Chapter 6: The IPCC experience and lessons for the IPBES Daniel Compagnon, Wolfgang Cramer Large-scale environmental assessments involve the participation of many scientific disciplines, even more so when repeated throughout the decades and be implemented in institutionalized context such as the IPCC and the IPBES. Although the purported function of these international organizations is to provide a working interface between science and the decisionmaking circles at international level, the "boundary work" concerns as much the coordination between scientific disciplines not used to work together and lacking a common language, as it defines the interactions between the world of politics and that of scientists. The development of common standards for scientific evidence and plausibility is crucial, both within and then between the natural and social sciences. The challenge is to produce statements of greater clarity than otherwise found in scientific publications for the use of policymakers, without compromising the established scientific standards.

The institutional similarities between these two international organizations are deliberate and originate from the same willingness of the UN member states to retain some control over potentially far-reaching scientific assessments (see chap. 2 above). Institutional differences are largely a consequence of more than two decades of experience between the establishment of the IPCC and that of IPBES, and the creation of a more favourable political "climate" during this time:

Today, many member states recognise more clearly the potential for valuable policy support provided by science-policy interfaces as compared to the 1980s. Likewise, lobbying groups from all kinds of political background sense the need to actively relate to the existing science-policy interface.

The development of IPBES will hopefully benefit from the IPCC experience in both areas, interaction with policymakers and complex scientific coordination. Therefore, it is important to understand the nature of the IPCC and its limits and shortcomings, and also to understand the differences between the two institutions.

The first section of this chapter reviews the social sciences literature devoted to the IPCC and its reports without endorsing all the opinions expressed, to reflect on the kinds of problems and challenges the IPBES will have to face. The second section, based on the personal experience of the second co-author, both in the IPCC and in the nascent IPBES, illustrates these problems and emphasises the specificities of the IPBES in its early stages.

The ambiguities of the IPCC "model"

Originally seen as a specialized and obscure institution, the International Panel on Climate Change (IPCC) has become the iconic example of a strong influence of science in the construction of international environmental regimes, even though this picture is not totally accurate. Its status in the public opinion was greatly enhanced by the 2007 Nobel Peace Prize (shared with Al Gore). There is a large literature on the IPCC, only part of which is in political science/international relations (for a review, [START_REF] Hulme | Climate change: What do we know about the IPCC?[END_REF]. The intergovernmental nature of the new organization underlined that many governments wanted to exert some effective control over the production/dissemination of credible, legitimate policy-relevant knowledge on climate change. The US and British conservative governments in particular did not trust UNEP, which they perceived as partial to the NGOs and the developing countries, and whose Executive Director Mostafa Tolba had been a strong advocate of a climate convention since 1985 [START_REF] Agrawala | Context and early origins of the intergovernmental panel on climate change[END_REF]. As the different US government agencies were divided on the issue (in particular the Department of Energy vs. the Environmental Protection Agency), and huge business interests (fossil fuel related industries) were at stake, some US decision makers saw it expedient to claim that acquiring more knowledge was a prerequisite for action. To them, acting on the basis of the precautionary principle (such as in the Ozone Layer regime for example) was not an option. Establishing an international assessment mechanism under the United Nations control could serve to postpone policy action until the scientific basis was deemed robust enough. Others who supported the creation of IPCC hoped, on the contrary, that an independent, scientifically grounded organization would counterbalance the influence of business lobbies in policy-making and stimulate international policy cooperation. In this context, the definition of the "science" on climate change was also a political question from the onset, although the organization's motto made it clear that its work was "policy-relevant and yet policy-neutral, never policy-prescriptive". It is quite ironic that in the late 1990s and early 2000s the IPCC stood accused by critics from the neo-conservative and climate denialist camp of having "politicized" the science of climate change. It is the intergovernmental nature of the IPCC that makes its assessments acceptable as a basis for policy-making by government negotiators in the UNFCCC. "However, having an intergovernmental status has imposed significant costs also: IPCC assessment summaries are widely regarded as being politically negotiated, which has, at times, undermined their credibility" (Agrawala, 1998: 611). Instead of just "speaking truth to power", the traditional function devoted to science (Price, 1965), climate scientists are involved in an political role through a process that produces, for the sake of policy relevance, what some people call "imperfect science" [START_REF] Malnes | Imperfect science[END_REF].

The IPCC is indeed an intergovernmental cooperation organization, with nearly universal membership from 195 countries, and the Panel plenary is the core decision-making body attended by governments' representatives. Since the Panel plenary elects the IPCC chairperson and the members of the Bureau (including the co-chairs of the three working groups and the task force on GHG inventories) from candidates proposed by member states, the whole structure of the IPCC is kept under the close supervision of states. 1 The fact that all these officials have a robust scientific track record does not diminish the influence of the states. In addition, the plenary also adopts the structure and mandate of IPCC Working Groups and Task Forces, the work plan of the IPCC, the scope and outline of IPCC reports, and it approves, at the end of the assessment process, the Working Group reports and the Synthesis reports rather formally, and the respective Summaries for policymakers on a line-by-line basis, until a consensus is reached. Therefore, the intervention of the member states is by no means limited to the final phase (see diagram below). IPCC works on a double consensus: scientific and political.

1 The original division in three working groups was proposed by UNEP executive director Mostafa Tolba, a compromise owing more to politics than to science (Bolin, 2007: 49-50).

Figure 1 IPCC process from scoping to publication of report (Scoping is the preparatory meeting that proposes a scope and outline for the future report; see http://www.ipcc.ch/index.htm)

The USA, which had played an important role in WCP, had accumulated in the 1970s and 1980s the largest body of knowledge on climate change through its national research programs, and was also then the leading emitter of greenhouse gases (GHGs) per country and per capita. The US government kept a close eye on the proceedings of the IPCC. In 2002, for example, the US government pressed for the replacement of British scientist Robert Watson, chairperson of the IPCC since 1996 and a former Associate Director for Environment in the Office of the US President, perceived as a "bête noire" of the US oil industry by the Bush administration as too outspoken on the anthropogenic causes of climate change [START_REF] Mackenzie | Too hot for head of climate panel[END_REF]. The US government also objected to what it considered politically charged language in the Third Assessment Report (2001) (Haas, 2005:394). Governments from Russia and Saudi Arabia criticized the IPCC reports when seen as undermining their interests, but were opposed by other (notably European) governments who insisted on clear language and scientific credibility of these reports. These dynamics contributed to the politicisation of the process. Some analysts see the control of states over the IPCC organization and the contents of the assessment reports diminishing over the years [START_REF] Siebenhüner | The changing role of nation states in international environmental assessments-the case of the IPCC[END_REF]. Due to the implication of a growing community of scientists in the assessments and the writing of the three working group reports, the successive reports have gained in credibility. The IPCC produces no new research but collects material published in scientific and publicly accessible, mostly peer reviewed journals and books, and to a lesser extent from other sources, evaluated according to their relevance and scientific credibility. It also improved its internal procedures to control errors, following the controversy that developed over some errors contained in the Fourth Assessment Report (2007). The Inter-Academy Council was tasked to audit the IPCC procedures and make some recommendations in a context politically loaded by the release of climate scientists' e-mails (the "climategate" affair) just before Copenhagen in 2009. Following its report in August 2010 (IAC, 2010), a number of significant improvements to the IPCC procedures were introduced, although the main structures were maintained as they were. The reform stopped short of addressing "IPCC's relationship to public policy and to its various global 'publics ' [and] its normative commitments in terms of accountability, political representation, and legitimacy" (Beck & al, 2014:82).

Indeed, the legitimacy of the IPCC is still contested by critics who emphasize the overrepresentation of scientists from the affluent West. Being a G8 recommendation the establishment of the IPCC was originally seen with suspicion by the developing countries, which were concerned that an organization dominated by Western scientists would foster the political and economic interests of the industrialized countries. Despite the Chair's and Secretariat's efforts to increase involvement of experts from the Global South and the election of an Indian chairperson Rajendra Kumar Pachauri in 2002, and 2008 again, the proportion of OECD/non OECD authors and reviewers remained stable through the years (Haas, 2005: 395;[START_REF] Hulme | Climate change: What do we know about the IPCC?[END_REF]. Another perceived weakness of the IPCC process is the perceived bias against social sciences, regarded as lacking the rigor of the hard sciences, with the exception of economics where quantitative assessments are used in a way that can be understood also by the climate scientists [START_REF] Bjurström | Physical and Economic Bias in Climate Change Research: A Scientometric Study of IPCC Third Assessment Report[END_REF]. Some observers see the same trend developing in the nascent IPBES (Morin et al. forthcoming). Many of the climate scientists having contributed to the IPCC reports are aware of these shortcomings, but the production of interdisciplinary knowledge stumbles on acute epistemological and methodological problems.

Social sciences still debate over the characterization of the IPCC: a scientific lobby, a political organization or both, an epistemic community or a "boundary" organization. The notion of "epistemic community" as defined by Peter Haas emphasizes the role of consensual, policy-relevant in regime making, however, defined as "a transnational network of professionals with recognized expertise and competence in a particular domain and an authoritative claim to policy-relevant knowledge within that domain or policy area" [START_REF] Haas | Introduction: Epistemic Communities and International Policy Coordination[END_REF] does not fit with the structure of the IPCC. In Haas' case study-the Mediterranean Action Plan-the experts were embedded in the decision-making process and coordinated transnationally by UNEP, whereas the IPCC is a formal international organization and professes "policy-neutral" and certainly "not policy prescriptive" activities.

Although the First Assessment Report ( 1990) is said to have played a major role in the signature of the UN Framework Convention for Climate Change (UNFCCC), and the Second Assessment Report (1995) provided major inputs in the negotiation of the Kyoto Protocol, the publication of the reports did not coincide closely with the negotiation process, leading some observers to question their impact on policy-making (Haas, 2005: 396;[START_REF] Victor | Embed the social sciences in climate policy[END_REF]. The IPCC's major contribution has been to establish beyond doubt the reality of climate change, its predominant anthropogenic causes and the seriousness of its expected impacts, producing a "shared scientific understanding of the climate issue" (Lidskog & Sundqvist, 2015:12). However, it failed to shape policy in any significant way, with the exception of EU policies or in countries like Norway, in the absence of "usable knowledge" at a scale (national and sub-national) where policy-makers need to take decisions.

The goal to limit global warming at 2° C above pre-industrial levels is often presented as an outcome of its reports is not in fact a IPCC recommendation at all but a political target. It appears for the first time in a decision of the European Council in 1996, resurfaces in a report of the German scientific advisory institute WBGU in 2007 as the limit of acceptable warming, it is then adopted by the EU as an comprehensible policy objective for the public, and is pushed in the negotiations until it is endorsed by the "Copenhagen Agreement" and the Conference of the Parties (COP) 16 held in Cancun November 2010 [START_REF] Aykut | Le régime climatique avant et après Copenhague: sciences, politiques et l'objectif des deux degrés[END_REF]. Its centrality as a policy objective and legitimate discourse on climate change is not backed by overwhelmingly scientific evidence. Science has always, throughout all IPCC reports, recognized that significant damages occur already now (at 1° C, roughly), and that they will grow incrementally. This example underlines how difficult it might be to endorse the interpretation of the IPCC as a "boundary organization" between the world of science and the world of politics as understood in the Science and Technologies Studies (STS). Far from being pure and neutral, science is embedded in social institutions and discourses. Scientific "truths" are relative to a social-historical context and knowledge is always contested. Because scientists respond to social demands, in particular from the policy-makers who largely set the research agenda, policy-relevant science is always a co-production [START_REF] Jasanoff | Science and Norms in Global Environmental Regimes[END_REF], or what Bruno Latour calls "hybridization" to reflect the blurred boundaries between science and policy. "Coproduction happens when policy influences the production and stabilization of knowledge, while knowledge simultaneously supports and justifies policy" (Lidskog & Sundqvist, 2015:6). Whatever research findings are offered, policy makers will tend to be selective about what use and do not use. In return, IPCC contributed actively to the construction of climate change science as policy-oriented knowledge, with certain epistemic and methodological limits.

However, if the boundary organization creates a network of relations between scientists and decisions-makers, the Subsidiary Body for Scientific and Technological Advice (SBSTA) of the UNFCCC qualifies better than the IPCC in this respect, as it is officially tasked with translating science findings into policy advice and allowing government officials to mix with scientists on a regime implementation agenda, it engages in "hybrid management" [START_REF] Miller | Challenges in the Application of Science to Global Affairs: Contingency, Trust, and Moral Order[END_REF]. It is in Convention committees such as the SBSTA that discussions over policy issues2 take place as noted by the first IPCC chairman who attended many of its meetings [START_REF] Bolin | A History of the Science and Politics of Climate Change: The Role of the Intergovernmental Panel on Climate Change[END_REF] at a time when the IPCC was kept separated from the UNFCCC. That the SBSTA decided to "act as the interface between the IPCC and the climate regime, with the IPCC submitting its reports to SBSTA" (Miller, 2001: 260) and to create a joint liaison group between the two institutions, underlines that the IPCC in itself is not the real science-policy interface as often suggested. Although, being a typical UN forum following the consensus rule of procedure, the SBSTA has moved at a slow pace and its proceedings were heavily politicized although allowing a greater participation from smaller countries and attention to their worldviews (Ibid: 268).

Besides, does the IPCC really sit "on the boundary"? The analysis of its structure and rules of procedure briefly sketched above rather suggests that it is fully inter-governmental, with the original intent "to keep science on a tight leash" (Haas 2005: 396). Yet, we should differentiate within the IPCC the large community of scientists who are contributors and lead authors to the various chapters of the working group reports, and the Panel of government representatives. In retrospect, it seems that through this duality-IPCC and SBSTA-the principals (governments) carefully circumscribed the role of science in the climate change regime in order to retain a large autonomy of political decision.

They could not, however, prevent the IPCC reports from gaining a wider audience in the media and civil society, which in turn has exerted a certain pressure on decision-makers. Thus, it is through the public space (in the sense of Habermas and Dryzek) that IPCC reports have influenced policy-making in a broader sense.

From the history of the IPCC, there are significant lessons to be drawn for the IPBES. Largely inspired by the IPCC [START_REF] Larigauderie | The Intergovernmental Science-Policy Platform on Biodiversity and Ecosystem Services: Moving a Step Closer to an IPCC-Like Mechanism for Biodiversity[END_REF], the platform on biodiversity and ecosystem services is also an intergovernmental organization where the Plenary is the main decision-making body. It is positioned in relation to the CBD very much in the way the IPCC is in relation with the UNFCCC, outside the negotiation process, and IPBES relies on member states for implementation (see chapter 3 above). The CBD also has a Subsidiary Body on Scientific, Technical and Technological Advice (SBSTTA) to provide policy-relevant advice to the parties. In addition, the status of IPBES assessments in relation to the Global Biodiversity Outlook (GBO) regularly published by the CBD Secretariat is yet to be decided. The same challenges of involving more scientists form the global South, or of balancing different knowledge systems are facing the IPBES too. Once again, at the beginning of IPBES, some observers pointed at the quasi-absence of social scientists in the governance of the organization, when social sciences could help develop a more self-critical and reflexive approach to knowledge [START_REF] Lemos | The Co-Production of Science and Policy in Integrated Climate Assessments[END_REF]. However, since 2015 and the more explicit integration of non-scientific systems of knowledge, including local knowledge from communities playing a key role in conservation of natural resources and habitats, four social scientists are involved in the MEP3 .

The next section will explore the similarities between the IPCC and the IPBES from the personal experience of a scientist, Wolfgang Cramer, who was involved in both.

IPCC and IPBES from within

Participating in science-policy interface work is a challenge for most scientists and can be a life-changing experience for some. Arguably, no other field of science than climate (and now biodiversity science) offers the same opportunity for intensive exchange about key issues between decision makers in international politics and the scientific community. A guiding principle is mutual respect, captured prominently by the principle "policy-relevant but not policy-prescriptive", which initially poses problems for some scientists who believe that it is their role to convince the policy maker of the need to act. So the first lesson to learn is not only to speak clear and jargon-free language, but also to listen, and to understand the professional context within which the policy maker plays his/her role.

What is rarely recognised by outsiders of the process is that the participation of researchers in science-policy interface work is on a voluntary basis -both for the IPCC and IPBES. There is no remuneration and there is the expectation that a large amount of time needs to be invested. Indeed, most scientists consider it an honour to be invited to contribute, and they invest significant amounts of their spare time in this work. Most roles are assigned by invitation, involving a selection procedure that gives preference to scientific credentials, but then also strives to involve "political" elements such as the goal to increase the participation of colleagues from low-income countries, an adequate gender balance and also a rich participation from different scientific disciplines. While this process makes a great point of departure for the building of an international expert community, it also creates an obstacle to wide participation because many scientists do not work in positions where they can, at their own discretion, allocate a part of their time (during or outside working hours) to the IPCC or IPBES. I will look, from my personal perspective, at a few of the different roles participation in the IPCC can imply for an individual scientist -many of these roles exist in similar ways for IPBES. Clearly the most accessible role is that of a reviewer. For the Fifth Assessment Report, invitations to review chapter drafts were sent out very broadly, and any interested expert was given access to draft documents in order to provide comments on them. The role of these reviewers is hugely important for the quality of the assessment, but it provides little to no benefits to them in terms of enhanced reputation or career advantages. It is therefore often hard to motivate scientists to carefully review report drafts. Yet their contributions are more and more important, given that, due to the growing amount of published scientific knowledge, the full amount of information can hardly be assessed by an individual report author any more. Even if a reviewer reads only some section, for whatever reason related to personal interest, the contribution will be useful and appreciated by report authors. I first reviewed draft IPCC chapters for the Second Assessment Report, in 1991, spending many days with them. While reading, I learned a lot about scientific studies I was unaware of, but also found gaps and mistakes. My sometimes detailed responses led to an invitation to one of the author meetings, and I so became more deeply involved in the process. In particular, there was an intensive and unresolved debate within one of the author teams where my comments helped resolve the issue.

The process of writing chapters has changed substantially during the past 25 years, because the amount of scientific literature keeps increasing, and the ambition still is to provide a comprehensive overview of the available knowledge. Without active reviewers, this can no longer be achieved, and it will be important to increase the number of experts participating in the process.

Other roles for the IPCC or IPBES are usually assigned after some nomination and selection procedure. National "focal points" in most countries assist their governments in this process, usually with the aim of encouraging participation from their country in the platform. Being a "lead author" is a rather important role. These experts are asked to summarize the knowledge for some part of the science covered by a chapter, to provide text for inclusion, and to consider the comments that have been made by the reviewers. As lead author in various chapters of the Third and Fourth Assessments of the IPCC, I had the opportunity to work closer than otherwise with some leading scientists in my discipline, and several highly useful professional collaborations have arisen from these exchanges.

Being a lead author also inevitably broadens your view of your own field of science, as you must also assess literature from topics outside your "scientific comfort zone". Authorship at this level, as member of a small international team, is a great opportunity to sharpen one's abilities to identify "key findings" and present them in a way that communicates them clearly to non-specialists. During the construction of the overall assessment, one can then watch, and engage with, the process of synthesising key findings into an overall message that is perhaps later found in the summaries of the chapter, the working group volume or even the overall assessment report.

A basic but permanent problem for authors is the limitation in the space they are given.

Much of the time, authors find their topic extremely important and they struggle to provide text that contains precise information while still being short enough to be accessible to the reader who typically looks for information on a large number of topics at the same time. Another problem is that "assessment" implies not only to report facts that are documented in the literature, but also to provide statements of their reliability. In ecosystems, for example, species often come and go, due to natural fluctuations, due to direct human interference, or due to climate change. In an assessment of these changes, the author is expected to provide clear guidance on the relative role for these different "drivers". When there is no definitive answer (a situation arising frequently), then the author must communicate just how uncertain a given conclusion is, considering all available evidence. To help authors achieve this task, the IPCC produces fairly detailed guidance documents. In recent reports, authors have used what is referred to as "calibrated uncertainty language", a useful tool but one that requires training and therefore additional time investment for authors.

A particularly significant task for the IPCC is that of a "coordinating lead author" -I have had this role in the recent (fifth) assessment, co-leading a chapter on "detection and attribution of observed impacts" for its Working Group II [START_REF] Cramer | Detection and Attribution of Observed Impacts[END_REF]. The coordinating task obviously implies the maintenance of a good communication process with the lead author's team which by necessity is a global team, involving colleagues from similar disciplines but different geographic and cultural backgrounds, in our case from Germany, USA, Sweden, Brazil, South Africa, and the Philippines. Given the great distances, such teams can meet physically only rarely, in our case only five times during the six years of work, at quasi-annual meetings, and it becomes therefore imperative that good electronic communications are put to use.

In our specific case, we identified a stronger-than-usual necessity to interact not only with each other but also with a large number of other chapters under preparation for the same report.

This was because, in different regions and different scientific domains, the notion of what constitutes a scientifically plausible detection of some climate change impact from observations may differ. To ensure consistency, it was necessary to first develop clearer terminology and then train colleagues from different domains and regions in its use. Our objective was that all chapters treating regional issues or specific domains (like ecosystems, agriculture, settlement and others) would use the same methods for detection and attribution, thereby allowing our chapter to provide a synthesis on the basis of the others. To approach this goal, we created a "social network", connecting members of our own team electronically to contact persons in the 20 sectoral/regional chapters of the same report.

With these individuals, we not only exchanged e-mails, but we also organized web conferences where we discussed our approach and its consequences for the assessment. During one author's meeting, in Buenos Aires (Argentina), we even rented a restaurant in order to strengthen our social links to the other authors, and this turned out to be successful.

What emerges from this experience of being a coordinating lead author is, firstly, that the task is much beyond the editorial effort of aggregating pieces of text into a coherent document. The early definition of the shared objectives, and the continuous monitoring of the process are crucial, internally for the chapter but also for the connection between the chapter and all others in the same working group report. The task therefore becomes a social process, demanding a substantial effort by the coordinator, as well as by everyone else involved. Increasingly, governments recognize that, due to the volume of available information, coordinating lead authors need support for carrying out this task. In our case, with support from the German government, the work was supported by an expert in climate change issues in her own right, who worked full-time for the chapter teamwithout her, the result would have been a mere shadow of what the chapter is now.

IPCC reports strive to accessible syntheses by adding, to the chapters, executive summaries, and to the working group reports, a technical summary and a summary for policymakers. The preparation of the latter represents a particular challenge for the coordinating lead authors, as they must confront delegations of all governments in one week-long event called the "approval plenary".

During this session, the full draft of the summary is projected on great screens, and every sentence of it is opened to questions by government delegates. For any statement relating to a particular chapter, the coordinating lead author must be ready to explain the underlying scientific evidence.

While the situation itself may feel like an academic examination (something most of us had thought we would never again have to face in our professional life), this is also a satisfying opportunity to explain the findings, to seek understanding for them, and, if necessary, to modify them for further improve their clarity. It has sometimes been claimed that this is a situation of "negotiation", and it is true that one cannot help but feel that some governments raise issues with the sole purpose to weaken some statement. But the principles are clear and undisputable: the report is under the responsibility of the scientists only, and changes are made with the principal objective of reaching an unambiguous and scientifically sound text. There simply is no mechanism that could introduce, on government request, any statement that could be scientifically incorrect. Sometimes, text is deleted by the scientists chairing the discussion, because a significant number of government delegations considered that it could lead to misunderstandings on critical issues. Since this map could only document cases where some impacts had not only been observed but also attributed to climate change on the basis of a published scientific study, it was evident to all parties in the room that many impacts must have occurred that are not actually shown on our map. This of course was no omission; it was a clear consequence of the scientific method being applied to any part of the assessment made by the IPCC. For a number of country delegation, this appeared nevertheless as misleading, since the map apparently showed "no impacts" where in fact impacts were likely to have occurred. During some time, I tried to meet these concerns but failed to convince the delegates. The chairman of the IPCC then decided to call for a spontaneous meeting, in the corridors of the conference centre, with these delegations, myself and any other interested party. As a result of this constructive discussion in the corridor, we further strengthened the statements made in the explanatory text of the map, such as to make sure that no erroneous conclusions about the absence of observed impacts could be drawn from it. For me, this exchange was personally rewarding: it became clear to me that government delegates were very serious and wanted clear and unambiguous messages, and that the concern was that some important impacts might end up not being recognized.

A further step in the crafting of clear statements from scientific evidence, made by the IPCC, is the development of a separate Synthesis Report, aiming at drawing together the most important conclusions from the three working group reports into a shorter document. Participating, as I did, in the work of the core writing team for this report is also highly rewarding because, in an even broader setting, one has to face the challenge to debate with colleagues the relation between different findings and their importance for the policy maker (specific aspects of this process, still concerning the map from our chapter, are elaborated a commentary by [START_REF] Hansen | Global distribution of observed climate change impacts[END_REF].

In closing, I want to mention a last organizing principle for assessment work such as that of the IPCC and IPBES: throughout the various layers of summary-writing and synthesis, it is extremely important that any top level statement can be traced back to the original scientific publication and even its underlying data. In principle, this is ensured by appropriate references, and all steps of quality control need to verify that no errors or omissions occur during the editorial process. In the case of a heavy political debate that arose about the correct interpretation of some summary findings of the Fourth Assessment Report, concerning the numbers of people potentially affected by agricultural yield losses in Africa, we actually documented the "line of sight" from the summary statement back to the underlying scientific reference in detail [START_REF] Müller | Climate change risks for African agriculture[END_REF]. Doing so was useful, for two reasons. First, it demonstrated that no actual errors had been made during the assessment and subsequent synthesis. But second, it also highlighted that the underlying scientific information urgently requires a greater research effort in the future.

In the end, despite the growing availability of scientific information, it becomes undisputable that our knowledge is limited. While, in the case of changing climate and declining biodiversity, the time is well over-due to take stronger action for the preservation of the environment than what has been seen so far, it also demonstrates that for better adaptation to the inevitable, a stronger research effort is needed. Being part of this effort of research and communication has been a very enriching experience in my professional and personal life.

Conclusion

Both the 27 years of IPCC existence and the more recent IPBES experience emphasise the importance of inter-scientific dialogue and the necessity to broaden the scope beyond the core natural sciences.

Although this objective is largely shared by a great majority of scientists and some significant cooperation was achieved so far, there is still room for improvement. In particular social sciences (or local and indigenous knowledge to that matter) are not there just to inform policy once the diagnosis has been produced by hard sciences, they must also be part of the definition of the problem form the onset.

The IPBES will benefit most from the IPCC experiment if it encourages the circulation of scientists between to two processes. Robert Watson's involvement in the IPBES structure is a case in point, but the participation of authors and lead authors such as the second author of this chapter are equally important. They will not only facilitate the articulation between the two series of assessments for the many overlapping issues between climate change and biodiversity loss. While IPBES certainly should avoid making the same errors as the IPCC, some "memory" from previous experiences will still be useful.

Finally, the history of building theses assessment processes through a trial and error approach shows that the debate on neutrality of "pure" science vs. politicisation of science has not been very productive. Both totally independent assessments and embedded scientific bodies can succeed or fail in making "science speak the truth to power". As social constructs, these assessment bodies will always be challenged and criticised, but such debates should not undermine the bodies' legitimacy, hence reducing their actual capacity to influence policy. This is particularly important for the IPBES future, when strong policy commitments are expected from megadiverse countries, some of which were initially sceptical about the establishment of the Platform altogether.

  The IPCC was established in November 1988, on the request of the G8 countries, under the authority of the World Meteorological Organization (WMO) and the United Nations Environmental Program (UNEP), a move condoned by UN member states in a General Assembly Resolution 43/53 of 6 December 1988. The Toronto conference organized by UNEP in June 1988, when discussions about IPCC were under way, underlined both the importance of the issue and the lack of scientific consensus (in particular within the US scientific establishment) on both the reality and the causes of climate change. It was the outcome of a decade of attempts to tackle the scientific challenges of climate change, in particular the World Climate Programme (WCP) a research initiative under WMO, UNEP and International Council of Scientific Unions (ICSU), set up in 1979.

  As coordinating lead author for the chapter on detection and attribution of impacts of recent climate change that have occurred already, I faced an unexpected situation during the approval plenary for the IPCC's Fifth Assessment Report, in Yokohama in March 2014, while presenting a global map of observed impacts of climate change.

Figure 2

 2 Figure 2 CRAMER, W., YOHE, G., AUFFHAMMER, M., HUGGEL, C., MOLAU, U., DIAS, M. S. & LEEMANS, R. 2014. Detection and attribution of observed impacts. Climate Climate Change 2014: Impacts, Adaptation, and Vulnerability. Cambridge University Press : p 1011.

The SBSTA concerns itself with other issues than climate risk assessment, such as technology transfers, standards and methodologies for measuring emissions and abatement efforts.

In 2015,

among the 25 MEP's members are social scientists (Marie Roué, (Anthropologist), Unai Pascual (Economist), Marie Stenseke (Human geography) and György Pataki (Management and communication)).


Eunok Yim

J.-M Chomaz

D Martinand

E Serre

Transition to turbulence in the rotating disk boundary layer of a rotor-stator cavity

Keywords: nonlinear instability, rotating flows, transition to turbulence Rotor Stator Stator

The transition to turbulence in the rotating disk boundary layer is investigated in a closed cylindrical rotor-stator cavity via direct numerical simulation (DNS) and linear stability analysis (LSA). The mean flow in the rotor boundary layer is qualitatively similar to the von Kármán self-similarity solution. The mean velocity profiles, however, slightly depart from theory as the rotor edge is approached. Shear and centrifugal effects lead to a locally more unstable mean flow than the self-similarity solution, which acts as a strong source of perturbations. Fluctuations start rising there, as the Reynolds number is increased, eventually leading to an edge-driven global mode, characterized by spiral arms rotating counter-clockwise with respect to the rotor. At larger Reynolds numbers, fluctuations form a steep front, no longer driven by the edge, and followed downstream by a saturated spiral wave, eventually leading to incipient turbulence. Numerical results show that this front results from the superposition of several elephant front-forming global modes, corresponding to unstable azimuthal wavenumbers m, in the range m ∈ [32, 78]. The spatial growth along the radial direction of the energy of these fluctuations is quantitatively similar to that observed experimentally. This superposition of elephant modes could thus provide an explanation for the discrepancy observed in the single disk configuration, between the corresponding spatial growth rates values measured by experiments on the one hand, and predicted by LSA and DNS performed in an azimuthal sector, on the other hand.

Introduction

Flows confined between two coaxial disks, one being at rest and the other rotating at rotation rate Ω D , are prototype flows of both practical relevance and fundamental interest [START_REF] Launder | Laminar, transitional, and turbulent flows in rotor-stator cavities[END_REF]. Industrial applications concern the increase in performances of rotating systems, from computer hard drives to aeronautical turbine stages, with issues related to a higher efficiency of the cooling [START_REF] Owen | Heat Transfer in Rotating-Disk System: Rotor-Stator Systems[END_REF] or to the control of rotor vibrations induced by a strong unsteadiness of the flow [START_REF] Dimarogonas | Flow-Induced Vibration of Rotating Shafts[END_REF].

Fundamental interest is instigated by the similitude of this flow with one of the few exact self-similar solutions of the nonlinear Navier-Stokes equations, that provides a well-defined theoretical framework. Assuming the cavity height H to be large with respect to the characteristic viscous length δ = (ν/Ω D ) 0.5 , where ν is the kinematic viscosity, the mean flow is indeed asymptotically close to the Batchelor solution between two infinite parallel disks [START_REF] Batchelor | Note on a class of solutions of the Navier-Stokes equations representing steady rotationally-symmetric flow[END_REF].

This flow presents the structure of an inviscid core rotating at constant angular velocity Ω F , bounded by two boundary layers near the disks, of von Kármán type on the rotating disk (von [START_REF] Kármán | Über laminare und turbulente Reibung[END_REF]) and of Bödewadt type on the stationary disk [START_REF] Bödewadt | Die drehströmung über festem grunde[END_REF]. These boundary layers have a three-dimensional profile, with the radial component of the velocity presenting an inflection point along the wall-normal direction.

The stability of the Batchelor flow is closely connected to that of the boundary layer flow above a single rotating disk [START_REF] Serre | Coupled numerical and theoretical study of the flow transition between a rotating and a stationary disk[END_REF]). An extensive literature now exists on the linear instabilities arising in these boundary layers, named Bödewadt-Ekman-von Kármán (BEK) flow after the different cases covered by different Rossby numbers (ratio of the jump in rotation rate between the disk and the bulk fluid rotation rate), from Ro = 1 (Bödewadt flow), Ro = 0 (Ekman flow) to Ro = -1 (von Kármán flow). These instabilities are usually referred to as type I (cross-flow) and type II (viscous) [START_REF] Lingwood | Absolute instability of the boundary layer on a rotating disk[END_REF]. A type III, was eventually identified in the upstream branch of the type I spatial mode, the coalescence of types I and III creating the type I absolute instability [START_REF] Lingwood | Absolute instability of the Ekman layer and related rotating flows[END_REF]. The stability properties of these instabilities are entirely known for self-similarity solutions obtained for infinite disks. Even though some results cannot be directly extended to cavity flow, they provide good reference data. In terms of the overall flow, these instabilities cause waviness in the streamlines, but as the disturbances have vorticity, they appear as travelling vortices rolling up around a circular or spiral axis when the mean flow is subtracted from the total [START_REF] Serre | Annular and spiral patterns in flows between rotating and stationary discs[END_REF]. The evolution to subsequent bifurcations leading to turbulence in the rotor-stator cavity has received much less attention. In the Batchelor regime, turbulence first occurs in the stator boundary layer, where transition mechanisms are difficult to identify due to the swift development of turbulence. To the best of our knowledge, experiments by [START_REF] Cros | Transition to turbulence of the batchelor flow in a rotor/stator device[END_REF] are the only ones to identify nonlinear interactions between circular and spiral modes, eventually leading to turbulence for moderate global Reynolds numbers Re = 73 890 (see § 2.2 for definition). In the rotor boundary layer, transition to turbulence spreads over a larger range of Reynolds numbers, allowing a more detailed analysis.

In an open cavity between two rotating disks with superimposed radial through flow, direct numerical simulations (DNS) of [START_REF] Viaud | The elephant mode between two rotating disks[END_REF][START_REF] Viaud | Transition to turbulence through steep global-modes cascade in an open rotating cavity[END_REF] showed a transition mechanism based on an a cascade of 'elephant' modes. Such a scenario had only been observed before on the Ginzbug-Landau model [START_REF] Couairon | Global instability in fully nonlinear systems[END_REF]. More precisely, the DNS showed a subcritical nonlinear saturated elephant global mode characterized by a stationary front located at the boundary where the flow transitions from local convective to local absolute instability as the fluid moves downstream (referred to as the c/a boundary in the forthcoming). This global mode is absolutely unstable to a secondary instability leading to turbulence. Due to the similitude of this flow with the von Kármán boundary layer, this result brought new insight to the long-term debate regarding the transition scenario in the case of a single rotating disk. Moreover, recent experimental results on a clean rotating disk by [START_REF] Imayama | On the laminar-turbulent transition of the rotating-disk flow: the role of absolute instability[END_REF] tend to support this scenario.

The debate regarding the transition scenario in the rotating disk boundary layer started in the middle of the nineties with [START_REF] Lingwood | Absolute instability of the boundary layer on a rotating disk[END_REF][START_REF] Lingwood | An experimental study of the absolute instability of the rotating-disk boundarylayer flow[END_REF][START_REF] Lingwood | Absolute instability of the Ekman layer and related rotating flows[END_REF]. The local linear stability analysis (LSA) of the flow showed that the critical local Reynolds number r δ (as defined in § 2.2) of the cross-flow absolute instability, r δ = 507, was not far below r δ = 513 characterizing the onset of turbulence in the experiments, thus suggesting the possibility of a direct route to turbulence. Since these pioneering works by Lingwood, numerous theoretical, numerical and experimental studies have been devoted to this problem. If further studies have confirmed these local linear stability results (see [START_REF] Davies | Global behaviour corresponding to the absolute instability of the rotating-disk boundary layer[END_REF][START_REF] Serre | Coupled numerical and theoretical study of the flow transition between a rotating and a stationary disk[END_REF]Appelquist et al. 2015, for example), no general agreement exists concerning their outcome in terms of global behaviour of the instabilities, as the development of the flow along the radial direction is taken into account. More specifically, this development could include the effects of a finite radius of the disk or, in an infinite domain, the competition between stabilizing non-parallel effects and destabilizing nonlinear ones. Theoretical work by [START_REF] Pier | Finite amplitude crossflow vortices, secondary instability and transition in the rotatingdisk boundary layer[END_REF] showed the possible existence of a global nonlinear elephant mode at the onset of local absolute instability, while numerical solutions of linearized Navier-Stokes equations [START_REF] Davies | Global behaviour corresponding to the absolute instability of the rotating-disk boundary layer[END_REF][START_REF] Davies | Global stability of the rotating-disk boundary layer[END_REF] and experimental investigation of [START_REF] Othman | Experimental investigation of absolute instability of a rotating-disk boundary layer[END_REF] using a low-amplitude initial pulse-jet excitation demonstrated that a convective behaviour eventually dominates even though regions of strong local absolute instability were present in the flow, therefore demonstrating its linear global stability. These two behaviours, the existence of a nonlinear global mode, while the flow is linearly globally stable, would imply the existence of a subcritical global bifurcation, which was demonstrated in an open cavity between two rotating disks by [START_REF] Viaud | The elephant mode between two rotating disks[END_REF] using DNS and later in the infinite single disk case by [START_REF] Appelquist | On the global nonlinear instability of the rotating-disk flow over a finite domain[END_REF] using linearized DNS.

These works led to further studies considering the effects of the roughness [START_REF] Othman | Experimental investigation of absolute instability of a rotating-disk boundary layer[END_REF][START_REF] Imayama | Experimental study of rotatind-disk boundary-layer flow with surface roughness[END_REF]) and the finite radius of the disk [START_REF] Healey | Model for unstable global modes in the rotating-disk boundary layer[END_REF][START_REF] Imayama | An experimental study of the edge effects on rotating-disk transition[END_REF]. For rough disks, recent experiments by [START_REF] Imayama | Experimental study of rotatind-disk boundary-layer flow with surface roughness[END_REF] showed that an alternative scenario is possible via a convectively unstable route occurring at a lower Reynolds number, leading to transition to turbulence below the onset of the absolute instability. Using the linearized complex Ginzburg-Landau equation in a spatially varying system of finite extent, [START_REF] Healey | Model for unstable global modes in the rotating-disk boundary layer[END_REF] showed that the flow could be linearly globally unstable before the occurrence of an absolutely unstable region at the edge of the disk and predicted, by adding a nonlinear term in the model, a stabilizing effect of the boundary condition. As the consequence, the local critical Reynolds number is expected to increase when the global Reynolds number at the edge is decreased. However, this behaviour was not confirmed experimentally by [START_REF] Imayama | An experimental study of the edge effects on rotating-disk transition[END_REF], who observed that the transition location remained unchanged in the vicinity of the disk edge. [START_REF] Pier | Transition near the edge of a rotating disk[END_REF] tried to reconciliate those contradictory results by suggesting to model the boundary condition at the edge such that it acts as a source of random noise in order to mimic the strong source of fluctuations observed in his experiments. DNS of [START_REF] Appelquist | On the global nonlinear instability of the rotating-disk flow over a finite domain[END_REF] showed that there is almost no effect of the edge geometry on the nonlinear global instability. Their results exhibited spatial growth rates of the perturbation in agreement with the LSA, but not with the experimental data of [START_REF] Imayama | On the laminar-turbulent transition of the rotating-disk flow: the role of absolute instability[END_REF]. The authors conjectured a possible effect of roughness of the disk surface leading to a roughness-driven nonlinear solution instead of a self-sustained global mode. Recently, [START_REF] Appelquist | Transition to turbulence in the rotating-disk boundary-layer flow with stationary vortices[END_REF] numerically studied the role of the roughness of the disk to bypass the linear growth of primary instabilities and directly trigger secondary global instabilities involved in the transition to turbulence.

No clear mechanism has been fully identified so far, regarding the late stage of the laminar-turbulent transition process. A secondary instability of the primary stationary vortices seems to be involved [START_REF] Kobayashi | Spiral vortices in boundary layer transition regime on a rotating disk[END_REF][START_REF] Pier | Finite amplitude crossflow vortices, secondary instability and transition in the rotatingdisk boundary layer[END_REF][START_REF] Viaud | Transition to turbulence through steep global-modes cascade in an open rotating cavity[END_REF][START_REF] Imayama | On the laminar-turbulent transition of the rotating-disk flow: the role of absolute instability[END_REF]) but its behaviour and its relation to the primary instability is not yet understood.

This paper focuses on the transition scenario in the boundary layer of the rotating disk of a closed rotor-stator cavity, using DNS and LSA. This paper is organized as follows: in § 2 the set-up of the simulation is described including the geometrical, mathematical and numerical modelling. Results are presented in § 3, and, finally, discussion and conclusion are provided in § 4 with the goal of bringing new insight on the transition process of the flow over a rotating disk.

Problem formulation

Geometry

The geometrical model corresponds to two horizontal disks enclosing a cylindrical domain of radius R, bounded by a shroud, i.e. a vertical cylinder at rest, of height 2H. The lower disk of the cavity is stationary (stator) and the other (rotor) rotates at uniform angular velocity Ω = Ω D e z , e z being the unit vector along the vertical axis. In all of the figures, seen from above, the disk rotates counter-clockwise. The cavity aspect ratio is defined by G = R/H, and is fixed here to G = 10.26 [START_REF] Peres | A 3d pseudospectral method for cylindrical coordinates. application to the simulations of rotating cavity flows[END_REF]. In contrast to open systems [START_REF] Viaud | The elephant mode between two rotating disks[END_REF][START_REF] Viaud | Transition to turbulence through steep global-modes cascade in an open rotating cavity[END_REF][START_REF] Appelquist | Global linear instability of the rotating-disk flow investigated through simulations[END_REF][START_REF] Appelquist | On the global nonlinear instability of the rotating-disk flow over a finite domain[END_REF][START_REF] Appelquist | Transition to turbulence in the rotating-disk boundary-layer flow with stationary vortices[END_REF], the present set-up does not require any artificial inlet and outlet boundary conditions to alleviate undesired spurious noise.

Mathematical model

The fluid motion is governed by the incompressible three-dimensional Navier-Stokes equations written in the cylindrical coordinates (r, θ , z) and in primitive variables (u, v, w, p). Quantities are made non-dimensional using [H, Ω -1 D , Ω D R] as characteristic scales of space, time and velocity, respectively. The global Reynolds number is defined as Re = Ω D R 2 /ν and has been varied here in the range [10 3 ,4 × 10 5 ]. Local stability parameters, Rossby (Ro) and local Reynolds numbers (r δ ), are defined according to [START_REF] Lingwood | Absolute instability of the Ekman layer and related rotating flows[END_REF]:

Ro(r) = Ω * (r) Ω * (r) , r δ (r) = r * δ , (2.1a,b)
where r * and r = r * /H denote the dimensional and non-dimensional radius, δ is the characteristic viscous length δ = √ ν/Ω * (r) and [START_REF] Lingwood | Absolute instability of the Ekman layer and related rotating flows[END_REF], where Ω * F was constant and estimated at z = ∞, Ω * F here varies with radius r, and is estimated by taking the azimuthal component of the velocity at mid-height. According to definition (2.1), r δ varies from 0 to r δ edge = R/ √ ν/Ω * . Note that the local Reynolds number Re δ used in Lingwood (1997) amounts to Re δ = r δ Ro.

Ω * (r) = (1/4)(Ω * F (r) + Ω D ) + ((1/16)(Ω * F (r) + Ω D ) 2 + (1/2) Ω * 2 (r)) 1/2 where Ω * (r) = Ω * F (r) -Ω D , with the fluid rotation Ω * F . Unlike
No-slip boundary conditions apply at each disk and side wall: u = w = 0, the azimuthal velocity v = 0 on the stationary disk and v = r * /R on the rotating disk. At the junction of the stationary shroud with the rotor, the azimuthal velocity profile is regularized using v = exp[(z -1)/0.006] as proposed in [START_REF] Serre | Annular and spiral patterns in flows between rotating and stationary discs[END_REF] where the value 0.006 was shown to accurately model the velocity profiles observed in experiments. This alleviates the singularity of the boundary conditions at the junction and obviates the need to model the thin gap between the edge of the rotating disk and the stationary shroud occurring in experiments.

Numerical methods

Direct numerical simulation (DNS)

The incompressible Navier-Stokes equations are solved over the domain

[-R/H, R/H] × [0, 2π] × [-1
, 1] using an efficient pseudo-spectral Chebyshev Fourier algorithm for a cylindrical geometry, as described and validated in [START_REF] Peres | A 3d pseudospectral method for cylindrical coordinates. application to the simulations of rotating cavity flows[END_REF]. The mesh is based on N r × N z Gauss-Lobatto collocation points in the (r, z) plane and N θ points in the azimuthal direction. The grid refinement is progressively increased with Reynolds number up to N r × N θ × N z = 226 × 336 × 71 for Re = 4 × 10 5 , with a corresponding time step δt = 10 -4 . To dissipate the energy cascading through the nonlinear evolution of the primary and the secondary instabilities at large Reynolds numbers, a spectral vanishing viscosity (SVV), introduced in Severac & [START_REF] Severac | A spectral vanishing viscosity for the les of turbulent flows within rotating cavities[END_REF], is used. Former SVV computations have shown the ability of this technique to accurately describe rotating disk cavity flows without affecting noticeably the characteristic parameters of the instability [START_REF] Severac | A spectral vanishing viscosity for the les of turbulent flows within rotating cavities[END_REF][START_REF] Viaud | Transition to turbulence through steep global-modes cascade in an open rotating cavity[END_REF][START_REF] Peres | A 3d pseudospectral method for cylindrical coordinates. application to the simulations of rotating cavity flows[END_REF]). The code is not massively parallelized and uses the OpenMP programming interface. For the grid sizes used in this paper, the typical computing time per iteration is 15 s on 4 CPUs. Typical times to get convergence after increase of Re vary between t = 5 and t = 20 depending on the Reynolds number value.

Local linear stability analysis (LSA)

LSA is also conducted on the mean flow obtained from DNS, as detailed in § 3.1. It is based on a Chebyshev pseudo-spectral collocation method. Under the parallel flow assumption, the perturbation to the mean flow is assumed to be a periodic function of r, θ and t, as the components of the state vector φ are sought after as φ = φ (z) exp(imθ + ikrωt). LSA of the BEK flow is also performed for comparison and the method is validated with respect to Lingwood's results for Ro = -0.8 and -0.6 (Lingwood 1997) (see table 1).

Results

The Reynolds number is increased step by step from zero up to Re = 4 × 10 5 . For each increment, the steady flow solution is assumed to be reached when

|U n+1 -U n |/δt 10 -5
, where U is the velocity vector, and the subscripts n + 1 and n correspond to time steps (n + 1)δt and nδt. No extra disturbances are added. Perturbations are then measured primarily by the magnitude of the axial component of the velocity, which varies about a zero mean within the boundary layers on the disks.

Steady and mean flow

The steady flow is axisymmetric with three velocity components, due to the circulation in the meridional (r, z)-plane, created by the centrifugal force associated with the rotation of the disk. In this meridional plane, the secondary flow is essentially radial in the two boundary layers along the disks, separated by a central core in nearly solid-body rotation, and where the azimuthal component of velocity is independent of z. As there is no radial flow in the central core and the cavity is closed, the radial outflow near the rotor is necessarily compensated by a radial inflow along the stator, as seen in figure 1. At high Reynolds number, the mean flow solution is obtained by averaging the solution, both in time and in the azimuthal direction.

Far from the rotor edge, the mean velocity profiles are very close to those of the von Kármán self-similarity solution (not shown). The Rossby number is Present LSA results at Re = 4 × 10 5 and comparisons with former results of [START_REF] Lingwood | Absolute instability of the Ekman layer and related rotating flows[END_REF] for BEK flow at various relevant Rossby numbers. Ro and r δ are the local Rossby and Reynolds numbers defined in § 2.2. Re δ = r δ Ro is the local Reynolds number used in [START_REF] Lingwood | Absolute instability of the Ekman layer and related rotating flows[END_REF]. k i is the spatial growth rate, m m is the most unstable azimuthal mode and ω c is the frequency at critical conditions for the absolute instability.

nearly constant over the radius, with variations of less than 2 % in the range of Ro ∈ [-0.73; -0.7]. Due to radial confinement, however, the differences between the BEK profiles and the mean flow grow when approaching the edge (growing r δ ). These differences are exemplified in figure 2 for r = 7.2 at Re = 4 × 10 5 . This departure of the velocity profiles from the BEK flow induces a decrease of r δ c/a for the same Rossby number, as shown in the table 1. The discrepancy in the local critical Reynolds numbers even grows as r is increased from 4.6 to 7.2. LSA of the mean flow shows how sensitive the stability properties are to variations in the velocity profiles. Indeed, the large reduction of the critical Reynolds number (r δ c/a = 492 → 425) as r is increased from 4.6 to 7.2 cannot be attributed to the small variation of the Rossby number (Ro = -0.72 → -0.71) governing the stability of the BEK flow. It actually turns out that this shifted critical Reynolds number can be accounted for by the small changes in the ū and v profiles rather than the changes in the w profiles. This is related to the different scalings prevailing for the radial and azimuthal components of the velocity, r * Ω * Ro, and for the vertical component, Ω * Ro, r * being of the order of O(10 2 ) near the critical limit (see [START_REF] Lingwood | Absolute instability of the Ekman layer and related rotating flows[END_REF]. 3.2. Unsteady flows and solution patterns in the cavity Typical solution patterns in the whole cavity are shown at the largest Reynolds number considered in this work (Re = 4 × 10 5 ) in figure 3. The stator boundary layer (which will be not discussed further) is already turbulent and characterized by a nearly axisymmetric pattern, except in a small flow region around the axis, where the flow remains stable (see [START_REF] Serre | Coupled numerical and theoretical study of the flow transition between a rotating and a stationary disk[END_REF]). This small region of stable flow damps any perturbation flowing inward along the stator, before it enters the rotor boundary layer.

The rotor boundary layer, which is more stable than the stator one, becomes unstable at large radii. Spiral arms form a regular pattern, followed downstream by a temporally and spatially disorganized state, characteristic of incipient turbulence, as seen in figure 3(a). As already observed in an open cavity by [START_REF] Viaud | Transition to turbulence through steep global-modes cascade in an open rotating cavity[END_REF], the transition to turbulence is characterized by a shortening of the radial wavelength of the vortices, from λ ∼ 25.5δ at r δ ∼ 420 to λ ∼ 15.6δ at r δ > 538, as seen in figure 3(b). This is supported by theoretical results, which predict a shift of approximately 30 %, from λ ∼ 29δ to λ ∼ 20δ, between the linear absolute instability analysis [START_REF] Lingwood | Absolute instability of the Ekman layer and related rotating flows[END_REF], only valid ahead of the front where the amplitudes of the fluctuations are small), and the saturated wave developing further downstream [START_REF] Pier | Finite amplitude crossflow vortices, secondary instability and transition in the rotatingdisk boundary layer[END_REF].

The temporal evolution of w in the rotor boundary layer is shown in figure 4, as the global Reynolds number is increased from rest. Each vertical black line in figure 4(a) (for t > 550) indicates a Reynolds number increment of 2 × 10 4 , from Re = 2 × 10 5 to 4 × 10 5 . As no visible change occurs until Re = 2 × 10 5 (t ∼ 550), the time scale has been reduced to save space. On the other hand, the time scale has been blown up on the last three time units (t = 670-673) to detail the unstable flow dynamics at Re = 4 × 10 5 .

The first instability pattern occurs at the edge, at Re = 2 × 10 5 (figure 4b). Although parameters (r δ , Ro) are in the range of characteristic values predicted for the local, type I, absolute instability of the corresponding BEK flow, instability occurs here at a much smaller local radius, r δ = 410, compared to r δ c/a = 530 predicted by LSA

= r δ c/a G/ ReΩ * Ω -1 D since r δ = r * /δ = r/G ReΩ * Ω -1 D .
For large Reynolds numbers (Re > 2.4 × 10 5 ), the spiral arms are followed downstream by incipient turbulence, as indicated by the white solid line in figure 4 (the definition of which is given in the caption). For the highest Re (figure 4e), turbulence is observed above r δ 538, corresponding to a non-dimensional radius r ∼ 9. Although still coherent, spiral arms at this Reynolds number exhibit some ripples suggesting interactions between azimuthal modes. This point is supported by the spectrum in the azimuthal wavenumber m at each r δ shown in figure 5 For each azimuthal wavenumber m, a specific LSA is conducted, based on the mean flow at the location of the primary front r δ front , where the spiral starts to develop (defined in the caption of figure 4). The variability of the flow is weak at this location, and the mean flow is assumed to depart weakly from the base flow. The m-specific c/a boundaries obtained by LSA are indicated by triangles. Compared to the spectral density plots, it appears that azimuthal modes grow upstream of the local c/a transitions. Although m = 82 spiral arms are observed in DNS for Re = 2 × 10 5 , the LSA of the mean flow leads to a convective instability. At Re = 3 × 10 5 (figure 5b), azimuthal wavenumbers grow and saturate close to r δ front = 420, whereas, according to LSA, the first mode becomes absolutely unstable at r δ c/a = 450. The flow is strongly convective at the front. This flow behaviour is similar to recent experimental measurements on a single disk where surface roughness is accounted for [START_REF] Imayama | Experimental study of rotatind-disk boundary-layer flow with surface roughness[END_REF]. However, when the global Reynolds number is increased (Re = 4 × 10 5 , figure 5c), the c/a boundary obtained by LSA of the mean flow matches the front location obtained by DNS, meaning that the mean flow is marginally absolute at the front. This feature is similar to what is observed in experiments with clean disks or in former DNS performed either on single disk [START_REF] Appelquist | On the global nonlinear instability of the rotating-disk flow over a finite domain[END_REF] or in an azimuthal sector of an open cavity [START_REF] Viaud | Transition to turbulence through steep global-modes cascade in an open rotating cavity[END_REF].

Steep-fronted global modes

These patterns of instability are now investigated in the framework of the frontforming, elephant, global modes. The turbulent kinetic energy E, corresponding to the total kinetic energy of the velocity perturbation (u , v , w ), is defined within the rotor boundary layer at z = 0.97 as

E(r) = r 2 2π 0 (u 2 + v 2 + w 2 ) dθ rotor . (3.1)
Using Parseval's identity, this energy can be split over the contribution of each azimuthal wavenumber m, denoted E m . Radial variations of log(E) for various global Reynolds numbers are plotted in figure 6 as a function of both the physical radius r and the local Reynolds parameter r δ , for 0 r δ 600. Note that the maximum of r δ , r δ edge , varies with Re since r δ edge ∝ r

√

ReΩ * . At large global Reynolds numbers, the rotation rate Ω * 0.91, and remains almost constant in space, except near the edge. At each Re, the value of the perturbation is taken at the last computing time, when the flow is steady.

For r δ edge 393 (Re < 1.7 × 10 5 ), the boundary layer remains undisturbed over most of the rotor surface, and the E-profiles only show small-amplitude peaks in the vicinity of the edge (approximately 4 times larger than the background noise). This indicates that the flow close to the rotor edge is unstable, and may act as a source of perturbations. At r δ edge 427 (see the curve plotted in figure 6 for Re = 2 × 10 5 ), the peak at the edge is still visible, but with a kinetic energy increased by about two orders of magnitude. This is related to the sharply peaked m = 82 azimuthal mode observed in figure 5(a), and characterized by spiral arms forming a regular front located in a very narrow region near the edge, as seen in figure 4(a). Its spatial growth rate, estimated from the corresponding energy profile in figure 6, is

k DNS i = -0.2.
For larger Reynolds numbers, r δ edge 447 (Re 2.2 × 10 5 ), the front moves inward (in term of physical radius r), as shown in figure 6(a). Whatever the Reynolds number, fluctuations start growing around r δ 400 (black circular symbols in figure 6b) and saturate around r δ 450, with an energy approximately two orders of magnitude larger. Thus, the front location predicted by DNS does not substantially change with r δ edge (figure 6b). The front may appear upstream of the local c/a boundary predicted by LSA, but this theoretical location gets progressively closer to the front when increasing the Reynolds number. At Re = 4 × 10 5 , the front location matches well with the critical Reynolds number for the local absolute instability predicted by LSA (r δ front 420 compared to r δ c/a = 425 in table 1). The mean w-profile at a radial location changes when increasing the Reynolds number, thus leading to more unstable local stability properties.

Compared to smaller Reynolds number cases (Re 2 × 10 5 ), the spatial growth rates of these fronts are smaller, and keep the same value k DNS i -0.07, whatever the r δ edge . This value is about two times smaller than the spatial growth rate of local absolute instability (k i predicted by LSA (table 1)) as well as the growth rate usually measured in the literature when computations are restricted to an azimuthal sector [START_REF] Viaud | Transition to turbulence through steep global-modes cascade in an open rotating cavity[END_REF][START_REF] Appelquist | On the global nonlinear instability of the rotating-disk flow over a finite domain[END_REF]). However, it matches very well the experimental measurements carried out on a single disk by [START_REF] Imayama | An experimental study of the edge effects on rotating-disk transition[END_REF], k EXP i -0.065, and reported in figure 4 of their paper. Although the Rossby numbers Ro in this study and the other studies differ, the spatial growth rate does not vary substantially from Ro = -1 to Ro = -0.7. To clarify this point, the flow dynamics has been decomposed into three sets of azimuthal wavenumbers, according to its energy level as computed in figure 5(c): the most energetic modes m 1 m m 2 and two other sets with lower energy, the low azimuthal wavenumber modes m m 1 -1, and the high azimuthal wavenumber modes such m m 2 + 1. The azimuthal wavenumbers Figure 7 shows the kinetic energies E m for some selected wavenumbers of the perturbation, chosen within the three sets, as well as the sums over the azimuthal wavenumber ranges for each set, compared to the total perturbation energy calculated on the whole spectrum. Results show that the low azimuthal wavenumber modes mostly provide energy at small and large radii, corresponding to r δ < 400 and r δ > 550, respectively (figure 7a). The most energetic modes clearly determine the slope of the steep front shown in figure 6. Interestingly, the slope of each front corresponds to a larger spatial growth rate (k DNS i ∼ -0.12), which is close to the value predicted by LSA for the local absolute instability of the mean flow (k i ∼ -0.157, see table 1). Although each azimuthal wavenumber exhibits a steep front, their summation presents a smoother growth since the energy of each mode starts growing at various radial locations. It is gratifying that this weaker spatial growth rate is in agreement with experiments. For the highest azimuthal wavenumbers m, the energies are low, but they still grow continuously with r δ , as depicted in figure 7(c).

Summary and concluding remarks

The laminar-turbulent transition of the incompressible rotating disk boundary layer has been investigated by DNS and LSA of the mean flow in a closed cylindrical rotorstator cavity. In the DNS, the global Reynolds number Re is increased incrementally from zero, and no artificial perturbation is superimposed to the flow. Unlike all former DNS studies in the literature, both in single disk and in cavity configurations, the simulations are not restricted to an azimuthal sector of the cavity and thus include all nonlinear interactions. The main findings of this study can be summarized as follows:

(i) Over a large radial extent, the steady boundary layers developing over the rotor of the cavity match the von Kármán self-similarity solutions calculated at corresponding Rossby numbers. The mean velocity profiles in the cavity, nevertheless, increasingly depart from the von Kármán solution as the edge is approached. These differences explain why the LSA yields a more unstable configuration in the case of the mean flow in the cavity than in the case of the self-similarity solution. Moreover, at the rotor edge, the flow is a radial wall jet impacting the shroud combined with a centrifugally unstable vertical boundary layer. That leads to a very unstable flow at the edge, associated with strong fluctuations observed at all investigated Reynolds numbers. (ii) At moderate Reynolds numbers, corresponding to r δ edge 426, strong fluctuations are observed at the edge. These fluctuations are associated with the occurrence of a steep front (corresponding spatial growth rate k DNS i -0.2), and are followed downstream by a spiral pattern presenting m = 82 arms. This pattern is observed at a local Reynolds number r δ smaller than the value r δ c/a predicted by LSA for the local convective absolute transition. This discrepancy suggests that the corresponding global mode is driven by the edge, with a scaling constant in r and varying with r δedge . It can be somewhat surprising that, although the mode properties are in favour of an edge-driven mode, the dominant wavenumber m = 82 is consistent with the edge-independent theory of [START_REF] Lingwood | Absolute instability of the Ekman layer and related rotating flows[END_REF]. (iii) At large Reynolds numbers (r δ edge 447), fluctuations still take the form of a steep front followed by a saturated spiral wave. The characteristics of theses fronts are now independent of the Reynolds number at the edge (r δ edge ), and this may be accounted for by a self-sustained rotor boundary layer global mode. The spatial growth along the radial direction of the energy of these fluctuations is quantitatively similar to that observed experimentally by [START_REF] Imayama | On the laminar-turbulent transition of the rotating-disk flow: the role of absolute instability[END_REF], and corresponds to a spatial growth rate two times smaller than theoretically predicted for local absolute instability threshold previously observed between sector DNS (and LSA) and experiments. A similar argument could also apply to explain the differences previously observed in the temporal frequencies. (iv) The azimuthal spectral analysis of the solutions at these Reynolds numbers shows that the spiral front results from the superposition of several absolutely unstable modes associated with azimuthal wavenumbers in the range m ∈ [32, 78]. Interestingly, the spatial growth rates for each m match the theoretical value of the LSA for m = 76 (given table 1). The slopes measured in figure 5 (v) At the large rotation rates presently studied, a secondary front is also conspicuously close to r DNS δ = 538 and immediately followed by incipient turbulence. The present results bring new insights to the way that turbulence occurs in the rotating disk boundary layer. In agreement with recent experimental observations of [START_REF] Pier | Transition near the edge of a rotating disk[END_REF] in a single disk configuration, two distinct mechanisms can be identified for the onset of turbulence. At small Reynolds numbers, before the flow transition from convective to absolute instability, fluctuations start rising at the edge, within a region where the flow is no longer similar to the von Kármán boundary layer, and where shear and centrifugal effects cause a strong instability. This region acts as a strong source of perturbations. This mechanism is related to finite-size effects of the cavity, and leads to a global mode which is obviously driven by the edge. At large Reynolds numbers, fluctuations take the form of a front the radial location of which is nearly fixed in r δ , independently of the flow conditions at the edge. This can be interpreted as a self-sustained global mode developing in the rotor boundary layer and exhibiting a steeper front. Such a behaviour had been previously observed in DNS in azimuthal sectors [START_REF] Viaud | The elephant mode between two rotating disks[END_REF][START_REF] Viaud | Transition to turbulence through steep global-modes cascade in an open rotating cavity[END_REF][START_REF] Appelquist | On the global nonlinear instability of the rotating-disk flow over a finite domain[END_REF]. But unlike previous DNS, the spatial growth rate of the front in the present DNS, which is approximately twice smaller than predicted by LSA, matches also well with experimental measurements of [START_REF] Imayama | An experimental study of the edge effects on rotating-disk transition[END_REF]. It is the superposition of various absolutely unstable modes with different azimuthal wavenumbers that accounts for this gentler front. These results can be viewed as a collection of elephant global modes whose the superposition would lead to a smoother integrated energy variation in r.

(k DNS i -0.07, k EXP i -0.065, k i -0.157).
This result explains the discrepancy observed in the growth rate values between LSA and DNS in azimuthal sectors on the one side, and experiments and the present simulations on the other side. It also shows that fully nonlinear simulations are required to reproduce experimental observations. It would seem, thus, that convective modes produced by steady roughness are not necessary to explain this disagreement. The fact that, in the present DNS, the m-dependent front associated with each individual azimuthal wavenumber develops upstream of the m-specific convective absolute boundary remains unexplained at this point. This could be related to the fact that the LSA is performed on the numerical mean flow instead of the steady base flow, or to a weak forcing through the inertial wave, and we are currently working along those lines.

To finally address the transition to turbulence, the scenario seems to involve a secondary instability, characterized by a second front occurring downstream of the first one. At Re = 4 × 10 5 , the highest Reynolds number of the present study, the secondary front rises in the lee of the first and it is immediately followed by incipient turbulence. This flow behaviour is similar to that observed in DNS of azimuthal sectors of a rotating open cavity [START_REF] Viaud | Transition to turbulence through steep global-modes cascade in an open rotating cavity[END_REF], or of a single disk [START_REF] Appelquist | On the global nonlinear instability of the rotating-disk flow over a finite domain[END_REF]. Further investigations at higher Reynolds numbers are however required to better characterize this secondary instability and its role in the transition scenario.

FIGURE 1 .

 1 FIGURE 1. Sketch of the rotor-stator cavity of radius R and height 2H. Only the top disk is rotating (anti-clockwise when seen from above). Steady velocity vectors projected in the (r, z)-plane show the secondary flow developing in the meridional plane. A typical axial profile of the mean radial velocity component is shown in the right panel, at r * = 0.7R.

FIGURE 2 .

 2 FIGURE 2. Comparisons of the velocities near the rotor between the mean flow solution --at r = 7.2 and Re = 4 × 10 5 , and the BEK self-similarity solution ---for Ro = -0.72 (mean Ro of the mean flow): (a) radial velocity u BEK and ū (b) azimuthal velocity v BEK and v and (c) axial velocity w BEK and w. The insets show the differences u = u BEK -ū. Here, the length and velocities are scaled as in Lingwood (1997), i.e. δ, r * Ω * Ro for u and v and Ω * Ro for w, where the superscript * indicates dimensional values, and Ω * and Ro are defined in (2.1).

FIGURE 3 .

 3 FIGURE 3. Instantaneous flow pattern in the whole cavity for Re = 4 × 10 5 . The stator is below. Iso-surfaces and iso-contours of the instantaneous axial velocity w. (a) Three-dimensional view. The rotor and stator boundary layers are shown for z = ±0.97H, respectively. (b) (r, z)-plane. The white dashed and solid lines in (b) show the position of the spiral pattern at r δ ∼ 420 and the beginning of the turbulent region for r δ > 538, respectively (see figure 4 for a definition).

  for the flows at three different global Reynolds numbers Re. Several azimuthal wavenumbers m become unstable at large radii, in a range m ∈ [75, 91] at Re = 2 × 10 5 , m ∈ [39, 75]

FIGURE 4 .

 4 FIGURE 4. Temporal evolution of the flow patterns in the rotor boundary layer (z = 0.97) for Re ∈ [2 × 10 5 , 4 × 10 5 ]. Stabilization of the solution is reached before the Reynolds number is increased again. (a) Spatio-temporal diagram at θ = 0, for r ∈ [5, 10.26]. The time axis is contracted for t 550 (Re < 2 × 10 5 ), where the flow remains stable, while it is expanded for t ∈ [670 673], to magnify the unstable solution. Vertical black lines indicate instants when Re is increased by 2 × 10 4 . (b-e) Instantaneous top views of axial velocity field w corresponding to four Re, at time instants as indicated on (a). The colour bar range in (b) is exaggerated 5 times than the one of (a) otherwise the colour bar ranges for (c-d) are the same as (a). The squares below correspond to zoomed plots. The white dashed (---) and solid (--) lines in (a-e) show the positions of the primary (1/2 log(E) > -4) and secondary fronts (first peak after the primary front where ∂E/∂r > 0.02), respectively, with E the kinetic energy of the perturbation defined in (3.1). The white dotted line (• • • ) in (a) is r = r δc/a G/ ReΩ * Ω -1 D with r δc/a = 425 from table 1.

FIGURE 5 .

 5 FIGURE 5. Logarithm of energy in the spectral space of the axial velocity component as a function of the local Reynolds number r δ at z = 0.97: (a) Re = 2 × 10 5 , (b) Re = 3 × 10 5 and (c) Re = 4 × 10 5 . The colour scales are normalized by the maximum amplitudes of the non-axisymmetric components. The arrow in (a) indicates m = 82 and the solid and dashed vertical lines correspond the primary and secondary fronts, as defined in figure 4. The triangles correspond to the m-specific c/a boundaries computed by LSA of the mean flow at the primary front locations. The horizontal lines (-•-) in (c) indicate the smallest and the largest modes, m 1 = 32 and m 2 = 78, respectively.

FIGURE 6 .

 6 FIGURE 6. Radial profiles of the turbulent kinetic energy E in the rotor boundary layer at z = 0.97, and for various global Reynolds numbers: • • • Re = 0.7 × 10 5 ; -• -Re = 1.7 × 10 5 ; ---Re = 2 × 10 5 ; -Re = 2.2 × 10 5 ; -• -Re = 2.4 × 10 5 ; • • • Re = 2.8 × 10 5 ; --Re = 3.2 × 10 5 ; ---Re = 3.6 × 10 5 and -Re = 4 × 10 5 . The symbols represent the locations of the primary ( E ) and the secondary fronts ( A ) as defined in figure 4. The arrows in (b) indicate the edge local Reynolds number r δ edge (maximum value of r δ ) for each Re.

FIGURE 7 .

 7 FIGURE 7. Kinetic energies of the perturbation for some selected azimuthal wavenumbers m in the ranges of (a) intermediate m ∈ [32, 78]: m = 41 ---, m = 65 --and m = 72 • • • (b) low m = [0, 32]: m = 0 --, m = 10 --and m = 30 ---, (c) large m > 78 : m = 90 --, m = 118 ---and m = 140 • • • at Re = 4 × 10 5 . The thick black lines show the total energy (1/2 log(E)) shown in figure 6 and lines with symbols show the total energy restricted to each range. The vertical solid line indicates the absolute instability transition r δc/a = 425 of the mean flow and dashed line represents the primary front of DNS defined in figure 4. The spatial growth rates are estimated as k DNS i,m41 = -0.18, k DNS i,m65 = -0.1, k DNS i,m72 = -0.13 and for low and large m, the spatial growth rates are approximately estimated as k DNS i,low -0.014 and k DNS i,larg -0.019.

  (b) are indeed almost similar for all m in the range m ∈ [32, 78], with k DNS i = [-0.18, -0.1], compared to the results of the LSA in range k LSA i ∼ [-0.2, -0.157].

TABLE 1 .

 1 Critical values for convective absolute (c/a) transition by linear stability analysis.

	Authors	Ro	r δc/a	Re δc/a	k i	m m	ω c
	LSA of the BEK flow
	Lingwood (1997) -0.8	543.5 434.8 -0.142 84.2 -0.0393
	Lingwood (1997) -0.6	575.7 345.4 -0.164 97.3 -0.0418
	Present results	-0.8	540	432.0 -0.143 83	-0.0391
		-0.72 529.2 381	-0.151 89.4 -0.042
		-0.71 529.6 376	-0.152 89.6 -0.042
		-0.6	565.3 339.2 -0.162 97.3 -0.044
	LSA of the mean flow at Re = 4 × 10 5
	(r = 4.6)	-0.72	492	354	-0.155	87	-0.051
	(r = 7.2)	-0.71	425	303	-0.157	76	-0.052

TABLE 2 .

 2 Table 2 summarizes present results, compared to existing experimental and numerical results in Imayama et al. (2014) and[START_REF] Appelquist | On the global nonlinear instability of the rotating-disk flow over a finite domain[END_REF]. Although the Rossby numbers differ, the present DNS over a full disk quantitatively retrieve the experimental spatial growth rate. Comparing the full disk DNS to LSA in our rotor-stator set-up also qualitatively explains by the coexisting azimuthal wavenumbers the differences Summary of present results in comparison to the experimental and DNS data for Ro = -1. Here, r δc varies depending on the definition of the characteristic location: for[START_REF] Imayama | On the laminar-turbulent transition of the rotating-disk flow: the role of absolute instability[END_REF][START_REF] Appelquist | On the global nonlinear instability of the rotating-disk flow over a finite domain[END_REF]) r δ end where the linear region ends and for r δ front for the present DNS. The spatial growth rate k i is estimated near the r δc . The frequency ω is defined as ω = ω * /Ω D and the theoretical value ω * c is the converted vale of ω c to the same scaling as ω * c = ω c Re δc/a -m m .

	Authors	Ro	r δc	k i	k i /k i,LSA m m	ω	ω ω * c
	Experiment (Imayama et al. 2014)	-1	510 -0.065	0.53	-∼ 40 0.79
	DNS m = 68 (Appelquist et al. 2016)	-1	582.8 -0.12	0.99	68 52.36 1.04
	Present DNS	-0.71 420	-0.07	0.45	66 ∼ 48 0.79

† Email address for correspondence: eric.serre@univ-amu.fr

Acknowledgements

This work was supported by the Labex MEC (ANR-10-LABX-0092) funded by the Investissements d'Avenir program of the French National Research Agency (ANR). This work was granted access to the HPC resources of Aix-Marseille University under the project Equip@Meso (ANR-10-EQPX-29-01) and of IDRIS, under the allocation i20170242 made by GENCI.


Richard M Lueptow

Denis Martinand

Eric Serre

Taylor Couette Flow with Imposed Radial and Axial Flows-A Weakly Nonlinear Analysis

The linear stability analysis of Taylor-Couette flows where axial and radial through-flows are superimposed is extended to consider the weakly nonlinear behavior of convective-type instabilities using a fifth-order amplitude equation and numerical simulations. Special attention is paid to the influence of the radius ratio η, particularly as the gap increases to become very wide (η increases from top to bottom in Figure 1), which magnifies the impact of the radial Reynolds number α. The instabilities take the form of pairs of counter-rotating toroidal vortices superseded by helical ones as the axial Reynolds number β is increased. Increasing the radial inflow (negative α) draws the vortices toward the inner cylinder, where they shrink relative to the annular gap when the gap is wide (Figure 1(d,g,h)). Strong axial and radial flows lead to steeply sloped helical vortices. Strong radial outflow (positive α) in a wide annular gap results in very large helical vortices (Figure 1(i)). In addition, radial inflows or outflows can change the supercritical bifurcation from laminar to vortical flow to become a subcritical bifurcation. The radial flow above which this change occurs decreases as the radius ratio decreases.

Figure 1 :

 1 Figure 1: Modes of instabilities for β = 20 and (a) η = 0.85 and α = -10, (b) η = 0.85 and α = 0, (c) η = 0.85 and α = 10, (d) η = 0.55 and α = -10, (e) η = 0.55 and α = 0, (f) η = 0.55 and α = 10, (g) η = 0.25 and α = -10, (h) η = 0.25 and α = 0, (i) η = 0.25 and α = 10. Isosurfaces of the radial velcoity of the instability are shown at 0.2 (red) and -0.2 (yellow) of the maximum value. The critical Taylor number and wavelength are indicated. Reprinted with permission [1].

Acknowledgements

This work was granted access to the HPC resources of IDRIS under the allocation 2016242 made by GENCI (Grand Equipement National de Calcul Intensif).


The transition to turbulence in the rotating disk boundary layer is investigated in a closed cylindrical rotor-stator cavity via direct numerical simulation using spectral vanishing vistocity method (DNS-SVV) and linear stability analysis (LSA). As shown in the Figure 1, the stator boundary layer (lower part) is already turbulent while the rotor boundary layer (upper part) remains stable until very high Reynolds number Re = U/νR then shows an organised behaviour followed by incipient turbulence. We focus on the global stability and the transition to turbulence of the rotor boundary layer. The mean flow in the rotor boundary layer is qualitatively similar to the von Kármán self-similarity solution [START_REF] Th | Über laminare und turbulenteReibung[END_REF]. The mean velocity profiles, however, slightly depart from theory which is developed on the infinite disk assumption [START_REF] Lingwood | Absolute instability of the Ekman layer and related rotating flows[END_REF] as the rotor edge is approached. Such meanflow modification near the edge seems to affect the stability behaviour of the system [START_REF] Healey | Model for unstable global modes in the rotating-disk boundary layer[END_REF][START_REF] Pier | Transition near the edge of a rotating disk[END_REF]. Shear and centrifugal effects lead to a locally more unstable mean flow than the self-similarity solution, which acts as a strong source of perturbations. Fluctuations start rising there, as the Reynolds number is increased, eventually leading to an edge-driven global mode, characterized by spiral arms rotating counter-clockwise with respect to the rotor (see Figure 1). At larger Reynolds numbers, fluctuations form a steep front, no longer driven by the edge, and followed downstream by a saturated spiral wave, eventually leading to incipient turbulence. Numerical results show that this front results from the superposition of several elephant front-forming global modes, corresponding to unstable azimuthal wavenumbers m, in the range m ∈ [32, 78]. The spatial growth along the radial direction of the energy of these fluctuations is quantitatively similar to that observed experimentally on the infinite single disk [START_REF] Imayama | On the laminar-turbulent transition of the rotating-disk flow: the role of absolute instability[END_REF]. This superposition of elephant-modes could thus provide an explanation for the discrepancy observed in the single disk configuration, between the corresponding spatial growth rates values measured by experiments on the one hand, and predicted by LSA and DNS performed in an azimuthal sector [START_REF] Appelquist | On the global nonlinear instability of the rotating-disk flow over a finite domain[END_REF], on the other hand.

Figure 1 :

 1 Figure 1: Instantaneous flow pattern in the whole cavity for Re = 4 × 10 5 . The stator is below. Iso-surfaces and iso-contours of the instantaneous axial velocity w. Three dimensional view. The rotor and stator boundary layers are shown for z = ±0.97H, respectively.

  Transition to turbulence in the rotating disk boundary layer of a rotor-stator cavity Eunok Yim 1,3 , J.-M. Chomaz 2 , D. Martinand 1 E. Serre 1 1 Aix-Marseille Univ., CNRS, Centrale Marseille, M2P2, France

2

LadHyX, CNRS-Ecole Polytechnique, F-91128 Palaiseau, France 3 current address LFMI, EPFL, Lausanne, Swiss

Acknowledgements

This work was supported by the Labex MEC (ANR-10-LABX-0092) funded by the Investissements d'Avenir program of the French National Research Agency (ANR). This work was granted access to the HPC resources of Aix-Marseille University under the project Equip@Meso (ANR-10-EQPX-29-01) and of IDRIS, under the allocation i20170242 made by GENCI.