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Vortex Solutions for the 2D Boussinesq Equations Under the Radial Gravity
Mahdi Kamandar , Behruz Raesi
Abstract:  
The main objective of this article is to find some vortex solutions of finite core size for plane Boussinesq equations under the radial gravity, coupled with a diffusive equation of temperature in a weighted subspace of L2(R2). Solutions are expanded into series of Hermite eigenfunctions. We find the coefficients of the series and show the convergence of them.
Keywords: Boussinesq equations, Vortex solution, Hermite functions.
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Type of Study: Research paper | Subject: Special
References
1. G. K. Batchelor, An introduction to fluid mechanics, Cambridge University Press, 1967.
2. M. Ben-Artzi, Global Solutions of Two-dimensional Navier-Stokes and Euler Equations, Archive for rational mechanics and analysis, 128, (1994), 329-358. [DOI:10.1007/BF00387712]
3. A. J. Bernoff, J. F. Lingevitch, Rapid Relaxation of an Axisymmetric Vortex, Physics of Fluids, 6, (1994), 3717-3723. [DOI:10.1063/1.868362]
4. S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability, Dover Publications, Inc., 1981.
5. P. Constantin, C. Doering, Heat Transfer in Convective Turbulence, Nonlinearity, 9, (1996), 1049-1060. [DOI:10.1088/0951-7715/9/4/013]
6. T. Gallay, Interaction of Vortices in Weakly Viscous Planar Flows, Archive for rational mechanics and analysis, 200, (2011), 445-490. [DOI:10.1007/s00205-010-0362-2]
7. T. Gallay, C. E. Wayne, Invariant Manifolds and the Long-time Asymptotics of the Navier-Stokes and Vorticity Equations on R2, Archive for rational mechanics and analysis, 163, (2002), 209-258. [DOI:10.1007/s002050200200]
8. A. E. Gill, Atmosphere Ocean Dynamics, Elsevier, 2016.
9. H. V. Helmholtz, Uber Integrale der Hydrodynamischen Gleichungen, ¨ Journal fur die reine und angewandte Mathematik, 55, (1858), 25-55. [DOI:10.1515/crll.1858.55.25]
10. T. Hou, C. Li, Global Well-posedness of the Viscous Boussinesq Equations, Discrete and Continuous Dynamical Systems, 12, (2005), 1-12. [DOI:10.3934/dcds.2005.12.1]
11. F. Jing, E. Kanso, P. K. Newton, Viscous Evolution of Point Vortex Equilibria: The Collinear State, Physics of Fluids, 22, (2010), 123102. [DOI:10.1063/1.3516637]
12. L. Kelvin, On Vortex Motion, Transactions of the Royal Society of Edinburgh, 25, (1868), 217-60. [DOI:10.1017/S0080456800028179]
13. H. Lamb, Hydrodynamics, Cambridge Univ Press, 1932.
14. M. Lappa, Thermal Convection: Patterns Evolutions and Stability, Wiley publication, 2010. [DOI:10.1002/9780470749982]
15. T. Ma, Sh. Wang, Rayleigh B'enard Convection: Dynamics and Structure in the Physical Space, Communications in Mathematical Sciences, 5(3), (2007), 553-574. [DOI:10.4310/CMS.2007.v5.n3.a3]
16. T. Ma, Sh. wang, Phase Transition Dynamics, Springer, 2014.
17. T. Ma, Sh. Wang, Tropical Atmospheric Circulations: Dynamic Stability and Transitions, Discrete and Continuous Dynamical Systems, 26(4), (2009), 1399-1417. [DOI:10.3934/dcds.2010.26.1399]
18. T. Ma, Sh. Wang, Astrophysical Dynamics and Cosmology, Journal of Mathematical Study, 47(4), (2014), 305-378. [DOI:10.4208/jms.v47n4.14.01]
19. A. Majda, Introduction to PDEs and Waves for the Atmosphere and Ocean, American Mathematical Soc., 2003. [DOI:10.1090/cln/009]
20. A. Majda, A. Bertozzi, Vorticity and Incompressible Flow, Cambridge University Press, 2002. [DOI:10.1017/CBO9780511613203]
21. L. M. Milne-Thomson, Theoretical Hydrodynamics, Dover Publications, 1968. [DOI:10.1007/978-1-349-00517-8]
22. R. J. Nagem, G. Sandri, D. Uminsky, C. E. Wayne, Generalized Helmholtz-Kirchhoff Model for Two-dimensional Distributed Vortex Motion, SIAM Journal on Applied Dynamical Systems, 8, (2009), 160-179. [DOI:10.1137/080715056]
23. R. J. Nagem, G. Sandri, D. Uminsky, Vorticity Dynamics and Sound Generation in Twodimensional Fluid Flow, The Journal of the Acoustical Society of America, 122, (2007), 128-134. [DOI:10.1121/1.2736513]
24. P. K. Newton,The N-Vortex Problem: Analytical Techniques, Springer-Verlag, 2001. [DOI:10.1007/978-1-4684-9290-3]
25. S. Ozer, T. S¸eng¨ul, Transitions of Spherical Thermohaline Circulation to Multiple Equi- ¨libria, Journal of Mathematical Fluid Mechanics, 20, (2018), 499-515. [DOI:10.1007/s00021-017-0331-8]
26. J. Pedlosky, Geophysical Fluid Dynamics I, II, Springer, 1981. [DOI:10.1007/978-3-662-25730-2]
27. L. Prandtl, Essentials of Fluid Dynamics, Hafner Pub. Co., 1952.
28. L. F. Richardson, Weather Prediction by Numerical Process, Cambridge University Press, 1922.
29. P. G. Saffman, Vortex Dynamics, Cambridge university press, 1992. [DOI:10.1017/CBO9780511624063]
30. M. Sharifi, B. Raesi, Vortex Theory for Two Dimensional Boussinesq Equations, Applied Mathematics and Nonlinear Sciences, 5(2), (2020), 67-84. [DOI:10.2478/amns.2020.2.00014]
31. S. G. L. Smith, R. J. Nagem, Vortex Pairs and Dipoles, Regular and Chaotic Dynamics, 18, (2013), 194-201. [DOI:10.1134/S1560354713010140]
32. L. Ting, C. Tung, Motion and Decay of a Vortex in a Nonuniform Stream, The Physics of Fluids, 8, (1965), 1039-1051. [DOI:10.1063/1.1761353]
33. D. Uminsky, C. E. Wayne, A. Barbaro, A Multi-moment Vortex Method for 2D Viscous Fluids, Journal of Computational Physics, 231, (2012), 1705-1727. [DOI:10.1016/j.jcp.2011.11.001]
34. J. Wu, The 2D Incompressible Boussinesq Equations: Lecture Notes, Oklahoma State University, 2012.
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