Snow and Ice-Symposium-Neiges et Glaces (Proceedings of the Moscow Symposium, August 1971; Actes du Colloque de Moscou, août 1971): IAHS-AISH Publ. No. 104, 1975. The formation of ablation moraines as a function of the climatological environment G. E. Glazyrin Abstract. A simple mathematical model of the formation of ablation moraine is given. The model was constructed on the assumptions that moraine products are deposited only along a narrow strip around the glacier; that there is no 'mixing' of the ice in glaciers; and that the motion of moraine fragments relative to the ice is very small. The model allows some determination of some characteristics of the life of the glacier. The model was tested on Ayutor-2 glacier, West Tien Shan. Résumé. On propose un modèle mathématique simple pour la formation d'une moraine d'ablation sur un glacier de montagne. Le modèle est construit à partir des hypothèses suivantes: (1) Le matériau morainique entre sur le glacier dans une zone relativement étroite située à l'amont du glacier (2) Il n'y a pas de 'turbulence' de la glace dans le glacier (3) Le mouvement du matériau morainique relativement à la glace voisine est très lent. Le modèle donne la possibilité de définir quelques caractéristiques de 'la vie' du glacier. Certains de ses aspects ont été vérifiés sur le glacier Aiutor-2 dans le Tian-Chan Occidental. The material used in the formation of surface moraines emerges on glaciers in three ways: (1) It is transferred by wind. This source is negligible (Nasyrov and Sadykov, 1966) even for glaciers of Middle Asia, where the content of dust in the air is very great, so it is not taken into account. (2) As a result of erosion of the glacier bottom; but the thickness of the ice layer near the bottom loaded with the moraine material is quite small even for ice caps and for the valley glaciers of Middle Asia it is even smaller (Boulton, 1970). (3) From the slopes bordering the glaciers. This source is of great importance for the glaciers whose accumulation area is surrounded by steep slopes, which are partially free from snow and ice. These glaciers which are nourished by avalanches, are widespread in Middle Asia and have the name 'Turkestan-type glaciers' (Kalesnik, 1937). Thus the rock material is transferred from the surrounding slopes to the comparatively narrow zone of glacier surface. Then as a result of glacier motion, this material moves downwards. Since this process takes place in the accumulation area, where accumulation predominates over ablation, the snow layer containing the moraine material (later we name it 'moraine layer') is Covered with snow layers, free from stones, and 'plunges' into the glacier body. When it reaches its maximum value at the altitude of the firn line, the thickness of snow and ice which covers the moraine layer begins to decrease downstream. In the points where it becomes zero, the ablation moraine appears (Fig. la). Now we introduce some assumptions: (1) The formation of moraines is a uniform and continuous process. This assumption is very inappropriate, as the stones which were deposited in the ice in the upper part of the glacier do not reach the surface on the tongue until after some tens of years. Thus, there is no necessity to take into account changes of ablation and intensity of the stone deposition and snow accumulation on the glacier surface year after year. This allows us to eliminate time as a variable from equations.
The formation of ablation moraines 107 bottom of glacier FIGURE 1. Diagram showing ablation moraine formation on a mountain glacier, (a), The longitudinal section; (b), Plan view. (2) Turbulent intermixing of the ice does not take place. Hence we may consider the processes which take place along isolated flow lines to be a function of altitude ' only and to use the simple differential equations. (3) The rock material which is deposited on the glacier does not move relative to the ice surrounding it. We introduce the following definitions: z is the vertical coordinate; X(z) is the long-term mean annual precipitation at an altitude z ; P{z) is the mean annual surface ablation at an altitude z; h(z) is the thickness of the snow or ice layer (in waterequivalent) covering the moraine layer; t is time; s is a distance along the glacier downwards; V(z) is the glacier velocity along the flow lines containing material which will form the future moraine; d(z) is the glacier slope; z x is the altitude of the glacier zone where the rock material from the surrounding slopes is accumulated; z^ is the firn line altitude; z 2 is the altitude where the ablation moraine appears. Now we define the simple relations. or thus ds dh = (X-P)dt, dt = V ' ds dt = dh = X-P and dh = -(X-P) sm a, KSina This simple equation leads to the following interesting conclusions: (1) If the accumulation of moraine material on the glacier surface in its upper part takes place in rather narrow space and the lower glacier is covered with ablation moraine for a large distance, then the glacier must be retreating. (2) The following relation exists between the upper limit of the glacier and the upper limit of the ablation moraine: the lower the altitude of the upper limit of the glacier, the closer it is to the firn line and the higher is the corresponding limit of ablation moraine, along a given flow line. In Fig. lb, for example, the points a and c at the upper limit of the glacier correspond to a x and c x at the upper limit of the ablation moraine. If the upper limit is the glacier is below the firn line, an unbroken moraine cover may occur. (3) The well-known definition of the firn line altitude as the average value of the altitudes of the lowest point of the bergschrund and the upper point of the ablation moraine may be obtained. Now we shall consider equation 1 from another point of view. Shumsky (1947) (1)
108 G. E. Glazyrin introduced the important concept characterizing the intensity of the glacier 'life'. It was 'energy of glaciation': *4 X(z)-P(z) If we assume in the first approximation that X(z) and P(z) are linear functions within the interval from z x to z 2, and that z a = Zf then, integrating equation 2, we find X P = E(z - Zf '). Substituting this expression in equation 1 we obtain: dh _ E(z ~~ Zf) Thus, the velocity of immersion and the rise of the moraine material are proportional to E. It means that if the other conditions are constant, the larger E is, the shorter is the path of moraine material in the ice. Different ways of using equation 1 were shown by Glazyrin (1969). We present some of these here. Until the present, the depth under the glacier surface at which the moraine layere is located was unknown, and this information is of great importance for the carrying out of observations with geophysical methods. If functions X, P, V and a are known then integrating equation 1 with the initial conditions z 0 =z l and h(z 0 ) = h(zi ) = 0 we have, (2) h = - I J V sin a As noted before, Fis the velocity of the ice motion along a flow line stretching along the upper boundary of the moraine layer. The equation may be applied if one assumes equality of the surface velocity and the velocity along the flow lines. The true depth of the immersion of the moraine layer is H = hp where p is average vertical ice or firn density. Now we consider the process of the formation of the surface moraine below the point z 2. Let m(z) be the specific moraine mass (the mass per unit area) at the altitude z;r(z) is the moraine thickness; c{z) is the mean concentration of the moraine material in the ice layer, which emerges on the surface at the altitude z; P t is the mean relative intensity of ice melting, i.e. the annual ablation of pure ice. We consider the moraine section situated at the altitude z. The changes in the moraine thickness per unit of distance along the moraine flow line are determined by the intensity of ice melting under the moraine (P m ) and by the concentration of the moraine material in the ice. If the glacier moves with the velocity V(z) then for the time dt it will pass a distance ds = Vdt. As ds = sin a we find = V sin a dt (3) During this time the ice melting will be P m dt. The ice-covered moraine is melting more slowly, so the layer of moraine cover is thicker. Thus the melting velocity under the moraine is P m = P x (fir) where <>(r) is the coefficient of the melting decrease by moraine with thickness r. Subsequently during the time dt the ice layer P m dt =Pi(j)(r)dt will melt under moraine and the specific mass of the moraine will
The formation of ablation moraines 109 increase by the value dm = cp m dt = cp x (z)dt. Combining equation 3 with this expression we obtain dm _ cpi <p{r) V sin a Note that we consider only the ice melting and P x is not the mean annual intensity of snow and ice melting. In most cases the specific moraine mass is not measured, but the thickness is measured. The two are related as m = ar where a is the moraine density. Equation 4 may be replaced by equation 5: dm _ d (ar) dr da _ cp $(r) Assume that we know the dependence of P x, V, a, a, r, on the altitude and hence da, dr -r" and In this case using equation 5 we may calculate the moraine material concentration in ice layers appearing on the surface at the different altitudes. Such an attempt was made for the glacier Ayutor-2 (West Tien-Shan). Shown in Fig. 2a-d are the values (4) (5) V cmcay 8 4- r s v' f cm 12 v> s. (4 O 3.6 3.16 3.SO 322 to- Q,6- S (?) SlniX A o,l- *"-- Ql I K> 3, to 3.SQ 3,22 -~Znm l» M O S.6 -i ' ZKm 320 3.22 FIGURE 2. Some characteristics of the Ayutor-2 glacier along its medial moraine, (a), The mean annual velocity; (b), the moraine thickness; (c), the ice melting depression caused by a moraine layer <t>(r); (d), the glacier inclination; (e), the moraine material content in the glacier ice.
110 G. E. Glazyrin which are necessary for the computation. Mean annual ice ablation was determined by the sums of positive daily mean air temperatures on the tongue, which are given by Schetinnikov (1968). Mean annual daily intensity of the ice melting relating to the whole year is Pi = 0.96 cmday. We introduce some additional assumptions: (a) Moraine densification is negligible, i.e.? = 0; (b) As the altitude range in the moraine area is not very large we may assume that the intensity of pure ice melting does not depend upon z. Now we obtain from equation 5: dh _ cpi 4>(r) V sin a or a dh c= KW) The average value of a is 1.5 gcm 3. In Fig. 2e the values of c in the ice layer appearing at the different altitudes are shown. We see that as the depth increases, the concentration of moraine material increases too. There is a source of error in the present model. As was shown by Dushkin (1964), moraine spreading takes place down glacier in a direction perpendicular to the glacier motion. This phenomenon may be disregarded only if the moraine cover is rather thin or the ice motion velocity is high. REFERENCES Boulton, G. S. (1970) On the origin and transport of englacial debris in Svalbard glacier.. Glaciol, 9, No. 56. Dushkin, M. A. (1964) Formirovanie sovremennych moren na koncevom pole lednika Boljshoy Aktru. Glaciologija Altaja, No. 3. Glazyrin, G. E. (1969) Ablacionnye morey kak istochnik inforraacii o processach, proiskhodjashchich v verchovijach lednikov. Meteorologija igidrologija, No. 2. Kalesnik, S. V. (1937) Gornye lednikovye rajony SSSR. Nasyrov, M A. and Sadykov, K. G. (1966) Opyt sostavlenija balansa moren (na primere lednika Imat). Gornoe oledenenie Uzbekistana i smezhnych territory. Schetinnikov, A. S. (1968) Raschet abljacii na lednike Ajutor-2. 'Sb. rabot Tashkentskoj gidrometobservatorii', No. 3. Shumsky, P. A. (1947) Energija oledenenija i zhizn' lednikov.