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

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

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.

Citation: Susanna V. Haziot. Study of an elliptic partial differential equation modelling the Antarctic Circumpolar Current. Discrete and 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.

[2]

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

[3]

H. BerestyckiL. 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.

[4]

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

[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.

[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.

[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.

[8]

A. ConstantinW. 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.

[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.

[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.

[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.

[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.

[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.

[14]

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

[15]

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

[16]

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

[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. 

[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.

[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.

[20]

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

[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.

[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.

[23]

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

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

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

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.

[2]

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

[3]

H. BerestyckiL. 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.

[4]

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

[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.

[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.

[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.

[8]

A. ConstantinW. 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.

[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.

[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.

[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.

[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.

[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.

[14]

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

[15]

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

[16]

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

[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. 

[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.

[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.

[20]

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

[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.

[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.

[23]

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

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

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

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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