• Previous Article
    Persistence and convergence in parabolic-parabolic chemotaxis system with logistic source on $ \mathbb{R}^{N} $
  • DCDS Home
  • This Issue
  • Next Article
    Stability of hyperbolic Oseledets splittings for quasi-compact operator cocycles
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. AliS. 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. CaoJ. 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. CaoJ. 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. GaoS. 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. KathirgamanathanR. 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. MarsaleixF. 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. ProdanovaJ. L. PerezD. SyrakovR. San JoseK. GanevN. 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. WangW. ZhouD. 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. ZhukT. T. TchrakianS. MooreR. 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. AliS. 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. CaoJ. 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. CaoJ. 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. GaoS. 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. KathirgamanathanR. 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. MarsaleixF. 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. ProdanovaJ. L. PerezD. SyrakovR. San JoseK. GanevN. 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. WangW. ZhouD. 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. ZhukT. T. TchrakianS. MooreR. 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.

Figure 1.  The boundary structure of $ \Omega $ for the case of weak solutions
[1]

Reinhard Racke, Jürgen Saal. Hyperbolic Navier-Stokes equations II: Global existence of small solutions. Evolution Equations and Control Theory, 2012, 1 (1) : 217-234. doi: 10.3934/eect.2012.1.217

[2]

Peter E. Kloeden, José Valero. The Kneser property of the weak solutions of the three dimensional Navier-Stokes equations. Discrete and Continuous Dynamical Systems, 2010, 28 (1) : 161-179. doi: 10.3934/dcds.2010.28.161

[3]

Daniel Coutand, J. Peirce, Steve Shkoller. Global well-posedness of weak solutions for the Lagrangian averaged Navier-Stokes equations on bounded domains. Communications on Pure and Applied Analysis, 2002, 1 (1) : 35-50. doi: 10.3934/cpaa.2002.1.35

[4]

Daniel Pardo, José Valero, Ángel Giménez. Global attractors for weak solutions of the three-dimensional Navier-Stokes equations with damping. Discrete and Continuous Dynamical Systems - B, 2019, 24 (8) : 3569-3590. doi: 10.3934/dcdsb.2018279

[5]

Fang Li, Bo You, Yao Xu. Dynamics of weak solutions for the three dimensional Navier-Stokes equations with nonlinear damping. Discrete and Continuous Dynamical Systems - B, 2018, 23 (10) : 4267-4284. doi: 10.3934/dcdsb.2018137

[6]

Jian-Guo Liu, Zhaoyun Zhang. Existence of global weak solutions of $ p $-Navier-Stokes equations. Discrete and Continuous Dynamical Systems - B, 2022, 27 (1) : 469-486. doi: 10.3934/dcdsb.2021051

[7]

Xiaoyu Chen, Jijie Zhao, Qian Zhang. Global existence of weak solutions for the 3D axisymmetric chemotaxis-Navier-Stokes equations with nonlinear diffusion. Discrete and Continuous Dynamical Systems, 2022  doi: 10.3934/dcds.2022062

[8]

Chérif Amrouche, María Ángeles Rodríguez-Bellido. On the very weak solution for the Oseen and Navier-Stokes equations. Discrete and Continuous Dynamical Systems - S, 2010, 3 (2) : 159-183. doi: 10.3934/dcdss.2010.3.159

[9]

C. Foias, M. S Jolly, I. Kukavica, E. S. Titi. The Lorenz equation as a metaphor for the Navier-Stokes equations. Discrete and Continuous Dynamical Systems, 2001, 7 (2) : 403-429. doi: 10.3934/dcds.2001.7.403

[10]

Vittorino Pata. On the regularity of solutions to the Navier-Stokes equations. Communications on Pure and Applied Analysis, 2012, 11 (2) : 747-761. doi: 10.3934/cpaa.2012.11.747

[11]

Tian Ma, Shouhong Wang. Asymptotic structure for solutions of the Navier--Stokes equations. Discrete and Continuous Dynamical Systems, 2004, 11 (1) : 189-204. doi: 10.3934/dcds.2004.11.189

[12]

John W. Barrett, Endre Süli. Existence of global weak solutions to Fokker-Planck and Navier-Stokes-Fokker-Planck equations in kinetic models of dilute polymers. Discrete and Continuous Dynamical Systems - S, 2010, 3 (3) : 371-408. doi: 10.3934/dcdss.2010.3.371

[13]

Fei Jiang, Song Jiang, Junpin Yin. Global weak solutions to the two-dimensional Navier-Stokes equations of compressible heat-conducting flows with symmetric data and forces. Discrete and Continuous Dynamical Systems, 2014, 34 (2) : 567-587. doi: 10.3934/dcds.2014.34.567

[14]

Changbing Hu, Roger Temam, Mohammed Ziane. The primitive equations on the large scale ocean under the small depth hypothesis. Discrete and Continuous Dynamical Systems, 2003, 9 (1) : 97-131. doi: 10.3934/dcds.2003.9.97

[15]

Alessio Falocchi, Filippo Gazzola. Regularity for the 3D evolution Navier-Stokes equations under Navier boundary conditions in some Lipschitz domains. Discrete and Continuous Dynamical Systems, 2022, 42 (3) : 1185-1200. doi: 10.3934/dcds.2021151

[16]

Hongjun Gao, Šárka Nečasová, Tong Tang. On weak-strong uniqueness and singular limit for the compressible Primitive Equations. Discrete and Continuous Dynamical Systems, 2020, 40 (7) : 4287-4305. doi: 10.3934/dcds.2020181

[17]

Jingrui Wang, Keyan Wang. Almost sure existence of global weak solutions to the 3D incompressible Navier-Stokes equation. Discrete and Continuous Dynamical Systems, 2017, 37 (9) : 5003-5019. doi: 10.3934/dcds.2017215

[18]

Oleg Imanuvilov. On the asymptotic properties for stationary solutions to the Navier-Stokes equations. Discrete and Continuous Dynamical Systems, 2021, 41 (5) : 2301-2340. doi: 10.3934/dcds.2020366

[19]

Minghua Yang, Zunwei Fu, Jinyi Sun. Global solutions to Chemotaxis-Navier-Stokes equations in critical Besov spaces. Discrete and Continuous Dynamical Systems - B, 2018, 23 (8) : 3427-3460. doi: 10.3934/dcdsb.2018284

[20]

Bingyuan Huang, Shijin Ding, Huanyao Wen. Local classical solutions of compressible Navier-Stokes-Smoluchowski equations with vacuum. Discrete and Continuous Dynamical Systems - S, 2016, 9 (6) : 1717-1752. doi: 10.3934/dcdss.2016072

2020 Impact Factor: 1.392

Metrics

  • PDF downloads (192)
  • HTML views (134)
  • Cited by (0)

Other articles
by authors

[Back to Top]