The Existence of Weak Solutions for the Vorticity Equation Related to the Stratosphere in a Rotating Spherical Coordinate System
In this paper, based on the Euler equation and mass conservation equation in spherical coordinates, the ratio of the stratospheric average width to the planetary radius and the ratio of the vertical velocity to the horizontal velocity are selected as parameters under appropriate boundary conditions....
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MDPI AG
2022-07-01
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| Online Access: | https://www.mdpi.com/2075-1680/11/7/347 |
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| author | Wenlin Zhang Michal Fečkan Jinrong Wang |
| author_facet | Wenlin Zhang Michal Fečkan Jinrong Wang |
| author_sort | Wenlin Zhang |
| collection | DOAJ |
| description | In this paper, based on the Euler equation and mass conservation equation in spherical coordinates, the ratio of the stratospheric average width to the planetary radius and the ratio of the vertical velocity to the horizontal velocity are selected as parameters under appropriate boundary conditions. We establish the approximate system using these two small parameters. In addition, we consider the time dependence of the system and establish the governing equations describing the atmospheric flow. By introducing a flow function to code the system, a nonlinear vorticity equation describing the planetary flow in the stratosphere is obtained. The governing equations describing the atmospheric flow are transformed into a second-order homogeneous linear ordinary differential equation and a Legendre’s differential equation by applying the method of separating variables based on the concepts of spherical harmonic functions and weak solutions. The Gronwall inequality and the Cauchy–Schwartz inequality are applied to priori estimates for the vorticity equation describing the stratospheric planetary flow under the appropriate initial and boundary conditions. The existence and non-uniqueness of weak solutions to the vorticity equation are obtained by using the functional analysis technique. |
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| format | Article |
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| language | English |
| last_indexed | 2024-03-09T10:23:46Z |
| publishDate | 2022-07-01 |
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| spelling | doaj.art-fee2dd0c77694cfb9f592030b9d322132023-12-01T21:53:29ZengMDPI AGAxioms2075-16802022-07-0111734710.3390/axioms11070347The Existence of Weak Solutions for the Vorticity Equation Related to the Stratosphere in a Rotating Spherical Coordinate SystemWenlin Zhang0Michal Fečkan1Jinrong Wang2School of Mathematics and Statistics, Liupanshui Normal University, Liupanshui 553004, ChinaDepartment of Mathematical Analysis and Numerical Mathematics, Faculty of Mathematics, Physics and Informatics, Comenius University in Bratislava, Mlynská dolina, 842 48 Bratislava, SlovakiaDepartment of Mathematics, Guizhou University, Guiyang 550025, ChinaIn this paper, based on the Euler equation and mass conservation equation in spherical coordinates, the ratio of the stratospheric average width to the planetary radius and the ratio of the vertical velocity to the horizontal velocity are selected as parameters under appropriate boundary conditions. We establish the approximate system using these two small parameters. In addition, we consider the time dependence of the system and establish the governing equations describing the atmospheric flow. By introducing a flow function to code the system, a nonlinear vorticity equation describing the planetary flow in the stratosphere is obtained. The governing equations describing the atmospheric flow are transformed into a second-order homogeneous linear ordinary differential equation and a Legendre’s differential equation by applying the method of separating variables based on the concepts of spherical harmonic functions and weak solutions. The Gronwall inequality and the Cauchy–Schwartz inequality are applied to priori estimates for the vorticity equation describing the stratospheric planetary flow under the appropriate initial and boundary conditions. The existence and non-uniqueness of weak solutions to the vorticity equation are obtained by using the functional analysis technique.https://www.mdpi.com/2075-1680/11/7/347stratospherevorticitystream functionweak solution |
| spellingShingle | Wenlin Zhang Michal Fečkan Jinrong Wang The Existence of Weak Solutions for the Vorticity Equation Related to the Stratosphere in a Rotating Spherical Coordinate System Axioms stratosphere vorticity stream function weak solution |
| title | The Existence of Weak Solutions for the Vorticity Equation Related to the Stratosphere in a Rotating Spherical Coordinate System |
| title_full | The Existence of Weak Solutions for the Vorticity Equation Related to the Stratosphere in a Rotating Spherical Coordinate System |
| title_fullStr | The Existence of Weak Solutions for the Vorticity Equation Related to the Stratosphere in a Rotating Spherical Coordinate System |
| title_full_unstemmed | The Existence of Weak Solutions for the Vorticity Equation Related to the Stratosphere in a Rotating Spherical Coordinate System |
| title_short | The Existence of Weak Solutions for the Vorticity Equation Related to the Stratosphere in a Rotating Spherical Coordinate System |
| title_sort | existence of weak solutions for the vorticity equation related to the stratosphere in a rotating spherical coordinate system |
| topic | stratosphere vorticity stream function weak solution |
| url | https://www.mdpi.com/2075-1680/11/7/347 |
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