In A rational explanation of cross-profile morphology for glacial ualleys and of glacial valley development, Hirano and Aniya (1988) present an explanation for the apparent catenary form of glaciated-valley cross profiles based on the assumption that glacial erosion works towards minimizing friction between a glacier and its channel. However, as there is as yet no convincing justification for the application of such extremum principles to this problem, it is not clear how useful this type of analysis might be. In more detail, Hirano and Aniya's analysis includes assumptions about patterns of basal water pressures and the controls on friction inconsistent with other work in this area. Furthermore, as Hirano and Aniya (1989) point out, the catenary solution they derive is the maximum rather than the minimum friction solution. The underlying reason for this surprising result is that the variational principle Hirano and Aniya use is inappropriately structured for the problem being addressed, as it does not involve a cross-section area, i.e. ice discharge, constraint.
Hirano and Aniya also discuss the progressive development of glacial-valley cross profiles, but their analysis is hindered by the fact that most of the data they present are purely morphological, and thus cannot be used to assess form development as a function of either time or extent of glacial erosion. The only possible exceptions are Graf's (1970) data, which give morphological parameters as a function of valley order, but these do not unequivocally support Hirano and Aniya's conclusions, and are used with implicit assumptions about the relationship between valley order and time of glacial occupation or magnitude of glacial erosion that are incorrect or uncorroborated, respectively. However, if we accept the common assumption that glacial erosion results in a progression from initial V-shaped to broadly U-shaped (parabolic or catenary) forms, Hirano and Aniya's argument that there are two distinct types of form development (incision in alpine situations versus valley widening under ice sheets) is justified, and is an important observation that needs to be addressed in future models of glacial-valley development.