_The MAR model description comes from Section 2.1 from [Christoph Kittel's PhD thesis](https://orbi.uliege.be/handle/2268/258491), with some minor format changes. This is Section 2.1.3.1._
The model includes a cloud-microphysical scheme solving conservation equations for the concentration of five water species (cloud droplets $q_{w}$, ice crystal $q_{i}$, rain drops $q_{r}$, snow particles $q_{s}$, and specific humidity $q_{v}$ firstly described by Gallée (1995)[^1]) and the ice crystal number $n_{i}$ (Massager et al., 2004[^2]). MAR solves the following conservative equation for every horizontal pixel and vertical-$\sigma$ layer:
where the three first terms represent the 3D advection by the wind following the $x, y$ and $\sigma$ directions, $F_{q \alpha}$ the turbulent flux divergence of the hydrometeor specie, and $P_{sed}$ a source term. $P_{sed}$ is an additional term that describes the sedimentation of precipitating hydrometeors (rain drops, snow particles, and ice crystals) depending on their specific falling velocities. The source term in the above equation represents the 21 microphysical processes detailed in Table 1 originally based on Kessler (1969)[^3] parameterisations in addition to the sedimentation towards the surface of the three precipitating hydrometeors. Since graupels are not (fully yet) included in the model, all accretion processes that should result in graupel following Lin et al. (1983)[^4] lead to snowflake formation assuming a Marshall-Palmer size distribution (Gallée, 1995)[^1].
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<!-- JFG note: rewrote mixing ratio symbols to make up for GitLab limitations w.r.t. the number of mathematical expressions; e.g. $q_{v}$ became q<sub>v</sub>, $q_{i}$ became q<sub>i</sub>, etc. -->
| Microphysical processes | References |
|------------------------|------------|
| Nucleation by cloud droplet solidification (q<sub>v</sub> to q<sub>i</sub>, if $T \leq -35^{\circ} \mathrm{C}$) | Emde and Kahlig (1989)[^5]; Levkov et al. (1992)[^6] |
| Deposition and condensation-freezing nucleation (q<sub>v</sub> to q<sub>i</sub>, if $T \leq -35^{\circ} \mathrm{C}$) | Meyers et al. (1992)[^7]; Levkov et al. (1992)[^6] |
| Contact-freezing nucleation or depositional growth of cloud ice (q<sub>v</sub> to q<sub>i</sub>) | Meyers et al. (1992)[^7]; Prenni et al. (2007)[^8]; Levkov et al. (1992)[^6] |
| Ice crystal sublimation (q<sub>i</sub> to q<sub>v</sub>) | Emde and Kahlig (1989)[^5]; Levkov et al. (1992)[^6] |
| Ice crystal melting (q<sub>i</sub> to q<sub>w</sub>, if $T \geq 0^{\circ} \mathrm{C}$) | Levkov et al. (1992)[^6] |
| Water vapor condensation (q<sub>v</sub> to q<sub>w</sub>, if $T \geq -35^{\circ} \mathrm{C}$) | Emde and Kahlig (1989)[^5] |
| Cloud droplet evaporation (q<sub>w</sub> to q<sub>v</sub>) | Lin et al. (1983)[^4]; Sundqvist (1988) [^9] |
| Cloud droplet autoconversion (q<sub>w</sub> to q<sub>r</sub>) | Lin et al. (1983)[^4]; Sundqvist (1988) [^9] |
| Depositional growth of snow (q<sub>v</sub> to q<sub>s</sub>) | Levkov et al. (1992)[^6] |
| Ice crystal aggregation (q<sub>i</sub> to q<sub>s</sub>) | Levkov et al. (1992)[^6] |
| Accretion of cloud droplet by rain (q<sub>w</sub> to q<sub>r</sub>) | Lin et al. (1983)[^4]; Emde and Kahlig (1989)[^5] |
| Accretion of cloud droplet by snow (q<sub>w</sub> to q<sub>s</sub>) | Lin et al. (1983)[^4]; Locatelli and Hobbs (1974) [^10] |
| Accretion of ice crystal by snow (q<sub>i</sub> to q<sub>s</sub>) | Lin et al. (1983)[^4]; Levkov et al. (1992)[^6] |
| Accretion of ice crystal by rain (since no graupel, q<sub>i</sub> to q<sub>s</sub>, if $T \leq 0^{\circ} \mathrm{C}$) | Lin et al. (1983)[^4]; Levkov et al. (1992)[^6] |
| Accretion of rain by ice crystal (since no graupel, q<sub>r</sub> to q<sub>s</sub>, if $T \leq 0^{\circ} \mathrm{C}$) | Lin et al. (1983)[^4] |
| Accretion of rain by snow (since no graupel, q<sub>r</sub> to q<sub>s</sub>, if $T \leq 0^{\circ} \mathrm{C}$) | Lin et al. (1983)[^4]; Emde and Kahlig (1989)[^5] |
| Accretion of snow by rain (since no graupel, q<sub>s</sub> to q<sub>r</sub>, if $T \leq 0^{\circ} \mathrm{C}$) | Lin et al. (1983)[^4]; Emde and Kahlig (1989)[^5] |
| Rain freezing (since no graupel, q<sub>r</sub> to q<sub>s</sub>, if $T \leq 0^{\circ} \mathrm{C}$) | Lin et al. (1983)[^4]; Emde and Kahlig (1989)[^5] |
| Rain evaporation (q<sub>r</sub> to q<sub>v</sub>) | Lin et al. (1983)[^4] |
| Deposition on snow (q<sub>v</sub> to q<sub>s</sub>) or sublimation (q<sub>s</sub> to q<sub>v</sub>) | Lin et al. (1983)[^4] |
| Snow melting (q<sub>s</sub> to q<sub>w</sub>, if $T \geq 0^{\circ} \mathrm{C}$) | Lin et al. (1983)[^4] |
| Rain sedimentation | Emde and Kahlig (1989)[^5] |
| Snow sedimentation | Emde and Kahlig (1989)[^5]; Levkov et al. (1992)[^6]; Locatelli and Hobbs (1974) [^10]; Fettweis et al. (2017)[^11] |
| Ice crystal sedimentation | Levkov et al. (1992)[^6] |
**Table 1:** Microphysical processes represented in MAR and associated references as firstly described by Gallée (1995)[^1] and modified afterwards.
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Table 1 also lists the parameterisations that have been recently included in MAR since the original description by Gallée (1995)[^1]. In particular, the ice crystal nucleation used by Lin et al. (1983)[^4] (and based on Fletcher (1962)[^12] overestimates ice crystal concentration leading to the model underestimation of the downwelling solar radiation towards the surface, the convective available potential energy (CAPE), and rain (Massager et al., 2004[^2]). It has thus been replaced by Meyers et al. (1992)'s parameterization[^7] later improved by Prenni et al. (2007)[^8]. Ice crystal sedimentation is not neglected anymore by adding a prognostic equation for ice crystal number according to Levkov et al. (1992)[^6]. Furthermore, the conversion rate of cloud droplets to rain particles takes into account an adapted parameterisation from Sundqvist (1988) [^9]. It relies on two parameters: a critical cloud water mixing ratio ($q_{wo}$) enabling the rainfall formation and a characteristic time scale for auto-conversion processes ($C_{o}$). Note that a low fraction of cloud droplets can be converted to rain even if $q_{w}$ is lower than $q_{wo}$ (Delobbe and Gallée, 1998[^13]). In [MARv3.11](https://gitlab.uliege.be/especes/mar/stable/-/releases/v3.11.5), $q_{wo}$ and $C_{o}$ values are respectively fixed to $1 \cdot 10^{-3} (\text{kg kg}^{-1})$ and $1 \cdot 10^{-4}$. Finally, other subtle adjustments such as an increase in snowfall sedimentation velocity or cloud lifetime (Fettweis et al., 2017[^11], 2020[^14]) have been made to tune the model in order to more accurately reproduce clouds over the polar ice sheets.
[^1]:Gallée, H., Fontaine de Ghélin, O., and van den Broeke, M. R.: Simulation of atmospheric circulation during the GIMEX 91 experiment using a meso-γ primitive equations model, Journal of climate, 8, 2843-2859, 1995.
[^2]:Massager, C., Gallée, H., and Brasseur, O.: Precipitation sensitivity to regional SST in a regional climate simulation during the West African monsoon for two dry years, Climate Dynamics, 22, 249-266, 2004.
[^3]:Kessler, E.: On the distribution and continuity of water substance in atmospheric circulations, in: On the distribution and continuity of water substance in atmospheric circulations, pp. 1-84, Springer, 1969.
[^4]:Lin, Y.-L., Farley, R. D., and Orville, H. D.: Bulk parameterization of the snow field in a cloud model, Journal of Applied Meteorology and climatology, 22, 1065-1092, 1983.
[^5]:Emde, K. D. and Kahlig, P.: Comparison of the observed 19th July 1981, Montana thunderstorm with results of a one-dimensional cloud model using Kessler parameterized microphysics, in: Annales geophysicae. Atmospheres, hydrospheres and space sciences, vol. 7, pp. 405-414, 1989.
[^6]:Levkov, L., Rockel, B., Kapitza, H., and Raschke, E.: 3D mesoscale numerical studies of cirrus and stratus clouds by their time and space evolution, Contributions to atmospheric physics, 65, 35-58, 1992.
[^7]:Meyers, M. P., DeMott, P. J., and Cotton, W. R.: New primary ice-nucleation parameterizations in an explicit cloud model, Journal of Applied Meteorology, 31, 708-721, 1992.
[^8]:Prenni, A. J., Harrington, J. Y., Tjernström, M., DeMott, P. J., Avramov, A., Long, C. N., Kreidenweis, S. M., Olsson, P. Q., and Verlinde, J.: Can ice-nucleating aerosols affect Arctic seasonal climate?, Bulletin of the American Meteorological Society, 88, 541-550, 2007.
[^9]:Sundqvist, H.: Parameterization of condensation and associated clouds in models for weather prediction and general circulation simulation, in: Physically-based modelling and simulation of climate and climatic change, pp. 433-461, Springer, 1988.
[^10]:Locatelli, J. D. and Hobbs, P. V.: Fall speeds and masses of solid precipitation particles, Journal of Geophysical Research, 79, 2185-2197, 1974.
[^11]:Fettweis, X., Box, J., Agosta, C., Amory, C., Kittel, C., Lang, C., van As, D., Machguth, H., and Gallée, H.: Reconstructions of the 1900-2015 Greenland ice sheet surface mass balance using the regional climate MAR model, The Cryosphere, 11, 1015-1033, 2017.
[^12]:Fletcher, N. H.: The physics of rainclouds, Cambridge University Press, 1962.
[^13]:Delobbe, L. and Gallée, H.: Simulation of marine stratocumulus: Effect of precipitation parameterization and sensitivity to droplet number concentration, Boundary-layer meteorology, 89, 75-107, 1998.
[^14]:Fettweis, X., Hofer, S., Krebs-Kanzow, U., Amory, C., Aoki, T., Berends, C. J., Born, A., Box, J. E., Delhasse, A., Fujita, K., Gierz, P., Goelzer, H., Hanna, E., Hashimoto, A., Huybrechts, P., Kapsch, M.-L., King, M. D., Kittel, C., Lang, C., Langen, P. L., Lenaerts, J. T. M., Liston, G. E., Lohmann, G., Mernild, S. H., Mikolajewicz, U., Modali, K., Mottram, R. H., Niwano, M., Noël, B., Ryan, J. C., Smith, A., Streffing, J., Tedesco, M., van de Berg, W. J., van den Broeke, M., van de Wal, R. S. W., van Kampenhout, L., Wilton, D., Wouters, B., Ziemen, F., and Zolles, T.: GrSMBMIP: intercomparison of the modelled 1980-2012 surface mass balance over the Greenland Ice Sheet, The Cryosphere, 14, 3935-3958, 2020.
