a note on cross-profile morphology for glacial valleys
TRANSCRIPT
Note on cross-profile morphology for glacial valleys 513
Copyright © 2005 John Wiley & Sons, Ltd. Earth Surf. Process. Landforms 30, 513–514 (2005)
Earth Surface Processes and LandformsEarth Surf. Process. Landforms 30, 513–514 (2005)Published online in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/esp.1220
Short Communication
A note on cross-profile morphology for glacialvalleysFrank Morgan*Department of Mathematics and Statistics, Williams College, Williamstown, MA 01267, USA
AbstractWe provide an improvement to the Hirano–Aniya catenary model for the cross-profile morpho-logy of a glacial valley. Copyright © 2005 John Wiley & Sons, Ltd.
Keywords: glacial valley; cross-profile morphology
Hirano and Aniya (1988) derive a catenary shape for the cross-profile of a glacial valley by minimizing friction J = ∫yds proportional to depth y and arclength, for given total arclength. The arclength constraint has no physical meaning,and the catenary is actually a maximum rather than a minimum (Hirano and Aniya, 1989), with the minimum frictionof 0 approached by a wiggly profile near the surface. (For further discussion see Harbor (1990) and Hirano and Aniya(1990).)
We consider a more appropriate constraint: fixed area A, representing a fixed volume of glacial ice. The Eulerequation for J − λA with Lagrange multiplier λ > 0 has first integral:
′ =−
−yy
y C2
2
21
( )
λ
A smooth, symmetric solution on [−x0, x0] with y′(0) = 0 must have λ > 1 and C > 0. Integration yields:
(λ2 − 1)3/2 | x | = C wλ 1 2 − + C arccos w
where w = y(λ2 − 1)/C − λ, −1/λ ≤ w ≤ 1, C/λ ≤ y ≤ C/(λ − 1), as pictured in Figure 1. A curious feature of the solu-tions is that they exist only for strictly positive depth. One interpretation is that on an inclined plane, the glacier wouldjust spread out. The top must be held in by some non-solution constraints.
As λ approaches infinity, the solution approaches a semicircle. As λ approaches 1, the solution gets wide or shallow.It would be interesting to know whether this shape models real glacial valleys well.
AcknowledgementsWe would like to thank Mark Greenwood for bringing this problem to our attention, and the National Science Foundation for partialsupport.
*Correspondence to:F. Morgan, Department ofMathematics and Statistics,Williams College, Williamstown,MA 01267, USA. E-mail:[email protected]
Received 6 October 2004;Accepted 4 January 2005
514 F. Morgan
Copyright © 2005 John Wiley & Sons, Ltd. Earth Surf. Process. Landforms 30, 513–514 (2005)
Figure 1. Minimizing friction for fixed area yields a model for the shape of a glacial valley.
References
Harbor JM. 1990. A discussion of Hirano and Aniya’s (1988, 1989) explanation of glacial valley cross profile development. Earth SurfaceProcesses and Landforms 15: 369–377.
Hirano M, Aniya M. 1988. A rational explanation of cross-profile morphology for glacial valleys and of glacial valley development. EarthSurface Processes and Landforms 13: 707–716.
Hirano M, Aniya M. 1989. A rational explanation of cross-profile morphology for glacial valleys and of glacial valley development: a furthernote. Earth Surface Processes and Landforms 14: 173–174.
Hirano M, Aniya M. 1990. A reply to ‘A discussion of Hirano and Aniya’s (1988, 1989) explanation of glacial valley cross profiledevelopment’ by Jonathan M. Harbor. Earth Surface Processes and Landforms 15: 379–381.