Unique Solvability of a System of Ordinary Differential Equations Modeling a Warm Cloud Parcel

We analyze the solvability of a system of ordinary differential equations modeling a warm cloud. A unique feature of this model is the automatic onset of nucleation when the moist air parcel becomes supersaturated; this is made possible by a non-Lipschitz right-hand side of the differential equation, which allows for nontrivial smooth solutions. Here we prove under mild assumptions on the external forcing that this system of equations has a unique physically consistent solution, i.e., a solution with a nonzero droplet population in the supersaturated regime.

  • 1.  R.P. Agarwal and  V. Lakshmikantham , Uniqueness and Nonuniqueness Criteria for Ordinary Differential Equations , World Scientific , Singapore , 1993 . Google Scholar

  • 2.  S.R. Bernfeld R.D. Driver and  V. Lakshmikantham , Uniqueness for ordinary differential equations , Math. Syst. Theory , 9 ( 1976 ), pp. 359 -- 367 . CrossrefGoogle Scholar

  • 3.  J.M. Bownds and  J.B. Diaz , On restricted uniqueness for systems of ordinary differential equations , Proc. Amer. Math. Soc. , 37 ( 1973 ), pp. 100 -- 104 . CrossrefISIGoogle Scholar

  • 4.  M. Coti Zelati A. Huang I. Kukavica R. Temam and  M. Ziane , The primitive equations of the atmosphere in presence of vapour saturation , Nonlinearity , 28 ( 2015 ), pp. 625 -- 668 . CrossrefISIGoogle Scholar

  • 5.  W.W. Grabowski , Untangling microphysical impacts on deep convection applying a novel modeling methodology , J. Atmos. Sci. , 72 ( 2015 ), pp. 2446 -- 2464 . CrossrefISIGoogle Scholar

  • 6.  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 ), pp. 3676 -- 3718 . CrossrefISIGoogle Scholar

  • 7.  H.G. Kaper and  M.K. Kwong , Uniqueness for a class of nonlinear initial value problems , J. Math. Anal. Appl. , 130 ( 1988 ), pp. 467 -- 473 . CrossrefISIGoogle Scholar

  • 8.  S. Kichenassamy, Fuchsian Reduction, Birkhäuser, Boston, 2007.Google Scholar

  • 9.  S. Kichenassamy and  W. Littman , Blow-up surfaces for nonlinear wave equations, I , Comm. Partial Differential Equations , 18 ( 1993 ), pp. 431 -- 452 . CrossrefISIGoogle Scholar

  • 10.  S. Kichenassamy and  W. Littman , Blow-up surfaces for nonlinear wave equations, II , Comm. Partial Differential Equations , 18 ( 1993 ), pp. 1869 -- 1899 . CrossrefGoogle Scholar

  • 11.  Y. Kogan and  W. Martin , Parameterization of bulk condensation in numerical cloud models , J. Atmos. Sci. , 51 ( 1994 ), pp. 2957 -- 2974 . CrossrefISIGoogle Scholar

  • 12.  J. Li and  E.S. Titi , A tropical atmosphere model with moisture: Global well-posedness and relaxation limit , Nonlinearity , 29 ( 2016 ), pp. 2674 -- 2714 . CrossrefISIGoogle Scholar

  • 13.  D.M. Murphy and  T. Koop , Review of the vapour pressures of ice and supercooled water for atmospheric applications , Q. J. R. Meteorol. Soc. , 131 ( 2005 ), pp. 1539 -- 1565 . CrossrefISIGoogle Scholar

  • 14.  N. Porz M. Hanke M. Baumgartner and  P. Spichtinger , A model for warm clouds with implicit droplet activation, avoiding saturation adjustment , Math. Clim. Weather Forecast. , 4 ( 2018 ), pp. 50 -- 78 . CrossrefGoogle Scholar

  • 15.  H.R. Pruppacher and  J.D. Klett , Microphysics of Clouds and Precipitation , Springer , Heidelberg , 2010 . Google Scholar

  • 16.  A.D. Rendall and  B.G. Schmidt , Existence and properties of spherically symmetric static fluid bodies with a given equation of state , Classical Quantum Gravity , 8 ( 1991 ), pp. 985 -- 1000 . CrossrefISIGoogle Scholar

  • 17.  R.R. Rogers and  M.K. Yau , A Short Course in Cloud Physics , 3 rd ed., Pergamon Press , Oxford , 1989 . Google Scholar

  • 18.  A. Seifert and  K.D. Beheng , A two-moment cloud microphysics parameterization for mixed-phase clouds. Part I: Model description , Meteorol. Atmos. Phys. , 92 ( 2006 ), pp. 45 -- 66 . CrossrefISIGoogle Scholar

  • 19.  P. Spichtinger , private communication , 2019 . Google Scholar

  • 20.  G. Teschl , Ordinary Differential Equations and Dynamical Systems , Amer. Math. Soc. , Providence , RI , 2012 . Google Scholar

  • 21.  W. Walter , Ordinary Differential Equations , Springer , New York , 1998 . Google Scholar

  • 22.  D.V.V . Wend, Existence and uniqueness of solutions of ordinary differential equations , Proc. Amer. Math. Soc. , 23 ( 1969 ), pp. 27 -- 33 . CrossrefISIGoogle Scholar