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This encyclopdia entry is forthcoming for the Ashgate Encyclopedia of Literary and Cinematic Monsters, ed. by Jeffrey A. Weinstock. Full publication details will be added here in due course, and can also be found at ~ ~ ~ Elves The word elf comes from Common Germanic, the ancestor-language of English, German, and the Scandinavian languages. Our earliest solid evidence for beliefs about elves comes from the second half of the first millennium A.D., in texts written by Churchmen in Germanic languages.
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- christmas traditions
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Viscous Plastic Model for the Glaciers Flow (Fedchenko’s Glacier as an Example)Farshed H.Karimov Institute of Geology, Earthquake Engineering and Seismology of the Academy of Sciences of the Republic of Tajikistan Dushanbe, Republic of Tajikistan, Aijni str., 121 Abstract The descrition of thelacier flow has been carried out in the model of viscouslastic bod onthe examle of the Fedchenkolacier on PamirsTa ikistan.The movement of the bod isconsidered as a su er osition of its movement as a whole, and as a result of dis lacementof arallella ersfrom slidinsurface toward the uer lasticboundar .In the ro osed modelthe s ecific estimates for the coefficients of friction and cohesion at the boundar ofthe lacierwith the surface of slidinhave been made. The seed of a laer is risin buadratic arabolawith the heiht of the laer under the slidinsurface. Estimates 13 for the viscositof laciersubstance within the values 10Pa∙s, characteristic for ice solids exhibitin viscous-lastic flow, have shown the velocity values close to the real data 0,7 m per day at an average. Theeffect of seismic vibrations on the velocitof thelacier, that reresents articular si nificancefor studies of the dynamics of glaciers in Central Asia in terms of climate change, has been considered. Seismic vibrations can accelerate the movement of glaciers in dependence of the amplitude, frequency and polarization of seismic waves. Introduction Inaccordance with theh sicalre resentationsthe laciersare solid, viscous-lastic bodies movinon an inclinedlane under the influence ofravitational forces. The uer art of thelacier bodis hard enouh and can be rearded as solid elastic medium. But its lower arts are under theressure of uer laers because of theravitational force imact and so, if the uer laer is thick enouh, theressure transfers the solid elastic state of the innerlacier art into the viscous-lastic one. The model of viscouslastic bodfor thelacier rovides tools to analze and estimate the dnamical arametersof thelacier flow. The results are im ortant fortakin intoaccount the reularities inlacier flow and arranging engineering protection measures in controlling the glaciers’ phenomena. The model and research methods Inaccordance with theh sicalre resentationsthe laciersare solid, viscous-lastic bodies movinon an inclinedlane under the influence ofravitational forces[1]. Let’s consider thelacier flow on the examle of the Fedchenkolacier, one of the biest laciers 2 of the world, located on the Pamirs in Ta ikistan. Thelacier area is eual to 600 km , the 3 volume of the bodis 130 kmmean thickness is about 500 min some sites reaches about 1 3 km, boddensit is900 k/m 1,6, flow rate is about 0.7 mer daat an averae, sloe radient is 0.06 - 4.5 km throuh 70 km. Maximal flow rate for the big glaciers are estimated lying in the range from 1 cm to 100 m in order of magnitudes [1-5]. The larger is a glacier, the faster it moves. The higher rates are in summers, the lower are in winters [1]. The middle part
of the Fedchenko glacier is moving faster with the rate about 0.9 m per day, the outer parts – about 0,27 m per day [1]. The “warm ice” and accumulation of water is observing in the Fedchenko glacier’s bottom [6]. The so called limit of hardness of the ice body under the 2 quasi static deformations is estimated in 4,5-5,0 MPa, i.e. 45-50 ton/m[2-5]. The ice‘s fluidity limit composes 55-65% of the compression strength, which is equal to 2,5-3,0 MPa -13 (look in the [7]). The viscosity coefficient is accepted as 10Pa∙s [2]. Takingin mind these physical parameters of the glacier body let’s compose the model for the flow.The pressure of the upper layers on lower ones,, can be estimated by means of thefollowing well known elementary expression pρg h, (1) whereρthe gravityg isis the thickness of the upper layer,is the body density, acceleration constant. Ifto insert the physical parameters into (1) one can find out that the critical thickness about 250 m is the threshold for the body substance transition from elastic state to plastic viscous. Besides the critical thickness for loosing the body’s compression strength is equal to about 600 m. By the way it follows, that as the matter of fact stable glaciers on the Earth can’t have thickness exceeding about 600 m, like mountains, which can’t be thicker than about 10 km [8]. Once they are becoming thicker, the bearing rocks in the bottom turned up mechanically broken and the object subsides maintaining the maximal affordable thickness. Sothe model for the glacier can be represented my means of two layers, hard elastic one as the upper part and viscous plastic as the lower part (Fig.1). The inclination angle isα. The body thicknesssliding forceF andaxis. Gravity forceP createsh isshown at the 0 oppositely pointed friction forceF. The hard part is shown in gray filled block; the viscous Tp plastic block is with no color filled. Fi .1: Glacier bodflow model9 10.
h F h0
0
PF α Themovement of the viscous plastic layers will be described by means of well known Navier-Stox’s equation for the layer with the mass– 2 dv dv m× 1m×g×sinα#η× ×S, (2) 2 dt d where isthe speed of this layer,t isthe time coordinate,ηis the viscosity coefficient,S is the layer’s slide surface square [11].
Calculations for the flow glacier’s velocity Theevenness of the glacier’s set stable movement at the infinitesimal low inclination angle 0.06 leads to the conclusion that there isn’t sliding between ending border layer and bedding plain. So it can be assumed that the large compression at the bottom of glacier and cohesion between the glacier body and bedrock inclination plain make the body’s boundary speed equal to zero: v |0. (3) h 0 dvρg 1%×h×sinα#c, (4) dhη wherecis the integrand constant. Thesolution for the upper rigid body’s part is the following – dvρ×g 1 ×H×sinα. h1h dhη 0 (5) Combiningconditions (4) , (5) with (3), one can obtain the following expression for the speed of a layer at the height: ρg h v1 ×h×sinα×H#h%. 0 η2 (6) Subtitutionof the physical values for the Fedchenko glacier into (6) leads to the upper glacier speed, equal to 0.7 m/day, nicely coinsiding with the really observed values. Conclusions Fairywell coinsidence of the values of upper glacier’s speed, obtained both following by the viscous plastic body and really observed one for the Fedchenko glacier confirms the rightfulness of the model. The leading role of viscous plastic effects in the dynamics of big glaciers is proving true. References ([1-7, 9-11 – in Russian], [8] – in English):
1.Tajikistan. The Nature and natural resourses, (1982). Dushanbe, „Donish“, 602 pp. 2.Geological glossary. Ed. Pfaffenholz, K.N., (1993). Moscow (M.): „Nedra“, v.1, 488 pp. 3.Paterson W.S., (1984). Physics of glaciers, M.: „Mir“, 312 pp. 4.Physical properties of rocks and mineral resourses.Chief ed. Fedynskii V.V. (1976). M.: „Nedra, 527 pp. 5.Vyalov S.S., (2000). Ice soil reology. M.: „Sirojizdat“, 464 pp. 6.Malkhasyan E.G., Rudich K.N., (1987). Volatile look of the Earth. M.: „Nedra“, 140 pp.
7.Lobanov B.A., (2006). Ice’s modeling in the finit element problems. E-magazin „Diffequations and management processess“, No4. 8.Mach E., (1897). The form of liquids, in „Popular scientifc lectures“. La Salle: „The Open court publishing company“, 382 pp. 9.Karimov F.H., (2011). Seismogenic landslides on the Tajikistan’s territory: from hazard assessment to risk reduction. Dushanbe: „Contrast“, 78 pp. 10.Karimov F.H., (2009). Factor of seismicity at the stability of snow covers and ice dambs. Dushanbe: „Koinot“, p. 74-79. 11.Landau L.D., (1988). Theoretical Physics, v.6. Hydrodynamics, M.: „Nauka“, 736 pp.
