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August  2019, 39(8): 4771-4781. doi: 10.3934/dcds.2019194

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

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

Received  October 2018 Revised  December 2018 Published  May 2019

Fund Project: The author is supported by the WWTF research grant MA16-009.

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: Geophysical flow, nonlinear boundary-value problem, global existence, uniqueness, positive-definite function.
Mathematics Subject Classification: Primary: 34B15; Secondary: 86A05.
Citation: Kateryna Marynets. Study of a nonlinear boundary-value problem of geophysical relevance. Discrete and Continuous Dynamical Systems, 2019, 39 (8) : 4771-4781. doi: 10.3934/dcds.2019194
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.

show all references

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