Study of a nonlinear boundary-value problem of geophysical relevance

Kateryna Marynets

doi:10.3934/dcds.2019194

Article Contents

2019, Volume 39, Issue 8: 4771-4781. Doi: 10.3934/dcds.2019194

This issue Previous Article On stratified water waves with critical layers and Coriolis forces Next Article On an exact solution of a nonlinear three-dimensional model in ocean flows with equatorial undercurrent and linear variation in density

Study of a nonlinear boundary-value problem of geophysical relevance

Kateryna Marynets^,

Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria

^* Corresponding author: Kateryna Marynets
^* Corresponding author: Kateryna Marynets

The paper is for the special theme: Mathematical Aspects of Physical Oceanography, organized by Adrian Constantin

Received: September 30, 2018

Revised: November 30, 2018

Published: May 2019

The author is supported by the WWTF research grant MA16-009

Abstract Full Text(HTML) Figure(3) Related Papers Cited by

Abstract

We present some results on the existence and uniqueness of solutions of a two-point nonlinear boundary value problem that arises in the modeling of the flow of the Antarctic Circumpolar Current.

Keywords:

Mathematics Subject Classification: Primary: 34B15; Secondary: 86A05.

Citation:

Full Text(HTML)

Figure 1. Depiction of the azimuthal and polar spherical coordinates $ \varphi \in [0,2\pi) $ and $ \theta \in [0,\pi] $ of a point $ P $ on the spherical surface of the Earth: $ \theta = 0 $ corresponds to the North Pole and $ \theta = \pi/2 $ to the Equator

Download: Full-size image PowerPoint slide

Figure 2. The unit vectors of the coordinate system on the eastward rotating spherical Earth, with $ e_{\varphi} $ pointing in the azimuthal direction

Download: Full-size image PowerPoint slide

Figure 3. Depiction of the stereographic projection $ P \mapsto P' $ from the North Pole to the equatorial plane, illustrated for a location that corresponds to the region where the Antarctic Circumplolar Current is encountered

Download: Full-size image PowerPoint slide

Related Papers

Cited by

References

[1]	A. Constantin, Global existence of solutions for perturbed differential equations, Ann. Mat. Pura Appl., 168 (1995), 237-299. doi: 10.1007/BF01759263.
[2]	A. Constantin, Nonlinear Water Waves with Applications to Wave-Current Interactions and Tsunamis, CBMS-NSF Regional Conference Series in Applied Mathematics, 81, SIAM, Philadelphia, PA, 2011. doi: 10.1137/1.9781611971873.
[3]	A. Constantin and R. S. Johnson, An exact, steady, purely azimuthal flow as a model for the Antarctic Circumpolar Current, J. Phys. Oceanogr., 46 (2016), 3585-3594. doi: 10.1175/JPO-D-16-0121.1.
[4]	A. Constantin and R. S. Johnson, Large gyres as a shallow-water asymptotic solution of Euler's equation in spherical coordinates, Proc. Roy. Soc. London A, 473 (2017), 20170063, 17 pp. doi: 10.1098/rspa.2017.0063.
[5]	A. Constantin and R. S. Johnson, Steady large-scale ocean flows in spherical coordinates, Oceanography, 31 (2018), 42-50. doi: 10.5670/oceanog.2018.308.
[6]	A. Constantin and S. G. Monismith, Gerstner waves in the presence of mean currents and rotation, J. Fluid Mech., 820 (2017), 511-528. doi: 10.1017/jfm.2017.223.
[7]	A. Constantin, W. Strauss and E. Varvaruca, Global bifurcation of steady gravity water waves with critical layers, Acta Mathematica, 217 (2016), 195-262. doi: 10.1007/s11511-017-0144-x.
[8]	W. A. Coppel, Stability and Asymptotic Behavior of Differential Equations, D. C. Heath and Co., Boston, Mass. 1965.
[9]	J. A. Ewing, Wind, wave and current data for the design of ships and offshore structures, Marine Structures, 3 (1990), 421-459. doi: 10.1016/0951-8339(90)90001-8.
[10]	S. V. Haziot and K. Marynets, Applying the stereographic projection to the modeling of the flow of the Antarctic Circumpolar Current, Oceanography, 31 (2018), 68-75.
[11]	H.-C. Hsu and C. I. Martin, On the existence of solutions and the pressure function related to the Antarctic Circumpolar Current, Nonlinear Anal., 155 (2017), 285-293. doi: 10.1016/j.na.2017.02.021.
[12]	D. Ionescu-Kruse, Local stability for an exact steady purely azimuthal flow which models the Antarctic Circumpolar Current, J. Math. Fluid Mech., 20 (2018), 569-579. doi: 10.1007/s00021-017-0335-4.
[13]	K. Marynets, On a two-point boundary-value problem in geophysics, Applicable Analysis, 98 (2019), 553-560. doi: 10.1080/00036811.2017.1395869.
[14]	K. Marynets, A nonlinear two-point boundary-value problem in geophysics, Monatsh Math., 188 (2019), 287-295. doi: 10.1007/s00605-017-1127-x.
[15]	K. Marynets, Two-point boundary-value problem for modeling the jet flow of the Antarctic Circumpolar Current, Electronic J. Diff. Eq., 56 (2018), Paper No. 56, 12 pp.
[16]	O. G. Mustafa and Y. V. Rogovchenko, Global existence of solutions for a class of nonlinear differential equations, Appl. Math. Letters, 16 (2003), 753-758. doi: 10.1016/S0893-9659(03)00078-8.
[17]	R. Quirchmayr, A steady, purely azimuthal flow model for the Antarctic Circumpolar Current, Monatsh. Math., 187 (2018), 565-572. doi: 10.1007/s00605-017-1097-z.
[18]	K. Schrader and P. Waltman, An existence theorem for nonlinear boundary value problems, Proc. Amer. Math. Soc., 21 (1969), 653-656. doi: 10.1090/S0002-9939-1969-0239176-0.
[19]	D. W. H. Walton, Antarctica: Global Science from a Frozen Continent, Cambridge University Press, Cambridge, 2013. doi: 10.1017/CBO9780511782299.