Session C04 Cryosphere: The role of microstructure and layering on the physical properties, metamorphosis, and deformation of snowpacks, AGU Fall Meeting, San Francisco, CA, Dec. 2002
Relation Between Grain Size and Correlation Length of Snow Christian Mätzler, Institute of Applied Physics, University of Bern, Switzerland,
[email protected]
Summary This work aims at a more complete characterization of snow texture. The usual Grain Size Dmax (Colbeck et al. 1990) as the maximum extent of prevailing grains has almost exclusively been used so far. For the description of processes at the grain surface (e.g. light scattering), a size parameter related to the specific surface s is needed, such as the Optical Grain Size Do or the
Correlation Length pc. In future snow measurements, Grain Size observations should be complemented by measurements of either Do or pc. Field methods can be envisaged from results found here. As a rule of thumb, Do is close to the minimum extent of the grains, e.g. the thickness of thin plates, or the diameter of needles, thus a complement to Dmax.
Different particle shapes with same Do and similar spectral albedo Different shapes of scatterers in Figures 1a and 1b with the same Do, pc, density and thus the same specific surface show similar reflectance and transmittance spectra, see Figures 2a and 2b.
Fig. 1b: Pack of irregularly spaced spheres.
Fig. 1a: Stack of irregularly spaced ice lamellae
Infinitely Deep Pack of Single-Size Scatterers
Thickness=0.2m 1
0.9 a: Mie, Delta-Edd. a: Lamella Model t: Mie, Delta-Edd. t: Lamella Model
0.8
a0: Mie, Delta-Eddington a0: Lamella Model
0.9 0.8
0.7
0.7
0.6 0.6
0.5 0.5
0.4 0.4
0.3
0.3
0.2
0.2
0.1
0 0.5
0.1
0.6
0.7
0.8
0.9 1 1.1 W avelength, micron
1.2
1.3
1.4
0 0.5
1.5
Figure 2a: Spectral albedo a and transmittance t of a 20cm deep snowpack, density 100kg/m3, incidence angle 53°. Comparison of spheres (D=0.8mm) with lamella pack for d=D/3: lamella thickness, blue: MieDelta-Eddington model of Wiscombe and Warren (1980), black: “Simple” model of Mätzler (2000).
0.6
0.7
0.8
0.9 1 1.1 W avelength, micron
1.2
1.3
1.4
1.5
Figure 2b: As Figure 2a, but for a deep snowpack with illumination adapted to the respective geometry: vertical incidence for lamella pack, diffuse illumination for pack of spheres. The spectral albedos of the two situations agree almost perfectly.
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Concept of the Optical Grain Size (see e.g. Grenfell and Warren, 1999): Optical Grain Size Do Diameter of the spheres with the same total surface area and the same volume and thus the same mass as the real grains.
Correlation length pc Equivalent ways to determine pc of an isotropic granular two-component medium: 1. Slope of normalised (A(0)=1), spatial Autocorrelation Function A(x), with x= displacement:
dA( x ) pc = − dx
−1
(1) x=0
2. Specific surface s and volume fraction v:
pc =
4v(1 − v) s
(2)
3. Mean intercept lengths in ice Li and air La (stereological parameters):
La ⋅ Li La + Li 2 pc = (1 − v) Do 3
pc = 4. Relation to Do:
(3) (4)
Remark: A(x) often has an exponential shape: A( x ) ≅ exp(− x / pex ) where pc=pex. However, when measured functions A(x) are fitted to exponentials, it is found that pex ≤ pc (Mätzler 2002) when pc is determined by (1).
Example of Autocorrelation Functions: Spherical Shells, a Model for Depth Hoar Figure 3: A(x) of spherical shells, diameter D, versus displacement x normalised to shell thickness d for different ratios, b=2d/D, solid lines, from top to bottom: b = 0.01, 0.5, 0.7, 1 (=full sphere), from Mätzler, 1997. Slope at x=0 of curves for small b (dashed line), crosses x axis at x=pc=2d.
d
1.0
0.8
0.6 A 0.4
0.2
0.0 0.0
0.5
1.0
1.5
2.0 x/d
Spherical shell: diameter D, thickness d d
2
2.5
3.0
3.5
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Dry Snow Table 1: Selected snow data with size information: Visually estimated Dmax, exponentially fitted pex and pc measured by using Equation (1) for dry-snow types (Mätzler, 2002). Density
Snow Type
Dmax
pex
pc
[kg/m3] 109
new snow + +
[mm] [mm] [mm] 1 0.061 0.070
177
nearly new / /
0.2-0.5 0.069 0.090
244
hard, densified snow / /
0.2-0.3 0.071 0.101
384
very hard slab / /
0.1-0.5 0.090 0.127
240
coarse grains
1
0.161 0.170
332
coarse grains
1
0.156 0.203
270
depth hoar ∧∧
279
depth hoar ∧∧
Sub-sample of snow section 2 × 2 mm2
Dry-snow data from Weissfluhjoch, Switzerland, adapted and extended from Wiesmann et al. (1998) with sub-samples of snow sections used to determine A(x), pex and pc.
1-1.5 0.219 0.204
2-3
0.327 0.325
Conclusions The Optical Grain Size Do and equivalently, the Correlation Length pc are important parameters of granular media through the connection to the specific surface. Processes with interactions at the ice surface (vapor deposition, light scattering, sound absorption, heat conduction, terminal fall velocity) are described by these parameters. The classical Grain Size Dmax gives different information, e.g. it can describe the progress of densification and destructive metamorphosis in new snow, where Dmax is much larger than Do. Measurements of Do or pc consist of noting the size of the minor 3
axis of the prevailing grains. Table 2 serves as a guide for different particle shapes to relate visible size parameters to pc and Do. Measurements require a microscope and a length scale. For particles with extreme axial ratios, for example in case of surface hoar, the visual estimate of the minor axis may not be possible. In such cases pc might be determined by visually estimating their surface area, and by measuring the diameter of the resulting droplet of the melted particles to get the volume, or else by measuring the weight. There exist a number of alternative methods: • Measurement of microwave brightness temperatures of snow slabs. Appropriate calibration data to determine pc exist from Wiesmann et al. (1998). Optimum frequencies are from 35 to 50 GHz or from 70 to 100 GHz. • Acoustic transmission experiments (Buser and Good, 1987). • At the snow surface, near-infrared reflectance measurements (Nolin and Dozier, 2000). Table 2: Relations between Grain Size Dmax, Optical Grain Size Do, Correlation Length pc and volume fraction v for different grain shapes. Particle Type
Dmax
Dmax /Do
Do
Do(1-v)/pc
Sphere, diameter D
D
1
D
1.5
Spherical Shell, shell thickness d >1
3d
1.5
Thin Plate, thickness d >1
3d
1.5
Long Needle, diameter d >1
1.5d
1.5
Oblate Spheroid, minor axis d >1
2d
1.5
Prolate Spheroid, minor axis d >1
4d/π
1.5
Snowpacks
Table 1
1.5
References 1.
2.
3.
4. 5. 6. 7. 8. 9.
Buser O., and W. Good. “Acoustic, geometric and mechanical parameters of snow”, Internat. Symp. on Avalanche Formation, Movement and Effects, Davos, Switzerland, 14-19 September 1986, in Salm B. and H. Gubler (eds.), IAHS Publ. 162, 61-71, 1987. Colbeck, S.C., E. Akitaya, R. Armstrong, H. Gubler, J. Lafeuille, K. Lied, D. McClung and E. Morris. "The international classification for seasonal snow on the ground", Internat. Comm. of Snow and Ice of the Int. Assoc. of Sci. Hydrol. and Internat. Glaciol. Soc., distr. by CIRES, Box 449, Univ. of Colorado, Boulder, CO 80309, USA, 1990. Grenfell, T., and S.G. Warren. “Representation of a non-spherical ice particle by a collection of independent spheres for scattering and absorption of radiation”, J. Geophys. Res., 104, 31697-31709, 1999. Mätzler, "Autocorrelation functions of granular media with free arrangement of spheres, spherical shells or ellipsoids", J. Applied Physics, Vol. 81 (3), pp.1509-1517, 1997. Mätzler, C. “A simple snowpack/cloud reflectance and transmittance model from microwaves to ultraviolet: the ice-lamella pack”, J. Glaciology, 46(152), 20-24, 2000. Mätzler C., “Relation between grain size and correlation length of snow”, J. Glaciology, 48(152), 461466, 2002. Nolin, A.W., and J. Dozier. “A hyperspectral method for remotely sensing grain size of snow”, Remote Sensing of Environment, 74, 207-216, 2000. Wiesmann, A., C. Mätzler and T. Weise. “Radiometric and structural measurements of snow samples”, Radio Science, 33(2), 273-289, 1998. Wiscombe W.J. and S.G. Warren. “A model for the spectral albedo of snow I: pure snow”, J. Atmosph. Sci. 37(12), 2712-2733, 1980.
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