Discontinuous differential equation for modelling the Antarctic Circumpolar Current

Michal Fečkan; Shan Li; JinRong Wang; Michal Fečkan; Shan Li; JinRong Wang

doi:10.3934/cam.2024036

Communications in Analysis and Mechanics

2024, Volume 16, Issue 4: 836-857. doi: 10.3934/cam.2024036

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Research article

Discontinuous differential equation for modelling the Antarctic Circumpolar Current

1.
Department of Mathematical Analysis and Numerical Mathematics, Faculty of Mathematics Physics and Informatics, Comenius University in Bratislava, Mlynská dolina, 842 48 Bratislava, Slovakia, and Mathematical Institute, Slovak Academy of Sciences, Štefánikova 49,814 73 Bratislava, Slovakia
2.
Department of Mathematics, Guizhou University, Guiyang, Guizhou 550025, China and Supercomputing Algorithm and Application Laboratory of Guizhou University and Gui'an Scientific Innovation Company, Guizhou University, Guiyang, Guizhou 550025, China

Received: 17 July 2024 Revised: 08 August 2024 Accepted: 08 August 2024 Published: 06 November 2024
34B15, 34B27, 34B60

In this paper, we were concerned with the existence of the solution related to the discontinuous differential equation, which corresponded to the stratification phenomenon in the Antarctic Circumpolar Current (ACC). By considering the piecewise vorticity function, we demonstrated the existence of solution corresponding to the discontinuous differential equation using Green's function, fixed point theory, and topological degree theory. This primarily included cases with piecewise constant vorticity, piecewise linear vorticity, and piecewise nonlinear vorticity. Additionally, we provided some examples to verify our results.
- Antarctic Circumpolar Current,
- stratification,
- Green's function,
- discontinuous differential equation,
- topological degree theory
Citation: Michal Fečkan, Shan Li, JinRong Wang. Discontinuous differential equation for modelling the Antarctic Circumpolar Current[J]. Communications in Analysis and Mechanics, 2024, 16(4): 836-857. doi: 10.3934/cam.2024036

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Abstract

In this paper, we were concerned with the existence of the solution related to the discontinuous differential equation, which corresponded to the stratification phenomenon in the Antarctic Circumpolar Current (ACC). By considering the piecewise vorticity function, we demonstrated the existence of solution corresponding to the discontinuous differential equation using Green's function, fixed point theory, and topological degree theory. This primarily included cases with piecewise constant vorticity, piecewise linear vorticity, and piecewise nonlinear vorticity. Additionally, we provided some examples to verify our results.

References

[1]	R. Karsten, H. Jones, J. Marshall, The role of eddy transfer in setting the stratification and transport of a circumpolar current, J. Phys. Oceanogr., 32 (2002), 39–54.
[2]	R. H. Karsten, J. Marshall, Testing theories of the vertical stratification of the ACC against observations, Dyn. Atmos. Oceans., 36 (2002), 233–246. https://doi.org/10.1016/S0377-0265(02)00031-3 doi: 10.1016/S0377-0265(02)00031-3
[3]	J. Marshall, H. Jones, R. Karsten, R. Wardle, Can eddies set ocean stratification? J. Phys. Oceanogr., 32 (2002), 26–38.
[4]	A. Constantin, On the modelling of equatorial waves, Geophys. Res. Lett., 39 (2012). https://doi.org/10.1029/2012GL051169 doi: 10.1029/2012GL051169
[5]	A. Constantin, R. S. Johnson, The dynamics of waves interacting with the equatorial undercurrent, Geophys Astrophys Fluid Dyn., 109 (2015), 311–358. https://doi.org/10.1080/03091929.2015.1066785 doi: 10.1080/03091929.2015.1066785
[6]	A. Constantin, R. S. Johnson, A nonlinear, three-dimensional model for ocean flows, motivated by some observations of the Pacific Equatorial Undercurrent and thermocline, Phys. Fluids., 29 (2017), 056604. https://doi.org/10.1063/1.4984001 doi: 10.1063/1.4984001
[7]	J. P. McCreary Jr, Modeling equatorial ocean circulation, Ann. Rev. Fluid Mech., 17 (1985), 359–409. https://doi.org/10.1146/annurev.fl.17.010185.002043 doi: 10.1146/annurev.fl.17.010185.002043
[8]	A. Constantin, R. S. Johnson, An exact, steady, purely azimuthal equatorial flow with a free surface, J. Phys. Oceanogr., 46 (2016), 1935–1945. https://doi.org/10.1175/JPO-D-15-0205.1 doi: 10.1175/JPO-D-15-0205.1
[9]	A. Constantin, R. S. Johnson, An exact, steady, purely azimuthal flow as a model for the Antarctic Circumpolar Current, J. Phys. Oceanogr., 46 (2016), 3585–3594. https://doi.org/10.1175/JPO-D-16-0121.1 doi: 10.1175/JPO-D-16-0121.1
[10]	D. Henry, C. I. Martin, Free-surface, purely azimuthal equatorial flows in spherical coordinates with stratification, J Differ. Equ., 266 (2019), 6788–6808. https://doi.org/10.1016/j.jde.2018.11.017 doi: 10.1016/j.jde.2018.11.017
[11]	D. Henry, C. I. Martin, Azimuthal equatorial flows with variable density in spherical coordinates, Arch. Ration. Mech. Anal., 233 (2019), 497–512.
[12]	D. Henry, C. I. Martin, Stratified equatorial flows in cylindrical coordinates, Nonlinearity, 33 (2020), 3889. https://doi.org/10.1088/1361-6544/ab801f doi: 10.1088/1361-6544/ab801f
[13]	C. I. Martin, Azimuthal equatorial flows in spherical coordinates with discontinuous stratification, Phys. Fluids., 33 (2021), 026602. https://doi.org/10.1063/5.0035443 doi: 10.1063/5.0035443
[14]	C. I. Martin, R. Quirchmayr, Exact solutions and internal waves for the Antarctic circumpolar current in spherical coordinates, Stud. Appl. Math., 148 (2022), 1021–1039. https://doi.org/10.1111/sapm.12467 doi: 10.1111/sapm.12467
[15]	C. I. Martin, R. Quirchmayr, Explicit and exact solutions concerning the Antarctic circumpolar currentwith variable density in spherical coordinates, J. Math. Phys., 60 (2019), 101505. https://doi.org/10.1063/1.5120627 doi: 10.1063/1.5120627
[16]	K. Deimling, Multivalued Differential Equations, Berlin, New York: De Gruyter, 1992. https://doi.org/10.1515/9783110874228
[17]	K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, Berlin, 1985. https://doi.org/10.1007/978-3-662-00547-7
[18]	M. Fečkan, Topological Degree Approach to Bifurcation Problems, Springer, Berlin, 2008. https://doi.org/10.1007/978-1-4020-8724-0

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