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August  2019, 39(8): 4415-4427. doi: 10.3934/dcds.2019179

Study of an elliptic partial differential equation modelling the Antarctic Circumpolar Current

Susanna V. Haziot

Fakultät für Mathematik, Universität Wien, Oskar-Morgenstern-Platz 1, Wien 1090, Austria

Received  July 2018 Revised  September 2018 Published  May 2019

We consider the ocean flow of the Antarctic Circumpolar Current. Using a recently-derived model for gyres in rotating spherical coordinates, and mapping the problem on the sphere onto the plane using the Mercator projection, we obtain a boundary-value problem for a semi-linear elliptic partial differential equation. For constant and linear oceanic vorticities, we investigate existence, regularity and uniqueness of solutions to this elliptic problem. We also provide some explicit solutions. Moreover, we examine the physical relevance of these results.

Keywords: Spherical coordinates, elliptic equation, gyres.
Mathematics Subject Classification: Primary: 35Q35, 76U05.
Citation: Susanna V. Haziot. Study of an elliptic partial differential equation modelling the Antarctic Circumpolar Current. Discrete & Continuous Dynamical Systems, 2019, 39 (8) : 4415-4427. doi: 10.3934/dcds.2019179
References:
[1]

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, Dover Publications, INC., New York, 1965. Google Scholar

[2]

G. E. Andrews, R. Askey and R. Roy, Special Functions, Cambridge University Press, 1999. doi: 10.1017/CBO9781107325937.  Google Scholar

[3]

H. Berestycki, L. Nirenberg and S. R. S. Varadhan, The principal eigenvalue and maximum principle for second-order elliptic operators in general domains, Comm. Pure Appl. Math., 47 (1994), 47-92.  doi: 10.1002/cpa.3160470105.  Google Scholar

[4]

H. Brézis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, 2011.  Google Scholar

[5]

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 Ser. A, 473 (2017), 20170063, 17pp. doi: 10.1098/rspa.2017.0063.  Google Scholar

[6]

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.  Google Scholar

[7]

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.  Google Scholar

[8]

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.  Google Scholar

[9]

D. Daners, The Mercator and stereographic projections, and many in between, Amer. Math. Monthly, 119 (2012), 199-210.  doi: 10.4169/amer.math.monthly.119.03.199.  Google Scholar

[10]

P. J. Dellar, Variations on a beta-plane: Derivation of non-traditional beta-plane equations from Hamilton's principle on a sphere, J. Fluid Mech., 674 (2011), 174-195.  doi: 10.1017/S0022112010006464.  Google Scholar

[11]

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.  Google Scholar

[12]

S. V. Haziot, Explicit two-dimensional solutions for the ocean flow in arctic gyres, Monatshefte für Mathematik, (2018), 1–12. doi: 10.1007/s00605-018-1198-3.  Google Scholar

[13]

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.  Google Scholar

[14]

C.W. Hughes, Nonlinear vorticity balance of the Antarctic Circumpolar Current, Journal of Geophysical Research, 110 (2005). doi: 10.1029/2004JC002753.  Google Scholar

[15]

I. G. Jonsson, Wave-current interactions, in The Sea, B. Le Méhauté, D.M. Hanes (Eds.), Ocean Eng. Sc., 9 (1990), 65-120. Google Scholar

[16]

E. Kamke, Differentialgleichungen: Lösungensmethoden und Lösungen, Neunte Auflage. Mit einem Vorwort von Detlef Kamke. B. G. Teubner, Stuttgart, 1977.  Google Scholar

[17]

K. Marynets, Two-point boundary problem for modeling the jet flow of the Antarctic Circumpolar Current, Electronic Journal of Differential Equations, 56 (2018), 1-12.   Google Scholar

[18]

M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Springer-Verlag, New York, 1984. doi: 10.1007/978-1-4612-5282-5.  Google Scholar

[19]

R. Quirchmayr, A steady, purely azimuthal flow model for the Antarctic Circumpolar Current, Monatshefte für Mathematik, 187 (2018), 565-572.  doi: 10.1007/s00605-017-1097-z.  Google Scholar

[20]

B. Riffenburgh, Encyclopedia of the Antarctic, Routledge, 2007. Google Scholar

[21]

S. R. Rintoul, Southern Ocean currents and climate, Papers and Proceedings of the Royal Society of Tasmania, 133 (2000), 41-50.  doi: 10.26749/rstpp.133.3.41.  Google Scholar

[22]

G. Szego, Inequalities for the zeros of Legendre Polynomials and related functions, Transactions of the American Mathematical Society, 39 (1936), 1-17.  doi: 10.1090/S0002-9947-1936-1501831-2.  Google Scholar

[23]

G. P. Thomas, Wave-current interactions: An experimental and numerical study, J. Fluid Mech., 216 (1990), 505-536.   Google Scholar

[24] G. K. Vallis, Atmosphere and Ocean Fluid Dynamics, Cambridge University Press, 2006.   Google Scholar
[25] D. W. H. Walton, Antarctica: Global Science from a Frozen Continent, Cambridge University Press, Cambridge, 2013.  doi: 10.1017/CBO9780511782299.  Google Scholar
[26]

V. Zeitlin, Geophysical Fluid Dynamics: Understanding (Almost) Everything with Rotating Shallow Water Models, Oxford University Press, 2018. Google Scholar

show all references

References:
[1]

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, Dover Publications, INC., New York, 1965. Google Scholar

[2]

G. E. Andrews, R. Askey and R. Roy, Special Functions, Cambridge University Press, 1999. doi: 10.1017/CBO9781107325937.  Google Scholar

[3]

H. Berestycki, L. Nirenberg and S. R. S. Varadhan, The principal eigenvalue and maximum principle for second-order elliptic operators in general domains, Comm. Pure Appl. Math., 47 (1994), 47-92.  doi: 10.1002/cpa.3160470105.  Google Scholar

[4]

H. Brézis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, 2011.  Google Scholar

[5]

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 Ser. A, 473 (2017), 20170063, 17pp. doi: 10.1098/rspa.2017.0063.  Google Scholar

[6]

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.  Google Scholar

[7]

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.  Google Scholar

[8]

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.  Google Scholar

[9]

D. Daners, The Mercator and stereographic projections, and many in between, Amer. Math. Monthly, 119 (2012), 199-210.  doi: 10.4169/amer.math.monthly.119.03.199.  Google Scholar

[10]

P. J. Dellar, Variations on a beta-plane: Derivation of non-traditional beta-plane equations from Hamilton's principle on a sphere, J. Fluid Mech., 674 (2011), 174-195.  doi: 10.1017/S0022112010006464.  Google Scholar

[11]

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.  Google Scholar

[12]

S. V. Haziot, Explicit two-dimensional solutions for the ocean flow in arctic gyres, Monatshefte für Mathematik, (2018), 1–12. doi: 10.1007/s00605-018-1198-3.  Google Scholar

[13]

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.  Google Scholar

[14]

C.W. Hughes, Nonlinear vorticity balance of the Antarctic Circumpolar Current, Journal of Geophysical Research, 110 (2005). doi: 10.1029/2004JC002753.  Google Scholar

[15]

I. G. Jonsson, Wave-current interactions, in The Sea, B. Le Méhauté, D.M. Hanes (Eds.), Ocean Eng. Sc., 9 (1990), 65-120. Google Scholar

[16]

E. Kamke, Differentialgleichungen: Lösungensmethoden und Lösungen, Neunte Auflage. Mit einem Vorwort von Detlef Kamke. B. G. Teubner, Stuttgart, 1977.  Google Scholar

[17]

K. Marynets, Two-point boundary problem for modeling the jet flow of the Antarctic Circumpolar Current, Electronic Journal of Differential Equations, 56 (2018), 1-12.   Google Scholar

[18]

M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Springer-Verlag, New York, 1984. doi: 10.1007/978-1-4612-5282-5.  Google Scholar

[19]

R. Quirchmayr, A steady, purely azimuthal flow model for the Antarctic Circumpolar Current, Monatshefte für Mathematik, 187 (2018), 565-572.  doi: 10.1007/s00605-017-1097-z.  Google Scholar

[20]

B. Riffenburgh, Encyclopedia of the Antarctic, Routledge, 2007. Google Scholar

[21]

S. R. Rintoul, Southern Ocean currents and climate, Papers and Proceedings of the Royal Society of Tasmania, 133 (2000), 41-50.  doi: 10.26749/rstpp.133.3.41.  Google Scholar

[22]

G. Szego, Inequalities for the zeros of Legendre Polynomials and related functions, Transactions of the American Mathematical Society, 39 (1936), 1-17.  doi: 10.1090/S0002-9947-1936-1501831-2.  Google Scholar

[23]

G. P. Thomas, Wave-current interactions: An experimental and numerical study, J. Fluid Mech., 216 (1990), 505-536.   Google Scholar

[24] G. K. Vallis, Atmosphere and Ocean Fluid Dynamics, Cambridge University Press, 2006.   Google Scholar
[25] D. W. H. Walton, Antarctica: Global Science from a Frozen Continent, Cambridge University Press, Cambridge, 2013.  doi: 10.1017/CBO9780511782299.  Google Scholar
[26]

V. Zeitlin, Geophysical Fluid Dynamics: Understanding (Almost) Everything with Rotating Shallow Water Models, Oxford University Press, 2018. Google Scholar

Figure 1.  The Mercator Projection for the ACC: We project the polar angle onto the $ x $-axis and the azimuthal angle onto the $ y $-axis. The South Pole is situated at $ x = -\infty $
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