WET SNOW PENDULAR REGIME: THE AMOUNT OF WATER IN RING-SHAPEDCONFIGURATIONS

A. Denoth*University of Innsbruck, A-6020, Austria

ABSTRACT: The water saturation of a natural snow cover varies, in general, from zero to approxi­mately 20% of the pore volume, whereby two essentially different types of water geometry - pendu­lar mode and funicular mode - can be observed. The pendular mode covers the low-saturation range(typically S :s; 7% for old coarse grainded Alpine snow) and it is assumed that the water component isarranged in isolated menisci or pendular rings around a contact zone between spheroidal ice grains.However, the water rings or menisci are, thennodynamically, in a very critical - may be in an unstable- configuration. As water rings or menisci represent closed electrically conducting loops, they may beresponsible for an induced diamagnetic behaviour of snow, especially in the microwave regime; andthis offers a way to measure the amount of water stored in this geometrical configuration. From acarefull analysis of the measured dielectric and magnetic penneability in the microwave C to K-bandsof moderate wet coarse grained Alpine snow results, that water rings seem only to exist at saturationslower than ~8%. For saturations exceeding this critical value, water rings begin to merge fonningclusters, whereby the number of ring-like geometries decreases in favour of larger but open-endedstructures.

KEYWORDS: snow physics, snow electromagnetic properties, snow water content

1. INTRODUCTION

Snow, as any other porous granular materi­al, has three distinct regimes of water satura­tion: the pendular regime at low saturationswith an isolated water phase and a continuousair path throughout the pore space, the funi­cular regime at higher saturations with conti­nuous liqUid paths throughout the pore spaceand a more or less isolated and trappedgaseous phase, and the regime of complete

~..._ -saturation (Scheidegger, 1974; Colbeck, 1982).-t' A relative simple method to obtain infonnation

on the geometry of the individual componentsof wet snow (ice grains, air and water) is themeasurement of electromagnetic parameters:dielectric function, reflection- and transmissioncoefficients, or wave propagation function.Data analysis based on an appropriate di­electric mixing model, whereby the model ofPolder and van Santen (1946) has proofed itsefficiency, allows a geometric characterizationof the solid and liquid components in tenns of

Corresponding author address: A. Denoth,Institute of Experimental Physics, Innsbruck,A-6020; email: annin.denoth@uibk.ac.at

243

three-axial ellipsoidal bodies. Various measu­rement series with snow in different stages ofmetamorphism indicate a significant change inwater geometry at saturations ranging from 7to approx. 14% of the pore volume (Colbeck,1982; Denoth, 1982a; Hallikainen et al.,1984).Due to capillary forces and surface tension,water in the pendular saturation regime mayexist in the fonn of isolated spheroids and alsomenisci around the contact zone of ice grains.Recently, a pore size characterization and acalculation of the water volume contained inmenisci has been given by Frankenfield(1996). As water menisci represent closedelectrically conducting loops, they may beresponsible for an induced diamagnetic beha­viour of snow, especially in the microwave re­gime; and this offers a way to measure theamount of water stored in this special geome­trical configuration. Quantitatively, the 'in­duced' diamagnetism depends on the cross­sectional area of the ring-shaped water bodies:so, grain-size will be a dominant parameter.Indications of the existence of water rings havebeen reported by Matzler (1988) andAchammer (1996).

In this study, high precision measurementsof complex magnetic penneability in the micro-

wave C- to K-band, 6 up to 16 GHz, arepresented.

2. METHODS AND MATERIALS

In the current study, a free-space measure­ment technique is used, with the snow sampleplaced in between transmitting and receivinghom antennas. Polystyrene lenses have beenused with high-gain hom antennas to convertthe spherical wavefront to a planar wavefrontat the sample location. The principle of oper­ation is shown in Fig.1: H1, H2: hom antennas,SH: sample holder, I, R, T: incident, reflectedand transmitted signal, Ref: reference signalfor phase measurements.

CH<~NS~w.r1

I~tR

HIY~RefFigure 1. Principle of free-space measurementtechnique: R-T-method

Additionally - and as a controlling procedure - astandard ellipsometric method (Azzam and

,,-.. Bashara, 1979) has been applied: the reflectioncoefficients for incident 1t and cr polarizedwaves have been measured at 10° incidentangle intervals (Brewster-method). The prin­ciple of operation is shown in Fig.2: H1, H2:hom antennas, SH: sample holder, I, R:incident and reflected signals.

Magnitude and phase of the reflected andtransmitted signals, respectively, are measuredwith a high-precision computer-controlled net­work-analyzer, model HP8510A. Frequency ofoperation is in the 6 up to 16 GHz range. Sys­tem characteristics (interference effects, signalattenuation) of both the R-T-system and theBrewster-system have been determined by ca­libration procedures using dielectric test mate­rials of known properties. Based on Fresnel's

Figure 2. Principle of free-space measurementtechnique: Ellipsometric method

formulae (Guenther, 1990) snow dielectric per­mittivity, s =s' - Ls", and magnetic permeabi­lity, J.l. =J.l.' - LIl", have been deduced from themeasured total reflection and transmission co­efficients by an iterative technique, wherebymultiple beam interference effects within thesnow sample have to be considered. Due tothe limited accuracy of the measurement sys­tem and additional errors introduced by datareduction, the lower measurement frequencyhas been set to f =6 GHz.

In contrast to the intrinsic natural diamagne­tism, the 'induced' diamagnetism depends onthe cross-sectional area of the ring-shapedwater bodies (if there are any), and, conse­quentely, depends on the grain size. Therefore,measurements have only be made with coarseup to extremely coarse grained old Alpinesnow and tim-type snow. The snow samplesselected have been characterized according towetness (Wtcx, [Vol%]), porosity (<1>, [%]), andmean grain size (r, [mm]). Wtot, the total watercontent, has been measured by a calibrated di­electric probe (Denoth, 1994a; Denoth, 1994b),<1> has been calculated from snow density andWtot, and r has been derived from a photo­graphical analysis of the individual snowsamples (Denoth, 1982b). All the measure­ments have been carried out in a cold roomwith controlled ambient temperature, withmaximum variations from -0.5 °c up to 0 °c.

3. WET SNOW DIAMAGNETIC FUNCTION

For 5 selected snow samples, typical mea­surements of magnetic suszeptibility, X=J.l.' - 1,

244

The electric current I in the rings is driven bythe induced electric field (according to Fara­day's Law of induction), and depends on theactual conductivity and self-inductance of thecurrent loop. For the present model calcu­lations it is assumed, that conductivity cr isdominated by a", the dielectric loss of waterwith a maximum in the X-band regime; theionic contribution to the total conductivity, how­ever, can be neglected: cr == (£).8o.a", (£) the fre­quency in rad/s. The magnitude of the total 'in­duced' magnetic dipole density p is given by:p = 'Y2.Wr.l.alA, Wr the fraction of total watervolume collected in pendular rings, a the ringradius, and A the ring cross-section. Assumingan isotropic distribution of pendular water rings,

water may be arranged in pendular menisci asshown in Fig.4a; for simplicity of mathematicalcalculations, water menisci in the snow pendu­lar regime have been replaced by toroidalshaped rings, Fig.4b.

Figure 4a. Arrangement of water menisci in acluster of ice grains

Figure 4b. Model of snow pendular regime:Water rings in a cluster of ice grains

16

16

.... a•

r[mm]1.51.31.51.10.7

•• . a.................... ,~- ... '" .......•..•......

<I> [%]4652464748

8 10 12 14

frequency (GHz)

8 10 12 14

frequency (GHz)

6

6

sample Wtot[%]a 4.0b 4.7c 5.1d 2.2e 2.0

x 15 .

iii 10I/)

.E

.g 5CI>co0)tvEO··

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~ 0x,.., -5

~ -10 .;§Q. -15

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245

and magnetic loss, fJ.", in the frequency rangefrom 6 GHz up to 16 GHz are shown in Fig.3aand Fig.3b, respectively. Characteristic data ofthe snow samples, samples a .... e, are givenin Table 1. For sample a, a typical error bar isshown for a frequency of 13 GHz.

Table 1. Characteristic data of 5 selected snowsamples

Model calculations for snow magnetic permea­bility have been done based on the concept ofelectrically conducting water rings around con­tact zones of ice grains. Due to surface tension

Figure 3a. Frequency dependence of magneticsuszeptibility for 5 selected snow samples

Figure 3b. Frequency dependence of magneticloss ~" for 5 selected snow samples

o 10 20 30 40 50 60 70

ring size cp

Figure 5. Ratio of ring-water volume to the ice­grain volume.

the 'induced' complex magnetic permeabilitycan be calculated from p according to:

Wr [%]4.11.01.32.2

r[mm]1.31.21.51.1

<1> [%]52454647

Wtot [%]4.75.55.12.2

samplebfcd

Table 2. Characteristic data of selected snowsamples, samples b, c, d and f.

been extended from 8 up to 16 GHz or from 6up to 16 GHz, respectively. Model calculationsof permeability fl. have been done according toEqu.(1), whereby the fraction of total waterwhich is arranged in pendurar rings, Wr, hasbeen used as a fitting parameter. Indications ofdrastic effects of small changes in snow watercontent on water geometry can be seen inFig.6: magnetic loss for 2 selected snowsamples, sample band f, which differ only inwater saturation, is shown in the 8 to 16 GHzrange together with a model calulation (solidlines). The appearant significant difference inmagnetic response may be explained by a dif­ference in water geometric arrangement:sample b seems to be well within the pendularregime, Wr ~ Wtot, sample f, however, may bein a transition between the pendular and funi­cular zone, Wr « Wtot. In contrast, samples cand d (Fig. 3b) differ significantely in the totalwater content, the difference in the magneticlosses, fl.", however, is very low: This may alsobe explained by a specific difference in watergeometry and the difference in' grain size.Characteristic data of these snow samples aregiven in Table 2.

(1)fl. =31 [3 - LN 1 (1-LM)]

with N = ''kWr• &".[a.colc]2 , andM = 2.1t.[f/c]2.&".A.[ln(8.alb)-2], a and b thetorus radii (Achammer, 1996).

A calculation of the water volume Vr con­tained in menisci around contact zones ofspherical ice grains of radius r has recently begiven by Frankenfield (1996):Vr =(4/3).1t.f'l.f(q>, e). The function f(q>, e) de­pends on the ring size q>, and the angle of con­tact between ice and water, e. Ring water con­tent Wr can be calculated according to:Wr = (1 - <1». f(q>, e), <1>: snow porosity. A gra­phical representation of f(q>, e), the ratio ofring-water volume to the ice-grain volume isgiven in Fig.5 for 2 limiting values of contactangle e, e = 0° and e =10° (Hobbs, 1974).

......_~ 0,20e = 10~/

j :.:: ':'::~:~~:~~~I.i.:~;;~,~:~.»~-~;.~~~ . ,~

E.g 0.05-1·····:·················· .>~ 0.00.---"""'"

.;;:

C') 15<o..- 12 ..... ....x

The ring volume limit (maximum ring size)is defined by zero capillary pressure differenceaeeross the wetting and non-wetting boundary,and this allows an estimation of the upper limitfor the existence of pendular water rings:f(q>, e) ::;; 0.09, Wr < 5% for 50% porosity.

Consequentely, it is expected that magneticeffects may only be observed with low tomoderate wet and coarse grained snow.

=::i. 9

1 6o

~ 3C)1\1E 0

s?:!::I.lII! _

b

: pendular-funicular. transition' .

•••• _ ••mm~ •••

• .. I :f

4. EXPERIMENTAL RESULTS

8 10 12 14

frequency (GHz)

16

The magnetic permeability of a total of 31snow samples has been measured in thefrequency range of 12 up to 16 GHz, forselected samples the measurement range has

Figure 6. Magnetic losses at the pendular­funicular transition. Data points are showntogether with model calculations (solid lines).

246

5. CONCLUSION

ACKNOWLEDGEMENTS

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Measurements of the magnetic response ofmoderate wet coarse-grained Alpine snowsamples confirm the existence of closed ring­shaped water bodies in the pendular regime.The pendular regime extents to a saturation ofapproximately 8% of the pore volume, and thetransition to the funicular mode occurs in a re­latively narrow transitional zone where thewater bodies re-arrange. Compared to the di­electric response of wet snow, the magneticresponse is relatively weak: it can only be ot>­served at low wetness and with large icegrains. So, for practical applications, the con­sideration of magnetic effects in wet snow maybe neglected.

Figure 8. Dependence of water fraction in ring­like structures on total water saturation.

o 2 4 6 8 10 12 14 16total water saturation Stot(%)

The 'Osterreichische Forschungsgemein­schaft' is thanked for supporting in part presen­tation through grant no. 06/5255. My sonDietmar is thanked for drawing Figures 4a and4b, an artist-like view of snow pendular regime.

Achammer, T., 1996: Untersuchungen ueberdas elektromagnetische Verhalten poroe­ser, koemiger Materialien im Mikrowellen­bereich. Thesis, Univ. Innsbruck.

REFERENCES

water content up to approximately Wtot ~ 4.5%,more or less all the free water may be ar­ranged in ring-like structures, although devi­ations can be observed for W tot > 3%. Forwater contents exceeding a critical value of ~4.5%, magnetic losses decrease significantelyand the material snow approaches its intrinsicvery low natural diamagnetism. The decreasein magnetic response may be due to a mergingof water rings forming closed spheroidal waterbodies or forming a more or less continuouswater phase. In Fig.8, the fraction of waterwhich is arranged in ring-like structures, Sr, isshown against total water content, Stot, where­by, for comparison, water content has beenrelated to snow porosity: Sr = Wr !<'P, and Stot =Wtot !<'P. The zone, where most of the watercomponent may be arranged in ring-shapedconfigurations, the pendular zone, extents fromzero saturation to approximately Stet ~ 8%, it isfollowed by a transitional zone in the range of8% ;5; Stet ;5; 13% and by the funicular zone athigher saturations. This compares favorablywith dielectric (Colbeck, 1982; Denoth, 1982a)and also hydraulic measurements (Denoth,1980).

247

Figure 7. Dependence of normalized magneticloss on total water content.

o 1 234 5 6 7

total water content Wid. (%)

The dependence of magnetic loss on total wa­ter content is show in Fig.7 for a selected fre­quency of 14 GHz. For comparison, measuredmagnetic data have been normalized to a stan­dard mean grain-size of r = 1mm: M" := J.L" ! rzThe solid line represents a model calculationbased on the assumption Wr= Wtot . For a total

,'.

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248