Either because the balances have not been measured over a whole glacier, or in order to check them, or in order to estimate the area of cross-sections, the glaciological balance is very often calculated for part of a valley glacier.

Fig. 1.

Let X, Y be fixed geographical coordinates, Z(t) the altitude of the surface of the glacier at this point (assumed to be ice, not firn) and ΔZ its variation during the year (t _i, t _i + 1). The (net) balance during the same year, in equivalent height of ice will be denoted b, and the annual discharge at two cross-sections AB and CD (Fig. 11), q _AB and q CD. The mass balance between both cross-sections may be written (in volume of ice):

In this relation b is defined at a point of given coordinates (X, Y), Actually b is normally measured at stakes moving with the ice. The slightly different value so measured will be denoted b’. In order to introduce the measured values b’ into Equation (1), a correction must be made.

A longitudinal section of the glacier is sketched in Fig. 1. After one year the vertical planes AB and CD have become the curved surfaces A’B’ and C’D’, and the new surface is E’F’. We must then compare the volume ABCD, equal to the volume A’B’C’D’, and the volume E’B’C’F’:

ABB’A’ and DCC’D’ are the discharges q AB and q CD. The distance between A’D’ and E’F’ is the measured balance b’ (negative in the case of Fig. 1). The distance between E’F’ and EF is the variation of altitude ΔZ (positive in the case of Fig. 1). Therefore:

denoting a sum along a cross-section (not the same in each case) and S the distance in this transversal direction. So it appears that in Equation (1), b’ may be used instead of b, if the discharges are measured at two cross-sections A and D. and if the sum of (ΔZ-b’) is extended between lines half-way between A and D and their position one year after. If for instance A and D are 1 km apart and the mean velocity along a cross-section is 100 m/a at A, 40 m/a at B, the surface over which the sum must be done is diminished by 3%.

Whichever may be the definition adopted for the balance (at a fixed point or at a stake), a second correction arises from the fact that the area of the cross-section fluctuates within the year. In general the topographic work is done at instants t _i which are near the end of the ablation season, when the area of a cross-section is near a minimum. Actually, during the remainder of the year the area is bigger, the surface being raised (in equivalent height of ice) by b(X. Y, I), the transient balance. The true annual discharge is then:

where U and V denote average values through a vertical of the velocities in the X and Y direction. U _s and V _a the surface velocities and H _i, the ice thickness at the point X, Y.

Assuming a linear increase with time of the transient balance b(X, Y, t) during the accumulation season (t _i, t _i+t ₁),and next a linear decrease with time during the ablation season (t _i + t ₁. + t _i +1); b _m denoting the maximum transient balance, at instant t _i + t ₁, and t ₂ = 1 -t ₁, the length of the ablation season (in years):

b’ and b’ _m, measured at moving stakes, may be used instead of b and b _m, since this approximation introduces a second-order correction only. Therefore:

In order to estimate the importance of this correction, a result found by Reference NyeNye (1965) will be used. Assuming either no sliding on the rock-bed, or Weertman’s law of friction, this author has shown that the discharge in a parabolic channel is. within a few per cent, equal to

Thus due to the second correction H _i should be changed to

In the Alps, b’ _m is of the order of 2 m of ice. Near the snout b’ can reach -6 m or more and t ₂ ≈ ⅓. When dealing with small glaciers, the mean thickness of which is 30 to 100 m, these corrections are not negligible.

MS. received 29 January 1979 and in revised form 25 June 1970