Simon Reality

Jonathan Saunders

Adrian Barrett

David Kent

Wallace

Simon Saunders

Jonathan Barrett

Adrian Kent

David Wallace

Hilary Greaves

Jenann Ismael

Wayne Myrvold

Worlds Galore? Essay Review of: Many Worlds? Everett, Quantum Theory

The 50 th anniversary of Everett"s 1957 paper has brought in its wake the publication of a number of important books on Everett"s life, work and theory. This book provides arguably the most vivid and comprehensive treatment of both state-of-the-art developments within and criticism of the Everett interpretation (hence the question mark in the title!).

basis" in Everett (or, more aptly, of the democracy of relative states) has achieved a solution commanding a broad consensus. 3 Wallace"s chapter is already a classic, and arguably the place to look if one wants to understand the modern-day approach to the ontology of the Everett theory, which is best described by borrowing Dennett"s term of "real patterns" as applied to the structures arising from decoherence within the universal wavefunction. (Throughout this review, when I say "Everett interpretation" or "Everett theory" I shall generally mean the decoherence-based approach to Everett"s ideas.)

Hartle"s and Halliwell"s chapters review theoretical aspects of decoherence in the context of the "decoherent histories" formalism (some of which are highly informative and not much discussed in the philosophy literaturethe present author not excepted). This formalism is particularly useful if one is discussing diachronic aspects of decoherence. (A review of a different line of developments originating from decoherence theory is given by Wojciech Zurek in Chapter 13.) Hartle"s main concern is with the derivation of quasi-classical realms within quantum theory (including the emergence of classical spacetime). Halliwell"s is with the notion of decoherence as arising from conservation laws, as opposed to the more familiar notion of environmentally-induced decoherence. The former notion gives rise to a structure of largely parallel worlds within the universal wavefunction, which is somewhat in contrast with the branching structure usually associated with the latter (and which is arguably the main tool in Wallace"s discussion). It seems to me that environmental interactions give rise to much more robust histories and much more interesting ones. There is a sense in which nothing happens in worlds arising from conservation-based decoherence, so rather than structure one has perhaps identified lack thereof. (Even the emergence of classical spacetime may be problematic, if it does not contain any concrete events.) Thus, perhaps, further ingredients are required to provide a fully satisfactory analysis (e.g. Zurek"s notion of "quantum Darwinism", see below).

When above I mentioned "broad consensus", I meant consensus among Everettians. Indeed, Part II is devoted to criticisms of an ontological picture that is based solely on the wavefunction (or even on a state in an abstract Hilbert space) and patterns within it (Chapters 5 and 6, by Tim Maudlin and John Hawthorne, followed by James Ladyman"s reply to Hawthorne and by a partial transcript of discussions relating to Parts I and II). Some of the arguments discussed in these chapters are relevant also to the opposition between Everett and alternative approaches to the foundations of quantum mechanics (in particular pilot-wave theories, an opposition well represented in Part V by Antony Valentini"s Chapter 16 and Harvey Brown"s reply4 ).

Maudlin is very much aware of the kind of game that Everettians (but by no means only Everettians) are engaging in. Given a theory (in this case unitary quantum mechanics), we need to analyse what structures in it play what role, in particular whether the theory contains any structures that behave like the things we wish to model. (Is there something in the theory that behaves like a tiger?) Even granting this structuralist strategy, Maudlin believes that not enough attention has been paid to actually showing that this strategy works in the case of a theory that is solely about the behaviour of rays in an abstract Hilbert space (whether or not this behaviour is unitary or includes a collapse mechanism), and he is sceptical about its chances of success, so that further structure needs to be postulated. I agree that more work could be done on the structural analysis of Hilbert-space, in particular on the issue of the emergence of classical spacetime and of configuration space (e.g. the constitution of spacetime events, the concept and role of quantum reference frames etc.). Indeed, it is no more respectable to help oneself uncritically to the configuration-space representation of the quantum state as it is to help oneself uncritically to some particular decomposition of a density matrix. On the other hand, I believe that crucial aspects of this work are already present in the literature, many of them in the literature on decoherence. 5 If I understand Maudlin correctly, however, he goes even further and disagrees with a purely structuralist strategy, requiring even of structures that are present (perhaps emergently) in the theory that one postulate which ones are of fundamental ontological import (so-called "primitive ontology"). 6 This is a deep disagreement with Everettians and with structuralists in general (cf. pp. 167-168): how could e.g. the potentials in the Schrödinger equation ever be structured as if they were functions on 3-dimensional space, if this space has no independent existence? (Or turning intuitions around: how could an independently existing space ever determine the form of equations defined on Hilbert space?) I do believe, however, that even if one postulates a primitive ontology one needs to do the same kind of careful structural analysis in order to show that this ontology indeed plays the role that it is supposed to play (as Maudlin himself states: it must not be "invisible and intangible"), a partial example of which would be the discussion of subquantum measurements in pilot-wave theory in Valentini"s Chapter 16.

Hawthorne"s chapter is rich in interesting points (taken up in later discussions), but the metaphysical background against which they are formulated is perhaps typical of a general Everett and spontaneous collapse theories is however more commonly seen as straightforwardly empirical (as part of the alternative between collapse and no-collapse). 5 A few relevant aspects are: the significance of certain operators as generators of the symmetry group of the fundamental equations (whether Galilei or Lorentz), representations of the fundamental symmetry group, s ste s of i p i iti it , te h i ues fo d essi g s ste s, te h i ues fo o st u ti g ohe e t states, the role of such states in decoherence, the emergence of classicality from decoherence, the emergence of spacetime from causal structure (perhaps applied to the concrete events of decoherence), the abstraction of spacetime from the behaviour of classical (or quasiclassical!) physical systems, i.e. reference frames, Poincaré s insights into the nature of space and their generalisation by Harvey Brown, the work of Julian Barbour, and so forth. 6 For instance, in the GRW theory the Schrödi ge e olutio is i te upted hits ultipli atio s ith Gaussians defined on 3-d space). According to Maudlin, one needs to specify a primitive ontology for the theo , a d his fa ou ed optio is to p o ote the hits to the status of p i iti e o tolog G'W theo ith flash o tolog . Fo a st u tu alist, flashes a e already part of the structure of the theory (they are part of the structure of the equations), and are thus already available (among other things!) as material for analysing whether the theory contains structure that behaves in the right way to describe the everyday world.

disconnection with science in contemporary metaphysics, an Achilles" heel exploited by James Ladyman in his reply.

With Parts III and IV (Chapters 6-9, by Simon Saunders, David Papineau, David Wallace, and Hilary Greaves and Wayne Myrvold, and Chapters 10-12, by Adrian Kent, David Albert, and Huw Price, followed by a partial transcript of the discussions) we turn to the most debated topic of recent years, namely probability in the Everett theory. Two developments have dominated this debate: first and foremost the Deutsch-Wallace derivation of the Born rule from decision-theoretic axioms for an Everettian agent, second the question of whether there is a sensible theory of confirmation that could be applicable to Everett and non-Everett theories alike.

We shall discuss some aspects of these questions in further detail below, but here is a brief orientation for the reader. At least prima facie there is a serious problem of how to make sense of probabilities in Everett, since all outcomes of an experiment actually occur. The intuition behind the work of Deutsch and Wallace, described in Chapter 8 (and of Greaves and Myrvold, described in Chapter 9) is that it still makes sense for an Everettian agent to have a preference ordering over bets; this leads to a Savage-style representation theorem in which an agent"s preferences are captured by maximising expected utility, with a measure over branches playing the role of the credence measure. Deutsch and Wallace go even further, providing axioms that constrain this measure to be uniquely given by the Born rule (the squared amplitude of the branches). Thus, while for Savage (and arguably in the context of Greaves and Myrvold"s approach), the probability measure in the representation theorem could be purely subjective, Deutsch and Wallace claim there is something in the physical world that provides a rational guide for our choice of measure in maximising expected utilities. In the terminology of David Lewis"s Principal Principle, such a thing is objective chance.7

Simon Saunders" Chapter 6 discusses the problem of probability precisely as the problem of whether one can identify objective chances in Everett, arguing that this task has been solved in Everett at least as well as in the classical case. The main stress in Saunders" chapter, however, is on the question of whether the measure arising in the representation theorems can be thought of as expressing genuine credences (i.e. degrees of belief in the presence of genuine ignorance), or merely the degree to which the agent cares about the different future branches. 8 Many workers adopt the "caring measure" approach (which fits better with a "fission picture", i.e. with "splitting" agents), while Saunders argues that adopting an "ignorance picture" is both consistent and does a better job at making sense of everyday talk (bearing in mind that if the Everett theory is true, many everyday alternatives are in fact cases of Everettian branching). 9 David Papineau in Chapter 7 takes a slightly different tack, arguing that the situation in the philosophy of classical probability is in such terrible shape that Everett is not so badly off after all. More specifically, he argues that the availability of an ignorance picture does not help even in the classical case (so that Everettians can easily do without it), and that the rationality of letting objective chances (which he reads as propensities) guide credences cannot in principle be shown even in the classical case (so that Everettians need not worry even should the Deutsch-Wallace results not be as relevant as claimed 10 ). The problem of making sense of probabilities is touched upon also in Part V, in Rüdiger Schack"s Chapter 15, and Part VI, both in Max Tegmark"s Chapter 19 and Lev Vaidman"s Chapter 20.

The nitty-gritty of the current Everettian approaches to probability is then spelled out in Chapter 8 by David Wallace, and Chapter 9 by Hilary Greaves and Wayne Myrvold. Wallace gives a fully axiomatised version of the Deutsch-Wallace theorem (with the advantage, in Wallace"s view, of highlighting the locus of disagreement with proponents of serious or even just polemic alternatives to the use of the Born rule in Everett). The final gloss Wallace gives to his derivation is very suggestive: in the Everettian case, symmetry considerations about probability go through to the very end, while in the classical case they are ultimately frustrated by something that breaks the symmetry (and makes only one alternative happen). Greaves and Myrvold discuss their own representation theorem (which allows for more general measures than the Born rule to be used in a many-worlds setting) and their confirmation theory, which applies uniformly to many-worlds and one-world theories. (More below on this issue.)

Criticism is then heaped on the decision-theoretic approaches and on the theory of confirmation in Chapters 10-12 by Kent, Albert and Price. Readers impressed by the beauty and power of the arguments and derivations in the previous chapters will be shaken by the eloquence and persuasiveness of the counterarguments in these further chapters. Kent provides a veritable broadside of counterarguments (some based on ingenious toy examples of many-worlds theories). I think Kent"s arguments against the ignorance picture are very strong, as are the ones arguing that the meaning of the branch weights is crucial to the possibility of their use in confirmation theory. 11 David Albert argues in his usual vivid way that (in the absence of a viable ignorance picture, insightfully discussed in his section 3.2), the Everett theory lacks the power to explain our empirical experience (of the Born rule), regardless of the fact that it may give us a rational guide for behaviour in betting scenarios. 9 I have similar worries with the ignorance picture as with decoherence based on conservation laws: the structure is there in the mathematics, but it does not seem to play any interesting role, at least in terms of the physics. In the case of the ignorance picture, there is an obvious role it can play in the metaphysical picture, ut is that e ough to ake it a eal patte ? Tha ks to Alistai Wilso fo dis ussio of this poi t. The situation is different in the two-vector formalism proposed by Lev Vaidman in Chapter 20 (see below). 10 And they might even be in a slightly better position, since according to Papineau they need not explain why maximising expected utility should be a rational strategy for an agent aiming for gains in a single actual world. 11 Others may be defused more easily: e.g., Kent points out that in a theory without objective branch weights, Greaves-Myrvold observers would end up confirming a different theory about branch weights in each world (all equally wrong). But this is just Greaves and Myrvold s o point about the incoherence of the actual frequentist approach to probabilities in a many-worlds setting.

According to Albert, this vitiates the use of Everettian probabilities in confirmation theory (but see our discussion below). Nor does he believe the Deutsch-Wallace approach makes a compelling case for the rationality of the Born strategy (as he funnily illustrates with his alternative "fatness" measure). Huw Price in the following chapter sketches a very appealing alternative picture of probabilities, namely a pragmatist view (i.e. a subjectivist view). He then goes one step further in the criticism of the Deutsch-Wallace approach: the issue, he argues, is not about agreement or disagreement about single decision-theoretic axioms, but about how to set up the decision-theoretic framework in the first place. In a many-worlds theory, agents (before a branching event) have no reason to limit their preferences over possible actions to ones defined in terms of their preferences for outcomes within the future branches. Such agents have more things to have preferences about (specifically, properties of the global quantum state after branching), and this changes the perspective from which to judge the rationality of decision-theoretic alternatives to the Born rule. The published transcripts highlight both the subtlety and the sensitivity of issues surrounding probability (Itamar Pitowsky on p. 405: "There"s no sense in which you"re unlucky, that"s the point"). I would like to emphasise in particular Myrvold"s take on the Deutsch-Wallace approach (p. 400), namely that, in a specific sense, the Born weights are the only natural ones that can be defined using nothing but Hilbert-space structure (so the theorem is conceptually akin to Gleason"s theorem), but that this falls short of showing the rationality of adopting them even in a many-worlds setting.

Let me add a few remarks on probabilities in Everett. First of all, I think the evidential problem can be reduced to the decision-theoretic one (even in the fission picture), irrespective of whether the decision-theoretic probabilities are genuinely explanatory. Indeed, assume that I believe in the Everett theory, and that the truth of the Everett theory gives me compelling rational reasons to adopt the Born rule as the optimal caring measure for my successors, and I arrange things for my successors accordingly. In worlds in which things do not work out (e.g. I keep losing money), I will start doubting whether the strategy I have been following in caring for my successors was indeed the rationally optimal one. By modus tollens, I shall start doubting the truth of the Everett theory (and might start believing more in the pilot-wave theory with a non-equilibrium distribution). Thus, caring measures (despite the fact that they are not claimed to be explanatory of this or that outcome) can be relevant to updating good old degrees of belief (in the assumed theory), at least if we can uphold the strong connection between truth of the theory and adoption of a specific rational betting strategy as in the Deutsch-Wallace approach.

Especially in view of the criticisms by Kent, Albert and Price, however, maybe the link between the probabilities used in decision theory and in confirmation theory should be severed. In the context of confirmation there is probably scope for using Vaidman"s notion of ignorance (the one of the post-experiment observer who has not yet seen the result). Indeed, in confirmation theory we can focus on whether existing evidence makes a difference to our belief in a theory. This post-experiment in-world perspective could justify the assumptions (or analogous ones) needed for the existing representation theorems, but now applied to confirmation-theoretic probabilities. The decision-theoretic problem could then be treated independently, allowing one to take fully into account the additional possibilities emphasised by Price.

There is a somewhat startling consequence one can draw from the Greaves-Myrvold scheme. Recall that (as we shall see below when discussing Valentini"s chapter), the predictions of the Everett theory and of pilot-wave theory under the assumption of equilibrium are the same, so that from the purely empirical point of view the two theories in this case are indistinguishable. The Greaves-Myrvold framework allows for the use of non-Born (quasi)credence functions while retaining a many-worlds setting. (At least) some such (quasi)credence functions can be generated by taking non-Born distributions and plugging them into de Broglie"s guidance equation. Thus the case of non-equilibrium distributions in pilot-wave theory becomes empirically indistinguishable from (and confirmed or disconfirmed to the same degree as) non-Born many-worlds theories in Greaves and Myrvold"s framework. The upshot is that, if one adopts their framework, the empirical underdetermination between pilot-wave theory and Everettian many-worlds is generalised also to the non-equilibrium case. One could take the resulting epistemological mess as an argument against Greaves-Myrvold. What I believe saves the day, however, are the nonempirical criteria of theory choice that would be used to break the tie. In the strict equilibrium case, Everettians might conceivably have an advantage of "economy" over the pilot-wave theorist (see below). But in the non-equilibrium case, the probabilistic predictions that turn out to be confirmed are not given by the quantum state, and it is explanatorily more powerful to assume that there is additional structure present, namely ensembles of systems in configurations distributed (more or less) according to the observed frequencies. This brings us to the issue of whether the decision-theoretic probabilities in Everett are at all explanatory of observed frequencies in our (initial segment of?) world, which conforms to the Born probabilities to an extremely high degree. (In this sense, our world is not just "like all the others".) I shall take the lead from David Papineau, comparing the situation in Everett with that in the classical deterministic case, but I shall focus on the role of typicality (rather than on analysing probabilities in terms of frequencies or propensities). In the classical deterministic case (think of coin tossing, or of classical statistical mechanics), we start with a subjectivist analysis of probabilities and seek to establish whether there are also objective chances in the sense of the Principal Principle. And it would seem that even the most committed subjectivist is happy to use certain short-cuts in setting their priors: in the case of coin tossing we might want to check whether the weight distribution in the coin seems loaded, in the case of statistical mechanics we have theoretical and dynamical considerations we bring to bear on the problem (temporal invariance, ergodic-type considerations etc.). These are all things that are "out there" in the world (whether as properties of objects or as properties of the dynamics). From the pragmatic standpoint, we are certainly justified in using these short-cuts: they are relatively easy to implement (perhaps adding a conditionalisation on assumed initial entropy levels), and they generally lead to good results (in which case updating on new evidence gives us no reason for substantially modifying our priors). If this justification counts as rational in the sense of the Principal Principle, we are even justified in calling these probabilities objective chances (although this is clearly open to doubt, cf. the problem of induction). This picture translates straightforwardly to the Everett case, at least as long as we take probabilities as guides for our in-world expectations (cf. both Price"s discussion of the pragmatist view of probabilities,12 and his critique of preferences based on in-world rewards in Chapter 12). We make an educated guess at setting our priors according to the Born rule (perhaps guided by considerations of symmetry or naturalness -cf. Myrvold"s take on the Deutsch-Wallace theorem), and as long as it works for us (in our world or initial segment thereof), we have no reason to change our "theory of chance".

But are these "objective" chances (or these pragmatic short-cuts) predictive or explanatory of our continuing or past success in our world? This is where Everettians are most hard-pressed (even independently of the question of whether the Deutsch-Wallace approach provides compelling rational reasons for adopting the Born rule). But they might be able to argue also in this case that Everett is no worse off than a classical determinist. For even in the classical case, no matter what good reasons we might have to adopt a certain probability measure, there will always be initial conditions that lead to deviant relative frequencies. In order to be able to predict or explain "good" long-run behaviour, we have to assume that the detailed initial conditions are not "atypical". One could argue that it would be irrational to adopt a probabilistic model if one were not to assume also that the actual world is typical, but that also seems to be a (merely) pragmatic consideration. If all typicality means is the assumption that the initial conditions are such that they lead to the desired behaviour, then it is hardly explanatory of that behaviour. This is passing a rather harsh judgement on the typicality approach to probability, but Everettians could push that line in arguing that they are actually no worse off than in the classical case. (Indeed, on p. 393 Saunders seems to be using the "t" word precisely in this context: "that"s the language in which the amplitudes are explanatorythey quantify "typicality"".)13 It seems to me that this line of thought, de-emphasising as it does the explanatory role of probabilities, on the whole favours a pragmatist reading of probabilities in Everett.

The subsequent Part V is entitled "Alternatives to Many Worlds" (although the results described by Wojciech Zurek in Chapter 13 are largely compatible with Everett). Chapter 14 by Jeff Bub and Itamar Pitowsky (with a reply by Chris Timpson) and Chapter 15 by Rüdiger Schack (possibly grouped with each other -and together with Zurek"s chapter -because of a common interest in information), are clearly more critical, as is Antony Valentini"s Chapter

(with a reply by Harvey Brown).

Zurek"s chapter is a guide through recent results that have developed out of the research on decoherence pursued since the 1980s in his group at Los Alamos. They are presented as going beyond decoherencewhere Zurek uses the term in the more restrictive sense of loss of coherence of the (reduced) state of a quantum systembut they are all about the consequences of entanglement between a quantum system and its environment, specifically: the phenomenon of environment-induced superselection (or "einselection"), an extremely elegant alternative derivation of the Born rule probabilities (but see also Albert"s comments on pp. 362-363), and the insights of "quantum Darwinism", i.e. the idea that certain einselected states acquire a particular degree of "objective existence" because they can be observed indirectly and without disturbing them via the redundant information about them stored in the environment. At one point Zurek is critical of the usual identification of Everett worlds via decoherence (cf. Wallace"s Chapter 2) because, as he puts it, decoherence requires the notion of tracing over the environment, i.e. of averaging implicitly using the Born rule, so that Everett worlds cannot be defined in this way prior to deriving the Born rule itself. However, I believe this is an artefact of Zurek"s focusing on the restricted sense of decoherence. If one looks at the definition of worlds using the decoherent histories formalism, I would argue that the vanishing of interference terms is at most a formal adequacy condition for the "probabilities" thus defined, and that the task of showing that they indeed play the role of probabilities still lies ahead.

Bub and Pitowsky present the case for viewing the structure of the Hilbert space as providing a probabilistic kinematics embodying certain information-theoretic principles, and they suggest that one can hold fast to a realistic one-world picture, if one gives up on the two "dogmas" that one must not treat measurements as primitive and that the quantum state represents physical reality. They emphasise the analogy with special relativity, in which they claim the construction of a Lorentz-invariant dynamics (bottom-up rather than top-down) is just a consistency check. But what they take to be the analogous step in quantum mechanics, namely the emergence of Booleanity via decoherence, seems to me more analogous to the derivation of the Galilei transformations as a limiting case of the Lorentz transformations. The crucial "consistency check" of providing a dynamical theory that will fit the kinematical framework, and is furthermore empirically adequate, is left out. After all, even Einstein worked for more than a decade after special relativity in order to show that also a theory of gravity (and not just a theory of electromagnetism) could be constructed that was compatible with relativity, and ended up modifying the picture of relativity in the process. While admiring Bub and Pitowsky"s work, I thus broadly agree with Chris Timpson"s spirited reply. 14 Chapter 15 by Rüdiger Schack argues that if knowledge of the universal quantum state is to be gained only through observation (note there may also be theoretical reasons for postulating 14 I a ot su e a out Ita a s ie s o this poi t a d e a sadl o lo ge ask hi , ut Jeff Bu i other writings suggests adopting the p efe ed o se a le app oa h he developed with Rob Clifton. Thus I believe he would agree that the kinematic analysis he provides with Pitowsky is not stand-alone.

a state of a certain form), then Deutsch-Wallace objective chances lose any practical advantage over a fully subjectivist view of the quantum state as a compendium of subjective probabilities. Note that this is a more extreme view of subjectivism than the one I was contemplating above, because Schack sees it as casting doubt on the utility of taking the quantum state as real. Schack"s ontology (as in the picture ostensibly painted in Bub and Pitowsky"s paper), while containing rational obsevers and data, is up for grabs. One could instead contemplate a subjectivist approach to probability within the Everett theory, where the ontology is supplied by the quantum state itself.

We then get to the polemic between pilot-wave theory and Everett theory, with Valentini"s Chapter 16 and Brown"s reply. The target of Valentini"s paper is the claim in the Everettian literature that configurations in pilot-wave theory are epiphenomenal, and that the theory already contains Everettian many worlds (most of them miscategorised as "empty waves").

Valentini has good arguments to bring against this claim and against some of the accompanying rhetoric (which is largely based on historical nonsense). Indeed, pilot-wave theory predicts a whole range of new phenomena that are merely obscured by the prevalent equilibrium conditions (i.e. configurations being currently largely distributed according to the Born probabilities). This observation is underlined by Valentini"s analysis of the theory of measurement in pilot-wave theory, which in principle is completely different from the theory of measurement in quantum theory (measurement is theory-laden!). WhileValentini does not generally deny that there is structure also in the empty waves, he denies that this mathematical structure has physical significance. To use Wallace"s terminology, he denies that these patterns have any predictive or explanatory usefulness, and thus denies them the status of "real patterns". At one point, however, Valentini erroneously denies that such structure is even present: while in the example of WKB states (which occurs even in Hartle"s Chapter 2) he correctly points out that there is no localised wavefunction "stuff" one could identify with emergent classical trajectories, he mistakenly states the same for his most realistic example (the case of the decoherent analysis of chaotic systems). 15 However, neither Valentini nor Brown, it seems to me, entirely manage to see all the subtleties in each other"s position. To state as Brown does that the decision-theoretic approach provides strong arguments for the Born rule regardless of the possibility of nonequilibrium in pilot-wave theory misses the point thatunless one begs the question about their observabilitythe actual configurations in pilot-wave theory break the symmetry assumption so crucial for the Deutsch-Wallace derivation (cf. Wallace"s gloss at the end of Chapter 8). And while in non-equilibrium the case would be clear-cut in favour of pilot-wave theory (as recognised by Wallace, p. 406), I believe the Everettian has a better case to make in the equilibrium case (even if equilibrium is purely contingent) than Valentini admits. Indeed, assume that (as appears to be the case), our evolution has taken place in a (contingent) regime of quantum equilibrium. Then our perceptual and cognitive capacities will not have evolved to exploit the possibilities provided by the sub-quantum measurements analysed by Valentini. 16 Indeed, we will arguably have evolved the same capacities whether we are made of configurations or of wavefunction "stuff", and thus the Everettian can at least argue that there is a profound underdetermination between pilot-wave theory and Everett theory for the case of equilibrium, even if this has just been an approximate and contingent feature of the period on Earth when life evolved. Whether or not the Everett theory has some non-empirical advantages over pilot-wave theory in this case I wish to leave open. What I believe is that ultimately the choice between these different approaches to quantum mechanicsas between collapse and no-collapse approacheswill be empirical (at least given an appropriate theory of confirmation). Equilibrium on all scales and in all regimes is itself fantastically unlikely in pilot-wave theory. Thus, just as an eventual success in predicting non-equilibrium will count as a resounding confirmation of pilot-wave theory, continued failure to find evidence of non-equilibrium will count as an empirical disconfirmation of the theory (cf. p. Peter Byrne"s talk was a highlight of the Oxford conference (and I assume of the Perimeter conference, too) relating the recent discoveries about the genesis and very early reception of Everett"s ideas, in particular the tragic head-on collision between Everett and Bohr and his associates, which Wheeler in a sense brought on while desperately trying to avoid it. David Deutsch discusses some visionary ideas about future advances that might come from further developing Everett"s ideas. Some of his remarks, however, set him apart from the other Oxford Everettians, in his readiness to talk about worlds even beyond the established framework of decoherence (but see p. 605). (This explains also part of his rhetoric against pilot-wave theories.) Max Tegmark reviews different levels on which the concept of multiverse is or could be useful in physics, namely regions beyond the cosmic horizon, postinflation bubbles, Everettian many -worlds, and other mathematical structures. The last of these is arguably rather speculative, and the others can in fact be seen as variants on Everett (because of their origin in quantum fluctuations and spontaneous symmetry breaking). ButTegmark"s point is that several of the issues that arise in the discussion of Everett arise also in other contexts, even were one to consider them in purely classical terms.

Finally, Lev Vaidman reviews Aharonov"s two-vector formalism of quantum mechanics, and proposes that it be used instead in an Everettian context. Vaidman"s main proposal is to use two-vectors as a useful descriptive tool for Everett worlds, but he also wonders about giving them ontological significance, specifically by taking a mixed backwards-evolving state. Note that if the backwards-evolving state has indeed ontological significance, this would provide a physical basis for the ignorance picture of branching favoured by Saunders. 17 This volume is a near-comprehensive guide to the state of the art in research on the Everett interpretation. The only aspect that appears to be underemphasised is the relation between Everett and locality, despite the fact that it is taken to be a selling point of the Everett theory (since it is just unitary quantum mechanics, it can be straightforwardly applied to relativistic versions of the theory). I take it the main options to understanding the relation between Everett and locality are (what I understand is) Myrvold"s position of thinking of branching as a hypersurface-dependent concept and my own suggestion of thinking of branching along future lightcones. The choice between these alternatives is in some aspects analogous to the choice between the ignorance and fission picture of branching, but disanalogous in that nothing of ontological import seems to hinge on it, so that it is arguably largely a matter of convention.

Related to the issue of locality is the one sad omission from the volume, namely discussion of the work of another champion of Everett: Hans-Dieter Zeh, one of the pioneers of decoherence and arguably the earliest researcher who emphasised its diachronic aspect. While worlds are global patterns arising from decoherence, Zeh emphasises the significance of the local patterns that decoherence gives rise to in the brain (or whatever the supervenience basis of mentality might be), thus focussing on the "many-minds" aspect of Everett, rather than the "many-worlds" aspect emphasised throughout the book. While some versions of many-minds theories are now recognised even by their authors as merely conceptually useful toy models (as one might gather from David Albert"s remark on p. 357), I regard the alternative between a global and local perspective to also be largely conventional, so that discussion of many minds in Zeh"s sense is in fact discussion of the functionalist foundations of the modern-day Everett programme.

As a final remark: should one say "Everett interpretation" or "Everett theory"? While other fundamental approaches to quantum mechanics (pilot-wave theories and spontaneous collapse) are clearly distinct theories, the terminology of "Everett theory" adopted by several authors in this book does not seem inappropriate, sincewith the notable exception of Schrödinger -, Everett was arguably the first to suggest that the pure unitary theory of the wavefunction should be taken seriously as a fundamental and universal theory. This volume is a fitting tribute to his genius.

  600) -although to give the theory a fair chance one might have to invest as much into the search for non-equilibrium as in the search for the Higgs boson! The final Part VI of the book contains further contributions by Peter Byrne (Chapter 17), David Deutsch (Chapter 18), Max Tegmark (Chapter 19) and Lev Vaidman (Chapter 20), together with transcripts form discussions relating to Parts V and VI.

De ohe e e is u ial i this espe t hethe o e thi ks of E e ett p i a il i te s of o lds , i.e. sta le st u tu es at a glo al le el, o of i ds , i.e. i te s of the lo al de ohe i g a ia les o hi h i ds arguably supervene (see the further remarks at the end of this review).

It is equally relevant to the opposition between Everett and spontaneous collapse theories, which is not t eated e pli itl i this olu e ut see the e a ks o ollapse theo ies i Maudli s hapte . The e is i fact a line of criticism of collapse theories that the Everettian can adopt (a worrying version of the so-alled tails p o le that is uite a alogous to thei iti al sta e to a ds pilot-wave theories. The opposition between

Wallace in Chapter

among others states that the Deutsch-Wallace approach derives the Principal Principle within the Everett theory. It may be more accurate to state that the principle is accepted as a definition of objective chances and that what is derived is the existence of the latter. 8 Note that this is distinct from the question of what utilities to attach to rewards in different branches, as ade lea fo i sta e i Da id Al e t s discussion in Chapter 11 (p. 361): the question is not about what utility to attach to winning or losing 1$ per se, but about whether and how much it should matter that I win or lose 1$ in various kinds of branches (in particular branches with various squared amplitudes).

As pointed out to me in personal communication by Price, his discussion in Chapter 12 is contrasting pragmatist probabilities and Everettian ones. What I am taking to be the Everettian translation of the pragmatist strategy is to take the Born rule as a guide for our in-world expectations even though the numbers it yields may not be subjective degrees of belief (if one rejects the ignorance picture).

Note that Saunders advocates the ignorance picture of branching, where the analogy is closer. At least one disanalogy, however, remains between the Everett case and the classical deterministic case: in the latter (in the language of the old D-N model of explanation) one can pack more of the explanatory work into the dynamical laws, and push any residual need for explanation on to the initial conditions. The effective laws in Everett are indeterministic, however, so that typicality may be thought to carry a heavier explanatory burden.

Indeed, the point of the analysis by Zurek and co-workers is precisely to establish the existence of localised wavefunction components within the overall approximately Liouville distribution defined by the reduced state the coherence length of the components being much shorter than the overall spread of the state.

At least if there is sufficiently strong mixing behaviour in the full component of the wavefunction, say, of the brain after it has effectively disentangled from the part of the environment being perceived. (Note that even mechanisms evolved under these conditions would be able to recognise statistical deviations from the Born rule when presented with them.)

Vaidman notes that his proposal of an ontological two-state does not fully restore time symmetry, because of the pure-mixed asymmetry of the past and future states. On the other hand, given the patent asymmetry of access we have to the two states (which needs to be better understood), one might consider both states to be mixtures, with us inhabiting (being aware of) only one component of the past state.


Anne-Gaëlle G Bader
email: ag.bader@brgm.fr

S Thibeau

O Vincké

Florence Delprat-Jannaud

S Saysset

G H Joffre

F M Giger

M David

H Gimenez

A Dieulin

Delprat Jannaud

M Gimenez

D Copin

CO 2 Storage Capacity Evaluation in Deep Saline Aquifers for an Industrial Pilot Selection. Methodology and Results of the France Nord Project

Keywords: CO2, Paris Basin, geological storage, storage capacity, saline aquifer

CO2 storage capacity evaluation in deep saline aquifers

for an industrial pilot selection. Methodology and results of the France Nord project

Introduction

The Paris Basin is the largest on-shore French sedimentary basin. It has been identified as a major basin to store CO 2 in saline aquifers. Since the 90's, the CO 2 storage capacity in saline aquifers was estimated through a succession of European projects: first evaluation was performed during the Joule project in 1996 [START_REF] Holloway | The underground disposal of carbon dioxide: final report of Joule 2[END_REF]; a second evaluation was performed during the Gestco project in 2003 ( [START_REF] Larsen | Assessing European potential for geological storage of CO2 -the GESTCO project[END_REF] and [START_REF] Bonijoly | Feasibility of CO2 storage in geothermal reservoirs, example of the Paris Basin -France[END_REF]), which was updated during the EU Geocapacity in 2009 ( [START_REF] Vangkilde-Pedersen | Assessing European capacity for geological storage of carbon dioxidethe EU GeoCapacity project[END_REF], [START_REF]Assessing European Capacity for Geological Storage of Carbon Dioxide[END_REF]). The estimations of the saline aquifer CO 2 storage capacity ranged from 800 Mt up to 27 Gt of CO 2 (a factor of 30 between lowest and highest estimations).

The France Nord project is a Joint Industry Project that has grouped 4 public research institutes (BRGM, IFPEN, INERIS and Eifer) and 7 industrial partners (Total, GDF SUEZ, Storengy, EDF, Air Liquide, Lafarge and Vallourec) from 2008 to 2012. The project included aspects related to CO 2 capture, CO 2 transportation and CO 2 storage in saline aquifers of Paris basin. The current paper is reviewing the main findings concerning the CO 2 storage evaluations.

The first step of the CO 2 storage part of the "France Nord" project was to qualify and select a site for CO 2 storage in saline aquifer in the Paris basin on the basis of available data. This site had to store the industrial emissions concentrated in the northern part of France. Its capacity had thus to be at least 200 Mt of CO 2 in order to store up to 5 Mt of CO 2 per year during 40 years.

The second step was to implement a CCS demonstration pilot of 100 kt of CO 2 , in a target identified during the previous step.

This paper presents the methodology used for the first step (site selection and capacity evaluation) of the project and the main results of this site assessment.

Due to the impossibility to identify a CO 2 storage target with a capacity larger than 200 Mt of CO 2 , the demonstration pilot was not implemented.

Aquifer selection

The aquifers investigated are located in the sedimentary formation of the Paris Basin. These reservoir formations were assessed from geological data and synthesis available at BRGM -the French Geological Survey ( [START_REF] Beccaletto | Overview of the Subsurface Structural Pattern Of The Paris Basin (France): Insights From The Reprocessing And Interpretation of Regional Seismic Lines[END_REF], [START_REF] Debrand-Passard | Synthèse géologique du Bassin de Paris, II, Atlas[END_REF]), published studies ( [START_REF] Guillocheau | Meso-cenozoic geodynamic evolution of the Paris Basin : 3D stratigraphic constraints[END_REF], [START_REF] Maget | Potentiel géothermique « basse température » en France[END_REF], [START_REF] Delmas | Petroleum potential[END_REF]) and additional data provided for the project by the partners (TOTAL, GDF SUEZ and IFPEN) Target formations were selected using the following criteria:

 the reservoir must have geographical, geological and petrophysical properties for storing large volumes of CO 2 safely;  the reservoir has to be deeper than 1000 meters to minimize the volume of the storage;  the aquifer has to be undrinkable and unsuitable for agricultural use due to its salinity (salinity > 10 g/l);  it should not have any risk of interaction with potential third surface or subsurface activity.

The criteria defined above allowed selecting 5 stratigraphic targets (displayed in red in Fig. 1 In order to select one or more sites with at least 200Mt CO 2 storage capacity, these 5 stratigraphic targets were subject to 3 regional models (Buntsandstein, Keuper and Dogger).

An identical approach was applied on each model to identify potential sites and allow comparison.

The methodology can be summarized as follows:

 selection of a common structural scheme (selection of major faults) in the whole Paris Basin (Fig. 3);  geometric modeling of reservoirs from available data (well logs, depth and thickness maps..) at basin scale;  petrophysical filling at basin scale: o from properties maps (porosity, permeability) when available (Buntsandstein, Keuper); o from porosity measured on logs, establishment of porosity-permeability laws and propagation of petrophysical properties in the model (Dogger);  selection of suitable injection sites considering the geological storage criteria (depth, salinity) and the properties of reservoir formation (thickness, porosity, permeability);  definition of the boundary conditions of aquifers (stocked, outfalls) to compute the initial pressure and model the pressure variations during the injection period and after;  refinement of the model in the suitable injection sites by a review of all available wells in order to simulate the injection of 200 Mt of CO 2 with vertical injectors perforated on the total height of the reservoir, 7 inches diameter and an overpressure authorized in the wells of maximum of 50% of the initial pressure;  and finally, sites comparison, taking into account the geological uncertainties, the number of injectors necessary, their spacing, the extension of the pressurized zone and the potential interactions with other users in the study area.

During this phase of the project, five sites with a 200 Mt CO 2 potential storage capacity were identified: Concerning the model of the Dogger formation, matrix porosity and permeability were computed from available log data (sonic, GR NPHI and RHOB on 160 wells), but, due to a complex diagenesis history, a dual porosity/permeability system occurs in the Dogger of the Paris Basin with presence of pathways of very high permeability. Actually, our knowledge of the Dogger does not allow any predictive localization of these drains. This means that we are not able to control the evolution of the CO 2 plume and its migration. Moreover, injection of 200 Mt of CO 2 in the permeable part of the Dogger results in an overpressure around 2 MPa that would impact the geothermal activity.

Because of these uncertainties on the Dogger geological properties and the possibility of an interaction with geothermal energy development within this level, this stratigraphic target was discarded.

Finally, only three sites (located in Keuper and Buntsandstein) were selected for further more detailed evaluation (Fig. 3).  In the Keuper, the southern site (Keuper Sud) is considered as the most appropriate target despite some expected problems due to the number of faults in the area;  The Northern site in Keuper formation (Keuper Nord) is structurally simpler but less efficient petrophysical properties (porosity and permeability) require a large number of injectors;  Injection in the Buntsandstein presents a significant risk of conflict of use with the production of drinkable water from the same aquifer because CO 2 can migrate laterally from the deep salted part of the aquifer to the shallower fresh water production zone.

A more detailed modeling of these 3 sites resulted in more accurate assessments of the storage capacity by:  refining geological and dynamic modeling, especially in the injection areas,  testing several scenarii for commissioning of CO 2 injectors,  investigating the behavior of the CO 2 with time.

Site modeling

The aim of this last step of site selection consisted in the elaboration of new fine models focused on the 3 most favorable sites defined earlier. The objective is to optimize the number of injection wells to reach 200Mt of injected CO 2 in the reservoir over 40 years. This second step of the project performed by Storengy team was conducted in two phases: a refinement of the geological and petrophysical modeling and the dynamic modeling.

In order to evaluate the sensitivity of the results regarding the parameters and the associated uncertainties, different models were computed.

Results for each site are described below.

Keuper Sud

Injection in the site "Keuper Sud" concerns the totality of the triasic formations. From base to top, it can be described as follows:

 Donnemarie formation : sandstones to conglomeratic sandstones with some silty layers  Grès intermédiaires formation: shales and sandstones  Chailly-Chaunoy formation: interbedded sandstones and shales.  Chalain formation: continental deposits consisting of a shaly environment with some unconnected channels (sandstones).  The top of the Trias consisting of the Rhetian marine deposits considered in the model as the caprock.

The geological model of "Keuper Sud" is based on the regional geological model produced by IFPEN in the previous stage. The fine model has to match the regional model in order to benefit from the boundary conditions. Properties of regional model (K, Phi, net thickness, salinity, T, P...) are used as input data; the structural schema has been completed with secondary faults pattern and complementary well data were added for a better definition of surfaces and petrophysical properties. Finally, the grid definition is 1000m x 1000m with 28 stratigraphic layers (to be compared with the regional model 3000m x 3000m x 8 layers, Fig. 4).

The model refinement led to a significant reduction of the net thickness compared with the regional model. This implies a lowering of the estimated storage capacity.

Two different petrophysical models were built:

 the first one includes porosity, from logs measurements, and permeability calculated using a Phi/K relationship.  the second one includes a petrophysical facies modeling, based on a study done by Total [START_REF] Cussey | Keuper du basin de Paris (zone ouest), synthèse sédimentologique : distribution latérale et vertical des depots et des reservoirs[END_REF]. Then, permeability and porosity were modeled for each facies, in order to have a more representative geographical distribution of the properties.

The simulation computed on the first model is considered as an optimistic case because the reservoir is considered with homogeneous layers. The simulation indicates a potential injection of 140 Mt with 15 injection wells (Fig. 4). The Donnemarie sandstones represent a large part of the total storage potential as well as the Chalain formation; however there are very limited geological data on this formation, which implies a high uncertainty on its storage capacity estimation.

The second model based on facies analysis provides a pessimistic case study because the channelized layers are thus not homogeneous and some layers have disappeared during the upscaling process. This model gives a potential of 54 Mt for the same injection design as the first model simulation. The Donnemarie formation concentrates the majority of storage potential. Note that modeling water production did not improve significantly the capacity due to the challenging connectivity of the sandstones.

Keuper Nord

The "Keuper Nord" zone is a priori of less interest than the "Keuper Sud" because:  Chalain formation is almost absent,  Grès intermédiaires formation are present on the half of the zone surface,  Donnemarie sandstones are also almost absent,  The only layer of interest in the zone is the Chailly-Chaunoy formation which shows good permeability (> 100mD).

The geological model of "Keuper Nord" was updated in the same way as "Keuper Sud"; the grid was finetuned (from 3000 m in the regional model to 300 m) and wells and logs analysis provided new petrophysical infill in the site area. As for the model "Keuper Sud", this work induced a significant reduction of the global Net to Gross ratio with a new estimation of the NTG to 0.2. The evaluation of the storage capacity is 40 Mt with 20 wells and its assessment was not carried further.

Buntsandstein

On this zone, the geometry and petrophysics of the reservoir were completed with some additional well data provided by Total. The static model has been locally refined but the properties of the local model are sensibly identical to those of the regional model (Fig 5 .).

Two areas were assessed for injection, located respectively east and west of the Marne Fault (Fig. 3 and5). The potential of the eastern zone has been reassessed to 133 Mt with 4 injectors but the risk of gas migration to the production zone of drinkable water was confirmed. It was thus decided to simulate the behavior of the reservoir further to the west in an area which does not present such risk owing to the Marne fault; however, petrophysical properties, thickness of the reservoir and lateral extension are significantly lower and injection simulations show a potential of only 87.5 Mt with 23 wells with the necessity to produce brine in order to keep a reasonable pressure in the reservoir. The produced brine has to be reinjected east of the fault to form a hydraulic barrier and prevent the migration of the CO 2 (Fig. 5.).

Summary of the storage capacity identified in the France Nord project

The objective of the storage capacity identification task in the France Nord project was, using the same methodology on several site, to optimize the number of wells required to inject 200Mt of CO 2 over 40 years (5Mt/year). Some alternative cases were performed in order to evaluate the uncertainties of our results. All these result are largely under the objectives of the France Nord project. It was thus decided not to go further through the second step of the project which was to implement a CCS demonstration pilot of 100 kt of CO 2 , in one of the 3 targets identified previously.

Comparison of the aquifer storage capacity with previous estimates

A first assessment of the saline aquifer CO 2 storage capacities was performed during the Joule II project (1996).

The capacity was evaluated at 810 Mt of CO 2 , which is a relatively low number. This can be explained by various assumptions  CO 2 can only be stored in structural traps,  only 3% of the aquifer pore volume is considered to be in a structural trap (number derived from an Utsira evaluation),  6% of the water in these traps can be replaced by CO 2 as these traps are considered to be "open", in pressure communication with the rest of the aquifer,  The global storage efficiency is hence reduced, of 0.18%,  Note also that in this study, the potential conflicts with potable water led to removing Buntsandstein from the review.

A second study assessment was performed during the GETSCO project (2003) where a range of capacity was established:  a high case, assuming open aquifer on the full volume of water with a storage coefficient of 6%,  a low case, analogue to the Joule approach where the storage coefficient accounts for structural traps with a value of 0.18% for Buntsandstein and Keuper and a value reduced to 0.01% in Dogger as most traps are considered filled with hydrocarbon reservoirs and are hence considered as unavailable for aquifer storage.

This second assessment leads to a range of capacity from 668 Mt up to 27 Gt of CO 2 .

The GESTCO evaluation for the Paris Basin was updated during the EU Geocapacity report (2009), with two estimates:

 a high case, entitled storage capacity estimation for aquifers, identical to the GESTCO approach,  a low case, entitled conservative capacity estimation for aquifers, where storage capacity is taken at 2% for open aquifers (instead of 6% previously) and 0.1% for closed aquifers (applying to Keuper only).

This third assessment leads to a range between 7.9 Gt to 27 Gt of CO 2 in saline aquifers of the Paris Basin.

Conclusion

The assessment phase of the project has shown that it was not possible to identify a single site to store 200 Mt CO 2 over 40 years on the basis of available data. The best identified site is the "Keuper Sud" site. Its storage capacity ranges between 54 and 140 Mt, with the necessity to drill about 15 injection wells over a large area (about 3000 km 2 ).

This demonstrates that high storage coefficients of 2% or 6% as used during previous GESTCO and EU Capacity projects do not apply to the saline aquifers we have studied. The constraint of the acceptable overpressure is the main obstacle to a massive injection of CO 2 in saline aquifers regardless the boundary condition considered (open aquifer). The injection rates are directly driven by this overpressure phenomenon. Do not take into account this phenomenon leads to an overestimation of storage capacity as shown in comparison with previous capacity estimates.

As shown by Thibeau et al. [START_REF] Thibeau | Using Pressure and Volumetric Approaches to Estimate CO2 Storage Capacity in Deep Saline Aquifers[END_REF], this result is not specific to the area we have considered, and applies to a very large variety of saline aquifers.

Fig. 1 -

 1 Fig. 1-Selection of the stratigraphic target suitable for CO2 storage. Buntsandstein sandstones, Donnemarie, Chaunoy and Boissy sandstones (Keuper) and Dogger limestones. These targets are represented on the Paris Basin log (modified from Guillocheau et al., 2000 [8]).

 2

 2 sites in the Keuper aquifer, one site North of Paris and one site Southeast of the basin;  2 sites in the Dogger, both north of Paris;  1 site in Buntsandstein, at the far east of the Paris Basin (in the Lorraine region). The three regional models have an East -West cumulative extension of 550 km and a North-South extension of 400 km, as illustrated on Fig.2.

Fig. 2 -

 2 Fig. 2-Map view of the models, Easting and Northing is in meters. The color scale shows the depth of the models, with contours every 500 m.

Fig. 3 -

 3 Fig.3-Study area in the northern part of France with the common structural pattern chosen for the project (red lines). The 3 sites selected at the end of the regional modeling are represented by circles.

Fig. 4 -

 4 Fig. 4-"Keuper Sud" model with grid refinement around the injection zone and along the faults. The 15 wells necessary to inject from 54 to 140 Mt are represented with their stratigraphic target.

Fig. 5 -

 5 Fig. 5-Site model of Buntsandstein site with grid refinement. Wells configuration as for the injection west of the Marne Fault ("Faille de la Marne" represented by the red line).

Table 1 .

 1 Summary of the storage capacity identified in the France Nord project

	Site	Potential of	Number of wells	Comments
		injection
	Keuper Nord	40 Mt	20 wells
	Keuper Sud	140 Mt	15 wells	Without facies modeling
				(optimistic case)
	Keuper Sud	54 Mt	15 wells	With facies modeling
				(pessimistic case)
	Buntsandstein	157 Mt	21 wells	Injection east of the Faille de la Marne.
				Risk of CO2 migration towards the drinkable part
				of the aquifer was considered as too important.
	Buntsandstein	87 Mt	23 wells	Injection west of the Faille de la Marne
				Aquifer properties significantly degraded
				compared to the previous case

Table 2

 2 summarizes these assessments and compares them to France Nord results. For France Nord Keuper, both sites (Keuper Nord and Keuper Sud) are stacked.The table clearly reflects that storage coefficients of 2% or 6% do not apply to the Paris Basin as overpressure constraints impede storing such volumes of CO 2 , despite the modeling of open boundary conditions or water production.

Table 2 .

 2 Summary of various CO2 storage capacity assessments of Paris Basin saline aquifers for each estimate (column) the methodology is mentioned on the second line, and for each aquifer the storage efficiency E is reminded

		JOULE II		Projets GESTCO (2003) et		France Nord
		(1996)		EU GEOCAPACITY (2009)		(2009-2011)
		Traps	Traps	Total	Conservative	Flow models
	Dogger	189 Mt	9 Mt	4320 Mt	1440 Mt	Potential Conflict
		(E=0.18%)	(E=0.01%)	(E=6%)	(E=2%)	with geothermal resources
	Keuper	529 Mt	130 Mt	4331 Mt	72 Mt	90-180 Mt
		(E=0.18%)	(E=0.18%)	(E=6%)	(E=0.1%)
	Buntsandstein	Conflict with	529 Mt	17640 Mt	5880 Mt	~ 90 Mt
		fresh water	(E=0.18%)	(E=6%)	(E=2%)
	Other fm.	91 Mt	-	845 Mt	530 Mt	-
	TOTAL	809 Mt	668 Mt	27136 Mt	7922 Mt	180-270 Mt

Acknowledgements

This work has been done within the France Nord project funded by ADEME (French Environment and Energy Management Agency) and industrial partners. The authors are grateful to N Thybaud from ADEME and to all the partners for permission to publish these results. The authors also thank all the technical contributors that have enabled to perform this work, namely: T. Fargetton, D. Dequidt, K. Djaouti, F. Delsante, P. Egermann and R. Nabil from Storengy; J.-P. Gely, and R. de Lannoy from GDF SUEZ, S. Gabalda, C. Kervevan C. Chiaberge and M. Gastine from BRGM, J.C. Lecomte, P. Houel, J.M. Daniel, A. Fornel and F. Roggero from IFPEN and D. Pourtoy, L. de Marliave and G. Moutet from Total. Thank you also to S. Grataloup for her constructive comments on this paper.


C Lagrue

F Azemar

A Besson

S Lamothe

A Lecerf

novel ligature methods For studying sublethal eFFects oF sit-and-wait predators: test using Cordulegaster boltonii (donovan, 1807) larvae (anisoptera: cordulegastridae)

A novel method of labial palp ligature was tested as a substitute for palp ablation for studying sublethal effects of larvae of C. boltonii on prey populations and their consequences for ecosystem functioning. Two alternative types of ligature were designed to test for neutral or aggressive, but non-lethal, predator-prey interaction effects. Ligature efficiency in preventing prey capture was very high and the effects on larval survival and emergence success were negligible. Potential advantages and drawbacks, compared to other methods, are discussed. The results indicate that this fully reversible method should be applied whenever possible, especially for naturally rare or endangered odon. spp.

INTRODUCTION

Predators can have indirect consequences on prey communities that extend beyond direct predator-induced mortality, although very few studies have compared the relative importance of lethal and sublethal effects of predators [START_REF] Lima | Nonlethal effects in the ecology of predator-prey interactions: what are the ecological effects of anti-predator decision-making?[END_REF][START_REF] Preisser | Scared to death? The effects of intimidation and consumption in predator-prey interactions[END_REF]. Thus, investigation of the sublethal consequences of prey responses to predators is essential to fully understand predator impacts on prey populations and ecosystem functioning [START_REF] Peckarsky | Sublethal consequences of stream-dwelling predatory stoneflies on mayfly growth and fecundity[END_REF]AN-HOLT & WERNER, 1998;[START_REF] Lima | Nonlethal effects in the ecology of predator-prey interactions: what are the ecological effects of anti-predator decision-making?[END_REF]. Until recently, very few studies have considered or focused on these "indirect" (i.e. sublethal) predator effects (PECK-ARSKY et al., 2008). For example, sublethal effects of odonate larvae on prey populations are often not measured [START_REF] Johnson | Prey depletion by odonate larvae: combining evidence from multiple field experiments[END_REF], or are assumed to be negligible compared with direct prey mortality through predation and consumption (WOODWARD & HILDREW, 2002a). However, predator avoidance can induce significant sublethal costs for aquatic invertebrates [START_REF] Peckarsky | Sublethal consequences of stream-dwelling predatory stoneflies on mayfly growth and fecundity[END_REF].

Studies considering sublethal effects of odonate larvae have often used caged animals that could not interact physically with their prey (TOUCHON & WARK-ENTIN, 2008;[START_REF] Stamper | The effects of perceived predation risk on pre-and post-metamorphic phenotypes in the common frog[END_REF]. However, by segregating prey and predator, this method prevents physical encounter and accounts mostly for chemosensory interactions, excluding or interfering with other important cues (tactile or visual for example) through which prey can detect predators [START_REF] Barbosa | Ecology of predator-prey interactions[END_REF]. Some experimental results have clearly highlighted the importance of investigating freely interacting predators and prey when studying potential effects of predator-prey physical interactions [START_REF] Hammond | Predator and prey space use: dragonflies and tadpoles in an interactive game[END_REF].

Some studies have assessed the effects of sublethal interactions between larval odonates and their prey by removing the labial palps ( [START_REF] Van Burskirk | Density-dependent cannibalism in larval dragonflies[END_REF][START_REF] Wissinger | Intraguild predation and competition between larval dragonflies: direct and indirect effects on shared prey[END_REF]. These structures are at the distal end of the mentum, the extendable mouthpart used to capture prey. Individuals with the labial palps ablated are unable to capture large prey ( [START_REF] Van Burskirk | Density-dependent cannibalism in larval dragonflies[END_REF][START_REF] Wissinger | Intraguild predation and competition between larval dragonflies: direct and indirect effects on shared prey[END_REF]. Such larvae are thus assumed to interact physically with prey without causing fatality of the latter, thereby allowing sublethal effects to be tested.

However, there are several major issues to labial palp removal. It is ethically questionable and it increases larval mortality significantly ( [START_REF] Van Burskirk | Density-dependent cannibalism in larval dragonflies[END_REF][START_REF] Wissinger | Intraguild predation and competition between larval dragonflies: direct and indirect effects on shared prey[END_REF]. Also, it is irreversible and compromises larval growth (moulting) and emergence into the adult [START_REF] Wissinger | Intraguild predation and competition between larval dragonflies: direct and indirect effects on shared prey[END_REF]. This clearly becomes problematic when considering rare or endangered species. Finally, this method is scientifically debatable as larvae with labial palps removed seem to adjust and switch to smaller prey that they are still able to capture, even without their mandibles ( [START_REF] Van Burskirk | Density-dependent cannibalism in larval dragonflies[END_REF]. This clearly induces a confounding factor when studying the predator's sublethal effects.

Here we propose and describe an alternative method that allows predator-prey physical interactions without prey fatality and thus enables the study of the sublethal effects of odonate larvae on prey populations. This method uses a single ligature to prevent odonate larvae from capturing and killing prey. It is also adjustable to the type of predator-prey interaction under study, either antagonistic (predator aggression toward prey) or neutral (predator presence without aggression).

STUDY SITE AND MODEL SPECIES 1 Animals were collected in the Montagne Noire, southwestern France, a 1,450-km 2 highland region covered by a mixed broad leaf forest. It is drained by a high density of structurally similar first and second-order permanent streams [START_REF] Lecerf | Riparian plant species loss alters trophic dynamics in detritus-based stream ecosystems[END_REF]. Forestry management is the only major anthropogenic disturbance, without however any marked alterations of stream habitats [START_REF] Laitung | Response of aquatic hyphomycete communities to enhanced stream retention in areas impacted by commercial forestry[END_REF]. These habitats are highly favourable to cordulegastrid dragonflies [START_REF] Askew | The dragonflies of Europe[END_REF][START_REF] Ferreras-Romero | The life cycle of Cordulegaster boltonii (Donovan, 1807) (Odonata: Cordulegastridae) in the Sierra Morena Mountains (southern Spain)[END_REF] and contain high densities of larvae of Cordulegaster boltonii (Donovan). Larvae used in our experiments were collected from a single 500 m reach in the Peyreblanque stream (43°25'N, 2°13'E; elevation, 750 m a.s.l.). Animals were captured using hand-nets (0.5 mm mesh size) and dredging the fine sediments of depositional zones where C. boltonii larvae burrow (FERRERAS-ROME- RO & CORBET, 1999). Larvae were brought back to the laboratory and maintained at 14°C in 50 L coolers filled with aerated stream water until required. Two centimetres of sand (1mm < Ø < 2 mm) were added to allow the larvae to burrow.

LIGATURES 1 Ligatures were done using a fine surgical needle and Fig. 1. Pictures of a type I ligature blocking the labial palps in the closed position but allowing full extension of the labium (prementum and submentum): 1 (A) ventral view of the ligature holding the labial palps closed; 1 (B) dorsal view of the same ligature; 1 (C) lateral view of the ligatured odonate larva's head with its labium fully extended and the labial palps maintained closed by the ligature (indicated by the black arrow). 1 [In the pictures, for visibility purposes, white cotton thread was used for the ligature instead of microbraided black nylon filament] microbraided black nylon filament (Ø = 0.1 mm). Head-mounted magnifiers were used for precision. During ligature, the labium was gently extended to avoid wounding the larva's head with the needle; no chemical anaesthetic was needed during the operation. The filament was then passed through the first labial palp by piercing the cuticle with the surgical needle and back through the second palp to form a loop with the filament. The loop was loosely tightened to avoid injuries to the larva but still prevent predation. Finally, ligatures were secured with a surgeon's knot. Two alternative ligature types were used. A first method (ligature type I) was designed to block the mandibles (i.e. labial palps) in the closed position (Fig. 1A and 1B) but allow normal extension of the labium (Fig. 1C). The second ligature method (type II) prevented both extension of the labium and opening of the mandibles; the filament going over the head-labium joint thus preventing prementum-submentum joint extension (Fig. 2A and2B). When completed, both ligature types gave a superficially similar aspect to C. boltonii larvae (Fig. 2C). SURVIVAL AND FEEDING TESTS 1 To test for ligature durability and efficiency in preventing Fig. 2. Pictures of a type II ligature preventing both extension of the labium and opening of the labial palps: (A) dorsal view of the ligature going over the head-labium joint and preventing prementum-submentum joint extension (the black arrow indicates the position of the oesophagus); 1 (B) lateral view of the odonate larva's head with the labium maintained folded (resting position) and the mandibles closed by the ligature going through each labial palp and around the back of the headlabium joint (ligature indicated by black arrows); 1 (C) frontal view of the ligatured larva's head. 1 [Note that both ligature types appear superficially similar when completed. In the pictures, for visibility purposes, white cotton thread was used for the ligature instead of microbraided black nylon filament] larvae from feeding, as well as possible ligature-induced mortality, control and ligatured C. boltonii larvae were kept in enclosures at the study site. For uniformity between treatments, only final instars of C. boltonii larvae with no external signs of metamorphosis were used in ligature tests (FERRE-RAS- [START_REF] Ferreras-Romero | The life cycle of Cordulegaster boltonii (Donovan, 1807) (Odonata: Cordulegastridae) in the Sierra Morena Mountains (southern Spain)[END_REF]. Thirty control (i.e. not ligatured) and 54 ligatured (27 with type I and 27 with type II ligatures) individuals were held individually in small 10 mm plastic mesh cages (10 × 10 × 10 cm). Cages were anchored to the stream substrate and contained 2 cm of sediment to allow burrowing. A third of the cages were retrieved after each of 7, 14 and 25 days to assess for temporal ligature effects on C. boltonii survival. Survival was compared in a pair-wise manner between treatments (control, type I and type II ligature) and treatment durations (7, 14 and 25 days) using Fisher's exact tests.

At the end of the survival test, C. boltonii larvae were returned to the laboratory and ligatures were removed using fine clippers. Larvae were kept individually in 400 mL containers filled with stream water. Animals were checked daily for a week or until faeces were released. Odonate larvae produce compact faeces enclosed in a strong peritrophic membrane, forming a faecal pellet allowing diet examination [START_REF] Lawton | Feeding and food energy assimilation in larvae of the damselfly Pyrrhosoma nymphula (Sulz.) (Odonata: Zygoptera)[END_REF]. These pellets were collected, fixed in 70% ethanol and examined under a compound microscope. Presence of sclerotized animal parts (mandibles or claws for example) in faecal contents indicated that larvae were able to kill and eat prey. Larvae were then offered four amphipods [Gammarus fossarum (Koch); Amphipoda: Gammaridae], which are large and fast moving prey, to test for post-ligature effects on C. boltonii predation abilities. Finally, larvae retrieved after 7 and 14 days were released at the study site. Individuals held in cages for 25 days were kept in the laboratory for emergence tests.

EMERGENCE TEST 1 Twenty-seven larvae held in cages at the study site for 25 days were kept in the laboratory to test for post-ligature effects on C. boltonii emergence success. Ten control and 17 previously ligatured (9 type I and 8 type II; one type II ligatured larva died before the cages were retrieved) individuals were kept in three separate 15 L aquaria (one per treatment group) filled with aerated stream water and 2 cm of sand. Emergence cages (5 mm diameter plastic mesh) were anchored to the substrate in the middle of the aquaria, allowing emergence [START_REF] Purse | Emergence of the damselflies, Coenagrion mercuriale and Ceriagrion tenellum (Odonata: Coenogrionidae), at their northern range margins, in Britain[END_REF]. Larvae were fed ad libidum with live G. fossarum. Aquaria were checked daily for a month for newly emerged individuals (tenerals). Tenerals were retained until their body hardened, enabling flight. Emergence was considered successful for animals released alive and capable of flying. Deaths during emergence or deformities preventing flight were recorded. Furthermore, the labium of each newly emerged larva was checked for ligature-induced problems. Proportions of individuals that successfully emerged (i.e. emergence success rates) were compared in a pair-wise manner between treatments (control, type I or type II ligature) using Fisher's exact tests.

RESULTS

SURVIVAL AND FEEDING TESTS

Two individuals (one in type I and one in type II ligature treatments) lost their ligature during the experiment, but whether they actively removed it or it was not properly secured could not be determined.

Overall, mortality was very low and there was no significant ligature-induced mortality in caged animals; only one ligatured (type II ligature) larva died. Survival rates were not significantly different between treatments: control (100%), type I ligatures (100%) and type II ligatures (96.3%) ligatures (Fisher's exact tests in pair-wise comparisons; all P > 0.05). Similarly, mortality rates of ligatured individuals were not significantly different between treatment durations (Fisher's exact tests; all P > 0.05). Contrary to labial palp ablation (WISSINGER & Mc-GRADY, 1993), ligatures did not cause unsuccessful moulting or metamorphosis. Ligatures induced minimal damage and were effectively fully reversible. Indeed, after ligature removal, all animals resumed feeding and consumed the four Gammarus fossarum within 24 hours regardless of treatment origin (control, type I or type II ligatures).

All but one of the 30 larvae from the control treatment released faecal pellets and had thus been feeding hours before being brought back to the laboratory. Accordingly, all 29 pellets contained sclerotized parts of larval insects (Plecoptera, Ephemeroptera, Trichoptera and Coleoptera) or of the crustacean G. fossarum. In contrast, a third of ligatured larvae (8 type I and 9 type II ligatured individuals) did not released faecal pellets. Faecal pellets released by the other ligatured larvae did not contain animal remains but constituted an empty peritrophic membrane. Ligatured larvae were thus effectively unable to capture and eat their usual prey or even smaller ones. However, the two larvae that had lost their ligatures had indeed resumed feeding and released faecal pellets containing larval insect parts.

EMERGENCE TEST

After a month, two control, two type I and one type II individuals were still alive but yet to attempt emergence and were thus released at the study site. All remaining larvae (8 control and 7 of each ligature type) had made an attempt at emerging and were included in the analyses. Three individuals, one in each category, died during emergence. In all three cases, ecdysis was incomplete and tenerals were found dead, still partly embedded in the exuviae. Additionally, one individual from the control treatment was deformed, having crumpled wings and a bent abdomen; it was clearly not viable and was not considered as successfully emerged. In pair-wise comparisons, emergence success rates were not significantly different between control (75%), type I (87%) and type II (87%) ligatured C. boltonii larvae (Fisher's exact tests; all P > 0.05). Furthermore, the labium of newly emerged individuals did not bear any effects, such as scars, from the ligatures.

DISCUSSION

Prey often appear unable to locate odonate larvae and, following encounter, seem physically too slow to evade the typically rapid dragonfly larva attack (<20 ms; [START_REF] Pritchard | Prey capture by dragonfly larvae (Odonata: Anisoptera)[END_REF]. However, previous experiments showed that up to 40% of C. boltonii attacks were unsuccessful (WOODWARD & HILDREW, 2002b). Odonate larvae can thus interact with their prey without an obligatory fatal output but potentially have sublethal effects (stress or increased emigration rate, for example) on prey populations. Sublethal components of predator effects on prey need to be decoupled from direct prey consumption to fully understand the topdown effects of these predators on aquatic ecosystems.

By permitting predator-prey physical interactions but preventing odonate larvae from actually capturing and eating their prey, ligatures allow testing for predator sublethal effects. This could be done through field and/or laboratory microcosm studies, combining type I (presence with aggression but without prey consumption) and type II (presence without aggression towards prey) ligatures as well as non-ligatured predators and a control treatment without the predator. This novel approach could substantially advance our understanding of the impacts of sit-and-wait invertebrate predators for which sublethal effects have been considered minor so far (WOODWARD & HILDREW, 2002a, 2002b), especially when compared to active predators [START_REF] Peckarsky | Sublethal consequences of stream-dwelling predatory stoneflies on mayfly growth and fecundity[END_REF]PECK-ARSKY & COWAN, 1995;[START_REF] Sih | Emergent impacts of multiple predators on prey[END_REF]. Furthermore, odonate larvae are comparatively large-bodied invertebrates, potentially influencing smaller invertebrate species independently of predation, through competition for space and/ or biotic disturbance, during burrowing phases for example. Overall, type I and type II ligatures should help shed light on the relative importance of predation, sublethal antagonistic interactions (type I ligature) and biotic disturbance (type II ligature) in determining prey distribution.

Compared to labial palp ablation, the method described herein is fully reversible and caused only minor lesions, i.e. a small puncture hole in each labial palp. As a result, ligatures were associated with extremely low mortality in C. boltonii larvae and had no effect on metamorphosis and emergence success. Furthermore, preliminary experiments in laboratory microcosms indicate that ligatured animals still behave as sit-and-wait predators, spending long periods of immobility in a buried position interrupted by short periods of active search for new hunting locations. This suggests that ligatures do not greatly affect animal behaviour, thus confirming the validity of our approach to address mechanistic issues in top-down effects of C. boltonii in stream food webs. We also believe that contrasting type I and type II ligatures will help shed light on the relative importance of sub-lethal predation due to aggressive behaviour (type I ligature) and biotic disturbance to the substrate (type II ligature) in determining prey distribution.

As animals used in ligature experiments can be released, this novel method is ethically less arguable than palp ablation and is absolutely necessary when using endangered or protected species. However, the method presents some shortcomings. Notably, it is not applicable to all odonate species; for example, larvae with hook-shaped labial palps, such as in the Aeshnidae, are unsuitable. Furthermore, there may be a size limitation in the application of our ligature methods, depending on odonate species and instar classes. Preliminary experiments on C. boltonii showed that larvae as small as F-2 instars could be ligatured (see FERRERAS-ROMERO & CORBET, 1999 for instar size classes). In our study, some C. boltonii larvae were also able to remove the ligature. This could probably be prevented by using stronger ligatures. Despite these drawbacks, ligature methods should be considered whenever possible as the preferred alternative to labial palp ablation.

acKnowledgments

  This study was funded by CNRS (INSU "EC2CO MicrobiEn" research action) and ONEMA (riparian-stream linkage research program) research grants to AL.


Mauro Ettorre
email: mauro.ettorre@univ-rennes1.fr

Massimiliano Casaletti
email: massimiliano.casaletti@univ-rennes1.fr

Anthony Grbic
email: agrbic@umich.edu

Power Link Budget for Propagating Bessel Beams

The power link budget for a system that transmits and receives propagating Bessel beams is studied. The transmitter and receiver are separated by a distance D and consist of leaky radial waveguides. Full-wave simulations are used to compute the admittance-matrix representation of the system. The resonances of the coupled transmitter and receiver are then derived using classical network theory. For comparison purposes, a second configuration with its transmitter and receiver connected by a circular waveguide is considered. In contrast to the open system, such a configuration is closed and does not radiate. It is found that within the non-diffractive range of the Bessel beam, both closed and open systems exhibit the same resonances within an error of 0.6%. Calculations show that the power efficiency of the open system can exceed 85% within the non-diffractive range. The proposed system may find application in areas such as wireless power transfer, near-field communication and non-destructive evaluation.

I. INTRODUCTION

In optics, propagating Bessel beams are defined as solutions to the scalar wave equation that remain confined and do not undergo diffractive spreading [START_REF] Durnin | Exact Solutions for nondiffracting Beams. I. The scalar theory[END_REF]. They are of the form: E(ρ, z) = J 0 (k ρ ρ)e -jkz z , where k 2 ρ +k 2 z = k 2 0 . The free-space propagation constant is denoted by k 0 , (ρ, z) are coordinates of the cylindrical system, and J 0 is the zeroth-order Bessel function of the first kind. The spatial spectrum of Bessel beams consists of a single ring in the spectral domain. Therefore, propagating Bessel beams are characterized by their transverse wavenumber:

0 ≤ k ρ ≤ k 0 .
Recently, two papers by the authors [START_REF] Ettorre | Generation of Propagating Bessel Beams using Leaky-Wave Modes[END_REF], [START_REF] Ettorre | Generation of Propagating Bessel Beams using Leaky-Wave Modes: experimental validation[END_REF] have shown that it is possible to generate propagating Bessel beams in the microwave frequency range using planar structures. In particular, propagating Bessel beams are generated using a leaky radial waveguide. The waveguide consists of an impedance sheet above a ground plane, fed by a coaxial probe (see Fig. 1). The resulting structure is simple to fabricate, planar and extremely compact in contrast to earlier structures [START_REF] Monk | The Generation of Bessel beams at millimetre-wave frequencies by use of an axicon[END_REF]- [START_REF] Salem | Microwave Bessel beams generation using guided modes[END_REF].

In the present work, a system consisting of two identical leaky radial waveguides (transmitter and receiver) is considered. The link budget associated with this system, capable of generating and receiving propagating Bessel beams, is found. The transmitter and receiver are axially aligned and separated by a distance D (see Fig. 1) in free space. Full-wave simulations are used to characterize the system in terms of twoport network parameters. This is in contrast to earlier works on Bessel beams which were based on approximate order-ofmagnitude calculations [START_REF] Durnin | Comparison of Bessel and Gaussian beams[END_REF]. For comparison purposes, a second configuration of identical, leaky radial waveguides is also considered. It consists of two leaky radial waveguides directly connected by a circular waveguide (with radial dimension ρ 1 ). The same resonances are found for the two configurations within the non-diffractive range of the generated Bessel beam. The power efficiency of the original system is evaluated using an accurate network approach. Efficiencies of greater than 85% at 10 GHz are possible within the non-diffractive range with a simultaneous conjugate match at receiver and transmitter, and proper Bessel beam tuning. Such high efficiencies are attributed to the focusing behavior and reduced radiation losses of propagating Bessel beams.

II. LEAKY-WAVE RADIAL WAVEGUIDE

The design procedure for the leaky-wave radial waveguides (transmitter and receiver) is provided in [START_REF] Ettorre | Generation of Propagating Bessel Beams using Leaky-Wave Modes[END_REF], [START_REF] Ettorre | Generation of Propagating Bessel Beams using Leaky-Wave Modes: experimental validation[END_REF]. As shown in these references, the zeroth-order Bessel profile is imposed on the z component of the electric field. Due to the symmetry of the structure only transverse magnetic-polarized (TM) Bessel beams (with respect to z in Fig. 1) are considered. The design parameters for the leaky radial waveguide are the frequency of operation f 0 , the transverse complex wavenumber k ρ = β ρ -jα ρ of the desired Bessel beam and the non-diffractive range. The value of the impedance sheet (Z s ), height (h 1 ) and radius (ρ 1 ) of the circular waveguide are then derived (see Fig. 1). In the present case, we assume: f 0 = 10 GHz and k ρ = (0.83 -j0.022)k 0 and the non-diffractive range is z < 2λ 0 . Therefore, it follows that Z s = -25j Ω, h 1 = 1 mm and ρ 1 = 2.96λ 0 = 88.89 mm (λ 0 and k 0 are the free space wavelength and wavenumber at f = 10 GHz, respectively). The behavior of the radial waveguide as a Bessel beam launcher was verified through full-wave simulations [START_REF]Comsol Multiphysics version 4[END_REF]. The E z component of the electric field was derived within the non-diffractive range of the leaky radial waveguide, as shown in Fig. 2. For small distances (z < 1.25λ 0 ), the shape of the Bessel beam profile remains nearly unchanged with distance, as expected for a Bessel beam. Further away from the aperture, the Bessel beam shape is preserved only in the main beam and increased side lobes appear.

III. FREE RESONANCE AND POWER EFFICIENCY OF THE SYSTEM

The link budget for the system shown in Fig. 1 is evaluated. The ports of the system correspond to the coaxial feeds. Fullwave simulations are used to derive the admittance matrix (Y-matrix) of the system for different separation distances D [START_REF]Comsol Multiphysics version 4[END_REF] of transmitter and receiver. It should be noted that the Ymatrix representation takes into account the radiation losses associated with the system. A closed structure consisting of two leaky radial waveguides directly connected by a circular waveguide is also considered for comparison purposes. The circular waveguide is chosen to have the same radial dimension, ρ 1 , as the leaky radial waveguides (see Fig. 1). This ensures that the circular waveguide wall coincides with a zero of the E z Bessel profile generated by the leaky-radial waveguides. Therefore, the circular waveguide should not perturb the propagation of the Bessel beams. It is worth noting that the closed configuration eliminates radiation loss and therefore represents the most efficient power transfer possible between the two Bessel beam launchers. The equivalence between the two configurations is verified by deriving their free resonances. In particular the zeros of the determinant of the Ymatrix are derived for the open structure, whereas a transverse resonance technique is adopted for the closed case. The same resonances are found for the two configurations within an error of 0.6% for distances within the non-diffractive range of the generated Bessel beam. This confirms that that both configurations are equivalent thanks to the accurate generation of Bessel beams by the leaky radial waveguide. To further validate this observation, the power efficiency of the open configuration is computed from the Y-matrix using standard network theory. The power efficiency is defined as the ratio of power delivered to the load to the power available from the source [START_REF] Pozar | Microwave Engineering[END_REF]. A simultaneous conjugate match is imposed on the two ports for maximum power transfer [START_REF] Pozar | Microwave Engineering[END_REF], [START_REF] Ettorre | A transponder-based, nonradiative wireless power transfer[END_REF]. The efficiency can exceed 85% at the operating frequency of 10 GHz, within the non-diffractive range of the Bessel beam. However, proper tuning of the Bessel launcher is needed to take into account the frequency shift associated with the coupling between transmitter and receiver. This high efficiency is due to the focusing behavior and reduced radiation losses of propagating Bessel beams.

IV. CONCLUSION

The power link budget for propagating Bessel beams has been studied. The propagating Bessel beams are generated using leaky radial waveguides. Full-wave simulations and classical network theory are used to evaluate the power efficiency and resonances of the system. It is found that the system is equivalent to a closed system consisting of a circular waveguide connecting two leaky radial waveguides. In addition, an efficiency in excess of 85% is reported within the nondiffractive range of the generated Bessel beam. The proposed system consisting of a Bessel beam transmitter and reciever may find application in areas such as wireless power transfer, near-field communication and non-destructive evaluation. ACKNOWLEDGEMENT M. Ettorre would like to thank the University of Rennes 1, Rennes, France, for its support and financial contribution. A. Grbic would like to acknowledge the support of the Presidential Early Career Award for Scientists and Engineers (FA9550-09-1-0696).

Fig. 1 .

 1 Fig. 1. Schematic of the propagating Bessel beam system. The transmitter and receiver are leaky radial waveguides. The leaky radial waveguide, with a metallic rim at ρ 1 , consists of an impedance surface above a ground plane. The transmitter and receiver are axially aligned and separated by a distance D in free space. The Bessel profile for the Ez component is graphically shown in the figure.

Fig. 2 .

 2 Fig. 2. Normalized Ez component at different distances above the leaky radial waveguide: z = 0.5-2λ 0 .