Journal of Glaciology, Vol. 31, No. 108. 1985 MIXING FORMULAE AND EXPERIMENTAL RESULTS FOR THE DIELECTRIC CONSTANT OF SNOW By ARI SIHYOLA, EBBE NYFORS, and MARTTI TIURI (Radio Laboratory, Helsinki University of Technology, SF-02150 Espoo, Finland) ABSTRACT. This paper discusses dielectric properties of snow according to various dielectric models and compares them with experimental results . The complex permittivity of wet snow is assumed to consist of two parts, being the sum of the permittivity of dry snow (a mixture of ice and air) and the excess permittivity due to liquid water (resulting from the dielectric mixture of water and air). In particular the effect of liquid water is considered. Exponential models and structure-dependent models based on mixture theories by Taylor and Tinga and others are applied. It is shown that the assumption that water inclusions have the form of either randomly oriented discs or needles, or of spheres do, not get empirical confirmation but the inclusions are preferably prolate ellipsoids (ellipticity 0.16) or oblate ellipsoids (ellipticity 0.12), dry snow being a dielectric mixture of randomly oriented di sc-shaped ice particles and air. RESUME. La constant dieleclrique de la neige, for mules de melange el resullals experimenlaux. On examine les proprietes dielectriques de la neige deduites de different s modeles dielectriques et on les compare aux observations. On suppose que la permitivite complexe de la neige humide est la somme des permitivites de la neige seche (melange de glace et d'air) et de la permitivite due a I'eau liquide (melange d'eau et d'air). L'influence de I'eau liquide est particulierement examinee. Des modeles exponentiels et des modeles structurodependants bases sur les theories de INTRODUCTION The effect of liquid wa ter upon the dielectric characteristics of snow has been a puzzle for glaciologists during the last few decades. Several paper s dealing with the electrical properties of snow have been publi shed. However, the results for wet snow are not always consistent with one another. Snow can be treated as a three-<:omponent mixture consisting of air, ice, and water. A special case is dry snow which contains no liquid (free) water. The dielectric properties of these constituents are well known, including their frequency depend ence (see, for example, Hall ikai nen, 1977). Experimental formulae for the complex permittivity of dry snow have also been pre se nted (Nyfors, 1982; Tiuri and others, 1984). Numerous formulae explaining and predicting the dielectric characteristics of wet snow h ave been presented . These may be mixing formulae that have the permittivities of air, ice, and water as parameters, or they may even be linearized functions of density and wetness. More rigorous mixing theories tak e into account the microscopic structure of snow and the liquid water distribution. In this case, the resulting formulae u sua ll y co ntain additional parameters (for example, depolarization factors of the ice and water particles) melange de Taylor, Tinga et autres sont appliquees. On montre que les hypotheses d'inclusions liquides ayant tant la forme de disques ou d'aiguilles orientes au hasard que de spheres ne sont pas en accord avec I'experience mais que les inclusions ont plutOt la forme d'ellipsoides allonges (ellipsite 0,16) ou ap[atis (ellipsite 0,[2); [a neige sec he etant un melange dielectrique de particu[es de glace en forme de disques orientes au hasard et d'air. ZUSAMMENFASSUNG . Mischformeln und experimelllel/e Ergebllisse fiir die Dieleklriziliilskollstanle des Schllees. Die[ektrische Eigenschaften des Schnees, wie sie aus verschiedenen die[ektrischen Modellen hervorgehen, werden mit experimentellen Ergebnissen verglichen. Es wird angenommen, da ss die komp[exe Permittivitat des feuchten Schnees sich aus zwei Teilen zusammensetzt: sie ist die Summe der Permittivitat trockenen Schnees (eines Gemisches von Eis und Luft) und der iiberschiissigen Permittivitat info[ge fliissigen Wassers im Schnee (als Ergebnis der die[ektrischen Mischung von Wasser und Luft). Insbesondere wird der Einfluss des freien Wassers untersucht. Des weiteren werden exponentielle Modelle von Taylor und Tinga u.a. angewandt. Die Hypothese, dass freies Wasser die Form von scheiben-, nade[- oder kuge[formigen Einsch[iissen mit zufalliger Orientierung annehme, wird nicht be statigt. Die Einsch[iisse sind meist Hing[iche Ellipsoide (ElIiptizittit 0,16) od er abgeplattete ElIipsoide (ElIiptizittit 0, [2). Trockener Schnee erweist sich als eine dielektrische Mischung von scheibenformigen Eiskornern mit zufalliger Orientierung und Luft . that are determined by the type of snow. In this paper it is presumed that the effects of ice and water on the permittivity of wet snow can be superposed. In other words, the three-<:omponent mixing formula (I) is separable in the form (2) where ES is the complex permittivity of snow, p is th e dry density (see below), and W is the wetness of snow. This means that wet snow is treated as a two-<:omponent mixture consisting of dry snow and liquid water. And dry snow is for its part a dielectric mixture of air and ice. There are papers employing this separation approach, for example Ambach and Denoth (1972), Denoth and Schittelkopf (1978), Ambach and Denoth (1980), Matzler and others (1984), Tiuri and others (1984); see also Stiles and U[aby (1981). The first term in Equation (2) is th e permittivity of dry snow, the density of which is equal to the "dry density" of the wet snow in question, i.e. the den si ty which the s now sample would have if all the liquid water were replaced by air. Because the dielectric 163 https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0022143000006419 Downloaded from https://www.cambridge.org/core. IP address: 168.151.1.126, on 23 Sep 2017 at 16:58:36, subject to the Cambridge Core terms of use, available at