変分原理にもとづく氷河U字谷の形態と発達過程に関する考察

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The fact that the cross-profile of glacial valley can be well approximated by parabolas (Y=aX2) is explained by the variation principle, assuming that the glacier erosion works towards minimizing the bottom friction under the constant length of contact surface. The variation principle proves that the ideal or fully-developed profile of the glacial valley be a catenary, if the friction is proportional to the ice thickness. Maclaurins expansion of the catenary shows that a parabola is a very good approximation of the catenary. Hence, the good approximation of the cross-profile by parabolas is explained together. Different catenaries are generated by changing the form ratio FR (depth to rim width) and are then approximated by Y=aXb by the method of least squares. The obtained b values for catenaries bocome fractionally larger than 2.0 with increasing form ratios up to 1.0, indicating that b value would practically range between 1.0 and about 2.0. The relationship is plotted on the b-FR diagram with FR in the abscissa and b in the ordinate, respectively. Two types of developmental succession of glacial valleys are distinguished in the diagram. The one means that glacier valleys started from shallow V-valleys get larger b value in pace with increase of FR. The other shows the opposite trend starting from rather deep V-valleys. The first type is called the Rocky Mountain model here after its source of data, and represents over-deepening of the glacial valley as it develops. The second type is called the Patagonia-Antarctica model likewise, which represents widening instead of deepening process of the profile. The difference is attributed to the nature of glaciers which produced these valleys, i.e., alpine glaciers and continental ices. The variation principle also shows that the circular profile gives the maximum area for a constant perimeter length. This means that the glacier ice be most easy to flow down as a psuedo-plastic material under a constant boundary shear. However, it seems unreasonable to consider that the boundary shear proportional simply to shear rate within the ice and not to the weight of it would control the erosional process of the glacier valley. The morphology of cross-valley profiles also suggests that a catenary gives better figuration for U-shaped glacier valley than an arc.
社団法人東京地学協会の論文

変分原理にもとづく氷河U字谷の形態と発達過程に関する考察

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