for the NLP system whose outputs our enjambment detection relies on. 15 A common source of error was hyperbaton: the displacement of phrases triggers constituency and dependency parsing errors. Prepositional phrase (PP) attachment also posed challenges: Verbal adjuncts get mistaken for PPs complementing nouns or adjectives. 16 Creating a reparsing module to manage hyperbaton and improve PP attachment results may be fruitful future work.

Scholarly results and discussion

The system's goal is detecting enjambment to help literary research on the phenomenon, via providing systematic evidence for its analysis.

We consider our untyped enjambed-line detection results helpful, given an F1 of ca. 80% on the diachronic test-set. As an example application, we examined the distribution of enjambment according to position in the poem, particularly in positions across a verse-boundary (lines 4-5, 8-9 and 11-12). Comparing the results for the 15 th -to-17 th centuries vs. the 19 th century (Table 6 and Figure 1), we see that enjambment across the tercets increases clearly in the 19 th century, with a small increase of enjambment across the quatrains (lines 4-5) and across the octavesestet divide (lines 8-9). 17

The value of the tool is helping perform such analyses on a large corpus. This opens the door for scholars to assess the literary relevance of the findings, and search for the best interpretation. ) show that phrase-bounded enjambment detection is 10 points of F1 lower in SonnetEvol than in Cantos20th. Also, phrase-bounded enjambment results for the 15 th -17

Enjambed line positions

Scholarly

th period (with more archaic language) are 10 points of F1 lower than in the 19 th century. 16 This is a common problem in syntactic parsing, even for contemporary languages (see Agirre et al, 2008, for English). For historical varieties, [START_REF] Stein | Old French dependency parsing: Results of two parsers analyzed from a linguistic point of view[END_REF] results for verbal adjuncts and prepositional complements in Old French also suggest the difficulties posed by prepositional phrases. 17 Given the manageable data volume, we validated the counts for enjambment across a verse boundary (Table 6) manually (but not the more voluminous data for all other positions).

Outlook

The characterization of enjambment in Spanish literary theory has unclear points. Systematically obtaining enjambment examples is helping us find additional evidence to analyze these unclear points. Moreover, we are not aware of a systematic large-sample study of enjambment across periods, literary movements, or versification types in Spanish, or other languages. Automatic detection can help answer interesting questions in verse theory, which would benefit from a quantitative approach, complementing small-sample analyses. e.g.: To what an extent is enjambment used differently in free verse vs. traditional versification? Students in our metrics classes are currently annotating enjambment for 450 sonnets. These annotations will permit inter-annotator agreement computation. We will also examine the possibility of using supervised machine learning to train a sequence labeling and classification model to complement our current detection rules.

FIGURE

  FIGURE 1: PERCENTAGE OF ENJAMBMENTS PER POSITION IN THE 15 TH -17 TH CENTURIES VS. THE 19 TH . The y-axis represents line-positions; the x-axis is the percentage of enjambed line-pairs for a position over all enjambed line-pairs in the period. Enjambment across quatrains and across the octave-sestet divide is very rare, with a small increase in the 19 th century. The division between the tercets blurs in the 19 th century, in the sense that enjambment across them is clearly higher than in the previous period

TABLE 1 : DISTRIBUTION OF SONNETS AND AUTHORS PER PERIOD. * Exact dates of birth and death are available for a minority of authors; often only the century was provided in the corpus sources. Periods ending in ".5" cover authors who lived in two centuries. E.g. period "15.5" covers authors born in the 15th and deceased in the 16th century

 1

TABLE 2 : DISTRIBUTION OF ENJAMBMENT TYPES IN THE MANUALLY ANNOTATED REFERENCE CORPORA, providing counts and each type's percentage of the total enjambments per corpus. Counts refer to pairs of enjambed lines

 2

			Test-Corpus
		SonnetEvol	Cantos20th
	Enjambment Types *	Count	%	Count	%	SonnetEvol Test-corpus
	Total Phrase-Bounded adj_adv	104 2	40.00 0.77	175 1	63.18 0.36	Period ** Total 15 th -17	th	Sonnet Count 72
	adj_noun	29	11.15	54	19.49	14.5	3
	adj_prep	14	5.38	11	3.97	15	2
	adv_prep	0	0	3	1.08	15.5	2
	noun_prep	39	15.00	85	30.69	16	14
	relword	1	0.38	2	0.72	16.5	21
	verb_adv	5	1.92	7	2.53	17	27
	verb_cprep verb_chain Total Cross-Clause 12	9 5 23	3.46 1.92 8.85	2 10 31	0.72 3.61 11.19	17.5 Total 19 th 18.5	3 28 3
	Total Expansions	133	51.15	71	25.63	19	17
	dobj_verb	65	25.00	39	14.08	19.5	8
	subj_verb	68	26.15	32	11.55	Total	100
	Total All Types	260	100	277	100	All Periods

TABLE 3 : DISTRIBUTION OF SONNETS BY PERIOD IN THE MANUALLY ANNOTATED SONNETEVOL CORPUS. The 16th, 17th and 19th centuries cover ca. 30% of the corpus each, and the 15th century covers ca. 10% of the sonnets * The project site describes each enjambment type: http://sites.google.com/site/spanishenjambment/enjambment-types ** Exact dates of birth and death are available for a minority of authors; often only the century was provided in the corpus sources. Periods ending in ".5" cover sonnets for authors who lived in two centuries. E.g. period "15.5" covers sonnets for authors born in the 15th and deceased in the 16th century

 3

  Table5provides the per-type results on the diachronic test-corpus (SonnetEvol). The project's site contains more detailed results.14

	Corpus	Task	N	P	R	F1
	SonnetEvol	span-match 260 typed span-match	74.18 61.24	87.64 72.31	80.35 66.31
	Cantos20th	span-match 277 typed span-match	84.01 78.04	89.17 83.39	86.51 80.63

TABLE 4 : OVERALL ENJAMBMENT DETECTION RESULTS. Number of test-items, Precision (P), Recall (R) and F1 in our two test-corpora, for the span-match and typed span-match enjambment detection tasks

 4

	Enjambment or Expansion Type *	N	P	R	F1
	Phrase-Bounded (all types)	104 66.19	88.46	75.72
	adj_adv	2	100	50.00	66.67
	adj_noun	29 54.55	82.76	65.75
	adj_prep	14 58.82	71.43	64.52
	noun_prep	39 55.36	79.49	65.26
	relword	1	100	100	100
	verb_adv	5 50.00	100	66.67
	verb_cprep	9 83.33	55.56	66.67
	verb_chain	5	100	80.00	88.89
	Cross-Clause 12	23 76.00	82.61	79.17
	Expansions (all types)	133 61.54	66.17	63.77
	dobj_verb	65 60.00	69.23	64.29
	subj_verb	68 63.24	63.24	63.24

TABLE 5 : ENJAMBMENT DETECTION RESULTS PER TYPE.

 5

relevance 15 th -17 th cent. 19 th cent.

			Count	%	Count	%
	4-5	across quatrains	2	0.07	19	0.26
	8-9	across octave-sestet divide	2	0.07	12	0.16
	11-12	across tercets	20	0.72	147	2.01

TABLE 6 : PAIRS OF ENJAMBED LINES ACROSS VERSE BOUNDARIES IN THE 15 TH -17 TH VS. THE 19 TH CENTURIES: Counts of enjambed line-pairs and percentages over the total number of enjambed line-pairs for each period. An example of the types of analyses stimulated by automatic enjambment detection

 6

	15 Expansions get lower F1 than phrase-bounded types overall. But we do not think that the F1 difference
	between SonnetEvol and Cantos20th is due to the higher proportion of expansions in SonnetEvol (Table 2): Results per-type (see project's site 14

Among others,[START_REF] Quilis | La estructura del encabalgamiento en la métrica española: Contribución a su estudio experimental[END_REF][START_REF] Domínguez Caparrós | Métrica española[END_REF],[START_REF] Paraíso | La métrica española en su contexto románico[END_REF],[START_REF] Spang | Ritmo y versificación. Teoría y práctica del análisis métrico[END_REF] for a description of enjambment, and Alarcos (1966),Senabre (1982[START_REF] Luján Atienza | Desde las márgenes de un río: la poesía coral de Diego Jesús Jiménez[END_REF][START_REF] Martínez Fernández | La voz entrecortada de los versos[END_REF] for case-studies on a single author.

http://sites.google.com/site/spanishenjambment contains samples for the corpus and results, and other details.

From Biblioteca Virtual Cervantes, http://www.cervantesvirtual.com/

About 30% of the 15th to 17th century authors had exact dates of birth and death, for the rest only the centuries were available. Among the 19th century authors, ca. 45% had exact dates of birth and death.

The full list of types identified by the system is at the project's site: https://sites.google.com/site/spanishenjambment/enjambment-types

https://sites.google.com/site/spanishenjambment/annotation-and-result-format

In other words, if there is an enjambment between lines 1 and 2, we consider that as "pair of enjambed lines" in the reference corpus.

Balancing across periods does not apply to the Cantos20th test-corpus: it covers the 20 th century only.

We did not define subtypes for cross-clause enjambment

Acknowledgements

Pablo Ruiz Fabo was supported by a PhD scholarship from Région Île-de-France. We also thank Clara Martínez Cantón's and Borja Navarro's metrics students for their ongoing sonnet annotation work, and Borja Navarro for introducing his students to the annotations required.


PhD Don C Des Jarlais

Huong Duong

Oanh Khuat

MD Thi Hai

Khuê Pham

MD Giang Hoang Thi

Thanh Nham

BA Thi Tuyet

Kamyar Arasteh

Jonathan Feelemyer

PhD Theodore Hammett

Marianne Peries

Laurent Michel

Vinh Vu

PhD Marie Jauffret-Roustide

PhD Jean-Pierre Moles

MD Didier Laureillard

MD Nicolas Nagot

Prospects for ending the HIV epidemic among persons who inject drugs in Haiphong, Vietnam

Keywords: HIV, Persons who inject drugs (PWID), Vietnam, Respondent driven sampling (RDS), Sex workers, Men

Background-To examine the prospects for "ending the HIV epidemic" among persons who inject drugs (PWID) in Haiphong, Vietnam. Reaching an incidence of < 0.5/100 person-years at risk (PY) was used as an operational definition for "ending the epidemic." Methods-A respondent driven sampling study of 603 PWID was conducted from September to October 2014. Current heroin use (verified with urine testing and marks of injection) was an eligibility requirement. A structured questionnaire was administered by trained interviewers to obtain demographic, drug use, and risk behavior data; HIV counseling and testing and HCV testing was also conducted. Two methods (by assuming all new injectors were HIV negative at first injection and by slope of prevalence by years injecting) were used for estimating HIV among persons injecting for < 5 years ("new injectors"). Comparisons were made to the HIV epidemic among PWID in New York City and modeling of the HIV epidemic in Can Tho province.

Introduction

"Combined prevention and care" for HIV, including needle/syringe programs (NSP), medication-assisted treatment (MAT) for substance use disorders and antiretroviral treatment (ART) for HIV infection, has dramatically reduced HIV transmission among persons who inject drugs (PWID) in a number of high-income settings. Examples include Amsterdam (Van den Berg, Smit, [START_REF] Van Den Berg | Full participation in harm reduction programmes is associated with decreased risk for human immunodeficiency virus and hepatitis C virus: evidence from the Amsterdam Cohort Studies among drug users[END_REF], France [START_REF] Desenclos | Surveillance of infectious diseases: principles and organisation in France in 2005[END_REF], Vancouver [START_REF] Montaner | diagnoses: an update on the 'treatment as prevention' experience in British Columbia[END_REF] and New York City [START_REF] Jarlais | Reducing HIV infection among new injecting drug users in the China-Vietnam Cross Border Project[END_REF]. HIV infection has not been eliminated in these areas, but HIV prevalence and incidence have been reduced to sufficiently low levels that one may conclude that the "epidemics" of HIV among PWID in these locations have ended. For the purposes of this paper, we consider an HIV incidence of less than 0.5/100 person-years (PY) at risk to be the "end of an HIV epidemic" among PWID. This value is derived from a recent review of HIV incidence in high-income settings with combined prevention for PWID. HIV incidence in these settings ranges from 0.01 to 0.3/100 PY (Des Jarlais 2016). There were no sites in which HIV transmission was completely eliminated, but it appears that combined prevention can reduce HIV transmission among PWID to a very low endemic level (< 0.5/100 PY), thus "ending an HIV epidemic." Whether these prevention successes can be replicated with PWID (and with other high-risk populations, such as MSM and commercial sex workers) in resource-limited settings is a critical question for HIV prevention. There is considerable evidence that NSP (Des [START_REF] Jarlais | High coverage needle/syringe programs for people who inject drugs in low and middle income countries: a systematic review[END_REF] and MAT for opiate dependence [START_REF] Feelemyer | Retention of participants in medication-assisted programs in low-and middle-income countries: an international systematic review[END_REF] can be effective in resource-limited settings. Such settings, however, typically have multiple barriers to implementation/scaling up of combined prevention and care for PWID. In addition to the lack of financial resources, there is often a lack of appropriately trained health workers, and severe stigmatization of both injecting drug use and HIV infection. The severe stigmatization may keep policy leaders from providing the needed interventions, keep healthcare workers from appropriately implementing interventions, and keep PWID from utilizing the interventions that are available.

In this report, we examine the prospects for ending the HIV epidemic among PWID in Haiphong, Vietnam through 1) a review of public policy for prevention of HIV among PWID in Vietnam, and 2) data from a recent respondent driven sampling (RDS) study of PWID in Haiphong, with a focus on "new injectors" (persons injecting for < 5 years) to estimate HIV incidence.

Methods

Policy Analysis

The policy analysis was based on a thorough review of relevant legal and policy documents, which we identified through interviews with key informants in government agencies and civil society organizations and searches of a comprehensive database of Vietnamese legal documents (THƯ VIỆN PHÁP [START_REF] Viện | Legal Documents: THƯ VIỆN PHÁP LUẬT Retrieved[END_REF].

RDS study

Subject Recruitment-Respondent driven sampling (RDS) [START_REF] Heckathorn | Respondent-Driven Sampling: A New Approach to the Study of Hidden Populations[END_REF] was used as the primary method of recruiting participants. In general, we followed standard RDS procedures for recruiting and training "seeds," aiming for a diverse group of with different ages, genders, and HIV statuses. The seeds were recruited by the peer support group members, participated in all study procedures, and were then instructed on how to recruit new study participants using the RDS coupons. The study name, logo and research site were printed on the RDS coupons. We were particularly interested in recruiting MSM-ID and FCSW-ID, as they might be at very high risk of acquiring HIV and at high risk of transmitting HIV to persons who do not inject drugs. We set targets of at least 30 MSM-PWID and 30 FCSW-PWID. These targets represent oversampling for MSM-PWID and FSCW-PWID at more than twice the estimated percentages of these groups in the overall PWID population in Haiphong. We included 3 seeds each from community support group (Virgin Flowers for FCSWs, White Sands for MSM, and Friendship Arms for PWID) among the initial 9 seeds. We encouraged the seeds from Virgin Flowers and White Sands (and their subsequent recruits) to recruit other MSM-PWID and other FCSW-PWID. The RDS procedures did not produce the desired oversampling of MSM-PWID and CSW-PWID, so that additional MSM-PWID and FCSW-PWID had to be recruited directly by the specific groups in the field.

Eligibility criteria-Persons were eligible to participate in the study if they were 18 or older, capable of giving informed consent, and had injected drugs in the past month. Staff visually inspected participants' arms for indications of injecting and an on-site urine test was used to confirm recent drug use.

Questionnaire-After eligibility had been confirmed and written informed consent obtained, a structured interview lasting approximately 40 minutes was administered by a trained interviewer. The questionnaire was based on instruments previously used by the research team, with adaptation and pre-testing for use in Haiphong. Drug use and sexual behavior questions covered the three months prior to the interview.

HIV and HCV counseling and testing-After completing the interview, each participant received counseling and testing for HIV and HCV. HIV antibody and HCV antibody testing were conducted at the laboratory of the Haiphong Provincial HIV/AIDS Center using Alere Determine 12.0 (Abbott) and HCV ELISA 3.0 (SD Bioline). HIV confirmation tests were done according to the National guidelines and used HIV1/2 3.0 rapid test (SD Bioline) plus the MUREX HIV Ag/Ab Combination test (Diasorin).

Estimating HIV incidence among "new injectors"-The questionnaire included items for age of first injection and date of birth. These items and the date of the interview permitted us to calculate the length of time since first injection for all participants. We used two methods for estimating HIV incidence among participants who had begun injecting less than 5 years before the time of their interview ("new injectors"). This cutoff point was used so that we would have sufficient numbers of new injectors to examine possible differences by gender/sexual behavior. First we assumed that all new injectors were HIV negative when they began injecting, and that HIV seropositive new injectors became infected halfway between the date of first injection and the date of the interview. The total number of HIV seropositive new injectors would then represent the number of incident HIV infections among these new injectors. The total time at risk would be the total years injecting for the HIV seronegative new injectors plus one half of the total years injecting for the HIV seropositives. The assumption that all participants would be HIV seronegative at the time of first injection might be questionable for participants at high risk of acquiring HIV through sexual transmission-MSM and FCSW. We thus restricted this estimate of HIV incidence to male participants who did not report any MSM behavior.

The second method for estimating HIV incidence among new injectors was to calculate the slope of the HIV prevalence by years injecting among the new injectors using ordinary least squares (OLS) regression. With an assumption of no differential loss of HIV seropositives and HIV seronegatives to the injection population during the short time period of being a new injector, the increase in prevalence by years injecting would correspond to the incidence (again, this estimation was restricted to male participants who did not report MSM behavior).

Honoraria-Each participant received VND 150,000 (Vietnamese dong, approximately US $ 7.50) plus VND 50,000 (approximately USD$2.50) for each peer they successfully recruited. These honoraria included compensation for time and effort to participate in the study and for expenses to travel to the study site. Participants received an additional VND 50,000 (approximately USD$2.50) for transportation to return to the study site to obtain their HIV and HCV test results.

Data Analyses-The RDS Analysis Tool (RDSAT) [START_REF] Volz | Respondent-Driven Sampling Analysis Tool (RDSAT)[END_REF] and STATA (STATA Corp, 2012) were used for data analysis.

Ethics Committee/Institutional Review Board Approvals-The study was reviewed and approved by the Institutional Review Boards of the Haiphong University of Medicine and Pharmacy and Mount Sinai Beth Israel.

Results

Policy context

The prospects for ending the HIV epidemic among PWID in Haiphong need to be considered within the evolving drug use and HIV policy frameworks in Vietnam. There are approximately 200,000 (PWID) in Vietnam and this is the focal population for the HIV epidemic in the country (UNODC, 2013).

The history of this legal and policy framework comprises two strands: one related to harm reduction programs for HIV prevention among PWID and the other related to substance use treatment. From 2002 to 2008, these two legal and policy strands appeared to develop in contradictory directions (T. M. [START_REF] Hammett | Social evils' and harm reduction: the evolving policy environment for human immunodeficiency virus prevention among injection drug users in China and Vietnam[END_REF]. The former became more progressive as reflected in the country's HIV/AIDS law of 2006 and its implementing decree 108 in 2007 and increasingly widespread implementation of needle/syringe and peer outreach programs with support from the government and police, many of which showed positive results (Theodore M. [START_REF] Hammett | Controlling HIV Epidemics among Injection Drug Users: Eight Years of Cross-Border HIV Prevention Interventions in Vietnam and China[END_REF]. At the same time, substance use "treatment" was becoming more repressive as the view of drug use as a "social evil" and moral failing solidified and the related system of compulsory drug rehabilitation/detention centers for "06 centers" derived from the re-education centers of the 1970s expanded (T. M. [START_REF] Hammett | Social evils' and harm reduction: the evolving policy environment for human immunodeficiency virus prevention among injection drug users in China and Vietnam[END_REF]. At its peak in about 2008, there were more than 120 "06" centers with more than 100,000 residents. There were 169,000 admissions to 06 centers from 2006-2010 [START_REF] Amon | Compulsory drug detention centers in China, Cambodia, Vietnam, and Laos: health and human rights abuses[END_REF]. The regimen employed in the "06" centers -brief and 'crude' detoxification, "moral education," and labor -has no basis in evidence, proved highly ineffective with relapse rates in excess of 90% after release from the centers, and violates human rights [START_REF] Amon | Compulsory drug detention in East and Southeast Asia: evolving government, UN and donor responses[END_REF]Watch, 2011).

In 2008, the repressive strand of substance use treatment policy began to change [START_REF] Vuong | Drug policy in Vietnam: A decade of change?[END_REF]. Pilot methadone maintenance programs were begun in Ho Chi Minh City and Haiphong (MOH Decisions 5073 and 5076 [2007]). These pilot programs revealed positive results in terms of relapse rates, health care utilization, quality of life, family re-integration, life stability, and cost-effectiveness [START_REF] Nguyen | Methadone maintenance therapy in Vietnam: an overview and scaling-up plan[END_REF][START_REF] Tran | Impact of methadone maitenance on health utility, health care utilization and expenditure in drug users with HIV/AIDS[END_REF]. Consequently, the government committed to a large and fairly rapid expansion of the methadone program to 30 provinces and 245 clinics, with an optimistic target of 80,000 clients by 2015. Thus far, the methadone program has been largely dependent on donor support, primarily from PEPFAR, but recently instituted patient co-payments of 10,000 Vietnamese Dong (VND) (approximately $0.50) per day to partially defray the cost. In addition, enrollment requirements for documentation of residence and police certification of status as registered drug users pose problems for many PWID who would like to access methadone treatment (Decree 96 [2012]).

A number of other laws and policies on substance treatment have been adopted in recent years [START_REF] Kamarulzaman | Compulsory drug detention centers in East and Southeast Asia[END_REF]. Drug use was decriminalized (Law Amending Penal Code, 2009). The 2012 Law on Handling Administrative Violations ended compulsory detention of sex workers in "05 centers" and required court decisions to commit PWID to 06 centers. In 2013, the Prime Minister officially approved (Decision 2596/QD-TTg, December 27, 2013) a Renovation Plan developed by the Ministry of Labor, Invalids, and Social Affairs (MOLISA) with input from other ministries, civil society organizations, and other stakeholders that explicitly recognized drug addiction as a chronic, relapsing condition with both biological and psychosocial causation and committed to a transition away from the system of 06 centers to a system based on voluntary, community-based, and evidence-based substance use treatment. The numbers of 06 centers and residents has already been reduced. By 2015, there were 15,500 total residents in 95 centers.

Still, the legal and policy frameworks on drug control and drug treatment in Vietnam remain inconsistent (Vietnam, 2013). Much of the punitive policy based on "social evils" and compulsory detention, such as the Law on Drug Prevention andControl (2000, revised in 2008) remains on the books despite the recent adoption of more progressive approaches. Also, an upsurge in street crime in Ho Chi Minh City thought to be perpetrated by PWID led the government in October 2014 to re-institute summary commitment to centers in the absence of a court decision. More than 400 individuals without residence documentation have been sent to "social centers" pending a court determination as to whether they should be further committed to 06 centers. (Resolution 77/2014/QH13; Resolution 98/NQ-CP, December 26, 2014). Clearly, progress toward a new evidence-based system of substance use treatment in Vietnam will continue to be halting and subject to periodic setbacks.

The local situation in Haiphong

In our estimation, Haiphong has generally been in the lead in the movement towards evidence-based substance use treatment and HIV prevention in Vietnam, partly due to the involvement of community based organizations that played a major role as a critical stakeholder of interventions among PWID. Despite this policy evolution in favor of harm reduction, "06" Centers for drug users and "05" Centers for commercial sex workers still exist in Haiphong. As noted above, a methadone maintenance pilot program was begun in Haiphong in 2008, and there are now 11 methadone maintenance treatment clinics with approximately 3,200 patients in the city, for an estimated 6,000 to 10,000 PWID (Around 7500 are registered with authorities). Needle/syringe programs, including syringe exchange and pharmacy sales of injection equipment to drug users were implemented in 2005. Currently, the great majority of PWID obtain injection equipment through purchasing at pharmacies (over 97%)(PSI Vietnam/Metrics, 2013). ART is offered free of charge to all HIV seropositive persons with CD4 cell counts <350 cells/mm 3 at 13 in clinics in Haiphong and currently serves approximately 4800 HIV positive persons (USAID, 2012).

New ART guidelines and goals

Vietnam has recently issued new guidelines for ART. All HIV seropositive members of "key populations" (PWID, MSM and commercial sex workers) should be offered ART regardless of their CD4 cell counts (Ministry of Health, 2015) and Vietnam has adopted the 90-90-90 goals for the provision of ART.

RDS study

Characteristics of the total sample-603 participants were recruited between September and October 2014; 581 through respondent driven sampling and 22 additional MSM and FCSW participants through direct recruiting by staff. Table 1 presents the demographic characteristics, drug use and sexual behaviors, and HIV and HCV prevalence for the entire sample. The sample was predominantly non-MSM male, had a mean age of 37 years, all were currently using heroin (an eligibility criterion), and pharmacies were the predominant source of new injection equipment. With the exception of unsafe sex (defined as less than consistent condom use) with a primary partner, frequencies of self-reported injecting risk behaviors and sexual risk behaviors were quite low.

Table 2 shows the increasing HIV and HCV prevalence by years injecting for the total sample. The mean number of years injecting was 9, with a median of 8 years. The HIV prevalence was relatively low among persons who had begun injecting over the last five years while the HCV prevalence was significantly higher. 3 shows demographic characteristics, risk behavior, HIV and HCV prevalence for the 178 participants who reported that they had first injected less than 5 years prior to the study interview. The HIV prevalence among new injectors was 9/178 = 5.1%, well below the 25% for the total sample while HCV prevalence was 43%, which was also lower than the 67% for the total sample. Note that the new injectors comprised a very substantial proportion (30%) of the total sample. As with the sample as a whole, the frequencies of self-reported injecting risk behaviors and sexual risk behaviors were low. There were substantial differences in HIV prevalence among non-MSM male new injectors (2/136, 1.5%), MSM new injectors (3/21, 14%), and female new injectors (4/21, 19%). The difference in HIV prevalence between non-MSM male new injectors and the combined MSM and female new injectors (6/39, 15.4%) was highly significant (p = 0.004 by Fisher's exact test).

HIV and HCV prevalence among new injectors-Table

We estimated injecting-related HIV incidence among non-MSM new injectors using the two methods described above. As MSM-PWID and female new injectors may have acquired HIV through sexual transmission, we excluded them from these estimates. Dividing total number of HIV seropositive new injectors by total years of injecting among HIV seronegatives plus one half of years injecting among HIV seropositives, the estimated HIV incidence was 1.2/100 person-years at risk (95% CI -0.24, 3.4). By the slope of prevalence by years injecting, the estimated HIV incidence was 0.8/100 person-years at risk (95% CI -0.9, 2.5). These two estimates of HIV incidence among non-MSM male new injectors are quite consistent.

Given the very modest number of MSM and female new injectors (18 MSM-PWID new injectors and 21 female PWID new injectors), we were not able to estimate HIV incidence for these groups. However, since HIV prevalence was considerably higher in these groups (15.4% for combined MSM and FCSW new injectors) than for non-MSM male new injectors (2.2%), it is likely that HIV incidence is much higher among MSM and FCSW new injectors than among non-MSM male new injectors and it is also likely that some MSM and FCSW were HIV seropositive before they began injecting.

Discussion

Haiphong experienced a high HIV seroprevalence epidemic among PWID in the early to mid-2000s. Integrated Bio-Behavioral Surveys (IBBS) of PWID in Haiphong, documented an HIV prevalence of 66% in 2006 and 48% in 2009. In the present study, we observed an HIV prevalence of 25% (95% CI 22%, 30%) among PWID in Haiphong. This suggests a "declining epidemic," with greater loss of HIV seropositives-due to death, disability or cessation of injecting-than new HIV infections in the population of persons actively injecting drugs. Entry of HIV seronegative persons into the injecting population would also contribute to a decline in HIV seroprevalence.

A more detailed comparison of the present study with the 2009 IBBS in Haiphong may be useful. First, several methodological differences should be noted. The IBBS used timelocation sampling rather than the RDS employed in the present study. The IBBS included only male PWID, while the present study included female PWID. The IBBS had sample sizes of 300 each, while the RDS study had a sample of 603. The 2009 study did have a category of "new injectors," but used a different definition of "new injector" (injecting for < 2 years), and found only 17 "new injectors," and but did not present HIV prevalence for this group. Thus, it is not possible to compare HIV prevalence and estimated incidence among "new injectors" in the present study to prevalence and estimated incidence among "new injectors" in the 2009 IBBS. Despite these differences there are several interesting comparisons of the 2009 IBBS study to the present study. First, there is a clear reduction in HIV prevalence from 48% in the 2009 IBBS to 25% in this RDS study. This would appear to be a continuation of the decline in HIV prevalence from the 66% observed in the 2006 IBBS.

There is also a pattern of low rates of reported injecting risk behavior. Among the 300 PWID recruited from Haiphong province in the 2009 IBBS, 21% reported sharing syringes in the previous 6 months, which was the lowest by far among PWID in the 12 provinces included in the survey. The reported injecting risk behavior among subjects in the RDS study was quite low, with 5% reporting receptive sharing and 3% reporting distributive sharing. Social desirability bias may figure in these reports of injecting risk behavior, but the low rates of injecting risk behavior are consistent with declining HIV prevalence overall and the low HIV prevalence (2.2%) and low estimated incidence (approximately 1/100 PY) among non-MSM male new injectors in the RDS study.

The mean years injecting in the 2009 IBBS was 7.4 years compared to 9 years in the RDS study. The fact that HIV prevalence declined substantially while the average time at risk (years injecting) increased modestly suggests reduction of HIV seropositives in the active injecting population. Death or disability due to HIV related illness and entry of HIV seropositives into methadone treatment (and cessation of injecting) would be plausible mechanisms for the reduction of HIV seropositives in the active injecting population.

Despite the methodological differences, the comparison of the 2009 IBBS and the 2014 RDS study does indicate a "declining HIV epidemic" in the active drug injecting population in Haiphong.

HCV prevalence was 67% for the PWID in our sample, and 43% among new injectors. These levels are much higher than the HIV prevalence among PWID in Haiphong (25% and 5% for all PWID and new injectors respectively). As Vietnam has expanded access to and use of antiretroviral medication to reduce infectiousness of HIV positive persons, thereby reducing the rates of transmission of the virus, blood-borne HCV transmission continues to be problematic. New direct acting antiviral agents (DAA) can help to reduce further HCV transmission, but this treatment has not expanded in Vietnam in conjunction with expanded ART treatment. While the IBBS 2009 did not report HCV prevalence among PWID or new injectors, a study published by Nadol et al reported HCV prevalence from stored samples of male PWID tested of HIV as part of the 2009 IBBS, with HCV prevalence of 75.8%, slightly higher than the 67% reported for our PWID sample [START_REF] Nadol | Findings from integrated behavioral and biologic survey among males who inject drugs (MWID) -Vietnam, 2009-2010: evidence of the need for an integrated response to HIV, hepatitis B virus, and hepatitis C virus[END_REF]. This data indicate that HCV prevalence has reduced slightly from the previous IBBS participants but still remains much higher than HIV prevalence. [START_REF] Ahmed | Drug injecting and HIV risk among injecting drug users in Hai Phong, Vietnam: a qualitative analysis[END_REF][START_REF] Jarlais | HIV incidence among injection drug users in New York City, 1992-1997: evidence for a declining epidemic[END_REF]National Institute of Hygiene and Epidemiology, 2011). Both cities experienced very high prevalence epidemics, with prevalence exceeding 50% in the peak years. Both cities then implemented HIV prevention measures and saw declines in their epidemics. The prevalence and incidence data in Table 4 suggests that Haiphong is behind New York City in "ending the epidemic" among PWID. This is consistent with the current state of interventions for PWID in the two cities. Both cities have needle/syringe programs (exchange and pharmacy sales) that provide good access to sterile injection equipment and both cities have large methadone maintenance treatment programs (with methadone treatment positions for about 25% of heroin injectors). The major difference in current prevention and care services is that New York City has conducted more HIV counseling and testing (98% of PWID have been tested versus 75% in Haiphong) and since 2011 New York City has had a policy to provide ART for all HIV seropositives (New York City Department of Health and Mental Hygeine, 2011) while Vietnam adopted a policy of providing ART to members of "key populations" (including PWID) four years later in 2015 (Organization, 2015).

Kato and colleagues [START_REF] Kato | The potential impact of expanding antiretroviral therapy and combination prevention in Vietnam: towards elimination of HIV transmission[END_REF] modeled HIV prevention for Can Tho province, Vietnam. Like Haiphong, Can Tho has a concentrated HIV epidemic focused in PWID. Kato concluded that maintaining the 2011 level of interventions in Can Tho would reduce HIV incidence to < 0.001/100 PY for the adult population by 2050 but that scale up of combined interventions (HIV counseling and testing, immediate ART for HIV seropositives, methadone maintenance for persons with opiate use disorders, enhanced condom distribution) for key populations (PWID, MSM, commercial sex workers) would produce HIV incidence <0.001/100 PY by 2025 and would be cost saving by 2029.

Thus, both the historical comparison with New York City and the modeling of a similar concentrated HIV epidemic in Can Tho province suggest that maintaining current interventions in Haiphong would lead to an "end of the HIV epidemic" among PWID in several decades, but that further scale up of interventions would reduce the time to the end of the epidemic by a decade or more. Given the current high coverage for access to sterile injecting equipment and methadone treatment, scaling up HIV counseling and testing and providing ART to all HIV seropositive PWID would be the highest priority. Reaching the 90-90-90 goals (UNAIDS, 2014) for provision of ART for HIV seropositive PWID should create a Treatment as Prevention environment for PWID and greatly reduce HIV transmission within this population.

The high HIV prevalence among the FCSW and MSM new injectors in this study is also of concern. These FCSW and MSM new injectors may have higher prevalence due to greater sexual risk (higher HIV prevalence among their sexual partners), greater injecting risk (including more HIV seropositives in their injecting networks) and/or less access to HIV prevention services. Additional research to understand the high HIV prevalence among these groups is needed, followed by focused interventions to address the reasons for the higher observed HIV prevalence. In addition, there is significant potential for heterosexual transmission from HIV seropositive male PWID to their non-injecting female sexual partners. Individual and couples counseling on sexual risk reduction, communication and disclosure of HIV status should be expanded and improved (Theodore M Hammett, et al., 2015).

Several limitations of this study should be noted. First, our analysis of prospects for ending the epidemic among PWID in Haiphong is based primarily upon data from the recent RDS survey. Future surveys will be needed to monitor actual progress towards ending the epidemic. Second, the self-reported rates of injecting risk behaviors and sexual risk behaviors were quite low, and may reflect some degree of social desirability bias. Third, our classification of "new injectors" was based on a question for age at first injection. Recall error for this question would have caused misclassification of new injectors. The relatively high HIV prevalence among MSM new injectors and female new injectors suggests an increasing importance of sexual transmission among PWID in Haiphong. We did not have large enough numbers of MSM and FCSW new injectors to estimate HIV incidence for these groups. Finally, while there was HIV prevalence data available from previous IBBS surveys that we used to compare to the HIV prevalence from our sample of PWID, we were not able to evaluate the same trends in HCV prevalence for all previous IBBS studies as this data was not collected, and we were only able to make comparisons to 2009 IBBS data among male PWID participants.

Predicting the future of HIV transmission among PWID in Vietnam requires considerable caution. Policies towards injecting drug use are still in flux, and international donor support is being reduced. Stigmatization of drug use and of HIV and HCV infection persists. Nevertheless, it is apparent that the HIV epidemic among PWID in Haiphong is in a declining phase, and that further scale up of interventions, particularly HIV counseling and testing and provision of ART to all seropositive PWID, should greatly accelerate the decline and that an end to the epidemic among PWID in Haiphong is an attainable public health objective. This will require new strategies to increase the coverage of HIV testing among PWID and to ensure good linkage to care, and good adherence to ART. Provision of direct acting antivirals (DAA) for HCV infection can help reduce spread of infection and should also be implemented and scaled up in Haiphong, along with strategies to ensure high uptake and adherence to treatment regimens.

Table 4

 4 presents an historical comparison of the HIV epidemics among PWID in Haiphong and in New York City

Table 2

 2 Human Resource Watch. The Rehab Archipelago: Forced Labor and other Abuses in Drug Detention Centers in Southern Vietnam. Human Resource Watch. 2011 HIV and HCV status by years injecting among PWID, Haiphong, Vietnam, 2014

	Years injecting HIV-N (%) HIV+ N (%) HCV-N (%) HCV + N (%)
	All	451 (74.79)	152 (25.21)	200 (33.17)	403 (66.8)
	< 1 year	44 (97.78)	1 (2.22)	37 (82.22)	8 (17.78)
	1-2 years	62 (93.94)	4 (6.06)	37 (56.06)	29 (43.94)
	3-4 year	63 (94.03)	4 (5.97)	28 (41.79)	39 (58.21)
	5-6 years	59 (84.29)	11 (15.71)	23 (32.86)	47 (67.14)
	7-10 years	83 (69.17)	37 (30.83)	34 (28.33)	86 (71.67)
	11+ years	140 (59.57)	93 (40.49)	39 (16.81)	193 (83.19)

Table 4

 4 Historical Comparison of HIV infection among PWID in New York City, US, and Haiphong, Vietnam Estimated HIV incidence (new injectors) = 1/100 PY, incidence among MSM and female new injectors likely to be much higher. Estimated incidence among long-term injectors not available Int J Drug Policy. Author manuscript; available in PMC 2017 June 01.

	Year New York City HIV Prevalence Haiphong HIV Prevalence
	1992	60%
	1998	25%
	2006	13% *	66%
	2009	10% *	48%
	2014	6% **	25% ***
	* Estimated HIV incidence among all PWID = 1/100 PY
	** Estimated HIV incidence among all PWID = 0.1/100 PY
	***

Int J Drug Policy. Author manuscript; available in PMC 2017 June 01.

Acknowledgments

The research reported here was supported by grants from the ARNS (France) and NIDA P30 DA11041-185 (US). The funding agencies had no role in designing the research, data analyses and preparation of the report.

Highlights

• We assessed prospects for ending the HIV epidemic among persons who inject drugs in Haiphong Vietnam

• HIV prevalence declined from 68% in 2006 to 25% in 2014 among persons who inject drugs

• Self-reported injecting "risk behavior" was quite low


Alexandros Alexakis

Marc-Etienne Brachet

Energy fluxes in quasi-equilibrium flows

 to the forced and dissipated flows of the spectrally truncated Navier-Stokes (TNS) equations. In both of these idealized systems a finite number of Fourier modes is kept contained inside a sphere of radius k max but while the first conserves energy in the second energy is injected by a body-force f and dissipated by the viscosity ν. For the TNS system stochastically forced with energy injection rate I E we show, using an asymptotic expansion of the Fokker-Planck equation, that in the limit of small k max η (where η = (ν 3 /I E ) 1/4 the Kolmogorov lengthscale) the flow approaches the absolute equilibrium solution of Kraichnan with such an effective "temperature" so that there is a balance between the energy injection and the energy dissipation rate. We further investigate the TNS system using direct numerical simulations in periodic cubic boxes of size 2π/k 0 . The simulations verify the predictions of the model for small values of k max η. For intermediate values of k max η a transition from the quasi-equilibrium "thermal" state to Kolmogorov turbulence is observed. In particular we demonstrate that, at steady state, the TNS reproduce the Kolmogorov energy spectrum if k max η 1. As k max η becomes smaller then a bottleneck effect appears taking the form of the equipartition spectrum E(k) ∝ k 2 at small scales. As k max η is decreased even further so that k max η (k 0 /k max ) 11/4 the equipartition spectrum occupies all scales approaching the asymptotic equilibrium solutions found before. If the forcing is applied at small scales and the dissipation acts only at large scales then the equipartition spectrum appears at all scales for all values of ν. In both cases a finite forward or inverse flux is present even for the cases where the flow is close to the equilibrium state solutions. However, unlike the classical turbulence where an energy cascade develops with a mean energy flux that is large compared to its fluctuations, the quasi-equilibrium state has a mean flux of energy that is subdominant to the large flux fluctuations observed.

Introduction

Turbulence is a classical example of an out of equilibrium system. In steady state, energy is constantly injected at some scale in while it is dissipated at smaller scales ν by viscous forces. This process requires a finite flux of energy from the former scale in to the latter ν that is provided by the well known Kolmogorov-Richardson cascade. Despite the out of equilibrium nature of turbulence there are circumstances where equilibrium dynamics become relevant. This has been claimed to be the case for the scales larger than than the injection scale > in . At these scales the energy flux is zero and can possibly be modeled using equilibrium dynamics [START_REF] Dallas | Statistical equilibria of large scales in dissipative hydrodynamic turbulence[END_REF][START_REF] Cameron | Effect of helicity on the correlation time of large scales in turbulent flows[END_REF][START_REF] Alexakis | On the thermal equilibrium state of large scale flows[END_REF]. Furthermore, at the smallest scales of the inertial range a so-called 'bottleneck' manifests where the power-law slope of the energy spectrum becomes less steep [START_REF] Donzis | The bottleneck effect and the kolmogorov constant in isotropic turbulence[END_REF][START_REF] Falkovich | Bottleneck phenomenon in developed turbulence[END_REF][START_REF] Martinez | Energy spectrum in the dissipation range of fluid turbulence[END_REF][START_REF] Lohse | Bottleneck effects in turbulence: Scaling phenomena in r versus p space[END_REF]. This has been interpreted by [START_REF] Frisch | Hyperviscosity, galerkin truncation, and bottlenecks in turbulence[END_REF] as an 'incomplete thermalization' that becomes asymptotically at equilibrium for hyper-viscous flows when the order of the hyper-viscosity tends to infinity. Equilibrium dynamics become also relevant in the presence of inverse cascades in finite domains where large scale condensates form [START_REF] Kraichnan | Inertial ranges in two-dimensional turbulence[END_REF][START_REF] Robert | Statistical equilibrium states for two-dimensional flows[END_REF][START_REF] Naso | Statistical mechanics of two-dimensional euler flows and minimum enstrophy states[END_REF][START_REF] Bouchet | Statistical mechanics of two-dimensional and geophysical flows[END_REF][START_REF] Shukla | Statistical theory of reversals in two-dimensional confined turbulent flows[END_REF]. Finally understanding equilibrium dynamics is important for systems that display a transition of from forward to an inverse cascade [START_REF] Alexakis | Cascades and transitions in turbulent flows[END_REF][START_REF] Sahoo | Discontinuous transition from direct to inverse cascade in three-dimensional turbulence[END_REF][START_REF] Benavides | Critical transitions in thin layer turbulence[END_REF][START_REF] Sozza | Dimensional transition of energy cascades in stably stratified forced thin fluid layers[END_REF][START_REF] Deusebio | Dimensional transition in rotating turbulence[END_REF][START_REF] Seshasayanan | On the edge of an inverse cascade[END_REF] since the large scale flows in these systems transition from an equilibrium state to an out of equilibrium state. Besides the possible applications, understanding the equilibrium dynamics in turbulence is also a much needed step required before understanding its much harder out-of-equilibrium counterpart. This has led many researchers [START_REF] Lee | On some statistical properties of hydrodynamical and magneto-hydrodynamical fields[END_REF][START_REF] Hopf | Statistical hydromechanics and functional calculus[END_REF][START_REF] Kraichnan | Inertial ranges in two-dimensional turbulence[END_REF][START_REF] Kraichnan | Helical turbulence and absolute equilibrium[END_REF][START_REF] Orszag | Statistical Theory of Turbulence[END_REF] to investigate the equilibrium state of the truncated Euler equations (TEE) where only a finite number of Fourier modes is kept and are given by:

∂ t u + P K [u • ∇u + ∇p] = 0, ∇ • u = 0. (1.1)
Here u is the incompressible velocity field, p is the pressure and P K is a projection operator that sets to zero all Fourier modes except those that belong to a particular set 'K' (here chosen to be all wavenumbers inside a sphere centered at the origin with radius k max ). These equations conserve exactly two quadratic invariants

the Energy E = 1 2 |u| 2 dx 3 and the Helicity H = 1 2 u • ∇ × udx 3 .

(1.2)

In Fourier space these invariants are distributed among the different modes which are quantified by the energy and helicity spherically averaged spectra E(k), H(k) respectively defined as

E(k) = 1 2k 0 k |k|<k+k0 |ũ k | 2 and H(k) = 1 2k 0 k |k|<k+k0 ũ-k • (ik × ũk ). (1.3)
Here ũ is the Fourier transform of u and we have assumed a triple periodic cubic domain of size 2π/k 0 . The spectra have been divided by the smallest non-zero wavenumber k 0 so that they have units of energy and helicity density respectively. At late times this system reaches a statistically steady state whose properties are fully determined by these two invariants. Using Liouville's theorem and assuming ergodicity [START_REF] Lee | On some statistical properties of hydrodynamical and magneto-hydrodynamical fields[END_REF] predicted that at absolute equilibrium this system will be such that every state u of a given energy E is equally probable. This is equivalent to the micro-canonical ensemble in statistical physics and it leads to equipartition of energy among all the degrees of freedom (ie among all Fourier amplitudes) and an energy spectrum given by E(k) ∝ k 2 . [START_REF] Kraichnan | Helical turbulence and absolute equilibrium[END_REF] generalized these results including helicity and assuming a Gaussian equipartition ensemble

P(u) = Z -1 exp [-αE -βH]
(1.4)

where P(u) is the probability distribution for the system to be found in the state u, E is the energy and H the helicity given in 1.2 and Z a normalization constant. The parameters α and β are the equivalent of an inverse temperature and inverse chemical potential in analogy with statistical physics. We note that for the TEE system eq.1.4 is not exact! This is because eq. 1.4 allows for fluctuations of the energy E which are not allowed for the TEE system. However it becomes closer to the true distribution as the number N of fourier modes becomes larger. For no helicity β = 0 this leads to the energy spectrum E(k) ∝ k 2 predicted by Lee. In the presence of helicity one obtains

E(k) = 4παk 2 α 2 -β 2 k 2 , H(k) = 4πβk 4 α 2 -β 2 k 2 .
(1.5)

The coefficients α and β are determined by imposing the conditions

E = k 0 k E(k) and H = k 0 k H(k) (1.6)
and E and H are the initial energy and helicity respectively. The predictions above have been verified for the truncated Euler system in numerous numerical simulations [START_REF] Orszag | Numerical simulation of turbulence[END_REF][START_REF] Cichowlas | Effective dissipation and turbulence in spectrally truncated euler flows[END_REF][START_REF] Krstulovic | Cascades, thermalization, and eddy viscosity in helical Galerkin truncated Euler flows[END_REF][START_REF] Dallas | Statistical equilibria of large scales in dissipative hydrodynamic turbulence[END_REF][START_REF] Cameron | Effect of helicity on the correlation time of large scales in turbulent flows[END_REF][START_REF] Alexakis | On the thermal equilibrium state of large scale flows[END_REF].

In the TEE however there is no exchange of energy with external sources or sinks and it is thus harder to make contact with the more realistic systems mentioned in the beginning of the introduction such as the scales larger than the forcing scale in a turbulent flow. It was shown recently in [START_REF] Alexakis | On the thermal equilibrium state of large scale flows[END_REF] that, in some cases, although the large scales are close to an equilibrium state there is still exchange of energy with the smaller turbulent and forcing scales generating energy fluxes (from the forced scales to the large scales and from the large scales to the turbulent scales). It appears thus that, even in the presence of sources and sinks, equilibrium dynamics can still be relevant. In this work we examine further this possibility by looking at the truncated Navier-Stokes (TNS) equations where there is constant energy injection and dissipation like in the regular Navier-Stokes Equation but the system is limited to a finite number of Fourier modes as in TEE. We show analytically in the next section that for weak energy injection and weak viscosity so the k max η (k 0 /k max ) 11/4 (where η the kolmogorov lengthscale) the system indeed reaches a quasi-equilibrium state whose probability distribution P(u) can be calculated. We verify and extend these results using direct numerical simulations in section 3. Our conclusions are presented in the last section.

Asymptotic expansion

We consider the truncated Navier-Stokes (TNS) equations

∂ t u + P K [u • ∇u + ∇P ] = f + ν∆u. (2.1)
where u is the incompressible velocity field, P is the pressure, ν is the viscosity and f is a forcing function. The domain is a 2π periodic cube so that the smallest non-zero wavenumber is k 0 = 1. The projection operator P K sets to zero all Fourier modes with wavenumbers outside the sphere of radius k max . In total there are N 4π 3 (k max /k 0 ) 3 Fourier wavenumbers inside this sphere. In order to proceed it helps to write the truncated Navier Stokes equation in Fourier space using the Craya-Lesieur-Herring decomposition [START_REF] Craya | Contributiona l'analyse de la turbulence associéea des vitesses moyennes[END_REF]; [START_REF] Lesieur | Décomposition d'un champ de vitesse non divergent en ondes d'hélicité[END_REF]; [START_REF] Herring | Approach of axisymmetric turbulence to isotropy[END_REF]) where every Fourier mode is written as the sum of two modes one with positive helicity and one with negative helicity ũk = ũ+

k h + k + ũ- k h - k .
The two vectors h ± k are given by:

h s k = k × (ê × k) √ 2|k × (ê × k)| + i s ê × k √ 2|ê × k| (2.2)
where ê is an arbitrary unit vector. The sign index s = ±1 indicates the sign of the helicity of h s k . The basis vectors h s k are eigenfunctions of the curl operator in Fourier space such that ik × h

s k = s|k|h s k . They satisfy h s k • h s k = 0 and (h s k ) * • h s k = 1, where the complex conjugate of h s k is given by (h s k ) * = h -s k = h s -k .
They form a complete base for incompressible zero-mean vector fields. This decomposition has been extensively used and discussed in the literature [START_REF] Cambon | Spectral approach to non-isotropic turbulence subjected to rotation[END_REF][START_REF] Waleffe | The nature of triad interactions in homogeneous turbulence[END_REF][START_REF] Chen | The joint cascade of energy and helicity in three-dimensional turbulence[END_REF][START_REF] Biferale | Inverse energy cascade in three-dimensional isotropic turbulence[END_REF][START_REF] Moffatt | Note on the triad interactions of homogeneous turbulence[END_REF][START_REF] Alexakis | Helically decomposed turbulence[END_REF][START_REF] Sahoo | Discontinuous transition from direct to inverse cascade in three-dimensional turbulence[END_REF]. Note that since every ũk is described by two complex amplitudes ũ± k that satisfy (ũ ± k ) * = ũ± -k there are in total 2N independent degrees of freedom. The truncated Navier-Stokes using the helical decomposition can then be written as

∂ t ũs k = V s k -νk 2 ũs k + f s k (2.3)
where the nonlinear term V s k is written as the convolution

V s k = p+q=k sp,sq C s,sq,sp k,q,p ũsp p ũsq q (2.4)
and the tensor C

s k ,sq,sp k,q,p is given by C s k ,sq,sp k,q,p = 1 2 (s q q -s p p)(h s k -k • h sq q × h sp p ). The nonlinearity V s k satisfies the following relations s,k ũs -k V s k = 0, s,k s k ũs -k V s k = 0 (2.5)
that correspond to the energy and helicity conservation respectively and

s,k ∂ ∂ ũs k V s k = 0. (2.6)
This last relation indicates that phase space volume is conserved by the nonlinearity (ie it satisfies a Liouville condition). We will assume that the forcing is written as

f s k = s k ξ s k
where ξ s k are random complex amplitudes that are statistically independent, normally distributed and delta-correlated in time such that ξ s k (t)ξ s -q (t ) = 2δ s,s δ q,k δ(t -t ). With this choice each forcing mode injects energy to the system on average at rate s k . Then the Fokker-Plank equation for the probability density P(u) in the 2N-dimensional space of all complex amplitudes ũ± k is given by

∂ ∂t P + s,k ∂ ∂ ũs k (V s k P) = ν s,k ∂ ∂ ũs k (k 2 ũs k P) + s,k s k ∂ ∂ ũs k ∂ ∂ ũs -k P, (2.7)
where the sum is over all 2N modes ũs k .

Multiplying by E = 1 2 s,q |ũ s q | 2 , integrating over the phase-space volume and using integration by parts we obtain the energy balance equation

∂ ∂t E = -D E + I E (2.8)
where

D E = ν s,k k 2 |ũ s k | 2 is the energy dissipation, I E = s,k s
k is the injection rate and the brackets stand for the average f ≡ f PdU where dU stands for the phase space volume element dU = s,k dũ s k . We are interested in the limit that the energy injection and dissipation rate are a small perturbation to the thermalized fluctuations. We thus set ν k = δν k , s k = δ s k where δ 1 is a small parameter. We then expand P in power series of δ as P = P 0 + δP 1 + . . . . We are going to also consider the long time limit so we can neglect the time derivative. To zeroth order we then have

s,k ∂ ∂ ũs k (V s k P 0 ) = 0.
(2.9)

The equation above implies that P 0 is constant along the trajectories in the phase space followed by solutions of the truncated Euler equations. These trajectories are expected to be chaotic for large N and since this is such a high dimensional space we can also conjecture that these trajectories are space filling (ie ergodic) in the subspace constrained by the invariants of the system. In other words we assume that the trajectory will pass arbitrarily close to any point that has the same energy and helicity as the initial conditions. In this case P 0 is determined by the energy and helicity of the system

P(u) = f (E, H).
For the present work however we are going to neglect the second invariant the helicity and assume dependence only on the energy. We then write the solution of eq. (2.9) as

P 0 = f (E) = f   1 2 s,k |ũ s k | 2   .
(2.10)

To next order we then get

s,k ∂ ∂ ũs k (V s k P 1 ) = ν s,k ∂ ∂ ũs k (k 2 ũs k P 0 ) + s,k s k ∂ ∂ ũs k ∂ ∂ ũs -k P 0 (2.11)
Substituting eq. (2.10) and using the chain rule for the derivatives

∂ ∂ ũs k f (E) = ∂E ∂ ũs k ∂f ∂E = ∂ ∂ ũs k 1 2 ss,q ũsq -q ũsq q ∂f ∂E = ũs -k ∂f ∂E (2.12)
we obtain for the function f (E)

s,k ∂ ∂ ũs k (V s k P 1 ) = ν s,k f + |ũ s k | 2 ∂f ∂E k 2 +     s,k s k   ∂f ∂E + k s k |ũ s k | 2 ∂ 2 f ∂E 2   .
(2.13) To obtain a closed equation for f (E) we average eq. (2.13) over the volume dU E of all points in phase space of energy between E and E + dE. This consists of a spherical shell in the 2N-dimensional phase space of radius √ 2E. Averaging over this volume leads the sum in the left hand side to drop out because it is a divergence and the trajectories determined by V s k stay inside the shell. The volume integrals of terms independent of ũs k are proportional to the shell volume dU E = S 2N (2E) N -1 dE where S 2N is the surface of an unit radius 2N-dimensional sphere. Terms proportional to

|ũ s k | 2 result due to symmetry dU E |ũ s k | 2 dU = (2N ) -1 dU E s,k |ũ s k | 2 dU = (2N ) -1 S 2N (2E) N . This leads to ν k |k| 2 f + E N ∂f ∂E +   s,k s k   ∂f ∂E + E N ∂ 2 f ∂E 2 = 0. (2.14) If we set K 2 = s,k |k| 2 8πk 5 max /(5k 3 0 ) and I E = s,k s
k then by multiplying by E N -1 the equation simplifies to

∂ ∂E ν K 2 E N f + I E E N ∂f ∂E = 0 (2.15)
that has the bounded solution:

f (E) = Z -1 exp - νK 2 I E E (2.16)
where Z is a normalization constant that imposes P(u)du = 1 and is given by:

Z = S 2N 2 N -1 I E νK 2 N Γ (N ) (2.17)
with Γ being the Gamma function. We have thus recovered the Kraichnan distribution of eq. (1.4) with β = 0 and inverse temperature given by α = νK 2 /I E . Note that α depends only on the ratio of ν and I E and thus is independent of δ and we have thus dropped the primes.

It worth restating that

P(u) = f 1 2 s,k |ũ s k | 2
given in eq.2.16 expresses the probability that the system finds itself in the particular state u with energy E = 1 2 s,k |ũ s k | 2 . If we would like to find the probability P (E) of finding the system in any state u of energy E we need to average over all states u that have energy E. This leads to the chi-distribution for the energy

P (E) = S 2N (2E) N -1 Z exp - νK 2 I E E .
(2.18)

For large N the distribution P (E) in 2.18 is highly peaked at the mean energy

E = N I E νK 2 . (2.19)
As N tends to infinity P (E) becomes asymptotically a delta function centered at E . The mean energy of any mode ũs k is given by 1 2 |ũ s k | 2 = E /2N Averaging over spherical shells then leads to the thermal equipartition spectrum

E(k) = 4πI E νK 2 k 3 0 k 2 (2.20)
Finally using eq. (2.16) one can calculate the energy dissipation

D E = νZ -1 s,q q 2 |ũ s q | 2 exp   - 1 2 α s,k |ũ s k | 2   dU = ν 2N Z s,q q 2 (2E) exp (-αE) dU = νK 2 N E = I E (2.21)
and verify that the energy balance relation in eq. (2.8) is satisfied. The results indicate therefore that for small viscosity the truncated system will converge to the absolute equilibrium solutions of such "temperature" 1/α so that the viscous dissipation balances the energy injection rate!

Numerical simulations

In this section we test the results of the previous section and extend our investigation beyond the asymptotic limit using direct numerical simulations of the TNS system of eq. (2.1). The simulations were performed using the ghost code [START_REF] Mininni | A hybrid MPI-OpenMP scheme for scalable parallel pseudospectral computations for fluid turbulence[END_REF]) that is a pseudospectral code with 2/3 de-alliasing and a second order Runge-Kutta. For all runs the energy injection rate was fixed to unity and the integration times were sufficiently long so that steady states were reached. The forcing used is random and white in time as Figure 1: Left panel: The probability distribution functions P (E) from the results of the 32 3 -grid numerical simulations compared with the theoretical prediction. Right panel: The probability distribution functions for the amplitudes of u s k (real and imaginary part) for k = (0, 2, 0), (0, 4, 0), (0, 8, 0), and for s = ±1.

the one discussed in the previous section. It is limited to a spherical shell of wavenumbers satisfying (k F |k| k F ). Three cases were examined. In the first case small resolution runs were performed on a cubic domain with N G = 32 grid points in each direction. These runs due to their small size allow to some extend a direct investigation of probability distribution function P(u). In the second case the simulations were performed on a larger grid N G = 256 numerical grid that, after de-aliasing, leads to k max = 85 and was forced at large scales (k F = 1, k F = 2). These simulations demonstrate the transition from a forward cascade to a quasi-equilibrium state predicted in the section before. In the third case the simulations were designed to demonstrate the presence of an inverse flux in the thermalized state. A smaller grid was used N G = 128 with k max = 42. The energy injection was at large wavenumbers (k F = 31, k F = 35) while we replaced ν∇ 2 u by the modified viscous term ν∇ 2 P Q D [u] that acts only a particular spherical set of small wavenumbers Q D satisfying (1 |k| 4). With this setup energy is forced to be transported inversely from the forced wavenumbers to the dissipation wave numbers.

Small resolution runs

For these small resolution runs the energy injection rate was fixed to unity and the viscosity was set to ν = 10 -4 . The maximum wavenumber for the grid N G = 32 that was used was k max = 10. Despite the small value of k max the total number of Fourier modes in this system is still quite high N = 5040. It is thus still impossible to verify the predictions for P(u) in full detain for such a high dimensional space. Nonetheless in the left panel of figure 1 we plot the probability distribution function for the energy P (E) based on the direct numerical simulations plotted with solid brown line and based on the analytic predictions of 2.18. The two curves overlap indicating that both the mean value and fluctuations around it are correctly captured. Note that for this large value of N the distribution is highly peaked however there is still a finite variance of E indicating that E is a fluctuating quantity at difference with the TEE where E is fixed by the initial conditions.

To further explore the validity of the result 2.16 on the right panel of figure 1 we plot the pdfs of the real and imaginary part of the modes ũs k for the three wavenumbers k = (0, 2, 0), (0, 4, 0), (0, 8, 0), and for s = ±1. The results of the previous section predict that these amplitudes follow a Gaussian distribution with the same variance. There Figure 2: Left panel: Energy spectra for six different runs of the TNS equations with k max = 85, I E = 1 and from dark to bright ν = 10 -1 , ν = 10 -2 , ν = 10 -3 , ν = 10 -4 , ν = 10 -5 , ν = 10 -6 , ν = 10 -7 , ν = 10 -8 , ν = 10 -9 , ν = 10 -10 , The forcing was restricted in the small wavenumbers and the dissipation in the large wavenumbers as indicated. Right panel: The wavenumber k tran where the Kolmogorov spectrum k -5/3 transitions to the thermal spectrum k 2 as a function of the Kolmogorov lengthscale η = (ν 3 /I E ) 1/4 . Here k tran for the simulations (diamonds) is estimated as the wavenumber that E(k) obtains its minimum. The dashed line gives the prediction in eq. (3.1). are 16 curves in total in the left panel of 1 that have the same variance and perfectly overlap with the Gaussian shown by the dashed line in agreement the prediction of eq. 2.16. Further more we calculated the elements of the co-variance matrix for these modes Σi, j = X i X j (where X i stand for the mode amplitudes (ũ s k ) and (ũ s k )). The offdiagonal elements i = j are two orders of magnitude smaller than the diagonal elements i = j. This indicates that the Fourier amplitudes ũs k are independent variables with Gaussian distribution further verifying the predictions of the previous section.

Quasi-equilibrium state and forward flux

In the previous case although it was possible to verify some of the predictions of the previous section on P(u). However the limited range of wavenumbers did not allow to test the predictions on the energy spectrum. To that end we used a series of simulations on a larger grid (N G = 256) with I E = 1 varying the viscous coefficient ν from ν = 2 • 10 -1 to ν = 10 -10 . We must note here that the time to reach saturation from zero initial conditions is proportional to T ∝ E/I E and can be very large for small values of ν. For this reason the runs with small ν started with random initial conditions with energy close to the one predicted. The same runs were repeated with slightly smaller or large energy to make sure all runs converged to the same point. For the large values of ν the simulations are 'well resolved' in the sense that one can observe clearly the dissipation range where there is an exponential decrease of the energy spectrum. For ν = 0.01 and ν = 0.001 (second and third dark line from the bottom) one can also see the formation of an inertial range that displays a negative power-law close to the Kolmogorov prediction E(k) = C K I 2/3 E k -5/3 (where C K 1.6 is the Kolmogorov constant [START_REF] Sreenivasan | On the universality of the kolmogorov constant[END_REF]; [START_REF] Donzis | The bottleneck effect and the kolmogorov constant in isotropic turbulence[END_REF]). As the value of ν is decreased a bottleneck at large wave numbers appears and energy starts to pile up at the smallest scales of the system. As the value of ν is decreased further this bottleneck appears to take the form of a positive power-law close to the thermal equilibrium prediction

E(k) ∝ A k 2 . Laminar k k ( max 0 ) + 1 ( k k 0 max ) 11/4
(k max η) Thermal Kolmogorov Thermal Kolmogorov This thermal spectrum occupies more wavenumbers as the viscosity is decreased until all wavenumbers follow this scaling. At the smallest value of ν the result is compared with the asymptotic result obtained in the previous section. The proportionality coefficient A can be estimated from eq. 2.20 to be A = 5I E /(2νk 5 max ) that guaranties that the energy balance condition 2ν kmax 0

k 2 E(k)dk = I E is satisfied. Matching the thermal with the Kolmogorov spectrum we obtain that the transition occurs at the wavenumber

k tran = k max 2C K 5 3/11 (k max η) 4/11 (3.1)
where η = (ν 3 /I E ) 1/4 is the Kolmogorov length-scale. The right panel of fig. 2 shows a comparison of this estimate with k tran measured from the spectra as the wavenumber at which E(k) obtains its minimum. The scaling agrees very well with the results from the simulations.

In summary for values of k max η larger than k 0 /k max (ie k 0 η 1) the flow is laminar displaying an exponential spectrum. For 1 k max η k 0 /k max there is the formation of an inertial range where the Kolmogorov spectrum dominates followed by the dissipation range. If (k 0 /k max ) 11/4 k max η 1 (so that k tran k 0 ) then the dissipative range no longer exist and there is coexistence of the Kolmogorov spectrum followed by a thermalized spectrum at large wavenumbers. Finally for k max η (k 0 /k max ) 11/4 the system is in the quasi-equilibrium thermal state predicted in the previous section. These result are summarized in the sketch in fig. 3.

The transition from a Kolmogorov spectrum to a thermal one resembles a lot the time evolution of the TEE studied in [START_REF] Cichowlas | Effective dissipation and turbulence in spectrally truncated euler flows[END_REF]. For the TEE at early times a k -5/3 energy spectrum develops as energy is transferred to larger and larger wavenumbers. When the maximum wavenumber k max is reached, the thermalized energy spectrum starts to develop displaying at intermediate times both spectral slopes k -5/3 and k 2 . The difference with the present runs is that in the Euler case the transition occurs as time is increased while in the present case we only consider the steady state, and vary the value of k max η.

Similarities can also be found with the recent work on a time-reversible version of the Navier-Stokes [START_REF] Gallavotti | Equivalence of dynamical ensembles and navier-stokes equations[END_REF]. Using shell models [START_REF] Biferale | Equivalence of nonequilibrium ensembles in turbulence models[END_REF] and three dimensional simulations [START_REF] Shukla | Phase transition in time-reversible navier-stokes equations[END_REF]. of the time-reversible Navier-Stokes these authors found similar transitions from a Kolmogorov to a thermal quasi-equilibrium with the formation of both spectra depending on the parameter regime. The results were interpreted in terms of a phase transition, a possibility that could further explored for the present work as well.

We note that the time averaged flux from large scales to small scales is constant in the inertial range. What varies as we change the value of k max η is the amplitude of the fluctuations of the flux around this mean value of the flux. This is displayed in figure 4 where the mean flux is shown with dark line together with the instantaneous fluxes at different times with bright colors for four different values of ν. For ν in the

Quasi-equilibrium state and inverse flux

In most three dimensional turbulent flows the scales larger than the forcing scale are close to an equilibrium state [START_REF] Alexakis | On the thermal equilibrium state of large scale flows[END_REF]. In many instances however there is a change of dynamics at large enough scales and the flow is constrained to two dimensional dynamics (like for example turbulence in thin layers or in rotating flows) where energy tends to cascade inversely [START_REF] Alexakis | Cascades and transitions in turbulent flows[END_REF]. There is thus a transition from a thermal state with zero energy flux, to a state that has a finite inverse flux of energy. Close to such transition (unless the transition is discontinuous) the system has to be close to the equilibrium state with an inverse energy flux. With this motivation we examine the case where the forcing is located at small scales while the dissipation is limited in the large scales so that there is an net inverse transfer of energy.

We have thus performed simulations on a N G = 128 numerical grid that leads to a k max = 48 forcing at wavenumbers in the range (k F = 31, k F = 45) with unit energy injection rate. At difference with the previous simulations the dissipation limited only to small wavenumbers that satisfy 1 |k| k = 4. The energy is thus injected at small scales and dissipated at large scales forcing that way an inverse transfer of energy.

The spectra for four different values of ν are shown in the left panel of fig. 6. The forcing and dissipation shells are indicated in the graph. In this case a spectrum E(k) = Ak 2 forms for all values of ν with small changes at the dissipation wave numbers for large values of ν. For small values o ν the amplitude of the spectrum A is such so that the energy balance is satisfied that leads to A 5I E /(2νk 5 ) where k is the maximum wavenumber where the dissipation acts. The dashed line shows this prediction for the smallest value of ν. The flux fluctuations shown in the right panel of the same figure (fig. 6) are always dominant. We note however that their averaged value leads to a constant negative flux in the k 2 inertial range where no forcing or dissipation are present. It is worth noting that in these quasi-equilibrium states the direction of the transfer of energy is not determined by the nonlinear term as in Kolmogorov turbulence but only by the location of the energy source and sink. If regular viscosity is used, although a thermalized state is still reached as predicted by the previous section, there is no net inverse flux of energy. This is demonstrated in fig. 7 where the energy spectra and the energy flux is shown from simulations with regular viscosity forced at small scales.

Conclusions

In this work we examined the spectrally truncated Navier-Stokes equations flows that are close to equilibrium. We showed analytically that in the limit of small viscosity the statistically steady state of these flows converge to the [START_REF] Kraichnan | Helical turbulence and absolute equilibrium[END_REF] solutions. We note that our prediction is not just for the spectrum in eq. (2.20) but for the full probability distribution P(u) for the system to find itself in a state u given in eq. (2.16).

The derivation was based on two major assumptions. First we assumed ergodicity for the solutions of the TEE. This assumption appears in most calculations of classical statistical physics that although it can only be proved in very few systems it appears to be a plausible one for many systems with large numbers of degrees of freedom. For the TEE it appears at least to be in agreement with the results of numerical simulations. The second assumption we made was to neglect the effect of the second invariant: the helicity. This assumption was made in order to simplify the (rather involved) calculation. Had we kept the effect of helicity then the zeroth order solution P 0 (u) would have been reduced to a function of two variables f (E, H) and we would have ended up with an elliptic partial differential equation to solve for f . However the presence of helicity would break the spherical symmetry in phase space that allowed us to calculate the involved integrals. The calculation is still feasible but much more lengthier and we leave it for future work.

The numerical investigation verified our analytical results and demonstrated that the Fourier amplitudes ũs k indeed become independent Gaussian variables, and that the energy distribution and energy spectrum approach that of the analytic predictions as the asymptotic limit is reached. Furthermore, the numerical investigation also shed light on how the TNS system transitions from the classical Kolmogorov turbulence state to the thermalized solutions of [START_REF] Kraichnan | Helical turbulence and absolute equilibrium[END_REF] as the k max η is varied and led to precise predictions on when the transition takes place. The transitions are compactly summarized in fig. 3 where the three different states "laminar", "turbulent" and "thermal" as well as the transitions from one state to the other are clearly marked. Finally it was also demonstrated that these quasi equilibrium states are present when the dissipation is localized in the small wave numbers and the forcing at large wavenumbers forming an inverse flux of energy.

It is worth noting that a fixed amplitude flux (positive or negative) was always present in our simulations and that it was determined by the injection rate. However, as the quasi-equilibrium state is approached, the amplitude of the velocity fluctuations increases. These velocity fluctuations then lead to fluctuations of the energy flux to have variance that is much larger then the mean value making this mean flux a sub-dominant. Furthermore, the mean energy in the quasi-equilibrium state scales like E ∝ 1/ν (see eq. 2.19) and therefore in the small viscosity limit the energy dissipation normalized by u 3 rms k 0 lim ν→0 I E 2E 2/3 k 0 = 0 becomes zero. Thus, opposed to Kolmogorov turbulence, there is no finite (normalized) energy dissipation rate in the zero viscosity limit. We have thus to distinguish the processes of energy transfer in these quasi-equilibrium states (sometimes referred as "warm cascades") from the energy cascade that is met classical turbulent flows. These states are dominated by fluctuations and the direction of the energy transfer is not determined by the properties of the non-linearity but only the location of the sources and sinks of energy and in general is a non-local process (see fig. 14 in [START_REF] Alexakis | On the thermal equilibrium state of large scale flows[END_REF]). There are many directions in which the present results can be pursued further. First of all including the effect of helicity is crucial to have a complete description of the system. Moreover carrying out the calculation at the next order so that statistics of the fluxes can also be calculated would be equally desirable. Finally, extending these results to two-dimensional flows, where the equilibrium states can take the form of large scale condensates, is another possible direction. Such calculations, although considerably longer than the ones presented here, should still be feasible and we hope to address them in our future work.

  The resulting energy spectra are shown in the left panel of figure 2 for ten values of ν. Dark colors indicate large values of ν while bright values indicate small values of ν.

Figure 3 :

 3 Figure 3: Turbulence behavior as a function of k max η.

Figure 4 :

 4 Figure 4: Time average energy fluxes (dark line) and instantaneous energy fluxes (bright lines) for three different values of the viscosity ν = 10 -3 (top left panel), ν = 10 -5 (top right panel), ν = 3 • 10 -5 (bottom left panel), ν = 3 • 10 -6 (bottom right panel). The dashed line shows the variation amplitude of the flux. (Note the change in the scale of the y-axis in the bottom figures)

Figure 5 :

 5 Figure 5: Left panel: probability distribution function of the energy flux for ν = 10 -5 at four different wavenumbers. Right panel: probability distribution function of the energy flux at k = 20 and different values of viscosity. The dashed line shows the variation amplitude of the flux.

Figure 6 :

 6 Figure 6: Left panel: Energy spectra from simulation forced at small scales and dissipated at large scales as indicated. For all runs I E = 1 and from dark to bright ν = 10, ν = 1, ν = 0.1, ν = 0.01. Right panel: Time average energy flux (dark line) and instantaneous energy fluxes (bright lines) for ν = 1.

Figure 7 :

 7 Figure 7: Left panel: Energy spectra from simulation forced at small scales and dissipated by regular viscosity. For all runs I E = 1 and from dark to bright ν = 0.01, ν = 0.003, ν = 0.001, ν = 0.0003, ν = 0.00001. Right panel: Time average energy flux (dark line) and instantaneous energy fluxes (bright lines) for ν = 0.00001.

† Email address for correspondence: alexakis@lps.ens.fr

This work was granted access to the HPC resources of MesoPSL financed by the Region Ile de France and the project Equip@Meso (reference ANR-10-EQPX-29-01) of the programme Investissements d'Avenir supervised by the Agence Nationale pour la Recherche and the HPC resources of GENCI-TGCC & GENCI-CINES (Project No. A0050506421) where the present numerical simulations have been performed. This work has also been supported by the Agence nationale de la recherche (ANR DYSTURB project No. ANR-17-CE30-0004). This work was also supported by the research Grant No. 6104-1 from Indo-French Centre for the Promotion of Advanced Research (IFCPAR/CEFIPRA).