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. I ts solution is reduced to the solution of a system of recurrent nonuniform 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. RESUME. Modele mathematique d'un glacier a trois dimensions nonisotherme . Le modele mathematique est construit pour des glaciers terrestres d'epaisseur tres inferieure aux dimensions horizon tales et que les rayons de courbures des grandes irregularites du fond, par la methode de la fine couche limite en coordonnees orthogonales sans dimensions. La dynamique est decrite par un systeme d'equations determine statiquement, ainsi est resolu le probleme des contraintes. Pour le cas general non isotherme, le calcul des distributions interdependantes de la vitesse et de la temperature est realise par iteration des solutions pour la vitesse et la temperature. La distribution de la temperature est determinee par une equation parabolique avec un petit parametre pour la derivee seconde. La solution se ramene a la resolution d'un systeme d'equations differentielles non uniforme reccurentes du premier ordre par le biais d'un developpement en serie du petit parametre. Une couche limite conductrice relativemen t mince s'ajoute aux surfaces superieures et inferieures du glacier, jouant le role d'un egalisateur de temperature dans le zone d'ablation . Pour les cretes de glaces, un probleme statiquement indetermine est resolu , si bien que les resultats sur les contraintes dependent de la distribution de la temperature. ZUSAMMENFASSUNG. Mathematisches Modell eines dreidimensionalen nicht-isothermen Gletschers. Fur Land-Gletscher, deren Dicke bedeutend geringer ist als ihre horizontalen Abmasse und die Krummungsradien grosser Unregelmassigkeiten am Untergrund, wird mit Hilfe der Methode der dunnen Grenzschicht ein mathematisches Modell in dimensionslosen orthogonalen Koordinaten entworfen. Die Bewegungsvorgange werden durch das st<\tisch bestimmte Gleichungssystem erfasst, wobei sich die Losung fur den Spannungszustand ergibt. Fur den allgemeinen nicht-isothermen Fall werden die gegenseitig abhiingige Geschwindigkeit und ...
Modelling the thermodynamics of a large ice sheet is in essence a problem of the deformation of a non-isothermal and relatively thin viscous layer. Reasons for forecasting the behaviour of a large ice sheet are: (a) to specify the relationship between the existence and development of the ice sheet and regional climatic and tectonic conditions; (b) to study the influence of variations in surface temperature, mass balance, and geothermal flux (as functions of coordinates and time) on the evolution of the thickness of the ice sheet. Hence the principal requirements for modelling are: 1.The model must be evolutionary; in particular, various stationary stages should be derived from it (rather than postulated).2.The model must be non-isothermal because temperature variations amount to some tens of degrees.3.Two-dimensionality of the model is essential if we are to use it for making realistic forecasts.4.Conditions at the boundary between land ice and the sea, which are the basic factors controlling the location of the margin of the ice sheet, should be taken into account.5.Finally, the model must be acceptable for digital analysis by computer.Contemporary mathematical models of the dynamics of the Antarctic and Greenland ice sheets are too simplified and do not fulfil these requirements. Our mathematical model of the evolution of the Antarctic ice sheet is based on temperature parameterization. The parameterization correctly describes the distribution of temperature and its dependence on surface temperature, advection of ice, geothermal flux, and heat losses. The problem reduces to Cauchy’s problem for a non-linear parabolic partial differential equation with three-step dependence of ice temperature on depth. A linear viscous flow law is assumed. Tests treat the whole problem of the thermo-hydromechanics of the ice sheet. The ice sheet is assumed to be bounded by a grounding line or ice wall where the ice is in hydrostatic equilibrium with the sea. In this case, the problem reduces to a non-linear parabolic partial differential equation with a complicated boundary condition at the moving edge. A digital representation of the model is presented first. Tests demonstrated the possibility of using an explicit three-layered scheme. Calculations for the East Antarctic ice sheet indicated that the ice margin should retreat slightly under contemporary climatic conditions (which corresponds to the field evidence). In future the boundary should tend towards a stationary state. A forecast of the behaviour of the ice sheet is derived. The East Antarctic ice sheet is found to be stable during various changes in climatic factors. The calculated hydromechanical characteristics of the ice sheet agree with the observed field data.
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