The Khanty (Ostyaks in the historical accounts) are hunters-fishermen who live on the eastern bank of the Ob river, in the Northwest taigas of Siberia. Mentioned since the eleventh century in medieval Russian chronicles, they have been living in the shadow of Russia since the conquest of Siberia in the sixteenth century, its colonization in the seventeenth century, Christianization in the eighteenth and nineteenth centuries, and Sovietisation and industrialization in the twentieth and twenty-first centuries. Their interactions with the Russian world are numerous, especially with respect to their religious system. For example, the simple hunting ritual described in seventeenth century little gradually came to be performed as an "existential territory", with the bear-prey, a celestial son come down to earth, put to death by men. Prohibited by the early Soviet power, the Bear Games continued to be performed in part clandestinely before reappearing publicly in the Russian Federation, after "Gorbachev gave [us] permission to shamanize" (Mandelstam Balzer, 1999: 197).

Firstly, I will briefly present the Ostyak ceremonies associated with the Bear through an examination of pre-revolutionary sources and then discuss the Bear Games as they have been organized since l994 in the Kazym region. My analysis is based on contemporary Indigenous discourse expressed in literature1 and amateur videos as well as on my fieldwork and interviews with northern and eastern Khanty informants (2004,2005,2006,2007), such as Timofey Moldanov and Zoya Loz'jamova (Kazym Khanty), Raysa Nettina (Ob Khanty), Eremey Ajpin (Agan Khanty), Agrafena Sopočina (Pim Khanty). If one emphasises the uniqueness of their individual worldview -a phenomenon that is highly enhanced by the complex history of Russia -above and beyond the scepticism of some observers and the uncertainties of the Khanty themselves, each of these informants, in his own way, aspires, like the Bear Games which replay the Creation, to recreate a new communal impulse, a future in the face of the dangers that encircle the Siberian taigas, where the bear is no longer as frightening as the oil and gas worker (Samson Normand de Chambourg, 2003a : 43), where "already, the big Russian man with the iron head and belly and the big Russian woman in a red caftan, like Ulyb and Salla escaped from fiction, devour everything in their path and make shadows in the sky" (SNdC, 2003b : 73.).

The Bear's Transfiguration

The Bear Games have been described since the nineteenth century, especially by Hungarian and Finnish ethnographers and linguists, visiting their "Siberian relatives". But how did the taiga dwellers turn the game hunt into Bear Games, the bear into a celestial son?

In his sovereign will to integrate the minorities of Russia, Peter the Great shaped geographical (1701) as well as spiritual space (1700, 1706) by a wide range of decrees: henceforth "there was only one God in heaven and one Tsar on earth." This concentration of political and ecclesiastical powers engendered an empire where these two aspects complemented and opposed each other on the field, depending on the period. In the Siberian taigas, Christianity laboured to establish itself in the face of the supposed "superstitions" of illiterate societies. Russian power was still imperfectly established; the clergy had often gone astray as the Metropolite of Tobolsk already lamented in 1622, and the Ob-ugrian population remained quick to defend a shamanism that is synonymous with local survival. The word "shamanize" -recently introduced into Russian through the horror-filled writings of the old believer and archpriest Avvakum exiled in Siberia (1672-1675) -symbolises a dark and infernal force illustrated in the Brief Chronicle of Siberia (1655-1700), by the "sorcerer" that Ermak met in Chandyr' in about 1580: "pierced by [vogul] sabres and knives", drunken and smeared with blood shed in such a manner, his body bears not a single wound at the conclusion of the séance of shamanic prediction (Karjalainen, 1996: 217).

Despite official Christianization of the Ob-Ugrians by the Metropolite of Tobolsk and Siberia, Filofey Leščinskiy2 , even the Russian administration did not take needless risks when it came to oath practices commonly used in Imperial Russia. Many ancient sources document that the oaths sworn to the Russian sovereign or to Russian justice were not sworn upon the New Testament Gospels or upon the cross as any Christian subject of the Empire would have done, but upon the bear. Thus, according to Jean-Luc Lambert's hypothesis, which overturns the traditional Finno-Ugric notion, in order to become established the oath practices imposed by the colonizing Russian power seem to have relied upon the pre-existing bear hunting rituals of the animistic Ob fishers and hunters3 . The formidable king of the forest was thus regarded as a transcendental and immanent dispenser of justice: indeed, were there not some neophytes who passed to the Metropolite Filofey the story of a perjurer buried in 1713 and then dug up by a bear in 1718 in order to be punished in the place where he had sinned 4 ?. Paradoxically, for their part the Ostyaks assimilated the Russians into their holistic vision of the world by apologizing to the killed animal through "devil games 5 ", "grand honours, (…) numerous scowls of feigned lamentations 6 ", "hymns, (…) much courtesy 7 ", "every possible tribute 8 ", as well as dances. In reality, the Ostyaks were not the ones "who forged the iron that pierced it", "the feather that hasted the path of the arrow was of a foreign bird & they did nothing more than let it go 9 ". The Russians and their arms are the guilty ones.

A potential figure of idealization, the bear gradually stops being the original prey as in other parts of Siberia, in order to crystallize mirror games between two opposed societies: "the simplest, the purest humanity", unaware of those "crude perversions, so common, even among educated nations", and the monasteries, where according to the Metropolite of Tobolsk (1622), a life prevails such that "one would think only pagans and people who are totally unaware of God live there 10 ". The bear reunites the two opposing worlds in a new ritual, but the colonizer's strategy stops where the Bear Games would soon start. The response of the colonized who appropriated the oath of allegiance, was to set it gradually into a mise en abyme, set it to songs and celebrations and enhance it with their own world perception. By neglecting to attribute any evolution to Indigenous societies, except their fatal degeneration in the face of the bitter fruits of colonisation (alcohol, smallpox and syphilis), many a traveller and scholar of the Empire wrongly saw the Ostyaks as ossified in time; the fierce peoples disarmed by the Pax Rusica would no longer be but children subjected to a possessed, ungrateful and cruel mother-like Nature and to an all-mighty and improbable Empire. Yet, far from the "poor and dark land 11 ", the "cold climate, injurious to the already frail health of the missionary father Makariy 12 ", the attributive adjectives given by the Kazym Khanty to a land not their own are inspired by its fecundity: "sacred roundness in the shape of a circle" (kusylak kerlum najan muhal) like the woman's belly, " silky and downy"(suhaŋ naj aŋki, putaŋ/pumtaŋ naj aŋki) like the game that ensures their existence, "earth with sable knees", "earth shared by the seven deities, earth shared by the six deities" (ļaptyevə tərumə g ortum mŭvievə, hətyevə g ortum mŭvievǎn), and so on. Did not the bear ask his Father to let him descend to earth, when from the sky -into which he had poked holes with his father's golden stick -she saw "the happy earth covered with yellow cloth, the happy earth covered with green cloth"? Dreaming about earth that you take into your hands does it not mean, among the Kazym Khanty, that Mother-Earth (Muv aŋki) is coming to help you 13 ? In addition to the wealth of the firm land, the Khanty also celebrate the foster-rivers whose names they bear and that they populate in the first spirits' immemorial wake: Kasum imi ("The Kazym woman"), As tyj iki ("The man of the Ob upper waters"), Vejt iki ("The man from Vjtehovo"), Lev kutup iki (« The man of the Middle of the Sos'va ») and so on. Khanty land is sacred because its primal inhabitants are, according to oral tradition, the celestial Tōrum' children, because they had to fight for a long time against the terrible taiga spirits, so that men could come to daylight in the Middle World. Life flows in fertile rivers which nourish men and lend their name to each Ostyak group and to the spirits.... From this "grandiose still life" which plunged the missionaries into a sense of unease, solitude, and their own insignificance, the Ostyak created a living organism, a sacred land."

In similar fashion, the religious system continued to change. Faced with the landscape of the taiga, people had to find their own place within it. At the outset, the need to get food meant that they developed a complex system of Indigenous concepts which detected and progressively defined a network of complementary relationships between all the inhabitants of the forest. When all is said and done, the existence of all depends upon this network created with the other forms of life with which human beings come in to contact on a daily basis and call "spirits" because they are modelled upon human relations; the end of a species would limit the symbolical alliances and the possibilities of exchange. In fact, through the rituals (purifications, offerings, apologies, averted accusation, and so on) that establish a relationship with the other species, humans negotiate the game like the animal spirit negotiates the sacrifice of one of their own. In this way, in the Siberian north, the hunter does not kill, he obtains prey that gives itself up to him; the bear does not kill, but tears apart the flesh of a traitor; the river does not kill, but drowns the one who offended it. It is a matter of ethics and partnership. Shamanism is a strategy, a strategy for good luck in hunting, a strategy for survival. Like the celestial horseman As tyj iki, one of whose arms remains hidden during his measured dance at the Bear Games, for fear that the living earth will deviate from its orbit, the hunter-fishers must comply with certain ethics if they want their universe to be perpetuated.

From the nineteenth century onward, in combination, this "efficiency" of ancestral shamanism, the transcendence of the Bear established by the imperial Russian power and the Good Word spread and sowed by the missionaries in the taigas, has contributed to the development of the Siberian hunting ritual into an Ob-ugrian Bear Dance or Bear Games (pūpi jāk, voj jāk14 ). The animal skin on which the Ostyaks had to swear "is nothing but a mask" that, as Castrén observes in his Travel Notes (1845)(1846)(1847)(1848)(1849), hides the double nature of the bear: human and divine. The oral literature collected by the Hungarian, Antal Reguly, and his successors (Bernát Munkácsi, József Pápay, etc.) sheds light on the Bear's transfiguration: how he became the beloved son of the celestial god Tōrum and gave birth to the first Bear Games by coming down to earth along an iron chain, where his heavenly father intended him to act as his "clawed arm", by having mercy on the just and chastising the perjurer 15 and by being put to death by men. In the same way as the fatal crown of hunters encircles the den and the recumbent figure of brown gold is laid low by the arrow, the side arm or shotgun responds to a vital undertaking. Both bears and hunters promise compensation within the ritual where the future existence of each one is at stake, that is, the return of the guest's soul to his heavenly father and good fortune in hunting for "the numerous men of the village". And once this night-dance of the Bear is accomplished 16 , the hunter who defeated the animal will be denounced: the symbolic father of the "holy beast" and his household will go into mourning for 4 to 5 weeks in the Kazym region. No one in the house will venture to walk bare-footed any more, to joke or to curse, to name things other than in bear language17 . A good death perpetuates life.

The Finn K.F. Karjalainen studied numerous authors (Witsen, Ahlqvist, Gondatti, Infant'ev) and undertook fieldwork among the Ostyaks (1898)(1899)(1900)(1900)(1901)(1902); he even attended a "bear ceremony" near Surgut. Although the descriptions of the nineteenth century differ according to the groups studied, the organization of the ritual seems to be almost the same: the obtaining of a bear by the hunter(s); the return to the village of the hunters who in litany-like fashion enumerate for the bear the various placesrecognized personally or described to them -that they passed through; the four or five days (depending on the sex of the animal) of ritual at the "onset of the evening" and into the depths of the night in the seasonal settlements "with the little girls, small girls, the little boys, small boys, for the games with the little girls,/ for the games with the little boys,/ in the house with a roaring fire, with a fire, in the house made ready by the Ostyak 18 " ; the mourning (pèsy).

As far as the structure of the Games is concerned, it is shared for the most part by a number of Ostyak rivers: seated in a place of honour in the house of the hunter who "brought it down" (from the forest), the bear is combed out and adorned as a male or a female according to its sex; its eyes are closed by silver or copper coins; each newcomer is cleansed with water before bowing down before the "holy beast", kissing it on the head or on the muzzle (as in the above mentioned taking of the oath) and bringing it an offering or a gift. The host or a musician is seated next to the heavenly host and the spectators on the bedstead, while the floor serves as a stage for the songs, plays and dances that will follow one another daily in codified order, with each game being accounted for by notching a piece of wood. Finally, during the fourth or fifth night, the most sacred according to a number of sources, the great spirits appear before the bear, invited by greeting songs "in order to be the witnesses to the good care lavished" on the furry icon; then a procession of threatening animals arrives (including the gadfly, the crane, the eagle-owl, the fox) which will set free the soul of the heavenly son; and a divination, destined to make clear who will be the next fortunate hunter, ends the ritual.

Thus the bear who « hears and sees everything » is honoured by the hunters in order to cast aside its anger and chase away "the arrow, made by the Russians, the boar-spear, made by the Russians" that ravished its soul and to achieve its ascension towards its heavenly father after leaving behind some final sung/chanted precepts. Each night, the "bear songs" (pūpi arəχh, voj ār) performed a capella by three (five, nine) men dressed in ordinary clothes, open the festival, evoking the heavenly and earthly life of Num-Tōrəm's son, the interdicts given by the father and broken by the son, the killing of the bear by the hunter, the actions of the animal, the pre-colonial golden age, and so on.; these episodes all form a Genesis. The ritual being thus founded, the dishonourable death, and the mission of formidable justice being affirmed by these (sung) chanted myths, "the beast of the forest" that lived among men can be acknowledged and welcomed y the Ostyak people.

Before the Bear, masculine and feminine dances are performed that leave their mark on the very name of the Games: pūpi jāk. Dressed in their finest clothes, all the guests have to dance for the Bear; men, first, then the women. Despite their collective and simultaneous performances, each man and each woman dances on his or her own, repeating a choreography composed of leaps to the rhythm of the music, of head and hand gestures, of twirls. Besides the Bear Dance's illustration of the actions of the "sacred beast," there are dances consisting of abrupt hand movements for the men and of tempered ones for the veiled women, whose precise significance already seem to have been lost by the time of Karjalainen's fieldwork at the end of the nineteenth and at the very beginning of the twentieth centuries, but remain as reminders of "a means of attack and defence19 " according to Karjalainen, of "eroticism, like the French cancan », according to Ahlqvist20 .

The little plays, sung or not, follow one other, describing daily life with its hunting scenes, with its (unpleasant) dealings with the Russian merchants and officials, its triviality -comic or tragic -and its harshness and poetry. Lev Tolstoy, who had read "an account about the theatre of the wild Vogul people" and a tragic hunting play, wrote some time later: "As far as I'm concerned, from this description alone I felt that this scene was an authentic work of art 21 ". Sometimes with hyperbole, derision, double meanings, and open ribaldry, male "actors" wearing masks made of birch bark and holding a stick (when playing males) or long coloured scarves (when playing females) portray their lives within the space of a performance and bring their existence into play through the successful proceedings of the ritual. They draw from a repertoire that has been passed down to them, but they are free to embroider and to improvise on the theme and the action of the chosen Ostyak "snapshot"; this oral patrimony is not set in stone since it continuously grows richer with each adventure or misadventure. A dozen plays may be staged each evening (Karjalainen mentions sixty plays and songs performed from 5 pm to 2 am). Some examples of Ob-ugrian plays are documented by the Finnish linguist and his compatriot August Ahlqvist: three Ostyak hunters who decided to go to town and sell a fox fur in exchange for a shotgun finally spend their money on alcohol and, when they regain consciousness, blame one another for having forgotten to buy the weapon; a mother revisits in her song the suitors (an Ostyak, a Samoyed, a Tatar) of her daughter who begins dancing at the mention of the latter, thus indicating the one she wants to marry; a woman who is "working" a sable skin, relates through song how this animal lives, how it raises its little ones, teaching them how to find their food and avoid the hunter's trap; for lack of ever being able to see the face of the maid he is courting, a young man prays to Numi-Tōrəm to make the wind blow and raise the maiden's scarf, but once his prayer is fulfilled, the boy runs back home and scorns himself, not forgetting to thank the Deity for having saved him from such an ill-favoured girl; a mother, accompanied by her daughter, goes to the forest to collect berries and soon loses sight of her and finds hertoo late -raped by a forest spirit; a drunken man reeling through the lunar taiga, mistakes his shadow for someone else, and considers his echo an answer, loses his temper and by using his stick, makes the "opponent" step backwards into a dark corner and vanish, leaving the drunkard to dance for joy on his way back home; after whistling and unbridled dancing, a boisterous couple of menkv (forest spirits) falls onto the matrimonial bed, and so on. The last night, after each greeting song of the "sorcerer", the spirits make their entrance (by rank of importance in the local group) into the house. The actors, dressed this time in rich fabric and furs, follow one another in order to bow before the heavenly son, perform a hieratic sacred dance, recieve a little vodka from the hunter-host, and greet the participants before leaving the room. In the Vogul taigas, even Saint Nicholas, flanked by a cross and a candle, performs a Russian dance as a token of respect to the Bear Games22 . In conclusion, a living and threatening bestiary terrifies the "powerless" audience and chases out the bear's soul which is then ready to return to the paternal "unfathomable seventh heaven".

At last, the divination may occur and the hunter-host may murmur a question to the bear. Reciting in its ear the names of the hunters, he lifts its sacred head until it grows lighter: the Bear thus reveals which hunter will receive good luck and will host the Bear Games the next time. According to the Ugrian groups the cooked meat of the celestial son is consumed during or after the Games with some parts being put aside until the bear skin is carried off and taken out through a window. And sometimes, before the mortal remains of the bear, an old woman about to die pretends to be simple and children lie, according to their parents' instructions, about improbable crows accused of having hollowed out the bear body with their beaks, before flying away 23 . The host-hunter may choose to give the bear skin as an offering to a spirit or keep it and sell it so that he can cover the cost of the Games (the vodka, the wine and the beer have to be conveyed from Russian villages often several hundreds versts24 away). Linked to Russian eschatology, other hunters used to keep the bear skin forty days after the ritual and celebrate the deceased again on the ninth, sixteenth and thirty-sixth days25 . This is how men mourn for the Bear. Thus, a substantial ritual expressed through sung myths, living sketches, dances relating the sacred history of the Bear and the voyage of the spirits (near or far) to "the wooden house, that reverberates the shouts of the river gulls of the surging pond, of the Ob' gulls" provides a response to the supremacy of the Russian State and to the "exact sciences" of the Book and the orthodox missionaries who celebrate its sacred history. This is a living tradition, as Jean-Luc Lambert noted, brought up to date with the flowing of the rivers for Indigenous communities who "historicize" the colonization in their own way -it is a trompe l'oeil Northern Christ whose kenos is a promise of salvation for Christians and whose good death is a token of good fortune for the Ostyaks' hunt.

Serving as a way to highlight the way in which the religion of the hunt, Christianity and oral tradition, all mingle to reinforce the social cohesion and resilience of holistic societies, forced to reconsider the world in the violent light of the chaos of colonisation, the Bear dance turns away the kiss of death and embraces the living universe.

Atheism, Folkorisation and Memory

The Soviet period was not only a disaster for the taiga communities, it also presented new challenges. The authoritarian nature of the political system certainly helped to save a part of the Indigenous heritage that might have naturally disappeared in an open political system. Along with the folklorisation of the living culture, the enforced secrecy imposed on the performance of the ritual partly preserved its symbolical power.

The bear was banished. The campaigns of Christianisation were followed by the atheist propaganda of the new soviet power. The shamans, who may have still been members of the local Soviets at the end of the 1920's, were soon declared "enemies of the people26 ": the authorities intended to break up the Indigenous societies by depriving them of their economic (the kulak) and spiritual intelligentsia.

(…) Twelve years ago, I've been disqualified from voting for observing a religious custom in my family, yet without ever going from one yurt to another in order to shamanize, as shamans usually do. I have never been a shaman, I made a living out of my own work: since I was a child, I've been employed by merchants, then I had my own exploitation which provided me my own livelihood as well as that of my people, that is to say some sixteen persons. I, Semën Pakin, aged 67 years, am not able to work any longer (I enclose a medical certificate); I now depend on my son David, but he and his wife Leljan have been deprived of their rights too, because their livelihood is made on my activity, they and look after me. Is this treatment fair ? After depriving me of my rights to be a shaman, the local authorities now treat me as a kulak exploiteur, which I've never been. My assets (and that of sixteen people) are: a house, four horses, three cows and one animal). I'm even deprived of the right to complain: the authorities wanted to confiscate the certificate signed by fellow citizens on 14 th july 1934, without checking their signature, and then demanded 43 rubles that I have not got, for the authentification. They have forbidden me, Semën Pakin, to go to the district capital of Samarovo, to the hospital, for me to be treated 27 .

For Russian authorities long time the shaman has long symbolized a potentially subversive counterpower, the obscure force of holistic societies. The howl that was heard as a (spirit) figure was suppressed at the command of Vassiliy Levin, the priest of the Ostyak yurts of Zamosovskiy, whose « hands since then have been shaking so much that he could hardly write 28 », as he himself reported to the French Academy Delisle in 1740. No longer could the echo of the « wild howl of the shaman [which] hurt so much the ear, that against my will, I began to shake" be heard, according to the statement of the father Irinarkh, describing his visit to an Ostyak shaman on the Poluy river, « a hoary old man and almost blind old man, [who] began to jump as high as an acrobat29 » ? During the Empire, the Russian civilizer and the colonized shamanist learned to distrust one other. While, Soviet propaganda and films depicted shamans in very simple terms, identifying their characteristics in real life is far more difficult. This is because shamanism has had to wear many masks; in Ugra, the party was looking for great shamans, like in other Siberian societies, but the Ugrian experience of the world has had over of the centuries become distributed in a constellation of ritual specialities. Thus, in these Ugrian communities, all the descriptive terms not only serve to create confusion about who is or is not a shaman, but also shed light on the fields of knowledge of each ritual specialist taken in a specific context.

Against a of atheistic propaganda, the Indigenous « wars » in response to sovietisation, naturally relied on shamanism (respected ritual specialists, ritual assemblies and divinations). They were quickly described as « counterrevolutionary movements » and their "players" were rudely disqualified as « shamans with a drum », « shaman with an axe » or « shaman of the dark hut ». Because the holistic society was discredited by the positivist ideology of the new historical « soviet » community, the rapid propaganda of planned progress and atheism and the often brutal local agents of sovietisation, it soon responded to what it felt to be a declaration of war, with its own war, that turned into an apparent defeat (mainly in the Kazym, then in the Num-to area). This clampdown on the Ostyaks and Samoyeds of the region, in winter 1933-1934, provoked a series of arrests of « counter-revolutionary elements », twenty-seven out of fifty-two of whom were dubbed « shamans », followed by a parody of justice and ruthless repression that raged until 1935: punitive commando groups armed with larch truncheons appeared in the taigas for the manhunt 30 .

Thus, the shaman was accused by the Soviet authorities of conducting « an obstinate propaganda among the Indigenous against the schooling and the boarding of the children, arguing that educated children would be sent to big towns, enlisted in the army and denied the possibility of traditional activities 31 ». For example, Evdokiya Semënovna Nikishkina, a « shaman with a drum and of the dark hut » of the Vanzevat yurts, was denounced by an activist of the literacy campaign (likbez) because of her objections to the opening of a boarder-school and her statement about a « Soviet power that does nothing good, arrests the rich ones and the shamans, then puts them into jail and make them starve 32 ». From then on the shamans formed a social class to be eradicated. Thus, Evdokiya Nikishkina of the Indigenous soviet from Polnovat, born in 1877, was arrested on 22 nd October 1937, judged on 5 december 1937, shot on11 janvier 1938 and not rehabilitated until 27 th July 1989 33 . But beyond the physical elimination of the shamans, shot or deported to camps, shamanism remained out of reach of the Soviet power, because it is a worldview which inhabits the whole society and survives in the people, as in the past "in the Irtysh region, in spite of the absence of sorcerers, the vanished powerful Tatar influence and later the overwhelming present Russian influence34 ".

In many cases, socialist emulation lead to arbitrary actions. From 1937 to 1939, across the whole country and through repressive acts as rapid as definitive, the State hunted down « internal ennemies » who kept haunting it, such as Konstantin Grishkin and Ksenofont Seburov, arrested in February 1937 for organizing Bear Games. Indeed, in the of the authorities, a ritual where about a hundred were gathered amounted to a counter-revolutionary assembly:

We lived peacefully until my father and his brothers, Ivan and Efim, were arrested again for having gathered people for the Bear Dance; indeed it was forbidden to organize that kind of things at that time. Our families remained without hunters 35 .

Most of the men taken (raflés) by the « Reds » are sentenced by a troika on 5 december 1937 (without even a trial being conducted) and shot on 11 january 1938 in Ostyako-Vogulsk, today Hanty-Mansiysk). Human quotas have to be met too.

When the Great Patriotic War which continued to weaken the North, the victory led to the ever-increasing inflow of alien men subject to Party line (partijnost') and to their quest of a better life in the North : I let a Russian man marry my daughter. My son-in-law liked drinking; He drank away my herd of reindeer. He had kisy [hautes bottes de fourrure], He drank his kisy as well. And after that, he drank away my kisy. And now Iam here, on your holiday, barefooted. Now listen: Be careful choosing your future son-in-law. Don't let such men marry your daughters 36 ! Far away in the taiga, with no kolkhozes (State farms) nor five-year plans, the Games, or fragments as often as not, were practiced as a form of dissidence, until the evil might be overcome, as predicted by the old Khanty Stepan Nikolaevič Aypin, prisonner n° 135, shot in Ostyako-Vogulsk on 21 rst of January 1938.

Meanwhile, the Bear cult persisted. It sometimes expressed itself differently, as in the story, told by the inhabitants of the village of Tugyany of an incident that occurred at the beginning of the 1950s. A little girl from a kindergarten vanished in the forest while walking with a group of children. One week later, a villager, Anastasiya Grishkina, and her son, while rowing on the river, saw the little girl bathing and brought her back to her parents', despite her desperate calls to her mother. Three days later, in spite of the presence of her mother, little Nina was still calling for her mother. At nightfall, a female bear came grunting and scratching at the door. Nina finally explained that "the bear was just like her mother; she was caring for her, feeding her with berries and letting her bathe". Anastasija Grishkina died shortly afterwards, and her son soon thereafter. They had been taken instead of the little girl. That was the explanation of the elders and it was confirmed by a shamanic séance in a "dark hut 37 ".

When the Soviet Union was about to collapse, the Bear festival which had survived in spite of its prohibition, in spite of the threat of its participants being dismissed from work or summoned before a court, was rightfully reinstated. After seventy stolen years, the Kazym men again called their deities to come "to the top of the small trees, to the middle of the tall trees". And the spirits that had vanished on the thresholds of schools, abruptly shrinking the perception of the Khanty world, reappeared in the land of "the Great Ob strode along by twenty reindeers", "in the playing house" "filled hup with the cries of the river" and purified by smoke: in Yuil'sk, on 5-11 January 1991, at Pëtr Ivanovic Sengepov's settlement, in Polnovat in March 1993, in Kazym in March 1996, in Syunyugan in December 1998, in Kazym in December 2002, and so on. Today, despite the irremediable losses 38 and the changes of meaning accompanying the new urban and country lifestyles of numerous Khanty, but also in the activist disposition of a younger intelligentsia, the great spirits have returned to the Kazym. Just in time for the generation that was born in the late 1920s and the early 1930s to pass on this abused heritage: "S. TARLIN: My father is Kazym Goddess elder brother ... -Look, you get the wrong tune; you started the song of Kunovat Goddess... S. TARLIN: -It is the song of the Middle Sosva Goddess I want to sing! I am going to visit my aunt and bring her a sable skin as a gift. I was sitting home on the bed, covered with a mat made of sedge, I got tired of sewing and went outside... 39 Today, from thirty to forty spirits, depending on the rivers, come down to humankind. They give back to men, one after another, the luck they deserve: "an even path young girls' feet, an even path under the young boys'feet" (Vèyt iki, 1987); "the dance that wards off war, the dance that wards off diseases" (Khin iki's song, 1989); 37 Slepenkova, 2002: 66. 38 Among the Mansi whose Bear festival has been more widely described by the scientific literature of the XIX th century, the islets of living traditional culture are rarefying: "The hunters gather into brigades of three or four men. They take their sleds. One of them goes to the bear, the others join him close to the lair. They kill the bear, and on their way back, their gunshots alert the villagers that the hunting has been successful. How would it happen exactly? I don't know, I don't remember, I never saw that. I just heard about it. But I saw a Bear Dance. Men wore different masks of birch bark. The feast lasted three or four days, and if the bear was a male, five days. One says that hereby live shamans. I've been told that they sacrify reindeers. But I've never been there, saw nothing of that kind" (Péter Erderley, Bekódolt Szibéria/La Sibérie décodée, Duna Televízió, 2004). A Bear festival has been organised in Pomapaul' (rajon of Ivdel) in april 2005 by the three Pakin brothers, whose benjamin, Vladimir (born in 1962), good sangvyltap player, "hosted" a bear. 39 Medvezh'ie igrishcha na Obi, Studia O.K., 2005, 28 mn. This Olga Kornenko's film is essentially based upon the montage of Timofej and Tat'jana Moldanov, Stas Kovalenko's amateur videos.

"the lofty dance that rides the waters' backs" (Kheymas' song, 1989); "the dance that shadows war, the dance that shadows misfortune" (Em vož iki's song, 1993); "the dance that fights discord, the dance that keeps war away" (song of Yuilsk protective spirit, 1993); "the dance that breeds fish, that breeds game" (song of the Goddess of Kazym, 1993); "the great dance that turns aside war, the great dance that turns aside diseases" (Lev kutup iki, 1993), "the dance that protects girls, the dance that protects boys" (As tyj iki's song, 1991); "The eternal dance of the young women's long life, the eternal dance of the young men's long life"" (Kaltašč's song, 199340 ).

The Goddess of Kazym was revived by collective memory and collected songs. Men call her "the father's grandmother" (shchashchi). Tōrum's daughter is touchytempered and tattoo-handed, her long braids are snakes and her clasps are living lizards. Beyond numerous tales and legends regarding "The Wife of Num to" (Lor yuran nenye), regarding "the one who is a lucky hunter" and who "wears a coat of mail that looks like the scales of a little fish", there is one of her songs sung during the Games that tells, for instance, of the journey of "the wife from the upstream" (vuyt imi) who regulated the world of the Kazym. She left from the Septentrion Sea coast, where she had vented her anger at her Nenets husband -grabbing "a saber like the Ob clear waters" cutting "the slightest blade of grass that showed, the slightest shrub that grew" as well as her husband's legs. She flew as a gold-winged duck to her celestial father who forgave her and agreed that she should become the protective spirit of "the world of the men to be, the world with the face of dolls to be". As she proceeds on her thousand-reindeer's herd, she animates and consecrates the space through her children to whom she allocates the territories and "her seven white-dressed servants" whom she allocates to the river mouths. Then she "sanctifies" the space with a stone nass (net for fishing) to keep away "the betula-barked small crafts" sent by the spirit of diseases and "the small crafts of war".

The lake that is named Tōrum, the name of the Bright one: In its centre, I, the Grand Goddess, will "enthrone" majestically! The goddess who expands her daughters' life expectancy The goddess who expands her sons' life expectancy As the goddess, I will "enthrone" majestically! The dance that breeds fish, The dance that breeds game, The dance that brings luck to reindeers, The dance of the girls' longevity, The dance of the boys' longevity, The dance that keeps battlefields and war away, I'm going.

In her wake followed Kheymas, "the man who leads the fish to the spawn area", the vert [male spirit], grand fish supplier", from the mouth of the Ob river, where "the big and shredded wood shavings" that fill up his house adorned with fish scales turn into many fish in summer and many game in winter (reindeers, squirrels, sables). Dressed for the Games in a most beautiful water garment (green or blue), Kheymas repeats a ritual move that recalls wood working, "he shakes his braids filled with lake fish", and, before going away, splashes with water or sprays with snow each participant so that they will catch fish profusely.

Vèjt iki, Tōrum's third son, "the warden of threatening clouds and winds", the grand vert leader of the elements, the weather master, left his residence in the Berezovo area along the Ob's river banks, right in the middle of the spring-thawoverflooded meadows, right in the middle of a betula copse, in a golden house with a hidden door. He wears a white hat, has winged arms, and is followed by two servants. He protects the men from opposite elements as well as the youngest male children. Em vozh iki, the warrior rider with eyes "as big as the moon", who watches the frontier between the dead and the living, is asked by a singer to appear and to annihilate any mischievous spirit. The mask-faced spirit is dressed in black like the coat of his dark mount. He chases the assaults of the evil spirits away and redirects the diseases toward the Ural where he gets rid of them by lapidating their bones. He holds the bears as a sacred icon -one of the deity's names is consequently "the vert whose traits resemble those of a toothed or a clawed beast".

Lev kutup iki is "the man from the Sosva middle stream", who has dark waters abundant with fish, the Gardian and protector of the Sacred Stone's foothills (the Ural), the vert who brings luck to the reindeer herds, the one who is "older than gold" and who lives in a house as "colourful as the golden undulating waters, a house resembling a wave, resembling waters blown away". He is preceeded by a singer who announces the imminence of the nightly sound of their hooves -that actually belong to the five colourful reindeers of Lev kutup iki's team. He comes majestically on a "happy nest covered by a black sheet" to pay tribute to the bear and to show his face to the humans as he dances his sacred dance, his face "encircled by a ray and by numerous braids".

As tyy iki, "the man from the Ob river's upstream", "the vert sympathizing with the young girls, the vert sympathizing with the young boys" is a celestial rider who has a mount that has "the colour of a spring squirrel, the colour of an autumn squirrel" and a light sabre as undulating as a wave. He rides around the world, easing in his wake the fears and the pains of humankind, fighting for good, defeating evil, succeeding sometimes in changing the thoughts of humans; he often appears dressed in nothing but white, a fox fur hat on his head, a piece of fox fur in his right hand -his left hand empty, for if the celestial rider were to use both his hands, the earth would stray from its orbit, provoking the end of the world. Welcomed by the two ort (chosen people who oversee how well the Games proceed), the divinity sketches a fragile and majestic dance, where well-placed moves and a slow rythm seem to constitute the very revolution of the earth.

Finally, "the grand silver-haired nay [female spirit or goddess] who allocated the place of each spirit in the Middle World and who decides each newborn's life expectancy on earth, ends the grand spirit procession. With the jerky rythm of an ironspeared arrow, Kaltashch (a man dressed like the goddess) hammers as recommandations, "her goodbye words, numerous and bright" addressed to the parents who want to see their children grow up "sound and strong", before women cover her with numerous scarves as an offering and she executes "the eternal dance of the young women's long life, the young men's long life eternal dance".

In the Russian Federation, sometimes the ritual can no longer limit itself to the sir or the community of a precise territory. Connoisseurs of the repertoire have to be sought out throughout the whole autonomous district so that the children of the Kazym ethnographic settlement Numsang ëx ("The thinking people") are able to reenact what has been saved by the elders. Already, Danil Nikolaevich Tarlin has taken with him to his grave one of the sacred songs of Kaltashch, and Nikolay Lozyamov, worn out gisant, has stopped dancing in the night. For Juriy Vella, the Forest Nenets, reindeer herder and poet, by now the Games are nothing but a funeral evening, the pale image of a broken communal momentum:

In the banqueting hall of the University of Debrecen, our Kazym Khanty are performing the Bear Games. The American and Japanese delegations are sitting there and listening carefully.

Tartu, Tallin and Helsinki, scientists and students from Budapest and Debrecen are eager and pleased to do the same. Even our people from Syktyvkar and Yoshkar-Oly are thrilled by what is being performed on stage. Moscow herself, conceited and bossy, sits there, compelled to listen as the occasion requires. But one could have cried, when our Khanty and Mansi fellows rudely break in on the performance with perfectly misplaced advice, when none should be given! Why? Because the West knows that THIS belongs to the past, that what is being played here on the stage is bound to die, that IT vanishes at every second with the last initiated men... But we, on the other hand, still kid ourselves 41 .

Semën Tarlin, however, as the repository of his great-grand-father's long song, hurries to go to the ethnographic settlements to pass on fragments of the Bear Games before his memory, grown old, plays up.

Games as Self-Expression

How should one decipher those Bear ritual games? Far from being fixed, Ugrian societies have never stopped constructing themselves through a number of interactions, especially religious ones, to which the Bear Games testify. Representing a real challenge to colonisation, they re-enact the orally transmitted experiences of the taiga communities, from the times of the Empire to the Federation Era, becoming an identity marker of hope.

Traditionally, the zither is also used for rituals of divination, as recalled by Karjalainen42 ; indeed, the drum that was still used among the most northern Ostyaks who had remained rebels in the campaigns of Christianization in the XVIIIth century seems to have vanished from that time onwards in other areas, such as the Irtysh, Vakh, Vasyugan, and so on, in favour of the zither (nars-yukh), which was less compromising in front of Russian authorities. Thus, « the ordinary musical instrument, as soon as it's in the hands of the sorcerer, gets so sacred that one can nor put it on the ground, nor carry it openly from one house to another43 ». The musician, his head turned to the sacred corner of the house, with the zither put on his right knee, calls the deities with a melody appropriate for each one. Meanwhile other "actors" of the ritual, in specific dress, headgears and ornamented gloves, feature as the divinities and their journey in the Bear's Game. So the musician (singer44 ) and the dancer (singer) seem to divide the actions of the shaman who usually concentrates on invoking the spirits/gods and tracing their journey. This particular "redeployment" of the shaman's art is not surprising, considering the nature of the instrument and the strategic tricks that microsocieties were forced to use at that time in order to escape from the Russifying ogre; without the drum, the shamanic journey came to an end, the spirits and deities themselves now come to visit a shaman who is fixed in place and whose practise is masked by his songs and his musical instrument.

The figure of the bear, the zither and the offerings progressively had to echo the (spirit) figures, the drumbeats and the sacrifices, too visible and suspect in the eyes of the colonizer. Through these games which work well as a big collective ritual aiming to bring good luck to hunters and to herders of northwest Siberia, Khanty managed, in spite of centuries of uneven contact with the Russian world, to remain "Others", by responding ritually to the Russian colonizer.

Today the Khanty have invested in urban space. Some of them are forced to leave their ancestral territory to resettle in places with an unobstructed view over "the big Russian man with the iron head and belly" (the derrick) and the "big Russian woman in a red gown" (the flare): others are seduced for others by the city lights. The asfal'tnye hanty as Timofey Moldanov defines himself 45 , are gradually introducing their sense of the sacred to the city. Since the creation of a park-museum in 1987, rituals and many cultural events have been organized to educate others about the Indigenous cultures of the epynomic peoples of the okrug. Thus, the ritual Games sacrilised the space at the edge of Khanty-Mansiysk and of the forest, three days in a row in July 2008, at the V th International Festival of Finno-Ugric Peoples. Because of the refusal of the bear to be celebrated in the capital (through divination the bear is now asked about which games she wants), a smaller "old man with teeth, old man with claws" has been invited from Kishyk and celebrated by people. The emergence of this urban ritual remains to be carefully studied, based on the initiative of the Khanty intelligentsia and thanks to a grant of the Department of the Indigenous affairs of the district, in such a way that avoids an intellectual appropriation of Games that are traditionally community ones. The Russian authorities, after forbidding the Bear Games in over fifty years of folklorisation, is trying to reify them for local tourism and presents them in the official leaflets as if they were an ancient celebration of a "small Switzerland".

Nevertheless, through this occupation of public space, which the Khanty and Mansi make sacred, they remind the state of their presence and of the difficulty of reconciling the natural with the human resources in the district. Two worldviews confront one another around the Bear. The Federation runs Siberia as a land of plenty, prisoner of short-term economic logic and of alien workers little concerned about the North. In contrast, the Khanty take care not to hurt the earth, not to take out of the nature more than is necessary to them (what a number of authors of the XIXth century qualified as inconsequential) and to leave behind them the least possible, so as to prevent evil spirits from doing them any wrong. This holistic vision of the world gives a new legitimacy to the games when it is confronted with industrial exploitation considered as "vain action lacking any spirituality", and a "negation of the Indigenous relation with Creation established across centuries". As during the Empire, it is a matter of survival for Indigenous peoples to reinvest in these games. For instance, the traditional behaviour code adopted in the forest or on the rivers that was transmitted through songs (like an echo of the commandments addressed by Num-Tōrum to his child before he/she came down on earth) is now related to Khanty identity and ecological claims. Similarly, the song of the Master of the Spirits of the Forest reminds humans of their alterity and calls to the deaf and blind civilization beyond the circle of Khanty, gathered for the Bear Dance.

When you are in the taiga, you walk among us, but your eyes don't see us. If we did not want to move aside from your way, we would bump into one another permanently 46 .

As the Soviet power tried to break the Weltanschauung of the communities, it turned them into one "people' while they used to define themselves in terms of their rivers. Today, Khanty use this status to be heard, but also to construct an image of themselves, their own image. Contemporary games participate in the process of selfconstruction in the post-Soviet space and enable Khanty people to reaffirm roots for the future.

An ethno-historical perspective shows that the Bear Games are not remanents of an archaic totemism, but the resilient expression of microsocieties confronted with the shock of colonization. The original hunting ritual has returned in a subtle war against the Russian-making machine and the political and religious weapons used by the colonizing power. At the heart of the ritual, as early as the XVIIIth century, Indigenous peoples accused the coloniser of being guilty of killing the "guarantor for the oath". Therefore they implicitely compelled the new power to succumb to the animal revenge of the celestial child. So long as ritual games perpetuate life and the balance of Creation, Khanty people will inhabit the Land

From

  FIG.1-Jeux de l'Ours (campement d'A.A. Ernyhov, cantonde Belojarsk, 2004)

FIG. 2 -

 2 FIG. 2 -Tente ostyak en écorce de bouleau (Poluj, début du XX e siècle; © Complexe muséal ethnographique de Salekhard))

Voir Moldanov, 1999 ; Material V jugorskih čtenij « Medved' v kul'ture obsko-ugorskih narodov,

;Kravčenko, 2004.

Voir SNdC, 2010.

Cf. Lambert, 2007-2008 : 19-43.

Patkanov, 2003: 26.

Šemanovskij (Irinarh), 1905: 25-26

Moldanova, 2001: 292.

Lambert, 2003Lambert, -2004 : 391-396. : 391-396.

Patkanov, 2003: 209.

Le nom même des deux rituels, l'un sporadique (voj jāk), lié à l'obtention d'un ours à la chasse, l'autre périodique (lènh jāk), célébré traditionnellement tous les sept ans à Vežakora et à Tegi en alternance, reflète le caractère sacré de la danse aux yeux de l'ours qui « se souviendra de chacune », de celle « dans son maigre vêtement en écorce de sapin » à celle « en caftan de drap, duveteux comme la fourrure de l'écureuil, en bottes ornées de perles de verres, pareilles aux coudes de l'écureuil »(Kravčenko, 2004 : 12).

L'ours qui voit et entend tout s'offusquerait qu'on lui manquât de respect par une parole ou un geste ; dans cet esprit, selon la terminologie de la langue de l'ours (soit plus de 500 termes dont 132 pour désigner le seul animal), les chasseurs ne l'écorchent pas, mais « ôtent sa fourrure », « dévêtent l'ancien » ; tout le corps de « l'homme de la forêt », du coeur (« le lieu sacré ») à la graisse (« la sente du couteau ») en passant par ses cuisses (« la cité »), a également un nom rituel. De même cette langue secrète est utilisée dans la maison pour désigner les animaux et les objets le temps du deuil : le banc est « celui qui a des jambes », le sel « la chose délicieuse », la tasse « l'arbre qui puise », l'eau, « la chose à boire », « manger » se dit « cueillir », etc.(Karjalainen, 1996 : 173-174).

Ibid., 1996: 165-166.

Ahlqvist, 1999: 119.

Karjalainen, 1996: 164.

Patkanov, 2003: 211.

Ancienne mesure russe équivalant à 1 067 mètres.

Patkanov, 2003: 213.

Voir SNdC, 2007-2008: 115-195.

GAHMAO [Gosudartvennyj Arhiv Hanty-Mansijskogo Avtonomnogo Okruga], fonds 111, inventaire 3, dossier 9, feuillets 85-86.

Šemanovskij (Irinarh), 1909: 403-410.

Voir Eremej Ajpin, Bož'ja Mater' v krovavyh snegah, Ekaterinburg, Pakrus. L'ouvrage consacré à ces événements a été traduit en français : La Mère de Dieu dans des neiges de sang, traduit du russe par A.-V. Charrin et A. Coldefy, Paris, Paulsen.

TOCDNI, fond 107, inventaire 1, dossier 111, feuillets 25-26.

Naša obščaja gor'kaja pravda, 2004:171-172.

Kniga rasstreljannyh, 1999: 122.

Karjalainen, 1996: 184.

GAHMAO.422. 17.1.1-4. Memories of Anna Grigor'evna Yumina (born in 1922) in Tugjany, rajon of Berezovo, that have been collected in 2002 by T.S. Seburova and archived as a manuscript at the State Archives of the Khanty-Mansi autonomous okrug.

This song has been performed by Nikolaj Loz'jamov of Kišik as an element of the fourth part of the Games (or Lŭļŋǎltup) which has a humoristic complexion and stages throughout the art of singing the relationships bearmen, men-men, men-nature.

Other deities leave "the dance that gives abundance of fish, the dance that gives abundance of game"(Kazym Goddess' song, 1993) etc "the dance that conjures up the strokes of war, the dance that conjures up the strokes of disease"(Hin iki' song, 1989), "the dance that guards from discord, the dance that protects from the war" (Juil'sk protective spirit, 1993).

Karjalainen, 1996: 204, 206, 230.

Ibid., 1996: 207.

Les termes vernaculaires désignant l'instrument et le musicien sont respectivement naras-juh (« l'arbre qui chante ») est narasti ho (« l'homme qui joue d'un instrument »).


Louis Moresi

Frederic Dufour

Hans Mi Ihlhaus

Viscoelastic formulation for modeling of plate tectonics

The Earth's tectonic plates are strong, viscoelastic shells which make up the outermost part of a thermally convecting, predominantly viscous layer. In order to build a more realistic simulation of the planet's evolution, the complete viscoelastic convection system must be included. A particle-in-cell finite el ement method is demonstrated which can simulate very large deformation viscoelasticity. This is applied to a plate-deformation problem. Numerical accuracy is demonstrated relative to analytic benchmarks, and the characteristics of the method are discussed.

INTRODUCTION

Underneath the lithospheric plates of the Earth lies the mantle (Figure I). Approximately 3000km deep, it is composed of solid rock that is warm enough to deform like a viscous fluid, albeit at incredibly slow speeds of a few centimetres per year. The plates move because the mantle is forever stirring as heat gen erated by natural radioactive decay struggles to es cape via thermal convection. The plates which form the ocean floors are part of this circulation and are sucked down when they become old, cold and dense.

The continental crust is formed by lower density rock which remains buoyant despite being cold. In the lithosphere the rocks are significantly cooler and be have as a viscoelastic, brittle solid. In regions of high stress, brittle failure gives rise to earthquakes. This picture of the Earth's interior is widely ac cepted by geophysicists. It clearly indicates that the fundamental process is thermal convection; plate tec tonics is the manner in which the system organizes.

Therefore, a consistent model of plate behaviour must contain a description of the convection system of which the plate is a part.

There are some fundamental problems which need to be addressed before the routine application of en gineering principles to the lithosphere. The principle issues is that plate tectonics is itself only a kinematic description of the observations: a fully consistent dy namic description of the motion of the plates is still

\i.u=f (1)
where u is the stress tensor and f a force term. As we are interested only in very slow deformations of highly viscous materials, (infinite Prandlt number) we have neglected all intertial terms in (I). It is conve nient to split the stress into a deviatoric part, r, and an isotropic pressure, p,

<T = T-pi ( 2
)
where I is the identity tensor.

Viscoelasticity

There are a number of different viscoelastic models, we will use the Maxwell model which has been used in previous studies of lithospheric deformation where viscous and elastic effects are important such as post glacial rebound [START_REF] Peltier | The impluse response of a Maxwell Earth[END_REF]. This model assumes that the strain rate tensor, D, defined as:

D ;i = � (8V; + 8\!j) 2 OXj OX; ( 3
)
is the sum of an elastic strain rate tensor D • and a viscous strain rate tensor Dv. The velocity vector, V, is the fundamental unknown of our problem and all these entities are expressed in the fixed reference frame x;. Now we decompose each strain rate tensor and l .

Dv = 3tr(Dv)I + D,.

where K. is the bulk modulus and ( is the bulk vis cosity. P = p as it p is a scalar.

:;.= + + rW-Wr

(8)
where W is the material spin tensor, W ; j = � (av; _ ai1 ) 2 OX j OX;

(9)

The W terms account for material spin during advec tion which reorients the elastic stored-stress tensor. We note that the form of equation ( 7) is unsuited to conventional fluids as the material has no long term resistance to compression. This behaviour is, how ever, relevant to the simulation of the coupled porous flow, matrix deformation problem. Here it is common to ascribe an apparent bulk viscosity to the matrix material in order to model compaction effects (e.g. [START_REF] Mckenzie | The generation and compaction of partially molten rock[END_REF], particularly for large scale geologi cal systems where the details of the pore network can not be measured directly.

Numerical implementation

As we are interested in solutions where very large de formations may occur -including thermally driven fluid convection, we would like to work with a fluid like system of equations. Hence we obtain a stress I strain-rate relation from ( 6) by expressing the Jau mann stress-rate in a difference form: (10) where the superscripts t, t + 2'.t indicate values at the current and future timestep respectively. ( 6) and ( 7 where a = 77/ µ is the shear relaxation time and , 8 = UK. is the bulk relaxation time. We can simplify the above equations by defining an effective viscosity T/eff and an effective compressibility (elf:

At T/eff = '1 2'.t + a
At and �.ff = e 2'.t + /3

Then the deviatoric stress is given by

( • t+t.t r' W1r1 r'W' ) T/eff D + --+----- µ !J.. t µ µ
and the pressure by

P t+ t. t = -" ( Dt+ .cl. t -L ) �•ff kk 2'.t K. ( 13
) (14) (15)
To model an incompressible material K. and ( are made very large such that Dkk � 0.

Our system of equations is thus composed of a quasi-Newtonian viscous part with modified mate-rial parameters and a right-hand-side term depend ing on values from the previous timestep. This ap proach minimizes the modification to the viscous flow code. Instead of using physical parameters for viscos ity and bulk modulus, we use effective material prop erties (13) to take into account elasticity. Then during computations for the force term, we add elastic inter nal stresses from the previous timestep or from initial conditions.

F'•1 = � ':-"" -�a':-"'' I h',fll•' µ flt IJ,J (16)
We solve ( 15) and ( 14) and obtain a solution for v•+t>.t. From this solution we compute the new stress state due to the velocity field and previous stored stresses.

Stability

The approach outlined above is unconditionally stable only if the timestep is larger than the relaxation time for the material, i.e.

flt< !l µ (17)
in the case of the shear moduli. Alternatively, this means a Deborah number, D e < 1, indicating that the method is appropriate to the viscous, rather than the elastic, limit.

One difficulty is that the tirnestep is not necessar ily chosen to match the physical problem, but by the Courant condition for the chosen mesh. This means that a convergence demonstration for arbitrarily small elements may not be possible for the general case. We are currently addressing this issue.

In practice, however, for our area of research, vis cous flow drives the plate motions, and the litho spheric plates are embedded in a highly viscous ma terial. This may produce a situation where a system wide relaxation time is more important than the re laxation times of individual materials, since loading and unloading of the elastic materials happens almost exclusively through a low-viscosity medium. Under these circumstances, the relaxation time of an indi vidual layer (such as the lithosphere) may be much larger than the Courant timestep, but stresses are ei ther balanced, or relaxed by driving a flow in one of the viscous materials.

COMPUTATIONAL METHOD 3.1 Choice of Numerical Scheme

In fluid dynamics, where strains are generally very large, but not important in the constitutive relation ship of the material, it is common to transform the equations to an Eulerian mesh and deal with convec tive terms explicitly. Problems arise whenever advec tion becomes strongly dominant over diffusion since an erroneous numerical diffusion dominates. In our case, the advection of material boundaries and the stress tensor are particularly susceptible to this nu merical diffusion problem. Mesh-based Lagrangian formulations alleviate this difficulty, but at the ex pense of remeshing and the eventual development of a less-than optimal mesh configuration. This increases complexity and can hinder highly efficient solution methods such as multigrid iteration. The Natural El ement Method eliminates remeshing difficulties but is associated with considerable complexity of imple mentation, particularly in 3D.

A number of alternatives are available which dis pense with a mesh entirely: smooth particle hydro dynamics and discrete element methods are common examples from the fluid and solid mechanics fields respectively. These methods are extremely good at simulating the detailed behaviour of highly deform ing materials with complicated geometries (e.g. free surfaces, fracture development), and highly dynamic systems. They are, in general, formulated to cal culate explicitly the interactions between individual particles which ultimately means that a great many timesteps would be required to study creeping flow where the timescales associated with inertial effects are very many orders of magnitude smaller than typi cal flow times.

We have therefore developed a hybrid approacha particle in cell finite element method which uses a standard Eulerian finite element mesh (for fast, im plicit solution) and a Lagrangian particle framework for carrying details of interfaces, the stress history etc.

The Particle in Cell Approach

Our particle-in-cell finite element method is based closely on the standard finite element method, and is a direct development of the material point method of [START_REF] Sulsky | Application of a particle-in-cell method to solid mechanics[END_REF]. The standard mesh is used to discretize the domain into elements, and the shape functions interpolate node points in the mesh in the usual fashion. The problem is formulated in a weak form to give an integral equation, and the shape func tion expansion produces a discrete (matrix) equation. Equation ( 1) in weak form, using the notation of (2) becomes l N(i,j)Tijdnl N,;pdn = l Nif;dn (18) where the trial functions, N, are the shape functions defined by the mesh, and we have assumed no non zero traction boundary conditions are present. For the discretized problem, these integrals occur over sub domains (elements) and are calculated by summation over a finite number of sample points within each el ement. For example, in order to integrate a quantity, In standard finite elements, the positions of the sam ple points, xp, and the weighting, wp are optimized in advance. In our scheme, the Xp 's correspond pre cisely to the Lagrangian points embedded in the fluid, and wp must be recalculated at the end of a timestep for the new configuration of particles. Constraints on the values of wp come from the need to integrate poly nomials of a minimum degree related to the degree of the shape function interpolation, and the order of the underlying differential equation (e.g Hughes, 1987). These Lagrangian points carry the history variables which are therefore directly available for the element integrals without the need to interpolate from nodal points to fixed integration points. In our cas• e, the dis tribution of particles is usually not ideal, and a unique solution for wp cannot be found, or we may find we have negative weights which are not suitable for inte grating physical history variables. We therefore store an initial set of wp 's based on a measure of local vol ume and adjust the weights slightly to improve the integration scheme. Moresi et al. (2000) give a full discussion of the implementation of the particle-in-cell finite element scheme used here including full details of the inte gration scheme and its assumptions. They also dis cuss the specific modifications to the material point method required to handle a convecting fluid.

BENCHMARKS

We have benchmarked our numerical scheme against analytic solutions in order to characterize its strengths and weaknesses, and to quantify the likely level of ac curacy we can achieve with a given mesh/particle den sity. We first benchmark the purely viscous flow case to provide a baseline for comparison with viscoelastic cases.

Analytical solution

We study the spreading of a rectangular sample of ma terial under a constant downward velocity V applied on top (see Fig. 2).

The specified boundary conditions give:

{ ( \I ) 00 ( b h o ) ; (1 -f.t); -1 } In 1 --t + L -- ho ;;J aV i.i! (25)
We use this relationship to eliminate the pressure derivative from ( 21) and ( 7) and express the unknown The Eulerian mesh does not carry any information from timestep to timestep other than the boundary conditions. Therefore, when convenient, the mesh may be modified, replaced, and refined as necessary.

For this problem, compression is applied by a moving boundary condition which causes the mesh to com pact in one direction. For simplicity, the mesh is sim ply scaled to the new aspect ratio without altering the number of elements. In a more complicated situation, however, it would be possible to regrid completely without loss of accuracy. The only detail which needs to be observed is that the updating of the boundary node locations follows the same formulation as that of the particles (here, a second order Runge-Kutta inte gration procedure) to prevent the boundary conditions from drifting with respect to the stored information on the particles.

In the x-direction we have a free surface. In order to investigate the properties of a particle representaion of such interfaces, it is important not to simply use a mesh-based boundary condition. Instead we use a mesh (Fig. 2) larger than the specimen and fill the gap with a backgroun� material having (17 = 103 MPa.s and� = 105 MPa.s).

The square mesh is composed of 4096 elements.

Results

We have tested the code, ELLIPSIS, with three dif ferent type of materials: viscous and compressible 4). The numerical parameters for the viscous part of ( 15) and ( 14) are:

TJ = 106 MPa.s. and e = 4.106 MPa.s.

(

) 27
Due to the mesh size and the prescribed velocity on top, the Courant condition requires 6.t < 5.10-3 s .. For our problem we take 6.t = 10-3 s ..

As an indicator of accuracy we compare analytical and numerical x-velocity at the point I (Fig. 2). The steps in the numerical solutions are related to the the motion of the material interface relative to the element edges. When the interface between the specimen and the background material crosses into a new element there is an immediate discontinuous contribution to the element equations from the sample material. We have verified that jump tends to zero as we increase the element and particle densities.

In the viscoelastic case, in Fig. ( 4) we can see that, for different relaxation time, the error against the an alytical solution remains below 3%. Some steps are still present, but in the viscoelastic case the veloc ity is considerably more noisy. What is most clear is that the computations with larger relaxation time have greater fluctuation in accuracy. The problem be comes more acute in this case when the Courant time (decreasing due to the compression of the background mesh) becomes comparable to the relaxation time. This can result in a loss of stability which is entirely an artefact of the discretization. The most promising solution to this issue is to compute the time-derivative of the stress tensor over a physically relevant inter val, rather than that imposed by the mesh. This is a particular focus of our current research. 1) is a gravitational body force due to density changes. We assume that these arise, for any given material, through temperature effects:

v • r -V"p = gpo ( I -aT)z ( 28
)
where g is the acceleration due to gravity, Po is mate rial density at a reference temperature, a is the coef ficient of thermal expansivity, and T is temperature.

z is a unit vector in the vertical direction. We have also assumed that the variation in density only needs to be considered in the driving term (the Boussinesq approximation).

The equation of motion is then

( [ r1 W1r1 r1W']) V T/eff --+ ----- µ f:l.t µ µ (29)
The velocity field u and pressure at t + 6.t can be solved for a given temperature distribution and the stress history from the previous step. Motion is driven by the heat escaping from the inte rior. The energy equation governs the evolution of the temperature in response to diffusion of heat through the fluid. For a given element of fluid, (30) where" is the thermal diffusivity of the material.

So far, all equations have been written in a purely Lagrangian framework. The time derivate of temper ature and the Jaumann stress rate refer to a frame of reference which is carried by the fluid. In choosing a solution method, it is necessary to choose whether to honour the Lagrangian formulation, or to work with a fixed reference frame and introduce additional terms to compensate for the advection of temperature and stress by the fluid.

Brittle failure

As we discussed above, plate models need to in clude a description of the brittle nature of the cold est part of the lithosphere. Geologists use this term quite loosely to distinguish fault-dominated deforma tion which may result in seismic activity, from ductile creep which occurs at higher temperature and pres sure. In all recent studies of mantle convection where the brittle lithospheric rheology has been taken into account, the brittle behaviour has been parameterized using a non-linear effective viscosity which is intro duced whenever the stress would otherwise exceed the yield value Tyield• This approach ignores details of individual faults, and treats only the influence of fault systems on the large-scale convective flow.

To determine the effective viscosity we extend (6)

by introducing a von Mises plastic flow rule:

v T T T • • • • -2 + -;;-+ -X� I I =D,+Dv+Dp =D (31) µ -17 -T
where A is a parameter to be determined such that the stress remains on the yield surface, and lrl = The value of ,X or 171 is iterated to allow stress to redistribute from particles which become unloaded. The iteration is repeated until the velocity solution is unchanged to within the error tolerance required for the solution as a whole.

The value of the yield stress is, in principle, a func tion of strain, yield history, and temperature, and can be distinct for different materials.

PLATE MODELING

As a simple example, we demonstrate the compres sion of a viscoelastic-brittle layer which lies on top of a slightly less dense viscous fluid layer (Figure 5). This system is an analogue of the cool oceanic lithosphere which rests upon the warm asthenosphere (though we do not solve the temperature equation in this case). The viscoelastic layer is initially split to provide an initiation point for a model subduc tion zone. The vertical boundaries are free-slip, and the right hand edge is given a horizontal velocity to shorten the system. There is a layer of highly com pressible material of very low viscosity above the elastic layer which accomodates the volume change associated with shortening of the mesh, and mimics a free surface boundary condition on the upper surface of the elastic layer. As compression proceeds, the vis coelastic layer flexes and the viscous layer flows to accomodate the deformation. As stresses build up in the model lithosphere, a second failure point develops allowing one half of the material to fold up under the other half. Further compression forces the two halves of the lithosphere layer to slide past each other along a zone of material failure. After this point, the pres-ence of the bottom boundary begins to interfere with the evolution of the system. This particular simulation demonstrates the capa bility of the algorithm in the simulation of subduction zone geometry in the style of [START_REF] Melosh | Dynamic support of the outer rise[END_REF] or Gur nis et al, ( 1996). The fact that the algorithm is imple mented within a fluid-dynamics framework suggests that a viscoelastic analysis of convection with strong temperature dependence of viscosity and a yield stress is now possible.

DISCUSSION

The algorithm described above is designed to intro duce elastic effects into convection simulations where temperature-dependent viscosity and yielding domi nate the mechanical behaviour. The viscosity of the mantle and the mantle lithosphere is very strongly de pendent on temperature (several orders of magnitude variation over 1000°C) whereas the shear modulus is not strongly affected (there is only a modest change in seismic wavespeed due to temperature). Therefore, elastic effects become unimportant outside the cold thermal boundary layer where viscosity is extremely large.

The influence of elastic stresses is likely to be felt at the subduction zones where the lithosphere is bent into the interior of the Earth. In these regions stresses are typically close to the yield stress -a fact which allows the plates to move in the first place. The ad dition of elasticity is likely to complicate the simple picture presented by [START_REF] Tackley | Self-consistent generation of tec tonic plates in three dimensional mantle convection[END_REF] and Moresi & Solomatov (1998) for viscous materials with a yield stress.

Our methodology is limited to a coarse continuum description of the subduction zone system at a resolu tion of a few km. This may be able to give us valu able information into the nature of plate tectonics, the thermal conditions in and around subducting litho sphere, and the stress state of the system. However, the resolution is too coarse to say anything about the detailed mechanics of the failure of lithospheric fault zones and the conditions for major failure. to occur. For this we require a coupling of the large-scale code with an engineering-scale code (e.g. DEM or small deformation Lagrangian FEM) using the large-scale to provide boundary conditions for the small scale. The issue of scale-bridging is important in many ar eas of numerical simulation. Essentially the same dif ficulties arise in material science where the atomic scale is best treated by molecular dynamics codes but the large scale must be treated as a continuum (e.g. Bernholc, 1999).

  Figure I: A simplified cross section of the Earth with major layerings shown to scale except for the upper boundary layer which is exaggerated in thickness by a factor of roughly two.

  1

  the deviatoric part of D and tr(D) repre sents the trace of the tensor. Individually we express each deformation tensor as a function of the deviatoric stress tensor r and pres sure P, which finally gives a tensorial equation: . is the Jaumann corotational stress rate for an element of the continuum, µ is the shear modulus and 77 is shear viscosity. The isotropic part gives a scalar equation for the pressure:

  dJ over the element domain n• we replace the contin uous integral by a summation(19)

DFigure 2 :

 2 Figure 2: Geometry and boundary conditions of for the analytic solution.

  Dxx as a function of p and D zz. Consequently we ob tain a relation between the velocity Vx of a point and its coordinate .r. \.� = ___ x_ x ]

  Fig.(3.c and d}, viscous and incompressible Fig.(3.a and b), viscoelastic and compressible Fig.(

  Figure 3: Viscous deformation . results _ : a. viscou � in compressible, analytical, b. viscous mcompress1ble, numerical, c. viscous compressible, analytical and d. viscous compressible, numerical.

Figure 4 :

 4 Figure 4: Viscoelastic material a. Numerical solution, b. µ = 104 MPa., c. µ = 1Cl5 MPa.

(

  T; jTiJ/2)(1121. We again express the Jaumann stress rate in difference form (I 0) to give:t+ t. t [ l l .X ] (W'r'r'W') (32) 2µC...t 2µNo modification to the isotropic part of the problem is required when the von Mises yield criterion is used.At yield we use the fact that lrl = Tyield to writer t+ At = r/ [2f/ + t. t + -1-r' + �( W ' r' -r'W')]µC...t µ using an effective viscosity, 171 given by 1 l'JTyieidf!C...t 1'/ = ----�-�---- 1'] Ty i e Id + TyieldµC.,, t + A1']µC...I We determine .X by equating the value of lr•+A• I with the yield stress in (33). Alternatively, in this particular case, we can obtain 171 directly as 1 ]' = T yie ld/ lf>effl where (35)

Figure 5 :

 5 Figure 5: Example: compression of a viscoelastic plate with yield stress overlying a low viscosity fluid of equal density D.

  ) become respectively t+t.t 71 2'.t b' + .e. t a t _ _ (2'. t D '+.e. t + __ /3_ t

	T	=	a + 2'.t		+	a + 2'.t	r
			+	0:2'.t 2'.t +a	(W'r1 -r1W') (11)
	P	-/3 + At kk	/3 + At	p	(12)

and t+t.t

ACKNOWLEDGEMENTS

The authors gratefully acknowledge the suggestions of an anonymous reviewer which have helped to clar ify this paper.