Continuous secondary ice production initiated by updrafts through the melting layer in mountainous regions

. An accurate prediction of the ice crystal number concentration in clouds is important to determine the radiation budget, the lifetime, and the precipitation formation of clouds. Secondary ice production is thought to be responsible for the observed discrepancies between the ice crystal number concentration and the ice nucleating particle concentration in clouds. The Hallett-Mossop process is active between -3 °C and -8 °C and has been implemented into several models while all other secondary ice processes are poorly constrained and lack a well-founded quantiﬁcation. During two hours of measurements 5 taken on a mountain slope just above the melting layer at temperatures warmer than -3 °C, a continuously high concentration of small plates identiﬁed as secondary ice was observed. The presence of drizzle drops suggests droplet fragmentation upon freezing as the responsible secondary ice mechanism. The constant supply of drizzle drops can be explained by a recirculation theory, suggesting that melted snowﬂakes, which sedimented through the melting layer, were reintroduced into the cloud as drizzle drops by orographically forced updrafts. Here we introduce a parametrization of droplet fragmentation at high 10 temperatures when primary ice nucleation is basically absent and the ﬁrst ice is initiated by collision of drizzle drops with aged ice crystals sedimenting from higher altitudes. Based on previous measurements, we estimate that a droplet of 200 µm in diameter produces 18 secondary ice crystals when it fragments upon freezing. The application of the parametrization to our measurements shows high uncertainties, but the estimated number of splinters produced per fragmenting droplet land for the remote sensing measurements. We like to thank Alexander Beck for his support in the organization of the RACLETS campaign. We thank the Swiss Federal Ofﬁce of Meteorology and Climatology MeteoSwiss for providing us with meteorological measurements and installing the ceilometer and the weather station in Klosters as well as the radar wind proﬁler in Wolfgang. We also like to thank the paragliding club Grischna for providing us with meteorological measurements from their Holfuy station at Gotschnagrat. We like to thank Hannes Griesche for supplying us with turbulence data, Patric Seifert (TROPOS, Germany) for discussing their measurements, Benjamin 5 Walter for providing as with data from the snow drift station at Gotschnagrat and we like to thank Pila Bossmann (wetterboss.com) for discussing the general weather situation. AL, FR, JW and UL acknowledge funding from the Swiss National Science Foundation (SNSF) grant number 200021_175824. CM acknowledges funding from the SNSF grant number 200021_169620.

From the six SIP mechanisms, the rime-splintering process is the best constrained mechanism and has been implemented in some cloud microphysics schemes (e.g., Scott and Hobbs, 1977;Beheng, 1987;Phillips et al., 2001). There have been fewer attempts to include collisional breakup (e.g., Yano and Phillips, 2011) and droplet fragmentation (e.g., Lawson et al., 2015). Sullivan et al. (2018) modeled all three ice multiplication processes and showed that none of the three SIP mechanisms dominates the ICNC enhancement. While collisional breakup is important when low updrafts prevail and high nucleation rates 5 are present, droplet fragmentation and the rime-splintering process can also be important in INP limited regions. However, they pointed out that the processes are not very well constrained by laboratory and in situ data.

Persistent SIP immediately above the melting layer
A remarkable finding of the study by Korolev et al. (2020) is the observation of small pristine ice crystals persisting immediately above the melting layer in clouds. They calculated a spatial correlation time τ corr during which the environmental changes (e.g., 10 air temperature, humidity and cloud particle number concentration) are insignificant and the shapes of the ice crystals can still be associated with the environment they are growing in. Based on their estimation, τ corr is on the order of 60-120 s. Assuming water vapor saturation over liquid, an ice column at temperatures below -3°C can reach a length between 50 and 150 µm during τ corr depending on its aspect ratio. A relative humidity close to water saturation is a valid assumption in MPCs (Korolev and Isaac, 2006). It can be concluded that if external processes like blowing snow or the seeder-feeder process can be excluded, the 15 discrepancy between the INPC and the concentration of columns shorter than 50 to 150 µm in MPCs must emerge from SIP. Korolev et al. (2020) suggested a recirculation process through the melting layer to be a possible explanation for the observed high concentration of small pristine ice crystals. Ice crystals, which fall through the melting layer as precipitation, melt into drizzle sized droplets. If convection or turbulence causes large updrafts, these droplets can be reintroduced into the cloud. Whenever these droplets freeze by the collision with ice crystals, the droplet fragmentation upon freezing process can become 20 active and produce high amounts of secondary ice particles.
2 Experimental setup

RACLETS campaign
The measurements used in this study were collected during the RACLETS campaign, which took place in February and March 2019 in the region of Davos in the Swiss Alps. The objective of this campaign was to improve the understanding of the influence 25 of topography and aerosols on the development of clouds. For this goal, a set of instruments was deployed at different locations. Fig. 1 shows the locations and instruments, which were used for the analysis of the presented case study.
Measurements of the in-cloud properties such as the cloud particle concentration as well as their size distribution and shape were taken from the HoloGondel platform (Beck et al., 2017) on the Gotschnabahn, which is described in more detail in section 2.2. As part of the MeteoSwiss observation network, a ceilometer (Vaisala, Model CL31) was installed in Klosters 30 (1200 m) to determine the cloud base height (Hervo, 2020a) as well as a wind profiler (Vaisala,Model Lap3000,Finland) in Wolfgang to measure the horizontal wind field (Hervo, 2020b). The general wind pattern on the ground was determined by data from different MeteoSwiss stations, the Snow and Avalanche Research SLF and the Holfuy station of the paragliding club Grischna. Temperature measurements were taken from the MeteoSwiss station in Klosters and the snow drift station installed at Gotschnagrat (Walter et al., 2020). For the analysis of the whole clouds, a vertically-pointing cloud radar (Model Mira-36, METEK GmbH, Germany; Görsdorf et al. (2015)) was installed in Wolfgang. INPCs were measured in the valley in  2014)) at Weissfluhjoch as described in Mignani et al. (2020). The detection limit of the INPC (lowest concentration 10 measurable) was calculated according to Vali (1971) considering the concentration obtained if only one (first) drop froze. A detection limit of 5.2·10 −4 L −1 and 4.9·10 −4 L −1 has been determined for Wolfgang and Weissfluhjoch, respectively.

HoloGondel
The HoloGondel platform consists of the HOLographic Imager for Microscopic Objects (HOLIMO 3G in Beck et al. (2017)), which records the concentration, size distribution and shapes of cloud particles, a temperature and relative humidity sensor 15 (HygroMet4, Rotronic) in a ventilated housing (RS24T, Rotronic) and a pressure sensor (Fig. 2). During the RACLETS campaign, the platform was installed on one of the gondolas running at the upper section between the Gotschnagrat mountain station (2280 m) and the middle station at Gotschnaboden (1790 m) covering a horizontal distance of about 830 m. To avoid any influence by the gondola stations on the measurements, data was only used if the difference of the pressure measured on the gondola and the pressure measured at the stations was more than 1.5 hPa, corresponding to a vertical distance of more than 20 15 m and a total distance of about 30 m between the gondola and the stations. The gondola runs with a maximum speed of about 6 m s −1 leading to a total measurement time of about 140 s per ride. To avoid influences from the gondola and its swing arm, only measurements of uphill rides were analysed, when the setup was in front of the gondola in the direction of travel.
HOLIMO 3G is an open-path instrument, which uses digital in-line holography. Holograms of the sample volume between the two towers (see Fig. 2b) are recorded of which 13.7 cm 3 were considered for the analysis, producing a 3D distribution of the 25 cloud particles between the two towers and a 2D image of each cloud particle. A more detailed description of the measurement principle can be found in Beck et al. (2017) and Henneberger et al. (2013). The pixel size is 3.1 µm, which allows us to observe cloud particles larger than 6.2 µm (Beck et al., 2017). The differentiation between ice and liquid is based on the shape of the particles (circular vs. non-circular). This is possible for particles larger than approximately 25 µm, depending on the shape of the ice particles (Henneberger et al., 2013).

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A neural network for particles larger than 25 µm (Touloupas et al., 2020) and a decision tree for particles smaller than 25 µm in their major axis size were used to separate cloud droplets from ice crystals and sort out artifacts, which are falsely identified as cloud particles by the software. All ice crystals were manually confirmed after the automated classification. Therefore, the uncertainty in the concentration of ice particles can be estimated with ± 5 % for ice crystals larger than about 100 µm and https://shop.swisstopo.admin.ch/de/products/height_models/dhm25200, last access: 9 March 2020.
± 15 % for ice crystals smaller than 100 µm (Beck, 2017). All ice particles were also manually classified into the habits plates, columns, irregular, aged ice (rimed particles and aggregates) and unidentified referring to ice crystals, which could not be classified because they were too small or because their orientation did not allow a decision on their habit. For the uncertainty of the different habits, the counting uncertainty ( √ N /V ; N : number of crystals, V : measurement volume) was added, because of their relatively low number compared to the measurement volume. The uncertainty of cloud droplets is estimated to be ± 6 % 5 as determined for the classification with the neural network in Touloupas et al. (2020). Again, for droplets larger than 40 µm the counting uncertainty was added due to their relatively small numbers. A trough over Russia reached relatively far to the south supporting the rise of air masses over Germany, Austria and eastern Switzerland, which were all covered by stratiform clouds. From the cloud radar measurements (Fig. 3), it can be anticipated 5 that the cloud top height was about 4 km in the Davos region referring to a cloud top temperature of around -8°C as measured by a radiosounde profile launched at 10:25 UTC from Wolfgang. Between 5 and 9 UTC, the cloud radar showed a midlevel cloud above the stratiform cloud, reaching from about 4.5 km to about 6.5 km (Fig. 3 a)) that might have acted as a seeder cloud in the early morning.
As can be inferred from the ceilometer data, light precipitation started at around 7 UTC in Klosters, where the valley station 10 of the Gotschnabahn is located (Fig. 3 b). The cloud base was located slightly above the height of the Gotschnaboden during the precipitation period, which lasted until about 10:30 UTC when the cloud began to dissipate. Measurements with HoloGondel were taken between 8 and 10 UTC. The rides used for the analysis are shown as grey bars in Fig. 3. During the measurement period, the temperature in Klosters increased from about 1.5°C to 3.5°C and the temperature at Gotschnagrat stayed relatively constant slightly below -2°C. The temperature at Gotschnaboden was derived from measurements on the gondola and is, 15 therefore, only available when the gondola was in operation and close to Gotschnaboden. During the measurement period it remained close to 0°C. The melting layer can be inferred from the dark band of the ceilometer data (Sassen et al., 2005) below Gotschnaboden ( Fig. 3 b)). A total of 9 complete measurement profiles were taken between 0°C and -2.7°C.
The main wind direction was from north to northeast as can be inferred from the wind profiler measurements taken in Wolfgang in Fig. 4 b). However, the ground measurements show that the wind direction was strongly influenced by the orography (Fig. 4 a)). The valley north of the Gotschnabahn forced the wind direction to northwest as can be seen from the wind mea-5 surements on Gotschnagrat. Thus, air masses measured on Gotschnabahn were pushed up the valley coming from northwest and must have been lifted through the melting layer before reaching the measurement site.
The slope inclination in the wind direction measured at Gotschnagrat is about 7°(averaged over Gotschnagrat and 1 km northwest). Assuming that the measured horizontal wind of about 5 m s −1 (Fig. 4) was blown up the slope without friction, the vertical wind speed reached about 0.6 m s −1 . The eddy dissipation rate, which is a measure for turbulence, was calculated from 10 the wind profiler and cloud radar measurements as described in Griesche et al. (2019) and reached values between 20 cm 2 s −3 and 90 cm 2 s −3 between 1800 m and 2300 m over Wolfgang between 8 and 10 UTC. Such values were reported in stratiform clouds (Borque et al., 2016) as well as in cumulus clouds with weak updrafts (Siebert et al., 2006). Since no measurements of turbulence are available over Klosters, we assume that similar turbulence is present at the measurement site.

15
Between 8 UTC and 10 UTC, a total volume of 43.7 L was measured by HOLIMO, distributed over nine uphill rides, each contributing between 2.6 L and 8.2 L. The difference in the measurement volumes is due to different onsets and offsets of the automated recording. HOLIMO observed aged ice crystals, here defined as aggregates and rimed particles, irregular particles, plates and very few small columns (see Fig. 5). About 70% of of all ice crystals were classified as unidentified because they were too small or the particle orientation made a decision on its habit inconclusive. As discussed in section 4.1, unidentified 20 particles as well as plates smaller than 93 µm are referred to as small plates. They are marked with dots in the histogram plot of the ice crystal size distribution in Fig. 5.
The concentrations of ice crystals and cloud droplets stayed relatively constant over the measurement period (see Fig. 6).
Taking the whole measurement period into account, a mean cloud droplet number concentration (CDNC) of 156 cm −3 ±9 cm −3 and a mean ICNC of 6.0 L −1 ±0.9 L −1 was measured. Between the different rides the CDNC varied between 67 cm −3 and 25 215 cm −3 and the ICNC between 3.4 L −1 and 18.0 L −1 . The most common observed ice particle habit was plates with a mean concentration of 0.8 L −1 ±0.3 L −1 varying between 0 L −1 and 1.4 L −1 between the different rides. Note that this is a lower estimate, since the software has difficulties in detecting transparent particles and depending on the size and orientation, plates may be hard to identify. Besides the ice particle habits, a mean concentration of droplets larger than 40 µm in diameter of 0.3 L −1 ±0.1 L −1 (see Fig. 7 for their size distribution), varying between 0 L −1 and 0.8 L −1 between the different rides, 30 was observed. It should be mentioned that the missing observations of plates or large droplets during single rides could be a stochastic effect due to the relatively low concentrations compared to the measurement volumes of each ride, i.e. for the smallest measurement volume of 2.6 L −1 , one observed particle results in a particle concentration of 0.4 L −1 . In the following,  temperatures higher than -6°C at both sites. Therefore, the upper estimate of the INPC at the measurement site (T>-3°C) is 5 equal to the detection limit and on the order of 5· 10 −4 L −1 for both sites, which lies several orders below the measured ICNC.

Estimation of the secondary ice crystal number concentration
Since the INPC at the measurement location lies several orders of magnitude below the ICNC, the contribution from primary ice nucleation will be neglected. Therefore, the concentration of secondary ice crystals can be estimated from the concentration  of ice crystals that have newly formed and grown at the same location. The measurements were taken at temperatures warmer than -3°C. In this temperature regime, newly formed ice crystals grow into plates (Libbrecht, 2005;Bailey and Hallett, 2009). Figure 5 shows the shapes and the size distribution of the ice crystals divided logarithmically into 9 size bins. The histogram plot shows that the majority of the classified particles smaller than 93 µm were plates (54%; Fig. 5, unidentified particles excluded), while aged crystals can only be found above that size. Hence, we take 93 µm as a threshold to divide between newly 5 formed ice crystals and those, which sedimented from above.
About 70% of the ice crystals smaller than 93 µm (from now on referred to as "small ice") could not be classified into a certain habit because the resolution was too low. Yet, a transparent part observed in the middle of many of these particles (see Fig. 5) supports the assumption based on the temperature regime that these ice crystals are actual plates. In the following, small ice classified as plates including the unidentified particles is referred to as "small plates".

10
A plate with a maximum dimension of 93 µm has a fall velocity of about 0.06 m s −1 (using the equations given in Pruppacher (2010)) which is one order of magnitude lower than the estimated updraft of about 0.6 m s −1 . This supports the assumption that small plates are unlikely to have sedimented from above. About 4% of the small ice was classified as columns. The few observed columns have an aspect ratio close to 1 (see Fig. 5), which is typical for columns growing at temperatures just below -3°C at low supersaturation (Libbrecht, 2005;Bailey and Hallett, 2009) and therefore, must have sedimented from slightly 15 higher altitudes. This is possible because columns have a higher fall velocity than plates (a column with a length of 93 µm falls with about 0.3 m s −1 using the equations given in Pruppacher (2010)). About 9% of the small ice was classified as irregular particles. These could be large secondary ice splinters. However, the fall velocity of irregular particles is hard to assess and it remains unclear if they have fallen from above or formed at the measurement site by SIP. Based on these assumptions, the lower estimate of the secondary ice concentration, which has formed and grown at the measurement site, is equal to the concentration of small plates as defined above and is on the order of 2.6 L −1 ±0.6 L −1 .

5
The time a plate needs to grow to 93 µm is hard to assess at temperatures close to 0°C because small variations in the environmental conditions, e.g., a temperature fluctuation of only 0.5°C, can change the growth time on the order of several minutes. To calculate the growth time, we use the general equation of the ice particle growth by vapor diffusion given in Fukuta and Takahashi (1999). Using the mass-size relation of hexagonal plates from Mitchell et al. (1990) and assuming water vapor saturation over liquid at a temperature of -2°C, a splinter of 5 µm needs about 9 minutes to grow to a size of 93 µm and about   12 https://doi.org/10.5194/acp-2020-986 Preprint. Discussion started: 13 October 2020 c Author(s) 2020. CC BY 4.0 License. 5 minutes to grow to a size of 60 µm. Thus, for any given time, a plate of 93 µm is 4 minutes older than a plate of 60 µm if the temperature is constant at -2°C and the water vapor is saturated over water. Therefore, plates with sizes between 60 µm and 93 µm must have newly formed by SIP within a time span of 4 minutes. The same calculation can be done for a bigger size interval, e.g. plates between 39 µm and 93 µm, which must have newly formed by SIP during a time span of about 6 minutes.
Taking temperature variations in the measurement volume into account, i.e. varying the temperature between -2°C and -1°C, 5 plates between 60 µm and 93 µm were newly formed by SIP during a time span of 4 to 6 minutes, while plates between 39 µm and 93 µm were newly formed by SIP during a time span of 6 to 10 minutes depending on the temperature. The observed concentration of plates with sizes between 60 µm and 93 µm was 0.9 L −1 ±0.3 L −1 , whereas between 39 µm and 93 µm the concentration of plates was 2.2 L −1 ±0.5 L −1 averaged over the two-hour measurement period. Thus, 0.9 L −1 ±0.3 L −1 of secondary ice has formed within 4 to 6 minutes and 2.2 L −1 ±0.5 L −1 within 6 to 10 minutes. Taking all named uncertainties 10 into account, the rate of secondary ice production during our case study lies between 0.17 L −1 min −1 and 0.3 L −1 min −1 .

Contribution of different SIP mechanisms
Droplets larger than 40 µm up to a size of 380 µm in diameter were observed between 0°C and -2.7°C (Fig. 7). Korolev et al.
(2020) explained the occurrence of large droplets just above the melting layer with a recirculation process (see section 1.2).
Ice particles fall through the melting layer and melt into drizzle drops, which can be reintroduced into the cloud by sufficiently 15 high updrafts. The estimated updraft in this case study is about 0.6 m s −1 , which is equal to the fall speed of a 150 µm droplet (Rogers and Yau, 1989). Most of the observed droplets were smaller than 150 µm (Fig. 7), while the remaining ones could have brought into the cloud by local turbulences. Also note that the updraft velocity is a very rough estimate, increasing it by 0.2 m s −1 is already enough to lift 93% instead of 79% of the observed droplets.
Droplet fragmentation requires the presence of large droplets (>≈40 µm, (e.g., Lawson et al., 2015;Korolev et al., 2020)) 20 and has no temperature constraint towards high temperatures. It is therefore a possible mechanism to explain the observed secondary ice. Collisional breakup cannot be completely ruled out but is not expected to produce high secondary ICNCs at warm temperatures as outlined in the introduction. Furthermore, the missing observation of fragment-like ice particles (e.g., broken-off branches) supports that collisional breakup was not very active. Moreover, fragments from collisional breakup are not necessarily small particles and therefore, could be miscounted as aged or irregular ice in this study.

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The rime-splintering process as well as ice fragmentation during thermal shock can both be excluded from being active because they require lower temperatures. MPCs are supersaturated with respect to ice, therefore, also the requirements for ice fragmentation during sublimation are not fulfilled (Korolev et al., 2020). Korolev et al. (2020) argued that INP activation in transient supersaturation around freezing drops could not be shown to be active in the atmosphere.
Therefore, we expect that most of the small plates emerged from droplet fragmentation and that the orographically-induced 30 updraft serves as a constant supply of new droplets larger than 40 µm in diameter, which originate from melted ice crystals.
A schematic of this recirculation process in mountainous regions is shown in Fig. 8. Because new droplets are continuously provided, we can assume that our measurements reflect a steady-state such that we observe freshly produced as well as aged 13 https://doi.org/10.5194/acp-2020-986 Preprint. Discussion started: 13 October 2020 c Author(s) 2020. CC BY 4.0 License. particles simultaneously. The combination of SIP above the melting layer and high updrafts on the windward slope will transport the secondary ice crystals to higher altitudes where they influence the cloud microphyisics.

Parametrization of SIP by droplet fragmentation at warm temperatures
Here we derive a parametrization of the SIP by droplet fragmentation at temperatures close to 0°C when primary ice nucleation can be neglected and droplets freeze only by the collision with ice crystals, which either sedimented from above or formed by 5 SIP. Like Korolev et al. (2020), we assume that only droplets larger than 40 µm are likely to contribute to the SIP by droplet fragmentation. The average number of splinters generated per second per droplet larger than 40 µm (G sp ) is the product of the The probability that a droplet with diameter d and a fall velocity v(d) collides with an ice crystal and freezes is equal to the collision efficiency E. It needs to be multiplied by the combined cross section of the ice crystal and the droplet and the 5 relative velocity between those two to obtain the collection kernel. If we divide the ICNC in i = 1, 2, ..., N size bins, we can approximate the probability that a droplet collides with any ice crystal by summing up the collection kernels of each size bin multiplied by the ICNC per size bin (ICN C i ). For simplification, the average fall velocity of the ice crystals in each size bin v i and the average diameter of the ice crystals in the respective size bin d i is used in the equation.
The probability of droplet fragmentation is size dependent (Takahashi and Yamashita, 1969;Kolomeychuk et al., 1975;Lauber et al., 2018). For a fragmentation to occur, the surface energy has to be overcome, which is proportional to d 2 . Assuming a fragmentation probability of 40% for droplets with d = 300 µm at temperatures larger than -2.5°C (i.e. p df (d=300 µm) = a·d 2 = 0.4), as measured by Keinert et al. (2020), the droplet fragmentation probability per freezing droplet p df can be estimated as The concentration of splinters produced per fragmentation event could not be quantified until now because many of the splinters being produced during a fragmentation event might be too small to be observed with the available measurement techniques (Lauber et al., 2018;Keinert et al., 2020). Here we consider the maximum number of splinters observed during a breakup event for different droplet sizes, as given in Lauber et al. (2018) taken from different studies as the best estimate available. The data points suggest a linear correlation of the number of splinters being produced per droplet N sp and the droplet 20 diameter d. Applying a linear regression, N sp can be estimated as: The number of splinters produced per second by each droplet larger than 40 µm at temperatures close to 0°C in the absence of INPs in MPCs is subsequently given by: In this section, the parametrization derived for SIP by droplet fragmentation for temperatures close to 0°C in the absence of INPs is applied to the presented case study. For simplicity, we assume that the ICNC as well as the size and shape distribution of the ice crystals stays constant over the whole measurement period of two hours, meaning that any ice crystal, which leaves the measurement volume is immediately replaced by a new one of the same size and shape. The same is valid for cloud droplets, which leave the measurement volume or freeze and potentially produce secondary ice splinters. Thus, we assume a constant production of secondary ice. Furthermore, we expect that droplets larger than 40 µm have a collision efficiency of E = 1.
Because of the broad size and shape distribution of the ice crystals, a rough assumption of their fall velocity for the estimation 5 of the relative fall velocity between the droplets and ice crystals has to be made. Therefore, we divide the ice crystals in two size bins (N = 2) and assume that half of the ice crystals (ICN C 1 = 3 L −1 ) are plates with a size of d 1 = 50 µm, while the other half (ICN C 2 = 3 L −1 ) is lump graupel with a size of d 2 = 300 µm. Plates with L ≈ 50 µm fall with a terminal velocity of about v 1 = 0.03 m s −1 (Pruppacher, 2010), while lump graupel with a diameter of 300 µm fall with about v 2 = 0.7 m s −1 (Locatelli and Hobbs, 1974). The terminal velocity of the single droplets is calculated with the equations given in Rogers and 10 Yau (1989) and are shown in Table 1 together with the calculated parameters for each droplet.
To calculate the production rate of secondary ice by droplet fragmentation, the average amount of splinters produced by all large droplets G sp (d) has to be multiplied by their concentration (0.3 L −1 ±0.1 L −1 ). Taking the single sizes of the droplets observed in this case study (Fig. 7) into account, this yields a production rate of 0.10 L −1 min −1 ± 0.03 L −1 min −1 of secondary ice, which is below the estimated production rate of secondary ice of 0.17 L −1 min −1 to 0.3 L −1 min −1 derived from the 15 observations.
The most uncertain parameter in eq. (5) is the number of splinters produced during a droplet fragmentation event. Taking the uncertainties into account and assuming that the proportionality of the number of splinters and the droplet diameter is correct, N sp needs to lie in the range of N sp ≈ 1.2 · 10 5 · m −1 · d to N sp ≈ 4.1 · 10 5 · m −1 · d to produce the measured concentrations of small secondary ice particles. This is equivalent to a range of 24 to 82 splinters produced by a fragmenting droplet of 200 µm 20 in diameter and is up to 5 times higher than the first assumption of N sp .
Apart from the average amount of splinters produced, also the probability that a droplet fragments when it freezes might be different in the atmosphere from what has been measured in the laboratory. Keinert et al. (2020) showed that droplet fragmentation is significantly higher in moving air than in stagnant air. Therefore, it may be reasonable to assume that p df is even higher in turbulent conditions. Taking the uncertainties into account and assuming that all droplets larger than 40 µm will 25 fragment upon freezing (p df (d) = 1), N sp needs to lie between N sp ≈ 9 · 10 4 · m −1 · d and N sp ≈ 2.2 · 10 5 · m −1 · d to explain the measured secondary ice concentration. This is equivalent to a range of 18 to 43 splinters produced per fragmenting droplet of 200 µm in diameter. Thus, the first assessment of N sp lies in the range of uncertainty if we expect that all droplets larger than 40 µm fragment when they freeze. Taking the average of the upper and lower estimate of N sp , the number of splinters produced per droplet larger than 40 µm in diameter at temperatures close to 0°C when turbulence or strong wind speeds are 30 present would change to:  (5) and eq. (6) and the estimated fall velocity based on the equation given in Rogers and Yau (1989). Note that pdf = 100% for all droplets in eq. (5)) Nsp (#) (eq. (5), eq. (6)) Gsp (min −1 ) (eq. (5), eq. (6) The calculated values for each parameter of eq. (5) and eq. (6) are shown in Table 1 for the single droplets. The contribution of the single droplets is highly unbalanced and the largest observed droplet (380 µm) alone is responsible for 97% and 79% of the produced secondary ice concentration using eq. (5) and eq. (6) respectively.

Caveats of the parametrization and its application to the case study
The first main caveat is the parametrization of N sp . The proportionality of N sp to d was based on only four data points, which 5 were derived from three different studies using different measurement techniques and only show the maximum number of observed fragments as discussed in Lauber et al. (2018). Apart from this, there is no physical basis for this correlation. There are so far no reliable measurements of the average number of fragments being produced per fragmentation and recent work by Kleinheins et al. (2020) provides an indication that a majority of possible fragment ejections during freezing could not be observed by the applied measurement techniques. The application to the case study suggests that the number of splinters may be 10 up to 5 times higher than assumed in eq. (4) but critically depends on the largest observed droplet. Until further measurements can constrain the concentration of fragments produced per fragmentation, N sp remains highly uncertain.
The second caveat is that the contribution of the different droplets is highly unbalanced. For example, the largest droplet (∼380 µm) in the present case study contributes 97% to the total amount of the produced concentration of secondary ice when using the generally derived parametrization (eq. (5)) and 79% when using the tuned parametrization (eq. (6)) (see Table 1). This imbalance might be a real possibility and very few large droplets may be enough to explain high concentrations of secondary ice, while the contribution of droplets smaller than about 100 µm may be negligible. However, the observation of a single droplet with a certain size is statistically insignificant and more observations are needed to determine the real size distribution of the cloud droplets.
Thirdly, we assume a continuous flow over two hours without any vertical gradient for the application to the case study.
Since the secondary ice splinters are expected to be very small, they will be lifted up with the updraft faster than the drizzle drops while they are growing to an observable size. The moment they leave the measurement volume before they reach a size of 93 µm, they will not be counted as secondary ice splinters anymore even though they were produced inside the measurement volume. Therefore, we expect that the secondary ice concentration is slightly underestimated. Drizzle drops can also produce secondary ice when they are outside the measurement volume but the secondary ice particles can be lifted into the measurement. 10 However, this is not very likely because measurements were taken very close to 0°C and droplets are thus unlikely to freeze before they reach the measurement volume.
Lastly, the determination of the concentration of secondary ice was based on rather rough assumptions, e.g. small ice crystals, which could not be classified, were expected to be plates and the growth time was determined for specific environmental conditions, which in reality changed with time and position inside the measurement volume. Apart from this, splinters were 15 assumed to have a specific size, while they could vary in reality. Moreover, only splinters, which were small enough to grow into plates, were considered as secondary ice in this study.

Summary
On 22 February 2019, wind from northwest pushed air masses up a mountain slope where measurements were taken just above the melting layer on a gondola (see Fig. 3). The measurements showed relatively constant conditions during the measurement 20 period of two hours (Fig. 6) with a CDNC of about 160 cm −3 and an ICNC of about 6 L −1 , which exceeded the measured INPC by several orders of magnitude. The majority of the observed small ice crystals (L<93 µm) were identified as plates (Fig.   5). As this is the preferred ice crystal habit at the temperatures in the measurement volume (T> -3°C), ice crystals smaller than 93 µm are assumed to have newly formed at the same environmental conditions. At such warm temperatures, primary ice nucleation can be neglected and the concentration of small plates (L<93 µm) most likely emerged from SIP.

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Remarkable was the observation of relatively large droplets between 40 µm and 380 µm in diameter (Fig. 7) above the melting layer. The appearance of cloud droplets larger than about 40 µm is often connected to SIP by droplet fragmentation (Korolev et al., 2020). The rime-splintering process can be excluded to be active and only a small contribution by collisional breakup is assumed, leaving droplet fragmentation as the mainly responsible secondary ice process. A recirculation theory proposed by Korolev et al. (2020) can explain these observations and can in general be applied to mountainous regions when a melting 30 layer is present and sufficiently large updrafts are produced on the windward side by the local topography. Aged ice crystals fall through the melting layer as precipitation and melt into drizzle drops. If sufficiently large updrafts are present, these drops are blown up a mountain slope, lifted through the melting layer and refreeze if they collide with aged ice crystals. Due to the pressure build-up during freezing, they will fragment and create secondary ice crystals, which again can initiate the freezing of another drizzle drop (see schematic in Fig. 8). The secondary ice crystals will be transported to higher altitudes where they influence the cloud microphyisics and subsequently the radiation budget, the lifetime and the precipitation pattern of the cloud.
A parametrization was introduced in section 4.3 for the generation of secondary ice particles by droplet fragmentation at temperatures close to 0°C when primary ice nucleation is basically absent (eq. (5)). Based on limited available measurements of 5 former laboratory studies, it is assumed that the amount of splinters produced per droplet is linearly correlated with their diameter and that a droplet of 200 µm produces 18 splinters on average when it fragments. Applying the presented parametrization to our measurements could not explain the estimated concentration of secondary ice. However, if we assume that all droplets larger than 40 µm fragment when they freeze, the estimated generation of secondary ice lies in the range of uncertainty (i.e.; a droplet of 200 µm in diameter produces between 18 and 43 splinters upon fragmentation). This assumption may be reasonable