We begin with a parcel model that uses the Community Aerosol–Radiation–Microphysics for Atmospheres (CARMA) code to resolve particle size distributions (PSDs) without any assumptions regarding PSD shape for three particle classes: unactivated aerosol, water drops, and pristine ice particles. The prognostic variables in the model are potential temperature, water vapor mixing ratio, and the number concentration of particles within a uniform number of size bins for each of the three particle classes. For hydrometeors the mass concentration of aerosol within each size bin is also prognostic. The size bins correspond to a geometric progression of total particle mass and the number of bins in the model is flexible. The mass bins for ice particles are matched to those for water drops, and thus changes to the assumed mass-dimensional relation, which do not change during a simulation, result in changes to the corresponding ice particle densities and sizes. All simulations include adiabatic expansion, droplet activation, diffusional growth of hydrometeors, and homogeneous freezing of activated water drops.

The vertical profile of updraft speed w is specified as a function of height z above the surface, and the height of the parcel is incremented by Δz=wΔt each time step of duration Δt. Parcel expansion is treated by assuming dry adiabatic ascent and iterating 3 times on parcel air pressure, temperature, and density assuming hydrostatic conditions and using the ideal gas law. All prognostic particle concentrations are then rescaled by the new air density. Latent heat released by water phase change is applied to the air temperature of the parcel using the time step for the process involved (described in next section).

Uptake of water by unactivated aerosol is neglected and activation of aerosol particles to water droplets is computed following . Diffusional growth of hydrometeors from water vapor is treated with the piecewise polynomial method of where the Courant–Friedrichs–Lewy condition is met on the mass grid, and first-order upwind advection elsewhere. Growth rates for water drops are computed from Eq. (3) of for water drops, and for ice particles the capacitance method described in Sect. 13.3 of is used assuming spheroids. The accommodation coefficient for diffusional growth is assumed to be unity. Radiative heating effects on activation and growth rates are neglected. Homogeneous freezing of water drops is computed following .

For parcel simulations that omit gravitational collection we include only one ice class, representing vapor-grown ice particles. For the baseline model configuration, we treat the ice as low-density spheres in which the density is computed from the mass-dimensional relation that found for aggregates of unrimed radiating assemblages of plates, side planes, bullets, and columns. Like , we apply this mass-dimensional relation to ice particles smaller than the 1 mm lower limit of , but unlike we use the diameter of a sphere with equivalent cross sectional area as done by and as used in the analysis of Airbus nephelometer data. For area-equivalent diameters Deq less than about 100 µm this relation implies an ice particle density exceeding that of bulk ice; for those sizes we assume spherical ice particles with a fixed density of 0.92 gcm-3 . We simplistically refer to these vapor-grown ice particles as “fluffy” ice.

Here, we use 150 size bins corresponding to spherical diameter for aerosol particles ranging from 10 nm to 14 µm and for water drops from 2 µm to 6.5 mm.

The aerosol are treated as ammonium bisulfate with properties as given by . For the initial aerosol size distribution (see Table ), we use the trimodal lognormal fit to TWP-ICE measurements at 500 m altitude from .

All processes are computed on the master time step of Δt=1 s with the exception of droplet activation, droplet homogeneous freezing, and hydrometeor diffusional growth, which are solved on 0.1 s sub-steps. Parameters that depend on pressure and temperature are updated every 15 s. Such parameters include particle terminal fall speeds, the gravitational collection kernel, and coefficients for droplet activation and hydrometeor diffusional growth.

The parcel simulations start at 500 m altitude with initial conditions from the 21:00 UTC TWP-ICE sounding on 23 January 2006 with air pressure at 944 hPa, temperature 296.8 K, and relative humidity 98 %.

For the assumed vertical profile of updraft speed, seen in upper left panel of Fig. , we use a coarse spline fit to the median profile of maximum retrieved updraft speeds during a period of about 4.5 h as a large monsoonal MCS passed over Darwin during TWP-ICE. For the baseline profile the updraft speed rapidly increases to about 8 ms-1 at 4 km, increases at a reduced rate to about 11 ms-1 just above 10 km, and falls off at an intermediate rate above, reaching 6 ms-1 at 15 km, which lies above the highest levels we consider here; for the spline fit we use updraft speeds of 0, 7, 11, and 0 ms-1 at respective altitudes of 0, 3, 11, and 18 km. This profile resembles the mean profile of maximum updraft speeds in oceanic deep convection from the airborne Doppler retrievals of , with values in the median profile here generally about 2 ms-1 less than in that mean profile. For many scenarios we consider two alternative updraft profiles, uniformly increasing and decreasing baseline updraft speeds by 50 %, as seen in Fig. . The time for a parcel to ascend from cloud base to -40 ∘C is just over 25 min for the baseline updraft.

First we consider only adiabatic expansion, droplet activation, diffusional growth, and homogeneous freezing of water drops. As seen in Fig. , stronger updrafts activate more aerosol by driving a greater supersaturations near cloud base (not shown), which drives greater number concentrations of ice upon homogeneous freezing of the water drops between about -35 and -36 ∘C. Since the liquid water content (LWC) that freezes does not vary with updraft strength, the freezing of more numerous droplets in the faster updrafts simply produces smaller ice particles. It is seen in the comparison of ice particle mass distributions that homogeneous freezing produces substantially smaller ice particles and greater IWC than in the Airbus measurements. For the baseline and strong updraft, the PSDs overlap with the hint of an upturn in the Airbus PSD envelope at the small end of the PSD measurements, suggesting that this smaller mode, if real and not a shattering artifact, could correspond to homogeneously frozen water drops, which would be expected to be more favored in stronger updrafts. As noted above, there is observational evidence for homogeneous freezing in strong updrafts e.g.,. In these simplified simulations, the size of homogeneously frozen drops best matches observations for the strong updraft.

Discontinuities are seen in the profiles of droplet number concentration in Fig.  and in later profiles. Such discontinuities result from discretization of the aerosol size distribution and treatment of activation of droplets from aerosol by size bin as a nearly instantaneous process. Solutions to reduce such an artifact could be devised, but we lack evidence that this artifact materially affects results or conclusions here.

Our treatment of heterogeneous ice freezing nuclei (IFN) considers freezing of activated water drops in the immersion and condensation modes using the approach described by . IFN activation follows the temperature-dependent fit provided in Fig. 2 of , without extrapolating beyond their sampled temperature range of -9 to -35 ∘C. The IFN are treated prognostically, assumed equally distributed among unactivated aerosol and droplets, and consumed when activated. Adding such IFN has a negligible impact on our results even when the total number available is tenfold that of the fit. Only when the concentration is increased by a factor of 100 do the IFN affect the results, with a very weak second mode developing at larger sizes as seen in Fig. , with Deq∼200 µm. This factor of 100 corresponds to an IFN concentration of 1 stdcm-3 at -35 ∘C, which is about 5 times greater than the greatest measured value contributing to the fit reported by . At warmer temperatures the factor of 100 provides modest numbers of IFN, corresponding to about 0.04 stdcm-3 at -10 ∘C.

Given the mild response to even a hundredfold increase in IFN active in the immersion mode relative to measured IFN concentrations, we next consider a source of ice particles effective at much warmer temperatures: ice multiplication from production of ice splinters associated with riming of supercooled water . Lacking a source of graupel entering the parcel from above, we crudely represent the Hallett–Mossop process with a “pseudo” version in which ice particles are introduced in the smallest size bin between temperatures of -3 and -8 ∘C and the water drop PSD rescaled to conserve total moisture. The potential ice embryos are treated prognostically and consumed when activated. The temperature dependence of their availability follows the triangular form of Eqs. (16)–(71) of and peaks at -5 ∘C.

measured production of about 350 splinters per milligram of rime accreted by graupel at -5 ∘C see also. The parcel air density at -5 ∘C is about 0.6 kgm-3, where a concentration of 1 stdcm-3 is equivalent to about 0.5 cm-3. Thus, 1 stdcm-3 of splinters would require about 1.4 gm-3 of accreted rime, or roughly 20 % of the supercooled water available in the parcel at that temperature. On the basis of laboratory measurements , p. 358 also note that approximately one splinter is produced for every 100 to 250 water drops larger than 24 µm diameter accreted by graupel at -5 ∘C, and thus 0.5 cm-3 (∼1 stdcm-3) of splinters would require 50 to 125 cm-3 of such drops. At levels corresponding to the Hallett–Mossop temperature range, effectively all drops are larger than 24 µm in diameter, and the required drop number concentrations (Nd) exceed Nd at that level for the slow updraft, and bracket Nd for the baseline and strong updrafts. However, our crude representation of splinter production does not consume drops as real riming would, and a substantial sink of drops can readily drive supersaturations that activate new drops (as seen in a number of simulations below), which might provide sufficient numbers of additional drops if riming were represented more physically, as well as providing a supply of droplets smaller than 13 µm diameter that are also required for ice production from rime splintering p. 358.

As seen in Fig.  formation of copious ice at such warm temperatures provides ample time for diffusional growth, and the resulting second mode is far more substantial than that produced by IFN alone, and is comprised of somewhat larger particles closer to the observational target. At temperatures colder than homogeneous drop freezing, however, the mass and numbers of ice particles are still dominated by homogeneously frozen drops, and their MMDeq is unaffected.

The implication that the dominant mode of mass in the Airbus size distributions results from ice particles that form at relatively warm temperatures and are largely vapor-grown is consistent with the common appearance of capped columns in the Airbus particle imagery presented in Fig. . Such a habit is consistent with the formation of splinters that grow as columns at temperature warmer than -10 ∘C where the Hallett–Mossop process is active, and at colder temperatures where plates are favored subsequent diffusional growth occurs through capping plates. Such polycrystalline features are consistent with “the response of a column growing in a platelike growth regime” described by in the context of cirrus crystals with well developed columnar forms developing polycrystalline plate or side-plane components while falling through plate growth regimes below. The Hallett–Mossop process also has been observationally associated with the generation of large number concentrations of pristine columnar crystals e.g.,. However, without a more careful analysis of abundance and contribution to total mass, the possibility of capped columns growing while falling in the vicinity of an updraft rather than rising through it e.g.,, or the dominance of some other process, cannot be ruled out.

Strengthening the updraft reduces the amount of time for diffusional growth, and the second mode is seen in Fig.  to be diminished relative to the baseline updraft profile. Weakening the updraft conversely provides more time for ice formed at warm temperatures to grow from vapor, with the second mode surpassing the modal Deq of the observational target. And while this mode develops a few grams per cubic meter of ice in the weak updraft at the expense of supercooled water, homogeneous freezing of drops still dominates the mass (seen in IWC profile) and numbers (seen in Ni profile) of ice particles, and thus their MMDeq is only modestly affected at cold temperatures aloft.

The second mode of the ice PSD has more than 3 times the total mass in the weak updraft relative to the baseline updraft, but the ascent time is only twice as long, and can directly account for only a factor of 2, though there is a positive feedback in which more vapor deposition leads to greater ice surface area and thus more deposition. Additionally, fewer water drops in the weaker updrafts (see Sect. ) provide less competition with the ice for vapor growth, since distributing the same LWC over a smaller number of water drops results in less total surface area available for condensation. Above 8 km altitude or so there is even less LWC to distribute, owing to a positive feedback in which greater depositional growth of ice results in less condensational growth of water drops, further diminishing competition from the drops. The resulting reduction in vapor competition with decreasing updraft strength is seen in Fig.  in terms of the fraction of hydrometeor diffusional growth occurring as vapor deposition on fluffy ice: fvap=max⁡(gf,0)/[max⁡(gl,0)+max⁡(gf,0)], where gf and gl are the respective vapor mass exchange rates with ice particles and water drops, in which we omit only the solute and curvature terms from the complete growth expressions used in the model (the g terms are much like Eqs. 1 and 2 of ). Thus, at an altitude of about 8 km there is effectively no competition for vapor from the water drops in the weak updraft, while in the baseline and strong updrafts condensation of water drops accounts for about half and three quarters of the diffusional vapor sink, respectively.

Hereafter, pseudo-Hallett–Mossop ice formation is included and heterogeneous IFN neglected by default, except for sensitivity to drop shattering considered further below.

Representing particle sedimentation in a parcel is problematic because in principle a parcel is zero-dimensional. However, it is implausible to ignore sedimentation during the duration of such deep ascent: the time for the parcel to climb from cloud base to -40 ∘C is just over 25 min for the baseline updraft and twice that for the weak updraft. Our approach is to treat the parcel as having a finite depth Δz on the order of 1 km and calculate sedimentation as an implicit loss rate for hydrometeor concentration ψ in each bin from dψ/dt=-ψvf/Δz, where vf is the terminal fall speed for the hydrometeors in the bin. Baseline terminal fall speeds for hydrometeors are computed following with a modification to the drag coefficient as described by .

For the parcel depth Δz, we consider the analysis of updraft thermals in CRM simulations of cumulus congestus driven by a surface heat source 80 km across. They report parcel sizes of about 1–2 km and derive a characteristic scale of about 2 km, which they note is equivalent to the boundary-layer depth in their setup. Shallower boundary layers associated with maritime deep convection might be expected to result in smaller characteristic parcel sizes, so we consider values of 0.5, 1, and 2 km here. Note that the parcel concept is a convenient idealization and, as such, its dimension is not particularly well posed and does not obviously imply a representative updraft width that is comparable.

Terminal fall speeds for pristine ice particles are seen in Fig.  to be only about 0.1–0.6 ms-1 for particles across the mass mode Deq of about 100–500 µm in the Airbus measurements, more than an order of magnitude smaller than updraft speeds in the baseline profile. (The discontinuity in slope near Deq=100 µm corresponds to the transition from spheres with bulk density of ice to the mass-dimensional relation described earlier.) Thus, the effect of sedimentation is seen in Fig.  to be modest for such particles, and at -40 ∘C the IWC is reduced from 5.9 gm-3 in a simulation without sedimentation to 5.7, 5.5, and 5.1 gm-3 for Δz=0.5, 1, and 2 km, respectively, and no change in MMDeq (Δz sensitivity not shown). After considering additional microphysical processes we will return to the sensitivity of our results to the Δz assumed, but in the meantime fix its value at 1 km.

Weaker updrafts provide greater ascent time, which favors the source of condensate from diffusional growth as seen already, but also favors the removal of condensate by sedimentation. Thus, comparing Figs.  and , the greatest impact on IWC is seen for the weak updraft: at -40 ∘C sedimentation reduces IWC by about 25 % relative to the corresponding case without sedimentation. For the strong updraft, sedimentation reduces IWC by less than 2 % relative to the corresponding case without sedimentation.

Hereafter, sedimentation is included in all simulations.

Hydrometeor growth from particle collisions is treated with the semi-implicit method of on the master time step of 1 s. The collection efficiency is given by the product of collision and coalescence efficiencies per Eq. (6) of ; for collisions with ice particles instead of “coalescence efficiency” we refer to “sticking efficiency”. The gravitational collision kernel is computed for hydrometeors following . The formulation of is used for riming of columns or when the maximum dimension of the larger particle is at least 10 times greater than that of the smaller particle and the terminal fall speed Reynolds number of the smaller particle is less than unity. The coalescence efficiencies of for self-collection of water drops are combined with those of for accretion, with a lower limit of 0.6 in the size range between accretion and self-collection per and only using their expression for self-collection when the smaller particle diameter is greater than 200 µm. These coalescence efficiencies are blended with those of following . Collision-induced breakup of water drops is based on , incorporating corrections from and , and as an explicit scheme is sub-stepped with a 0.05 s time step for stability. The sticking efficiency of collisions between water drops and ice particles is assumed to be unity, and drop freezing is treated as instantaneous. Collisions between ice particles are expected to be inefficient at the temperatures for which ice is present in these parcel simulations, and are neglected by default for our parcel simulations, but considered in sensitivity tests discussed in Sect. .

For simplicity, raindrops that freeze with equivalent spherical diameter larger than 200 µm are treated as ice spheres with the same mass but with the bulk density of ice (0.92 gcm-3), referred to hereafter as dense ice to distinguish it from the fluffier ice properties that are consistent with agreement of Airbus PSD and IWC measurements, as discussed in Sect. 3 of Part 1. Raindrop freezing can occur either through IFN activation within a water drop or through coagulation of a drop with an ice crystal of lesser mass. Sufficiently heavy riming of a fluffy ice crystal is also assumed to contribute a fraction of mass to dense ice following the approach of . Below a cutoff size of 200 µm in maximum dimension, fluffy ice particles are assumed unable to collect water drops, following the treatment of collisions between plates and water drops by . Various studies support the existence of such a threshold size that is habit-dependent, generally increasing with crystal branching, from about 50 µm for capped columns to about 800 µm for dendrites and references therein. In the absence of a size cutoff, low collision efficiencies on the order of 10 % are otherwise computed, which could be reconcilable with threshold interpretations ; we remove the cutoff as a sensitivity test that is discussed below.

Using the described treatment for gravitational collection and raindrop breakup leads to substantial loss of condensate mass and substantially larger ice particles, as seen in Fig. . Collision–coalescence dramatically reduces water drop concentrations below the melting level. There is a slight recovery from the initial dip in drop concentrations from the activation of aerosol around 3 km altitude, but by 4 km altitude the aerosol reservoir is depleted and droplet concentrations continue their decline. At about the same level sedimentation by larger raindrops is already desiccating the parcel (relative to the case without gravitational collection), at temperatures warmer than the Hallett–Mossop range. Subsequent ice production glaciates the parcel with Δz=1 km by about -8 ∘C via collisions between ice splinters and raindrops. (We define glaciation throughout as the level at which LWC < 0.1 gm-3.) While the number concentrations of fluffy ice particles are about the same with and without gravitational collection, their mass increases from diffusional growth much more appreciably in the presence of gravitational collection. This increased diffusional growth results from depletion of aerosols, droplets, and ultimately droplet surface area that otherwise (in the absence of gravitational collection) compete for vapor and limit the saturation ratio to little more than unity. The reduced competition for vapor is evident in the profile of fvap, extended from Eq. () to also include competition from dense ice (in the denominator). In the absence of gravitational collection, the competition for vapor from condensation on water drops corresponds to greatly reduced values of fvap below about 10 km altitude. It is the lack of such competition that results in much greater fluffy IWC at altitudes below the homogeneous freezing level for parcels with gravitational acceleration. The lack of such competition is also seen in the fluffy ice PSDs aloft to result in a mode of vapor-grown ice that is substantially larger in terms of modal Deq as well as total mass (corresponding to the area under the curve in this PSD plotting convention). The lack of such vapor competition with supercooled water drops also corresponds to the lack of a smaller mode in the fluffy ice PSD aloft, which is attributable to homogeneous drop freezing in the absence of gravitational collection.

Gravitational collection is seen to result in substantial supersaturations at levels above which the aerosol and then droplets are depleted and below which there is sufficient surface area of ice, corresponding to altitudes just below the melting level to about 2 km above it. Notable supersaturations above the melting level are also realized in CRM simulations of deep convection, as discussed by and in references therein. As discussed below, entrainment of aerosol serves to reduce these supersaturations substantially. We note that breakup of raindrops contributes to the large supersaturations, as when that process is omitted, sedimentation depletes LWC faster, fewer raindrops collect smaller water drops, and more smaller water drops increase competition for water vapor, thereby reducing supersaturations (not shown).

The results are only mildly sensitive to an increase or decrease by a factor of 2 in the parcel depth Δz used for sedimentation. Fluffy IWC at -40 ∘C is seen to change by less than 30 % in response to changing Δz by a factor of 2 either way, and MMDeq at that level barely responds to such variations in Δz. Perhaps the greatest impact seen above the melting level induced by an increase in Δz, corresponding to less efficient sedimentation, is an increase in dense IWC, the primary source of which is freezing of rain. Thus, less efficient sedimentation of rain yields greater dense IWC. Note that this dense ice, with a fall speed of about 5 ms-1 at a particle diameter of 1 mm as seen in Fig. , would not be expected to persist in anvil outflow, unlike the more slowly falling fluffy ice.

A so-called sedimentation efficiency, the inverse e-folding depth of fluffy IWC from sedimentation: Esed=P/(wΔz IWCf), where P is sedimentation flux of fluffy ice and IWCf the fluffy IWC, is seen to respectively increase and decrease by about a factor of 2 in response to decreasing and increasing Δz at mid-levels, as expected from its formulation. The tendency of Esed to increase with height results not only from the increasing size of the fluffy ice particles but also from an increase in terminal fall speeds with decreasing air density. Computing instead a combined sedimentation efficiency for all hydrometeors (not shown), by vertically averaging liquid-equivalent sedimentation fluxes P and (wΔz[LWC+IWC]) between cloud base and -40 ∘C, the resulting averages for Δz of 0.5, 1, and 2 km are approximately 0.25, 0.17, and 0.11 km-1 for the baseline updraft strength. The magnitude of Esed would be a challenge to constrain from observations; we simply note that the vertical averages here for all hydrometeors are comparable to the constant value of 0.15 km-1 used by . Their approach considered a convective plume with simplified cloud and precipitation microphysics, and their Esed value was empirically determined to produce reasonable results in the context of their other model assumptions. We assume Δz=1 km for sedimentation hereafter.

Scaling up the pseudo-Hallett–Mossop source strength affects little other than the number concentration, size, and fluffy IWC. As seen in Fig. , a greater number of ice splinters results in smaller ice particles aloft, which would be expected if a fixed IWC were simply distributed over a greater number of ice particles. However, smaller ice particles fall more slowly and thus there is a modest sedimentation feedback in that somewhat more IWC is lofted when their numbers are so increased.

Note that if 350 ice splinters are produced from 1 mg of accreted rime, as discussed above, a pseudo-Hallett–Mossop source of 2 stdcm-3 would require riming nearly 3 gm-3 of supercooled water, equivalent to more than half the LWC at -5 ∘C in the baseline updraft. A source of 3 stdcm-3 would require riming all the supercooled water in the baseline updraft, which can be considered an upper limit on the pseudo-Hallett–Mossop source.

Given the overlap with the measured PSD mode at Deq≈300 µm for the simulation with a pseudo-Hallett–Mossop source of 2 stdcm-3, we next vary the updraft strength using that pseudo-Hallett–Mossop source strength. As seen in Fig. , the baseline updraft best matches the observed ice PSD aloft.

In the case of the strong updraft, the parcel reaches the melting level before there is enough time for warm rain to deplete the parcel of liquid water. Glaciation is delayed until about -21 ∘C, at which point frozen raindrops predominantly produce dense ice. Competition for vapor with water drops and then the dense ice, as evident in the reduced values of fvap, reduces the diffusional growth of fluffy ice relative to the baseline and weak updrafts, resulting in substantially less fluffy IWC aloft and smaller MMDeq for the strong updraft.

In Sect.  it was noted that the greater time of ascent in weaker updrafts provided for greater diffusional growth of fluffy ice particles. In the absence of gravitational collection, parcels were saturated with respect to liquid water and, not coincidentally, supercooled water was present all the way to the homogeneous freezing level, as seen in Figs.  and . In that limiting case, a longer ascent time provides a longer exposure to water saturation, and thus ascent time is a leading factor determining diffusional growth of ice. When gravitational collection is included, however, the limiting behavior of all parcels maintaining water saturation up to the homogeneous freezing level is no longer realized. Instead the updraft-strength dependence of fluffy ice diffusional growth relates primarily to competition for vapor. When gravitational collection is included, perhaps the most direct effect of greater ascent time in weaker updrafts is that it provides more time for the warm rain process to deplete LWC.

The fluffy ice PSD aloft is seen to be prominently bimodal for the weak updraft. Earlier times in the evolution of hydrometeor PSDs (not shown) reveal that the larger mode forms on early Hallett–Mossop splinters, which form during supersaturated conditions in the weak updraft because collision–coalescence has time to sweep out all cloud droplets. The early splinters thus grow rapidly enough from vapor for the latent heat release to warm the parcel, delaying temperature-dependent production of further splinters until the parcel cools enough from adiabatic expansion. The smaller ice particle mode forms on those later splinters. While this production of a bimodal fluffy ice PSD makes sense in terms of the model physics, the treatment of Hallett–Mossop ice production in the parcel model is rather contrived to overcome a lack of riming graupel particles entering the parcel from above. In nature or in a more realistic modeling framework there may well be other feedbacks that render such a bimodal feature an artifact of our parcel framework.

Although the focus of this study is the fluffy ice expected to persist in convective outflow, we note that the IWC of dense ice is seen in Fig.  to increase with updraft strength, from sedimentation of raindrops first and then of the dense ice that results from raindrops freezing.

As mentioned earlier, by default we assume that water drops are not collected by fluffy ice with a maximum dimension less than 200 µm. If we relax that assumption and instead use our computed collision efficiencies, the only notable change for the baseline updraft (not shown) is the parcel glaciates at about -5 ∘C instead of about -8 ∘C when we impose the size cutoff. The warmer glaciation corresponds to dense ice appearing about 1 km lower than in the baseline case, but fall speeds for raindrops and dense ice are comparable and there is little difference in results colder than about -5 ∘C.

With gravitational collection included, if we allow collisions between fluffy ice particles with a sticking efficiency of 0.015, the results (not shown) are indistinguishable from those in which such collisions are ignored. The value of 0.015 represents rather aggressive ice aggregation in 1-D column simulations described in Sect. 5 of Part 1 and in Sect.  below. While such a sticking efficiency results in substantial ice aggregation in the column simulations, the timescale over which collisions modify the ice PSD is about 2 orders of magnitude smaller in an updraft parcel framework; for stratiform precipitation the relevant speed for the transit timescale is set by ice particle terminal fall speeds on the order of 10 cms-1 while in the parcel it is set by the updraft speed on the order of 10 ms-1.

If we instead follow and use a temperature-dependent sticking efficiency of exp⁡(0.09TC) where TC is air temperature in degrees Celsius, the resulting fluffy ice particles do become larger in the baseline updraft, with MMDeq of 380 instead of 310 µm at -40 ∘C, but the fluffy IWC is unchanged (not shown). However, our 1-D column simulations suggest that such a sticking efficiency formulation is implausibly aggressive, as discussed below in Sect. . Although there are reasons to expect aggregation to be more efficient at warmer temperatures, the observational basis for such an exponential dependence is unclear and current literature offers no alternative forms .

Given a null result in permitting collisions between fluffy ice particles with a plausible sticking efficiency in terms of our modeling frameworks, we neglect such collisions hereafter.

We next consider the ice-multiplication process of some drops shattering during freezing. We follow and allow ice splinters to form when a water drop larger than 50 µm in diameter collides with a smaller ice particle (with maximum dimension no greater than half the drop diameter) limited to temperatures between -5 and -15 ∘C. We allow Nsh splinters to form for 25 % of such collisions, resulting in a net multiplication factor of fsh=0.25Nsh (see and references therein for further description; note that for simplicity we adjust fsh so that the splinters formed fit evenly into one bin, which amounts to a miniscule adjustment for a grid with 150 bins). So as not to combine ice-multiplication processes but instead evaluate them separately, for the simulations with drop shattering we omit the pseudo-Hallett–Mossop source and instead revert to immersion freezing IFN as described in Sect. , at a concentration of 1 std cm-3, or 100 times that of the parameterization.

With a net multiplication factor of fsh=5 the freezing-induced drop shattering is seen in Fig.  to produce substantially more fluffy IWC than for simulations lacking it and a number of other processes, as seen in Fig. . However, the number concentration of fluffy ice particles is quite small (off scale in the figure) and the fluffy MMDeq aloft is nearly 1 mm, substantially larger than the Airbus target. Increasing that net multiplication factor tenfold is seen to produce results closer to those with a pseudo-Hallett–Mossop source of 2 std cm-3 but without drop shattering. The fluffy ice PSD aloft is seen to be monomodal with fsh=5 but the development of a second mode is suggested by a shoulder for fsh=50. In the latter the mode of smaller particles corresponds to shattering of water drops grown from diffusional growth and the mode of larger particles to shattering of raindrops grown from collision–coalescence, and in subsequent diffusional growth the small ice particles do not catch up with the larger ones despite their faster relative growth. In contrast there is greater diffusional growth of ice particles with fsh=5 and fewer ice particles, and because of less ice surface area and greater supersaturations, the smaller do catch up with the larger particles, producing a single size mode by -40 ∘C.

A net multiplication factor of fsh=50 is substantially greater than fsh<2, which is the maximum value supported by published laboratory studies cf.. However, recently combined microphysics measurements obtained within tropical cumulus clouds with a column model to derive an implied secondary ice particle production rate from drop shattering of 1000 mg-1 of freezing drops

Note that on p. 2442 of , the optimized fragmentation factor should be 109 kg-1 instead of 10-9 kg-1 as published (Paul Lawson, personal communication, 2015).

As discussed in Part 1, high IWC–low Ze conditions are evidently prevalent in mature convection with developed stratiform precipitation characteristic of the Airbus measurements. So far we have ignored any mixing with environmental air, which we address next. We treat entrainment as a source term for each prognostic variable ϕ: dϕdt=dzdtdϕdz=wϵ(ϕenv-ϕ), where w is updraft speed, ϵ entrainment rate in units of reciprocal distance, and ϕenv the environmental value of ϕ linearly interpolated to altitude z. The prognostic variables subject to entrainment are potential temperature, water vapor mixing ratio, and aerosol and hydrometeor concentrations in each size bin; heterogeneous IFN and potential pseudo-Hallett–Mossop embryos are not subject to entrainment.

Being difficult at best to constrain through observation, we base our entrainment rates on CRM studies of deep tropical convection. In a study of the transition from shallow to deep maritime convection, reported vertical mass fluxes to be dominated by plumes with effective ϵ≲ 0.1–0.2 km-1 in their deep cumulus regime. In a study of more intense, continental convection during a monsoon break period, reported the deepest ascent to be associated with ϵ≈0.2 km-1 in their control simulations, and tending toward larger values aloft, up to 0.5 km-1 in simulations with higher resolution. They suggest such strong entrainment rates might be reconcilable with those of by virtue of continental convection being more intense than maritime convection. Finally, consider convection driven by a surface heating source on the order of 100 km across and report a dominant ϵ≃0.5 km-1 for the cumulus congestus regime they simulated in 3-D, while noting that for the strongest updrafts ϵ is reduced, falling to 0.2 km-1. Focused on a deep cumulus regime for maritime conditions, we use a vertically uniform ϵ=0.1 km-1 as our baseline, but consider greater ϵ in sensitivity tests.

Here we use steady-state results from 1-D column simulations of stratiform precipitation as described in Sect. 5 of Part 1 as a mixing environment for parcels. In these column simulations the Airbus measurement target serves as upper boundary condition for fluffy ice and the profiles of temperature and water vapor mixing ratio are from TWP-ICE soundings during the passage of the MCS on 23 January 2006. Instead of using the column model solution from Sect. 5 of Part 1 with 50 size bins, we use 150 bins to match our parcel configuration, and as in Sect. 5 of Part 1 compare our results with S-band profiles of Ze and mean Doppler velocity (MDV) from TWP-ICE. (Note that there are no mixed-phase particles in the model so there is no possibility of a bright band.) A uniform sticking efficiency of 0.01 is seen in Fig.  to reasonably match Ze above the melting level. A uniform sticking efficiency of 0.015 is seen to better match observed Ze below the melting level and MDV throughout the column, but produces excessive Ze above. The temperature-dependent sticking efficiency from as adopted by , which yields a sticking efficiency already greater than 0.025 at -40 ∘C, is seen in to result in excessive Ze throughout the column and MDV below the melting level roughly 50 % greater than observed. Primarily concerned with entrainment of ice above the melting level, we use column model results derived with a sticking efficiency of 0.01 for the entrainment environment.

Weak entrainment is seen in Fig.  to reduce LWC below the melting level, and stronger entrainment reduces LWC even more, owing to LWC in the entrained stratiform precipitation being less than 1 gm-3, which is less than that in the parcel above about 1 km altitude. Parcel simulations that entrain the same environment but without hydrometeors also reduce LWC comparably (not shown). Glaciation starts a bit lower in the ascent of the parcel with stronger entrainment, since ice reaches the melting level in the environment. However, both entraining parcels are seen to loft supercooled water higher than the non-entraining parcel: the non-entraining parcel glaciates near -8 ∘C and the entraining parcels near -18 ∘C. This lofting of supercooled water, which also occurs in parcels that entrain the environment with no hydrometeors (not shown), results from the rapid activation of droplets on any entrained aerosol above the melting level, seen in the enhanced water drop concentrations, which quenches the supersaturation that otherwise develops below the Hallett–Mossop level. Thus, although the entraining parcels entrain ice from the environment, fluffy IWC in the non-entraining parcel is greater in the first few kilometers above the melting level because of greater diffusional growth of splinters caused by the much greater supersaturation at those levels. At colder temperatures fluffy IWC in the parcel with stronger entrainment surpasses that of the non-entraining parcel primarily from entrainment of environmental ice, seen in Fig.  to exceed 2 gm-3 above the melting level and to increase upward.

The -40 ∘C level occurs at a lower level than for the non-entraining parcel, by about 700 and 900 m for parcels entraining at 0.1 and 0.2 km-1, primarily attributable to entrained air being colder than the parcel. The fluffy ice PSDs for the entraining parcels are seen to be comprised of (1) a broad mode corresponding to the entrained ice, populated at larger sizes in the column through aggregation, and (2) a narrow mode from diffusional growth of pseudo-Hallett–Mossop splinters. The narrow mode of fluffy ice diminishes in total mass and modal Deq as ϵ is increased; increasing ϵ to 0.5 km-1 is enough to effectively eliminate the narrow mode (not shown). The decreases in both the mass and modal Deq of the narrow mode result from a reduction in diffusional growth of pseudo-Hallett–Mossop splinters from decreased supersaturation and associated greater vapor competition in the first few kilometers above the melting level for the entraining parcels. Ultimately it is therefore entrainment of environmental aerosol that leads to the smaller modal Deq of the narrow mode in the fluffy ice PSD aloft, a point we shall revisit below.

For weakly entraining parcels the modal Deq of the narrow mode of the fluffy ice mass PSD aloft is seen in Fig.  to decrease with increasing updraft strength, attributable to increased competition with water drops and then dense ice, as evident in the profile of fvap and discussed in Sect.  for non-entraining parcels. The broad mode of entrained ice is largely unresponsive to changes in updraft strength, explained to first approximation by the entrainment source being independent of updraft speed: dϕ/dz=ϵ(ϕenv-ϕ). That is, the degree to which a stronger updraft in this framework entrains more rapidly is canceled by the shorter time over which the entrainment occurs. The response of entraining parcels to changes in updraft strength is insensitive to modest changes in entrainment, insofar as the response is comparable for parcels with twice the baseline entrainment rate (not shown).

For weakly entraining parcels, the modal Deq of the narrow mode in the fluffy ice mass PSD aloft is seen in Fig.  to decrease with increasing numbers of pseudo-Hallett–Mossop splinters, attributable to increased competition for vapor among ice particles. The greater fluffy IWC aloft is consistent with slower sedimentation for the smaller ice particles.

So far we have treated ice as spheres with a mass-dimensional relation from for aggregates of unrimed radiating assemblages of plates, side planes, bullets, and columns (see Sect. ). A slight variation is to use the same mass-dimensional relation but instead of spheres treat the particles as oblate spheroids, done by numerically inverting the dependence of randomly projected cross sectional area on spheroid volume to determine the dependence of particle aspect ratio on Deq. The particle geometries are then used in the capacitance shape factors for diffusional growth as well as for terminal fall speeds. For a given ice volume an oblate spheroid falling with its axis of rotation parallel to the flow falls slower than a sphere, as seen in Fig. . This reduction has very little impact on parcel simulations that include gravitational collection (not shown).

However, the mass-dimensional relation used for spheres and oblate spheroids corresponds to rather large densities for vapor-grown ice particles. As an alternative we treat fluffy ice particles with a maximum dimension greater than 15 µm as hexagonal plates by adopting the area- and mass-dimensional relations of and treating them as spheres at smaller sizes. Terminal fall speeds are seen in Fig.  to be substantially reduced for Deq from tens to hundreds of micrometers. This reduction in fall speeds does impact our results, if subtly; although profiles are little affected (not shown) it is seen by comparing Figs.  and that there is some effect on the fluffy ice PSDs aloft. The changes are principally attributable to particle geometry alone, in which Deq of such a plate is somewhat greater than for a fluffy sphere in our baseline treatment.

A multiplicity of variations in aerosol populations are possible, including any of the parameters characterizing the initial multimodal size distribution in the parcel and assumed chemical composition, as well the vertical profile of aerosol entrained by the parcel. Here we merely decrease and increase initial aerosol number concentrations in all three modes by a factor of 25, in both the initial condition of the parcel and in the environment. These aerosol changes are seen in Fig.  to induce respective decreases and increases of less than a factor of 10 in cloud droplet number concentration (Nd) near cloud base. This less than linear response of Nd to changing aerosol number concentrations results from a negative feedback in which increasing Nd leads to increased total droplet surface area, which quenches peak supersaturations and thereby increases the size of the smallest aerosol activated near cloud base e.g.,. The effects of such changes in Nd on warm rain process are seen to be modest in entraining parcels. The parcel with access to the fewest aerosol particles is seen to become depleted in Nd well below the melting level, leading to greatly enhanced supersaturation and greater diffusional growth of ice splinters when they appear, which in turn results in a narrow mode of fluffy ice aloft that has more mass and a greater modal Deq.

Entrainment serves to buffer the response of the parcel to changes in aerosol number concentration, as non-entraining parcels respond in the same sense but with a greater magnitude of change to the fluffy ice PSD aloft, as seen in Fig. . In the parcel with access to the most aerosol particles, the so-inhibited warm rain process barely affects the LWC profile below the melting level, which is thus nearly adiabatic (cf. Fig. ). Fewer aerosol particles result in fewer droplets and less LWC, which both contribute to less competition for vapor by supercooled water, and thus fluffy IWC and the modal size of the dominant mode of fluffy ice PSD aloft increase with decreasing aerosol numbers. The bimodal PSD aloft for the non-entraining parcel with the fewest aerosol particles is similar to that discussed above for a non-entraining parcel in a weak updraft (see Sect. ), in which lack of droplet surface area leads to enough diffusional growth on the first splinters to warm the parcel and delay further production of splinters, which come to comprise the smaller of the two modes aloft.

Reducing initial relative humidity of the parcel from 98 to 80 % raises cloud base from about 600 to 1000 m (not shown). Since the updraft speed increases with altitude at those levels, such an increase in cloud base results in an increase in Nd. The Nd increase is far less substantial than that induced by increasing aerosol numbers by a factor of 25, and does not persist above the Hallett–Mossop zone, where entrained aerosol greatly influence Nd. Thus, the changes induced aloft are minimal in response to a rather substantial increase in cloud-base altitude within this modeling framework.