# 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  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 & Continuous Dynamical Systems - A, 2021, 41 (1) : 131-167. doi: 10.3934/dcds.2020219
##### References:
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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.  Google Scholar [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. Google Scholar [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.  Google Scholar [8] A. Bensoussan and J. L. Lions, Nouvelles méthodes en contrôle impulsionnel, Appl. Math. Optim., 1 (1974/75), 289-312.  doi: 10.1007/BF01447955.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [12] H. Brézis, Problèmes unilatéraux, J. Math. Pures Appl., (9) 51 (1972), 1–168.  Google Scholar [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.  Google Scholar [14] C. Cao and E. S. 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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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [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.  Google Scholar [8] A. Bensoussan and J. L. Lions, Nouvelles méthodes en contrôle impulsionnel, Appl. Math. Optim., 1 (1974/75), 289-312.  doi: 10.1007/BF01447955.  Google Scholar [9] A. Bensoussan and J. L. 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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.  Google Scholar [15] J. Céa, Lectures on Optimization: Theory and Algorithms, Springer, Berlin ew York, 1978.  Google Scholar [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.  Google Scholar [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.  Google Scholar [18] M. Coti Zelati and R. Temam, The atmospheric equation of water vapor with saturation, Boll. Unione Mat. Ital., 5 (2012), 309-336.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [26] M. Frémond, Non-Smooth Thermomechanics, Springer-Verlag, Berlin, 2002. doi: 10.1007/978-3-662-04800-9.  Google Scholar [27] A. E. Gill, Atmosphere-Ocean Dynamics, I nternational Geophysics Series, Volume 30, Academic Press, 1st Edition, 1982,662 pages. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [33] G. J. Haltiner, Numerical Weather Prediction, John Wiley and Sons, New York, 1971. Google Scholar [34] G. J. Haltiner and R. T. Williams, Numerical prediction and dynamic meteorology, John Wiley and Sons, New York, 1980. Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [38] D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and their Applications, Academic Press Inc., Harcourt Brace Jovanovich Publishers, New York, 1980.  Google Scholar [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.  Google Scholar [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.  Google Scholar [41] A. S. Kravchuk and P. J. Neittaanmki $\ddot{a}$, Variational and Quasi-Variational Inequalities in Mechanics, Solid Mechanics and its Applications, 147. 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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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