# American Institute of Mathematical Sciences

January  2021, 41(1): 131-167. doi: 10.3934/dcds.2020219

## Mathematical analysis of a cloud resolving model including the ice microphysics

 1 Department of Mathematics and The Institute for Scientific Computing, and Applied Mathematics, Indiana University, Bloomington, IN 47405-7106, USA 2 Climate and Global Dynamics Lab, National Center for Atmospheric Research, Boulder, CO 80307-3000, USA

* Corresponding author: Roger Temam

Received  January 2020 Published  January 2021 Early access  May 2020

We extend our study for the warm cloud model in [13] to the analysis of a more general cloud model including the ice microphysics in [28]. The moisture variables comprise water vapor, cloud condensates (cloud water, cloud ice), and cloud precipitations (rain, snow), with respective mass ratios $q_v$, $q_c$ and $q_p$. A typical assumption in [13] for the calculation of condensation rate is that the warm clouds are exactly at water saturation with no supersaturation in general. When the ice microphysics are included, the situation becomes more complicated. We have to consider both the saturation mixing ratio with respect to water ($q_{vw}$) and the saturation with respect to ice ($q_{vi}$) when the temperature $T$ is below the freezing point $T_w$ but above the threshold $T_i$ for homogeneous ice nucleation. A remedy, acceptable from the physical and mathematical viewpoints, is to define the overall saturation mixing ratio $q_{vs}$ as a convex combination of $q_{vw}$ and $q_{vi}$. Under this setting, supersaturation can still be avoided and we have the constraint $q_v \le q_{vs}$ with $q_{vs}$ depending itself on the state. Mathematically, we are led to a system of equations and inequations involving some quasi-variational inequalities for which we prove the global existence and regularity of solutions.

Citation: Yining Cao, Chuck Jia, Roger Temam, Joseph Tribbia. Mathematical analysis of a cloud resolving model including the ice microphysics. Discrete and Continuous Dynamical Systems, 2021, 41 (1) : 131-167. doi: 10.3934/dcds.2020219
##### References:
 [1] K. Adamy, A. Bousquet, S. Faure, J. Laminie and R. Temam, A multilevel method for finite volume discretization of the two dimensional nonlinear shallow-water equations, Ocean Modelling, 33 (2010), 235-256. [2] A. Bensoussan and A. Friedman, On the support of the solution of a system of quasi variational inequalities, J. Math. Anal. Appl., 65 (1978), 660-674.  doi: 10.1016/0022-247X(78)90170-1. [3] A. Bensoussan and J. L. Lions, Impulse Control and Quasivariational Inequalities, Translated from the French by J. M. Cole. $\mu$. Gauthier-Villars, Montrouge; Heyden $&$ Son, Inc., Philadelphia, PA, 1984. [4] A. Bensoussan and J. L. Lions, Inéquations quasi variationnelles dépendant d'un paramètre, Ann. Scuola Norm. Sup. Pisa Cl. Sci., (4) 4 (1977), 231–255. [5] A. Bensoussan and J. L. Lions, Contrôle impulsionnel et contrôle continu. Méthode des inéquations quasi variationnelles non linéaires, (French)[Impulse control and continuous control, methods of the nonlinear and stationary quasi-variational equations.] C. R. Acad. Sci. Paris Sér. A, 278 (1974), 675–679. [6] A. Bensoussan and J. L. Lions, Contrôle impulsionnel et inéquations quasi-variationnelles stationnaires, (French)[Impulse control and continuous control, methods of the nonlinear quasi-variational inequalities, ] C. R. Acad. Sci. Paris Sér. A-B, 276 (1973), A1279–A1284. [7] A. Bensoussan and J. L. Lions, Nouvelle formulation de problèmes de contrôle impulsionnel et applications.(French)[New Formulations of the Impulse control problems and applications], C. R. Acad. Sci. Paris Sér. A-B, 276 (1973), A1189–A1192. [8] A. Bensoussan and J. L. Lions, Nouvelles méthodes en contrôle impulsionnel, Appl. Math. Optim., 1 (1974/75), 289-312.  doi: 10.1007/BF01447955. [9] A. Bensoussan and J. L. Lions, On the support of the solution of some variational inequalities of evolution, J. Math. Soc. Japan, 28 (1976), 1-17.  doi: 10.2969/jmsj/02810001. [10] A. Bousquet, M. Coti Zelati and R. Temam, Phase transition models in atmospheric dynamics, Milan J. Math., 82 (2014), 99-128.  doi: 10.1007/s00032-014-0213-y. [11] A. Bousquet, M. Chekroun, Y. Hong, R. Temam and J. Tribbia, Numerical simulations of the humid atmosphere above a mountain, Math. Clim. Weather Forecast, 1 (2015), 96-126. [12] H. Brézis, Problèmes unilatéraux, J. Math. Pures Appl., (9) 51 (1972), 1–168. [13] Y. Cao, M. Hamouda, R. M. Temam, J. Tribbia and X. Wang, The equations of the multi-phase humid atmosphere expressed as a quasi variational inequality, Nonlinearity, 31 (2018), 4692-4723.  doi: 10.1088/1361-6544/aad525. [14] C. Cao and E. S. Titi., Global well-posedness of the three-dimensional viscous primitive equations of large scale ocean and atmosphere dynamics, Ann. of Math., (2) 166 (2007), 245–267. doi: 10.4007/annals.2007.166.245. [15] J. Céa, Lectures on Optimization: Theory and Algorithms, Springer, Berlin ew York, 1978. [16] M. Coti Zelati, M. Fremond, R. Temam and J. Tribbia, The equations of the atmosphere with humidity and saturation: uniqueness and physical bounds, Physica D, 264 (2013), 49-65.  doi: 10.1016/j.physd.2013.08.007. [17] M. Coti Zelati, A. Huang, I. Kukavica, R. Temam and M. Ziane, The primitive equations of the atmosphere in presence of vapor saturation, Nonlinearity, 28 (2015), 625-668.  doi: 10.1088/0951-7715/28/3/625. [18] M. Coti Zelati and R. Temam, The atmospheric equation of water vapor with saturation, Boll. Unione Mat. Ital., 5 (2012), 309-336. [19] R. Courant, Variational methods for the solution of problems of equilibrium and vibrations, Bull. Amer. Math. Soc., 49 (1943), 1-23.  doi: 10.1090/S0002-9904-1943-07818-4. [20] J. I. Díaz, Mathematical analysis of some diffusive energy balance models in Climatology, in Mathematics, Climate and Environment (eds. J. I. Díaz and J.L.Lions), Research Notes in Applied Mathematics, Masson, Paris, 27 (1993), 28–56. [21] J. I. Díaz and L. Tello, On a nonlinear parabolic problem on a Riemannian manifold without boundary arising in Climatology, Collectanea Mathematica, Volum L, Fascicle 1, 50 (1999), 19–51. [22] G. Duvaut and J.-L. Lions, Inequalities in Mechanics and Physics, translated from the French by C.W. John Springer-Verlag, Berlin ew York, 1976. [23] I. Ekeland and R. Temam, Convex Analysis and Variational Problems, Translated from the French. Studies in Mathematics and its Applications, Vol. 1. North-Holland Publishing Co., Amsterdam-Oxford; American Elsevier Publishing Co., Inc., New York, 1976. [24] E. Feireisl, A note on uniqueness for parabolic problems with discontinuous nonlinearities, Nonlinear Anal., 16 (1991), 1053-1056.  doi: 10.1016/0362-546X(91)90106-B. [25] E. Feireisl and J. Norbury, Some existence, uniqueness and non-uniqueness theorems for solutions of parabolic equations with discontinuous nonlinearities, Proc. Roy. Soc. Edinburgh Sect. A, 119 (1991), 1-17.  doi: 10.1017/S0308210500028262. [26] M. Frémond, Non-Smooth Thermomechanics, Springer-Verlag, Berlin, 2002. doi: 10.1007/978-3-662-04800-9. [27] A. E. Gill, Atmosphere-Ocean Dynamics, I nternational Geophysics Series, Volume 30, Academic Press, 1st Edition, 1982,662 pages. [28] W. W. Grabowski, Toward cloud resolving modeling of large-scale tropical circulations: A simple cloud microphysics parametrization, Journal of the Atmospheric Sciences, 55 (1998), 3283-3298.  doi: 10.1175/1520-0469(1998)055<3283:TCRMOL>2.0.CO;2. [29] W. W. Grabowski, H. Morrison, S. Shima, G. C. Abade, P. Dziekan and H. Pawlowska, Modeling of Cloud Microphysics: Can We Do Better?, Bull. Amer. Meteor. Soc., 100 (2019), 655-672.  doi: 10.1175/BAMS-D-18-0005.1. [30] W. W. Grabowski and P. K. Smolarkiewicz, A multiscale anelastic model for meteorological research, Monthly Weather Review, 130 (2002), 939-956.  doi: 10.1175/1520-0493(2002)130<0939:AMAMFM>2.0.CO;2. [31] B. Guo and D. Huang, Existence of weak solutions and trajectory attractors for the moist atmospheric equations in geophysics, J. Math. Phys. 47 (2006), 083508, 23pp. doi: 10.1063/1.2245207. [32] B. Guo and D. Huang, Existence of the universal attractor for the 3-D viscous primitive equations of large-scale moist atmosphere, J. Differential Equations, 251 (2011), 457-491.  doi: 10.1016/j.jde.2011.05.010. [33] G. J. Haltiner, Numerical Weather Prediction, John Wiley and Sons, New York, 1971. [34] G. J. Haltiner and R. T. Williams, Numerical prediction and dynamic meteorology, John Wiley and Sons, New York, 1980. [35] S. Hittmeir, R. Klein, J. Li and E. S. Titi, Global well-posedness for passively transported nonlinear moisture dynamics with phase changes, Nonlinearity, 30 (2017), 3676-3718.  doi: 10.1088/1361-6544/aa82f1. [36] R. Kano, The existence of solutions for tumor invasion models with time and space dependent diffusion, Discrete Contin. Dyn. Syst. Ser. S, 7 (2014), 63-74.  doi: 10.3934/dcdss.2014.7.63. [37] E. Kessler, On the distribution and continuity of water substance in atmospheric circulations, in Meteorological Monographs, American Meteorological Society, Boston, MA, 10 (1969), 1–84. [38] D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and their Applications, Academic Press Inc., Harcourt Brace Jovanovich Publishers, New York, 1980. [39] R. Klein and A. J. Majda, Systematic multiscale models for deep convection on mesoscales, Theor. Comput. Fluid Dyn., 20 (2006), 525-551.  doi: 10.1007/s00162-006-0027-9. [40] G. M. Kobelkov, Existence of a solution 'in the large' for the 3D large-scale ocean dynamics equations, C. R. Math. Acad. Sci. Paris, 343 (2006), 283-286.  doi: 10.1016/j.crma.2006.04.020. [41] A. S. Kravchuk and P. J. Neittaanmki $\ddot{a}$, Variational and Quasi-Variational Inequalities in Mechanics, Solid Mechanics and its Applications, 147. Springer, Dordrecht, 2007. doi: 10.1007/978-1-4020-6377-0. [42] Y.-L. Lin, R. D. Farley and H. D. Orville, Bulk parameterization of the snow field in a cloud model, J. Climate Appl. Meteor., 22 (1983), 1065-1092.  doi: 10.1175/1520-0450(1983)022<1065:BPOTSF>2.0.CO;2. [43] J. L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires, Dunod, Paris, 1969. [44] F. Lenzen, J. Lellmann, F. Becker and C. Schnörr, Solving quasi-variational inequalities for image restoration with adaptive constraint sets, SIAM J. Imaging Sci., 7 (2014), 2139-2174.  doi: 10.1137/130938347. [45] J.-L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications, Vol. I, Springer-Verlag, New York, 1972. [46] J. L. Lions, R. Temam and S. Wang, New formulations of the primitive equations of atmosphere and applications, Nonlinearity, 5 (1992), 237-288.  doi: 10.1088/0951-7715/5/2/001. [47] M. Milasi, Existence theorem for a class of generalized quasi-variational inequalities, J. Global Optim., 60 (2014), 679-688.  doi: 10.1007/s10898-013-0114-6. [48] J. Mossino, Application des inéquations quasi-variationnelles à quelques problèmes non linéaires de la physique des plasmas, (French. English summary), Israel J. Math., 30 (1978), 14-50.  doi: 10.1007/BF02760826. [49] J. Mossino, Étude d'une inéquation quasi-variationnelle apparaissant en physique, in Convex Analysis and Its Applications (eds. A. Auslender), Lecture Notes in Econom. and Math. Systems, Vol. 144. Springer, Berlin, (1977), 139–157. [50] J. Mossino, Sur certaines inéquations quasi-variationnelles apparaissant en physique, C. R. Acad. Sci. Paris Sér. A-B, 282 (1976), 187-190. [51] J. Pedlosky, Geophysical Fluid Dynamics, 2$^{nd}$ edition, Springer-Verlag, New York, 1987. [52] M. Petcu, R. M. Temam and M. Ziane, Some mathematical problems in geophysical fluiddynamics, in Computational Methods for the Atmosphere and the Oceans, (eds.R. M. Temam and J. J. Tribbia), vol. 14 of Handbook of Numerical Analysis. Elsevier, Amsterdam, (2009), 577–750. doi: 10.1016/S1570-8659(08)00212-3. [53] E. Polak and A. L. Tits, A globally convergent, implementable multiplier method with automatic penalty limitation, Appl Math Optim., 6 (1980), 335-360.  doi: 10.1007/BF01442901. [54] R. R. Rogers and M. K. Yau, A Short Course in Cloud Physics, 3$^rd$ edition, Pergamon Press, Oxford, New York, 1989. [55] S. A. Rutledge and P. V. Hobbs, The mesoscale and microscale structure and organization of clouds and precipitation in midlatitude cyclones. XII: A diagnostic modeling study of precipitation development in narrow cold-frontal rainbands, J. Atmos. Sci., 41 (1984), 2949-2972.  doi: 10.1175/1520-0469(1984)041<2949:TMAMSA>2.0.CO;2. [56] R. Temam, Navier-Stokes Equations and Nonlinear Functional Analysis, 2$^{nd}$ edition, SIAM, Philadelphia, 1995. doi: 10.1137/1.9781611970050. [57] R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, AMS-Chelsea publication, Providence, Rhode Island, 2001. doi: 10.1090/chel/343. [58] R. Temam and J. Tribbia, The equations of moist advection: A unilateral problem, Quarterly Journal of the Royal Meteorological Society, 142 (2016), 143-146.  doi: 10.1002/qj.2638. [59] R. Temam and X. Wang, Approximation of the equations of the humid atmosphere with saturation, SIAM Journal on Numerical Analysis, 55 (2017), 217-239.  doi: 10.1137/15M1039420. [60] R. Temam and K. Wu, Formulation of the equations of the humid atmosphere in the context of variational inequalities, Journal of Functional Analysis, 269 (2015), 2187-2221.  doi: 10.1016/j.jfa.2015.02.010. [61] M. K. Yau and P. M. Austin, A model for hydrometer growth and evolution of raindrop size spectra in cumulus cells, Journal of the Atmospheric Sciences, 36 (1979), 655-668.  doi: 10.1175/1520-0469(1979)036<0655:AMFHGA>2.0.CO;2.

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##### References:
 [1] K. Adamy, A. Bousquet, S. Faure, J. Laminie and R. Temam, A multilevel method for finite volume discretization of the two dimensional nonlinear shallow-water equations, Ocean Modelling, 33 (2010), 235-256. [2] A. Bensoussan and A. Friedman, On the support of the solution of a system of quasi variational inequalities, J. Math. Anal. Appl., 65 (1978), 660-674.  doi: 10.1016/0022-247X(78)90170-1. [3] A. Bensoussan and J. L. Lions, Impulse Control and Quasivariational Inequalities, Translated from the French by J. M. Cole. $\mu$. Gauthier-Villars, Montrouge; Heyden $&$ Son, Inc., Philadelphia, PA, 1984. [4] A. Bensoussan and J. L. Lions, Inéquations quasi variationnelles dépendant d'un paramètre, Ann. Scuola Norm. Sup. Pisa Cl. Sci., (4) 4 (1977), 231–255. [5] A. Bensoussan and J. L. Lions, Contrôle impulsionnel et contrôle continu. Méthode des inéquations quasi variationnelles non linéaires, (French)[Impulse control and continuous control, methods of the nonlinear and stationary quasi-variational equations.] C. R. Acad. Sci. Paris Sér. A, 278 (1974), 675–679. [6] A. Bensoussan and J. L. Lions, Contrôle impulsionnel et inéquations quasi-variationnelles stationnaires, (French)[Impulse control and continuous control, methods of the nonlinear quasi-variational inequalities, ] C. R. Acad. Sci. Paris Sér. A-B, 276 (1973), A1279–A1284. [7] A. Bensoussan and J. L. Lions, Nouvelle formulation de problèmes de contrôle impulsionnel et applications.(French)[New Formulations of the Impulse control problems and applications], C. R. Acad. Sci. Paris Sér. A-B, 276 (1973), A1189–A1192. [8] A. Bensoussan and J. L. Lions, Nouvelles méthodes en contrôle impulsionnel, Appl. Math. Optim., 1 (1974/75), 289-312.  doi: 10.1007/BF01447955. [9] A. Bensoussan and J. L. Lions, On the support of the solution of some variational inequalities of evolution, J. Math. Soc. Japan, 28 (1976), 1-17.  doi: 10.2969/jmsj/02810001. [10] A. Bousquet, M. Coti Zelati and R. Temam, Phase transition models in atmospheric dynamics, Milan J. Math., 82 (2014), 99-128.  doi: 10.1007/s00032-014-0213-y. [11] A. Bousquet, M. Chekroun, Y. Hong, R. Temam and J. Tribbia, Numerical simulations of the humid atmosphere above a mountain, Math. Clim. Weather Forecast, 1 (2015), 96-126. [12] H. Brézis, Problèmes unilatéraux, J. Math. Pures Appl., (9) 51 (1972), 1–168. [13] Y. Cao, M. Hamouda, R. M. Temam, J. Tribbia and X. Wang, The equations of the multi-phase humid atmosphere expressed as a quasi variational inequality, Nonlinearity, 31 (2018), 4692-4723.  doi: 10.1088/1361-6544/aad525. [14] C. Cao and E. S. Titi., Global well-posedness of the three-dimensional viscous primitive equations of large scale ocean and atmosphere dynamics, Ann. of Math., (2) 166 (2007), 245–267. doi: 10.4007/annals.2007.166.245. [15] J. Céa, Lectures on Optimization: Theory and Algorithms, Springer, Berlin ew York, 1978. [16] M. Coti Zelati, M. Fremond, R. Temam and J. Tribbia, The equations of the atmosphere with humidity and saturation: uniqueness and physical bounds, Physica D, 264 (2013), 49-65.  doi: 10.1016/j.physd.2013.08.007. [17] M. Coti Zelati, A. Huang, I. Kukavica, R. Temam and M. Ziane, The primitive equations of the atmosphere in presence of vapor saturation, Nonlinearity, 28 (2015), 625-668.  doi: 10.1088/0951-7715/28/3/625. [18] M. Coti Zelati and R. Temam, The atmospheric equation of water vapor with saturation, Boll. Unione Mat. Ital., 5 (2012), 309-336. [19] R. Courant, Variational methods for the solution of problems of equilibrium and vibrations, Bull. Amer. Math. Soc., 49 (1943), 1-23.  doi: 10.1090/S0002-9904-1943-07818-4. [20] J. I. Díaz, Mathematical analysis of some diffusive energy balance models in Climatology, in Mathematics, Climate and Environment (eds. J. I. Díaz and J.L.Lions), Research Notes in Applied Mathematics, Masson, Paris, 27 (1993), 28–56. [21] J. I. Díaz and L. Tello, On a nonlinear parabolic problem on a Riemannian manifold without boundary arising in Climatology, Collectanea Mathematica, Volum L, Fascicle 1, 50 (1999), 19–51. [22] G. Duvaut and J.-L. Lions, Inequalities in Mechanics and Physics, translated from the French by C.W. John Springer-Verlag, Berlin ew York, 1976. [23] I. Ekeland and R. Temam, Convex Analysis and Variational Problems, Translated from the French. Studies in Mathematics and its Applications, Vol. 1. North-Holland Publishing Co., Amsterdam-Oxford; American Elsevier Publishing Co., Inc., New York, 1976. [24] E. Feireisl, A note on uniqueness for parabolic problems with discontinuous nonlinearities, Nonlinear Anal., 16 (1991), 1053-1056.  doi: 10.1016/0362-546X(91)90106-B. [25] E. Feireisl and J. Norbury, Some existence, uniqueness and non-uniqueness theorems for solutions of parabolic equations with discontinuous nonlinearities, Proc. Roy. Soc. Edinburgh Sect. A, 119 (1991), 1-17.  doi: 10.1017/S0308210500028262. [26] M. Frémond, Non-Smooth Thermomechanics, Springer-Verlag, Berlin, 2002. doi: 10.1007/978-3-662-04800-9. [27] A. E. Gill, Atmosphere-Ocean Dynamics, I nternational Geophysics Series, Volume 30, Academic Press, 1st Edition, 1982,662 pages. [28] W. W. Grabowski, Toward cloud resolving modeling of large-scale tropical circulations: A simple cloud microphysics parametrization, Journal of the Atmospheric Sciences, 55 (1998), 3283-3298.  doi: 10.1175/1520-0469(1998)055<3283:TCRMOL>2.0.CO;2. [29] W. W. Grabowski, H. Morrison, S. Shima, G. C. Abade, P. Dziekan and H. Pawlowska, Modeling of Cloud Microphysics: Can We Do Better?, Bull. Amer. Meteor. Soc., 100 (2019), 655-672.  doi: 10.1175/BAMS-D-18-0005.1. [30] W. W. Grabowski and P. K. Smolarkiewicz, A multiscale anelastic model for meteorological research, Monthly Weather Review, 130 (2002), 939-956.  doi: 10.1175/1520-0493(2002)130<0939:AMAMFM>2.0.CO;2. [31] B. Guo and D. Huang, Existence of weak solutions and trajectory attractors for the moist atmospheric equations in geophysics, J. Math. Phys. 47 (2006), 083508, 23pp. doi: 10.1063/1.2245207. [32] B. Guo and D. Huang, Existence of the universal attractor for the 3-D viscous primitive equations of large-scale moist atmosphere, J. Differential Equations, 251 (2011), 457-491.  doi: 10.1016/j.jde.2011.05.010. [33] G. J. Haltiner, Numerical Weather Prediction, John Wiley and Sons, New York, 1971. [34] G. J. Haltiner and R. T. Williams, Numerical prediction and dynamic meteorology, John Wiley and Sons, New York, 1980. [35] S. Hittmeir, R. Klein, J. Li and E. S. Titi, Global well-posedness for passively transported nonlinear moisture dynamics with phase changes, Nonlinearity, 30 (2017), 3676-3718.  doi: 10.1088/1361-6544/aa82f1. [36] R. Kano, The existence of solutions for tumor invasion models with time and space dependent diffusion, Discrete Contin. Dyn. Syst. Ser. S, 7 (2014), 63-74.  doi: 10.3934/dcdss.2014.7.63. [37] E. Kessler, On the distribution and continuity of water substance in atmospheric circulations, in Meteorological Monographs, American Meteorological Society, Boston, MA, 10 (1969), 1–84. [38] D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and their Applications, Academic Press Inc., Harcourt Brace Jovanovich Publishers, New York, 1980. [39] R. Klein and A. J. Majda, Systematic multiscale models for deep convection on mesoscales, Theor. Comput. Fluid Dyn., 20 (2006), 525-551.  doi: 10.1007/s00162-006-0027-9. [40] G. M. Kobelkov, Existence of a solution 'in the large' for the 3D large-scale ocean dynamics equations, C. R. Math. Acad. Sci. Paris, 343 (2006), 283-286.  doi: 10.1016/j.crma.2006.04.020. [41] A. S. Kravchuk and P. J. Neittaanmki $\ddot{a}$, Variational and Quasi-Variational Inequalities in Mechanics, Solid Mechanics and its Applications, 147. Springer, Dordrecht, 2007. doi: 10.1007/978-1-4020-6377-0. [42] Y.-L. Lin, R. D. Farley and H. D. Orville, Bulk parameterization of the snow field in a cloud model, J. Climate Appl. Meteor., 22 (1983), 1065-1092.  doi: 10.1175/1520-0450(1983)022<1065:BPOTSF>2.0.CO;2. [43] J. L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires, Dunod, Paris, 1969. [44] F. Lenzen, J. Lellmann, F. Becker and C. Schnörr, Solving quasi-variational inequalities for image restoration with adaptive constraint sets, SIAM J. Imaging Sci., 7 (2014), 2139-2174.  doi: 10.1137/130938347. [45] J.-L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications, Vol. I, Springer-Verlag, New York, 1972. [46] J. L. Lions, R. Temam and S. Wang, New formulations of the primitive equations of atmosphere and applications, Nonlinearity, 5 (1992), 237-288.  doi: 10.1088/0951-7715/5/2/001. [47] M. Milasi, Existence theorem for a class of generalized quasi-variational inequalities, J. Global Optim., 60 (2014), 679-688.  doi: 10.1007/s10898-013-0114-6. [48] J. Mossino, Application des inéquations quasi-variationnelles à quelques problèmes non linéaires de la physique des plasmas, (French. English summary), Israel J. Math., 30 (1978), 14-50.  doi: 10.1007/BF02760826. [49] J. Mossino, Étude d'une inéquation quasi-variationnelle apparaissant en physique, in Convex Analysis and Its Applications (eds. A. Auslender), Lecture Notes in Econom. and Math. Systems, Vol. 144. Springer, Berlin, (1977), 139–157. [50] J. Mossino, Sur certaines inéquations quasi-variationnelles apparaissant en physique, C. R. Acad. Sci. Paris Sér. A-B, 282 (1976), 187-190. [51] J. Pedlosky, Geophysical Fluid Dynamics, 2$^{nd}$ edition, Springer-Verlag, New York, 1987. [52] M. Petcu, R. M. Temam and M. Ziane, Some mathematical problems in geophysical fluiddynamics, in Computational Methods for the Atmosphere and the Oceans, (eds.R. M. Temam and J. J. Tribbia), vol. 14 of Handbook of Numerical Analysis. Elsevier, Amsterdam, (2009), 577–750. doi: 10.1016/S1570-8659(08)00212-3. [53] E. Polak and A. L. Tits, A globally convergent, implementable multiplier method with automatic penalty limitation, Appl Math Optim., 6 (1980), 335-360.  doi: 10.1007/BF01442901. [54] R. R. Rogers and M. K. Yau, A Short Course in Cloud Physics, 3$^rd$ edition, Pergamon Press, Oxford, New York, 1989. [55] S. A. Rutledge and P. V. Hobbs, The mesoscale and microscale structure and organization of clouds and precipitation in midlatitude cyclones. XII: A diagnostic modeling study of precipitation development in narrow cold-frontal rainbands, J. Atmos. Sci., 41 (1984), 2949-2972.  doi: 10.1175/1520-0469(1984)041<2949:TMAMSA>2.0.CO;2. [56] R. Temam, Navier-Stokes Equations and Nonlinear Functional Analysis, 2$^{nd}$ edition, SIAM, Philadelphia, 1995. doi: 10.1137/1.9781611970050. [57] R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, AMS-Chelsea publication, Providence, Rhode Island, 2001. doi: 10.1090/chel/343. [58] R. Temam and J. Tribbia, The equations of moist advection: A unilateral problem, Quarterly Journal of the Royal Meteorological Society, 142 (2016), 143-146.  doi: 10.1002/qj.2638. [59] R. Temam and X. Wang, Approximation of the equations of the humid atmosphere with saturation, SIAM Journal on Numerical Analysis, 55 (2017), 217-239.  doi: 10.1137/15M1039420. [60] R. Temam and K. 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In the first simulation, we have a single mountain of height 2500 meters. The amount of snow $q_s$ is measured in $\text{g}\cdot \text{kg}^{-1}$. Brighter color indicates higher quantity (i.e., more snow) in the contour plots. Solid deep blue represents total absence of snow. The dashed red line shows the separation between rain and snow. We can see that the area that receives the most snow is the part of the atmosphere slightly to the left of the peak above the mountain. Note that the air flows from left to right. These results are coherent with the physical context
In the second simulation, there are two mountains of heights 2500 m and 1500 m with the taller mountain on the left. The amount of snow $q_s$ is measured in $\text{g}\cdot \text{kg}^{-1}$. Brighter color indicates higher quantity of snow. Solid deep blue represents total absence of snow. The dashed red line shows the separation between rain and snow. The air flows from left to right as before. In this simulation, the taller mountain on the left blocks the passing of moist, resulting in less snow around the lower mountain on the right. These results are coherent with the physical context
In the third simulation, we have two mountains of heights 1500 m and 2500 m with the taller mountain on the \emph{right}. The amount of snow $q_s$ is measured in $\text{g}\cdot \text{kg}^{-1}$ and brighter color indicates higher quantity of snow. Solid deep blue represents total absence of snow. The dashed red line shows the separation between rain and snow. The air flows from left to right as before. In this simulation, there is snow in the atmosphere above both mountains, as the mountain on the left does not block as much moist as in the previous simulation. These results are also coherent with the physical context
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