Theory of structure formation in snowfields motivated by penitentes, suncups, and dirt cones

Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics

Volumn 63, Issue 5, 2001, Pages

  Betterton, M D ^a  

^a HARVARD UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

ASPECT RATIO; AVALANCHES (SNOWSLIDES); EIGENVALUES AND EIGENFUNCTIONS; GLACIERS; LIGHT REFLECTION; MATHEMATICAL MODELS; PERTURBATION TECHNIQUES; SNOW; SOLAR RADIATION; STABILITY; SUBLIMATION; WIND TUNNELS;

DIRT CONES;

SNOW MELTING SYSTEMS;

EID: 0035333464     PISSN: 1063651X     EISSN: None     Source Type: Journal    
DOI: 10.1103/PhysRevE.63.056129     Document Type: Article

References (34)

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17. S. Takahashi, Low Temp. Sci. Ser. A 37, 13 (1978).
18. The articles describing these experiments are in Japanese. Translated copies are available from S. G. Warren, University of Washington, Seattle.
19. F. B. Workman and W. H. Workman, Peaks and Glaciers of Nun Kun (Charles Scribner’s Sons, New York, 1909).
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22. C. L. Driedger, edited by P. W. Lipman and D. R. Mullineaux, The 1980 Eruptions of Mount St. Helens, Washington, Geological Survey Professional Paper 1250, (United States Government Printing Office, Washington, DC, 1981), pp. 757–760.
23. F. K. Ball, Weather 9, 322 (1954).
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25. Eric Nodwell and Thomas Tiedje, personal communication, 2000.
26. Rhodes, Armstrong, and Warren 2 report two “initial attempts at modeling” published in German. Currently Nodwell and Tiedje are investigating the formation of suncups (personal communication). They consider in detail the scattering of light in the snowpack, and deduce a range of allowable suncup sizes from this.
27. W. J. Wiscombe and S. G. Warren, J. Atmos. Sci. 37, 2712 (1980).
28. C. J. Van Der Veen, Fundamentals of Glacier Dynamics (A. A. Balkema, Rotterdam, 1999).
29. Because of the large penetration depth of visible wavelengths, one might think that snow thicknesses greater than 50 cm may be required for these arguments to hold. Although the visible wavelength albedo changes dramatically for snow depths (Formula presented) cm, the total amount of light absorbed does not change much down to thicknesses of a few cm—because the visible wavelengths have a low contribution to ablation. (See Ref. 27.) Note also that while the wavelength dependence of light scattering is dependent on snow grain size, old snow consistently shows rather uniform grain sizes of 1–1.5 mm 27.
30. One might think that diffusion of heat through the snow can be an important stabilizing factor. However, a simple calculation shows that this is not sufficient to provide a short-wavelength cutoff. I coupled temperature diffusion in the snow with heat conservation on the interface, assuming constant temperature in the air. This gives a dispersion relation with a q term from reflections, as in the text, and a term proportional to (Formula presented) from heat diffusion in the snow. As a result of this q dependence, at short wavelengths the stabilization by heat diffusion is controlled by a dimensionless parameter (Formula presented) (Formula presented), where (Formula presented) is the temperature difference between the surface of the snow and deep in the snow, C is the specific heat, and L the latent heat. In order for heat diffusion in the snow to stabilize short wavelengths, we must have (Formula presented). However, (Formula presented). Thus for a temperature difference of (Formula presented) the parameter (Formula presented), which is far too low to provide the cutoff.
31. If most melting takes place during the four hours of the day when the sun is most intense, this number gives 14 cm/day for the melting rate of the snow. It is useful to compare this to the field measurements of Kotlyakov and Lebedeva on penitente formation 6. If we divide their measurement of the total radiation received in one day by the solar intensity at noon, we would get 3.5 h estimated illumination time. They typically found snow height decreases of 6 cm/day. Thus our estimate may be too high, but is the correct order of magnitude. Kotlyakov and Lebedeva found 1.5-m high penitentes formed in 24 days.
32. When defined this way, (Formula presented) is an extinction length for reflections, which differs by a factor of 2 from the transmission extinction length.
33. M. G. Worster and J. S. Wettlaufer, edited by W. Shyy and R. Narayan, Fluid Dynamics at Interfaces (Cambridge University, Cambridge, England, 1999), page 339.
34. Drewry 21 deduces a value of 5.6(Formula presented) erg (Formula presented) (Formula presented) for debris covering dirt cones, which he compares to other measurements for sand around 2(Formula presented) erg (Formula presented) (Formula presented) The CRC Handbook, CRC Handbook of Chemistry and Physics, edited by David R. Lide (CRC, Boca Raton, 1994) gives the thermal conductivities of dirt, 1(Formula presented) erg (Formula presented) (Formula presented) sand, (Formula presented) erg (Formula presented) (Formula presented) and chalk, (Formula presented) erg (Formula presented) (Formula presented)