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<article xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:oasis="http://docs.oasis-open.org/ns/oasis-exchange/table" xml:lang="en" dtd-version="3.0"><?xmltex \makeatother\@nolinetrue\makeatletter?>
  <front>
    <journal-meta><journal-id journal-id-type="publisher">ACP</journal-id><journal-title-group>
    <journal-title>Atmospheric Chemistry and Physics</journal-title>
    <abbrev-journal-title abbrev-type="publisher">ACP</abbrev-journal-title><abbrev-journal-title abbrev-type="nlm-ta">Atmos. Chem. Phys.</abbrev-journal-title>
  </journal-title-group><issn pub-type="epub">1680-7324</issn><publisher>
    <publisher-name>Copernicus Publications</publisher-name>
    <publisher-loc>Göttingen, Germany</publisher-loc>
  </publisher></journal-meta>
    <article-meta>
      <article-id pub-id-type="doi">10.5194/acp-21-3855-2021</article-id><title-group><article-title>Continuous secondary-ice production initiated by updrafts <?xmltex \hack{\break}?> through the melting layer in mountainous regions</article-title><alt-title>Continuous secondary-ice production initiated by updrafts through the melting layer</alt-title>
      </title-group><?xmltex \runningtitle{Continuous secondary-ice production initiated by updrafts through the melting layer}?><?xmltex \runningauthor{A.~Lauber et al.}?>
      <contrib-group>
        <contrib contrib-type="author" corresp="yes" rid="aff1">
          <name><surname>Lauber</surname><given-names>Annika</given-names></name>
          <email>annika.lauber@env.ethz.ch</email>
        <ext-link>https://orcid.org/0000-0001-5731-2652</ext-link></contrib>
        <contrib contrib-type="author" corresp="yes" rid="aff1">
          <name><surname>Henneberger</surname><given-names>Jan</given-names></name>
          <email>jan.henneberger@env.ethz.ch</email>
        <ext-link>https://orcid.org/0000-0001-6979-3174</ext-link></contrib>
        <contrib contrib-type="author" corresp="no" rid="aff2">
          <name><surname>Mignani</surname><given-names>Claudia</given-names></name>
          
        <ext-link>https://orcid.org/0000-0001-9250-0587</ext-link></contrib>
        <contrib contrib-type="author" corresp="no" rid="aff1">
          <name><surname>Ramelli</surname><given-names>Fabiola</given-names></name>
          
        </contrib>
        <contrib contrib-type="author" corresp="no" rid="aff1">
          <name><surname>Pasquier</surname><given-names>Julie T.</given-names></name>
          
        <ext-link>https://orcid.org/0000-0002-2327-240X</ext-link></contrib>
        <contrib contrib-type="author" corresp="no" rid="aff1">
          <name><surname>Wieder</surname><given-names>Jörg</given-names></name>
          
        <ext-link>https://orcid.org/0000-0003-2858-686X</ext-link></contrib>
        <contrib contrib-type="author" corresp="no" rid="aff3">
          <name><surname>Hervo</surname><given-names>Maxime</given-names></name>
          
        <ext-link>https://orcid.org/0000-0003-3614-1297</ext-link></contrib>
        <contrib contrib-type="author" corresp="no" rid="aff1">
          <name><surname>Lohmann</surname><given-names>Ulrike</given-names></name>
          
        <ext-link>https://orcid.org/0000-0001-8885-3785</ext-link></contrib>
        <aff id="aff1"><label>1</label><institution>ETH Zurich, Institute for Atmospheric and Climate Science, Zurich, Switzerland</institution>
        </aff>
        <aff id="aff2"><label>2</label><institution>Department of Environmental Sciences, University of Basel, Basel, Switzerland</institution>
        </aff>
        <aff id="aff3"><label>3</label><institution>Federal Office of Meteorology and Climatology MeteoSwiss, Payerne, Switzerland</institution>
        </aff>
      </contrib-group>
      <author-notes><corresp id="corr1">Annika Lauber (annika.lauber@env.ethz.ch) and Jan Henneberger (jan.henneberger@env.ethz.ch)</corresp></author-notes><pub-date><day>15</day><month>March</month><year>2021</year></pub-date>
      
      <volume>21</volume>
      <issue>5</issue>
      <fpage>3855</fpage><lpage>3870</lpage>
      <history>
        <date date-type="received"><day>22</day><month>September</month><year>2020</year></date>
           <date date-type="rev-request"><day>13</day><month>October</month><year>2020</year></date>
           <date date-type="rev-recd"><day>29</day><month>January</month><year>2021</year></date>
           <date date-type="accepted"><day>2</day><month>February</month><year>2021</year></date>
      </history>
      <permissions>
        <copyright-statement>Copyright: © 2021 </copyright-statement>
        <copyright-year>2021</copyright-year>
      <license license-type="open-access"><license-p>This work is licensed under the Creative Commons Attribution 4.0 International License. To view a copy of this licence, visit <ext-link ext-link-type="uri" xlink:href="https://creativecommons.org/licenses/by/4.0/">https://creativecommons.org/licenses/by/4.0/</ext-link></license-p></license></permissions><self-uri xlink:href="https://acp.copernicus.org/articles/.html">This article is available from https://acp.copernicus.org/articles/.html</self-uri><self-uri xlink:href="https://acp.copernicus.org/articles/.pdf">The full text article is available as a PDF file from https://acp.copernicus.org/articles/.pdf</self-uri>
      <abstract><title>Abstract</title>
    <p id="d1e162">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 <inline-formula><mml:math id="M1" display="inline"><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">3</mml:mn></mml:mrow></mml:math></inline-formula> and <inline-formula><mml:math id="M2" display="inline"><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">8</mml:mn></mml:mrow></mml:math></inline-formula> <inline-formula><mml:math id="M3" display="inline"><mml:msup><mml:mi/><mml:mo>∘</mml:mo></mml:msup></mml:math></inline-formula>C and has been implemented into several models, while all other secondary-ice processes are poorly constrained and lack a well-founded quantification. During 2 h of measurements taken on a mountain slope just above the melting layer at temperatures warmer than <inline-formula><mml:math id="M4" display="inline"><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">3</mml:mn></mml:mrow></mml:math></inline-formula> <inline-formula><mml:math id="M5" display="inline"><mml:msup><mml:mi/><mml:mo>∘</mml:mo></mml:msup></mml:math></inline-formula>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 slightly sub-zero temperatures, where primary-ice nucleation is basically absent, and the first ice is initiated by the collision of drizzle drops with aged ice crystals sedimenting from higher altitudes. Based on previous measurements, we estimate that a droplet of 200 <inline-formula><mml:math id="M6" display="inline"><mml:mrow class="unit"><mml:mi mathvariant="normal">µ</mml:mi></mml:mrow></mml:math></inline-formula>m in diameter produces 18 secondary-ice crystals when it fragments upon freezing. The application of the parametrization to our measurements suggests that the actual number of splinters produced by a fragmenting droplet may be up to an order of magnitude higher.</p>
  </abstract>
    </article-meta>
  </front>
<body>
      

      <?xmltex \hack{\newpage}?>
<sec id="Ch1.S1" sec-type="intro">
  <label>1</label><title>Introduction</title>
      <p id="d1e233">Accurate weather forecasting in mountainous regions is more challenging than over flat topography as orography has a strong influence on the local weather; e.g., it creates local up- and downdrafts, which strongly impacts the development of clouds <xref ref-type="bibr" rid="bib1.bibx70 bib1.bibx26" id="paren.1"/>. Yet, orography is not very well resolved in numerical weather prediction models because the high resolution needed requires a lot of computational power that is often not available <xref ref-type="bibr" rid="bib1.bibx18 bib1.bibx21" id="paren.2"><named-content content-type="pre">e.g.,</named-content></xref>. Furthermore, microphysical processes in clouds, which affect the development of the concentration and size of ice crystals and cloud droplets, are still not completely understood <xref ref-type="bibr" rid="bib1.bibx15" id="paren.3"><named-content content-type="pre">e.g.,</named-content></xref>. However, the correct phase partitioning and concentration of cloud particles is important for the determination of the radiation budget, the lifetime, and the precipitation amount of clouds <xref ref-type="bibr" rid="bib1.bibx54 bib1.bibx26" id="paren.4"><named-content content-type="pre">e.g.,</named-content></xref>. This can be especially important for mixed-phase clouds (MPCs) consisting of ice crystals and liquid droplets since they have a major contribution to the total precipitation in the midlatitudes <xref ref-type="bibr" rid="bib1.bibx60" id="paren.5"/>.</p>
      <?pagebreak page3856?><p id="d1e257">MPCs are thermodynamically unstable because the saturation water vapor pressure over ice is lower than over liquid water, allowing ice crystals  to grow faster than liquid droplets. The particle growth reduces the water vapor pressure and eventually leads to the evaporation of cloud droplets when the water vapor pressure drops below water vapor saturation over liquid. This process is called the Wegener–Bergeron–Findeisen process <xref ref-type="bibr" rid="bib1.bibx90 bib1.bibx7 bib1.bibx16" id="paren.6"/> and can lead to rapid glaciation of clouds. Nevertheless, persistent MPCs have been frequently observed in mountainous regions, where the local topography produces sufficiently large updrafts to sustain a continuous condensate production <xref ref-type="bibr" rid="bib1.bibx40 bib1.bibx55" id="paren.7"><named-content content-type="pre">e.g.,</named-content></xref>. An updraft velocity of about 2 m s<inline-formula><mml:math id="M7" display="inline"><mml:msup><mml:mi/><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula> is often sufficient to maintain supersaturation with respect to water, which enables a simultaneous growth of cloud droplets and ice crystals <xref ref-type="bibr" rid="bib1.bibx40" id="paren.8"/>.</p>
      <p id="d1e283">MPCs can exist between 0 and <inline-formula><mml:math id="M8" display="inline"><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">38</mml:mn></mml:mrow></mml:math></inline-formula> <inline-formula><mml:math id="M9" display="inline"><mml:msup><mml:mi/><mml:mo>∘</mml:mo></mml:msup></mml:math></inline-formula>C. In this temperature range, ice-nucleating particles (INPs) are required for cloud droplets to freeze. The resulting so-called primary ice can create additional ice crystals (secondary ice), which again can fragment and produce more secondary ice by any kind of fragmentation, referred to as secondary-ice production (SIP) <xref ref-type="bibr" rid="bib1.bibx15" id="paren.9"><named-content content-type="pre">e.g.,</named-content></xref>. Assuming only primary ice to exist, one would expect that the ice crystal number concentration (ICNC) equals the activated ice-nucleating-particle concentration (INPC) in clouds. This is observed for example in shallow stratocumulus <xref ref-type="bibr" rid="bib1.bibx59" id="paren.10"/> or thin stratus clouds but often does not hold true in more convective clouds. A discrepancy between the INPC and the ICNC of up to 4 orders of magnitude has been observed in several studies <xref ref-type="bibr" rid="bib1.bibx38 bib1.bibx31 bib1.bibx12 bib1.bibx45 bib1.bibx44 bib1.bibx5" id="paren.11"><named-content content-type="pre">e.g.,</named-content></xref>. If surface-based processes like blowing snow <xref ref-type="bibr" rid="bib1.bibx5" id="paren.12"><named-content content-type="pre">e.g.,</named-content></xref> and the seeder–feeder process (hydrometeors which formed aloft precipitate into the cloud below; e.g., <xref ref-type="bibr" rid="bib1.bibx1 bib1.bibx67" id="altparen.13"/>) can be excluded as a potential ice source, this discrepancy can only be explained by SIP.</p>
      <p id="d1e327">On 22 February 2019, during the RACLETS (Role of Aerosols and CLouds Enhanced by Topography on Snow) campaign taking place in the region of Davos in the Swiss Alps <xref ref-type="bibr" rid="bib1.bibx56 bib1.bibx68 bib1.bibx67 bib1.bibx19" id="paren.14"/>, a discrepancy between the INPC and ICNC of several orders of magnitude was measured. In the following subsection different SIP mechanisms, which might be responsible for this high discrepancy, are described. Furthermore, a recirculation theory by <xref ref-type="bibr" rid="bib1.bibx43" id="text.15"/> is introduced in Sect. <xref ref-type="sec" rid="Ch1.S1.SS2"/>, which can explain the observations.</p>
<sec id="Ch1.S1.SS1">
  <label>1.1</label><title>Secondary-ice production mechanisms</title>
      <p id="d1e346">Six SIP mechanisms have been discussed over the past decades. They are expected to be active depending on the environmental conditions like temperature, cloud particle size distribution, and updrafts: (i) the rime-splintering or Hallett–Mossop process, (ii) collisional breakup, (iii) droplet fragmentation upon freezing, (iv) thermal-shock fragmentation, (v) fragmentation of sublimating or melting ice, and (vi) activation of INPs in transient supersaturation of freezing droplets <xref ref-type="bibr" rid="bib1.bibx42" id="paren.16"/>.</p>
      <p id="d1e352">The rime-splintering process describes the production of secondary ice during riming. Yet, the underlying physical mechanism of this process is not well defined <xref ref-type="bibr" rid="bib1.bibx15" id="paren.17"/>. The most common explanation for SIP during riming is that droplets, which freeze on an ice crystal, build up an internal pressure and break up upon freezing. In a laboratory study, <xref ref-type="bibr" rid="bib1.bibx24" id="text.18"/> observed that rime-splintering is active at temperatures between <inline-formula><mml:math id="M10" display="inline"><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">3</mml:mn></mml:mrow></mml:math></inline-formula> and <inline-formula><mml:math id="M11" display="inline"><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">8</mml:mn></mml:mrow></mml:math></inline-formula> <inline-formula><mml:math id="M12" display="inline"><mml:msup><mml:mi/><mml:mo>∘</mml:mo></mml:msup></mml:math></inline-formula>C when particles grow by riming. <xref ref-type="bibr" rid="bib1.bibx14" id="text.19"/> showed that at temperatures above <inline-formula><mml:math id="M13" display="inline"><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">3</mml:mn></mml:mrow></mml:math></inline-formula> <inline-formula><mml:math id="M14" display="inline"><mml:msup><mml:mi/><mml:mo>∘</mml:mo></mml:msup></mml:math></inline-formula>C, droplets spread out on the ice surface and do not build an ice shell, while at temperatures below <inline-formula><mml:math id="M15" display="inline"><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">8</mml:mn></mml:mrow></mml:math></inline-formula> <inline-formula><mml:math id="M16" display="inline"><mml:msup><mml:mi/><mml:mo>∘</mml:mo></mml:msup></mml:math></inline-formula>C the ice shell of rime might be too strong to break <xref ref-type="bibr" rid="bib1.bibx23" id="paren.20"/>. While several observations of secondary ice could be explained by the rime-splintering process <xref ref-type="bibr" rid="bib1.bibx61 bib1.bibx62 bib1.bibx25 bib1.bibx9" id="paren.21"><named-content content-type="pre">e.g.,</named-content></xref>, secondary ice was also observed in cases where these requirements were not fulfilled <xref ref-type="bibr" rid="bib1.bibx43" id="paren.22"><named-content content-type="pre">e.g.,</named-content></xref>, or the SIP by rime-splintering alone was expected to be too slow to explain the observed rapid glaciation <xref ref-type="bibr" rid="bib1.bibx32" id="paren.23"><named-content content-type="pre">e.g.,</named-content></xref>.</p>
      <p id="d1e451">Collisional breakup describes the break-up of an ice crystal due to a collision with another ice crystal. This process was shown to be active when large ice crystals (e.g., graupel or aggregates) are present <xref ref-type="bibr" rid="bib1.bibx42" id="paren.24"/>. Fragments of ice crystals were observed in several field studies <xref ref-type="bibr" rid="bib1.bibx33 bib1.bibx73" id="paren.25"><named-content content-type="pre">e.g.,</named-content></xref>. <xref ref-type="bibr" rid="bib1.bibx87" id="text.26"/> found that significant SIP occurs by collisional breakup when relatively large concentrations of rimed ice crystals are present in convective clouds. He concluded that collisional breakup cannot explain high secondary ICNCs at warm temperatures. The laboratory study of <xref ref-type="bibr" rid="bib1.bibx84" id="text.27"/> showed that a maximum SIP rate occurs at <inline-formula><mml:math id="M17" display="inline"><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">16</mml:mn></mml:mrow></mml:math></inline-formula> <inline-formula><mml:math id="M18" display="inline"><mml:msup><mml:mi/><mml:mo>∘</mml:mo></mml:msup></mml:math></inline-formula>C when ice spheres with different growth modes collide.</p>
      <?pagebreak page3857?><p id="d1e488">Droplet fragmentation during freezing can happen when liquid water gets trapped inside a freezing droplet after an ice shell formed around the droplet and expands. Due to the lower density of ice than liquid water, an internal pressure builds up. If the pressure reaches a critical point, different processes have been observed to release the pressure: a complete breakup in mainly two halves; opening and closing of cracks; bubbles forming and bursting on the ice shell; and streams being ejected from the shell, which may contain small ice fragments <xref ref-type="bibr" rid="bib1.bibx46" id="paren.28"/>. An average number of splinters being ejected per freezing droplet could not be quantified so far because many ejected splinters are expected to be too small, and the processes happen so fast that the used measurement techniques until today were not able to detect all splinters <xref ref-type="bibr" rid="bib1.bibx46 bib1.bibx34" id="paren.29"/>. <xref ref-type="bibr" rid="bib1.bibx34" id="text.30"/> showed that droplets freezing in moving air break up 1 order of magnitude more frequently than droplets freezing in stagnant air. Therefore, high wind speed and turbulence can be expected to increase the likelihood of droplet fragmentation. The highest fragmentation rate was observed around <inline-formula><mml:math id="M19" display="inline"><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">15</mml:mn></mml:mrow></mml:math></inline-formula> <inline-formula><mml:math id="M20" display="inline"><mml:msup><mml:mi/><mml:mo>∘</mml:mo></mml:msup></mml:math></inline-formula>C. It decreases to higher and lower temperatures, while it seems to increase again at temperatures close to 0 <inline-formula><mml:math id="M21" display="inline"><mml:msup><mml:mi/><mml:mo>∘</mml:mo></mml:msup></mml:math></inline-formula>C <xref ref-type="bibr" rid="bib1.bibx34" id="paren.31"/>. The larger the droplet, the more likely it will fragment and the more splinters it will likely produce <xref ref-type="bibr" rid="bib1.bibx39 bib1.bibx46" id="paren.32"/>. Many field studies showed that large droplets are present in clouds before ICNCs exceed the INPCs by orders of magnitude <xref ref-type="bibr" rid="bib1.bibx38 bib1.bibx10 bib1.bibx58 bib1.bibx32 bib1.bibx69 bib1.bibx49 bib1.bibx43" id="paren.33"><named-content content-type="pre">e.g.,</named-content></xref> and often explained these observations with SIP by droplet fragmentation. Even though large droplets are rare in clouds, they might start a cascading process of splinter production when produced splinters hit other droplets, which subsequently freeze and produce more splinters <xref ref-type="bibr" rid="bib1.bibx38 bib1.bibx11 bib1.bibx50" id="paren.34"/>.</p>
      <p id="d1e544">Thermal-shock fragmentation <xref ref-type="bibr" rid="bib1.bibx36" id="paren.35"/> and the activation of INPs in transient supersaturation of freezing droplets both require precipitation-sized droplets to be active, while the fragmentation of sublimating ice can be important if cloud regions are affected by entrainment of dry air <xref ref-type="bibr" rid="bib1.bibx42" id="paren.36"/>. <xref ref-type="bibr" rid="bib1.bibx63" id="text.37"/> observed no sublimation breakup for columnar and plate-like crystals, while breakup during melting only took place at relative humidities below 90 %.</p>
      <p id="d1e556">From the six SIP mechanisms, the rime-splintering process is the best-constrained mechanism and has been implemented in some cloud microphysics schemes <xref ref-type="bibr" rid="bib1.bibx74 bib1.bibx6 bib1.bibx64" id="paren.38"><named-content content-type="pre">e.g.,</named-content></xref>. There have been fewer attempts to include collisional breakup <xref ref-type="bibr" rid="bib1.bibx92 bib1.bibx65 bib1.bibx76 bib1.bibx78" id="paren.39"><named-content content-type="pre">e.g.,</named-content></xref> and droplet fragmentation <xref ref-type="bibr" rid="bib1.bibx50 bib1.bibx91" id="paren.40"><named-content content-type="pre">e.g.,</named-content></xref>. <xref ref-type="bibr" rid="bib1.bibx82" id="text.41"/> 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 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.</p>
</sec>
<sec id="Ch1.S1.SS2">
  <label>1.2</label><title>Persistent SIP immediately above the melting layer</title>
      <p id="d1e585">A remarkable finding of the study by <xref ref-type="bibr" rid="bib1.bibx43" id="text.42"/> is the observation of small pristine ice crystals persisting immediately above the melting layer in clouds. They calculated a spatial correlation time <inline-formula><mml:math id="M22" display="inline"><mml:mrow><mml:msub><mml:mi mathvariant="italic">τ</mml:mi><mml:mi mathvariant="normal">corr</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> during which the environmental changes (e.g., 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, <inline-formula><mml:math id="M23" display="inline"><mml:mrow><mml:msub><mml:mi mathvariant="italic">τ</mml:mi><mml:mi mathvariant="normal">corr</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> is on the order of 60–120 s. Assuming water vapor saturation over liquid, an ice column at temperatures below <inline-formula><mml:math id="M24" display="inline"><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">3</mml:mn></mml:mrow></mml:math></inline-formula> <inline-formula><mml:math id="M25" display="inline"><mml:msup><mml:mi/><mml:mo>∘</mml:mo></mml:msup></mml:math></inline-formula>C can reach a length between 50 and 150 <inline-formula><mml:math id="M26" display="inline"><mml:mrow class="unit"><mml:mi mathvariant="normal">µ</mml:mi></mml:mrow></mml:math></inline-formula>m during <inline-formula><mml:math id="M27" display="inline"><mml:mrow><mml:msub><mml:mi mathvariant="italic">τ</mml:mi><mml:mi mathvariant="normal">corr</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> depending on its aspect ratio. A relative humidity close to water saturation is a valid assumption in MPCs <xref ref-type="bibr" rid="bib1.bibx41" id="paren.43"/>. It can be concluded that if external processes like blowing snow or the seeder–feeder process can be excluded, the discrepancy between the INPC and the concentration of columns shorter than 50 to 150 <inline-formula><mml:math id="M28" display="inline"><mml:mrow class="unit"><mml:mi mathvariant="normal">µ</mml:mi></mml:mrow></mml:math></inline-formula>m in MPCs must emerge from SIP.</p>
      <p id="d1e663"><xref ref-type="bibr" rid="bib1.bibx43" id="text.44"/> suggested a recirculation process through the melting layer to be a possible explanation for the observed high concentration of small pristine ice crystals. Large 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 active and produce high numbers of secondary-ice particles.</p>
</sec>
</sec>
<sec id="Ch1.S2">
  <label>2</label><title>Experimental setup</title>
<sec id="Ch1.S2.SS1">
  <label>2.1</label><title>RACLETS campaign</title>
      <p id="d1e684">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 of topography and aerosols on the development of clouds. For this goal, a set of instruments were deployed at different locations. Figure <xref ref-type="fig" rid="Ch1.F1"/> shows the locations and instruments which were used for the analysis of the presented case study.</p>

      <?xmltex \floatpos{t}?><fig id="Ch1.F1" specific-use="star"><?xmltex \currentcnt{1}?><?xmltex \def\figurename{Figure}?><label>Figure 1</label><caption><p id="d1e691">Overview of the measurement location and the experimental setup. The geographical location of the Gotschnabahn and the surrounding topography is shown in <bold>(a)</bold>. An enlarged section of the measurement sites (black rectangle in <bold>a</bold>) and the instrument setup are shown in <bold>(b)</bold>. The elevation data were taken from the digital height model DHM25 of the Federal Office of Topography swisstopo: <uri>https://shop.swisstopo.admin.ch/de/products/height_models/dhm25200</uri> (last access: 9 March 2020).</p></caption>
          <?xmltex \igopts{width=341.433071pt}?><graphic xlink:href="https://acp.copernicus.org/articles/21/3855/2021/acp-21-3855-2021-f01.png"/>

        </fig>

      <?pagebreak page3858?><p id="d1e712">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 <xref ref-type="bibr" rid="bib1.bibx4" id="paren.45"/> on the Gotschnabahn, which is described in more detail in Sect. <xref ref-type="sec" rid="Ch1.S2.SS2"/>. As part of the MeteoSwiss observation network, a ceilometer (Vaisala, Model CL31) was installed in Klosters (1200 m) to determine the cloud base height <xref ref-type="bibr" rid="bib1.bibx29" id="paren.46"/> as well as a wind profiler (Vaisala, Model Lap3000, Finland) in Wolfgang to measure the horizontal wind field <xref ref-type="bibr" rid="bib1.bibx30" id="paren.47"/>. 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 Grischa paragliding club. Temperature measurements were taken from the MeteoSwiss station in Klosters <xref ref-type="bibr" rid="bib1.bibx28" id="paren.48"/> and the snow drift station installed at Gotschnagrat <xref ref-type="bibr" rid="bib1.bibx88 bib1.bibx89" id="paren.49"/>. For the analysis of the whole clouds, a vertically pointing cloud radar (Model Mira-36, METEK GmbH, Germany; <xref ref-type="bibr" rid="bib1.bibx20" id="altparen.50"/>) was installed in Wolfgang. INPCs were measured in the valley in Wolfgang (1630 m) and on the mountain top of Weissfluhjoch (2670 m), where ambient aerosol was sampled through heated inlets using a high-flow-rate impinger (Bertin Technologies, Model Coriolis <inline-formula><mml:math id="M29" display="inline"><mml:mrow class="unit"><mml:mi mathvariant="normal">µ</mml:mi></mml:mrow></mml:math></inline-formula>, France). The samples were analyzed for the INPC immediately on-site with a drop-freezing technique using DRINCZ (Drop Freezing Ice Nuclei Counter Zurich; <xref ref-type="bibr" rid="bib1.bibx13" id="altparen.51"/>) of the Swiss Federal Institute of Technology in Zurich (ETH) in Wolfgang and the University of Basel's LINDA (LED-based ice nucleation detection apparatus; <xref ref-type="bibr" rid="bib1.bibx80" id="altparen.52"/>) at Weissfluhjoch as described in <xref ref-type="bibr" rid="bib1.bibx56" id="text.53"/>. The detection limit of the INPC (lowest concentration measurable) was calculated according to <xref ref-type="bibr" rid="bib1.bibx86" id="text.54"/> considering the concentration obtained if only one (first) drop froze. A detection limit of <inline-formula><mml:math id="M30" display="inline"><mml:mrow><mml:mn mathvariant="normal">5.2</mml:mn><mml:mo>×</mml:mo><mml:msup><mml:mn mathvariant="normal">10</mml:mn><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">4</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:math></inline-formula> and <inline-formula><mml:math id="M31" display="inline"><mml:mrow><mml:mn mathvariant="normal">4.9</mml:mn><mml:mo>×</mml:mo><mml:msup><mml:mn mathvariant="normal">10</mml:mn><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">4</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:math></inline-formula> L<inline-formula><mml:math id="M32" display="inline"><mml:msup><mml:mi/><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula> has been determined for Wolfgang and Weissfluhjoch, respectively.</p>
</sec>
<sec id="Ch1.S2.SS2">
  <label>2.2</label><title>HoloGondel</title>
      <p id="d1e813">The HoloGondel platform consists of the HOLographic Imager for Microscopic Objects 3G (HOLIMO 3G in <xref ref-type="bibr" rid="bib1.bibx4" id="altparen.55"/>) – which records the concentration, size distribution, and shapes of cloud particles  – and a temperature and relative-humidity sensor (HygroMet4, Rotronic) in a ventilated housing (RS24T, Rotronic) as well as a pressure sensor (Fig. <xref ref-type="fig" rid="Ch1.F2"/>). 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 were only used if the difference between 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 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<inline-formula><mml:math id="M33" display="inline"><mml:msup><mml:mi/><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula>, 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 analyzed, when the setup was in front of the gondola in the direction of travel.</p>

      <?xmltex \floatpos{t}?><fig id="Ch1.F2" specific-use="star"><?xmltex \currentcnt{2}?><?xmltex \def\figurename{Figure}?><label>Figure 2</label><caption><p id="d1e835">The HoloGondel platform installed on the Gotschnabahn in <bold>(a)</bold> and an enlarged picture of the platform in <bold>(b)</bold>, showing HOLIMO 3G with its measurement volume between the two towers, two optical particle counters (not used in this study) next to HOLIMO, and the temperature and relative-humidity (RH) sensor. The sonic anemometer was turned off due to a limited supply of power.</p></caption>
          <?xmltex \igopts{width=341.433071pt}?><graphic xlink:href="https://acp.copernicus.org/articles/21/3855/2021/acp-21-3855-2021-f02.jpg"/>

        </fig>

      <p id="d1e850">HOLIMO 3G is an open-path instrument, which uses digital in-line holography. Holograms of the sample volume between the two towers (see Fig. <xref ref-type="fig" rid="Ch1.F2"/>b) are recorded, of which 13.7 cm<inline-formula><mml:math id="M34" display="inline"><mml:msup><mml:mi/><mml:mn mathvariant="normal">3</mml:mn></mml:msup></mml:math></inline-formula> was considered for the analysis, producing a 3D distribution of the 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 <xref ref-type="bibr" rid="bib1.bibx4" id="text.56"/> and <xref ref-type="bibr" rid="bib1.bibx27" id="text.57"/>. The pixel size is 3.1 <inline-formula><mml:math id="M35" display="inline"><mml:mrow class="unit"><mml:mi mathvariant="normal">µ</mml:mi></mml:mrow></mml:math></inline-formula>m, which allows us to observe cloud particles larger than 6.2 <inline-formula><mml:math id="M36" display="inline"><mml:mrow class="unit"><mml:mi mathvariant="normal">µ</mml:mi></mml:mrow></mml:math></inline-formula>m <xref ref-type="bibr" rid="bib1.bibx4" id="paren.58"/>. 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 <inline-formula><mml:math id="M37" display="inline"><mml:mrow class="unit"><mml:mi mathvariant="normal">µ</mml:mi></mml:mrow></mml:math></inline-formula>m, depending on the shape of the ice particles <xref ref-type="bibr" rid="bib1.bibx27" id="paren.59"/>.</p>
      <p id="d1e902">A neural network for particles larger than 25 <inline-formula><mml:math id="M38" display="inline"><mml:mrow class="unit"><mml:mi mathvariant="normal">µ</mml:mi></mml:mrow></mml:math></inline-formula>m <xref ref-type="bibr" rid="bib1.bibx85" id="paren.60"/>  and a decision tree for particles smaller than 25 <inline-formula><mml:math id="M39" display="inline"><mml:mrow class="unit"><mml:mi mathvariant="normal">µ</mml:mi></mml:mrow></mml:math></inline-formula>m in their major axis 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 <inline-formula><mml:math id="M40" display="inline"><mml:mrow><mml:mo>±</mml:mo><mml:mn mathvariant="normal">5</mml:mn></mml:mrow></mml:math></inline-formula> % for ice crystals larger than about 100 <inline-formula><mml:math id="M41" display="inline"><mml:mrow class="unit"><mml:mi mathvariant="normal">µ</mml:mi></mml:mrow></mml:math></inline-formula>m and <inline-formula><mml:math id="M42" display="inline"><mml:mrow><mml:mo>±</mml:mo><mml:mn mathvariant="normal">15</mml:mn></mml:mrow></mml:math></inline-formula> % for ice crystals smaller than 100 <inline-formula><mml:math id="M43" display="inline"><mml:mrow class="unit"><mml:mi mathvariant="normal">µ</mml:mi></mml:mrow></mml:math></inline-formula>m <xref ref-type="bibr" rid="bib1.bibx3" id="paren.61"/>. 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.<?pagebreak page3859?> Plates and columns are hexagonal prisms with the diameter of the hexagonal basis <inline-formula><mml:math id="M44" display="inline"><mml:mi>a</mml:mi></mml:math></inline-formula> and the prism height <inline-formula><mml:math id="M45" display="inline"><mml:mi>h</mml:mi></mml:math></inline-formula>. Plates have a dimension of <inline-formula><mml:math id="M46" display="inline"><mml:mrow><mml:mi>a</mml:mi><mml:mo>&gt;</mml:mo><mml:mi>h</mml:mi></mml:mrow></mml:math></inline-formula>, while columns have a dimension of <inline-formula><mml:math id="M47" display="inline"><mml:mrow><mml:mi>a</mml:mi><mml:mo>≤</mml:mo><mml:mi>h</mml:mi></mml:mrow></mml:math></inline-formula>. For the uncertainty in the different habits, the counting uncertainty (<inline-formula><mml:math id="M48" display="inline"><mml:mrow><mml:msqrt><mml:mi>N</mml:mi></mml:msqrt><mml:mo>/</mml:mo><mml:mi>V</mml:mi></mml:mrow></mml:math></inline-formula>; <inline-formula><mml:math id="M49" display="inline"><mml:mi>N</mml:mi></mml:math></inline-formula>: number of crystals; <inline-formula><mml:math id="M50" display="inline"><mml:mi>V</mml:mi></mml:math></inline-formula>: measurement volume) was added because of their relatively low number compared to the measurement volume. The uncertainty in cloud droplets is estimated to be <inline-formula><mml:math id="M51" display="inline"><mml:mrow><mml:mo>±</mml:mo><mml:mn mathvariant="normal">6</mml:mn></mml:mrow></mml:math></inline-formula> % as determined for the classification with the neural network in <xref ref-type="bibr" rid="bib1.bibx85" id="text.62"/>. Again, for droplets larger than 40 <inline-formula><mml:math id="M52" display="inline"><mml:mrow class="unit"><mml:mi mathvariant="normal">µ</mml:mi></mml:mrow></mml:math></inline-formula>m the counting uncertainty was added due to their relatively small numbers.</p>
</sec>
</sec>
<sec id="Ch1.S3">
  <label>3</label><title>Case study 22 February 2019</title>
<sec id="Ch1.S3.SS1">
  <label>3.1</label><title>Weather situation</title>
      <p id="d1e1068">On 22 February 2019, a ridge was located in front of Switzerland, which guided air masses from the north to the Davos region. 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. <xref ref-type="fig" rid="Ch1.F3"/>a), it can be anticipated that the cloud top height was about 4 km in the Davos region, referring to a cloud top temperature of around <inline-formula><mml:math id="M53" display="inline"><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">8</mml:mn></mml:mrow></mml:math></inline-formula> <inline-formula><mml:math id="M54" display="inline"><mml:msup><mml:mi/><mml:mo>∘</mml:mo></mml:msup></mml:math></inline-formula>C as measured by a radiosonde launched at 10:25 UTC from Wolfgang. Between 05:00 and 09:00 UTC, the cloud radar showed a midlevel cloud above the stratiform cloud, reaching from about 4.5 to about 6.5 km, that might have acted as a seeder cloud in the early morning.</p>

      <?xmltex \floatpos{t}?><fig id="Ch1.F3" specific-use="star"><?xmltex \currentcnt{3}?><?xmltex \def\figurename{Figure}?><label>Figure 3</label><caption><p id="d1e1094">Observations of the reflectivity measured by the cloud radar installed in Wolfgang are shown in <bold>(a)</bold>. The ceilometer data in <bold>(b)</bold> show the estimated cloud base with gray dots above Klosters. A dark band underneath the cloud base indicates the melting layer. Temperature and relative-humidity measurements at the highest (closest to Gotschnagrat) and lowest point (closest to Gotschnaboden) of the gondola measurements and Klosters during the time <?xmltex \hack{\mbox\bgroup}?>HoloGondel<?xmltex \hack{\egroup}?> was measuring are shown in <bold>(c)</bold> and <bold>(d)</bold>, respectively.</p></caption>
          <?xmltex \igopts{width=398.338583pt}?><graphic xlink:href="https://acp.copernicus.org/articles/21/3855/2021/acp-21-3855-2021-f03.png"/>

        </fig>

      <p id="d1e1120">As can be inferred from the ceilometer data, light precipitation started at around 07:00 UTC in Klosters, where the valley station of the Gotschnabahn is located (Fig. <xref ref-type="fig" rid="Ch1.F3"/>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 08:00 and 10:00 UTC. The rides used for the analysis are shown as gray lines in Fig. <xref ref-type="fig" rid="Ch1.F3"/>. The temperatures and relative humidities at Gotschnaboden and Gotschnagrat (Fig. <xref ref-type="fig" rid="Ch1.F3"/>c and d) are derived from measurements of the highest and lowest point of the measurements on the gondola and are therefore only available when the gondola was in operation and close to one of the stations. During the measurement period, the temperature in Klosters increased from about <inline-formula><mml:math id="M55" display="inline"><mml:mrow><mml:mo>+</mml:mo><mml:mn mathvariant="normal">1.5</mml:mn></mml:mrow></mml:math></inline-formula> to <inline-formula><mml:math id="M56" display="inline"><mml:mrow><mml:mo>+</mml:mo><mml:mn mathvariant="normal">3.5</mml:mn></mml:mrow></mml:math></inline-formula> <inline-formula><mml:math id="M57" display="inline"><mml:msup><mml:mi/><mml:mo>∘</mml:mo></mml:msup></mml:math></inline-formula>C at a relative humidity of about 80 %, while the measurements on the gondola were taken mainly between 0 and <inline-formula><mml:math id="M58" display="inline"><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">2.5</mml:mn></mml:mrow></mml:math></inline-formula> <inline-formula><mml:math id="M59" display="inline"><mml:msup><mml:mi/><mml:mo>∘</mml:mo></mml:msup></mml:math></inline-formula>C at relative humidities above 95 %. The melting layer can be inferred from the dark band of the ceilometer data <xref ref-type="bibr" rid="bib1.bibx72" id="paren.63"/> below Gotschnaboden (Fig. <xref ref-type="fig" rid="Ch1.F3"/>b), which was measured above Klosters and may slightly differ at Gotschnaboden.</p>
      <p id="d1e1184">The main wind direction was from north to northeast as can be inferred from the wind profiler measurements taken in Wolfgang in Fig. <xref ref-type="fig" rid="Ch1.F4"/>b. However, the ground measurements show that the wind direction was strongly influenced by the orography (Fig. <xref ref-type="fig" rid="Ch1.F4"/>a). The valley north of the Gotschnabahn forced the wind direction to the northwest as can be seen from the wind measurements 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.</p>

      <?xmltex \floatpos{t}?><fig id="Ch1.F4" specific-use="star"><?xmltex \currentcnt{4}?><?xmltex \def\figurename{Figure}?><label>Figure 4</label><caption><p id="d1e1193">The wind direction and speed at different weather stations averaged between 08:00 and 10:00 UTC in the Davos region are shown in <bold>(a)</bold>. The colors of the wind barbs show the wind speed according to the color bar, while stations with black lines only report wind direction. The wind profiler data, measured in Wolfgang, are shown in <bold>(b)</bold>. The elevation data were taken from the digital height model DHM25 of the Federal Office of Topography swisstopo: <uri>https://shop.swisstopo.admin.ch/de/products/height_models/dhm25200</uri> (last access: 9 March 2020).</p></caption>
          <?xmltex \igopts{width=341.433071pt}?><graphic xlink:href="https://acp.copernicus.org/articles/21/3855/2021/acp-21-3855-2021-f04.png"/>

        </fig>

      <?pagebreak page3860?><p id="d1e1211">The slope inclination in the wind direction measured at Gotschnagrat is about 7<inline-formula><mml:math id="M60" display="inline"><mml:msup><mml:mi/><mml:mo>∘</mml:mo></mml:msup></mml:math></inline-formula> (averaged over Gotschnagrat and 1 km northwest). Assuming that the measured horizontal wind of about 5 m s<inline-formula><mml:math id="M61" display="inline"><mml:msup><mml:mi/><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula> (Fig. <xref ref-type="fig" rid="Ch1.F4"/>) was blown up the slope without friction, the vertical wind speed reached about 0.6 m s<inline-formula><mml:math id="M62" display="inline"><mml:msup><mml:mi/><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula>. The eddy dissipation rate, which is a measure for turbulence, was calculated from the wind profiler and cloud radar measurements as described in <xref ref-type="bibr" rid="bib1.bibx22" id="text.64"/> and reached values between 20 and 90 cm<inline-formula><mml:math id="M63" display="inline"><mml:msup><mml:mi/><mml:mn mathvariant="normal">2</mml:mn></mml:msup></mml:math></inline-formula> s<inline-formula><mml:math id="M64" display="inline"><mml:msup><mml:mi/><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">3</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula> at an altitude of 1800 to 2300 m between  08:00 and 10:00 UTC over Wolfgang. Such values were reported in stratiform clouds <xref ref-type="bibr" rid="bib1.bibx8" id="paren.65"/> as well as in cumulus clouds with weak updrafts <xref ref-type="bibr" rid="bib1.bibx75" id="paren.66"/>. Since no measurements of turbulence are available over Klosters, we assume that similar turbulence is present at the measurement site.</p>

      <?xmltex \floatpos{t}?><fig id="Ch1.F5" specific-use="star"><?xmltex \currentcnt{5}?><?xmltex \def\figurename{Figure}?><label>Figure 5</label><caption><p id="d1e1282">The figure shows a randomly selected sample of ice crystals, which were observed in the cloud together with a stacked size distribution of the different habits. The total concentrations together with the uncertainties in each habit are shown in the legend of the histogram. Unidentified crystals as well as plates smaller than 93 <inline-formula><mml:math id="M65" display="inline"><mml:mrow class="unit"><mml:mi mathvariant="normal">µ</mml:mi></mml:mrow></mml:math></inline-formula>m in their major axis are referred to as “small plates” as discussed (see text for more details).</p></caption>
          <?xmltex \igopts{width=398.338583pt}?><graphic xlink:href="https://acp.copernicus.org/articles/21/3855/2021/acp-21-3855-2021-f05.png"/>

        </fig>

      <?xmltex \floatpos{t}?><fig id="Ch1.F6" specific-use="star"><?xmltex \currentcnt{6}?><?xmltex \def\figurename{Figure}?><label>Figure 6</label><caption><p id="d1e1302">The ICNC, CDNC, CDNC for droplets larger than 40 <inline-formula><mml:math id="M66" display="inline"><mml:mrow class="unit"><mml:mi mathvariant="normal">µ</mml:mi></mml:mrow></mml:math></inline-formula>m, and the concentration of plates averaged over each of the nine measurement rides are shown with the four different symbols. The corresponding lines show the mean concentration of the complete measurement volume with the shaded areas as the uncertainty in the mean concentrations.</p></caption>
          <?xmltex \igopts{width=312.980315pt}?><graphic xlink:href="https://acp.copernicus.org/articles/21/3855/2021/acp-21-3855-2021-f06.png"/>

        </fig>

</sec>
<sec id="Ch1.S3.SS2">
  <label>3.2</label><title>In situ measurements</title>
      <?pagebreak page3861?><p id="d1e1327">Between 08:00 and 10:00 UTC, a total volume of 43.7 L was measured by HOLIMO, distributed over nine uphill rides, each contributing between 2.6 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. <xref ref-type="fig" rid="Ch1.F5"/>). About 35 % 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 Sect. <xref ref-type="sec" rid="Ch1.S4.SS1"/>, unidentified particles as well as plates smaller than 93 <inline-formula><mml:math id="M67" display="inline"><mml:mrow class="unit"><mml:mi mathvariant="normal">µ</mml:mi></mml:mrow></mml:math></inline-formula>m are referred to as small plates. They are marked with dots in the histogram plot of the ice crystal size distribution in Fig. <xref ref-type="fig" rid="Ch1.F5"/>.</p>
      <?pagebreak page3862?><p id="d1e1344">The concentrations of ice crystals and cloud droplets stayed relatively constant over the measurement period (see Fig. <xref ref-type="fig" rid="Ch1.F6"/>). Taking the whole measurement period into account, a mean cloud droplet number concentration (CDNC) of 156 cm<inline-formula><mml:math id="M68" display="inline"><mml:msup><mml:mi/><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">3</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula> <inline-formula><mml:math id="M69" display="inline"><mml:mo>±</mml:mo></mml:math></inline-formula> 9 cm<inline-formula><mml:math id="M70" display="inline"><mml:msup><mml:mi/><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">3</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula> and a mean ICNC of 6 L<inline-formula><mml:math id="M71" display="inline"><mml:msup><mml:mi/><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula> <inline-formula><mml:math id="M72" display="inline"><mml:mo>±</mml:mo></mml:math></inline-formula> 1 L<inline-formula><mml:math id="M73" display="inline"><mml:msup><mml:mi/><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula> were measured. Between the different rides the CDNC varied between 67 and 215 cm<inline-formula><mml:math id="M74" display="inline"><mml:msup><mml:mi/><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">3</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula> and the ICNC between 4 and 18 L<inline-formula><mml:math id="M75" display="inline"><mml:msup><mml:mi/><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula>. The most commonly observed pristine ice particle habit was plates, with a mean concentration of 0.8 L<inline-formula><mml:math id="M76" display="inline"><mml:msup><mml:mi/><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula> <inline-formula><mml:math id="M77" display="inline"><mml:mo>±</mml:mo></mml:math></inline-formula> 0.2 L<inline-formula><mml:math id="M78" display="inline"><mml:msup><mml:mi/><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula>, varying between 0.1 and 1.5 L<inline-formula><mml:math id="M79" display="inline"><mml:msup><mml:mi/><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula> 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 <inline-formula><mml:math id="M80" display="inline"><mml:mrow class="unit"><mml:mi mathvariant="normal">µ</mml:mi></mml:mrow></mml:math></inline-formula>m in diameter of 0.3 L<inline-formula><mml:math id="M81" display="inline"><mml:msup><mml:mi/><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula> <inline-formula><mml:math id="M82" display="inline"><mml:mo>±</mml:mo></mml:math></inline-formula> 0.1 L<inline-formula><mml:math id="M83" display="inline"><mml:msup><mml:mi/><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula>, varying between 0 and 0.8 L<inline-formula><mml:math id="M84" display="inline"><mml:msup><mml:mi/><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula> between the different rides, 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<inline-formula><mml:math id="M85" display="inline"><mml:msup><mml:mi/><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula>, one observed particle results in a particle concentration of 0.4 L<inline-formula><mml:math id="M86" display="inline"><mml:msup><mml:mi/><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula>. In the following, we consider the whole measurement period and site as one measurement volume, where the cloud particle concentrations are constant.</p>
      <p id="d1e1556">The INPC was measured in the valley in Wolfgang (1630 m) and on the mountain top of Weissfluhjoch (2670 m) (see Fig. <xref ref-type="fig" rid="Ch1.F1"/> for the geographical positions) simultaneously at 08:00 and 10:00 UTC on 22 February 2019. No INP was detected for temperatures higher than <inline-formula><mml:math id="M87" display="inline"><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">6</mml:mn></mml:mrow></mml:math></inline-formula> <inline-formula><mml:math id="M88" display="inline"><mml:msup><mml:mi/><mml:mo>∘</mml:mo></mml:msup></mml:math></inline-formula>C at both sites. Therefore, the upper estimate of the INPC at the measurement site (<inline-formula><mml:math id="M89" display="inline"><mml:mrow><mml:mi>T</mml:mi><mml:mo>&gt;</mml:mo><mml:mo>-</mml:mo><mml:mn mathvariant="normal">3</mml:mn></mml:mrow></mml:math></inline-formula> <inline-formula><mml:math id="M90" display="inline"><mml:msup><mml:mi/><mml:mo>∘</mml:mo></mml:msup></mml:math></inline-formula>C) is equal to the detection limit and on the order of <inline-formula><mml:math id="M91" display="inline"><mml:mrow><mml:mn mathvariant="normal">5</mml:mn><mml:mo>×</mml:mo><mml:msup><mml:mn mathvariant="normal">10</mml:mn><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">4</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:math></inline-formula> L<inline-formula><mml:math id="M92" display="inline"><mml:msup><mml:mi/><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula> for both sites, which lies several orders below the measured ICNC.</p>
</sec>
</sec>
<sec id="Ch1.S4">
  <label>4</label><title>Discussion</title>
<sec id="Ch1.S4.SS1">
  <label>4.1</label><title>Estimation of the secondary-ice crystal number concentration</title>
      <p id="d1e1651">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 <inline-formula><mml:math id="M93" display="inline"><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">3</mml:mn></mml:mrow></mml:math></inline-formula> <inline-formula><mml:math id="M94" display="inline"><mml:msup><mml:mi/><mml:mo>∘</mml:mo></mml:msup></mml:math></inline-formula>C. In this temperature regime, newly formed ice crystals grow into plates <xref ref-type="bibr" rid="bib1.bibx51 bib1.bibx2" id="paren.67"/>. Figure <xref ref-type="fig" rid="Ch1.F5"/> shows the shapes and the size distribution of the ice crystals divided logarithmically into nine size bins. The histogram plot shows that the majority of the classified particles smaller than 93 <inline-formula><mml:math id="M95" display="inline"><mml:mrow class="unit"><mml:mi mathvariant="normal">µ</mml:mi></mml:mrow></mml:math></inline-formula>m were plates (54 % if unidentified particles are excluded; Fig. <xref ref-type="fig" rid="Ch1.F5"/>), while aged crystals can only be found above that size. Hence, we take 93 <inline-formula><mml:math id="M96" display="inline"><mml:mrow class="unit"><mml:mi mathvariant="normal">µ</mml:mi></mml:mrow></mml:math></inline-formula>m as a threshold to divide between newly formed ice crystals and those which sedimented from above.</p>
      <p id="d1e1697">About 70 % of the ice crystals smaller than 93 <inline-formula><mml:math id="M97" display="inline"><mml:mrow class="unit"><mml:mi mathvariant="normal">µ</mml:mi></mml:mrow></mml:math></inline-formula>m (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. <xref ref-type="fig" rid="Ch1.F5"/>) 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”.</p>
      <p id="d1e1710">A plate with a maximum dimension of 93 <inline-formula><mml:math id="M98" display="inline"><mml:mrow class="unit"><mml:mi mathvariant="normal">µ</mml:mi></mml:mrow></mml:math></inline-formula>m has a fall velocity of about 0.05 m s<inline-formula><mml:math id="M99" display="inline"><mml:msup><mml:mi/><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula> (using the equations given in <xref ref-type="bibr" rid="bib1.bibx66" id="altparen.68"/>), which is 1 order of magnitude lower than the estimated updraft of about 0.6 m s<inline-formula><mml:math id="M100" display="inline"><mml:msup><mml:mi/><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula>. 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. <xref ref-type="fig" rid="Ch1.F5"/>), which is typical for columns growing at temperatures just below <inline-formula><mml:math id="M101" display="inline"><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">3</mml:mn></mml:mrow></mml:math></inline-formula> <inline-formula><mml:math id="M102" display="inline"><mml:msup><mml:mi/><mml:mo>∘</mml:mo></mml:msup></mml:math></inline-formula>C at low supersaturation <xref ref-type="bibr" rid="bib1.bibx51 bib1.bibx2" id="paren.69"/>, and therefore they must have sedimented from slightly higher altitudes. This is possible because columns have a higher fall velocity than plates (a column with a length of 93 <inline-formula><mml:math id="M103" display="inline"><mml:mrow class="unit"><mml:mi mathvariant="normal">µ</mml:mi></mml:mrow></mml:math></inline-formula>m falls with about 0.3 m s<inline-formula><mml:math id="M104" display="inline"><mml:msup><mml:mi/><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula> using the equations given in <xref ref-type="bibr" rid="bib1.bibx66" id="altparen.70"/>). About 10 % 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<inline-formula><mml:math id="M105" display="inline"><mml:msup><mml:mi/><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula> <inline-formula><mml:math id="M106" display="inline"><mml:mo>±</mml:mo></mml:math></inline-formula> 0.6 L<inline-formula><mml:math id="M107" display="inline"><mml:msup><mml:mi/><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula>.</p>
      <p id="d1e1828">The time a plate needs to grow to 93 <inline-formula><mml:math id="M108" display="inline"><mml:mrow class="unit"><mml:mi mathvariant="normal">µ</mml:mi></mml:mrow></mml:math></inline-formula>m is hard to assess at temperatures close to 0 <inline-formula><mml:math id="M109" display="inline"><mml:msup><mml:mi/><mml:mo>∘</mml:mo></mml:msup></mml:math></inline-formula>C because small variations in the environmental conditions, e.g., a temperature fluctuation of only 0.5 <inline-formula><mml:math id="M110" display="inline"><mml:msup><mml:mi/><mml:mo>∘</mml:mo></mml:msup></mml:math></inline-formula>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 <xref ref-type="bibr" rid="bib1.bibx17" id="text.71"/>. Using the mass–size relation of hexagonal plates from <xref ref-type="bibr" rid="bib1.bibx57" id="text.72"/> and assuming water vapor saturation over liquid at a temperature of <inline-formula><mml:math id="M111" display="inline"><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">2</mml:mn></mml:mrow></mml:math></inline-formula> <inline-formula><mml:math id="M112" display="inline"><mml:msup><mml:mi/><mml:mo>∘</mml:mo></mml:msup></mml:math></inline-formula>C, a splinter of 5 <inline-formula><mml:math id="M113" display="inline"><mml:mrow class="unit"><mml:mi mathvariant="normal">µ</mml:mi></mml:mrow></mml:math></inline-formula>m needs about 9 min to grow to a size of 93 <inline-formula><mml:math id="M114" display="inline"><mml:mrow class="unit"><mml:mi mathvariant="normal">µ</mml:mi></mml:mrow></mml:math></inline-formula>m and about 5 min to grow to a size of 60 <inline-formula><mml:math id="M115" display="inline"><mml:mrow class="unit"><mml:mi mathvariant="normal">µ</mml:mi></mml:mrow></mml:math></inline-formula>m. Thus, for any given time, a plate of 93 <inline-formula><mml:math id="M116" display="inline"><mml:mrow class="unit"><mml:mi mathvariant="normal">µ</mml:mi></mml:mrow></mml:math></inline-formula>m is 4 min older than a plate of 60 <inline-formula><mml:math id="M117" display="inline"><mml:mrow class="unit"><mml:mi mathvariant="normal">µ</mml:mi></mml:mrow></mml:math></inline-formula>m if the temperature is constant at <inline-formula><mml:math id="M118" display="inline"><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">2</mml:mn></mml:mrow></mml:math></inline-formula> <inline-formula><mml:math id="M119" display="inline"><mml:msup><mml:mi/><mml:mo>∘</mml:mo></mml:msup></mml:math></inline-formula>C, and the water vapor is saturated over water. Therefore, plates with sizes between 60 and 93 <inline-formula><mml:math id="M120" display="inline"><mml:mrow class="unit"><mml:mi mathvariant="normal">µ</mml:mi></mml:mrow></mml:math></inline-formula>m must have newly formed by SIP within a time span of 4 min. The same calculation can be done for a bigger size interval, e.g., plates between 39 and 93 <inline-formula><mml:math id="M121" display="inline"><mml:mrow class="unit"><mml:mi mathvariant="normal">µ</mml:mi></mml:mrow></mml:math></inline-formula>m, which must have newly formed by SIP during a time span of about 6 min. Taking temperature variations in the measurement volume into account, i.e., varying the temperature between <inline-formula><mml:math id="M122" display="inline"><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">2</mml:mn></mml:mrow></mml:math></inline-formula> and <inline-formula><mml:math id="M123" display="inline"><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:mrow></mml:math></inline-formula> <inline-formula><mml:math id="M124" display="inline"><mml:msup><mml:mi/><mml:mo>∘</mml:mo></mml:msup></mml:math></inline-formula>C, plates between 60 and 93 <inline-formula><mml:math id="M125" display="inline"><mml:mrow class="unit"><mml:mi mathvariant="normal">µ</mml:mi></mml:mrow></mml:math></inline-formula>m were newly formed by SIP during a time span of 4 to 6 min, while plates between 39 and 93 <inline-formula><mml:math id="M126" display="inline"><mml:mrow class="unit"><mml:mi mathvariant="normal">µ</mml:mi></mml:mrow></mml:math></inline-formula>m were newly formed by SIP during a time span of 6 to 10 min depending on the temperature. The observed concentration of small plates with sizes between 60 and 93 <inline-formula><mml:math id="M127" display="inline"><mml:mrow class="unit"><mml:mi mathvariant="normal">µ</mml:mi></mml:mrow></mml:math></inline-formula>m was 0.9 L<inline-formula><mml:math id="M128" display="inline"><mml:msup><mml:mi/><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula> <inline-formula><mml:math id="M129" display="inline"><mml:mo>±</mml:mo></mml:math></inline-formula> 0.3 L<inline-formula><mml:math id="M130" display="inline"><mml:msup><mml:mi/><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula>, whereas between 39 and 93 <inline-formula><mml:math id="M131" display="inline"><mml:mrow class="unit"><mml:mi mathvariant="normal">µ</mml:mi></mml:mrow></mml:math></inline-formula>m the concentration of small plates was 2.1 L<inline-formula><mml:math id="M132" display="inline"><mml:msup><mml:mi/><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula> <inline-formula><mml:math id="M133" display="inline"><mml:mo>±</mml:mo></mml:math></inline-formula> 0.5 L<inline-formula><mml:math id="M134" display="inline"><mml:msup><mml:mi/><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula> averaged over the 2 h measurement period. Thus, 0.9 L<inline-formula><mml:math id="M135" display="inline"><mml:msup><mml:mi/><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula> <inline-formula><mml:math id="M136" display="inline"><mml:mo>±</mml:mo></mml:math></inline-formula> 0.3 L<inline-formula><mml:math id="M137" display="inline"><mml:msup><mml:mi/><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula> of secondary ice has formed within 4 to 6 min and 2.1 L<inline-formula><mml:math id="M138" display="inline"><mml:msup><mml:mi/><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula> <inline-formula><mml:math id="M139" display="inline"><mml:mo>±</mml:mo></mml:math></inline-formula> 0.5 L<inline-formula><mml:math id="M140" display="inline"><mml:msup><mml:mi/><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula> within 6 to 10 min. Taking all named uncertainties into account, the rate of secondary-ice production during our case study was 0.24 L<inline-formula><mml:math id="M141" display="inline"><mml:msup><mml:mi/><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula> min<inline-formula><mml:math id="M142" display="inline"><mml:msup><mml:mi/><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula> <inline-formula><mml:math id="M143" display="inline"><mml:mo>±</mml:mo></mml:math></inline-formula> 0.09 L<inline-formula><mml:math id="M144" display="inline"><mml:msup><mml:mi/><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula> min<inline-formula><mml:math id="M145" display="inline"><mml:msup><mml:mi/><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula>.</p>
</sec>
<sec id="Ch1.S4.SS2">
  <label>4.2</label><title>Contribution of different SIP mechanisms</title>
      <?pagebreak page3863?><p id="d1e2211">Droplets larger than 40 <inline-formula><mml:math id="M146" display="inline"><mml:mrow class="unit"><mml:mi mathvariant="normal">µ</mml:mi></mml:mrow></mml:math></inline-formula>m up to a size of 380 <inline-formula><mml:math id="M147" display="inline"><mml:mrow class="unit"><mml:mi mathvariant="normal">µ</mml:mi></mml:mrow></mml:math></inline-formula>m in diameter were observed between 0 and <inline-formula><mml:math id="M148" display="inline"><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">2.7</mml:mn></mml:mrow></mml:math></inline-formula> <inline-formula><mml:math id="M149" display="inline"><mml:msup><mml:mi/><mml:mo>∘</mml:mo></mml:msup></mml:math></inline-formula>C. <xref ref-type="bibr" rid="bib1.bibx43" id="text.73"/> explained the occurrence of large droplets just above the melting layer with a recirculation process (see Sect. <xref ref-type="sec" rid="Ch1.S1.SS2"/>). Ice particles fall through the melting layer and melt into drizzle drops, which can be reintroduced into the cloud by sufficiently high updrafts. The estimated updraft in this case study is about 0.6 m s<inline-formula><mml:math id="M150" display="inline"><mml:msup><mml:mi/><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula>, which is equal to the fall speed of a 150 <inline-formula><mml:math id="M151" display="inline"><mml:mrow class="unit"><mml:mi mathvariant="normal">µ</mml:mi></mml:mrow></mml:math></inline-formula>m droplet <xref ref-type="bibr" rid="bib1.bibx71" id="paren.74"/>. Most of the observed droplets were smaller than 150 <inline-formula><mml:math id="M152" display="inline"><mml:mrow class="unit"><mml:mi mathvariant="normal">µ</mml:mi></mml:mrow></mml:math></inline-formula>m and could be lifted up in the clouds by the updraft, while the remaining ones could have been 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<inline-formula><mml:math id="M153" display="inline"><mml:msup><mml:mi/><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula> is already enough to lift 86 % instead of 79 % of the observed droplets.</p>
      <p id="d1e2298">Droplet fragmentation requires the presence of large droplets (<inline-formula><mml:math id="M154" display="inline"><mml:mrow><mml:mo>&gt;</mml:mo><mml:mo>≈</mml:mo><mml:mn mathvariant="normal">40</mml:mn></mml:mrow></mml:math></inline-formula> <inline-formula><mml:math id="M155" display="inline"><mml:mrow class="unit"><mml:mi mathvariant="normal">µ</mml:mi></mml:mrow></mml:math></inline-formula>m; <xref ref-type="bibr" rid="bib1.bibx50 bib1.bibx43" id="altparen.75"><named-content content-type="pre">e.g.,</named-content></xref>) 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 it 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) is evidence 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.</p>
      <p id="d1e2326">The rime-splintering process and ice fragmentation during thermal shock can both be excluded from being active because they require lower temperatures. Additionally, INP activation in transient supersaturation around freezing drops is unlikely to be active in a relatively warm environment <xref ref-type="bibr" rid="bib1.bibx43" id="paren.76"/>. As MPCs are supersaturated with respect to ice, the requirements for ice fragmentation during sublimation are also not fulfilled <xref ref-type="bibr" rid="bib1.bibx43" id="paren.77"/>.</p>
      <p id="d1e2335">Ice crystals can partially melt and create fragments, which can be lifted back into the cloud. However, the ice crystals have to be sufficiently small to be lifted, while at the same time, small ice crystals are likely to completely melt before re-entering the cloud. Furthermore, the breakup rates of this process depend on temperature and humidity and largely on the initial shape of the ice crystals <xref ref-type="bibr" rid="bib1.bibx42" id="paren.78"/>. <xref ref-type="bibr" rid="bib1.bibx63" id="text.79"/> observed no sublimation breakup for columnar and plate-like crystals and breakup during melting only at relative humidities below 90 %. The shapes of ice crystals in our study are mostly solid particles that have columnar and plate-like shapes (Fig. <xref ref-type="fig" rid="Ch1.F5"/>), and the relative humidity on the gondola never dropped below 95 % (Fig. <xref ref-type="fig" rid="Ch1.F3"/>d). Taking all these considerations into account, we assume that ice fragmentation during melting has a negligible effect on the SIP.</p>
      <p id="d1e2349">Therefore, we expect that most of the small plates emerged from droplet fragmentation and that the orographically induced updrafts serve as a constant supply of new droplets larger than 40 <inline-formula><mml:math id="M156" display="inline"><mml:mrow class="unit"><mml:mi mathvariant="normal">µ</mml:mi></mml:mrow></mml:math></inline-formula>m in diameter, which originate from melted ice crystals. A schematic of this recirculation process in mountainous regions is shown in Fig. <xref ref-type="fig" rid="Ch1.F7"/>. 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 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 microphysics.</p>

      <?xmltex \floatpos{t}?><fig id="Ch1.F7"><?xmltex \currentcnt{7}?><?xmltex \def\figurename{Figure}?><label>Figure 7</label><caption><p id="d1e2364">Schematic of the recirculation process, which was introduced by <xref ref-type="bibr" rid="bib1.bibx43" id="text.80"/> and adapted for an updraft created by a mountain slope. Aged ice crystals sediment from the cloud and turn into drizzle-sized drops when they fall through the melting layer. These drops are reintroduced into the cloud by wind forced up the mountain slope (updraft), where they freeze when they collide with aged ice crystals and fragment upon freezing. The small fragments grow into plates due to the environmental conditions (<inline-formula><mml:math id="M157" display="inline"><mml:mrow><mml:mi>T</mml:mi><mml:mo>&gt;</mml:mo><mml:mo>-</mml:mo><mml:mn mathvariant="normal">3</mml:mn></mml:mrow></mml:math></inline-formula> <inline-formula><mml:math id="M158" display="inline"><mml:msup><mml:mi/><mml:mo>∘</mml:mo></mml:msup></mml:math></inline-formula>C) and may hit other droplets, which again freeze and possibly fragment.</p></caption>
          <?xmltex \igopts{width=236.157874pt}?><graphic xlink:href="https://acp.copernicus.org/articles/21/3855/2021/acp-21-3855-2021-f07.png"/>

        </fig>

</sec>
<sec id="Ch1.S4.SS3">
  <label>4.3</label><title>Parametrization of SIP by droplet fragmentation at warm temperatures</title>
      <?pagebreak page3864?><p id="d1e2407">Here we derive a parametrization of the SIP by droplet fragmentation at temperatures close to 0 <inline-formula><mml:math id="M159" display="inline"><mml:msup><mml:mi/><mml:mo>∘</mml:mo></mml:msup></mml:math></inline-formula>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 SIP. Like <xref ref-type="bibr" rid="bib1.bibx43" id="text.81"/>, we assume that only droplets larger than 40 <inline-formula><mml:math id="M160" display="inline"><mml:mrow class="unit"><mml:mi mathvariant="normal">µ</mml:mi></mml:mrow></mml:math></inline-formula>m are likely to contribute to SIP by droplet fragmentation. To calculate the splinter generation rate of a droplet with diameter <inline-formula><mml:math id="M161" display="inline"><mml:mrow><mml:msub><mml:mi>d</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> (<inline-formula><mml:math id="M162" display="inline"><mml:mrow><mml:msub><mml:mi>g</mml:mi><mml:mi mathvariant="normal">sp</mml:mi></mml:msub><mml:mo>(</mml:mo><mml:msub><mml:mi>d</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula>), the droplet freezing rate by collision (<inline-formula><mml:math id="M163" display="inline"><mml:mrow><mml:msub><mml:mi>f</mml:mi><mml:mi mathvariant="normal">col</mml:mi></mml:msub><mml:mo>(</mml:mo><mml:msub><mml:mi>d</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula>) has to be multiplied by the droplet fragmentation probability during freezing (<inline-formula><mml:math id="M164" display="inline"><mml:mrow><mml:msub><mml:mi>p</mml:mi><mml:mi mathvariant="normal">df</mml:mi></mml:msub><mml:mo>(</mml:mo><mml:msub><mml:mi>d</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula>) and the number of splinters per fragmenting droplet (<inline-formula><mml:math id="M165" display="inline"><mml:mrow><mml:msub><mml:mi>N</mml:mi><mml:mi mathvariant="normal">sp</mml:mi></mml:msub><mml:mo>(</mml:mo><mml:msub><mml:mi>d</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula>):
            <disp-formula id="Ch1.E1" content-type="numbered"><label>1</label><mml:math id="M166" display="block"><mml:mrow><mml:msub><mml:mi>g</mml:mi><mml:mi mathvariant="normal">sp</mml:mi></mml:msub><mml:mfenced open="(" close=")"><mml:mrow><mml:msub><mml:mi>d</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:mrow></mml:mfenced><mml:mo>=</mml:mo><mml:msub><mml:mi>f</mml:mi><mml:mi mathvariant="normal">col</mml:mi></mml:msub><mml:mfenced open="(" close=")"><mml:mrow><mml:msub><mml:mi>d</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:mrow></mml:mfenced><mml:mo>⋅</mml:mo><mml:msub><mml:mi>p</mml:mi><mml:mi mathvariant="normal">df</mml:mi></mml:msub><mml:mfenced close=")" open="("><mml:mrow><mml:msub><mml:mi>d</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:mrow></mml:mfenced><mml:mo>⋅</mml:mo><mml:msub><mml:mi>N</mml:mi><mml:mi mathvariant="normal">sp</mml:mi></mml:msub><mml:mfenced open="(" close=")"><mml:mrow><mml:msub><mml:mi>d</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:mrow></mml:mfenced><mml:mo>.</mml:mo></mml:mrow></mml:math></disp-formula>
          To obtain the total splinter generation rate (<inline-formula><mml:math id="M167" display="inline"><mml:mrow><mml:msub><mml:mi>G</mml:mi><mml:mi mathvariant="normal">sp</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula>) in a volume <inline-formula><mml:math id="M168" display="inline"><mml:mi>V</mml:mi></mml:math></inline-formula>, which contains <inline-formula><mml:math id="M169" display="inline"><mml:mrow><mml:msub><mml:mi>N</mml:mi><mml:mrow><mml:mi mathvariant="normal">d</mml:mi><mml:mo>&gt;</mml:mo><mml:mn mathvariant="normal">40</mml:mn><mml:mspace width="0.125em" linebreak="nobreak"/><mml:mrow class="unit"><mml:mi mathvariant="normal">µ</mml:mi></mml:mrow><mml:mi mathvariant="normal">m</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula> droplets with diameters larger than 40 <inline-formula><mml:math id="M170" display="inline"><mml:mrow class="unit"><mml:mi mathvariant="normal">µ</mml:mi></mml:mrow></mml:math></inline-formula>m, the sum of <inline-formula><mml:math id="M171" display="inline"><mml:mrow><mml:msub><mml:mi>g</mml:mi><mml:mi mathvariant="normal">sp</mml:mi></mml:msub><mml:mo>(</mml:mo><mml:msub><mml:mi>d</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula> over all droplets with diameter <inline-formula><mml:math id="M172" display="inline"><mml:mrow><mml:msub><mml:mi>d</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> (<inline-formula><mml:math id="M173" display="inline"><mml:mrow><mml:mi>n</mml:mi><mml:mo>=</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:mrow></mml:math></inline-formula>, 2, …, <inline-formula><mml:math id="M174" display="inline"><mml:mrow><mml:msub><mml:mi>N</mml:mi><mml:mrow><mml:mi mathvariant="normal">d</mml:mi><mml:mo>&gt;</mml:mo><mml:mn mathvariant="normal">40</mml:mn><mml:mspace linebreak="nobreak" width="0.125em"/><mml:mrow class="unit"><mml:mi mathvariant="normal">µ</mml:mi></mml:mrow><mml:mi mathvariant="normal">m</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula>) has to be divided by <inline-formula><mml:math id="M175" display="inline"><mml:mi>V</mml:mi></mml:math></inline-formula>:
<?xmltex \hack{\newpage}?><?xmltex \hack{\vspace*{-6mm}}?>
            <disp-formula id="Ch1.E2" content-type="numbered"><label>2</label><mml:math id="M176" display="block"><mml:mrow><mml:msub><mml:mi>G</mml:mi><mml:mi mathvariant="normal">sp</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mstyle displaystyle="true"><mml:mfrac style="display"><mml:mn mathvariant="normal">1</mml:mn><mml:mi>V</mml:mi></mml:mfrac></mml:mstyle><mml:munderover><mml:mo movablelimits="false">∑</mml:mo><mml:mrow><mml:mi>n</mml:mi><mml:mo>=</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:mrow><mml:mrow><mml:msub><mml:mi>N</mml:mi><mml:mrow><mml:mi mathvariant="normal">d</mml:mi><mml:mo>&gt;</mml:mo><mml:mn mathvariant="normal">40</mml:mn><mml:mspace width="0.125em" linebreak="nobreak"/><mml:mrow class="unit"><mml:mi mathvariant="normal">µ</mml:mi></mml:mrow><mml:mi mathvariant="normal">m</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:munderover><mml:msub><mml:mi>g</mml:mi><mml:mi mathvariant="normal">sp</mml:mi></mml:msub><mml:mfenced close=")" open="("><mml:mrow><mml:msub><mml:mi>d</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:mrow></mml:mfenced><mml:mo>.</mml:mo></mml:mrow></mml:math></disp-formula>
          The probability that a droplet with a fall velocity of <inline-formula><mml:math id="M177" display="inline"><mml:mrow><mml:mi>v</mml:mi><mml:mo>(</mml:mo><mml:msub><mml:mi>d</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula> collides with an ice crystal with an equivalent particle area diameter <inline-formula><mml:math id="M178" display="inline"><mml:mrow><mml:msub><mml:mi>a</mml:mi><mml:mi>j</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> and freezes is equal to the collision efficiency <inline-formula><mml:math id="M179" display="inline"><mml:mrow><mml:mi>E</mml:mi><mml:mo>(</mml:mo><mml:msub><mml:mi>d</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula>, <inline-formula><mml:math id="M180" display="inline"><mml:mrow><mml:msub><mml:mi>a</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula>. It needs to be multiplied by the combined cross-section of the ice crystal and the droplet (<inline-formula><mml:math id="M181" display="inline"><mml:mrow><mml:mn mathvariant="normal">0.25</mml:mn><mml:mo>⋅</mml:mo><mml:mi mathvariant="italic">π</mml:mi><mml:mo>⋅</mml:mo><mml:mo>(</mml:mo><mml:msub><mml:mi>d</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mi>a</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:msup><mml:mo>)</mml:mo><mml:mn mathvariant="normal">2</mml:mn></mml:msup></mml:mrow></mml:math></inline-formula>) and the absolute relative velocity between those two (<inline-formula><mml:math id="M182" display="inline"><mml:mrow><mml:mo>|</mml:mo><mml:mi>v</mml:mi><mml:mo>(</mml:mo><mml:mi>a</mml:mi><mml:mi>j</mml:mi><mml:mo>)</mml:mo><mml:mo>-</mml:mo><mml:mi>v</mml:mi><mml:mo>(</mml:mo><mml:msub><mml:mi>d</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo>)</mml:mo><mml:mo>|</mml:mo></mml:mrow></mml:math></inline-formula>) to obtain the collection kernel. To calculate the freezing rate of a droplet in volume <inline-formula><mml:math id="M183" display="inline"><mml:mi>V</mml:mi></mml:math></inline-formula>, the sum of the collection kernel between the droplet and each ice crystal in the same volume with an equivalent particle area diameter <inline-formula><mml:math id="M184" display="inline"><mml:mrow><mml:msub><mml:mi>a</mml:mi><mml:mi>j</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> (<inline-formula><mml:math id="M185" display="inline"><mml:mrow><mml:mi>j</mml:mi><mml:mo>=</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:mrow></mml:math></inline-formula>, 2, …, <inline-formula><mml:math id="M186" display="inline"><mml:mrow><mml:msub><mml:mi>N</mml:mi><mml:mi mathvariant="normal">ice</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula>; <inline-formula><mml:math id="M187" display="inline"><mml:mrow><mml:msub><mml:mi>N</mml:mi><mml:mi mathvariant="normal">ice</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula>: number of ice crystals) has to be divided by <inline-formula><mml:math id="M188" display="inline"><mml:mi>V</mml:mi></mml:math></inline-formula>:
            <disp-formula id="Ch1.E3" content-type="numbered"><label>3</label><mml:math id="M189" display="block"><mml:mtable class="split" columnspacing="1em" rowspacing="0.2ex" displaystyle="true" columnalign="right left"><mml:mtr><mml:mtd><mml:mrow><mml:msub><mml:mi>f</mml:mi><mml:mi mathvariant="normal">col</mml:mi></mml:msub><mml:mfenced open="(" close=")"><mml:mrow><mml:msub><mml:mi>d</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:mrow></mml:mfenced></mml:mrow></mml:mtd><mml:mtd><mml:mrow><mml:mo>≈</mml:mo><mml:mstyle displaystyle="true"><mml:mfrac style="display"><mml:mn mathvariant="normal">1</mml:mn><mml:mi>V</mml:mi></mml:mfrac></mml:mstyle><mml:munderover><mml:mo movablelimits="false">∑</mml:mo><mml:mrow><mml:mi>j</mml:mi><mml:mo>=</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:mrow><mml:mrow><mml:msub><mml:mi>N</mml:mi><mml:mi mathvariant="normal">ice</mml:mi></mml:msub></mml:mrow></mml:munderover><mml:mi>E</mml:mi><mml:mfenced close=")" open="("><mml:mrow><mml:msub><mml:mi>d</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>a</mml:mi><mml:mi>j</mml:mi></mml:msub></mml:mrow></mml:mfenced><mml:mo>⋅</mml:mo><mml:mo>|</mml:mo><mml:mi>v</mml:mi><mml:mfenced close=")" open="("><mml:mrow><mml:msub><mml:mi>a</mml:mi><mml:mi>j</mml:mi></mml:msub></mml:mrow></mml:mfenced><mml:mo>-</mml:mo><mml:mi>v</mml:mi><mml:mfenced open="(" close=")"><mml:mrow><mml:msub><mml:mi>d</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:mrow></mml:mfenced><mml:mo>|</mml:mo></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd/><mml:mtd><mml:mrow><mml:mo>⋅</mml:mo><mml:mstyle displaystyle="true"><mml:mfrac style="display"><mml:mrow><mml:mi mathvariant="italic">π</mml:mi><mml:mo>⋅</mml:mo><mml:msup><mml:mfenced open="(" close=")"><mml:mrow><mml:msub><mml:mi>d</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mi>a</mml:mi><mml:mi>j</mml:mi></mml:msub></mml:mrow></mml:mfenced><mml:mn mathvariant="normal">2</mml:mn></mml:msup></mml:mrow><mml:mn mathvariant="normal">4</mml:mn></mml:mfrac></mml:mstyle><mml:mo>.</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:math></disp-formula>
          The probability of droplet fragmentation is size-dependent <xref ref-type="bibr" rid="bib1.bibx83 bib1.bibx39 bib1.bibx46" id="paren.82"/>. For a fragmentation to occur, the surface energy has to be overcome, which is proportional to <inline-formula><mml:math id="M190" display="inline"><mml:mrow><mml:msubsup><mml:mi>d</mml:mi><mml:mi>n</mml:mi><mml:mn mathvariant="normal">2</mml:mn></mml:msubsup></mml:mrow></mml:math></inline-formula>. Therefore, we expect <inline-formula><mml:math id="M191" display="inline"><mml:mrow><mml:msub><mml:mi>p</mml:mi><mml:mi mathvariant="normal">df</mml:mi></mml:msub><mml:mo>(</mml:mo><mml:msub><mml:mi>d</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo>)</mml:mo><mml:mo>=</mml:mo><mml:mi>c</mml:mi><mml:mo>⋅</mml:mo><mml:msubsup><mml:mi>d</mml:mi><mml:mi>n</mml:mi><mml:mn mathvariant="normal">2</mml:mn></mml:msubsup></mml:mrow></mml:math></inline-formula> for some constant <inline-formula><mml:math id="M192" display="inline"><mml:mi>c</mml:mi></mml:math></inline-formula>. The constant <inline-formula><mml:math id="M193" display="inline"><mml:mi>c</mml:mi></mml:math></inline-formula> can be estimated by assuming a fragmentation probability of 40 % for droplets with <inline-formula><mml:math id="M194" display="inline"><mml:mrow><mml:msub><mml:mi>d</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mn mathvariant="normal">300</mml:mn></mml:mrow></mml:math></inline-formula> <inline-formula><mml:math id="M195" display="inline"><mml:mrow class="unit"><mml:mi mathvariant="normal">µ</mml:mi></mml:mrow></mml:math></inline-formula>m at temperatures higher than <inline-formula><mml:math id="M196" display="inline"><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">2.5</mml:mn></mml:mrow></mml:math></inline-formula> <inline-formula><mml:math id="M197" display="inline"><mml:msup><mml:mi/><mml:mo>∘</mml:mo></mml:msup></mml:math></inline-formula>C, i.e., <inline-formula><mml:math id="M198" display="inline"><mml:mrow><mml:msub><mml:mi>p</mml:mi><mml:mi mathvariant="normal">df</mml:mi></mml:msub><mml:mo>(</mml:mo><mml:msub><mml:mi>d</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mn mathvariant="normal">300</mml:mn><mml:mspace linebreak="nobreak" width="0.125em"/><mml:mrow class="unit"><mml:mi mathvariant="normal">µ</mml:mi></mml:mrow><mml:mi mathvariant="normal">m</mml:mi><mml:mo>)</mml:mo><mml:mo>=</mml:mo><mml:mn mathvariant="normal">0.4</mml:mn></mml:mrow></mml:math></inline-formula>, as measured by <xref ref-type="bibr" rid="bib1.bibx34" id="text.83"/>. This yields a droplet fragmentation probability per freezing droplet of
            <disp-formula id="Ch1.E4" content-type="numbered"><label>4</label><mml:math id="M199" display="block"><mml:mrow><mml:msub><mml:mi>p</mml:mi><mml:mi mathvariant="normal">df</mml:mi></mml:msub><mml:mfenced close=")" open="("><mml:mrow><mml:msub><mml:mi>d</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:mrow></mml:mfenced><mml:mo>≈</mml:mo><mml:mn mathvariant="normal">4.4</mml:mn><mml:mo>×</mml:mo><mml:msup><mml:mn mathvariant="normal">10</mml:mn><mml:mn mathvariant="normal">6</mml:mn></mml:msup><mml:mspace linebreak="nobreak" width="0.125em"/><mml:msup><mml:mi mathvariant="normal">m</mml:mi><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">2</mml:mn></mml:mrow></mml:msup><mml:mo>⋅</mml:mo><mml:msubsup><mml:mi>d</mml:mi><mml:mi>n</mml:mi><mml:mn mathvariant="normal">2</mml:mn></mml:msubsup><mml:mo>,</mml:mo></mml:mrow></mml:math></disp-formula>
          which differs from other studies, where the probability of droplet fragmentation follows a normal distribution centered at <inline-formula><mml:math id="M200" display="inline"><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">15</mml:mn></mml:mrow></mml:math></inline-formula> <inline-formula><mml:math id="M201" display="inline"><mml:msup><mml:mi/><mml:mo>∘</mml:mo></mml:msup></mml:math></inline-formula>C independent of droplet size, with a probability close to 0 % for slightly sub-zero temperatures <xref ref-type="bibr" rid="bib1.bibx82 bib1.bibx81 bib1.bibx77" id="paren.84"><named-content content-type="pre">e.g.,</named-content></xref>. However, the study of <xref ref-type="bibr" rid="bib1.bibx34" id="text.85"/> showed that droplet fragmentation is also likely to happen at temperatures close to 0 <inline-formula><mml:math id="M202" display="inline"><mml:msup><mml:mi/><mml:mo>∘</mml:mo></mml:msup></mml:math></inline-formula>C.</p>
      <p id="d1e3308">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 <xref ref-type="bibr" rid="bib1.bibx46 bib1.bibx34" id="paren.86"/>. Here we consider the maximum number of splinters observed during a breakup event for different droplet sizes, which were summarized by <xref ref-type="bibr" rid="bib1.bibx46" id="text.87"/> from different studies, to be the best estimate available. The data points suggest a linear correlation between the number of splinters being produced per droplet <inline-formula><mml:math id="M203" display="inline"><mml:mrow><mml:msub><mml:mi>N</mml:mi><mml:mi mathvariant="normal">sp</mml:mi></mml:msub><mml:mo>(</mml:mo><mml:msub><mml:mi>d</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula> and the droplet diameter <inline-formula><mml:math id="M204" display="inline"><mml:mrow><mml:msub><mml:mi>d</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula>. Applying a linear regression, <inline-formula><mml:math id="M205" display="inline"><mml:mrow><mml:msub><mml:mi>N</mml:mi><mml:mi mathvariant="normal">sp</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> can be estimated as
<?xmltex \hack{\newpage}?><?xmltex \hack{\vspace*{-6mm}}?>
            <disp-formula id="Ch1.E5" content-type="numbered"><label>5</label><mml:math id="M206" display="block"><mml:mrow><mml:msub><mml:mi>N</mml:mi><mml:mi mathvariant="normal">sp</mml:mi></mml:msub><mml:mfenced open="(" close=")"><mml:mrow><mml:msub><mml:mi>d</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:mrow></mml:mfenced><mml:mo>≈</mml:mo><mml:mn mathvariant="normal">9</mml:mn><mml:mo>×</mml:mo><mml:msup><mml:mn mathvariant="normal">10</mml:mn><mml:mn mathvariant="normal">4</mml:mn></mml:msup><mml:mspace width="0.125em" linebreak="nobreak"/><mml:msup><mml:mi mathvariant="normal">m</mml:mi><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:mrow></mml:msup><mml:mo>⋅</mml:mo><mml:msub><mml:mi>d</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo>.</mml:mo></mml:mrow></mml:math></disp-formula>
          The splinter generation rate in volume <inline-formula><mml:math id="M207" display="inline"><mml:mi>V</mml:mi></mml:math></inline-formula> at temperatures close to 0 <inline-formula><mml:math id="M208" display="inline"><mml:msup><mml:mi/><mml:mo>∘</mml:mo></mml:msup></mml:math></inline-formula>C in MPCs is subsequently given by
            <disp-formula id="Ch1.E6" content-type="numbered"><label>6</label><mml:math id="M209" display="block"><mml:mtable class="split" columnspacing="1em" rowspacing="0.2ex" displaystyle="true" columnalign="right left"><mml:mtr><mml:mtd><mml:mrow><mml:msub><mml:mi>G</mml:mi><mml:mi mathvariant="normal">sp</mml:mi></mml:msub></mml:mrow></mml:mtd><mml:mtd><mml:mrow><mml:mo>≈</mml:mo><mml:mn mathvariant="normal">9.9</mml:mn><mml:mo>×</mml:mo><mml:msup><mml:mn mathvariant="normal">10</mml:mn><mml:mn mathvariant="normal">10</mml:mn></mml:msup><mml:mo>⋅</mml:mo><mml:mi mathvariant="italic">π</mml:mi><mml:mspace width="0.125em" linebreak="nobreak"/><mml:msup><mml:mi mathvariant="normal">m</mml:mi><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">3</mml:mn></mml:mrow></mml:msup><mml:mo>⋅</mml:mo><mml:mstyle displaystyle="true"><mml:mfrac style="display"><mml:mn mathvariant="normal">1</mml:mn><mml:mrow><mml:msup><mml:mi>V</mml:mi><mml:mn mathvariant="normal">2</mml:mn></mml:msup></mml:mrow></mml:mfrac></mml:mstyle><mml:munderover><mml:mo movablelimits="false">∑</mml:mo><mml:mrow><mml:mi>n</mml:mi><mml:mo>=</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:mrow><mml:mrow><mml:msub><mml:mi>N</mml:mi><mml:mrow><mml:mi mathvariant="normal">d</mml:mi><mml:mo>&gt;</mml:mo><mml:mn mathvariant="normal">40</mml:mn><mml:mspace width="0.125em" linebreak="nobreak"/><mml:mrow class="unit"><mml:mi mathvariant="normal">µ</mml:mi></mml:mrow><mml:mi mathvariant="normal">m</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:munderover><mml:msubsup><mml:mi>d</mml:mi><mml:mi>n</mml:mi><mml:mn mathvariant="normal">3</mml:mn></mml:msubsup><mml:munderover><mml:mo movablelimits="false">∑</mml:mo><mml:mrow><mml:mi>j</mml:mi><mml:mo>=</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:mrow><mml:mrow><mml:msub><mml:mi>N</mml:mi><mml:mi mathvariant="normal">ice</mml:mi></mml:msub></mml:mrow></mml:munderover><mml:mi>E</mml:mi><mml:mfenced close=")" open="("><mml:mrow><mml:msub><mml:mi>d</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>a</mml:mi><mml:mi>j</mml:mi></mml:msub></mml:mrow></mml:mfenced></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd/><mml:mtd><mml:mrow><mml:mo>⋅</mml:mo><mml:mo>|</mml:mo><mml:mi>v</mml:mi><mml:mfenced open="(" close=")"><mml:mrow><mml:msub><mml:mi>a</mml:mi><mml:mi>j</mml:mi></mml:msub></mml:mrow></mml:mfenced><mml:mo>-</mml:mo><mml:mi>v</mml:mi><mml:mfenced close=")" open="("><mml:mrow><mml:msub><mml:mi>d</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:mrow></mml:mfenced><mml:mo>|</mml:mo><mml:mo>⋅</mml:mo><mml:msup><mml:mfenced close=")" open="("><mml:mrow><mml:msub><mml:mi>d</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mi>a</mml:mi><mml:mi>j</mml:mi></mml:msub></mml:mrow></mml:mfenced><mml:mn mathvariant="normal">2</mml:mn></mml:msup><mml:mo>.</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:math></disp-formula></p>
<sec id="Ch1.S4.SS3.SSS1">
  <label>4.3.1</label><title>Application of the parametrization to the case study</title>
      <p id="d1e3602">In this section, the parametrization derived for SIP by droplet fragmentation for temperatures close to 0 <inline-formula><mml:math id="M210" display="inline"><mml:msup><mml:mi/><mml:mo>∘</mml:mo></mml:msup></mml:math></inline-formula>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 2 h, 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 <inline-formula><mml:math id="M211" display="inline"><mml:mrow><mml:mi>E</mml:mi><mml:mo>(</mml:mo><mml:msub><mml:mi>d</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula>, <inline-formula><mml:math id="M212" display="inline"><mml:mrow><mml:msub><mml:mi>a</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:mo>)</mml:mo><mml:mo>=</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:mrow></mml:math></inline-formula> with any ice crystal.</p>
      <p id="d1e3646">To calculate the fall velocities of the ice crystals, we divide them into plates and lump graupel. The former includes the classes “plates” and “unidentified”, while the later includes all other ice crystals (see Sect. <xref ref-type="sec" rid="Ch1.S2.SS2"/> for a more detailed description of the classes). The fall velocity of plates was calculated with the following equation from <xref ref-type="bibr" rid="bib1.bibx66" id="text.88"/> (converted to SI base units) using the maximum dimension of plates <inline-formula><mml:math id="M213" display="inline"><mml:mrow><mml:msub><mml:mi>L</mml:mi><mml:mi mathvariant="normal">pla</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula>:
              <disp-formula id="Ch1.E7" content-type="numbered"><label>7</label><mml:math id="M214" display="block"><mml:mrow><mml:mi>v</mml:mi><mml:mfenced close=")" open="("><mml:mrow><mml:msub><mml:mi>L</mml:mi><mml:mi mathvariant="normal">pla</mml:mi></mml:msub></mml:mrow></mml:mfenced><mml:mo>≈</mml:mo><mml:mn mathvariant="normal">156</mml:mn><mml:mspace linebreak="nobreak" width="0.125em"/><mml:msup><mml:mi mathvariant="normal">m</mml:mi><mml:mn mathvariant="normal">0.14</mml:mn></mml:msup><mml:mspace linebreak="nobreak" width="0.125em"/><mml:msup><mml:mi mathvariant="normal">s</mml:mi><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:mrow></mml:msup><mml:mo>⋅</mml:mo><mml:msubsup><mml:mi>L</mml:mi><mml:mi mathvariant="normal">pla</mml:mi><mml:mn mathvariant="normal">0.86</mml:mn></mml:msubsup><mml:mo>.</mml:mo></mml:mrow></mml:math></disp-formula>
            To derive the fall speed of lump graupel with a maximum dimension of <inline-formula><mml:math id="M215" display="inline"><mml:mrow><mml:msub><mml:mi>L</mml:mi><mml:mi mathvariant="normal">gra</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula>, we use the equation provided by <xref ref-type="bibr" rid="bib1.bibx53" id="text.89"/> (again, converted to SI base units):
              <disp-formula id="Ch1.E8" content-type="numbered"><label>8</label><mml:math id="M216" display="block"><mml:mrow><mml:mi>v</mml:mi><mml:mfenced close=")" open="("><mml:mrow><mml:msub><mml:mi>L</mml:mi><mml:mi mathvariant="normal">gra</mml:mi></mml:msub></mml:mrow></mml:mfenced><mml:mo>≈</mml:mo><mml:mn mathvariant="normal">124</mml:mn><mml:mspace width="0.125em" linebreak="nobreak"/><mml:msup><mml:mi mathvariant="normal">m</mml:mi><mml:mn mathvariant="normal">0.34</mml:mn></mml:msup><mml:mspace width="0.125em" linebreak="nobreak"/><mml:msup><mml:mi mathvariant="normal">s</mml:mi><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:mrow></mml:msup><mml:mo>⋅</mml:mo><mml:msubsup><mml:mi>L</mml:mi><mml:mi mathvariant="normal">gra</mml:mi><mml:mn mathvariant="normal">0.66</mml:mn></mml:msubsup><mml:mo>.</mml:mo></mml:mrow></mml:math></disp-formula>
            This yields a splinter generation rate of 0.06 L<inline-formula><mml:math id="M217" display="inline"><mml:msup><mml:mi/><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula> min<inline-formula><mml:math id="M218" display="inline"><mml:msup><mml:mi/><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula> <inline-formula><mml:math id="M219" display="inline"><mml:mo>±</mml:mo></mml:math></inline-formula> 0.02 L<inline-formula><mml:math id="M220" display="inline"><mml:msup><mml:mi/><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula> min<inline-formula><mml:math id="M221" display="inline"><mml:msup><mml:mi/><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula> of secondary ice, which is about 1 order of magnitude below the estimated production rate of secondary ice of 0.24 L<inline-formula><mml:math id="M222" display="inline"><mml:msup><mml:mi/><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula> min<inline-formula><mml:math id="M223" display="inline"><mml:msup><mml:mi/><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula> <inline-formula><mml:math id="M224" display="inline"><mml:mo>±</mml:mo></mml:math></inline-formula> 0.09 L<inline-formula><mml:math id="M225" display="inline"><mml:msup><mml:mi/><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula> min<inline-formula><mml:math id="M226" display="inline"><mml:msup><mml:mi/><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula> derived from the observations.</p>

      <?xmltex \floatpos{t}?><fig id="Ch1.F8" specific-use="star"><?xmltex \currentcnt{8}?><?xmltex \def\figurename{Figure}?><label>Figure 8</label><caption><p id="d1e3883">The plots show the different parameters of the splinter generation rate <inline-formula><mml:math id="M227" display="inline"><mml:mrow><mml:msub><mml:mi>g</mml:mi><mml:mi mathvariant="normal">sp</mml:mi></mml:msub><mml:mo>(</mml:mo><mml:msub><mml:mi>d</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula> (Eq. <xref ref-type="disp-formula" rid="Ch1.E1"/>) for the first assumptions (Eqs. <xref ref-type="disp-formula" rid="Ch1.E3"/>–<xref ref-type="disp-formula" rid="Ch1.E5"/>; solid lines) and for the tuned version of <inline-formula><mml:math id="M228" display="inline"><mml:mrow><mml:msub><mml:mi>N</mml:mi><mml:mi mathvariant="normal">sp</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> in turbulent conditions that can explain the observations (Eqs. <xref ref-type="disp-formula" rid="Ch1.E9"/> and <xref ref-type="disp-formula" rid="Ch1.E10"/>; dashed lines).</p></caption>
            <?xmltex \igopts{width=341.433071pt}?><graphic xlink:href="https://acp.copernicus.org/articles/21/3855/2021/acp-21-3855-2021-f08.png"/>

          </fig>

      <?pagebreak page3865?><p id="d1e3935">The most uncertain parameter in Eq. (<xref ref-type="disp-formula" rid="Ch1.E6"/>) 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, <inline-formula><mml:math id="M229" display="inline"><mml:mrow><mml:msub><mml:mi>N</mml:mi><mml:mi mathvariant="normal">sp</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> has to be on the order of <inline-formula><mml:math id="M230" display="inline"><mml:mrow><mml:msub><mml:mi>N</mml:mi><mml:mi mathvariant="normal">sp</mml:mi></mml:msub><mml:mo>(</mml:mo><mml:msub><mml:mi>d</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo>)</mml:mo><mml:mo>=</mml:mo><mml:mo>(</mml:mo><mml:mn mathvariant="normal">4.9</mml:mn><mml:mo>×</mml:mo><mml:msup><mml:mn mathvariant="normal">10</mml:mn><mml:mn mathvariant="normal">5</mml:mn></mml:msup><mml:mo>±</mml:mo><mml:mn mathvariant="normal">3.1</mml:mn><mml:mo>×</mml:mo><mml:msup><mml:mn mathvariant="normal">10</mml:mn><mml:mn mathvariant="normal">5</mml:mn></mml:msup><mml:mo>)</mml:mo><mml:mspace width="0.125em" linebreak="nobreak"/><mml:msup><mml:mi mathvariant="normal">m</mml:mi><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:mrow></mml:msup><mml:mo>⋅</mml:mo><mml:msub><mml:mi>d</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> to produce the measured concentrations of small secondary-ice particles. This is equivalent to <inline-formula><mml:math id="M231" display="inline"><mml:mrow><mml:mn mathvariant="normal">99</mml:mn><mml:mo>±</mml:mo><mml:mn mathvariant="normal">62</mml:mn></mml:mrow></mml:math></inline-formula> splinters produced by a fragmenting droplet of 200 <inline-formula><mml:math id="M232" display="inline"><mml:mrow class="unit"><mml:mi mathvariant="normal">µ</mml:mi></mml:mrow></mml:math></inline-formula>m in diameter and is 2 to 9 times higher than the first assumption of <inline-formula><mml:math id="M233" display="inline"><mml:mrow><mml:msub><mml:mi>N</mml:mi><mml:mi mathvariant="normal">sp</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula>.</p>
      <p id="d1e4046">Apart from the average number 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. <xref ref-type="bibr" rid="bib1.bibx34" id="text.90"/> showed that droplet fragmentation is significantly higher in moving air than in stagnant air. Therefore, it may be reasonable to assume that <inline-formula><mml:math id="M234" display="inline"><mml:mrow><mml:msub><mml:mi>p</mml:mi><mml:mi mathvariant="normal">df</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> is even higher in turbulent conditions, and we expect here that every droplet fragments when it freezes:
              <disp-formula id="Ch1.E9" content-type="numbered"><label>9</label><mml:math id="M235" display="block"><mml:mrow><mml:msub><mml:mi>p</mml:mi><mml:mi mathvariant="normal">df</mml:mi></mml:msub><mml:mfenced close=")" open="("><mml:mrow><mml:msub><mml:mi>d</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:mrow></mml:mfenced><mml:mo>=</mml:mo><mml:mn mathvariant="normal">1</mml:mn><mml:mo>.</mml:mo></mml:mrow></mml:math></disp-formula>
            To explain the observed splinter generation rate of the measurements when turbulent conditions are expected, <inline-formula><mml:math id="M236" display="inline"><mml:mrow><mml:msub><mml:mi>N</mml:mi><mml:mi mathvariant="normal">sp</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> needs to be on the order of
              <disp-formula id="Ch1.E10" content-type="numbered"><label>10</label><mml:math id="M237" display="block"><mml:mrow><mml:msub><mml:mi>N</mml:mi><mml:mi mathvariant="normal">sp</mml:mi></mml:msub><mml:mfenced close=")" open="("><mml:mrow><mml:msub><mml:mi>d</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:mrow></mml:mfenced><mml:mo>=</mml:mo><mml:mfenced close=")" open="("><mml:mrow><mml:mn mathvariant="normal">2.2</mml:mn><mml:mo>×</mml:mo><mml:msup><mml:mn mathvariant="normal">10</mml:mn><mml:mn mathvariant="normal">5</mml:mn></mml:msup><mml:mo>±</mml:mo><mml:mn mathvariant="normal">1.3</mml:mn><mml:mo>×</mml:mo><mml:msup><mml:mn mathvariant="normal">10</mml:mn><mml:mn mathvariant="normal">5</mml:mn></mml:msup></mml:mrow></mml:mfenced><mml:msup><mml:mi mathvariant="normal">m</mml:mi><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:mrow></mml:msup><mml:mo>⋅</mml:mo><mml:msub><mml:mi>d</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo>.</mml:mo></mml:mrow></mml:math></disp-formula>
            This is equivalent to <inline-formula><mml:math id="M238" display="inline"><mml:mrow><mml:mn mathvariant="normal">44</mml:mn><mml:mo>±</mml:mo><mml:mn mathvariant="normal">26</mml:mn></mml:mrow></mml:math></inline-formula> splinters produced per fragmenting droplet of 200 <inline-formula><mml:math id="M239" display="inline"><mml:mrow class="unit"><mml:mi mathvariant="normal">µ</mml:mi></mml:mrow></mml:math></inline-formula>m in diameter and is only 1 to 4 times higher than the first assumption of <inline-formula><mml:math id="M240" display="inline"><mml:mrow><mml:msub><mml:mi>N</mml:mi><mml:mi mathvariant="normal">sp</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula>.</p>
      <p id="d1e4189">The calculated values for each parameter of Eq. (<xref ref-type="disp-formula" rid="Ch1.E1"/>) are shown in Fig. <xref ref-type="fig" rid="Ch1.F8"/> as a function of droplet size and for each measured droplet individually for the first assumptions (Eqs. <xref ref-type="disp-formula" rid="Ch1.E3"/>–<xref ref-type="disp-formula" rid="Ch1.E5"/>) as well as for the tuned version of <inline-formula><mml:math id="M241" display="inline"><mml:mrow><mml:msub><mml:mi>N</mml:mi><mml:mi mathvariant="normal">sp</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> in turbulent conditions that can explain the observations (Eqs. <xref ref-type="disp-formula" rid="Ch1.E9"/> and <xref ref-type="disp-formula" rid="Ch1.E10"/>).</p>
</sec>
<sec id="Ch1.S4.SS3.SSS2">
  <label>4.3.2</label><title>Caveats of the parametrization and its application to the case study</title>
      <p id="d1e4224">The first main caveat is that the parametrization was derived solely from laboratory measurements. However, the direct observation of SIP by droplet fragmentation is basically impossible as the process happens on a millisecond timescale <xref ref-type="bibr" rid="bib1.bibx46" id="paren.91"/>, and the secondary-ice splinters can be smaller than 10 <inline-formula><mml:math id="M242" display="inline"><mml:mrow class="unit"><mml:mi mathvariant="normal">µ</mml:mi></mml:mrow></mml:math></inline-formula>m <xref ref-type="bibr" rid="bib1.bibx43" id="paren.92"/>. Therefore, the laboratory measurements are the best estimate available.</p>
      <p id="d1e4241">The second main caveat is the parametrization of <inline-formula><mml:math id="M243" display="inline"><mml:mrow><mml:msub><mml:mi>N</mml:mi><mml:mi mathvariant="normal">sp</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula>. The proportionality of <inline-formula><mml:math id="M244" display="inline"><mml:mrow><mml:msub><mml:mi>N</mml:mi><mml:mi mathvariant="normal">sp</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> to <inline-formula><mml:math id="M245" display="inline"><mml:mi>d</mml:mi></mml:math></inline-formula> was based on only four data points, which were derived from three different studies using different measurement techniques and only show the maximum number of observed fragments as discussed in <xref ref-type="bibr" rid="bib1.bibx46" id="text.93"/>. 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 <xref ref-type="bibr" rid="bib1.bibx37" id="text.94"/> 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 up to 9 times higher than assumed in Eq. (<xref ref-type="disp-formula" rid="Ch1.E5"/>) but critically depends on the largest observed droplet. Until further measurements can constrain the concentration of fragments produced per fragmentation, <inline-formula><mml:math id="M246" display="inline"><mml:mrow><mml:msub><mml:mi>N</mml:mi><mml:mi mathvariant="normal">sp</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> remains highly uncertain.</p>
      <?pagebreak page3866?><p id="d1e4293">Thirdly, the contribution of the different droplets is highly unbalanced, which can be seen in Fig. <xref ref-type="fig" rid="Ch1.F8"/>. For example, the largest droplet (<inline-formula><mml:math id="M247" display="inline"><mml:mrow><mml:mo>∼</mml:mo><mml:mn mathvariant="normal">380</mml:mn></mml:mrow></mml:math></inline-formula> <inline-formula><mml:math id="M248" display="inline"><mml:mrow class="unit"><mml:mi mathvariant="normal">µ</mml:mi></mml:mrow></mml:math></inline-formula>m) in the present case study contributes 97 % to the total amount of the produced concentration of secondary ice when using the first assumptions of <inline-formula><mml:math id="M249" display="inline"><mml:mrow><mml:msub><mml:mi>p</mml:mi><mml:mi mathvariant="normal">df</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> (Eq. <xref ref-type="disp-formula" rid="Ch1.E4"/>) and <inline-formula><mml:math id="M250" display="inline"><mml:mrow><mml:msub><mml:mi>N</mml:mi><mml:mi mathvariant="normal">sp</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> (Eq. <xref ref-type="disp-formula" rid="Ch1.E5"/>) and 77 % when expecting turbulent conditions (Eq. <xref ref-type="disp-formula" rid="Ch1.E9"/>) with a tuned version of <inline-formula><mml:math id="M251" display="inline"><mml:mrow><mml:msub><mml:mi>N</mml:mi><mml:mi mathvariant="normal">sp</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> that can explain the observations (Eq. <xref ref-type="disp-formula" rid="Ch1.E10"/>). 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 <inline-formula><mml:math id="M252" display="inline"><mml:mrow class="unit"><mml:mi mathvariant="normal">µ</mml:mi></mml:mrow></mml:math></inline-formula>m 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.</p>
      <p id="d1e4366">As a fourth caveat, we assume a continuous flow over 2 h 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 <inline-formula><mml:math id="M253" display="inline"><mml:mrow class="unit"><mml:mi mathvariant="normal">µ</mml:mi></mml:mrow></mml:math></inline-formula>m, 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 volume. However, this is not very likely because measurements were taken very close to 0 <inline-formula><mml:math id="M254" display="inline"><mml:msup><mml:mi/><mml:mo>∘</mml:mo></mml:msup></mml:math></inline-formula>C, and droplets are thus unlikely to freeze before they reach the measurement volume.</p>
      <p id="d1e4387">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 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 to be secondary ice in this study.</p>
</sec>
</sec>
<sec id="Ch1.S4.SS4">
  <label>4.4</label><title>Other case studies with similar observations</title>
      <p id="d1e4399">Several other studies observed an increase in the concentration of small ice crystals in the presence of large supercooled drops in clouds <xref ref-type="bibr" rid="bib1.bibx79 bib1.bibx50 bib1.bibx35 bib1.bibx43 bib1.bibx52" id="paren.95"><named-content content-type="pre">e.g.,</named-content></xref>. Updrafts were made responsible for the origin of these large droplets in all of the studies. The measurements were taken mostly at temperatures lower than during our case study, where newly formed ice crystals grow into columns. The recirculation process through the melting layer described above (Fig. <xref ref-type="fig" rid="Ch1.F7"/>) is therefore also expected to play a role for droplet fragmentation in higher regions of the cloud if the updrafts are strong enough to lift drizzle drops high enough until they freeze. Besides droplet fragmentation, the rime-splintering process is expected to be active in the temperature regime between <inline-formula><mml:math id="M255" display="inline"><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">3</mml:mn></mml:mrow></mml:math></inline-formula> and <inline-formula><mml:math id="M256" display="inline"><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">8</mml:mn></mml:mrow></mml:math></inline-formula> <inline-formula><mml:math id="M257" display="inline"><mml:msup><mml:mi/><mml:mo>∘</mml:mo></mml:msup></mml:math></inline-formula>C, which makes the assignment of the observed secondary ICNC to a specific process more difficult. However, images of deformed frozen drops in all of the abovementioned studies are strong evidence that droplet fragmentation was active and should be accounted for besides the rime-splintering process.</p><?xmltex \hack{\newpage}?>
</sec>
</sec>
<?pagebreak page3867?><sec id="Ch1.S5" sec-type="conclusions">
  <label>5</label><title>Summary</title>
      <p id="d1e4448">On 22 February 2019, wind from the northwest pushed air masses up a mountain slope, where measurements were taken just above the melting layer on a gondola (see Fig. <xref ref-type="fig" rid="Ch1.F3"/>). The  measurements showed relatively constant conditions during the measurement period of 2 h (Fig. <xref ref-type="fig" rid="Ch1.F6"/>), with a CDNC of about 160 cm<inline-formula><mml:math id="M258" display="inline"><mml:msup><mml:mi/><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">3</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula> and an ICNC of about 6 L<inline-formula><mml:math id="M259" display="inline"><mml:msup><mml:mi/><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula>, which exceeded the measured INPC by several orders of magnitude. The majority of the observed small ice crystals (<inline-formula><mml:math id="M260" display="inline"><mml:mrow><mml:mi>L</mml:mi><mml:mo>&lt;</mml:mo><mml:mn mathvariant="normal">93</mml:mn></mml:mrow></mml:math></inline-formula> <inline-formula><mml:math id="M261" display="inline"><mml:mrow class="unit"><mml:mi mathvariant="normal">µ</mml:mi></mml:mrow></mml:math></inline-formula>m) were identified as plates (Fig. <xref ref-type="fig" rid="Ch1.F5"/>). As this is the preferred ice crystal habit at the temperatures in the measurement volume (<inline-formula><mml:math id="M262" display="inline"><mml:mrow><mml:mi>T</mml:mi><mml:mo>&gt;</mml:mo><mml:mo>-</mml:mo><mml:mn mathvariant="normal">3</mml:mn></mml:mrow></mml:math></inline-formula> <inline-formula><mml:math id="M263" display="inline"><mml:msup><mml:mi/><mml:mo>∘</mml:mo></mml:msup></mml:math></inline-formula>C), ice crystals smaller than 93 <inline-formula><mml:math id="M264" display="inline"><mml:mrow class="unit"><mml:mi mathvariant="normal">µ</mml:mi></mml:mrow></mml:math></inline-formula>m are assumed to have newly formed in the same environmental conditions. At such warm temperatures, primary-ice nucleation can be neglected, and the concentration of small plates most likely emerged from SIP.</p>
      <p id="d1e4533">Remarkable was the observation of relatively large droplets between 40 and 380 <inline-formula><mml:math id="M265" display="inline"><mml:mrow class="unit"><mml:mi mathvariant="normal">µ</mml:mi></mml:mrow></mml:math></inline-formula>m at a concentration of 0.3 L<inline-formula><mml:math id="M266" display="inline"><mml:msup><mml:mi/><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula> <inline-formula><mml:math id="M267" display="inline"><mml:mo>±</mml:mo></mml:math></inline-formula> 0.1 L<inline-formula><mml:math id="M268" display="inline"><mml:msup><mml:mi/><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula> above the melting layer. The appearance of cloud droplets larger than about 40 <inline-formula><mml:math id="M269" display="inline"><mml:mrow class="unit"><mml:mi mathvariant="normal">µ</mml:mi></mml:mrow></mml:math></inline-formula>m is often connected to SIP by droplet fragmentation <xref ref-type="bibr" rid="bib1.bibx43" id="paren.96"/>. The rime-splintering process and other SIP processes are not expected to be active and are thus excluded from consideration, 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 <xref ref-type="bibr" rid="bib1.bibx43" id="text.97"/> can explain these observations and can in general be applied to mountainous regions when a melting 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 and lifted through the melting layer, and they 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. <xref ref-type="fig" rid="Ch1.F7"/>). The secondary-ice crystals will be transported to higher altitudes, where they influence the cloud microphysics and subsequently the radiation budget, the lifetime, and the precipitation pattern of the cloud.</p>
      <p id="d1e4592">A parametrization was introduced in Sect. <xref ref-type="sec" rid="Ch1.S4.SS3"/> for the generation of secondary-ice particles by droplet fragmentation at temperatures close to 0 <inline-formula><mml:math id="M270" display="inline"><mml:msup><mml:mi/><mml:mo>∘</mml:mo></mml:msup></mml:math></inline-formula>C, when primary-ice nucleation is basically absent (Eq. <xref ref-type="disp-formula" rid="Ch1.E6"/>). Based on limited available measurements of previous laboratory studies, it is assumed that the number of splinters produced per droplet is linearly correlated with their diameter and that a droplet of 200 <inline-formula><mml:math id="M271" display="inline"><mml:mrow class="unit"><mml:mi mathvariant="normal">µ</mml:mi></mml:mrow></mml:math></inline-formula>m produces 18 splinters on average when it fragments. Applying the presented parametrization to our measurements could not explain the estimated concentration of secondary ice, and the number of splinters produced per fragmenting droplet has to be higher; i.e., a droplet of 200 <inline-formula><mml:math id="M272" display="inline"><mml:mrow class="unit"><mml:mi mathvariant="normal">µ</mml:mi></mml:mrow></mml:math></inline-formula>m in diameter has to produce <inline-formula><mml:math id="M273" display="inline"><mml:mrow><mml:mn mathvariant="normal">99</mml:mn><mml:mo>±</mml:mo><mml:mn mathvariant="normal">62</mml:mn></mml:mrow></mml:math></inline-formula> splinters upon fragmentation. This number can be reduced to <inline-formula><mml:math id="M274" display="inline"><mml:mrow><mml:mn mathvariant="normal">44</mml:mn><mml:mo>±</mml:mo><mml:mn mathvariant="normal">26</mml:mn></mml:mrow></mml:math></inline-formula> if we assume that all droplets larger than 40 µm fragment when they freeze. This assumption may be reasonable when strong wind speeds or turbulence are present. In either case, the application shows that droplets smaller than about 100 <inline-formula><mml:math id="M275" display="inline"><mml:mrow class="unit"><mml:mi mathvariant="normal">µ</mml:mi></mml:mrow></mml:math></inline-formula>m are negligible for SIP by droplet fragmentation. However, these results critically depend on a single large droplet next to other rather crude assumptions and should be considered with caution. Therefore, the parametrization should be implemented in a model and tested on other case studies, which observed high concentrations of small ice crystals in clouds with temperatures close to 0 <inline-formula><mml:math id="M276" display="inline"><mml:msup><mml:mi/><mml:mo>∘</mml:mo></mml:msup></mml:math></inline-formula>C. The implementation can improve the prediction of the ICNC in mountainous regions, where orographically forced updrafts through a melting layer are observed. A better prediction of the ICNC can improve the determination of the optical properties and the lifetime of clouds as well as precipitation formation from clouds containing ice <xref ref-type="bibr" rid="bib1.bibx54 bib1.bibx26" id="paren.98"><named-content content-type="pre">e.g.,</named-content></xref>.</p>
</sec>

      
      </body>
    <back><notes notes-type="codedataavailability"><title>Code and data availability</title>

      <p id="d1e4675">The data used in this study are available at <ext-link xlink:href="https://doi.org/10.5281/zenodo.4534382" ext-link-type="DOI">10.5281/zenodo.4534382</ext-link> <xref ref-type="bibr" rid="bib1.bibx47" id="paren.99"/> or referenced herein. The software to reproduce the figures and for calculations done for this study is available at <ext-link xlink:href="https://doi.org/10.5281/zenodo.4534536" ext-link-type="DOI">10.5281/zenodo.4534536</ext-link> <xref ref-type="bibr" rid="bib1.bibx48" id="paren.100"/>.</p>
  </notes><notes notes-type="authorcontribution"><title>Author contributions</title>

      <p id="d1e4693">AL analyzed and interpreted the data, created the figures, and wrote the manuscript with contributions from JH, CM, FR, JTP, JW, MH, and UL. AL, JH, CM, FR, JTP, and JW performed the measurements on the Gotschnabahn. MH operated the radar wind profiler and the ceilometer and processed the data.</p>
  </notes><notes notes-type="competinginterests"><title>Competing interests</title>

      <p id="d1e4699">The authors declare that they have no conflict of interest.</p>
  </notes><ack><title>Acknowledgements</title><p id="d1e4705">The authors would like to thank the whole RACLETS team for the installation and maintenance of the measurement setup as well as many fruitful scientific discussions. We would especially like to thank Michael Lehning (WSL/SLF, EPFL) and his whole team for their effort in supporting and realizing the RACLETS campaign. A special thanks also goes to the Davos Klosters Bergbahnen AG and the staff of the Gotschnabahn, especially the technical managers Andrea Margadant and the managing director Markus Good, for the permission to take measurements on one of their gondolas as well as the on-site support. We would also like thank Paul Fopp for providing his land for the remote-sensing measurements. We would 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 would also like to thank the Grischa paragliding club for providing us with meteorological<?pagebreak page3868?> measurements from their Holfuy station at Gotschnagrat. We would like to thank Hannes Griesche for supplying us with turbulence data, Patric Seifert (TROPOS, Germany) for discussing their measurements, and Benjamin Walter for providing us with data from the snow drift station at Gotschnagrat, and we would like to thank Pila Bossmann (<uri>https://wetterboss.com/</uri>, last access: 22 September 2020) for discussing the general weather situation. We would also like to thank our reviewers Alexei Korolev and Andrew Heymsfield for their constructive and helpful feedback.</p></ack><notes notes-type="financialsupport"><title>Financial support</title>

      <p id="d1e4713">This research has been supported by the Swiss National Science Foundation (grant nos. 200021_175824 and 200021_169620).</p>
  </notes><notes notes-type="reviewstatement"><title>Review statement</title>

      <p id="d1e4719">This paper was edited by Daniel Knopf and reviewed by Alexei Korolev and Andrew Heymsfield.</p>
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    <!--<article-title-html>Continuous secondary-ice production initiated by updrafts  through the melting layer in mountainous regions</article-title-html>
<abstract-html><p>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 and −8&thinsp;°C and has been implemented into several models, while all other secondary-ice processes are poorly constrained and lack a well-founded quantification. During 2&thinsp;h of measurements taken on a mountain slope just above the melting layer at temperatures warmer than −3&thinsp;°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 slightly sub-zero temperatures, where primary-ice nucleation is basically absent, and the first ice is initiated by the collision of drizzle drops with aged ice crystals sedimenting from higher altitudes. Based on previous measurements, we estimate that a droplet of 200&thinsp;µm in diameter produces 18 secondary-ice crystals when it fragments upon freezing. The application of the parametrization to our measurements suggests that the actual number of splinters produced by a fragmenting droplet may be up to an order of magnitude higher.</p></abstract-html>
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