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    Configuration in 3D of wet and dry parts in a hail particle during wet growth, just after the onset of wet growth. Continued wet growth should lead to larger radius and angular extent of the wet part owing to accretion. The wet part is in the tropics and both dry parts are centered on the poles. The equator bisects the wet part. While falling, the particle rotates around its axis through the poles, an axis that corresponds to the nearly horizontal minor axis of spheroidal hail. The axis wobbles around the horizontal direction and is rarely exactly horizontal. Flux of accreted water onto the particle surface is schematically depicted.

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    Schematic geometry of a graupel or hail particle for the special situation of wet growth being absent while there is internal freezing of trapped liquid in its interior. The surface is all ice. Simultaneously there is dry growth of the surface if liquid being accreted.

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    Schematic diagram of two vertical cross sections of a hail particle during wet growth: (left) perpendicular to the equatorial plane and (right) in the equatorial plane. Flows of heat in the wet component’s heat budget are shown both outside and inside the particle. (left) The vertical cross section bisects two spherical sectors of the dry part (lighter shading). (right) The section is through only the wet part and includes none of the dry part.

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    General scheme for time-dependent freezing for graupel or hail particles.

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    Iterative scheme for wet growth of hail and graupel. The boxes list quantities calculated. The box for the dry part applies to hail only.

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    Predicted (large triangles) and observed (asterisks) equatorial surface temperatures averaged between latitudes of −30° and +30°, from our offline simulations of a single hailstone observed in the wind-tunnel experiment by Garcia-Garcia and List (1992). Each simulation was performed with the freezing scheme for hail (Part I). The initial diameter is 2 cm. For the model, the averaging was performed by assuming a uniform surface temperature across each of the two components. When σd > ½, then ; otherwise . Also shown is the predicted average temperature of the surface of the dry part (plus symbols). The same range of LWC and T as observed for different hailstones was simulated. The eight panels are for air temperatures inside the tunnel ranging from −20° to −6°C with an interval of 2°C. Results from a spherically symmetric model (one-part model) are shown (small triangles). The spherically symmetric one-part model disagrees with the observations.

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    Two-part-model-predicted (large triangles) and observed (asterisks) ice fractions from the same offline simulation of a hailstone observed in the experiment by Garcia-Garcia and List (1992), plotted as in Fig. 6. Results obtained using a spherically symmetric (one part) model with parameters adjusted in an attempt to optimize agreement with observations are shown (small triangles). The spherically symmetric one-part model does not represent the observations well.

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    Instantaneous values of predicted skin thickness for all hailstones of the wind-tunnel experiment simulated here (Garcia-Garcia and List 1992) plotted against that diagnosed from a theoretical relation by List [2014a, his Eq. (26)]. This is for instantaneous data from simulations of the same set of experiments in ambient conditions (from −6° to −20°C and from 1 to 12 g m−3) plotted in Figs. 6 and 7. That theoretical relation relates the skin thickness to the surface temperature (liquid–air interface) and ice fraction of the accretion, which were both predicted by our simulations. Data were selected from times in our simulations during wet growth when (top) σd < 0.1, (middle) 0.4 < σd < 0.6, and (bottom) σd > 0.9. A line of best fit for data points is plotted (solid), as is the identity (1:1) line (dashed).

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    Timelines predicted by the two-part model for a single simulated hailstone of (from top to bottom) particle dimensions (mm) including the thickness of the outer liquid skin on the wet part and the sponge, the cumulative ice fraction all by accretion since the start, the rates of accretion and shedding of cloud liquid (g s−1), the area fraction σd, the wet-growth flag (1 for wet growth; 0 for dry growth), the density of sponge, temperatures of the surface of the wet part and of liquid at the inner sponge–liquid interface TL/i, and velocities of the ice–liquid and liquid–air interfaces in the wet part. This was for one of the simulations displayed in Figs. 6 and 7 with an ambient air temperature of −16°C and liquid water content of 12 g m−3. (top) Dark dotted line is a comparison of predicted skin thickness with that diagnosed from theory by Lozowski (1991) and List (2014a) for the same ambient conditions. Also shown are predictions by the one-part model (light thin lines). The observations for this simulation are shown in the plots for −16°C in Figs. 6 and 7 (right column, third row).

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    Predicted ice fractions for initial diameters of 0.5, 1, 2, and 4 cm from the same offline simulation of a hypothetical wind-tunnel experiment, plotted as in Fig. 7. A unique fall speed (Lesins and List 1986) for each initial size was used to determine the wind speed for each set of runs. The control run started with a diameter of 2 cm simulates the observations by Garcia-Garcia and List (1992). The same repressurization time of 1 min was used for all experiments.

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    Sequence of stages in soaking the particle during prolonged wet growth, depicted schematically. Shown are sections through both poles of the particle, including both wet and dry parts. Liquid or fully soaked regions are denoted with darker shading, while regions that are soaking are shown with lighter shading. Regions without any liquid are unshaded. (a)–(c) Soaking of the particle and the preliminary estimate of the liquid skin’s volume is valid for the volume of exterior liquid (). (d) The particle is fully soaked with surplus liquid accumulating in an outer layer of liquid thicker than the initial skin (). Initially, there is only a thin skin of liquid covering the wet part and the sponge is shallow, as shown in (a). Next, the sponge deepens in (b). The sponge has reached its maximum depth in (c), and the rest of the particle is now soaking (if the dry particle’s bulk density is less than 910 kg m−3). If the hail originally had a high density when dry (>910 kg m−3), then the core and dry part can never soak, so stage (c) is skipped. The sponge is always fully soaked. For graupel, the wet sponge covers the entire particle.

  • View in gallery

    Meridional section through the poles of a hailstone. The shallow layer (bounded by gray curves) is assumed to convey most of the heat flow (thick arrow) from wet to dry parts. Its depth is twice the maximum depth of the straight line (CD) between the surface points at representative latitudes of the wet and dry parts (C and D, respectively). Each representative latitude is located at the average of all latitudes in its component over a given hemisphere.

  • View in gallery

    (right) Schematic diagram of hypothetical volumes created by spherical extrapolation is shown. (left) A section through the wet part in a plane through the poles for the actual particle is shown. Also shown is the same section for the spherically extrapolated volumes [subscript ϕ of ice outside the core in (top right) the dry part and (bottom right) the wet part].

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Theory of Time-Dependent Freezing. Part I: Description of Scheme for Wet Growth of Hail

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  • 1 Department of Physical Geography and Ecosystem Science, University of Lund, Lund, Sweden
  • 2 Hebrew University of Jerusalem, Givat Ram, Jerusalem, Israel
  • 3 Program in Atmospheric and Oceanic Sciences, and Cooperative Institute for Climate Science, Princeton University, Princeton, New Jersey
  • 4 Hebrew University of Jerusalem, Givat Ram, Jerusalem, Israel
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Abstract

At subzero temperatures, cloud particles can contain both ice and liquid water fractions. Wet growth of precipitation particles occurs when supercooled cloud liquid is accreted faster than it can freeze on impact.

With a flexible framework, the theory of wet growth of hail is extended to the case of the inhomogeneities of surface temperature and of liquid coverage over the surface of the particle. The theory treats the heat fluxes between its wet and dry parts and radial heat fluxes from the sponge layer through the liquid skin to the air. The theory parameterizes effects of nonsphericity of hail particles on their growth by accretion. Gradual internal freezing of any liquid soaking the hail or graupel particle’s interior during dry growth (“riming”) is treated as well. In this way, the microphysical recycling envisaged by Pflaum in a paper in 1980 is treated, with alternating episodes of wet and dry growth.

The present paper, the first of a two-part paper, describes the scheme to treat wet growth, accounting for dependencies on condensate content, temperature, and particle size. Comparison with the laboratory experiments is presented.

Corresponding author address: Vaughan T. J. Phillips, Department of Physical Geography and Ecosystem Science, University of Lund, Sölvegatan 12, SE-223 62, Lund, Sweden. E-mail: vaughan.phillips@nateko.lu.se

Abstract

At subzero temperatures, cloud particles can contain both ice and liquid water fractions. Wet growth of precipitation particles occurs when supercooled cloud liquid is accreted faster than it can freeze on impact.

With a flexible framework, the theory of wet growth of hail is extended to the case of the inhomogeneities of surface temperature and of liquid coverage over the surface of the particle. The theory treats the heat fluxes between its wet and dry parts and radial heat fluxes from the sponge layer through the liquid skin to the air. The theory parameterizes effects of nonsphericity of hail particles on their growth by accretion. Gradual internal freezing of any liquid soaking the hail or graupel particle’s interior during dry growth (“riming”) is treated as well. In this way, the microphysical recycling envisaged by Pflaum in a paper in 1980 is treated, with alternating episodes of wet and dry growth.

The present paper, the first of a two-part paper, describes the scheme to treat wet growth, accounting for dependencies on condensate content, temperature, and particle size. Comparison with the laboratory experiments is presented.

Corresponding author address: Vaughan T. J. Phillips, Department of Physical Geography and Ecosystem Science, University of Lund, Sölvegatan 12, SE-223 62, Lund, Sweden. E-mail: vaughan.phillips@nateko.lu.se

1. Introduction

At subzero temperatures warmer than about −36°C, water can persist in a supercooled liquid state. At such temperatures, hydrometeors in clouds can contain both ice and liquid water fractions because of processes of drop freezing and wet growth of graupel and hail. This coexistence of liquid and ice in the same precipitation particle is influential for several reasons. First, freezing changes the shape, size, and surface properties of precipitation particles and affects rates of collection and fall velocities of particles. When surfaces of ice precipitation are wet, ice crystals were seen to be collected with a high “sticking efficiency” of 100% (Macklin 1961) (see also Musil 1970; Lin et al. 1983; Willis and Heymsfield 1989). Moreover, the presence of liquid water in particles affects radar reflectivity, differential reflectivity, and other radar properties of clouds (Ryzhkov et al. 2011; Kumjian et al. 2013). For example, during melting the shape of a hail particle changes and the differential reflectivity may increase if the resulting raindrop is more oblate, as it may be since the oblateness is due to different physical processes (preferential regions of riming for hail versus elastic deformation of surface by drag for rain). Since wet growth occurs preferentially for the largest ice precipitation, use of polarimetric radars to detect wet growth would be expected to improve the prediction of big hail. Furthermore, liquid on the largest particles, from either wet growth or melting, leads to shedding that generates new raindrops. Since intensity of drop freezing and wet growth depend on supercooled water content, which is higher for high concentrations of aerosol, these processes are part of the cloud–aerosol interaction.

Time dependence of the masses of ice and liquid during freezing (or melting) arises because any freezing can only progress as fast as its latent heat can be dissipated to (or drawn from) the ambient air. For larger particles, such as precipitation, the freezing takes longer.

There are two types of freezing with appreciable time dependence considered in this study. The first type is freezing of accreted water at the surface of graupel or hail. If all accreted water freezes immediately upon contact, the regime of hail/graupel growth is known as dry growth. If only a portion of accreted water freezes, so the particle surface remains wet, the regime of hail/graupel growth is known as wet growth. The second type is drop freezing, estimated by Pruppacher and Klett (1997) to take up to 1 or 2 min in natural clouds, as will be explained in Phillips et al. (2014, hereafter Part II).

Wet growth was studied in several wind-tunnel experiments, the first of which were by List (1959) and Macklin (1961). Lesins and List (1986) observed that not all unfrozen water is shed, and that it resides on the wet part of the stone, partly soaking a “sponge” of ice (a mesh of dendritic-like ice). Some of the unfrozen liquid forms an outer liquid “skin” of depth 0.1–1 mm, with strong temperature gradients across it. Lozowski (1991) theorized that the depth of this liquid skin controls the freezing rate and is influenced by dynamical processes of accretion and shedding. List (1990) showed how the surface of the skin must be supercooled in order to convey latent heat of freezing to the air, since heat can flow through the skin only toward cooler temperatures, whereas the liquid temperature at the base of the skin must be close to, and slightly below, 0°C.

In theoretical models, it is usually assumed that graupel or hail in wet growth are spherically symmetric and covered by a homogeneous film of water (Musil 1970; Rasmussen and Heymsfield 1987; Ferrier 1994; Pruppacher and Klett 1997; Zakinyan 2007; List 2014a,b). This assumption substantially simplifies the theoretical analysis. Yet laboratory experiments by List (1959), Macklin (1961), Mossop and Kidder (1962), Kidder and Carte (1964), Lesins and List (1986), and Garcia-Garcia and List (1992) show that water forms a torus around the particle equator during wet growth, while the poles remain dry and icy [reviewed by Rasmussen and Heymsfield (1987)]. Such inhomogeneity was attributed to less rime deposited per unit surface area at the “poles” than the “equator” of a falling hail particle, with the extra latent heating from freezing causing the particle’s “tropics” to be warmer (Garcia-Garcia and List 1992). The reason for this inhomogeneity is that large hail particles are usually oblate spheroids approximately. If spheroidal, they must fall with their equator oscillating around the vertical plane while rotating around their minor axis (“symmetric gyration”) (Knight and Knight 1970; Kry and List 1974a,b; Thwaites et al. 1977; Stewart and List 1983). The minor axis, passing through their poles, wobbles around the horizontal (nutation and precession). Hence, their tropics receive the brunt of the accreted condensate. By contrast, smaller particles, such as graupel, may tumble or may fall with a major axis horizontal.

There have been scarcely any attempts to reproduce observations of wet growth by simulating laboratory experiments using numerical models, as far as we are aware. During the present study, our attempts to reproduce laboratory observations of oblate-spheroidal hail from Garcia-Garcia and List (1992) with the assumption of a homogeneous film of water covering the entire particle were not successful. It was not possible to predict the observed gradual decrease of average ice fraction of hailstones with increasing environment liquid water content. The assumption of a homogeneous film exaggerates the contribution from the wet region to the overall ice fraction and makes the film too thin initially, biasing its temperature and heat loss.

Accordingly, a “two-part model” of such wet growth for oblate-spheroidal hail is proposed in the present study, with a dry and a wet region coexisting on each particle (except when completely wet after prolonged wet growth). Laboratory observations of the critical temperature of the dry surface for onset of wet growth are applied. The heat balance of each of the two regions is considered, treating the inhomogeneity of their surface temperatures. Both parts of the particle’s volume are thermally coupled by heat conduction through its interior.

There have been problems with previous models of wet growth (e.g., List and Dussault 1967; Musil 1970; Lesins and List 1986; Rasmussen and Heymsfield 1987; List 2014a,b). Early models were encapsulated by that from Pruppacher and Klett [1997, Eqs. (16-49)–(16-51) therein]. They all made the assumptions of 1) a surface temperature of 0°C both when dry at the transition to wet growth and when wet, 2) complete coverage of the entire surface by liquid when wet, and/or 3) an “ice fraction” of accreted liquid freezing that is prescribed somehow. Such assumptions were problematic. First, regarding assumption 3, there has been a lack of observations of the ice fraction for all sizes of hail. Assumption 1 was shown to be unrealistic when List (1990) observed appreciable supercooling of the liquid skin. Phillips et al. (2005) treated wet growth without assumption 1. Both problems were partly solved by the model from List (2014a,b), who replaced assumption 1 by prediction of the temperature of the liquid skin from the fall speed by introducing three new assumptions: 4) a quasi-constant thickness of the liquid skin, 5) its inner surface at the ice sponge being 0°C, and 6) sphericity of the particle. This enabled a prediction of the ice fraction by solving a heat budget equation for the hailstone, replacing assumption 3. However, as noted below, there may sometimes be significant errors from the approximations 4–6. Second, real hail particles are not always fully covered by liquid, as noted above. Although an improvement, the model by List still used assumption 2.

Paucity of laboratory observations has left certain dependencies of wet growth unobserved, such as that on size. Hence, in the present paper, we propose a model that treats such dependencies of wet growth and internal freezing (e.g., of liquid from prior wet growth) during dry growth. Assumptions 1–6 that impeded the realism of past models are avoided. A cost of our approach is that the model requires a prediction of the liquid skin thickness and a simplified representation of the internal structure of the stone in the absence of those past assumptions.

New procedures of wet growth of graupel and hail (this paper; Part I) and of detailed freezing of drops (Part II) have been implemented in the spectral bin microphysics scheme of the Hebrew University Cloud Model (HUCM). In contrast with theories by Lozowski (1991) and List (1990, 2014a,b), we make no assumption about approximate constancy of thickness of the liquid skin during wet growth over time. Effects on heat loss from the augmentation of surface area by roughness and oblateness of hail may be treated, optionally.

The structure of the paper is as follows. A new scheme for dry and wet growth of graupel and hail is outlined in section 2. Comparison with laboratory measurements is presented in section 3. Part I ends with a discussion. In Part II, applications of the scheme in simulations of natural clouds will be presented.

2. Scheme for dry and wet growth of graupel and hail

a. A two-part model for hail and the algorithm

In wet growth, the volume of each hail particle is assumed to be divided into

  • a “wet part” covered by a liquid surface in wet growth, centered on the particle’s equator, and
  • a “dry part” with an icy surface in dry growth.

The wet surface forms in the zones of preferential accretion near the plane containing the vector of particle-relative air velocity. To express heat budgets and other equations, we first approximate the spheroidal shape of each part (wet or dry) by part of an equivalent sphere, writing equations as for spheres. Effects from oblate-spheroidal shape are taken into account by implementation of inhomogeneous accretion over particle surface.

For hail, the geometry of two parts for hail during wet growth, as treated by the scheme, is illustrated schematically in Fig. 1. The dry part represents two conical volumes with a combined surface of relative area of about σd. Their axes of revolution pass through each pole of the particle. The wet part, of relative area of about 1 − σd, is the remaining volume of the particle “sandwiched” symmetrically between both sectors. The particle’s internal structure has mirror symmetry with respect to the “equatorial” plane and rotational symmetry about the minor axis (normal to that plane). The angular extent by latitude of the wet component in a given “hemisphere” is . In general, the wet and dry parts may each have a different radius, belonging to different concentric spheres, in the same particle. The differing radii are treated by the scheme, for some purposes, such as for dimensions and ice-front velocity in the wet part. The angular extent of the wet component varies between no coverage and complete coverage, depending on the liquid mass fraction of the hailstone (for a liquid fraction of about 20%, most of the surface is covered by liquid, while for a liquid fraction of 0%, none is covered), as observed (Garcia-Garcia and List 1992). Only the outer portion of the wet component is always wet (liquid skin and sponge), while the inner core may be dry.

Fig. 1.
Fig. 1.

Configuration in 3D of wet and dry parts in a hail particle during wet growth, just after the onset of wet growth. Continued wet growth should lead to larger radius and angular extent of the wet part owing to accretion. The wet part is in the tropics and both dry parts are centered on the poles. The equator bisects the wet part. While falling, the particle rotates around its axis through the poles, an axis that corresponds to the nearly horizontal minor axis of spheroidal hail. The axis wobbles around the horizontal direction and is rarely exactly horizontal. Flux of accreted water onto the particle surface is schematically depicted.

Citation: Journal of the Atmospheric Sciences 71, 12; 10.1175/JAS-D-13-0375.1

The heat balance of each of the two parts (wet and dry) is considered, treating the inhomogeneity of their surface temperatures. Both parts of the particle’s volume are thermally coupled by heat conduction through its interior. The wet growth of the wet part determines the liquid fraction of the particle and the fraction of its surface remaining dry. Both fractions influence the surface temperature of the dry part and its dry growth. Thus, both parts, wet and dry, are closely interdependent. Within each part, internal particle properties are assumed to be radially dependent. Their inhomogeneity around the particle is merely represented by the contrast between wet and dry parts. Both parts are described below.

b. Calculation of particle mass and liquid fraction due to accretion

Accretion of liquid and ice is performed by the cloud model during the time step outside the freezing scheme before it is called. Masses of the entire particle, its ice, and its liquid are mp, mi, and mw, respectively. Both mp and mw include the mass of liquid accreted during the time step. Just before performing freezing, the masses are
e1
e2
where and mw,0 are masses of water in a particle at the beginning of the time step and just before freezing, respectively, with analogous definitions for and mp,0. Also, is the rate of shedding of accreted condensate to the air from the particle. In Eqs. (1) and (2), and are the rates of accretion of cloud-liquid (before shedding) and cloud-ice mass, respectively, by the hail (or graupel) particle of equivalent spherical radius . Here and are size distribution functions of drops and ice crystals of radii, rd and rice, and masses, md and mice, respectively, while is the relative velocity between the hail particle and either the water drop or ice crystal. The nonsphericity of hailstone shape may be taken into account following Lesins and List (1986), with a size-dependent axial ratio s from Knight (1986). The variables E and Eice are collection efficiencies of droplets riming and crystals being accreted, respectively; Eice is the product of the sticking and collision efficiencies of the ice–ice collisions. Sticking efficiency can be small for collisions with a dry icy surface of graupel or hail (neglected currently in our cloud model, so = 0). During dry growth, = 0. During wet growth, the shedding rate is (Garcia-Garcia and List 1992)
e3
where KE = 100 − 4ΔT (m3 kg−1). The liquid water content (LWC) has a critical value , generalized by Levi and Lubart (1998) for all particle sizes; (kg m−3; a typing error in the paper by Levi and Lubart is corrected here), where is the supercooling of ambient air and is the radius of hailstone (m). Shedding only occurs at large sizes of hail, warm subzero temperatures, or high liquid water content. There is none in air colder than −25°C. Only hail larger than about 2 cm in diameter can shed at all subzero temperatures warmer than −25°C in the mixed-phase region (0° to −36°C) when in a region of high LWC of 7 g m−3. Graupel or hail smaller than smaller than 1 cm cannot shed at any temperature if in regions of LWC less than about 5 g m−3.

The liquid fraction is calculated as fL = mw/mp. Volumes of liquid and ice are denoted by Vw = mw/ρL and , respectively. Here ρi is the bulk density of the particle when completely dry and implies a hypothetical equivalent spherical radius ri.

c. Conditions for onset of wet growth

Wet growth occurs whenever the predicted surface temperature of the hypothetical particle with a completely dry surface exceeds an empirical critical threshold Tcrit. We propose a simple fit to values of Tcrit obtained in laboratory observations by Garcia-Garcia and List (1992) and Levi and Lubart (1998):
e4
We evaluated from the heat budget at the surface if it is assumed to be completely dry:
e5

The terms on the left-hand side of Eq. (5) are, respectively. the conduction through the ice shell to the surface of latent heat from freezing of any internal liquid and the latent heat released at the hypothetically dry surface from freezing of accreted liquid. The terms on the right-hand side of Eq. (5) are, respectively, losses of this heat to the air by sensible conduction and as latent heat from sublimation, as well as loss of heat in warming accreted liquid up to the surface temperature. Equation (5) does not treat wet growth per se and is only used to detect whether the growth is dry or wet. Here, fh and fυ are the ventilation coefficients of heat and vapor transfer, respectively, for hailstones and graupel determined by the Reynolds (NRe = υt2rpρ/η), Prandtl (), and Schmidt () numbers, using empirical expressions (Rasmussen and Heymsfield 1987, their Table 1). These were modified for NRe > 6000 to include observations by Zheng and List (1994): (to obtain a similar equation for fυ, replace by ), where s is the axial ratio (diameter of the minor axis divided by that of the major axis). Also, is a factor to parameterize the enhancement of surface area by the oblate-spheroidal shape and is the ratio of capacitances of the actual spheroidal particle and the sphere of radius with the same volume, and is the eccentricity. For spherical particles (e.g., graupel), . {For examples of hail simulated in section 3, a size-dependent formula for s = max[1 − 16.5Dmax (m), 0.5] was applied from observations by Knight (1986), where Dmax is the maximum diameter along the major axis; it yields s = 0.67 and for Dmax = 2 cm to match with the shape of the hailstone model used by Garcia-Garcia and List (1992).}

Equation (5) was solved iteratively for by the secant method (Kreyszig 1988). The heat flow rate from the interior to the particle’s outer surface accounts for the possibility of gradual freezing of internal liquid during dry growth, as proposed by Pflaum (1980) (section 2d). When there is no interior liquid () or hail grows in the regime of wet growth, then = 0 while are evaluated from the fall speed and NRe of the dry particle. Otherwise, , where is the outer ice shell thickness. Figure 2 shows the assumed structure of the particle during gradual internal freezing when there is no wet growth (e.g., during dry growth) with the ice shell of thickness . Here, is the average thermal conductivity of the ice shell, where is the density of the ice shell.

Fig. 2.
Fig. 2.

Schematic geometry of a graupel or hail particle for the special situation of wet growth being absent while there is internal freezing of trapped liquid in its interior. The surface is all ice. Simultaneously there is dry growth of the surface if liquid being accreted.

Citation: Journal of the Atmospheric Sciences 71, 12; 10.1175/JAS-D-13-0375.1

In Eq. (5), are evaluated from the dry particle’s fall speed and NRe. In the case of internal liquid, are evaluated partly also from the wet particle’s fall speed and Reynolds number (see section 2i). Also is the ratio of surface areas of the actual spheroidal particle and sphere of equivalent spherical radius. The internal freezing is assumed to happen deep inside the hailstone at the edge (at approximately T0 = 0°C) of an inner spongy core of radius containing all interior liquid (see section 2d). The particle size in Eq. (5) is evaluated from the equivalent spherical radius of the entire particle:
e6

The “soakable” volume of the entire particle’s ice is given by if < 910 kg m−3 and otherwise. Any soakable volume is the hypothetical volume of empty space when all liquid is removed from a particle, such that it becomes a mixture of pure ice and air. Also, ρI = 920 kg m−3 is the density of pure ice. [For evaluation of conditions for onset of wet growth with Eq. (6), any liquid on the surface is neglected during soaking. The two-part model of wet growth (see section 2h) will not use this assumption.]

Note that depends not only on ambient temperature and LWC but also on the size of the particle. Knowing the critical value of , one can calculate the critical size, ronset = rp ( = Tcrit, T, ), at which wet growth starts for given values of and T. It may be shown that in case all graupel or hail particles grow within the same environment, there is a unique solution for ronset because the rates of latent heating from complete freezing increase with size (∝D2.5) more strongly than the rate of dissipation of heat from the particle (∝D1.9 approximately; see section 3). Hence, intersects Tcrit at only one size, ronset.

d. Dry growth and gradual internal freezing of graupel or hail

When , dry growth occurs. The surface of the particle is then all ice (“dry”). All accreted liquid must freeze on impact immediately without shedding. Nevertheless, gradual internal freezing of liquid soaked in the particle’s interior may still occur during dry growth, as predicted by Pflaum et al. (1978) and Pflaum (1980).

Freezing of internal liquid during dry growth takes significant time and may even be incomplete (Pflaum 1980). It is because the outer ice shell thermally insulates the liquid, prolonging the freezing, which then lasts longer than the freezing of drops of the same mass as the internal liquid.

Dry growth is treated as follows. If , only the liquid accreted is frozen without any internal freezing. Otherwise, internal freezing occurs if loss of heat to the air and to warming accreted condensate is faster than the latent heating from freezing of accreted condensate.

In the absence of wet growth, any liquid in the interior of the particle freezes only gradually. All liquid inside the ice particle, of volume , is assumed to be concentrated in a core of spongy ice that is fully soaked. The spongy core is encased by an outer ice shell as shown in Fig. 2. The shell has the same bulk density as the dry ice particle and has outer and inner equivalent spherical radii [Eq. (6)] and [Eq. (8)]. This picture is a great simplification because according to observations (e.g., Levi and Lubart 1991, their Fig. 1) spongy ice may form several layers.

Such internal freezing is assumed never to happen during wet growth, even if liquid is in the interior. Generally, any freezing will tend to occur preferentially in the outermost part of the liquid of any particle as near to the particle’s outer surface as possible (Part II).

During internal freezing, the temperature of the icy surface, , is evaluated using Eq. (5) and is calculated as discussed in section 2c. The ice part of the sponge is assigned a bulk density (not including mass of internal liquid) midway between maximum and minimum values assumed by Rasmussen and Heymsfield (1987), (kg m−3). The spongy core has a volume
e7
yielding
e8

As before, Eq. (6) defines . Finally, the mass of ice frozen is predicted as .

e. Geometry of wet and dry parts of hail during wet growth

It is assumed that the dry part of the surface of the hail particle σd is related to the liquid fraction fL in a simple linear fashion:
e9

Since graupel is usually smaller than hail and is less likely to gyrate, for graupel in wet growth it is assumed that . From laboratory observations by Lesins and List (1986) and List (1990) the constant C1 [in Eq. (9)] is estimated as about 5. The assumed structure of the particle in the case of wet growth is shown schematically in Fig. 3. Note that the angular extent of the wet component increases from zero to complete coverage as the liquid mass fraction increases from 0% to about 20%, so angles shown in Fig. 3 depict only one stage of wet growth. All quantities related to the wet part are denoted by an asterisk. Below the particle’s surface, spongy ice is formed by dendritic propagation of ice into the surface water. There are the following zones within the wet part:

  • part of an inner core of equivalent spherical radius rcore and volume of Vi,core, consisting of ordinary ice with the same bulk density as the completely dry particle for all r < rcore, which may become soaked if < 910 kg m−3 and if the sponge has reached its maximum volume;
  • spongy ice having an empirical bulk density (Rasmussen and Heymsfield 1987) for all r in the range > r > rcore, and being always fully soaked, without any air spaces (r is the radial distance from the center of the particle and the outer liquid surface of the wet part has a radius of ); and
  • exterior liquid above the ice for all r in > r > , either as a thin skin during soaking of the particle or as a thicker layer later when it is fully soaked.
Fig. 3.
Fig. 3.

Schematic diagram of two vertical cross sections of a hail particle during wet growth: (left) perpendicular to the equatorial plane and (right) in the equatorial plane. Flows of heat in the wet component’s heat budget are shown both outside and inside the particle. (left) The vertical cross section bisects two spherical sectors of the dry part (lighter shading). (right) The section is through only the wet part and includes none of the dry part.

Citation: Journal of the Atmospheric Sciences 71, 12; 10.1175/JAS-D-13-0375.1

Each interface between these parts is some part of the surface of a concentric sphere (except that in the heat budget equations, the enhancement of its area by roughness and spheroidal shape, is parameterized).

The dry part (denoted by a tilde) and the core both have the same uniform properties and consist of ice with the same bulk density (without including any soaked liquid) as the completely dry particle. It overlaps with the same inner core noted above, which shares this bulk density. (Geometrically the core is shared by the dry and wet parts in 3D, but this division is trivial and of no consequence for the core’s properties.) During soaking, a minor portion of the particle’s liquid forms a thin skin over the entire wet part; the majority of it is in the interior of the ice particle. After the sponge has reached the maximum volume, then soaking of the rest of the interior of the particle including the dry part may occur (if the completely dry particle’s bulk density was less than 910 kg m−3). The entire particle, including the core and rest of dry component, soaks as one entity. Eventually the dry part may become fully soaked (unless too dense to soak). This is assumed to occur by the ice being a mixture of pure ice and air, with the possibility of all air spaces being filled by liquid. However, the dry part’s surface is assumed never to become wet. When dry (if soakable) and wet parts are fully soaked extra liquid can accumulate on the exterior of the wet part beyond the depth of the usual liquid skin. The liquid skin is thickened, extending outwards like a wedge.

The model presented is qualitatively similar in basic principles to that of Rasmussen and Heymsfield (1987). Yet it differs insofar as inhomogeneity of both wet and dry regions on a particle is included.

The assumed geometry of wet and dry parts (Figs. 1 and 3) is a simplification. In reality, there may be multiple mobile liquid skins on the particle. Multiple concentric internal spongy layers are often seen in real hailstones as well as cavities inside spongy layers produced by displacement of liquid by the airflow (e.g., Browning et al. 1968). These are not represented explicitly in Fig. 3. Yet a quenched section of a spongy hailstone sampled during its wet growth does resemble Fig. 3, with the outer spongy deposit in a wide band around the particle without completely covering its surface (Knight and Knight 1973, their Fig. 7a). Equally, spherically asymmetric distribution of cavities in spongy ice was seen in many hailstones sampled from the ground, implying a lack of perfectly random rotation during freefall in the cloud (Browning et al. 1968). This spherical asymmetry is consistent with the inhomogeneous distribution of accretion over each particle that caused wet and dry parts of a surface to coexist in the experiment by Garcia-Garcia and List (1992). It is consistent with the model’s structure.

The oblate spheroid, with an axial ratio declining from unity to about a half with increasing size, is the most common shape of hail (Macklin 1977; Knight 1986; List 1986; Straka et al. 2000). Such shapes were found to fall by symmetric gyration in numerical simulations (Kry and List 1974a,b) and by observations of freely falling replicas of hail (Knight and Knight 1970). Symmetric gyration is consistent with observations of the rotation rate of freely falling natural hail near the ground (Lozowski and Beattie 1979; Matson and Huggins 1980). There have been several successful icing experiments that have grown artificial hailstones resembling natural hail in oblate-spheroidal shape and structure by mimicking symmetric gyration (Macklin 1977; Thwaites et al.1977; Lesins and List 1986). Such fall behavior of hail in nature is expected to produce the same type of rotational symmetry about a minor axis, with spherical asymmetry and an oblate-spheroidal shape and with a wet component centered around the equator, as in the model (Figs. 1 and 3). Finally, it is the outer spongy layer of any real hailstone consisting of multiple internal layers that is most influential for current wet growth thermodynamically, since it is closest to the freezing near the surface. This motivates the design of our model’s assumed internal structure on a single “bulk” outer spongy layer to represent the net effect from multiple spongy layers. Despite its simplicity, the assumed geometry of our treatment of the internal structure of the hailstone yields results close to observations of ice fraction and surface temperature in laboratory experiments by Garcia-Garcia and List (1992) simulated offline, as shown below (see section 3).

For our model of wet growth, the internal structure is an essential part of the prediction of frozen fraction of accretion. First, as noted below (see section 2g), the thickness of the liquid skin governs the rate of dissipation of latent heat of freezing and, hence, the freezing rate. The skin thickness is in turn constrained by the soakable volume of the particle and by the total mass of liquid. This soakable volume is governed by the volume of the sponge, which is diagnosed from the advected scalars using some empirical assumptions. Second, the areal coverage by the wet component determines the internal heat flow through the stone.

Now consider the thermodynamics of both parts: dry and wet.

f. Rate of freezing onto dry part of particle

The equivalent spherical radius of the dry part is assumed to be the same as for a completely dry particle with the same mass of ice as the actual particle: . The average temperature of the icy surface of the dry part is prescribed
e10
Equation (10) effectively limits the rate of accretion onto the dry part, making it generally lower (per unit surface area) than that onto the wet part. The mass of ice frozen during the time step onto the dry part = Δ t is obtained from the heat budget of the dry part’s surface:
e11
where and are rates of collection of cloud liquid and ice by the dry surface (e.g., may be assumed for simplicity, optionally). The value of is calculated from Eq. (9). To apply Eq. (11), the heat flow from the wet part through the interior of the stone into the dry part (Fig. 3) is required during wet growth; its formula is specified below (see section 2g, step 2; appendix C).

g. Rate of freezing onto wet part of particle

To calculate the rate of freezing and then determine the structure of the wet part of hailstone, it is necessary to calculate the surface temperature as well as temperature of liquid at the sponge–liquid interface TL/i.

The equation system consists of equations of heat balance at the surface and sponge–liquid interface as well as the heat balance for the entire wet component. Much of the procedure involves estimating the change in dimensions of the ice sponge during the time step, for a given mass frozen, to infer the radial speed of the ice–sponge front υis, as well as TL/i. All balance equations for dry and wet parts are solved together using the iterative procedure.

To describe all terms of the balance equations, some additional relationships are required. The derivation of the entire equation system for the wet part is performed using six steps. Here we present the main equations. Additional ones are presented in appendix B.

The first balance equation (step 1) is written for the outer surface of the liquid skin and serves for calculation of the temperature of the liquid surface :
e12

The flow rates noted in Eq. (12) are depicted in Fig. 3 with arrows. The net rate of accretion of liquid onto the wet part is . The rate of accretion of ice by the wet part is . The temperature of shed water is assumed to be following List et al. (2014a).

The variables and are the area and depth of the liquid skin, respectively, while is the thermal conductivity of liquid. The dimensionless factor C3 ≈ 1.5 parameterizes how the spicules and protuberances on the real surface cause its area to exceed that for a perfectly smooth sector (Garcia-Garcia and List 1992, p. 2069); C3 is uncertain, so an intermediate value in the range of between 1 and 2 is selected from observations by Aufdermaur and Joss (1967) and Schuepp and List (1969a,b). Factor parameterizes the spheroidal shape of hail, as noted above, being the ratio of the actual capacitance of the spheroidal particle to . Note that heat flow through the interior of the hailstone is not presented in Eq. (12) because its source is the latent heat of freezing at the sponge–liquid interface. It does not pass through the liquid skin and accordingly does not affect the heat balance at the particle surface.

The heat budget [Eq. (12)] is rearranged to yield the temperature of the outer liquid surface of the wet part:
e13

It is solved iteratively for by the secant method, using a value of TL/i from the previous iteration of the wet-growth algorithm (see section 2h). Here, , and . The shedding rate is determined from Eq. (3).

The second main equation (step 2) is the heat budget of the entire volume of the wet part that is used to calculate the mass frozen in the wet part. The budget includes the latent heat from freezing and its conduction both through the outer liquid layer to the surface and through the interior of the stone into the dry sectors:
e14

The heat flow is through the interior of the hail particle, almost tangentially beneath the surface, from the ice front of the wet component toward the outer surfaces of the dry sectors. The flow rate is assumed to be proportional to , where is the average thermal conductivity of the hailstone consisting of thermal conductivities of the mixture of ice, liquid, and air for the entire particle. The term is the surface area of the liquid (geometric mean of inner and outer surface areas of the outer liquid layer).

On any real hailstone, there may be many liquid skins covering the sponge that are not of uniform depth. This must be especially true at the earliest moments of wet growth when small “puddles” of liquid must appear on the tropics of the particle. Freezing would be expected to prefer the shallower periphery of each puddle of the skin, where the thermal resistance to the flux of latent heat to the air–liquid surface is less (a “puddle effect”). (Also, any turbulent enhancement of the liquid’s thermal conductivity would depend on average thickness of the liquid skin, with lateral liquid velocity at the surface from friction with air creating a radial shear inside it.) Hence, we define an effective depth (Fig. 3), which is the hypothetical depth that the liquid skin would have if it were of uniform depth (and nonturbulent) covering the entire wet part but with the same actual heat flow to the air. Since such a puddle effect is expected to be greatest at the start of wet growth, the ratio of effective to actual average depths is parsimoniously assumed proportional to the liquid fraction of the particle whenever the ratio is less than unity. Liquid skins are assumed to have a minimum depth on the order of 0.01 mm (List 1990). To treat the puddle effect, we propose for the effective depth of the liquid skin. Both ζ = min[C5fL, 1] and C5 are dimensionless factors parameterizing effects from the spatial variability of liquid depth over the surface of the particle. Assuming the effective depth tends to the actual depth () as σd → 0 suggests that C5C1.

The heat flow through the interior of the hail particle (Fig. 3) is approximated (appendix C) by
e15

Here, the latitudes separating the wet and dry parts are at . Also, C4 ≈ 0.4 is a dimensionless coefficient.

The knowledge of as well as the sponge density allows calculation of the radius of the outer surface of the ice sponge at the end of the time step (appendix B: steps 3–5). The theory implies a velocity of ice sponge interface of
e16
where is the radius to the outer surface of the ice sponge at the start of the time step Δt. Equation (16) represents further development of the wet growth theory beyond that of List (1990, 2014a,b) and Lozowski (1991), who assumed that is equal to the rate of particle radius growth.
Finally, the third main equation (step 6) allows calculation of TL/i. An empirical relationship between (m s−1) and TL/i from Hobbs (1974), also used by List (1990, 2014a,b) and Phillips et al. (2005), is applied:
e17

Equation (17) relates TL/i to all thermodynamical and soaking processes expressed in the value of (m s−1). The value TL/i together with surface temperature determine the temperature gradient across the outer liquid layer. This gradient determines the rate of dissipation of latent heat to the air and the ice fraction.

h. Numerical method for wet growth

For time-dependent freezing of hail and graupel, the cloud model performs calculations according to the flow chart presented in Fig. 4. After the onset of wet growth, the wet and dry parts of hail are coupled thermally by heat flow through the interior of the particle. All equations of both parts, wet and dry, are solved together simultaneously as a single set. Figure 5 shows the iterative algorithm for wet growth of hail. At each time step, Eq. (17) is solved for TL/i by a modification of the secant method of numerical iterative solution, finding where the expression T0 − (υis/0.000 28)1/2.39TL/i = 0. During each iteration, the simultaneous equations for both dry [Eqs. (7)(8)] and wet [Eqs. (9)(17) and (B1)(B10)] parts are solved from the previous iteration’s estimate of TL/i. Each iteration yields estimates of masses of accreted liquid frozen in the time step, and , and of particle dimensions, liquid fraction and σd, informing the next iteration. After convergence on the solution of TL/i, the mass of ice acquired in the time step is
e18
Fig. 4.
Fig. 4.

General scheme for time-dependent freezing for graupel or hail particles.

Citation: Journal of the Atmospheric Sciences 71, 12; 10.1175/JAS-D-13-0375.1

Fig. 5.
Fig. 5.

Iterative scheme for wet growth of hail and graupel. The boxes list quantities calculated. The box for the dry part applies to hail only.

Citation: Journal of the Atmospheric Sciences 71, 12; 10.1175/JAS-D-13-0375.1

The mass of the bin mp is prescribed for a spectral bin microphysics model. The iterative procedure yields values of surface temperature for the wet and dry surface, and . These are used to alter mw and mi by evaporation and sublimation from the wet and dry parts, and the air temperature and vapor mixing ratio are adjusted accordingly (Fig. 4).

i. Terminal velocity of particles with liquid

At Reynolds numbers less than 4000, the skin friction partly determines the drag coefficient (Rasmussen and Heymsfield 1987) and is much reduced for a wet surface compared to a dry one. In that case, the fall speed is an average of fall speeds of two hypothetical particles of the same mass and size as the actual particle, the first covered by no liquid, υt,dry = υt,d(mp, ρa)rp,d/rp, and the second completely covered by liquid υt,wet. The average is weighted by area fraction: υt = σdυt,dry + (1 − σd)υt,wet, where υt,d and rp,d are, respectively, the fall speed and the size of a completely dry particle with no liquid anywhere either inside or on it. Now υt,wet is the fall speed of a just-wet sphere, using the smooth-sphere drag coefficient as a function of the Best number from Clift et al. (1978) [Rasmussen and Heymsfield 1987; Phillips et al. 2007, their Eq. (25)].

At higher Reynolds numbers (>4000), skin friction is unimportant and particle shape determines the drag coefficient. The completely wet fall speed has the drag coefficient of the same particle with a dry surface. In that case, we use υt = υt,dry. Finally, for all Reynolds numbers, when the particle is fully soaked and liquid is accumulating on the exterior, the fall speed is interpolated between values when just soaked and at equilibrium. This equilibrium corresponds to the maximum amount of exterior liquid, maintained at criticality by shedding.

3. Comparison with laboratory observations and sensitivity tests

A test bed was constructed to compare predictions by the wet-growth model for hail with the results of laboratory experiments by Garcia-Garcia and List (1992).

a. Description of laboratory experiment

Garcia-Garcia and List (1992) suspended single spheroidal “hailstone models” of artificial composition in cloudy air in a wind tunnel. Each hailstone model observed was initially 2 cm in major diameter and with a density of 915 kg m−3. Each was exposed to unique conditions of cloud-liquid content (from 1 to 12 g m−3) and temperature (from −20° to −5°C) held constant during each experiment until it doubled in size, though never for more than 5 min. The wind speed was selected to match the expected initial fall speed, while the pressure inside the tunnel was decreased with temperature in a realistic way (about 400–600 hPa). As for hail in freefall, the minor axis was almost normal to the airflow relative to the particle. Temperatures averaged over the tropics of the hailstone between −30° and +30° were observed. At the end of the experiment, the tunnel was repressurized and the fan was switched off. The period of repressurization was about 1 min (Lesins and List 1986). The ice accreted onto the model was then weighed before and after centrifuging away the liquid so as to infer the ice fraction. More details are given by Garcia-Garcia and List (1992) and Lesins and List (1986). An increase of up to 10% in the ice fraction due to extra freezing during repressurization (1 min) was seen (List et al. 1995, p. 172). This counts as a bias if the measured ice fraction is supposed to apply to natural hailstones.

Garcia-Garcia and List (1992, p. 2067) reported formation of “wheel like” shapes and “spike-like structures protruding perpendicularly from the equatorial region of the hailstone” for the final wet hailstones that they observed. Of course, roughness elements can occur on natural wet hail. But real hailstones are usually almost spheroidal in nature, not wheel-like, and have axial ratios usually larger than s = 0.5–0.6 (Knight 1986). The hailstone model used by Garcia-Garcia and List was initially with s = 0.67 but the eventual particle after doubling in size evidently produced a much lower axial ratio than this. Imperfections in mimicking the fall behavior of hail in the experiment by Garcia-Garcia and List (1992), by mounting the hailstone on a rod, would be expected to have artificially enlarged these spike-like structures and wheel-like shapes. So the hailstones in the laboratory experiments became much more nonspherical and rough during the wind-tunnel experiment than in nature.

b. Setup of offline simulation

The above scheme for wet and dry growth (section 2) with shedding and the possibility of gradual internal freezing, was applied offline to simulate the single hailstone replica grown in each laboratory experiment. The hailstone plastic replica used in the experiment was represented realistically as a spheroid with the same initial major axis and axial ratio as reported (2 cm and 0.67, respectively). Both periods of the laboratory experiment were simulated as closely as possible, prescribing all experimental conditions as noted above:

  • exposure to cloudy air in a constant wind speed of about 30 m s−1, for either 5 min or up to the time of doubling of the original size of the model (2-cm diameter), whichever is sooner;
  • exposure to subsaturated air at atmospheric pressure just after switching off the fan and repressurizing the chamber for 1 min.

By simulating the increase in ice fraction (by up to 10%) during repressurization, we were able to avoid any inconsistency between the prediction and observation. However, some aspects of the wind-tunnel experiment were not reported (e.g., rate of deceleration of the wind during repressurization, possible extra freezing during centrifuging) and other measurement biases exist [e.g., underestimate of liquid mass due to some liquid being entrapped by ice in the accretion according to Lesins and List (1986), biases of probes measuring liquid water content]. Thus, exact agreement is not to be expected between the theory and observations.

Finally, for simplicity of the offline comparison we prescribed , matching the initial radius of the hailstone model. This was because the hailstone plastic replicas used in the wind-tunnel experiments were artificially impermeable to liquid and the airspeed in the real experiments was not varied to match fall speed during growth. Therefore, Eq. (B3) could not be applied in the usual way.

The hailstone was more nonspherical and rough during the wind-tunnel experiment than in nature (see section 3a). Consequently, to initialize the wet-growth scheme appropriately for the experiment, the dimensionless factor C3 was chosen equal to 3. The factor accounts for boosting of surface area by extra nonsphericity [Eq. (12)]. This represents a doubling of the usual value (1.5) in the standard version of the scheme used in the cloud model.

c. Results from control simulation of hail growth in laboratory experiment

Figure 6 compares the prediction by our two-part model of hail growth for equatorial surface temperature, averaged between latitudes of −30° and +30°, as well as the difference between surface temperatures of the equatorial region and of dry part, with the laboratory observations by Garcia-Garcia and List (1992). The average error in the equatorial surface temperature for all permutations of LWC and T observed for different hailstones is less than about 0.4°C. Moreover, the predicted difference, between surface temperatures of the equatorial region and of the dry part, is less than a few degrees, which is consistent with that observed for equatorial and polar surface temperatures in another experiment (List et al. 1995). The dry part’s surface temperature is predicted to be usually less than the equatorial average, as expected from the poles being observed to be cooler than the equator and preferred by the dry regions.

Fig. 6.
Fig. 6.

Predicted (large triangles) and observed (asterisks) equatorial surface temperatures averaged between latitudes of −30° and +30°, from our offline simulations of a single hailstone observed in the wind-tunnel experiment by Garcia-Garcia and List (1992). Each simulation was performed with the freezing scheme for hail (Part I). The initial diameter is 2 cm. For the model, the averaging was performed by assuming a uniform surface temperature across each of the two components. When σd > ½, then ; otherwise . Also shown is the predicted average temperature of the surface of the dry part (plus symbols). The same range of LWC and T as observed for different hailstones was simulated. The eight panels are for air temperatures inside the tunnel ranging from −20° to −6°C with an interval of 2°C. Results from a spherically symmetric model (one-part model) are shown (small triangles). The spherically symmetric one-part model disagrees with the observations.

Citation: Journal of the Atmospheric Sciences 71, 12; 10.1175/JAS-D-13-0375.1

Figure 7 shows the predicted and observed ice fraction from the full two-part model. Below some temperature-dependent critical value of LWC, the ice fraction is unity and there is no wet growth. As LWC increases above the critical value among experiments at a given temperature, the ice fraction is reduced as wet growth becomes more prolific. The average absolute error in the ice fraction predicted by the two-part model is less than about 0.04 for all experiments. Thus, the accuracy of freezing predicted by the full model is adequate. Ice fractions in Fig. 7 less than about 75% involve complete coverage of the stone by liquid, and between about 75% and 100% are with partial coverage by the wet part.

Fig. 7.
Fig. 7.

Two-part-model-predicted (large triangles) and observed (asterisks) ice fractions from the same offline simulation of a hailstone observed in the experiment by Garcia-Garcia and List (1992), plotted as in Fig. 6. Results obtained using a spherically symmetric (one part) model with parameters adjusted in an attempt to optimize agreement with observations are shown (small triangles). The spherically symmetric one-part model does not represent the observations well.

Citation: Journal of the Atmospheric Sciences 71, 12; 10.1175/JAS-D-13-0375.1

Additionally, to demonstrate an importance of consideration of inhomogeneity of hail surface during wet growth, the model was artificially altered. Extra symbols (small triangles) in Figs. 6 and 7 indicate results of the simulations using our hail particle model with the assumptions of spherical symmetry of process of freezing with ; that is, the entire particle surface is wet (a “one part” or “spherically symmetric” model). In view of its uncertainty, the factor accounting for roughness elements boosting the surface area is adjusted in an attempt to maximize agreement of the ice fraction predicted by this one-part scheme with observations (C3 = 2.5). But even having done this, the spherically symmetric model still does not represent well the observations:

  1. At intermediate LWCs of wet growth, and especially those close to the onset, the surface is up to 2–3 K too warm for the one-part model as compared to the observations. The one-part model is especially inaccurate at the colder ambient air temperatures, for which wet growth is less prolific and the real surface drier.
  2. The critical LWC needed for onset of appreciable liquid on the surface is about 100%–200% too high for the one-part model as compared to the full one-part model.
  3. At small and intermediate values of LWC, the ice fraction decreases much more sharply with increasing ambient LWC than in the two-part model and observations.
  4. At the highest LWCs of wet growth, the ice fraction simulated by the spherically uniform model is typically lower than that observed in the laboratory experiments.

This behavior of the artificial one-part model is explicable as follows. Generally, any wet film tends to slow the process of freezing, partly by thermally insulating the region of freezing in the sponge and cooling the particle’s surface, inhibiting heat loss to the air. This inhibition of freezing, in turn, tends to thicken the wet film further. Thus, the liquid skin’s thickness controls (and is partly controlled by) the growth of the hailstone and is also partly controlled by the particle’s hydrodynamics (Lozowski 1991). Close to the onset of wet growth in the one-part model, the artificial spreading of any initial liquid film over the entire particle makes it unrealistically thin and liable to freeze completely, owing to insufficient thermal insulation and too warm a surface temperature relative to the actual wet part (Fig. 6). It makes the effective onset of wet growth occur at higher liquid fractions of the particle or at higher LWCs. Conversely, if the entirety of the particle is artificially covered with a liquid layer that is too thick for any reason, then the freezing is excessively slowed by too much thermal insulation and ice fraction becomes less than in the two-part model (and in the observations), exacerbating the excessive thickness (Fig. 7). By making the entire particle wet, the one-part model exaggerates the overall tendency for ice fraction to be extreme, either too high or too low, and so there is too sharp a transition in between. The full one-part model by List (2014a,b) would share this behavior of the ice fraction (too high or too low) qualitatively and its lack of treatment of surface roughness would greatly reduce its predicted ice fraction during wet growth (not shown).

Although the thickness of the liquid skin—a decisive quantity—has never been measured as far as we are aware, our prediction of it may be compared with that from an existing spherically symmetric theory. Figure 8 shows a scatterplot comparing the water skin thickness predicted by our full two-part spheroidal model and that diagnosed for the wet part using the particle heat budget from List’s one-part spherical model [List 2014a, his Eq. (26)]. To perform such a comparison the water skin thickness was predicted, together with different sets of temperature–ice fraction pairs, by our two-part model. The temperature–ice fraction pairs were chosen from simulations of wind-tunnel experiments by our two-part model over the ambient conditions (from −6° to −20°C and from 1 to 12 g m−3) in Figs. 6 and 7. Their average values were shown to agree well with the observations by Garcia-Garcia and List (1992), as noted above. The values of the water skin thickness in the two-part model and in the spherically symmetric List’s model were compared at the same values of the ice fractions determined by our two-part model. Since List’s theory only applies to spherical particles homogeneously covered by liquid, for the first comparison we selected data from times in our simulations when σd < 0.1 during wet growth. The skin thicknesses predicted by our two-part model and those diagnosed by List’s theory differ by no more than a factor of 2. This discrepancy indicates the difference in description of internal particle structure between the models, concerning the growth rates of radii of ice sponge and surface, temperature of the internal liquid–sponge interface, heat flux through the stone into the dry part, and surface roughness and effective nature of skin thickness (the above puddle effect). To reveal the differences between two models during the beginning of wet growth, two more comparisons were conducted for a partially wet surface (0.4 < σd < 0.6) and a surface just after the onset of wet growth (σd > 0.9). For such values of σd our two-part hail model behaves less like a one-part model, predicting a skin up to several times thicker than the one-part model. The difference between predicted and theoretically diagnosed thicknesses is much greater than for the first comparison, especially close to the onset of wet growth where some of the factors noted above (e.g., puddle effect, heat flux through stone) are more significant. This illustrates the impact from geometrical configuration of liquid and ice on the freezing process. Results of freezing from the two-part model are likely to be more realistic.

Fig. 8.
Fig. 8.

Instantaneous values of predicted skin thickness for all hailstones of the wind-tunnel experiment simulated here (Garcia-Garcia and List 1992) plotted against that diagnosed from a theoretical relation by List [2014a, his Eq. (26)]. This is for instantaneous data from simulations of the same set of experiments in ambient conditions (from −6° to −20°C and from 1 to 12 g m−3) plotted in Figs. 6 and 7. That theoretical relation relates the skin thickness to the surface temperature (liquid–air interface) and ice fraction of the accretion, which were both predicted by our simulations. Data were selected from times in our simulations during wet growth when (top) σd < 0.1, (middle) 0.4 < σd < 0.6, and (bottom) σd > 0.9. A line of best fit for data points is plotted (solid), as is the identity (1:1) line (dashed).

Citation: Journal of the Atmospheric Sciences 71, 12; 10.1175/JAS-D-13-0375.1

Figure 9 shows the evolution with time of several parameters of one of the hailstones in our simulation of the wind-tunnel experiment (Garcia-Garcia and List 1992), as seen from our theory. The hailstone was grown at an ambient air temperature of −16°C and LWC of 12 g m−3. The thickness of the outer layer of liquid is less than about 0.6 mm. According to theoretical inferences by List (1990, his Fig. 4) from laboratory observations of spheroidal hail particles, this is a realistic order of magnitude. As the particle grows by wet growth, liquid accumulates on the exterior of the particle and the area fraction of the dry part decreases from unity gradually to zero on a time scale of about a minute or so. Also, both rates of accretion and shedding increase with particle size steadily over time. The predicted thickness of the skin is higher by a factor of less than about 2 compared to that diagnosed from theory by Lozowski [1991, his Eq. (8)] and List [2014a, his Eq. (26)] who both assumed it has a quasi-steady state. As noted earlier, the diagnosis from such one-part models tend to underpredict the liquid skin’s thickness. Moreover, the shedding rate is predicted by the two-part model to be small compared with the freezing and accretion rates (Fig. 9). Hence, for this case Lozowski’s [1991, his Eq. (5)] argument about the liquid skin thickness over the entire particle being governed primarily only by shedding, and not by freezing, seems implausible. In our validated two-part model, the skin thickness is due mostly to a “dynamic equilibrium” between gain and removal of liquid by accretion and freezing, respectively.

Fig. 9.
Fig. 9.

Timelines predicted by the two-part model for a single simulated hailstone of (from top to bottom) particle dimensions (mm) including the thickness of the outer liquid skin on the wet part and the sponge, the cumulative ice fraction all by accretion since the start, the rates of accretion and shedding of cloud liquid (g s−1), the area fraction σd, the wet-growth flag (1 for wet growth; 0 for dry growth), the density of sponge, temperatures of the surface of the wet part and of liquid at the inner sponge–liquid interface TL/i, and velocities of the ice–liquid and liquid–air interfaces in the wet part. This was for one of the simulations displayed in Figs. 6 and 7 with an ambient air temperature of −16°C and liquid water content of 12 g m−3. (top) Dark dotted line is a comparison of predicted skin thickness with that diagnosed from theory by Lozowski (1991) and List (2014a) for the same ambient conditions. Also shown are predictions by the one-part model (light thin lines). The observations for this simulation are shown in the plots for −16°C in Figs. 6 and 7 (right column, third row).

Citation: Journal of the Atmospheric Sciences 71, 12; 10.1175/JAS-D-13-0375.1

In our two-part model simulation, the cumulative ice fraction increases with time initially as the accretion rate is weaker than later on. The wet part’s liquid skin is thin and warm, so freezing is not so inhibited. But eventually the cumulative ice fraction begins to decline as the liquid skin thickens and the wet surface cools, with the accretion rate growing to become too great for the dissipation-limited (heat loss to air) freezing rate. During wet growth, the liquid surface has a temperature intermediate between temperatures of the air (−16°C) and ice front (as cold as −0.75°C), as required to dissipate the latent heat of freezing to the surface and then to the air. The liquid surface temperature is usually warmer than the surface temperature of the dry part (−3°C in this simulation). At the end of the wind-tunnel experiment (140 s), wet growth ceases and gradual freezing of internal liquid begins, with a slow further increase of ice fraction.

Figure 9 also compares the velocities of the ice–liquid interface and wet surface . As assumed for the theory by List (2014a,b), both are similar, though initially there is a transient difference in the first 10 s or so. (For the simulations of some of the other hailstones for other ambient conditions, there might be sustained differences between both velocities, however.) A physical mechanism for maintaining both velocities at similar values involves adjustment of surface temperature. For instance, when there is a positive perturbation to for some reason (e.g., the rate of accretion of liquid increases suddenly), heat is dissipated away faster (warming of extra accreted condensate, greater surface area for heat loss) and so the surface cools, which tends to increase the radial temperature gradient across the skin, despite the initial thickening of it. Latent heat from freezing can be conveyed to the surface faster through the liquid skin and the freezing rate increases, restoring toward the new value of . Reductions of σd and with increasing liquid fraction tend to thin the liquid skin again, also facilitating the adjustment.

Finally, Fig. 9 compares the results of our new two-part model with those from the conventional one-part model (lighter lines). During the first minute of the simulation, the two-part model shows an appreciable fraction of the particle that is dry and the one-part model differs greatly from it. The ice fraction is about 0.8 for the one-part model compared to about 0.6 for the full model, before 30 s. The one-part model involves a liquid skin that is too thin (by a factor of 3–10 in the first 30 s) and, hence, a supercooling of the surface temperature that is too weak (by a factor of 3) relative to the two-part model. Its sponge density (not including liquid) is up to 100% larger. But at longer times beyond 1 min when the (two part) particle is almost completely covered by liquid, the two models (one and two part) converge, regarding ice fractions and other properties. Figure 9 illustrates the crucial role of the geometry of the liquid skin, which governs the thermal insulation of the region of freezing at the sponge, loss of latent heat to air, and overall wet growth.

d. Sensitivity tests of scheme for hail growth

The simulation of the laboratory experiment by Garcia-Garcia and List (1992) for wet growth of hail shown above was repeated with altered values of the initial sizes of the hailstone replica. We are not aware of wind-tunnel observations for these alternative sizes. For each new size, a corresponding value of terminal velocity was estimated, defining the simulated wind speed in the tunnel. The initial size determined the axial ratio of the stone using the empirical expression above for s. Growth continued until the size had doubled, though never for more than 5 min.

Figure 10 shows the predicted response of average ice fraction to alteration of the initial diameters of hail from 0.5 to 4 cm. A strong sensitivity is predicted both with respect to initial diameter of hail as well as to the ambient LWC. The average ice fraction increases with decreasing size generally at any given liquid water content. Moreover, the minimum liquid water content, for which any wet growth can occur, increases with decreasing size too. Qualitatively, this sensitivity of minimum LWC is consistent with that reported by the modeling study of Levi and Lubart (1998).

Fig. 10.
Fig. 10.

Predicted ice fractions for initial diameters of 0.5, 1, 2, and 4 cm from the same offline simulation of a hypothetical wind-tunnel experiment, plotted as in Fig. 7. A unique fall speed (Lesins and List 1986) for each initial size was used to determine the wind speed for each set of runs. The control run started with a diameter of 2 cm simulates the observations by Garcia-Garcia and List (1992). The same repressurization time of 1 min was used for all experiments.

Citation: Journal of the Atmospheric Sciences 71, 12; 10.1175/JAS-D-13-0375.1

There are implications from these sensitivities for hail in natural clouds. When younger, hail particles are smaller and grow by dry growth. Their liquid water fraction is zero. When more mature, the regime of wet growth begins when hail falls through a large mass of supercooled droplets. If there is too much tilt of the updraft away from the vertical, then the zones of location of hail and small cloud droplets do not coincide. In that case, wet growth does not start, or else soon terminates if active.

4. Conclusions

Most hail in nature is spheroidal. The spheroidal hail has a dominant type of orientation during its fall, with accretion and freezing of liquid concentrated in the equatorial zone, which further leads to a decrease in the aspect (or “axial”) ratio. The smaller axis is located almost horizontally, with the particle spinning around it, and the larger axis is almost vertical. (This “symmetric gyration” of oblate-spheroidal hail does not apply to all hail, and the smallest hail can be spherical or conical, while any hail in wet growth can develop lobes or other asymmetries, all causing alternative fall behaviors.) The conventional one-part approach, which assumed spherically symmetric hail growth, was unable to reproduce the laboratory observations of spheroidal hail growth. In the present study, the theory of wet growth is extended to the case of inhomogeneities of surface temperature and of liquid coverage over the surface of the hail particle.

Accordingly, a new two-part model of wet hail growth is proposed that takes into account the inhomogeneity of accretion over the particle surface. The two-part theory treats the heat fluxes between its wet and dry parts and from the sponge radially through the liquid skin to the air. The tenet is that both wet and dry parts may coexist on the surface of any given hailstone while it is growing by wet growth. Also, the effect from inhomogeneities of liquid skin depth on the heat fluxes though the liquid skin is parameterized. A puddle effect after the onset of wet growth is treated. Gradual internal freezing of liquid soaking the interior of the hail or graupel particle during dry growth (“riming”) is treated by solving the heat and mass balance equations for dry and wet parts. The inclusion of size dependence and detailed morphology of ice into our scheme of wet growth allows the time dependence of freezing to be represented more realistically.

In contrast to List (1990, 2014a,b), the proposed model of hail growth does not assume a priori any equality of the velocities of the ice sponge interface υis and the outer liquid surface υih. In the present paper the motion of the sponge–liquid interface is governed by dissipation of heat from the particle (usually to air mostly) while that of the outer liquid surface is governed by the mechanics of accretion—a different process. It is interesting that the example of simulations with the two-part model shows that υis and υih are actually similar since the adjustment of temperature of the liquid skin’s surface provides a mechanism for balancing υis and υih. At the same time, it is an open question whether this adjustment is always possible even at the most intense accretion rates and warmer subzero temperatures.

The conclusions of the present paper are as follows:

  1. The new two-part model of wet growth of hail shows good agreement with laboratory observations when a wind-tunnel experiment is simulated both for ice fraction and surface temperatures. Although the thickness of the liquid skin was not measured in the simulated experiments, its order of magnitude is consistent with estimates in the literature (<1 mm) and with two other models of wet growth.
  2. Artificially prohibiting the two-part treatment from our scheme in simulations of the experiment in 1 and replacing it by entire coverage of the particle with liquid during any wet growth yielded unrealistic results for ice fraction and/or surface temperatures. The two-part treatment is necessary for accuracy of the time evolution of wet growth.
  3. An offline simulation with the new two-part model of wet growth for a single set of conditions (−16°C and 12 g m−3) showed how the thickness of the liquid skin is almost constant over time, as assumed by List (2014a,b).
  4. Sensitivity tests with the two-part model show a strong sensitivity of the ice fraction of the deposit with respect to the size of the hail particle. There are reductions of the ice fraction (e.g., by about 0.1) and of the critical liquid water content (by up to 100%) for onset of wet growth when the hail size is doubled.

The most salient result is the strong dependency on hail diameter of the ice fraction predicted during wet growth. Size matters immensely for wet growth. Yet the dependency on size of wet growth has not been quantified in laboratory experiments hitherto—a remarkable gap in empirical knowledge. Our theory predicts the natural tendency for wet growth to be promoted by larger sizes of the rimed particle D. Indeed, this tendency explains why hail—the largest of all types of hydrometeor in clouds—is observed to display wet growth (e.g., Browning et al. 1968), while wet growth of small riming snow is practically never seen in nature. Physically this size dependence arises because the rate of dissipation of heat to the air determines the wet-growth rate and increases with size (∝D1.9 approximately if fhD0.9 for centimeter-sized hail) more slowly than does the rate of accretion of liquid mass (∝D2.5 approximately). Hence, the ratio of rates of accretion of liquid relative to its freezing must increase with increasing size. A key influence is from the accretion rate being proportional to the particle’s cross-sectional area (∝D2). Therefore, wet growth is more likely for larger particles.

The results of the new two-part scheme for time-dependent freezing of spheroidal hail, for a situation with wet growth followed by gradual internal freezing, turned out to be in good agreement with laboratory measurements. Dependencies of both surface temperature and ice fraction, on the environmental values of LWC, agree excellently with the results of the laboratory measurements within a wide range of ambient air temperature from −6° to −20°C. Of course, there are limitations to the realism of our model, especially with the simplified treatment of the internal structure. Sections of hailstones reveal multiple layers of clear and translucent ice, which are only treated by one or two layers in our model. In nature, freely falling hailstones might be expected to flip after prolonged growth so that their diameter becomes horizontal, preventing the axial ratio from becoming too small and promoting full coverage of the entire stone by liquid. Flipping was artificially prevented in the wind-tunnel experiments underpinning our model. Nevertheless, our two-part model only predicts partial coverage of the stone by liquid in the earlier stages of wet growth before such flipping is most likely, so this is not necessarily a problem. Also, the frequency of flipping in nature is not yet observed and so is not treated here.

In our two-part model of hail, explicit treatment of the time evolution of liquid skin thickness allows situations in which the skin is deepening or thinning to be simulated. Such situations could arise just after onset of wet growth or during changing ambient conditions. In particular, the two-part treatment is found to be of paramount importance for realism of the critical ambient conditions for onset of wet growth, because the thickness of the liquid skin depends on areal coverage of the wet part and controls the thermal insulation by the skin of the region of freezing at the sponge just below it. This thermal insulation determines the loss of latent heat through the liquid skin to the air. Hence, the geometry of the liquid skin, which in nature only covers a fraction of the surface of the hailstone initially, controls freezing during wet growth of hail.

A detailed comparison of the results of the two-part model and a conventional one-part model in reproduction of the laboratory results was conducted. A bias exists in the conventional one-part approach, which is that dry growth persists over the entire particle and any liquid skin immediately freezes, even when the surface temperature exceeds the empirical threshold for onset of wet growth. Thus, the one-part model is unrealistic and predicts the transition from dry to wet growth at LWC values twice as large as the observed threshold value. Moreover, the one-part model is unable to reproduce the observed surface temperatures at intermediate liquid water content of wet growth and displays a warm bias of up to 2–3 K, especially at the colder ambient temperatures of the simulated experiments. Nevertheless, a case study of the time evolution for intense wet growth (−16°C and 12 g m−3) shows that if the wet part occupies over 90% of the particle surface, the hail properties predicted by the one-part model eventually may converge with time toward those of the two-part model.

Thus the spherically symmetric model of hail growth is expected to produce realistic results in cases when either the LWC is very high or the duration of a hailstone in wet growth is long, so that all of the surface of hailstone can become wet. Note that the duration of such an “adaptation time” can be more than 1 min or so, during which, in a real situation, hailstones would typically fall by more than 1 km. During this adaptation time, wet growth could sometimes be replaced by dry growth again and the adjustment of one-part model to the two-part model would not be achieved. The bias at the beginning of wet growth inherent in the one-part spherically symmetric model would introduce substantial errors in the description of wet growth of hail in cloud models.

The laboratory experiments of wet growth have been very few. Those by Garcia-Garcia and List (1992) provided a perspicacious wealth of knowledge and were the unique basis for the present paper. Specifically, we recommend that the community invest in reconstruction of their horizontal wind tunnel with modern equipment and perform such laboratory observations of wet growth over a wider range of hail sizes and of ambient conditions of temperature, liquid and ice water content, and droplet mean size. Variables such as the ice sponge density (without liquid), distribution of liquid skin thickness over the particle, amounts and temperature of shed water, and values of constants in the present paper (C1C5) need to be observed. A problem with the experiment by Garcia-Garcia and List was that about 1 min elapsed between switching off the fan in the tunnel and extraction of the accretion for weighing its liquid and ice outside. There was partial refreezing, biasing their results. More prompt weighing of the liquid and ice, perhaps somehow in the tunnel, must be possible. Better mapping of the temperature and liquid skin thickness over the hailstone surface during wet growth, and of the cloud-droplet size distribution and liquid content in the ambient air, must be possible with modern sensors [e.g., visible and infrared high-speed video cameras (e.g. Bauerecker et al. 2008), fast cloud droplet and cloud particle imager (CPI) optical probes, ultrasound probes for the interior of the hail particle, radar, and microwave imagery] and digital technology (e.g., precise control of ambient conditions of temperature, humidity, and turbulence). Equally, another problem of the experiments by Garcia-Garcia and List was that the hailstone was mounted on a rod. Although symmetric gyration (rotation and nutation/precession) of the hailstone was mimicked by the rod’s motions, prolonged growth gave rise to unrealistic wheel-like shapes in the experiments, which occasional flipping during freefall would prevent in nature. A vertical wind tunnel with a high-speed flow (up to 50 m s−1), to suspend the hail particles freely, could be constructed somehow with modern technology (e.g., with automated control of the flow direction to keep the hail particle balanced and away from the walls). Though an engineering challenge, a resurgence of wind-tunnel experiments might provide breakthroughs of realism in the study of wet growth.

Finally, the possibility of extra phenomena, hitherto unforeseen, affecting wet growth cannot be ruled out. What happens to wet growth when ice is accreted or when ice is nucleated at the surface by accretion of (e.g., biological) ice nucleus aerosols or when chemically contaminated (organics, lower pH) cloud liquid is accreted are open questions completely unobserved so far. They too merit investigation in the laboratory.

In summary, the current paucity of laboratory observations of the size dependence of wet growth makes our new theory of time-dependent freezing quite applicable to treatment of clouds in atmospheric models. There is particular need for laboratory observations of size-dependent wet growth at the warmest subzero temperatures, which allow the smaller rimed particles to become wet. Aspects of wet growth requiring observations include surface roughness, any turbulent enhancement of thermal conductivity of water inside the liquid skin, and the effective depth for heat transfer when the liquid depth is inhomogeneous.

There are two possible ways to improve the theory. The first way is to get more information concerning the parameters of hail structure during the wet growth. It would allow us to choose parameters of the two-part model for various conditions and sizes. Equally, future experiments with especially large hailstones will extend the observational basis for the shedding parameterizations to large sizes in a more robust way. The second possible way is to solve extra differential prognostic equations for more variables characterizing the internal structure of hailstones. This way requires, however, significantly more computational time.

Acknowledgments

V. Phillips was supported by three COST Short-Term Scientific Mission (STSM) research grants from the European Union, by two awards related to ice microphysics from U.S. National Science Foundation (PDM program) (Award ATM-0852620) and U.S. Department of Energy [Office of Biological and Environmental Research (BER), ASR program (Award DE-SC0002383)], and by a subaward from Hebrew University (DE-SC0006788). The group of the Hebrew University is supported by the U.S. Department of Energy’s (DOE) Office of Biological and Environmental Research (DE-S0006788; DE-SC0008811) and by the Binational U.S.–Israel Science Foundation (Grant 2010446). The authors express their gratitude to Charles Knight and Andrew Heymsfield for advice on aspects of the scheme.

APPENDIX A

List of Symbols

Symbols used in the paper are described in Table A1.

Table A1.

Symbols used in this paper, along with their descriptions and units.

Table A1.

APPENDIX B

Estimate of Volume of Liquid Skin

To calculate the radius of the outer surface of the ice sponge at the end of time step and the velocity of ice sponge interface [Eq. (16)], it is necessary to consider processes of soaking and to calculate densities and volumes of different layers in the particle (steps 3–5).

a. Step 3: Preliminary estimate of depth of outer liquid skin

In step 3, the outer liquid skin is considered. In earlier studies of melting (Rasmussen and Heymsfield 1987; Phillips et al. 2007), all liquid during soaking was assumed only to accumulate inside the particle. The present approach differs because the liquid skin’s thickness needs to be predicted when simulating the heat flux through it to the air. While the particle is soaking but not yet fully soaked, the wet part has a liquid skin of volume . A preliminary estimate assumes it is a small fixed fraction C2 of the volume of all liquid in the skin and sponge:
eb1
where . The estimate is preliminary because it neglects the possibility of exterior liquid accumulating in the skin when the sponge has reached its maximum extent. That possibility will be accounted for in later steps. Equation (B1) expresses a hypothesis of self-similarity of partitioning of liquid between the sponge’s interior and exterior across a wide range of liquid volumes and rates of accretion in the presence of dynamical molding of the liquid. In reality, there may be limited dependencies of on particle size, LWC, and temperature. Nevertheless, although simple, Eq. (B1) gives rise to realistic results in the offline validation. The value is the hypothetical volume of liquid in the sponge when the sponge has reached its maximum possible radial extent for the particle size. In nature, radial sections through many particles in wet growth of only a single size would reveal that their sponge is never closer to the center than some minimum distance. The sponge-free central region varies in radius from particle to particle in such a monodisperse population, but it has a certain minimum size. This universal absence of sponge in some central region exists because there is a minimum size (e.g., a few millimeters or centimeters) for wet growth at each temperature and cloud-liquid content of ambient air. Thus, their precursor particles (e.g., small graupel or frozen drops) may have been previously in dry growth before starting wet growth. Hence, the sponge’s soakable volume (equal to the volume of liquid inside the sponge) cannot exceed , corresponding to the sponge’s maximum possible radial extent ().

Also in Eq. (B1) there is a dimensionless parameter, < 0.5. It is estimated as follows. While the particle’s sponge is still growing and the particle is not yet fully soaked, most but not all of Vw is in the particle’s interior. In the laboratory experiment by Garcia-Garcia and List (1992), a hailstone reaching 4 cm in diameter and riming at an LWC of 4 g m−3 in an air temperature of −10°C was observed to have a liquid fraction of about 0.12 (if the mass of the initial 2-cm-hailstone model is counted as ice), which implies . The observed supercooling of its wet surface of 2°C and the observed product of collection efficiency, ice fraction, and LWC (3 g m−3) imply a thickness of the water skin of about 0.3 mm (List 1990, his Fig. 4). The ratio of its masses, of the liquid skin of such thickness and all liquid, implies .

To predict in Eq. (B1), the maximum radial extent of the sponge is constrained by the density of the spongy ice and the minimum size of the core. First, the spongy ice of the wet part has a bulk density similar to that proposed by Rasmussen and Heymsfield (1987):
eb2

This may be derived by assuming the sponge is fully soaked with an ice fraction (by mass) of . Without a prognostic variable for mass of ice or liquid in the sponge, we use . [This bulk density in Eq. (B2) does not include the liquid and is not to be confused with the entire bulk density of the sponge including liquid, which varies little.] The purpose of using , which is only for the ice part, is that it is related to the ice-front propagation speed and TL/i. Now varies widely, between perhaps about 10% and almost 100% of the density of pure ice.

All wet growth is assumed to be spongy in our scheme. For simplicity, the sponge is assumed never to contain any air and is always fully soaked, as its ice almost never pierces the air–liquid interface of the skin (e.g., List 1990). So the total volume of sponge is determined by the amount of liquid in the particle. The sponge’s soakable volume always equals the volume of liquid soaking it. This soakable volume is simply the component of the sponge’s total volume that is not pure ice. The sponge’s radial extent, in turn, determines that of the inner spherical core of ice just beneath it.

The minimum possible size of the core, for a given particle size, is realized by those particles with a sponge at its maximum extent. This minimum size exists because wet growth only occurs at particle sizes greater than ronset, and so is estimated by
eb3

The minimum volume of the spherical core is .

Next it is assumed that the rates of freezing, and of dissipation of heat to the air (per unit solid angle or per unit area on a spherical surface), are approximately uniform over the entire surface of the particle. This implies the mass of ice at all depths beneath the outer surface, per unit of solid angle subtended at the particle’s center, is similar between both dry and wet parts (Fig. 3). The wet part has a wet surface chiefly because liquid is accreted too fast there for it all to freeze. The assumption implies that the “spherically extrapolated volume” (see appendix D) of the dry part’s ice outside the inner core (before freezing is evaluated) is
eb4
Since masses of ice in the core, sponge, and rest of dry part all sum to mi,0, this yields
eb5
The sponge attains its maximum radial extent when Vi,core = Vi,core,min. Equations (B4) and (B5) yield an expression for in terms of Vi,core, which implies a maximum soakable volume of sponge (when at maximum radial extent) at the end of the time step of
eb6

In summary, Eq. (B1) yields the preliminary estimate of the liquid skin’s thickness, using from Eq. (B6) and the mass frozen in the wet part during the time step from step 2. This thickness, in turn, will be used to infer the actual soakable volume of sponge in the next step. [A more accurate alternative to this approach, which is too computationally expensive for our cloud model currently, would be to introduce an extra advected scalar for the mass of ice in the sponge, , used for Eq. (B7); the sponge dimensions and liquid skin thickness could then be predicted without some of the assumptions inherent in Eqs. (B1)(B6).]

b. Step 4: Determination of soakable volume of entire particle (wet and dry parts)

Before the freezing in the time step is performed, the sponge has a volume of . At the end of the time step, the sponge has a density of . If it is not yet at maximum radial extent (), the preliminary estimate of the volume of liquid in the exterior layer is valid and the sponge contains a volume of liquid equal to . In general, the soakable volume of the sponge at the end of the time step is
eb7
Equation (B7) yields and , using . The total soakable volume of the entire particle in both dry and wet parts at the end of the time step is then
eb8

Here, is estimated from by Eq. (B4) and yields by Eq. (B5).

In summary, the soakable volume within the ice of the entire particle is calculated in Eq. (B8) as the sum of that of the sponge, core, and rest of the dry part. The sponge’s soakable volume from Eq. (B7) is evaluated using its maximum soakable volume from step 3.

c. Step 5: Determination of radii of ice sponge–liquid and liquid–air interfaces

The wet part’s total volume of ice, , and volume of exterior water, , at the end of the time step are
eb9
eb10

Combining Eqs. (B8)(B10) yields and . Radii to the outer surfaces of ice and liquid in the wet part are and , respectively. Also . Equations (B8) and (B10) express the assumption that liquid soaks the interior first of the wet part, creating a sponge, and then soaks the core and dry part (unless it is too dense) before finally accumulating on the exterior of the wet part in an outstanding wedge and then in a spherical shell, when the particle is fully soaked. The sequence of stages of soaking for a graupel or hail particle is shown in Fig. B1. Following Rasmussen and Heymsfield (1987), hail particles denser than 910 kg m−3 have no soakable volume (except for the sponge).

Fig. B1.
Fig. B1.

Sequence of stages in soaking the particle during prolonged wet growth, depicted schematically. Shown are sections through both poles of the particle, including both wet and dry parts. Liquid or fully soaked regions are denoted with darker shading, while regions that are soaking are shown with lighter shading. Regions without any liquid are unshaded. (a)–(c) Soaking of the particle and the preliminary estimate of the liquid skin’s volume is valid for the volume of exterior liquid (). (d) The particle is fully soaked with surplus liquid accumulating in an outer layer of liquid thicker than the initial skin (). Initially, there is only a thin skin of liquid covering the wet part and the sponge is shallow, as shown in (a). Next, the sponge deepens in (b). The sponge has reached its maximum depth in (c), and the rest of the particle is now soaking (if the dry particle’s bulk density is less than 910 kg m−3). If the hail originally had a high density when dry (>910 kg m−3), then the core and dry part can never soak, so stage (c) is skipped. The sponge is always fully soaked. For graupel, the wet sponge covers the entire particle.

Citation: Journal of the Atmospheric Sciences 71, 12; 10.1175/JAS-D-13-0375.1

In summary, and defining the dimensions of the outer layer of liquid covering the wet part are evaluated from Eqs. (B9) and (B10), which treat the volumes of sponge and interior liquid soaking the ice using the soakable volume of the particle from step 4.

APPENDIX C

Approximate Estimate of Heat Flow between Wet and Dry Parts

Equation (15) for the interior flow of latent heat from freezing at the surface of the wet part’s ice, through the stone to the surface of the dry part, is derived as follows. For the purpose of treating the interior heat transfer, the particle is assumed to be spherical with a heat flow rate from the wet to dry part that is purely “meridional” in direction (the particle has rotational symmetry around its axis through the poles). Also, the depth of the liquid skin is neglected for this purpose. Figure C1 illustrates how the heat flow through the entire particle may be approximated by that between two “representative latitudes” of the ice surface at temperatures TL/i and in the wet and dry parts, respectively (points C and D in Fig. C1). The extents by latitude of the wet and dry parts in a given hemisphere are 2β and 2α, respectively. Each representative latitude is at the average latitude (β and 2β + α) for its component in the hemisphere and both latitudes are separated by 45°. Much of the heat flow is assumed to be confined to a “shallow layer” just beneath the icy surface, in which the heat flow is shallow < . The layer is shallow because the source and sink of heat driving the flow are both near the outer ice surface of the particle and are not far apart (on average about 45°). The shallow layer’s heat flow is assumed to be a fixed fraction of the total value, so that = C7shallow. This shallow layer is of uniform depth (distance LM in Fig. C1) below the particle surface. Its depth (LM) is twice the maximum depth of the straight line (CD) between the representative latitudes (at C and D) at the particle surface on the meridional section.

Fig. C1.
Fig. C1.

Meridional section through the poles of a hailstone. The shallow layer (bounded by gray curves) is assumed to convey most of the heat flow (thick arrow) from wet to dry parts. Its depth is twice the maximum depth of the straight line (CD) between the surface points at representative latitudes of the wet and dry parts (C and D, respectively). Each representative latitude is located at the average of all latitudes in its component over a given hemisphere.

Citation: Journal of the Atmospheric Sciences 71, 12; 10.1175/JAS-D-13-0375.1

The shallow layer contains some of the shortest paths (least thermal resistance) that the heat can take between the two representative latitudes of wet and dry parts at the surface (Fig. C1). All such possible meridional paths are shorter than the shortest arc following the surface (arc CELD in Fig. C1) between both representative latitudes. The depth of the shallow layer below the particle’s surface is . In the meridional section, the shallow layer is oriented along the base (CD) of the isosceles triangle (OCD) created by both radial directions from the particle’s center (O) to the representative-latitude points (C and D) and is bisected by the conical surface (passing through line OLM in Fig. C1) of latitude, β + π/8. The shallow layer has cross-sectional area A on this conical surface (through OLM). The cross-sectional area has a depth of (1 – C6) below the particle surface and is a difference between two concentric areas on the conical surface, such that . Here, P is the radius of the latitude circle on the particle surface along the outer edge of A, where . Here, is the angular extent by “latitude” of the wet component in a given “hemisphere” (Fig. C1).

The heat flow rate in the shallow layer may be regarded as approximately uniform over A and normal to it. Integrating Fourier’s law of thermal conduction over A yields a heat flow (in watts) of shallow ≈ −KpAdT/dx, where dT/dx is the uniform temperature gradient normal to A and x is the tangential distance in the direction of heat flow. Here Kp is the thermal conductivity of the mixture of ice, air, and internal liquid soaking the particle and is the volume-weighted average of the thermal conductivities of these three components (Ki, Ka, KL). The average gradient in temperature normal to A may be approximated by dT/dx ≈ −(TL/i)/(π/4). (The shortest arc on the particle surface between both representative latitudes is π/4 in length.) Hence, = 8C4Kp(TL/i), where C4 = C7(1 − ) and C6 = 0.85. Parsimoniously, we assume the heat flow rate through the shallow layer of cross-section area is twice that passing from wet to dry parts below it, so that C7 ≈ 1.5 and C4 ≈ 0.4. This value of C4 yielded agreement of the predicted equatorial surface temperature with laboratory observations by Garcia-Garcia and List (1992) in offline simulations (see section 3). So the above assumptions seem adequate for the purpose of thermal coupling of wet and dry parts.

APPENDIX D

Concept of Spherical Extrapolation

A new geometrical concept of “spherical extrapolation” is defined as follows. Extrapolation of the angular extent (subtended at the center of the particle) of a component into all directions from the center to form a hypothetical volume (either a sphere or a spherical shell) with spherical symmetry of the same radius creates a “spherically extrapolated volume.” Such extrapolation is denoted by ϕ. (Each component has no dependence on direction of either its properties or radial extent when viewed from the particle center for all directions within the component’s angular extent.) Figure D1 illustrates the concept. For example, if the volume of ice in the dry part outside the core is , then its spherically extrapolated volume is a spherical shell of volume . Similarly, if the volume of spongy ice in the wet part outside the core is , then its spherically extrapolated volume is a spherical shell of volume, . The spherically extrapolated volume, either of a hemisphere or any other spherical sector, is its corresponding sphere.

Fig. D1.
Fig. D1.

(right) Schematic diagram of hypothetical volumes created by spherical extrapolation is shown. (left) A section through the wet part in a plane through the poles for the actual particle is shown. Also shown is the same section for the spherically extrapolated volumes [subscript ϕ of ice outside the core in (top right) the dry part and (bottom right) the wet part].

Citation: Journal of the Atmospheric Sciences 71, 12; 10.1175/JAS-D-13-0375.1

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