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June  2022, 42(6): 2859-2892. doi: 10.3934/dcds.2022002

## The primitive equations of the polluted atmosphere as a weak and strong limit of the 3D Navier-Stokes equations in downwind-matching coordinates

 1 Department of Information Engineering, Computer Science and Mathematics, University of L'Aquila, 67100 L'Aquila, Italy 2 Bolyai Institute, University of Szeged, H-6720 Szeged, Hungary

*Corresponding author: Donatella Donatelli

Received  June 2020 Revised  July 2021 Published  June 2022 Early access  January 2022

Fund Project: The research of DD and NJ leading to these results has received funding from the European Union's Horizon 2020 Research and Innovation Programme under the Marie Sklodowska-Curie Grant Agreement No 642768 (Project Name: ModCompShock)

A widely used approach to mathematically describe the atmosphere is to consider it as a geophysical fluid in a shallow domain and thus to model it using classical fluid dynamical equations combined with the explicit inclusion of an $\epsilon$ parameter representing the small aspect ratio of the physical domain. In our previous paper [14] we proved a weak convergence theorem for the polluted atmosphere described by the Navier-Stokes equations extended by an advection-diffusion equation. We obtained a justification of the generalised hydrostatic limit model including the pollution effect described for the case of classical, east-north-upwards oriented local Cartesian coordinates. Here we give a two-fold improvement of this statement. Firstly, we consider a meteorologically more meaningful coordinate system, incorporate the analytical consequences of this coordinate change into the governing equations, and verify that the weak convergence still holds for this altered system. Secondly, still considering this new, so-called downwind-matching coordinate system, we prove an analogous strong convergence result, which we make complete by providing a closely related existence theorem as well.

Citation: Donatella Donatelli, Nóra Juhász. The primitive equations of the polluted atmosphere as a weak and strong limit of the 3D Navier-Stokes equations in downwind-matching coordinates. Discrete and Continuous Dynamical Systems, 2022, 42 (6) : 2859-2892. doi: 10.3934/dcds.2022002
##### References:
 [1] R. A. Adams and J. J. Fournier, Sobolev Spaces, 2nd edition, Pure and Applied Mathematics (Amsterdam), 140. Elsevier/Academic Press, Amsterdam, 2003. [2] I. Ali, S. Kalla and H. Khajah, A partial differential equation related to a problem in atmospheric pollution, Math. Comput. Modelling, 28 (1998), 1-6.  doi: 10.1016/S0895-7177(98)00169-1. [3] J. D. Anderson and J. Wendt, Computational Fluid Dynamics, volume 206, Springer, 1995. [4] G. Andria, A. Lay-Ekuakille and M. Notarnicola, Mathematical Models for Atmospheric and Industrial Pollutant Prediction, In XVI IMEKO World Congress, Vienna, Austria, 2000. [5] P. S. Arya, Introduction to Micrometeorology, volume 79, Elsevier, 2001. [6] P. Azérad and F. Guillén, Mathematical justification of the hydrostatic approximation in the primitive equations of geophysical fluid dynamics, SIAM J. Math. Anal., 33 (2001), 847-859.  doi: 10.1137/S0036141000375962. [7] O. Besson and M. Laydi, Some estimates for the anisotropic Navier-Stokes equations and for the hydrostatic approximation, RAIRO Modél. Math. Anal. Numér., 26 (1992), 855-865.  doi: 10.1051/m2an/1992260708551. [8] C. Cao, J. Li and E. S. Titi, Local and global well-posedness of strong solutions to the 3D primitive equations with vertical eddy diffusivity, Arch. Ration. Mech. Anal., 214 (2014), 35-76.  doi: 10.1007/s00205-014-0752-y. [9] C. Cao, J. Li and E. S. Titi, Global well-posedness of the three-dimensional primitive equations with only horizontal viscosity and diffusion, Comm. Pure Appl. Math., 69 (2016), 1492-1531.  doi: 10.1002/cpa.21576. [10] C. Cao and E. S. Titi, Global well-posedness and finite-dimensional global attractor for a 3-D planetary geostrophic viscous model, Comm. Pure Appl. Math., 56 (2003), 198-233.  doi: 10.1002/cpa.10056. [11] 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., 166 (2007), 245-267.  doi: 10.4007/annals.2007.166.245. [12] C. Cao and E. S. Titi, Global well-posedness of the 3D primitive equations with partial vertical turbulence mixing heat diffusion, Comm. Math. Phys., 310 (2012), 537-568.  doi: 10.1007/s00220-011-1409-4. [13] H. E. de Swart, The Governing Equations and the Dominant Balances of Flow in the Atmosphere and Ocean, USPC Summerschool, Utrechts, 2016. [14] D. Donatelli and N. Juhász, Weak solution of the merged mathematical equations of the polluted atmosphere, Math. Methods Appl. Sci., 43 (2020), 9245-9261.  doi: 10.1002/mma.6618. [15] I. Kukavica and M. Ziane, On the regularity of the primitive equations of the ocean, Nonlinearity, 20 (2007), 2739-2753.  doi: 10.1088/0951-7715/20/12/001. [16] H. Gao, S. Necasova and T. Tang, On weak-strong uniqueness and singular limit for the compressible primitive equations, Discrete Contin. Dyn. Syst., 40 (2020), 4287-4305.  doi: 10.3934/dcds.2020181. [17] P. Goyal and A. Kumar, Mathematical Modeling of Air Pollutants: An Application to Indian Urban City, Air Quality-Models and Applications, InTech, 2011. doi: 10.5772/16840. [18] B. Hosseini, Dispersion of Pollutants in the Atmosphere: A Numerical Study, Master's thesis, Simom Fraser University, 2013. [19] P. Kathirgamanathan, R. McKibbin and R. McLachlan, Source term estimation of pollution from an instantaneous point source, Research Letters in Information Mathematic Science, 3 (2002), 59-67. [20] J. Li and E. S. Titi, The primitive equations as the small aspect ratio limit of the Navier–Stokes equations: Rigorous justification of the hydrostatic approximation, J. Math. Pures Appl., 124 (2019), 30-58.  doi: 10.1016/j.matpur.2018.04.006. [21] P.-L. Lions, Mathematical Topics in Fluid Mechanics: Volume 2: Compressible Models, volume 2, Oxford University Press on Demand, 1996. [22] P. Marsaleix, F. Auclair and C. Estournel, Considerations on open boundary conditions for regional and coastal ocean models, Journal of Atmospheric and Oceanic Technology, 23 (2006), 1604-1613.  doi: 10.1175/JTECH1930.1. [23] A. Monin, On the boundary condition on the earth surface for diffusing pollution, Advances in Geophysics, 6 (1959), 435-436. [24] J. Pedlosky, Geophysical Fluid Dynamics, Springer Science & Business Media, 2013. [25] M. Petcu and D. Wirosoetisno, Sobolev and Gevrey regularity results for the primitive equations in three space dimensions, Appl. Anal., 84 (2005), 769-788.  doi: 10.1080/00036810500130745. [26] M. Prodanova, J. L. Perez, D. Syrakov, R. San Jose, K. Ganev, N. Miloshev and S. Roglev, Application of mathematical models to simulate an extreme air pollution episode in the Bulgarian city of Stara Zagora, Applied Mathematical Modelling, 32 (2008), 1607-1619.  doi: 10.1016/j.apm.2007.05.002. [27] J. Simon, Compact sets in the space $L^p(0, t; B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96.  doi: 10.1007/BF01762360. [28] R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, AMS Chelsea Publishing, Providence, RI, 2001. doi: 10.1090/chel/343. [29] R. Temam and M. Ziane, Some mathematical problems in geophysical fluid dynamics, Handbook of Mathematical Fluid Dynamics, 3 (2004), 535-657. [30] Q. Wang, W. Zhou, D. Wang and D. Dong, Ocean model open boundary conditions with volume, heat and salinity conservation constraints, Advances in Atmospheric Sciences, 31 (2014), 188-196.  doi: 10.1007/s00376-013-2269-y. [31] G.-T. Yeh and C.-H. Huang, Three-dimensional air pollutant modelling in the lower atmosphere, Boundary-Layer Meteorology, 9 (1975), 381-390. [32] S. Zhuk, T. T. Tchrakian, S. Moore, R. Ordóñez-Hurtado and R. Shorten, On source-term parameter estimation for linear advection-diffusion equations with uncertain coefficients, SIAM J. Sci. Comput., 38 (2016), A2334-A2356.  doi: 10.1137/15M1034829.

show all references

##### References:
 [1] R. A. Adams and J. J. Fournier, Sobolev Spaces, 2nd edition, Pure and Applied Mathematics (Amsterdam), 140. Elsevier/Academic Press, Amsterdam, 2003. [2] I. Ali, S. Kalla and H. Khajah, A partial differential equation related to a problem in atmospheric pollution, Math. Comput. Modelling, 28 (1998), 1-6.  doi: 10.1016/S0895-7177(98)00169-1. [3] J. D. Anderson and J. Wendt, Computational Fluid Dynamics, volume 206, Springer, 1995. [4] G. Andria, A. Lay-Ekuakille and M. Notarnicola, Mathematical Models for Atmospheric and Industrial Pollutant Prediction, In XVI IMEKO World Congress, Vienna, Austria, 2000. [5] P. S. Arya, Introduction to Micrometeorology, volume 79, Elsevier, 2001. [6] P. Azérad and F. Guillén, Mathematical justification of the hydrostatic approximation in the primitive equations of geophysical fluid dynamics, SIAM J. Math. Anal., 33 (2001), 847-859.  doi: 10.1137/S0036141000375962. [7] O. Besson and M. Laydi, Some estimates for the anisotropic Navier-Stokes equations and for the hydrostatic approximation, RAIRO Modél. Math. Anal. Numér., 26 (1992), 855-865.  doi: 10.1051/m2an/1992260708551. [8] C. Cao, J. Li and E. S. Titi, Local and global well-posedness of strong solutions to the 3D primitive equations with vertical eddy diffusivity, Arch. Ration. Mech. Anal., 214 (2014), 35-76.  doi: 10.1007/s00205-014-0752-y. [9] C. Cao, J. Li and E. S. Titi, Global well-posedness of the three-dimensional primitive equations with only horizontal viscosity and diffusion, Comm. Pure Appl. Math., 69 (2016), 1492-1531.  doi: 10.1002/cpa.21576. [10] C. Cao and E. S. Titi, Global well-posedness and finite-dimensional global attractor for a 3-D planetary geostrophic viscous model, Comm. Pure Appl. Math., 56 (2003), 198-233.  doi: 10.1002/cpa.10056. [11] 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., 166 (2007), 245-267.  doi: 10.4007/annals.2007.166.245. [12] C. Cao and E. S. Titi, Global well-posedness of the 3D primitive equations with partial vertical turbulence mixing heat diffusion, Comm. Math. Phys., 310 (2012), 537-568.  doi: 10.1007/s00220-011-1409-4. [13] H. E. de Swart, The Governing Equations and the Dominant Balances of Flow in the Atmosphere and Ocean, USPC Summerschool, Utrechts, 2016. [14] D. Donatelli and N. Juhász, Weak solution of the merged mathematical equations of the polluted atmosphere, Math. Methods Appl. Sci., 43 (2020), 9245-9261.  doi: 10.1002/mma.6618. [15] I. Kukavica and M. Ziane, On the regularity of the primitive equations of the ocean, Nonlinearity, 20 (2007), 2739-2753.  doi: 10.1088/0951-7715/20/12/001. [16] H. Gao, S. Necasova and T. Tang, On weak-strong uniqueness and singular limit for the compressible primitive equations, Discrete Contin. Dyn. Syst., 40 (2020), 4287-4305.  doi: 10.3934/dcds.2020181. [17] P. Goyal and A. Kumar, Mathematical Modeling of Air Pollutants: An Application to Indian Urban City, Air Quality-Models and Applications, InTech, 2011. doi: 10.5772/16840. [18] B. Hosseini, Dispersion of Pollutants in the Atmosphere: A Numerical Study, Master's thesis, Simom Fraser University, 2013. [19] P. Kathirgamanathan, R. McKibbin and R. McLachlan, Source term estimation of pollution from an instantaneous point source, Research Letters in Information Mathematic Science, 3 (2002), 59-67. [20] J. Li and E. S. Titi, The primitive equations as the small aspect ratio limit of the Navier–Stokes equations: Rigorous justification of the hydrostatic approximation, J. Math. Pures Appl., 124 (2019), 30-58.  doi: 10.1016/j.matpur.2018.04.006. [21] P.-L. Lions, Mathematical Topics in Fluid Mechanics: Volume 2: Compressible Models, volume 2, Oxford University Press on Demand, 1996. [22] P. Marsaleix, F. Auclair and C. Estournel, Considerations on open boundary conditions for regional and coastal ocean models, Journal of Atmospheric and Oceanic Technology, 23 (2006), 1604-1613.  doi: 10.1175/JTECH1930.1. [23] A. Monin, On the boundary condition on the earth surface for diffusing pollution, Advances in Geophysics, 6 (1959), 435-436. [24] J. Pedlosky, Geophysical Fluid Dynamics, Springer Science & Business Media, 2013. [25] M. Petcu and D. Wirosoetisno, Sobolev and Gevrey regularity results for the primitive equations in three space dimensions, Appl. Anal., 84 (2005), 769-788.  doi: 10.1080/00036810500130745. [26] M. Prodanova, J. L. Perez, D. Syrakov, R. San Jose, K. Ganev, N. Miloshev and S. Roglev, Application of mathematical models to simulate an extreme air pollution episode in the Bulgarian city of Stara Zagora, Applied Mathematical Modelling, 32 (2008), 1607-1619.  doi: 10.1016/j.apm.2007.05.002. [27] J. Simon, Compact sets in the space $L^p(0, t; B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96.  doi: 10.1007/BF01762360. [28] R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, AMS Chelsea Publishing, Providence, RI, 2001. doi: 10.1090/chel/343. [29] R. Temam and M. Ziane, Some mathematical problems in geophysical fluid dynamics, Handbook of Mathematical Fluid Dynamics, 3 (2004), 535-657. [30] Q. Wang, W. Zhou, D. Wang and D. Dong, Ocean model open boundary conditions with volume, heat and salinity conservation constraints, Advances in Atmospheric Sciences, 31 (2014), 188-196.  doi: 10.1007/s00376-013-2269-y. [31] G.-T. Yeh and C.-H. Huang, Three-dimensional air pollutant modelling in the lower atmosphere, Boundary-Layer Meteorology, 9 (1975), 381-390. [32] S. Zhuk, T. T. Tchrakian, S. Moore, R. Ordóñez-Hurtado and R. Shorten, On source-term parameter estimation for linear advection-diffusion equations with uncertain coefficients, SIAM J. Sci. Comput., 38 (2016), A2334-A2356.  doi: 10.1137/15M1034829.
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