New Hailstone Physics. Part II: Interaction of the Variables

By List, Roland

Proquest LLC

ABSTRACT

The reduction of parameter dimensions in Part I is complemented by the compaction of parameter space in Part II. The range of diameters is 0.5 = D = 8 cm, and the assumed liquid water content varies within 1 = Wf = 3 for dry growth and Wf = 6 g m-3 for shedding. Entirely new data throw new light onto HMT and growth.

Results are as follows: (i) dry growth is unimportant, since most hailstones grow spongy; (ii) radial growth is slow for dry and fast for spongy growth because less latent heat of freezing needs to be discarded if a smaller portion of the accreted water is frozen: this growth with shedding is particularly effective if the product Y of the net collection efficiency and ice mass fraction of the deposit is 0.2 = Y = 0.6; (iii) the lowest possible surface temperature tS for dry growth is -32.3°C. For water-skin-covered, spongy particles tS > -5°C, and tS > -0.55°C for shedding from wet surfaces without water skins; and (iv) the interplay between water-skin thickness and surface temperature allows interconnection of all variables. However, new icing experiments are necessary to prove the proposed sphere growth by special gyration, to quantify the components of Y, and to address water-skin properties and growth.

Radically redesigned dynamic cloud models need to incorporate hail packaging and rain spectra evolution in clouds. The latter will connect hailstone shedding with a warm rain process that is parallel to and interacts with hail formation.

(ProQuest: ... denotes formulae omitted.)

1. Introduction

List (2014, hereinafter Part I) served to develop equations governing all aspects of heat and mass transfer (HMT) and growth of spherical hailstones. Part II presents the results of the calculations involving all variables, as classified in the three CASES [(i) dry; (ii) shedding fromwet surfaces, not covered by permanent water skins; and (iii) surfaces with water skins] (Part I). They are sequenced in different variable combinations and represent the main body of the new results. In addition, the parameter space is minimized by restricting the range of variables to the one of importance for growth.

The reduction from six ''old'' to four newly defined variables for HMT allows a much more compact representation of the results-without any loss of accuracy and without any additional assumptions. The six old variables are air temperature tA (°C) and hailstone surface temperature tS (°C), cloud liquid water content Wf (kgm23), net collection efficiency ENC (2), ice fraction of the spongy deposit If (2), and Reynolds number Re. The four new variables are X (gm23), Y (2), F (kgm23), and C (2), with X (5Wf Re3/4) replaced by a proportional factor X0 = WfD3/4, where D(m) is the diameter of the spherical hailstone and X0 is preferable over X because Re for free-fall needs to be split into its many components to solve the main equation. The quantity Y is the product of ENC and If, and F and C are functions only of tA and tS.

Previous experiments with gyrating, water-skin-covered spheroids limited tS to =25°C (Lesins and List 1986; Garcia-Garcia and List 1992; Greenan and List 1995; Zheng and List 1995). Shedding from wet surfaces that are not covered by water skins narrows the resulting temperature band, 08 = tS = 20.55°C. These temperature range reductions alone are equivalent to a reduction of parameter space by 87.5% and 98.5%, respectively. The growth speed of ice dendrites into water skins is a key to the growth of spongy ice. The hailstone's radial growth speed VR (ms21) is linked to the temperature gradient across the skin [(to 2 tS)/d)], with d (m) the water-skin thickness.With the temperature of the sponge-water skin interface, at to ' 0°C, the gradient can be reduced to tS/d. Supercooled water skins limit the HMT (CASE 3).

For CASE 1 the governing equations are simplified by setting the product of ENC and the fractional ice content If of the deposit equal to unity (Y = 1). In CASE 2 with heavy shedding, the group of four new variables is also reduced to three by setting tS ' 0°C. The variable reduction in HMT for CASE 3 is achieved by equating the speed of the ice sponge front, penetrating the water skin, to the growth speed of the skin- air interface (see Fig. 5 in Part I). This compensates for the addition of d.

One of the major inputs into growth calculations is the free-fall velocity V (m s21) of the rough, spherical hailstones, calculated with measured drag coefficients CD (2) (List et al. 1969). The spheres had various degrees of roughness, characterized by a ratio kr between roughness element height and a hailstone diameter of 0.07 [typical for collected hailstones (List 1958b)]. This value is chosen for the present calculations. Note that the flow regime even around the smallest spherical hailstones (D = 0.5 cm) is supercritical due to surface roughness. Thus, CD is assumed to be constant at 0.5 for D=2 cm, up to Re'400 000, the upper limit of the drag experiments (List et al. 1969). At tA5220°C the Re of hailstones with D = 8 cmis ;100 000. However, for D 5 0.5 cm CD is unity (List and Schemenauer 1971).

The most common small hail particles found as embryos in hailstones are water soaked but originally lowdensity conical graupel (List 1958a,b, 1961b, Part I). Small hail has a density between that of water and ice, with the mix determining the ice mass fraction If of the particle. Other embryos, such as frozen drops, are also possible, but rare. There is no substantiation that embryos start from frozen droplets. The density of deposits directly grown on hailstones is in the same range (915 , rH , 1000 kgm23, with an average of 958 kgm23, where rH is the hailstone density). The density of (frozen) hailstones is very close to ice density (Vittori and Di Caporiacco 1959; Macklin et al. 1960; List 1961b; List et al. 1970). Low-density shells and cores have been observed after the draining of water from grown spongy ice, thus causing densities as low as 0.1 , rH , 0.3 gcm23 (List 1978-79). Draining can occur either during contact with the ground or during free-fall by the Bernoulli effect. Such overlapping of sequential growth phases had first been explained by List (1961a). Thus, references in the literature of large low-density hailstones (Knight et al. 2008; Rasmussen and Heymsfield 1987a,b, c) need to be viewed with caution. They could have been caused by superposition of growth phases or plain size inflation by naming a D = 2-cm hailstone as large.

All calculations in this paper are carried out for clouds represented by the Beckwith (1960) soundings of Denver, Colorado, hailstorms, with supercooling to the limit of 240°C, which is neither being postulated nor is it generally observed. Depending on the properties of the available ice nuclei, any appropriate freezing level can be chosen to further restrict the growth region. For figures involving HMT and growth, that flexibility will be maintained in this paper. Every attempt was made to achieve optimal accuracy. Hence, drag coefficients were tailored to hailstone size (see above).

The calculations are restricted to one set of conditions describing hail clouds. They are limited to spherical hailstones. The equations represent only the template for other studies. There was no intent to solve all the problems.

The approximations for the different material constants and their dependence on temperature are listed in the appendix. No attempt has been made to make the equation dimensionally correct.

Nomenclature

In the text the quantities for D, Wf, the ''net accreted ice fraction'' Y, or any other quantity will often be specified by a suffix ''i,'' indicating the magnitude in centimeters, grams per cubic meter, or (2), respectively. A list of all symbols can be found in Part I.

The dimensions underlying all equations are in SI units, with the exception of the ones in the appendix. The dimensions used in the figures may be different from the SI units.

The sign ''.'' for temperature means ''warmer than.'' For temperature level ''.'' means ''higher'' or ''colder.''

2. The equations

There are six equations to be evaluated for the understanding of the growth and heat and mass transfer of spherical hailstones.

The growth equation, linking radial growth to net accreted liquid water content [(22) of Part I], is

... (1)

where VR and V are the radial growth and free-fall velocities (m s21), respectively, and EENC (2) is the product of collision and net collection efficiencies. The term E is set to unity in all other equations (however, it was carried in Figs. 1 and 10 as a warning for extrapolations for D , 0.5 cm).

The HMT is given by its most inclusive form [(18) of Part I]:

... (2)

The lhs of (2) was derived from the ratio of the sum of the heat fluxes by conduction and convection, and evaporation over the sum of the accretion-related terms. The variable product ENC If is the new variable Y.

In this equation Re1/2 for free-fall is substituted with

... (3)

with CRe linking Re1/2 and D3/4 by a fourth-root factor. At this time, information is required about the spherical hailstone's density rH and its CD. Note that the fourthroot factor will be of importance in error considerations. Term CRe depends on tA and air pressure pA. Using Denver hailstorm soundings by Beckwith (1960) removes pA as a factor. Assuming further that the hailstones are spherical, both rH and CD need to be known to solve (3).

The meanings of the newly added variables in (2) and (3) are: K is the experimentally determined nondimensional correction factor for HMT; k is the thermal conductivity of air (Wm-1 °C-1); Dwa is the diffusivity of water vapor in air (m2 s21); LE is the latent heat of evaporation from solids or liquids (J kg21); TA is the absolute air temperature (K); eS is the saturation vapor pressure at the surface (Pa); eA is the water vapor pressure in air (Pa); n is the kinematic viscosity of air (m2 s21); cw is the specific heat of water (J kg21 °C-1); LF is latent heat of fusion (J kg21); and g is gravity acceleration (m s22).

Here (2), which can be solved for X0 or Y (5ENCIf), is abbreviated by using two relationships for the auxiliary variablesFandC, quantities that are only dependent on air and surface temperatures:

... (4)

and

... (5)

Term F is a measure of the heat contributed by convection and conduction to freeze the net accreted water, whileCis the fraction of heat contributing to freezing by the supercooled accreted water.

With the definitions of F and C, (2) can be abbreviated

... (6)

Since both F and C are functions of tA and tS only, the four variables can also be given by the set (X, Y, tA, and tS). There is no algebraic solution for (4); thus, iteration procedures are required. The next equation links VR (m s21) with the driving temperature gradient across the water skin of thickness d:

... (7)

with C356.0343108 °Csm22 (rH is set for the skin with 1000 kgm23, the latent heat of fusion LF = 3.337 3 105J kg21 for to'0°C, and kw50.553Wm-1 °C-1). The other new symbol, x (2), represents the multiplier caused by turbulence in the water skin.

Illustrations of background information are presented in Figs. 1 and 2. Figure 1 shows VR as a function of the accreted liquid water content according to (1), with the terminal velocity substituted by (15) of Part I. The cutoffis for radial growth speed, its maximum, is set at 4mmmin21. The auxiliary functions F [(4)] and C [(5)] for dry growth and growth with shedding are displayed in Fig. 2. The function C is so close to linear that it was drawn as linear. Minor deviations are caused by the different temperature dependencies of the cw and LF.

3. Range restrictions of variables

In Part I, the HMT of a single hailstone has been reduced to the motion of a point in four-parameter space. The purpose of range restriction, stressed in this part, is to minimize the parameter volume, that is, to concentrate on the most probable growth conditions. In the following all the limiting conditions will be included starting with Fig. 3, even if some are substantiated only later. The reason for this step is simple: there will be no need for additional, upgraded diagrams. The reader is also faced with the real (reduced) range of conditions right from the beginning

The following bounds and ranges are introduced: 0.5 # D # 8.0 cm, 08 = tS = tA = 240°C. Further for CASE 1, 1 # Wf # 3gm3; rH = 915 kgm23; the upper limit of Wf for ice growth is set lower than the adiabatic value because of the depletion of cloud water by the hailstones that grow while competing for the available cloud droplets (List et al. 1968). For CASES 2 and 3, 1 # Wf # 6gm3, and 915 # rH # 1000 kgm23 or average (958 kgm23); the Wf limit is doubled because of shedding-and shed drops can be recollected by growing hailstones. For CASE 3, tS = 25°C, based on experiments with gyrating spheroids (Lesins and List 1986; Garcia-Garcia and List 1992; Greenan and List 1995; Zheng and List 1995) and supported later in section 5 (Fig. 10). Doubling could involve hail accumulation zones as envisaged by Kessler (1969) and as seen in the packaged hail, identified by frequency analysis of return signals from a vertically pointing Doppler radar. Luckily, the antenna was hit by hail [Thomson 1997; Thomson and List 1999; and, reprocessed in color by Thompson (List 2004)].

The upper limit for both ENC and If is unity, while the lower limit has to accommodate ''mushy'' hailstones with If as low as 0.1-0.3 (such hailstones maintain their physical integrity at free-fall speeds, as had been shown in icing experiments in wind tunnels). The limits Li ofX0, the product of Wf and D3/4, are L1 = 0.59 for W1 and D0.5, L3 = 1.78 for W3 and D0.5, L6 = 3.56 for W6 and D0.5, and Lmax = 28.5 for W6 and D8. Dimensions of L are gm23 cm3/4. There is a combined requirement, namely, that both lower limits of Wf (1 gm23) and D (0.5 cm) are adhered to at all times. In the figures with X0(5WfD3/4) as an abscissa, a gray band appears that contains those restrictions. Last but not least, a limit to VR of 4mmmin21 is suggested that is roughly in agreement with radar observations.

These restrictions may be changed by any user of the equation set.

4. The X0, Y, tA, tS domains

a. The tA-X0 domain

1) CASE 1 (FIG. 3)

The main properties of and differences between the three CASES are shown in Fig. 3. For CASE 1 the governing (2) is reduced by setting the product Y (5ENCIf) equal to unity. The tS = 0°C line represents the limit of dry growth at ice density. It is not the Schumann-Ludlam limit (SLL) because the SLL requires a wet surface due to the shedding of nonfreezing accreted water. Lower ice densities will require even lower surface temperatures. Higher air temperatures are less conducive for low-density ice growth. The lowest surface temperature for dry growth is 232.3°C. It is observed for D0.5 hailstones with W1. Dry growth (riming) of large hailstones at 240°C is possible up to X0 = 7.26. This is the low limit for cloud water supercooled to tA 5 240°C. Less supercooling moves tS to a higher temperature. With ... the correspondingWf is 1.67 gm23.

The tA-X0 domain is the key diagram for dry growth. It is further partitioned by giving the X0 ranges of Di hailstones. The horizontal Di bars cover the range that starts at the leftwithW1 and extends toW3. The bars are aligned to touch the tS50°C line. TheWi ends define the corresponding X0i . This limits the dry conditions for the Di bar chosen as long as tS # 0°C (as octagons). The possible growth regions are between X01 and ... hailstone can grow dry for W1 at tA = 22.45°C and for W3 at tA = 28.7°C; a D2 hailstone can grow dry for W1 at tA = 28.2°C, and for W3 at tA = 228.2°C; a D4 hailstone can grow dry for W1 at tA = 214.9°C and W3 at5241.4°C (for graphic purposes only). ForD4 the biggestWf for dry growth (at tA5240°C) is 2.6 gm23. A D8 hailstone reaches the limit of dry growth with W1 at 226.5°C; at 240°C, the upper limit moved to Wf 5 1.67 gm23. At 240°C and X0 = 7.26 gm23 cm3/4, all hailstones reach the uppermost limit for dry growth. However, X0 is reduced for tA.240°C, as Fig. 3 (CASE 1) clearly shows. Hence, the domain in which hailstones grow dry is within the range designated by octagons and the gray area. There is another point: The low limit for a D4 hailstone gives the upper limit for all Di hailstones with Di , D4 at that tA level; that is, all hailstones with sizes D , 4 cm grow dry. At every level a different maximum size is allowed and each size is coupled to a different Wf.

By definition X0 provides the link between the two pairs ftA; tSg and fWf ; Dg. The pairs are equal if they have the same X0. Each pair produces a multitude of combinations for one X0j , for every X0j . But choosing Wf or D starts to unravel a situation if tA or tS is chosen.

In CASE 1 the upper limit of dry growth at240°C is at X0 = 7.26 gm23 cm3/4, but it does not indicate the beginning of spongy growth. This is dealt with by CASES 2 and 3, where the hailstones are wet and involve the relevant latent heat of evaporation from the liquid phase and not from ice. Thus, the limit for wet growth at240°C is at X0 = 6.88 gm23 cm3/4.

One general observation about the occurrence of large hailstones: They must be products of the most effective icing conditions. Thus, it is unlikely that they grow at low Wf and close to 240°C because that would require extended residence time in hail growth regions. Maintaining continuous updrafts of up to ;70ms21 for sufficient time is not likely-which would kill the notion of low-density large hailstones.

The reason why large hailstones grow dry at high levels (low tA) is because all components of the HMT increase substantially with increasing size. According to (1), radial growth speed is proportional to both V and Wf. Higher HMT translates into faster freezing of accreted water (and less sponge) or higher If or both. Falling to the 0°C level will favor spongy growth above the tS = 0°C height level (Fig. 3, CASE 1)-unless the fall occurs outside the updraftin a cloud part with low Wf. Further fall from the 0°C level to the ground will lead to melting, with the meltwater being drawn into any remaining loose ice deposit-another reason for questioning references to finding low-density hailstones at the ground.

Proof by radar is quite difficult because of the ambiguity of the reflectivity of spongy, spherical hailstones (Joss and List 1963; Joss 1964). Considering that X0max 528:5gm23 cm3/4 (at W6 and D8), and that the limit to dry growth is at X0max #7:26gm23 cm3/4, dry growth covers only a small region of the whole growth domain. This region becomes even smaller if the nucleation temperature is moved from 240°C to lower height levels with higher tA.

This situation points to the importance of numerically exploring the size sorting mechanism and the related evolution of size spectra in pulsating updrafts.

2) CASE 2 (FIG. 3)

Figure 3,CASE2, shows the slightly supercooled bands (08 # tS # 20.55°C), together with the sponginess/ice fraction of the deposits-as given by Y. The lower temperature (20.55°C) represents the limit of the radial growth speed, set at 4mmmin21. This is in agreement with the postulated fastest radial growth speed of dendrites [(19) of Part I]. The hailstone surfaces are neither dry nor covered by a continuous water skin. They result from shedding of small partial surface skins, blown away and disintegrating into drop spectra immediately after shedding. All those narrow bands in Fig. 3, CASE 2, are indicative of the magnitude of Y, the product of ENC and If. As expected, smaller Y values are found at higher tA, and dry growth is moved into the top-leftcorner of the diagram with smaller X0 and low tA.

As an example of a more detailed analysis, CASE 2 is further explored by breaking up X0 into its components, as is displayed in the tA-Wf domain for different Di and Yi (Fig. 4). At given Wf and Yi the air temperature is markedly lower for larger hailstones (this figure could be extended to CASE 3 by expanding the bands from 08 . tS = 20.55°C to 08 . tS = 25°C.) For D8 the hailstone starts to grow dry above the tA = 229.2°C level, assuming that the cloud water is still unfrozen at this temperature. The requirement is less stringent for smaller hailstones.

3) CASE 3

Figure 3, CASE 3, represents hailstones with a permanent water skin for a bandwidth 08 = tS = 25°C. Their surface temperature theoretically covers the whole range of X0 and for'08 = tA= 240°C. For tS' 0°C the sponginess is about the same as in CASE 2. Any point (tA,X0) can be created for any tS by proper selection of Y. Note that the patterns of the tS groups at 08,20.558, and25°Cof different Yare similar to each other.They seem to be parallel shifted.

The SLL for CASES 2 and 3 is given by the line tS 5 0°C (Fig. 3).

4) THE SUBSET tA-Wf (FIG. 4)

In the discussion of the tA-X0 domain, a venue was shown that allows the breakdown of X0 (Fig. 3: case 1 vs cases 2 and 3). Something similar can be achieved by a different split of the variables. In Fig. 4 tA is plotted versus Wf, relegating D into a parameter position, together with Y and for different tS.As such, tS could also be extended to25°C, making the results also relevant to CASE 3. However, it was not done to avoid cluttering. The parameters chosen are presented for pairs of Di/Yi. The calculations show that the range of conditions for the limit of wet growth with tS5 0°C are confined to tA = 230.8°C (see below), depending on Y. Air temperatures for spongy growth are generally at much warmer tA. Temperatures for low ''accreted spongy fraction'' (Y , 0.2) are mostly at tA. 210°C.

b. The tS-X0 domain (Fig. 5)

1) CASE 1

The lower limit ... sets tS at 232.3°C (Fig. 5, top left, CASE 1). For easier interpretation X0 has been decomposed intoWf and D (bottom left). The limits for X0i are indicated in all panels of Fig. 5 and later, as long asX0 is a coordinate. The associated panel (bottom left) shows that D8 hailstones start to grow dry for the interval 4.77#X0 #7.26 gm23 cm3/4. The lower limit is for tA5 229.2°C; the upper is 240°C.

2) CASES 2 AND 3

The solutions for CASE 2 are restricted to the thin tS band, limited by 08=tS=20.55°C, at the bottomof Fig. 5a, while CASE 3 is for conditions within a band 08 = tS = 25°C that relates tS to X0 for different values of tA and Y. The general condition is tS . tA. Figure 5 also shows that very low tA can be tolerate easier, tS=25°C, ifYis closer to unity. At high air temperatures and low X0, tS is decreasing fast, while the decline is slowing with decreasing tA.

c. The Y-X0 display, CASES 2 and 3 (Fig. 6)

Since Y is equal to unity for CASE 1, the study of the Y-X0 domain is restricted to CASES 2 and 3. CASE 2 limits the surface temperature to the gray bands with 08 = tS = 20.55°C for every tA, whereas CASE 3 produces wider bands with 08 = tS = 25°C (octagons 1 gray). The results are represented by Fig. 6. The region ... is cut off, so is the hatched area at high values of Y and X0 because tA is limited by 240°C.

The hyperbolic character of the type X0Y is imposed by (2). Lowering tA increases the values of X0 and/or Y. The magnitude of Y can be reduced by shedding through decreasing ENC and/or a lower fractional ice content of the deposit. Actually, Lesins and List (1986), Garcia- Garcia and List (1992), Greenan and List (1995), and Zheng and List (1995) have shown that increasing ice fraction and shedding are strongly correlated for gyrating spheroids. Their diagrams for ENC and If versus Wf could be used to form products Y for spheroids. Such data would be suggestive for spherical hailstones.

d. The tS -tA domain (Fig. 7)

The quasi-linear relationship (5) between tA and tS, for constant Wf and D (or X0) is displayed in Fig. 7. There are three cutoffs: (i) at tS5232.3°C (CASE 1 atD0.5 and W1) becauseWf,W1 is not allowed; (ii) at tS=20.55°C (CASE 2); and (iii) at tS = 25°C (CASE 3). Increasing size and/or liquid water content (or X0) increases tS. Because of the lower limit of Wf, a D0.5 hailstone requires a tA colder than 22.45°C (CASE 1) or ,22.68°C (CASE 3) for growth with tS = 0°C at W1. For W3 and tS = 0°C, tAmax is 28.67°C. CASE 1 shows how tA and tS relate to different Di. Low tA and tS are favoring high Di. The D8 hailstones can only grow dry at cold temperatures, coupled with high tS, and most importantly, only at Wf above but close to W1.

For CASES 2 and 3, only three combinations of diameterDi and liquid water contentWi are given for three Y each.Depending onY,D8 hailstones can grow over a tA range covering 35°C in CASE 3, while D0.5 ice particles only barely cover 10°C. For CASE 2 the difference between the two ranges is 25°Cversus 5°C. The insert in Fig. 7 refers to this range assessment. This figure makes it quite clear how much more cooling is required to freeze larger proportions of the deposit, as expressed by lower tA. In general, increasing D orWf at a given tS will cool tA to the limit. Note that the relationship between X0 and tS is quite identical in CASES 2 and 3 because the former has no permanent water skin, while the latter does.

e. The VR-tA domains (Fig. 8)

1) CASE 1

In (1) relating VR toWf and V can also be connected to theHMTequations either through tA or tS. This provides valuable new insights about the differences between CASE 1 and the contrasting CASES 2 and 3, as well as the different parameter ranges and restrictions. CASE 1 is relatively simple because it is for dry growth, whereas CASES 2 and 3 have Y as an additional dimension. Figure 8, CASE 1, is a plot of the VR-tA domain for Y1.0. There are three wedge-type data arrays with origins Oi, where ''i'' gives Wf in grams per cubic meter. To visualize trends the origin O4.5 and the corresponding W4.5 wedge is also shown. Note that tA associated with the O1 is 22.45°C, caused by the warming due to the freezing of the accreted water with W1. The lines forming the wedges at Oi are the intersections of tS = 0°C and D0.5. The data are divided by lines for lower tS and larger Di. The highest growth rate for W3 is ,2mmmin21 for Dmax, slightly larger than 2 cm. It occurs at the lowest supercooling for dry growth. The D8 hailstones can only grow slightly above W1, with tS near 0°C and tA in the 230°C range. It is the size that lowers the hailstones' tS out of range-even at tA5 240°C.

2) CASE 3 (AND 2)

The VR-tA domain for the two shedding CASES has a more complex structure because of the added variable Y. This is demonstrated by Fig. 8, CASE 3, with the splitting of tS into bands with tA at different Y. Two groups of constant Wi are displayed, in contrast to the groups of Di of CASE 1. CASE 2 is not displayed, but its bands of 08 = tS = 25°C would shrink to 08 = tS = 20.55°C; that is, they would have a width of approximately one-tenth of that of CASE 3. The wedges of CASE 1 are becoming more open as Y1.0 decreases to Y0.2. Noteworthy is the shiftof the Yi lines for different Di. With decreasing Y the bands for 08 = tS = 25°C vary less with tA, and they move to higher air temperatures. There are two limiting features: the two data frames are for W1 and W6, each covering the range of D0.5-D8. Again, the cutoffwas set at VR 5 4mmmin21. In contrast to W6, the VR for W1 varies only within a restricted range up to ;1mmmin21. Figure 8 documents the strong dependences of VR on both Wf and Di for all CASES. CASE 3 shows why Y/1 is limiting sponge growth and also reduces the radial growth rate. Term VR is halted by the lowest tA possible. More heat needs to be removed when If increases.

There is an interesting conclusion, namely, that high growth rates are only possible for low net accreted ice fractions Y, that is, preferably within 0.2 # Y # 0.6.

The broken lines of Fig. 8 show that the dependencies on D have been linearized. In addition, some lines for constant D have been slightly shifted to cover up small differences in dependence of Y.

f. The VR-tS domains (Fig. 9)

1) CASE 1

The radial growth velocity for dry growth of spherical hailstones in the VR-tS domain is displayed in Fig. 9, CASE 1, as subdivided by D, Wf, and tA. Its abscissa is limited to tS = 232.3°C for tA = 240°C. The lowest trapezoidal array represents D0.5, bounded at the bottom by W1 and W3 at the top. The dashed second array represents the conditions for D2, while the third is for D8. With the larger sizes such as D8, the growth conditions with the highest VR (#3mmmin21) occur with warming tS and lowering tA. Augmenting Di increases the radial growth speed. The tS cutoffis at the end of the displayed scale.

2) CASES 2 AND 3

Shedding is always associated with spongy growth. Thus, as with Fig. 8, CASE 3, a Y substructure is added to the diagram (Fig. 9, CASE 3). There are two extreme frames in the VR-tS domain: one is forW1, the otherW6. The range limits at D0.5 and D8 are different, with the exception of the basis at the D0.5 trapezes. The trapezoidal arrays for Y1.0, Y0.6, and Y0.2 carve out the conditions for W6 (top array) and W1 (bottom array). This makes it clear that, for high radial growth speeds, Y, as in Fig. 9, has to be in the range of 0.6 = Y = 0.2. Both Figs. 8 and 9 suggest that the cutoffof VR, presently at 4mmmin21 could be moved to 5 or even 6mmmin21. This would lower the limiting surface temperature to tS520.68 or 20.65°C, respectively, which is equivalent to a very slight shiftof the VR cutoffaway from the tS 5 0°C line in CASE 2 of Figs. 3-9.

Figure 9, CASES 2 and 3, shows that most of the domain covering tS to 240°C is irrelevant, except that it better explains the data trends in the limited bands. It is a great example for showing the drastic restriction of the parameter space that is basically caused by shedding. For both CASES, VR increases with increasing Wf and D; VR is greatest for tS = 0°C and the warmest possible tA. As in Fig. 8 freezing a smaller fraction of the accreted water allows faster radial growth speeds because less latent heat of fusion needs to be removed. The consequence of further increasing Y is to reduce VR. Note that the scale of VR is different in Figs. 9a and 9b because dry growth is much slower for CASE 1. Most of this shiftis a direct consequence of the doubling of W3 to W6.

5. Overview

Figure 10 represents all CASES, thus allowing comparisons. The original hailstone is given by size and height level by a point in Fig. 10a, together with the resulting free-fall speed. Selecting an accreted liquid water content produces a VR in Fig. 10b. CASE 1 displays tA for W1 and W3 and different Di in Fig. 10c (a more detailed version of this display is given in Fig. 8, CASE 1). Figure 10b also covers CASE 2 by the display of the ice front temperature to, opposite the corresponding radial growth speed. This is also to underline that VR in CASE 2 should not differ by much from CASE 3. The story line for CASE 3 continues directly from Fig. 10b to Fig. 10d, where VR is related to the temperature gradient across the water skin, after choosing If. Neglecting the interface temperature to, this gradient is composed of the ratio 2tS over d (Fig. 10e). At this point, no information is available about waterskin thickness. Nevertheless, the possible range of conditions is highlighted (octagons). Thus, guesses have to be made based on the icing of spheroids (d may have a thickness of magnitude ;0.1 mm). The results are surprising, considering the suggestion of very large temperature gradients across the water skin of up to 40°Cmm-1. This value is, however, reduced for smaller hailstones, smaller Wf, and smaller If. CASE 3 is then developed beyond Fig. 10e, into Fig. 10f, which relates tS to X0. Figure 10f is an expansion of the bottom-leftcorner of Fig. 5, CASE 3. The complex matching of tA with tA in Fig. 10a is not attempted. The key parts of Fig. 10 serve for comparison of the different CASES, the relationship between tS and d, and the connection to the new variable X0.6. Summary and comments

a. General findings

The four new variables X, Y, F, and C, established in Part I, are fully representing the HMT of spherical hailstones over their whole size range 0.5 # D # 8 cm. The corresponding equations were developed without any restrictions or approximations. They are based on the recognition that the six variables of the HMT can be simply reduced to four new ones. It was inherently built into the six-variable equation. The possible simplifications were just not recognized before. The auxiliary variables F and C are functions of tA and tS only. Thus, the HMT is equally determined by X, Y, tA, and tS. The variable X0 contains the factors CD and rH in a fourthroot term. This considerably reduces specific errors. Most importantly, it allows a replacement of X by X0 according toX5WfRe1/25CReWfD3/45CReX0, that is, WfD3/4 = X0. Thus, the free-fall Re is being replaced by D. This step was made possible by assuming a coupling of tA and pA to height in Denver hailstorms. Errors are introduced by approximations (hailstone density, drag coefficient) and assumptions about 16 water-skin properties, gyration effects, etc.

All combinations of variables are explored, but only after limitation of the ranges of variables, such as 1 # Wf # 3 or 6gm23 (3gm23 for dry growth, 6 gm23 for shedding), 0.5 # D # 8 cm, and 08 = tS = tA = 240°C, and, most surprisingly, tS = 25°C for hailstones covered by water skins. While cooling cloud water to tA5240°C is extreme, occasional scavenging of all nuclei in an updraftis possible. However, the lowest W1 limits tS to .232.3°C for dry growth (CASE 1). All these tS limits allow a substantial compaction of the results to the most probable growth conditions, and further reduce the parameter volume covered by growing hailstones. This is in addition to the reduction of parameter space from six to four dimensions (CASE 2) or three in CASES 1 and 3 (see Part I). For CASE 2 a maximum (dendritic) growth speed of VR = 4mmmin21, corresponding to tS = 20.55°C, was assumed. This limit can be raised, but may necessitate an increase in Wf to .6gm23. The reader may redesign such limits.

The VR in CASE 3 is highly dependent on diameter; it is fastest for 0.2 , Y (5ENCIf) , 0.6. This expresses the fact that spongy growth is required for high radial growth speeds and the creation of large hailstones. The reason for this is that complete freezing (If = 1) requires a time-consuming transfer of a latent heat of freezing to the air, which slows radial growth. However, if less water is frozen in the deposit, VR is increased. The thickness of the hailstone-covering supercooled water skins has been related to tS andX0. In laboratory experiments with icing of gyrating spheroids, photography and fast-scanning infrared cameras did not reveal details of the water skin (such as thickness or turbulence-induced effects by impinging cloud droplets). Expected effects are estimated to be ,20%. This figure is supported by the combined effect of (air) turbulence, roughness, and gyration, which produced an overall correction factor of K51.29.

Experiment-based data on spheres, including limited icing studies, and results and implications from experiments with gyrating spheroids have been applied and extrapolated, thus creating a robust background for the calculations and conclusions.

b. Specific results

1) The four-variable equations are solved and all results for all parameter combinations are given; this also includes the breaking down of X0 intoWf and D. The separation of Y into ENC and If is possible but premature because of the lack of relevant experimental data.

2) Calculation of radial growth speed gave an amazing result: The surface temperature of a spherical hailstone that grows while shedding from a permanent water skins is warmer than 25°C (CASE 3) (in confirmation of previous icing experiments).

3) The surface temperature of fast-shedding spherical hailstones without water skins is deduced to be warmer than 20.55°C (CASE 2).

4) Most hail growth conditions in a thunderstorm are associated with spongy growth.

5) CASES 1 and 2 are relatively simple. In CASE 1 the surface is dry, but its (surface) temperature varies. In CASE 2 tS ' 0°C is assumed, while the hailstone's substrate is spongy with varying If . CASE 3 has both a varying surface temperature, tS # 0°C and a spongy substrate. Further, CASES 2 and 3 are very similar because their surface temperatures are arranged in bands of different width, both bordering 0°C # tS # 25°C. The biggest DtS of CASE 3 is ;10 3 the bandwidth of CASE 2.

6) Dry growth at tS = 0°C produces deposits with ice density (minus air bubbles caused by originally dissolved air); lower densities require tS,0°C. Large hailstones can only grow dry (with ice density) at high altitudes with lowWf. Such conditions, however, are not conducive to the growth of large hailstones. Further, if formed at high levels, ''low density'' hailstones would have to descend through fewer cold-air layers with tA still below 0°C and where dry growth is replaced by spongy growth. Then lowdensity particles would be compacted by accreted water, even at tA , 0°C.

7) The three types of experimentally determined effects by air turbulence, roughness, and gyration can be safely built into the HMT equations with one single multiplicative factor, K = 1.29.

8) CASE 2 is built up by bands with surface temperatures 08 = tS = 20.55°C over a background of Y that decreases at constant tA with increasing X0. CASE 3, the most complex CASE, is similar but allows a larger supercooling of the water skins' surface, with bands bracketed by 08 = tS = 25°C.

9) The gradients across the water skin are up to 40°C mm-1, a value that is reduced for smaller hailstone sizes, smallerWf, and smallermass fractions of ice in the deposit If . Water skins may have thicknesses of ;0.1 mm.

c. Errors

The errors in using an average CD = 0.75 in the calculation of the growth velocity VR and the heat and mass transfer components of growing hailstones are within 65%. However, this error is not random; it has a negative sign for smaller hailstones (#25%) because the fall velocities are smaller, whereas a positive sign applies for larger hailstones. For radial growth speed calculations in all figures of this Part II, the correct CD has been applied (1.0 for 0.5 cm stones and 0.5 for Di = 2 cm). The assumed factor kr = 0.07 (roughness element height 7% of diameter) is reflected in CD = 0.5 for D = 2. (The drag increases to CD = 0.67 for kr = 0.15.) Hailstone density errors are ,5%. There is an error expected in the calculation of the effective dendritic growth speed because accreted droplets penetrate the water skins and make it more turbulent. No effect of impinging drops has ever been observed in the laboratory, but their influence needs to be quantified in future experiments-they are far from being easy! Considering the magnitude of K, however, it is not expected that this skin turbulence effect exceeds 20%.

d. Main implication

The importance of shedding, quantized by ENC, and If of spongy deposits are clearly established as main factors controlling hailstone growth. These two variables are key to rapid growth of deposits of low ice fraction, because fast partial freezing requires less latent heat to be discarded. They are as important as Wf and D (or Re).

The surface temperatures tS and tA control the HMT of hailstones with X0 and Y through the two new variables F and C. In CASE 1 any two out of X0, tA, and tS will determine the value of the third.

For free-fall D can be isolated with a known CD, thus circumventing the problems associated with the application of Re.

Shedding, enhanced by spongy ice growth, can initiate and maintain a warm rain process parallel to hail formation [proposed by Joe et al. (1980) and Joe (1982)]. Efficient rain in midlatitudes is normally formed in convective clouds that contain the ice phase, that is, the embryos of hailstones, small hail or hailstones per se (=5mm).According to the field experience of the author in Malaysia and Indonesia, the convective clouds are the major ''vehicles'' for hail-coupled warm rain.

e. Outlook

1) The equations and data provided can serve as input into entirely new dynamic cloud models. Thereby, it is imperative that turbulence is not characterized as a mathematical continuum of singularities, evenly distributed over the whole updraft. Updrafts are rather consisting of vortices, two to four stacked on top of each other (a scenario supported by the radar-observed packaging of hailstones). They need to be treated by Navier-Stokes equations. Inspiration and guidance may be found in the rich literature on wakes. Extensive future vertical Doppler radar frequency soundings, as taken by Thomson (1997) and Thomson and List (1999) and refined by Thomson (List 2004), are necessary to obtain more sizespectrum information on hail evolution. The pulsing observed is similar to the one described for tropical rain by McFarquhar and List (1991). New dynamic cloud models, with packaging of hail and spectra evolution, need to be designed that also include shedding. A parallel evolution of warm rain [based on 100- and 50-kPa collision-coalescence-breakup experiments by List et al. (2009a,b)] may provide the link between hail and rain.

The quasi-steady state, conveyor belt-rigged giant hailstorms of the high plains of Colorado are different from the above-mentioned scenario.

2) The new HMT for spherical ice particles is perfectly suited to serve as the basis for an expansion to spheroidal and ellipsoidal hailstones.

3) There is a need for consolidation of the HMT and growth of ice or ice/water particles with sizes of 0.1- 1.0 cm. This needs to address the biggest questions and errors that occur in that size range. It would also shine more light onto the transition from ice crystals to graupel to small hail to hailstones, or drops to hailstones. Free-fall behavior, growth, and HMT of such embryo particles have been treated over the whole size range by List and Schemenauer (1971), Schemenauer and List (1978), Cober and List (1993), and Youk et al. (2006). There is also a vast literature on such particles. One key issue will involve the replacement of the unsatisfactory definitions of the ice particles in the Glossary of Meteorology (Glickman 2000).

4) Another major task is the improvement of the knowledge base of spinning tops, as described by Klein and Sommerfeld (1965). Many questions, such as the onset of gyration and dependence of spin and nutation/ precession frequencies on size and shape, are still open. Does gyration stop, as indicated in the latest growth stage of the giant hailstones (Knight and Knight 2005)? Most important, however, are experiments in icing wind tunnels, confirming that the special gyration proposed in Part I can produce spherical hailstones. Further, net collection efficiencies, coupled with ice contents of deposits, need to be measured, together with spectra of shed drops and water-skin properties.

5) The pioneering field study by Rasmussen and Heymsfield (1987a,b,c) stressed the role of water drops shed from hailstones acting as hail embryos when frozen. That work should be continued because the step to rain needs to be further substantiated in field experiments and refined by models.

Acknowledgments. The groundwork for this paper was laid by the author over a period of more than 60 years with the help of many graduate students, summer students, and associates. Their invaluable contributions are greatly appreciated. Both host institutions, the Swiss Federal Institute for Snow and Avalanche Research in Davos (1952-63) and the University of Toronto (1963-2012), provided substantial support, as did many other agencies: the Swiss Federal Department of Interior (Forestry), the Swiss National Fund, the Swiss Hail Insurance Company, the National Research Council of Canada, the National Science and Engineering Research Council of Canada, and last but not least the Canadian Meteorological Service. The criticism and suggestions by the reviewers considerably improved the manuscript. They were greatly appreciated. The author is also grateful to Raul Cunha for his very valuable assistance and to the Department of Physics, University of Toronto, which has provided a pleasant research environment. Over the past decade, these studies have been fully supported by the Roland List Foundation.

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ROLAND LIST

Department of Physics, University of Toronto, Toronto, Canada

(Manuscript received 12 June 2012, in final form 18 February 2014)

Corresponding author address: Prof. Roland List, Department of Physics, University of Toronto, 60 St. George Street, Toronto ON M5S 1A7, Canada.

E-mail: [email protected]

APPENDIX

Approximations

The following approximations have been used:

Thermal conductivity of air (Wm-1 °C-1):

k = 0.023 82 1 (7.118 3 1025) tA;

Dynamic viscosity of air (kgm-1s21):

...

Kinematic viscosity (m2 s21):

? = rH m-1, with rA (kgm23) as air density;

Diffusivity of water vapor in air (m2 s21):

...

Saturation pressure over water (Pa):

esw(t) = 10b, where b = 25.547 - 4.9283log10(273.15 1 t) 2 2937.4 3 (273.15 1 t)21;

Saturation pressure over ice (Pa):

esi(t) = 10b, where b = 12.55 - 2667 3 [2667 3 (273.15 1 t)21];

Latent heat of vaporization of water (J kg21):

Ly(t) = 2.5008 3 106 [273.15 3 (273.15 1 t)21]b, with b = 0.167 1 3.67 3 1024 3 (273.15 1 t);

Latent heat of sublimation of ice (J kg21):

Ls(t) = 2.8345 3 106 2 190 t;

Specific heat of water (J kg21 °C-1):

cw(t) = 4217.8 1 0.3471 t2;

The parameterizations of k, m, Dwa, Ly, Ls, and cw are found in Pruppacher and Klett (1978), whereas esw and esi are from Iribarne and Godson (1981). The value for Ls has been approximated by the author.

It is recommended that two master files are set up, one for hailstones with wet surfaces and the other for dry (ice) surfaces. As a check, it is desirable to have two quantities monitored: (i) the sum of all four heat transfer components (5 0) and (ii) the equality of the auxiliary functions F and C, as calculated directly with tA and tS or through (6).

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