A mathematical model is presented of non-stationary melting processes of ice including particles of morainic material. The problem is treated as a Stephen-type one with the phase boundary of ice melting being located under the debris cover. The main terms of the heat-balance equation for a glacier surface are solar radiation and convective heat transfer. The quantitative relationships characterizing the effect of glacier run-off augmentation from under a thin layer of debris cover are obtained for different bulk moraine concentrations inside the ice. The concept of equivalent time is introduced. It is defined as the time elapsed until the moment the sub-moraine ice-ablation rate becomes equal to the ablation rate of clean ice. This moment signifies the beginning of the shielding stage. Thus, a glacier can be considered as a self-controlling system with respect to its summer run-off. A series of numerical tests for Djankuat glacier, Central Caucasus, has been carried out. The dynamics of moraine-cover growth and alterations of seasonal ablation rate under debris show perfect agreement between the computed data and the results of 14 years of direct observations. Some practical recommendations concerning artificial blackening of a glacier surface for augmentation of liquid run-off are presented. Conditions promoting increase of run-off are: relatively high albedo, relatively low summer air temperature, and relatively small convective heat transfer between the air and the ice surface. The method of artificially blackening a glacier surface is by means of a durable thin dark polymer film. In conclusion, some further aspects of the problem are discussed.

ABSTRACT. A mathematical model is presented of nonstationary melting processes of ice including particles of morainic material. The problem is treated as a Stephen-type one with the phase boundary of ice melting being located under the debris cover. The main terms of the heat-balance equation for a glacier surface are solar radiation and convective heat transfer. The quantitative relationships characterizing the effect of glacier run-off augmentation from under a thin layer of debris cover are obtained for different bulk moraine concentrations inside the ice. The concept of equivalent time is introduced. It is defined as the time elapsed until the moment the sub-moraine iceablation rate becomes equal to the ablation rate of clean ice. This moment signifies the beginning of the shielding stage. Thus, a glacier can be considered as a self -controlling system with respect to its summer run-off. A series of numerical tests for Djankuat glacier, Central Caucasus, has been carried out. The dynamics of moraine-cover growth and alterations of seasonal ablation rate under debris show perfect agreement between the computed data and the results of 14 years of direct observations. Some practical recommendations concerning artificial blackening of a glacier surface for augmentation of liquid run-off are presented. Conditions promoting increase of run-off are: relatively high albedo, relatively low summer air temperature, and relatively small convective heat transfer between the air and the ice surface. The method of artificially blackening a glacier surface is by means of a durable thin dark polymer film . In conclusion, some further aspects of the problem are discussed . RESUME.

(lnstitut Mekhaniki , Moskovskiy Gosudarstvennyy Universitet im . M. V . Lomonosova , Michurinskiy Prospekt, Moscow V-234 , U.S.S.R. )!\ nSTRACT. For Aat externa l ice sheh·cs. expanding freely in a ll direc tions. the problem of thermodynamics is o ne-dimensional. In the affine dimensio nless system of coordina tes , equations of the dynamics together with the rheological eq u ation lead to th e non-lin ear integro-diffe rentia l equation in volving the reduced tempe rature. In the quasi-stead y case the boundary probl em fo r this equation is solved by mea ns of the me thod of combining asymptoti c expansions. I t is shown tha t if ice is coming from the upper and lower surfaces in the opposite direc tions the regime is u nsteady becau se of the internal h ea t accumulation.The integro-differential equation for the temperature in the case o f thinning internal ice shelves is more comp lica ted , but it can be solved by a method analogous to the one mentioned above. R ESUME. Modi/es mathi l1latiqlles de plateformes de glace. Po ur d es plateform es d e gla ce externes pl a tes. s'e te ndant librement dans toutes les direc ti ons le probl eme th erm odynamique es t uni-dimensionnel. Da ns un sys tem e de coord onn ees affine sa ns dim ensio ns, les eq uati ons de la dyn am iquc com binees avec I'equation rheologiquc conduisent it un e equation integ ro-diffe renti elle non-lin ea ire concern a nt la tempera ture red uite. Dans le cas d'un e tat quas i-stationn a ire. le pro bl e m e des limites p o ur ce u e equa ti on es t resolu en associant d es developpements asympto tiqu es. On mon trc que si la glace arrivan t a u voisinage d es surfa ces inferieures e t supe rie ures provi ent d e direc tions opposees , le regime es t insta bl e it ca u se de I'accumulati o n interne de ch aleur.L' equati on integro-differentielle pour la te mpe ra ture d a ns le cas de pla teform es de glace intern es a min cissa ntes est plus comp liquee. ma is la m a rch e d e la solution es t a na logue it cell e m e ntionnee ci-dess us. VO Il Scheifeisen. Fur Aa ch e, aussere Schelfe ise, die sich allsei ts fr ei ausdehnen konne n , genugt ein eindim ensiona les thermodynamisch es Modell. Im affinen , dimensionslosen System der Koordinaten fuhren di e d ynamischen G leichunge n zusammen mit der rheologischen G leic hu ng zur nichtlineare n Int egro-Diffe re nti a lgleichung fur di e reduzierte Tempe ratur. Im quasistationaren Falllasst sich das Randwertprobl e m fur di ese G leichung mit Hilfe der Nahtme thode. verbunde n mit asymptotischer Fortsetzung losen. Es wird gezeigt, dass fur den Fa ll d es Eiszustromes von der Ober-und Unterseite in entgege ngese tzt e r Ri chtung d as System infolge d e r Ansammlung innerer Warme instationiir wird . ZUSAMMENFASSUNG. Mathematische ModelleDie Integro-Differenti a lgle ichung fur di e Tempera tur im Fa ll e von ausdunnenden. inneren Schelfeisen ist verwi ckelter, doch lasst sie sich mit ahn lich e n Methoden losen wir die oben genannte. INTROD UCTIONFloating glaciers are Aat slabs which b ec...

For flat external ice shelves, expanding freely in all directions, the problem of thermodynamics is one-dimensional. In the affine dimensionless system of coordinates, equations of the dynamics together with the rheological equation lead to the non-linear integro-differential equation involving the reduced temperature. In the quasi-steady case the boundary problem for this equation is solved by means of the method of combining asymptotic expansions. It is shown that if ice is coming from the upper and lower surfaces in the opposite directions the regime is unsteady because of the internal heat accumulation.The integro-differential equation for the temperature in the case of thinning internal ice shelves is more complicated, but it can be solved by a method analogous to the one mentioned above.

A mathematical model is constructed for land glaciers with the thickness much less than the horizontal dimensions and radii of curvature of large bottom irregularities by means of the method of a thin boundary layer in dimensionless orthogonal coordinates. The dynamics are described by a statically determinate system of equations, so the solution for stresses is found. For the general non-isothermal case the interrelated velocity and temperature distributions are calculated by means of the iteration of solutions for velocity and for temperature. Temperature distribution is determined by a parabolic equation with a small parameter at the senior derivative. Its solution is reduced to the solution of a system of recurrent non-uniform differential equations of the first order by means of a series expansion of the small parameter. A relatively thin conducting boundary layer adjoins the upper and lower surfaces of a glacier, playing the role of a temperature damper in the ablation area. For ice divides, the statically indeterminate problem is solved, so the result for stresses depends on the temperature distribution.