THE MEANING OF LOVE:

A DIALOGUE BETWEEN VLADIMIR SOLOVYOV AND NIKOLAI FYODOROV

Françoise Lesourd

Over many years, beginning in 1881, Vladimir Solovyov engaged in a running dialogue with Nikolai Fyodorov. Their long discussions in the library of the Rumyantsev Museum where Fyodorov worked are well known, and were immortalized in a drawing by Leonid Pasternak, the father of the Nobel prize-winning poet. According to Nicolai Lossky, Fyodorov was even one of the precursors of Solovyov. While Lossky does not specify the 1 nature of the filiation between the two, it appears in essence to lie in the refusal of any form of thinking that is unconnected to action, and unconcerned with bringing about "what ought to be."

The work by Solovyov that most echoes Fyodorov's thought is The Meaning of Love, and it can even be said to both incorporate and develop the latter's ideas, but also on occasion to challenge them.

It is a work that contests with pungent irony the conventional thinking about carnal love, which-so Solovyov states-is not linked to the continuation of the species. It is simply an objective fact-and he cites a certain number of concrete examples which demonstrate, See Lossky, History of Russian Philosophy, 75-80. 1 paradoxically, that the stronger the amorous passion becomes, the less the multiplication of the species appears to be of importance; it is even, he states, the dissociation of the two that characterizes human love: "it is in man, compared to all the rest of the animal realm, that multiplication is least prevalent, when sexual love attains its fullest meaning and greatest power." 2 For him, the specific characteristic of human love is that it magnifies individuals in their absolute singularity, in what distinguishes them from the species. However, this also brings us back to the ethical postulate inherited from Kant and often restated by Solovyov, particularly in The Justification of the Good, which states that the human being is always an end and must never be considered a means: When I am pitiless or indifferent to others, consider myself at liberty to injure them and do not think it my duty to help them, they appear to me not what they really are. A being appears as merely a thing. 3 The very idea that "individual love" could be an "an instrument of service for the ends of the species" is in contradiction with the idea that man, as a superior being, represents "end 4 of the process of the creation of the world." 5 Solovyov, The Meaning of Love, 21. 2 Solovyov, The Justification of the Good, 71.

3 Solovyov, Ibid., 32. 4 Solovyov, Ibid., 36. While the individual cannot be a means of ensuring the continuation of the species, no more can it fulfill the same function within the framework of the historical process: "individuality possesses an independent significance, and cannot . . . be merely an instrument of ends of the historical process external to it." The rejection of the idea of progress flows, 6 therefore, from the same source as the original conception of carnal love developed above. In this respect, there is a striking convergence with the thinking of Fyodorov, who never ceases to assert with even more pungent irony the absurdity and, more particularly, the immorality of progress, in which the new generations feed upon their predecessors, and then proceed to eliminate them.

Of course-writes the latter-the educated can always argue that in the past the elderly were physically eliminated, whereas today they are merely objects of contempt. Is it not a sign of real progress that physical assassination has been replaced by elimination of the intellect? Progress has truly made great strides, one could say! 7 "From a biological standpoint, progress is when the young devour their elders, when sons oust their fathers; from a psychological standpoint, it replaces love for their fathers by cold superiority and contempt towards them, it is the moral, or rather, the amoral elimination of the fathers by their sons. Solovyov for his part considers that the love of parents for their children, which is commonly presented as a natural extension of the passion of love, is destroyed, rendered meaningless, by the succession of generations, that baleful infinity which he denounces: "the succeeding generation directly and quickly abrogates its predecessors and convicts them of the absurdity of their existence, to be straightaway, in its own turn, convicted of the same absurdity at the hands of its own offspring." 9 Progress, thus described, which is at once absurd and immoral, is opposed by the intimately linked forces of religion and memory in the commemoration of the dead. In The Justification of the Good, in which Solovyov seeks to identity "the fundamental facts of morality" among the three essential feelings that go to make up these fundamentals: shame, pity, and reverence; the last of which concerns everything that we assume to be above ourselves (pietas) and corresponds to the veneration of our fathers, or more simply the love of children for their parents: "every rational being comes to recognize its dependence upon something invisible and unknown." Such is the meaning of the hearth-home: the domestic 10 altar on which a flame is kept burning in memory of the forefathers. For his part, Fyodorov condemns "the transformation of the hearth, the altar to the forefathers, into a culinary instrument." According to him, "when the human race becomes that coming together of all 11 its sons to celebrate the memory of all their fathers, as though they were but one father, then the hearth-home of this family will be the sun itself." 12 Solovyov, The Meaning of Love, 48. 9 Solovyov, Justification, 76. 10 Fyodorov, "Vopros o bratstve," 67. 11 Fyodorov, Ibid., 138. Like love, the ancestor cult stands opposed to the absurd elimination of one generation by another and the triumph of the lie represented by a separate terrestrial existence of the living as much as of the dead. And yet, despite a shared lucidity of gaze on the absurdity of the mere automatic succession of generations, and a shared conception of the origins of religion, there also exists a latent though not always clearly apparent divergence between Solovyov and Fyodorov. It is well known that the great project of the latter was to unite the whole of humanity in a "common task" capable of resuscitating all past generations, "the forefathers" (and actually bring them back to life, through men's concerted action), the ultimate aim being to make all of humanity immortal. In Fyodorov's utopia, all attention is concentrated upon the forefathers. In The Philosophy of the Common Task, he writes: so that the ways of man may meet the ways of God . . . what is important is not the prediction but the joint, fraternal, common action-that bequeathed to us by our forefathers. The business of our time is to turn the hearts of the sons towards their fathers (Gospel of Saint Luke 1:17-this is the business of our time, instead of "turn[ing] the hearts of the fathers to the children," as in the Gospel )-it is to this 13 that God calls us. 14 When Solovyov asserts the need to overcome "the base, rectilineal form of limitless propagation, i.e., the repetition of one and the same organism in the monotonous replacement Here too Fyodorov modifies the text of the Gospel, as he is sometimes wont to do, but in this case quite 13 radically.

Fyodorov, "Vopros o bratstve," 68. of single temporal existences," he appears to be co-opting Fyodorov's central idea. But this 15 is merely an impression, as in reality the exclusively retrospective nature of the latter's conception is foreign to him: "parents cannot be for their children the end of their life, in the same sense that children are for their parents." Complete "reciprocity" of this sort is 16 impossible. In contrast, Fyodorov's "common task" consists precisely in making "fathers" and ancestors the aim of the whole existence of the succeeding generations.

Solovyov's attitude towards conjugal and family life (to which he refers using the profoundly disparaging definition of "everyday triviality", in Russian obydenshchina) stands as a sort of challenge to conventional thinking. This can be interpreted as an echo of his unhappy love for Martynova (in the context of which The Meaning of Love was written), visible in particular in the sentence: "a most powerful love is very frequently unshared," but 17 going beyond this biographical episode, it is clear that his acute sensitivity to everything that could be termed "everyday triviality" irrevocably damned earthly love: "When you look at the facts, instead of the poetry of a central, eternal reuniting, we merely find a superficial, more or less prolonged, but at all events temporary, more or less intimate, but nevertheless exterior, rapprochement of two beings constrained within the narrow limits of the prose of everyday life." 18 Where Solovyov and Fyodorov diverge is over the role and place of sexual love, which the latter ultimately aims to eliminate. Solovyov, on the other hand, considers that the power of love, and more precisely of carnal love-he insists on this point-is what Solovyov, The Meaning of Love, 110.

15 Solovyov, Ibid., 49. 16 Solovyov, Ibid., 25. 17 Solovyov, Ibid., 51. characterizes humanity. "There is only one force that can, in a very real way, undermine from within the very foundations of egoism at its roots, and this is precisely love and essentially sexual love." The love of which he writes is in no way disincarnate. Sexual love alone 19 possesses sufficient force to tear down the barriers of egoism. But this force, instead of serving the continuation of the species, the perpetuation of society and of culture is redirected "within" the human personality, thus contributing to its transfiguration and hence to its progression towards perfection: "The power of this spiritual-corporeal creativeness in human beings is only a transformation or directing inward of that same creative power which in nature, being directed outward, produces the [bad] infinity of physical propagation of organisms." Contrary to Fyodorov's assertions, the mere fact of the "simple abstinence from 20 the act of generation will by no means deliver [the human being] from death" -it is not a 21 first stage towards immortality, or the refusal of the bad infinity embodied in the succession of the generations.

There is a "false spirituality" that eludes the veritable question, that of the transformation within: the other forms of love, by which "a false spirituality and an impotent moralism would desire to replace sexual love," fail to satisfy "the two basic demands without which the resolute abrogation of [ipseity] . . . is an impossibility" -the first of these forms 22 being mystical love, in which there is an excessive disparity between the one that loves and the object of this love, and the second, as we saw above, being the love of parents for their children, which excludes any real reciprocity. Solovyov, Ibid., 45. Solovyov never loses sight of the Platonic ideal, the ultimate meaning of love, which consists in restoring the wholeness of the person, and in realizing the "free unity of the male and female principles, preserving their formal individualization but having surmounted their essential separateness." But here we find the specific terminology used by Fyodorov 23 (division, in Russian rozn'), and the trinitary model which is always present in his writing, the idea of "union without confusion," in which the notion of unitotality finds its source. Its effect is to abolish the limits of an individuality hitherto closed in on itself. It leads to an "abrogation of selfishness," draws mankind out of its "false self-assertion," and once more finds the 24 25 image of God in the loved one, or simply enables the latter to be seen as he or she truly is, by restoring their absolute value in the eyes of the Creator.

As with Fyodorov, Solovyov's ideal is not beyond reach, rather it represents an undertaking proposed to humanity, a task to be accomplished, which will transform humanity from within as he puts it: "but the task of love has never been stated consciously, and therefore never resolved, as it should have been." 26 This approach envisioning the ideal as a task to be accomplished represents the same type of thought, a point noted by Nicolas Lossky, as we mentioned earlier. Solovyov's precise preoccupation is the way things should be, both in The Meaning of Love and also, especially, in The Justification of the Good. In The Meaning of Love, he takes the example of speech, of everything it means for human culture and society, a faculty which is constantly being developed and perfected, and which has accompanied all human evolution: "it would be exceedingly sad to relate to it merely as a natural process . . . as if we speak as birds sing . . . but should never make of language an instrument for the coherent imparting of certain thoughts, a means for the attainment of rational ends we consciously set ourselves." 27 Language is at once a natural process and a product of human culture and man's conscious creative activity.

The same is true of love. It is a human capacity that must be developed; it too, therefore, represents a task to be accomplished, and must not remain "entirely in the obscure regions of tumultuous affections and involuntary attractions," but rather must be the object 28 of conscious transformations. Unlike language, love belongs to the sentiments and emotions;

in this respect, a conception close to Fyodorov's thinking can be observed: the idea that human nature itself is not immutable, that it can be the object of a truly conscious overhauling process.

Indeed, there is an essential concept current at the time that needs to be taken into account in this connection, that of evolution. Even if Fyodorov is fiercely hostile to Darwinism insofar as it further narrows the distance between man and animal, the influence of this theory cannot be overlooked. But the-originally anti-religious-idea of evolution came to be, as it were, inserted into the divine project for the world. There remained the idea that if human nature is capable of evolution, if it has been able to adapt increasingly effectively to its environment, then it is perfectible and-to extend this theory further-can indeed be transformed, through conscious action. Man thus becomes a co-participant in the work of God: "The psycho-physical process of the restoration of the Divine image in material and yet the latter reproached him with not developing his ideas as far as he should have done, and with failing to draw all the consequences from them: the need to involve all humanity in the common task of resuscitating the dead, the outcome of which would not be the end of the world, but rather escape from the Last Judgment. As Fyodorov puts it: "The realization of truth before the end of the world is impossible, says Solovyov, thereby demonstrating that he has devoted little thought to the common task which consists in averting the end, that of the earth and of the whole universe; the end is only inevitable for a nature abandoned to its blindness." 33 Subsequently, as Solovyov anticipated, the religious dimension gradually came to be eliminated from this great work. This voluntarism was to find its true expression after the revolution, in a number of artistic and literary currents, in particular, with all its ambiguity, in work such as that of Andrei Platonov. The fundamental transformation of human nature was 34 also the dream of Proletkult in the 1920s. But we can see that the roots of this dream are to 35 be found in a much earlier period, and Fyodorov's doctrine is an essential pointer to this.

When mention is made of a "common task" which is to lead humanity to perfection and immortality, it should be underscored that it is not a matter of a "superman" but of the human species taken as a whole.

In the area of pre-revolutionary thought, Ideal love, in Solovyov's eyes, possesses this creative potential: it is one form taken by this participation of man in the work of God. Ideal love, he argues, reinstates the image of God in every man, and makes possible the victory over death and the achievement of concrete immortality. In this respect, Solovyov's assertions are somewhat enigmatic: as true love recognizes the loved one's absolute signification in reality, it "really rescues us from the inevitability of death and fills out our life with an absolute content." This is a natural 38 evolution: nature itself tends to free us from the constraints of the species and its uniform perpetuation (to "produce posterity and then to die)." Thanks to all the spiritual potential 39 contained in love, it is possible to override the law of death, according to which there exists a "contradiction and antagonism between the species and the individual." Thus, the idea of 40 evolution extends as far as to include a victory over death-but what exactly does this victory consist of? Solovyov's position is complicated by the meaning he attributes to the term "genuine reality," by which he also means phenomena of a spiritual order.

For humanity to be able to free itself from death is therefore a legitimate historical process; as regards this point, Solovyov entirely adopts Fyodorov's position on immortality:

"If in the biological process nature is striving ever more and more to limit the law of death, then must not humans in the historical process completely abrogate this law ? Immortality, 41 then, is the end and the aim of the historical process: Solovyov appears to fully espouse Fyodorov's great project for immortality, both that of making humanity immortal and that of resuscitating all past generations: "Then every living thing on earth would recover from its requires that "a human being, having attained the highest perfection" would need "to snatch away from Death all its prey." If not, the most acceptable course of action, morally 43 speaking, would be to renounce such immortality.

Solovyov's position reproduces word for word Fyodorov's teachings, reflecting first and foremost his refusal of the idea of progress, with all the victims it implies, and which the men of the sixties were so ready to accept. Мichael Hagemeister reminds us that Sergei Bulgakov, in his article entitled "Apocalypse and Socialism," describes in a similar vein "this repugnant image of descendants partying on the tombs of their ancestors." But what is 44 called into question is as much Dostoevsky's idea of paradise (as expounded by Ivan Karamazov). The traditional representation is unacceptable if paradise is reserved for a certain number-the elect-while part of humanity would be condemned to eternal torment.

But at the same time, Solovyov poses a question that seems at first sight completely absent from Fyodorov: what, precisely, is worthy of immortality? It is here that we see his scathing condemnation of everyday triviality, which appears to be even more relevant today, accompanied by an irony that lends a particular dynamic quality to his presentation: "The incompatibility of immortality with such an existence is plain at the first glance." It goes 45 without saying that the name of Tolstoy, and The Death of Ivan Ilitch must be mentioned here. "For such a life," says Solovyov, "death is not only inevitable, but even extremely desirable.

Is it possible without dreadful anguish even to picture to oneself the existence, prolonged without end, of any society lady, or sportsman, or card-player?" 46

This, then, is Solovyov's first major objection: is it worthwhile resuscitating cannibals? In his letter to Fyodorov, he observers: "resuscitating humanity at the level of the cannibal would be both impossible and absolutely undesirable." Yet, Fyodorov's project, for 47 its part, aims to make humanity immortal in its totality and without exception, and he repeatedly states that salvation should be afforded to all, and that ultimately even Judas was saved, for he repented.

Immortality, for him, supposes that humanity has reached the stage of perfection; it therefore constitutes the end of a progression towards the renewal of humanity. Without perfection, immortality is meaningless. Ultimately, there is no real disagreement between them, and the condemnation of contemporary urban civilization in Fyodorov is even stronger than that of the "card-player" in Solovyov. The process of bringing ancestors back to life and defeating death is considered inseparable from that which leads to perfection.

All this must be the outcome of what Solovyov calls "the spiritual exploit." Moreover it is not unreasonable to suppose that it is specifically Fyodorov that he has in mind here, for the latter's whole existence represented a form of "spiritual exploit" (dukhovnyy podvig). It requires a mental effort to attain immortality: "Let us suppose," writes Solovyov, "that some human has so strengthened his spirit by consistent concentration of will and consciousness, Among the intelligentsia, thinkers and writers, this modest, totally disinterested man was considered an immense authority, and in particular by Vladimir Solovyov. In his letters, the latter refers to him as his "teacher" and "spiritual father." However, there are not many such letters: we only know of two, dating from the mid 1880s.

In the first of these, written shortly after they first made acquaintance, Solovyov states that he accepts "unconditionally without the need for any further conversation" Fyodorov's which is inescapably present even if the latter absolutely refuted the suggestion: not only does man become a co-creator, but he also becomes the equal of God. Whether it was his intention or not, his project was not religious in its principle, but what it leads to is the obsolescence of religion.

Solovyov was aware of certain hidden or unrecognized implications of Fyodorov's project: in particular that paradoxically, despite his negative attitude towards the "learned" as a separate social category (or rather because of it), he places science so high that it came to replace religion. He writes to Fyodorov: "our enterprise must have a religious, not a scientific, character, and must lean on the believing mass and not the reasoning intelligentsia." 58 This is something that Fyodorov repeats incessantly, without accepting the idea that in fact, according to his conception, man takes the place of God. Discovering therein the hidden potential of Fyodorov's theory, Solovyov also underscores the astonishing link between it and certain trends, certain currents of thought, that were to become fully apparent in the twenties, in particular those that revolve around the proponents of the current known as "God-building" (bogostroitelstvo). Solovyov both anticipated and announced these later developments.

The dialogue between the two thinkers was pursued over more than a decade, and there were several facets to it, but what perhaps is most revealing of its nature is the article by forth another equally false view that nature is lifeless matter and a soulless machine." And, 63 in revenge, "offended" nature responds by refusing to feed humanity! Which is why a truly living Christianity, envisaged as a universal task, implies a revision of our collective relationship to nature.

Indeed, 1891 was the year in which a discovery made by the American army made a particular impression on Fedorov: it claimed to have managed to trigger rainfall by firing cannon rounds up into the sky. Thus it was shown that the implements of destruction could be used for the salvation of humanity by regulating nature. The date is particularly relevant as that year nature had demonstrated just how much of a "blind force" hostile to man it could be, and that it needed to be studied, controlled and enlightened by reason. Instead of submitting himself to this blind force often identified with necessity, man needed to learn how to control it: for Fyodorov too, redemption had as much to do with nature as with humanity.

But for him, the redemption and salvation of humanity is played out in the present world, in "genuine reality." He explicitly excludes the idea of transcendence: "The search for meaning is the search for a goal, a cause, a common task. Those who do not accept this must . . . assume an external, transcendental existence (the religious philosophy of history)." 64 Or rather, somewhat astonishingly, according to him the resurrection of the dead must occur in immanence, that is, here on earth, as the outcome of human effort, without supposing the transition to another dimension, or some sort of end of the world: if the Apostle Paul speaks of the "longanimity of God," "does that not mean that the teaching of Christianity can produce an effect, that union may occur, and that in this case the transition to a new heaven, a new Solovyov, Ibid., 71. 63 Fedorov, What was Man Created for? 79.

For both writers, the task that humanity must accomplish vis-à-vis itself is precisely that of attaining perfection in genuine reality; nevertheless there remains an ambiguity, something unsaid. As Fyodorov points out: "On the matter of the common task of resuscitating the dead, not only does [Solovyov] nowhere state his position clearly and openly, but he has also undertaken to conceal his thoughts so carefully that, in the discussion prompted by his presentation, not even the slightest mention was ever made of the common Curiously, despite these fundamental disagreements the two thinkers remained very close, while continually returning to a sort of ever-unresolved misunderstanding. This may perhaps be explained by the extreme importance attached in those years to the human capacity

  task or of resurrection."71 This ambiguity doubtless originates in what Solovyov actually understands by the term "genuine reality": for him, it transcends the limits of empirical existence. This is something he underscores in The Meaning of Love: "The true being of the human being in general and of each human in particular is not exhausted by his given empirical manifestations." In this statement, he highlights a point with which Fyodorov is in total 72 disagreement: "True salvation is regeneration or a new birth; but new birth presupposes the death of the old false life-and no one wants to die." It was precisely this passage through 73 actual death, seen as a necessity for redemption and transfiguration, and for the resurrection of the flesh itself, that Fyodorov refused to accept, and against which he elaborated his grand project.

  has no means to perfect itself by itself, apart from us." Humanity must 29 "consistently carry . . . farther and right to the end what has already been given in germ, in the very nature" of the task, "in the very basis of the process."30 The universal regeneration promised by Christ cannot come about without us, moreover "it depends upon us that Christ should rise from the dead in His humanity," states 31 Solovyov, in this lecture "On the Collapse of the Medieval World-conception": "Christianity, as a divinely-human religion, presupposes divine action, but requires at the same time activity on the part of man." In this respect, his position appears to coincide with that of Fyodorov,

	32

Solovyov, Ibid., 57.

humanity

  ] an intensive spiritual life . . . He occupied a tiny room and slept not more than four or five hours a day on a bare hump-backed trunk, resting his head on some hard object . . . He wanted to possess no property, and never had even a warm overcoat.51

	and so purified his bodily nature by ascetic achievement," that he "attain[s] complete 48
	spiritualization and immortality." Although Solovyov adds the restriction that this 49
	supposition is "something completely fantastical," these terms cannot but remind us of 50
	Fyodorov's extraordinary mode of existence.
	It is well known, from numerous sources, including N. Lossky whom we may quote,
	that Fyodorov
	liv[ed
	Solovyov, Ibid., 67.

  Vladimir Soloviev and the Spiritualization of Matter, 108.mind that it "is not created by us." Solovyov highlights one aspect of Fyodorov's project,

		57
		52
	48	Solovyov, The Meaning of Love, 103.
		Solovyov, Ibid.

49 Solovyov, Ibid. 50 Lossky, History of Russian Philosophy, 75. 51 Solovyov to Fedorov, 1881. Quoted from Smith,

  Fyodorov entitled "A Project for the Union of the Churches," published in answer to the

59

Solovyov, The Meaning of Love, 105. 57

Solovyov to Fyodorov, mid 1880s (Solovyov, Pis'ma, 347)

. 58

Fyodorov, "Proekt soedinenija cerkvej," 370-87.

 Solovyov, Ibid., 117.

 Solovyov, Ibid., 73.

 Solovyov, Ibid., 47.

 Solovyov, Ibid.

Solovyov to Fyodorov, mid 1880s (Solovyov, Pis'ma, 346).

evils and maladies, would be free and immortal. But in order that they may be at the same time happy, one more condition is necessary: they must forget their parents, forget the actual authors of this new prosperity." This is what is unthinkable and morally unacceptable. Logic great project of resuscitating the dead and making humanity immortal, and that "since the appearance of Christianity" this project is "the first advance of the human mind along the way of Christ." 53 Yet, in The Meaning of Love, written at least ten years after these letters, Solovyov indirectly questions the type of perfection towards which humanity should aim, as described in Fyodorov's project. For the latter, human perfection is first of all linked to knowledge: mankind will need to possess such a vast wealth of knowledge that it allows him, in effective reality, to resuscitate the dead fathers. But, Solovyov objects, "omniscience" is incompatible with human liberty: it is "our ignorance about the interconnecting bond of concrete particulars in the unity of the whole" that "leaves us, notwithstanding, freedom of action, which, with all its consequences, has already entered from all eternity into the absolute and all-embracing design." It is man's 54 ignorance that guarantees his freedom of choice. Moreover, as was pointed out above, it is difficult not to see a Promethean temptation in Fyodorov's doctrine, even though Fyodorov himself completely rejected this possibility. According to Solovyov, in Fyodorov's future "psychocracy," humanity "put[s] a screen in front of divinity," and from this perspective, 55 "God would not really be visible behind mankind." 56 The progression of humanity towards immortality, the creative development of universal life, demands the active participation of man, and yet it must always be borne in Solovyov to Fedorov, mid-1880s (Solovyov, Pis'ma, 345). 53 Solovyov, The Meaning of Love, 105. 54 Solovyov to Fyodorov, mid 1880s (Solovyov, Pis'ma, 347). 55 Solovyov to Fyodorov, mid 1880s, Ibid. lecture given by Solovyov in 1891 entitled "On the decline of the medieval world-view." 60 The date is significant because 1891 was the year of a great famine caused by an exceptional drought, which claimed many victims, especially in regions far from the capitals. The date explains why this great undertaking aimed at ensuring the salvation of humanity foregrounds the relationship to nature, which is indeed of concern to all (and which is why this thinking is particularly relevant to today).

On becoming an official religion, Christianity took on certain aspects of paganism, writes Solovyov. It attached too much importance to the external manifestations of faith, that is, to ritual. Some who were aware of a major deformation of Christianity-which they no longer saw as a work of transfiguration, but one that had spread its efforts too widely in "works" aimed at ensuring salvation-withdrew from society. In a sentence that might be reminiscent of Fyodorov, Solovyov speaks of "the contrast between the paganism of the city and the Christianity of the desert." But the latter form of Christianity, through its "pseudo-61 Christian" individualism and its "false spiritualism," changed the relation with the world. Nature came to be considered an "evil principle," whereas it is not of itself evil, but only "passive and inert." Paradoxically, it is the progress of Modern Times, through the 62 smoothing of human relations, that has come closer to the Christian ideal, whereas it claimed to reject it. But the relationship with nature has not changed, far from it: "Against the pseudo-Christian spiritualism which regards nature as an evil principle, they [the modernists] put earth, will take place non-catastrophically, without judgment, that it will be the work of humanity itself, having become the docile instrument of the divine will . . .?" 65 He contrasts this form of resurrection, which he calls transcendent resurrection, with the evil resurrection, which he also calls "the resurrection of anger" (the anger of God at the inaction of men indifferent to His will), which will herald in hell and eternal torment, the necessary corollary to a paradise reserved for the chosen few: if "The human race . . . fails to unite in the task of learning about the blind force, but instead submits to it . . . [it] can only expect a transcendent resurrection, brought about not by ourselves, but from without, not through the action of our own will, but even against it, a resurrection of anger, of the last judgment, of the condemnation of some (the sinners) to eternal torments, and others (the just) to contemplation of the torments of the former." 66 The Last Judgment then assumes the full meaning of its Russian designation, the "terrifying judgment," as this resurrection would then be accomplished not by us, but in spite of us, and would be the punishment reserved for humanity if it fails to unite in a common undertaking aimed at universal salvation.

Fyodorov's relation to the concept of transcendence is somewhat obscure, ranging from pure and simple acceptance to a refusal to take into consideration domains that are in any case outside the human sphere, just as, in the Middle Ages, everything in the field of knowledge that manifested "vain curiosity" was condemned: "we set a limit to vain human Fyodorov, "Proekt soedineniya tserkvey," 372. 65 Fyodorov, "Supramoralizm ili vseobshchiy sintez," 402. curiosity, directed towards the transcendent, towards thought without action," extending as 67 far as the great project of a world and life without end or interruption.

Although the strongly retrospective character of Fyodorov's thinking is not to be found in Solovyov's presentation, their common ideal was to restore the world to its supposed pristine purity. In his book Solovyov and gnosticism, Alexei Kozyrev describes the links between Solovyov's thinking and Origen's conception of apocatastasis. Therein lies the 68 meaning of love: it is the force that not only restores the divine image in the loved one but also restores the world to its original purity and the human being to its masculine and feminine bi-unity. But Solovyov never ceases to remind us that perfection "is not of this world," moreover, in a letter to Fyodorov, he underscores that "the mere physical resurrection of the dead itself cannot be the goal in and of itself." 69 It is this that arouses Fyodorov's indignation, as he notes in quite violent terms after this presentation: "one might wonder whether Solovyov was actually aware of the meaning of what he was saying; whether, for instance, he understood that salvation, like redemption, is only the abstract form of the task of resuscitating all the dead, or whether he realized that extending salvation 'to the full circle of the life of man and the century' and, equally, 'beginning to realize salvation in our reality' means transforming the force of nature, which is the bearer of death, into a life-bearing force, in order to resuscitate the dead?" 70 Fyodorov, "Vopros o bratstve," 46.


Emmanuel Dormy
email: dormy@dma.ens.fr

Ludivine Oruba

Ludovic Petitdemange

Three Branches of Dynamo Action

Keywords: Dynamo action, Magnetostrophic balance, Dynamo bifurcation

published or not. The documents may come

Introduction

The Earth's magnetic field is sustained by self-exciting dynamo action in the liquid core of our planet. Part of the kinetic energy of the flow is transferred to magnetic energy. In fact, in the Earth's core, most of the energy is anticipated to be dissipated by electrical currents. We describe here the existing numerical models and show how they appear to fall into three distinct branches characteristic of different forces balances.

Since the first full numerical models of dynamo action (Zhang & Busse 1988, Zhang & Busse 1989[START_REF] Glatzmaier | [END_REF], many parameter space surveys have been performed. This has allowed to produce phase-diagrams which, depending on the controlling parameters, describe which dynamo states can be achieved. Wide parameters surveys (e.g. Christensen & Aubert 2006, Schrinner et al. 2012) clearly identified two branches of dynamo action. The first one is characterised by a dominant axial dipole, while the second one, at larger forcing, is largely multipolar, with a fluctuating dipolar component. It was shown (Simitev & Busse 2009) that the transition between these two dynamo states can be hysteretic if stress-free boundary conditions were considered.

Sadly, upon closer investigation none of these two states turned out to be relevant for the Geodynamo. The first state, characterised by a dominant axial dipole, was shown to be largely controlled by viscous effects (King & Buffett 2013, Oruba & Dormy 2014a), and the second one to involve significant inertial effects (Christensen & Aubert 2006, Schrinner et al. 2012, Oruba & Dormy 2014b).

The Earth rotates with one revolution per day, given the viscosity of liquid iron at these temperature and pressure (see de Wijs et al. 1998), we can easily conclude that viscous effects will only become relevant at very small scales. A naive dimensional analysis approach yields 2 ν ∼ ν/Ω. Asymptotic developments (e.g. [START_REF] Dormy | Mathematical Aspects of Natural Dynamos[END_REF]) reveal more elaborate scalings of the form α ν ∼ ν/Ω L α-2 (where L denotes the typical size of the Earth's core). The first relevant length scale for a vertical shear, for example, corresponds to α = 3 . For geophysically relevant estimates, these length scales would be of a few meters, less than a kilometer. Length scales which lie below the resolution of current numerical models.

Typical velocities in the Earth's core can be inferred from the secular variation of the magnetic field [START_REF] Holme | Treatise on Geophysics[END_REF], this yields U 10 -4 m s -1 . There again, dimensional analysis reveals the length scale at which inertial effects will be comparable to the effects of global rotation (the so-called Rossby radius), I ∼ U/Ω, again of the order of a few meters. So that inertial effects are not expected to play a significant role, on the time scale of secular variation, at the large spatial scale.

The relevant balance for the Earth's core is therefore one in which both viscous effects and inertial effects are negligible on the large scales. This is known as the magnetostrophic balance (see [START_REF] Moffatt | Magnetic Field Generation in Electrically Conducting Fluids[END_REF]. The issue of whether the limit system of equations (i.e. omitting both the viscous term and the inertial term in the governing equations) is well posed is a challenging one. First put forward by Taylor in 1963(Taylor 1963), this system proved extremely difficult to solve numerically. The well-posedness of this limit raises complicated mathematical issues [START_REF] Gallagher | INDAM Series[END_REF]. A first set of solutions in the case of an axi-symmetric configuration has however recently been achieved (Roberts & Wu 2014, Wu & Roberts 2015).

We here take the simpler point of view of retaining all the terms in the equations, but vary the parameters, so as to try and approach a magnetostrophic equilibrium. The crucial issue in doing so, is to assess that inertial and viscous effects are indeed small in the realised solution. To that end, a useful tool is to study the bifurcation diagram for various choices of the parameters, the different branches on this diagram naturally corresponding to different forces balances. The bifurcation diagram for dynamo action in the limit relevant to the Earth's core has been the focus of many analytical or mixed analytical-numerical studies [START_REF] Eltayeb | [END_REF], Childress & Soward 1972, Soward 1974, Fautrelle & Childress 1982, Roberts 1988). The main result is that the bifurcation diagram should consist of two branches. The first branch necessarily involves significant viscous effects and is referred to as the "weak-field" branch. On this branch, viscous effects are necessary to allow deviation from the Proudman-Taylor constraint (i.e. the tendency for the flow in a rapidly rotating reference frame to be independent on the coordinate in the direction of the axis of rotation). The flow will thus develop short length scales in the directions orthogonal to the axis of rotation. The most obvious of these length scales involving an E 1/3 dependence (see [START_REF] Dormy | Mathematical Aspects of Natural Dynamos[END_REF], for example). As the strength of convection increases, both the flow and the field gain in amplitude. A transition, characterised by a turning point, is anticipated when the Lorentz force becomes large enough. When this turning point is reached, the weak-field solution becomes unstable, and the magnetic field experiences a runaway amplification. Saturation will be achieved when the field reaches a strength sufficient for the Lorentz force to be comparable with the Coriolis force. This second branch is referred to as the "strong-field" branch. On this branch the amplitude of the Lorentz force is comparable to that of the Coriolis force.

The above description was so far disconnected from direct numerical simulations of spherical dynamos. Recently, however, Dormy (2016) pointed out the existence of a third dynamo state, numerically achievable at the cost of an under-estimated magnetic diffusivity. This regime appears to approach the relevant magnetostrophic force balance. This strong-dipolar dynamo state, described numerically, is characterised by an hysteresis with respect to the viscous-state. The transition occurs at a turning point, which is characterised by a runaway field growth. This bifurcation sequence establishes a first connection between direct numerical models and earlier asymptotic developments.

In this article, we will rapidly review the available results on dynamo states available from numerical simulations. We further investigate the numerical strong-dipolar (SD) branch, describe its relation with the earlier theoretical bifurcation sequence, and ponder on the relevance of the strong-dipolar state to the Geodynamo.

Governing equations

Let us start by introducing the standard mathematical model for the Geodynamo. The numerical simulations discussed in this paper are restricted Boussinesq models. The computational domain consists of a spherical shell with aspect ratio r i /r o = 0.35 . The flow is thermally driven, and a fixed difference of temperature is imposed between the inner and outer boundaries. It should be noted, that in the Earth's core, buoyancy effects are associated with both thermal and compositional effects. The simplest form of governing equations is however similar in both cases (see Braginsky & Roberts 1995, for a detailed discussion). All the simulations used in this work rely on no-slip mechanical boundary conditions as well as an insulating outer domain. The inner core is insulating in most simulations, and a few simulations involve a conducting inner core with the same conductivity as the fluid.

The governing equations in the rotating frame of reference can then be written in their non-dimensional form -using L = r o -r i as unit of length, L 2 /η as unit of time, ∆T as unit of temperature, and (ρµηΩ) 1/2 as unit for the magnetic field -as

E η [∂ t u + (u • ∇)u] = -∇π + E ∆u -2e z ×u + Ra q T r + (∇ × B)×B , (1) ∂ t B = ∇ × (u × B) + ∆B , ∂ t T + (u • ∇)T = q ∆T , (2)
with

∇ • u = ∇ • B = 0 . (3)
System (1-3) involves four independent non-dimensional parameters, which are the Ekman number E = ν/(ΩL 2 ) , the magnetic Ekman number E η = η/(ΩL 2 ) , the Roberts number q = κ/η , and the modified Rayleigh number Ra = αg∆T L/(κΩ) , in which ν is the kinematic viscosity of the fluid, α the coefficient of thermal expansion, g is the gravity at the outer bounding sphere (the gravity profile is linear in radius), κ its thermal diffusivity, and η its magnetic diffusivity. The modified Rayleigh number Ra , as defined above, differs from its most classical definition αg∆T L 3 /(νκ) , to which it is related via an Ekman factor. Whereas the later is the relevant parameter to measure energy input in the standard Rayleigh-Bénard setup, it is not any longer relevant in the magnetostrophic limit. The above four independent parameters are enough to fully define the system. We shall now consider the relevant values of these parameters for the Earth's core. The orders of magnitude of the dimensional coefficients outlined in the introduction reveal E 10 -15 , E η 10 -9 , q 10 -5 .

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The last non-dimensional parameter, Ra, controlling the strength of thermal convection is difficult to quantify in a Boussinesq formalism. While the heat gradient across the core is of the order of 10 3 K, most of the heat in the actual core is carried along the adiabat. Only the super-adiabatic gradient is relevant in the Boussinesq framework. This deviation is only of the order of 10 -3 K (see Gubbins 2001, Jones 2011), so difficult to estimate with great precision. This results in the following estimate for our definition of the Rayleigh number Ra 10 13 (Gubbins 2001). Obviously this value should be large enough, so that, even though q is a vanishing number, the product Ra = Ra q remains of order unity. The above estimate yields Ra 10 8 . The non-dimensional form chosen in (1-3), often referred to as the "strong-field scaling", highlights the primary magnetostrophic balance (order one terms) and the three vanishing parameters (4). The Taylor state introduced above (Taylor 1963) amounts to dropping all small terms in (1-3). The only parameter left to control this limit system is then the modified Rayleigh number Ra . So that the strength of the magnetic field in this limit should depend on this sole parameter.

It is worth pondering on the ratios of the small terms in (1-3). The ratio of the Ekman number to the magnetic Ekman number defines the magnetic Prandtl number Pm = E/E η , this defines the ratio of two small parameters (vanishing in the geophysically relevant magnetostrophic limit), but as we shall see controlling this ratio in the limiting process is essential. The ratio of the magnetic Prandtl number to the Roberts number defines the classical hydrodynamic Prandtl number, Pr = Pm/q. There again, both Pm and q are small numbers, but their ratio remains an important quantity.

The importance of the ratio of small parameters in a double (or even tripple in our case) limit, will not come as a surprise to the mathematical community. Besides, its importance has recently been stressed in physical applications when considering the saturation properties of MHD turbulence (Fromang & Papaloizou 2007, Fromang et al. 2007).

Our analysis is tested against a wide numerical database corresponding to some 300 direct numerical simulations. The data sample is composed of 180 runs kindly provided by U. Christensen, and of additional runs, either previously reported in Morin & Dormy (2009), Schrinner et al. (2012) and Dormy (2016), or presented in Table 1.

The Weak-Dipolar dynamo state

The first and most documented dynamo state is for obvious reason the dipolar state. Dynamos in this state have been reported since the very early days of dynamo modelling.

Owing to their dipolar nature, these models have even often been argued to be relevant to the Geodynamo. It is now evident that viscous effects are present at leading order in the forces balances.

This branch is the first dynamo mode produced as the modified Rayleigh Ra is increased away from its critical value for the onset of convection Ra c (which can itself be subcritical (see Guervilly et al. 2015)). The onset of dynamo action, or dynamo bifurcation, has been carefully investigated in Morin & Dormy (2009). Depending on the parameters being considered, they reported either super-critical, sub-critical or isola branches for the onset of dynamo action (see figure 1). It is important to stress that each point on these numerical bifurcation diagrams corresponds to a time averaged fully three-dimensional simulation.

The Ekman number was varied between 10 -3 and 10 -4 , and the bifurcation type over this range appears to depend only on the parameter E η . Super-critical bifurcations were obtained for both E = 3 × 10 -4 and E = 10 -4 , with E η < 5 × 10 -5 ; Sub-critical bifurcations for both E = 3 × 10 -4 and E = 10 -4 , with 5 × 10 -5 < E η < 2 × 10 -4 ; and Isola were obtained for both E = 10 -3 and E = 3 × 10 -4 with E η > 2 × 10 -4 . These simulations were all performed at fixed Pr = E E -1 η q -1 = 1 . The physical explanation for this change of behaviour has not been achieved so far, but the ordering highlighted above points to the importance of inertial effects in controlling the nature of the transition. §

The importance of the flow helicity on the dynamo generation mechanism for this dynamo state has been highlighted by Olson et al. (1999). Besides, Sreenivasan & Jones (2011) argue that kinematic helicity enhancement by the magnetic field could provide a mechanism for the occurrence of sub-critical dynamo branches.

The importance of viscous forces in this dynamo branch has long been overlooked. It was however recently pointed out (King & Buffett 2013) that the typical length scale of the flow exhibits a clear E 1/3 dependence, characteristic as explained above of the viscous-Coriolis dominant force balance.

We present in figure 2 three different length scales. The first one, upeak , is defined by considering the time averaged kinetic energy spectrum. It is defined as upeak = π/l peak where l peak corresponds to the spherical harmonic degree for the peak of the energy spectrum. The second length scale uCA06 corresponds to the length scale defined in Christensen & Aubert (2006) and used in King & Buffett (2013), it is defined as uCA06 = π/l CA06 where l CA06 corresponds to the mean value of the spherical harmonics degree in the time-averaged kinetic energy spectrum (see Christensen & Aubert 2006, equation (27)). The third length scale uvort , introduced by Oruba & Dormy (2014a), is defined as

u 2 vort = u 2 (∇ × u) 2 , (5)
where • denotes time and volume averaged quantities. Figure 2 highlights that the three typical length scales defined above follow an Ekman dependence characteristic of the viscous, E 1/3 , scaling.

Because viscous forces are important in this branch, and to attempt a link with the earlier asymptotic studies listed above, we will in the sequel refer to this branch as the weak-dipolar (WD) branch. Of course this branch is saturated, and non-linear effects are affecting the flow, both via the non-linear inertial term and via the Lorentz force.

From Weak-Dipolar dynamos to Fluctuating-Multipolar dynamos

As the forcing is increased, i.e. as the modified Rayleigh Ra is further increased away from Ra c , a transition to a fluctuating-multipolar (FM) dynamo state has been initially reported by Kutzner & Christensen (2002) and described in further details in Christensen & Aubert (2006). This transition was very early associated with the strength of inertial effects. Indeed Christensen & Aubert (2006) pointed out that the transition was controlled by the "local" Rossby number Ro = U (Ω u CA06 ) -1 based on the mean velocity length scale uCA06 , defined above (the denotes dimensional quantities). This clearly indicates that inertial effects become significant at the flow length scale when the weak-dipolar mode is lost. More recently, Oruba & Dormy (2014b) showed that rather than measuring the typical length scale of the realised flow, one could account for the transition with the parameter Ro E -1/3 , where Ro = U (ΩL) -1 , because of the above mentioned dependence of the flow length scale as E 1/3 L in the weak-dipolar branch.

The typical length scales are presented as a function of the Ekman number on figure 3. While the length scales uvort and uCA06 are by construction affected by the viscous dissipation length scale, the length scale upeak (corresponding to the energy spectrum peak) exhibits a much weaker dependence on the Ekman number (we stress however the large dispersion due to the small data sample and the difficulty to precisely estimate l peak , which is a small integer). This appears as a signature of inertial effects at the dominant scale of the flow. A way to further assess the importance of viscosity is to consider the fraction of energy being dissipated by viscous forces f ν . The ratio of the energy being dissipated by viscous forces to the total energy dissipation (viscous and ohmic)

f ν = E (∇ × u) 2 E (∇ × u) 2 + (∇ × B) 2 = ρν (∇ × u ) 2 ρν (∇ × u ) 2 + ηµ -1 (∇ × B ) 2 (6)
is reported on figure 4 as a function of the Rossby number based on the viscous scale (Oruba & Dormy 2014b). The relevance of this parameter Ro E -1/3 to distinguish weakdipolar and fluctuating-multipolar dynamos is evident. We should stress that Schrinner (2013) investigated the behaviour of f ν (actually f ohm = 1 -f ν ) and concluded that f ν was increasing with the local Rossby number, for which Ro E -1/3 offers a very good proxy in the weak-dipolar state.

The fraction of viscously dissipated energy is in general significant in both dynamo states (though some end members models of the weak-dipolar state, reach 10% of viscous dissipation). It is worth noting that this fraction is on average larger in the multipolar regime, for which the field is weaker and the ohmic dissipation is thus lower.

The transition from the weak-dipolar to the fluctuating-multipolar state is illustrated in figure 5. Here again the simulations were performed at fixed Pr = E E -1 η q -1 = 1 . The two branches are identified with different symbols. These are easily identified by measuring the strength of the dipolar component relative to the total field intensity. The dipolar state is indicated with circles, whereas the fluctuating-multipolar state is indicated with grey squares. In these graphs the magnetic energy density of the non-dimensional magnetic field is reported: Λ = B 2 /2 , this corresponds to the Elsasser number. In terms of dimensional variables, this amounts to Λ = B 2 / (2Ωρµη) .

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a. . Weak-dipolar (circles) and fluctuating-multipolar (grey squares) branches with no-slip boundary conditions for (a) E = 3 × 10 -5 , E η = 1.2 × 10 -4 , q = 0.25 , (b) E = 10 -4 , E η = 10 -4 , q = 1 , and (c) E = 3 × 10 -4 , E η = 10 -4 , q = 3 .

Figures 5a and 5b highlight the discontinuity in the dynamo branches. Figure 5c shows that for some parameters, the efficiency of the viscous dynamo state starts to decrease before the transition occurs. This highlights that although the primary force balance on this branch involves the viscous term, non-linear effects are important, in saturating the field growth, but also in modifying the flow (which, in this case, provides a lower saturation level at larger forcing).

Non-linear effects in a rotating flow are known to drive zonal flows through the Reynolds stress. Such zonal flows are very weakly damped. If their radial structure . Weak-dipolar (circles) and fluctuating-multipolar (grey squares) branches for E = 10 -4 , E η = 10 -4 , q = 1 , in the case of stress-free boundary conditions. is larger than E 1/4 L they are dominated by boundary layers dissipation [START_REF] Morin | [END_REF]. It results that the large-scale zonal flows will behave in a different manner in the case of stress-free boundary conditions (for which only the bulk viscous effects will be relevant). This will, of course be particularly true when the Ekman number is moderately small (as is the case in most numerical models). As the Ekman number decreases, the correction due to the boundary layer dissipation (via Ekman pumping) will become less and less important. For stress-free boundary conditions the possible bistability between the weak-dipolar state and the fluctuating-multipolar state was first highlighted by Simitev & Busse (2009). This is directly associated with the zonal flow, which once present prevents the formation of an organized large-scale field. So that the transition to the fluctuating-multipolar branch is hysteretic and once on this branch, the controlling parameter (the Rayleigh number) can be decreased below the transitional value without recovering the weak-dipolar state. Schrinner et al. (2012) further demonstrated that in this case the controlling parameter was still a local Rossby number, but which needed to be based on the amplitude of the convective flow, and not of the zonal flow itself. This bistability is illustrated by figure 6.

This second dynamo state is characterised by fluctuations of the dipolar component. For this reason, it has sometimes been argued that geomagnetic polarity reversals may be due to the fact the Geodynamo operates near this transition (Olson & Christensen 2006, Christensen 2010). Indeed, an estimation of the parameter Ro E -1/3 in the Earth's core would be very close to the critical value observed in numerical simulations (below, but close to 10 -1 ). It is however most unlikely that the Geodynamo operates on the viscous dipolar branch. The resulting viscous length scale would be extremely small, less than 100 m. The strong-dipolar state described previously is most likely the relevant one, and this transition is probably not relevant to the actual Geodynamo reversals (see discussions in Oruba & Dormy 2014b).

a.

The Strong-Dipolar dynamo state

Recently, Dormy (2016) has shown that if E η was small enough in the numerical simulations, the viscous-dipolar mode could exhibit a transition to a very different state, of stronger, yet still dipolar, magnetic field. The corresponding bifurcation diagram is illustrated in figure 7. Dormy (2016) pointed that a "cusp catastrophe" occurs in the bifurcation diagram as E η is decreased at fixed E . This catastrophe acounts for the transition between the single branch reported on figure 1a, yet characterised by a sudden increase near Ra/Ra c 3, and the turning points highlighted by figure 7.

The Elsasser number Λ is usually assumed to be of order unity in the strongdipolar state. As can be seen in figure 7, while the values are "of the order of unity" (compared to the extreme values of some parameters, such as those listed in (4)), they are significantly greater than unity. For this reason a modified Elsasser number

Λ = Λ L/(Rm B ) , (8)
was used in Dormy (2016), and was shown to offer a closer measurement of the balance between the Coriolis and the Lorentz forces. In the above definition, we used Rm = U L η , and

B 2 = B 2 (∇ × B) 2 . ( 9
)
Writting in dimensional form the ratio of the Lorentz to the Coriolis force yields

{(µρ) -1 ∇ × B × B} {2Ω × u} = B 2 2ΩµρU B . ( 10
)
The classical Elsasser number stems from U B η, which is a sensible approximation in an asymptotic sense, as the magnetic Reynolds number is neither very large nor very small in this problem. The modified Elsasser number offers a finer measure of this balance by writting U B /η Rm B /L .

Dormy ( 2016) has shown that this dynamo state corresponds to a primarily magnetostrophic balance, by comparing the radial components of the curl of the Lorentz a. force and the Coriolis force (in order to get rid both of the pressure gradient and of the buoyancy force).

The transition from the viscous-dipolar to the strong-dipolar state, characterised by a runaway field growth, as the turning point of the weak-dipolar state is reached, was also reproduced in Dormy (2016).

Geophysical relevance of the strong-dipolar state

We discuss here additional simulations performed in the range E ∈ [3 × 10 -4 , 10 -5 ] , E η ∈ [1.4 × 10 -6 , 2.5 × 10 -5 ] , q ∈ [5, 18] (see Table 1). In these simulations the parameters were chosen such that the strong-field state reported in the previous section was maintained in the limiting process of decreasing both E and E η . This corresponds to a distinguished limit, relating the small parameters, as introduced in Dormy (2016).

We can first consider, as we did for the first two dynamo states, the length scales dependency with the Ekman number. These are presented in figure 8. While viscosity clearly affects the small length scales of the flow, the length scale upeak (corresponding to the energy spectrum peak) appears reasonably independent on the Ekman number, consistent with a large-scale magnetostrophic balance.

In order to assess that these simulations offer a sensible approximation to the magnetostrophic limit, the Elsasser number Λ is reported in figure 9a as a function of the combination Ra = Ra q . We also report the modified Elsasser number Λ as a function of Ra in figure 9b. As the field increases with Rm in these simulations, the modified Elsasser number presents a narrower range of variations than the Elsasser number. The first important observation in the strong-dipolar state is that indeed, whereas the Elsasser number is larger than unity and exhibits a clear variation with Ra , the modified Elsasser number is much closer to unity for all the simulations in the strong-dipolar state (whereas a wide disparity can be observed in both plots for the weak-field state, though with a narrower range in the case of the modified Elsasser number).

The second essential information, is that despite the variations in E , E η , and q all the strong-dipolar points appear to sit on a single curve. They are only (or almost only) a. functions of Ra . This vindicates the scenario of Dormy ( 2016) that these dynamos are approaching a dominant magnetostophic balance.

A key prediction on the strong-dipolar branch is that the kinetic energy should be significantly lower than the magnetic energy. This contrasts with the strongly inertial regime in which equipartition is eventually expected. Because of the smallness of the magnetic Ekman number E η in (1), the magnetic energy E M should here be much larger than the kinetic energy E K . We report in figure 10 the evolution of the ratio of the kinetic energy E K over the magnetic energy E M as a function of the inverse Ekman number for the three dynamo states discussed in this article. This quantity varies significantly in the weak-dipolar state (see figure 10a). It is less than unity for most models, but no clear trend with the Ekman number can be emphasised. In the multipolar-fluctuating state, the field is generally weaker, while the driving by buoyancy is stronger, as a result, most of these dynamos are characterised by a ratio larger than unity (see figure 10b). In figure 10c, the strong-dipolar models exhibit a clear and systematic decrease of this ratio with the inverse Ekman number. They always correspond to lower values of this ratio than those achieved in the weak-dipolar state. For the smallest Ekman number considered here, this ratio reaches a value lower than on the two other branches. This contrasts with Yadav et al. (2016) who argue that largely super-critical dynamos are needed to decrease this ratio (a result which is however most likely correct on the weakfield branch).

Of course, viscous forces are still present in the strong-dipolar state simulations. Their importance can be quantified by considering, as we did for the weak and multipolar states, the fraction of viscous dissipation. The ratio of the energy being dissipated by viscous forces to the total energy dissipation (viscous and ohmic) should vanish as one approaches the magnetostrophic limit. Figure 11 presents the evolution of the ratio f ν as a function of various parameters. Figure 11a shows f ν as a function of the modified a. 10 3 Elsasser number Λ . It highlights that the strong-dipolar state simulations span over a wide range of f ν . While the viscous dissipation never amounts to more than 50% of the total dissipation, this is a very significant variation. It is then enlightening to represent f ν as a function of the inverse Ekman number. This graph, represented in figure 11b, shows that, for those dynamos that are in the strong-dipolar state, the fraction of viscous dissipation indeed vanishes as the Ekman number decreases. So that most of the energy is dissipated via ohmic dissipation for the smaller Ekman numbers considered.

E -1 E K E M c.
a. The relation between figure 10c and figure 11 is subtle. With our choice of nondimensional form, the kinetic and magnetic energy are expressed, in units of ΩρηL 3 , as

E K = E η 2 u 2 dV , E M = 1 2 B 2 dV . (11)
If our choice of units yields order one values for the integrals, then the ratio E K /E M will vanish as E η .

If we now form the energy equations, after integration by parts, we get dE

K dt = Ra q T u • r dV -∇ × (u × B) • B dV -E (∇ × u) 2 dV ,( 12
) dE M dt = ∇ × (u × B) • B dV -(∇ × B) 2 dV , (13)
The above two equations can be rewritten in a condensed form dE

K dt = P -L -ED u , dE M dt = L -D B . (14)
The first term on the right of ( 12) is the energy production term P . In non-magnetic hydrodynamics, this energy has to be entirely dissipated in a statistically steady state by the last lerm on the right-hand-side of (12), i.e. the viscous dissipation ED u . In magnetohydrodynamics, however, the second term on the right-hand-side of (12), L, allows a transfer of energy to the induction equation (it is equal and opposite to the first term on the right of ( 13)), where the energy can be ohmically dissipated via the last term in (13), D B . Equation ( 14) stresses that f ν = ED u /(ED u + D B ) vanishes in the limit of small Ekman numbers.

Conclusion

We discussed the different dynamo states obtained in numerical models of the Geodynamo. We show that at least three distinct states, characterised by different forces balances can be highlighted in the available databases of numerical dynamos. These are represented on a schematic phase-diagram at fixed Ekman number on figure 12. The question of the most relevant choice of E η at a given E is a difficult one. Dormy (2016) suggested that the ratio of these two small parameters should be determined through a distinguished limit, instead of systematically trying to maximise E η at a given E (i.e. minimise Pm).

It should be noted that the distinguished limit advocated above consists in relating the two small parameters E η and E in the limiting process, with the ratio E/E η ≡ Pm vanishing in the limit. This constrasts with an alternative approach, which consists in dropping the inertial term, either entirely, or for the non-zonal terms only [START_REF] Glatzmaier | [END_REF], Jones & Roberts 2000[START_REF] Hughes | [END_REF], which amounts to an infinite Pm limit.

The first dynamo state (WD), and probably the most documented one, corresponds to a state in which viscosity is entering the main balance, and influencing the convection length-scale. It does not mean to say that these dynamos correspond to a pure "VAC" (Viscous-Archemedian-Coriolis) balance. Other terms, such as inertial effects or Lorentz force, obviously affect the solution. The second state (FM), originally highlighted by Kutzner & Christensen (2002), corresponds to a state in which inertial forces became strong enough so that the Rossby radius became comparable with the convection length scale. In this state the dipolar component no longer dominates the solution, and the field is referred to as multipolar. The third state (SD), introduced by Dormy (2016), is characterised by a runaway field growth from the weak state, due to a cusp catastrophe in the bifurcation diagram. This state appears to correspond to the magnetostrophic force balance anticipated in the Earth's core. Of course, viscous effects, and inertial effects are still present, but they do not enter the leading order balance.

As the co-existence of the weak and the strong-dipolar states for a given set of parameters (bistability) is associated with the occurence of a fold in the bifurcation diagram due to a cusp catastrophe, some models obtained at values of Pm = E/E η below the appearance of this catastrophe will necessarily share some properties of the strong-dipolar state.

The strong-field state can only be achieved in numerical models so far by adopting a large value of Pm = E/E η . This is only due to computational limitations, and this ratio can be decreased, following a distinguished limit as advocated in Dormy (2016).

Key issues remain, such as: whether dynamo action can be observed for Ra/Ra c lower than unity. This analytical prediction on the strong-field branch has so far never been reproduced in numerical models -whether the point at which the cusp catastrophe occurs indeed decreases to lower values of Pm = E/E η as E is decreased -whether a transition from the strong-dipolar state to the fluctuating-multipolar state can be observed for larger forcing. Further studies will undoubtably be useful to address such open issues.

Appendix A. Numerical data

Figure 1 .Figure 2 .Figure 3 .

 123 Figure 1. Bifurcation diagrams obtained for E = 3 × 10 -4 and (a) E η = 5 × 10 -5 , (b)E η = 10 -4 , (c) E η = 2×10 -4 (• stable, × unstable, ⊗ meta-stable).These respectively correspond to a super-critical, a sub-critical and an isola bifurcation diagram. The same sequence of transitions between the various bifurcation diagrams were obtained at different values of E, for similar values of E η (seeMorin & Dormy 2009).

Figure 4 .

 4 Figure4. Fraction of viscous dissipation as a function of Ro E -1/3 . The transition from the weak-dipolar (circles) to the fluctuating-multipolar state (grey squares) is clearly controlled by Ro E -1/3 , characterising the importance of inertial effects at the flow length scale. The fraction of viscous dissipation is significant in both regimes and appears enhanced by inertial effects.

  Figure5. Weak-dipolar (circles) and fluctuating-multipolar (grey squares) branches with no-slip boundary conditions for (a) E = 3 × 10 -5 , E η = 1.2 × 10 -4 , q = 0.25 , (b) E = 10 -4 , E η = 10 -4 , q = 1 , and (c) E = 3 × 10 -4 , E η = 10 -4 , q = 3 .

  Figure6. Weak-dipolar (circles) and fluctuating-multipolar (grey squares) branches for E = 10 -4 , E η = 10 -4 , q = 1 , in the case of stress-free boundary conditions.

Figure 7 .

 7 Figure7. Weak-dipolar (circles) and strong-dipolar (bullets) branches for E = 3 × 10 -4 , E η = 1.7 × 10 -5 , q = 18 . Crosses indicate unstable solutions. The Elsasser number is represented in a linear scale on panel (a) and a log-scale on panel (b), thus highlighting the lower branch.

Figure 8 .

 8 Figure 8. Evolution of the typical length scales of the flow as a function of the Ekman number, same representation as in figure 2, but for numerical dynamos in the strong-dipolar state.

Figure 9 .

 9 Figure 9. Strength of the magnetic field, as measured by (a) the Elsasser number Λ , (b) the modified Elsasser number Λ , both as a function of the modified Rayleigh number Ra, in the weak-dipolar regime (circles) and in the strong-dipolar regime (bullets). The insets present an enlarged representation of dynamos in the strongdipolar state.

Figure 10 .

 10 Figure 10. Ratio of the kinetic energy E K over the magnetic energy E M as a function of the inverse Ekman number, for (a) weak-dipolar, (b) fluctuating-multipolar and (c) strong-dipolar dynamo states.

Figure 11 .

 11 Figure 11. The f ν parameter as a function of (a) the modified Elsasser number Λ and (b) the inverse Ekman number E -1 , in the weak-dipolar state (circles) and in the strong-dipolar state (bullets).

Figure 12 .

 12 Figure12. Phase-diagram illustrating, for a given Ekman number, the three different dynamo states observed in numerical simulations of the Geodynamo, namely the weakdipolar (WD), the fluctuating-multipolar (FM) and the strong-dipolar (SD) states. The cusp indicates the region of bistability between the WD and the SD states. The grey shaded region marks the explored strong-dipolar state.

Table A1 .

 A1 Direct Numerical Simulations performed in the strong-dipolar state.

	Ra Nu	uvort	uCA06	upeak	B	Rm	Λ E K /E M	f ν
		E = 3 × 10 -4 E η = 2.50 × 10 -5 q = 12
	1800 1.50 0.0696 0.4760 0.7854 0.0817 167 11.32	0.0312 0.34
	2100 1.60 0.0700	-	-0.0733 205 12.70	0.0410 0.35
	2400 1.68 0.0674 0.4586 0.7854 0.0712 228 15.00	0.0433 0.37
	2700 1.85 0.0647 0.4425 0.7854 0.0648 265 17.32	0.0503 0.38
	3000 1.92 0.0651 0.3653 1.0472 0.0617 300 18.22	0.0617 0.40
	3300 2.04 0.0638 0.3530 1.0472 0.0584 330 19.92	0.0691 0.41
	3840 2.23 0.0622 0.3653 0.7854 0.0526 389 21.35	0.0900 0.44
		E = 3 × 10 -4 E η = 1.67 × 10 -5 q = 18
	1890 1.23	-0.4987	-	-145 6.60	0.0262 1.00
	2250 1.38	-0.4553	-	-207 12.00	0.0289 1.00
	3150 1.55 0.0655	-	-0.0690 285 22.50	0.0295 0.37
	3600 1.71 0.0648 0.4189	-0.0595 348 23.60	0.0428 0.39
	4050 1.84 0.0632	-	-0.0555 396 27.27	0.0479 0.40
	4500 1.93 0.0622 0.3977 0.7854 0.0533 438 30.35	0.0527 0.41
		E = 1 × 10 -4 E η = 8.33 × 10 -6 q = 12
	2160 1.45 0.0528 0.3927 0.6283 0.0711 205 10.50	0.0165 0.26
	2400 1.60 0.0508	-	-0.0671 240 13.72	0.0175 0.27
	2640 1.76 0.0429	-	-0.0620 270 16.25	0.0148 0.27
	2880 1.83 0.0487 0.3452 0.6283 0.0603 298 18.20	0.0206 0.28
	3300 1.95 0.0478 0.3173 0.7854 0.0564 341 20.35	0.0248 0.29
	3840 2.21 0.0455 0.2964 0.6283 0.0502 422 23.05	0.0319 0.32
		E = 3 × 10 -5 E η = 3.00 × 10 -6 q = 10
	3000 1.81 0.0362 0.2732 0.6283 0.0496 335 11.25	0.0148 0.22
	3600 2.13 0.0350 0.2493 0.6283 0.0474 394 17.60	0.0136 0.20
	4200 2.40 0.0332 0.2212 0.6283 0.0467 455 24.40	0.0134 0.21
		E = 1 × 10 -5 E η = 2.00 × 10 -6 q = 5
	1800 2.00 0.0363 0.2513	-0.0638 198 8.20	0.0049 0.07
	2000 2.17 0.0267 0.2513	-0.0632 211 10.00	0.0044 0.11
		E = 1 × 10 -5 E η = 1.43 × 10 -6 q = 7
	2310 1.90 0.0327 0.2474	-0.0579 246 10.80	0.0040 0.08
	2520 2.12 0.0294 0.2310	-0.0546 285 12.75	0.0046 0.10
	2800 2.30 0.0265 0.2137	-0.0518 320 14.50	0.0051 0.12

§ Note that the original paper (Morin & Dormy 2009) uses a different ordering based on the magnetic Prandtl number at fixed Ekman number.

Acknowledgements

This work was granted access to the HPC resources of MesoPSL financed by the Region Ile-de-France and the project Equip@Meso (reference ANR-10-EQPX-29-01) of the programme Investissements d'Avenir supervised by the Agence Nationale pour la Recherche. Numerical simulations were also carried out at CEMAG and TGCC computing centres (GENCI project x2013046698).