Mathematical Models of Ice Shelves

Abstract

(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...

Ice sheet grounding line dynamics: Steady states, stability, and hysteresis

“…At the calving front, there is an imbalance between hydrostatic pressures in ice and water due to the buoyancy of ice. This imbalance must be compensated by a longitudinal stress [e.g., Shumskiy and Krass , 1976]: …”

Parameter Sensitivity Studies for the Ice Flow of the Ross Ice Shelf, Antarctica

“…In contrast, Clough and Hansen (1979) obtained a linear temperature profile and marine ice layer of 6 m thickness at station J9. The process of marine-ice accretion has a direct influence on the temperature profile, as deduced by analysis of the heat transport equation by Shumskiy and Krass (1976). Radio echo soundings (Neal 1979) and studies of bottom mass balance rates (Thomas 1978, Shabtaie andBentley 1979) gave evidence that at the ice/ocean interface of the Ross Ice Shelf mainly melting conditions prevail (in contrast to the Hells Gate, Amery and Filchner-Rønne Ice Shelves).…”

Parameter sensitivity studies for the ice flow of the Ross Ice Shelf, Antarctica

“…In contrast, Clough and Hansen [1979] obtained a linear temperature profile and marine ice layer of 6 m thickness at station J9. The process of marine ice accretion has a direct influence on the temperature profile, as deduced by analysis of the heat transport equation by Shumskiy and Krass [1976]. Radio echo soundings [ Neal , 1979] and studies of bottom mass balance rates [ Thomas , 1978; Shabtaie and Bentley , 1979] gave evidence that at the ice‐ocean interface of the Ross Ice Shelf mainly melting conditions prevail (in contrast to the Hells Gate, Amery and Filchner‐Rønne Ice Shelves).…”