August  2019, 39(8): 4519-4532. doi: 10.3934/dcds.2019185

Benjamin-Ono model of an internal wave under a flat surface

1. 

School of Mathematical Sciences, University College Cork, Cork, Ireland

2. 

School of Mathematical Sciences, Technological University Dublin, City Campus, Kevin Street, Dublin 8, Ireland

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

Received  August 2018 Revised  October 2018 Published  May 2019

Fund Project: This work was supported by EPSRC grant no EP/K032208/1. AC is also funded by SFI grant 13/CDA/2117.

A two-layer fluid system separated by a pycnocline in the form of an internal wave is considered. The lower layer is infinitely deep, with a higher density than the upper layer which is bounded above by a flat surface. The fluids are incompressible and inviscid. A Hamiltonian formulation for the dynamics in the presence of a depth-varying current is presented and it is shown that an appropriate scaling leads to the integrable Benjamin-Ono equation.

Citation: Alan Compelli, Rossen Ivanov. Benjamin-Ono model of an internal wave under a flat surface. Discrete & Continuous Dynamical Systems - A, 2019, 39 (8) : 4519-4532. doi: 10.3934/dcds.2019185
References:
[1]

T. B. Benjamin, Internal waves of permanent form in fluids of great depth, J. Fluid Mech., 29 (1967), 559-562.  doi: 10.1017/S002211206700103X.  Google Scholar

[2]

T. B. Benjamin and T. J. Bridges, Reappraisal of the Kelvin-Helmholtz problem, Part 1. Hamiltonian structure, J. Fluid Mech., 333 (1997), 301-325.  doi: 10.1017/S0022112096004272.  Google Scholar

[3]

T. B. Benjamin and T. J. Bridges, Reappraisal of the Kelvin-Helmholtz problem, Part 2. Interaction of the Kelvin-Helmholtz, superharmonic and Benjamin-Feir instabilities, J. Fluid Mech., 333 (1997), 327-373.  doi: 10.1017/S0022112096004284.  Google Scholar

[4]

T. B. Benjamin and P. J. Olver, Hamiltonian structure, symmetries and conservation laws for water waves, J. Fluid Mech., 125 (1982), 137-185.  doi: 10.1017/S0022112082003292.  Google Scholar

[5]

J. L. BonaD. Lannes and J.-C. Saut, Asymptotic models for internal waves, J. Math. Pures Appl., 89 (2008), 538-566.  doi: 10.1016/j.matpur.2008.02.003.  Google Scholar

[6]

A. Compelli, Hamiltonian formulation of 2 bounded immiscible media with constant non-zero vorticities and a common interface, Wave Motion, 54 (2015), 115-124.  doi: 10.1016/j.wavemoti.2014.11.015.  Google Scholar

[7]

A. Compelli, Hamiltonian approach to the modeling of internal geophysical waves with vorticity, Monatsh. Math., 179 (2016), 509-521.  doi: 10.1007/s00605-014-0724-1.  Google Scholar

[8]

A. Compelli and R. Ivanov, On the dynamics of internal waves interacting with the Equatorial Undercurrent, J. Nonlinear Math. Phys., 22 (2015), 531-539.  doi: 10.1080/14029251.2015.1113052.  Google Scholar

[9]

A. Compelli and R. Ivanov, The dynamics of flat surface internal geophysical waves with currents, J. Math. Fluid Mech., 19 (2017), 329-344.  doi: 10.1007/s00021-016-0283-4.  Google Scholar

[10]

A. Constantin, Nonlinear water waves with applications to wave-current interactions and tsunamis,, CBMS-NSF Regional Conference Series in Applied Mathematics, 81 Society for Industrial and Applied Mathematics, Philadelphia, 2011. doi: 10.1137/1.9781611971873.  Google Scholar

[11]

A. Constantin and J. Escher, Symmetry of steady periodic surface water waves with vorticity, J. Fluid Mech., 498 (2004), 171-181.  doi: 10.1017/S0022112003006773.  Google Scholar

[12]

A. Constantin and J. Escher, Analyticity of periodic traveling free surface water waves with vorticity, Ann.Math., 173 (2011), 559-568.  doi: 10.4007/annals.2011.173.1.12.  Google Scholar

[13]

A. Constantin and R. Ivanov, A Hamiltonian approach to wave-current interactions in two-layer fluids, Phys. Fluids, 27 (2015), 086603.  doi: 10.1063/1.4929457.  Google Scholar

[14]

A. ConstantinR. Ivanov and C. -I. Martin, Hamiltonian formulation for wave-current interactions in stratified rotational flows, Arch. Rational Mech. Anal., 221 (2016), 1417-1447.  doi: 10.1007/s00205-016-0990-2.  Google Scholar

[15]

A. ConstantinR. Ivanov and E. Prodanov, Nearly-Hamiltonian structure for water waves with constant vorticity, J. Math. Fluid Mech., 10 (2008), 224-237.  doi: 10.1007/s00021-006-0230-x.  Google Scholar

[16]

A. Constantin and R. S. Johnson, The dynamics of waves interacting with the Equatorial Undercurrent, Geophys. Astrophys. Fluid Dyn., 109 (2015), 311-358.  doi: 10.1080/03091929.2015.1066785.  Google Scholar

[17]

A. ConstantinD. Sattinger and W. Strauss, Variational formulations for steady water waves with vorticity, J. Fluid Mech., 548 (2006), 151-163.  doi: 10.1017/S0022112005007469.  Google Scholar

[18]

A. Constantin and W. Strauss, Exact steady periodic water waves with vorticity, Comm. Pure Appl. Math., 57 (2004), 481-527.  doi: 10.1002/cpa.3046.  Google Scholar

[19]

W. Craig and M. Groves, Hamiltonian long-wave approximations to the water-wave problem, Wave Motion, 19 (1994), 367-389.  doi: 10.1016/0165-2125(94)90003-5.  Google Scholar

[20]

W. CraigP. Guyenne and H. Kalisch, Hamiltonian long-wave expansions for free surfaces and interfaces, Comm. Pure Appl. Math., 58 (2005), 1587-1641.  doi: 10.1002/cpa.20098.  Google Scholar

[21]

W. CraigP. Guyenne and C. Sulem, Coupling between internal and surface waves, Nat. Hazards, 57 (2011), 617-642.  doi: 10.1007/s11069-010-9535-4.  Google Scholar

[22]

W. Craig and C. Sulem, Numerical simulation of gravity waves, J. Comp. Phys., 108 (1993), 73-83.  doi: 10.1006/jcph.1993.1164.  Google Scholar

[23]

A. V. Fedorov and J. N. Brown, Equatorial waves, in Encyclopedia of Ocean Sciences (ed. J. Steele), Academic, San Diego, Calif., 2009,271–287. doi: 10.1016/B978-012374473-9.00610-X.  Google Scholar

[24]

A. S. Fokas and M. J. Ablowitz, The inverse scattering transform for the Benjamin-Ono equation – a pivot to multidimensional problems, Stud. Appl. Math., 68 (1983), 1-10.  doi: 10.1002/sapm19836811.  Google Scholar

[25] R. S. Johnson, A Modern Introduction to the Mathematical Theory of Water Waves, Cambridge University Press, Cambridge, 1997.  doi: 10.1017/CBO9780511624056.  Google Scholar
[26]

I. G. Jonsson, Wave-current interactions, In: The sea 9 (1990), Wiley: New York, 65–120. Google Scholar

[27]

D. J. Kaup and Y. Matsuno, The inverse scattering transform for the Benjamin-Ono equation, Stud. Appl. Math., 101 (1998), 73-98.  doi: 10.1111/1467-9590.00086.  Google Scholar

[28]

D. Milder, A note regarding `On Hamilton's principle for water waves', J. Fluid Mech., 83 (1977), 159-161.   Google Scholar

[29]

J. Miles, On Hamilton's principle for water waves, J. Fluid Mech., 83 (1977), 153-158.  doi: 10.1017/S0022112077001104.  Google Scholar

[30]

F. Nansen, Farthest North: Volume I, Harper and Brothers, 1897. Google Scholar

[31]

H. Ono, Algebraic solitary waves in stratified fluids, J. Phys.Soc. Japan, 39 (1975), 1082-1091.  doi: 10.1143/JPSJ.39.1082.  Google Scholar

[32]

D. H. Peregrine, Interaction of water waves and currents, Adv. Appl. Mech., 16 (1976), 9-117.   Google Scholar

[33]

A. F. Teles da Silva and D. H. Peregrine, Steep, steady surface waves on water of finite depth with constant vorticity, J. Fluid Mech., 195 (1988), 281-302.  doi: 10.1017/S0022112088002423.  Google Scholar

[34]

G. P. Thomas and G. Klopman, Wave-current interactions in the near-shore region, In: Gravity Waves in Water of Finite Depth, Advances in Fluid Mechanics, Computational Mechanics Publications, 1997. Google Scholar

[35]

E. Wahlén, A Hamiltonian formulation of water waves with constant vorticity, Lett. Math. Phys., 79 (2007), 303-315.  doi: 10.1007/s11005-007-0143-5.  Google Scholar

[36]

V. Zakharov, Stability of periodic waves of finite amplitude on the surface of a deep fluid,, Zh. Prikl. Mekh. Tekh. Fiz., 9 (1968), 86–94 (in Russian); J. Appl. Mech. Tech. Phys., 9 (1968), 190–194 (English translation). doi: 10.1007/BF00913182.  Google Scholar

show all references

References:
[1]

T. B. Benjamin, Internal waves of permanent form in fluids of great depth, J. Fluid Mech., 29 (1967), 559-562.  doi: 10.1017/S002211206700103X.  Google Scholar

[2]

T. B. Benjamin and T. J. Bridges, Reappraisal of the Kelvin-Helmholtz problem, Part 1. Hamiltonian structure, J. Fluid Mech., 333 (1997), 301-325.  doi: 10.1017/S0022112096004272.  Google Scholar

[3]

T. B. Benjamin and T. J. Bridges, Reappraisal of the Kelvin-Helmholtz problem, Part 2. Interaction of the Kelvin-Helmholtz, superharmonic and Benjamin-Feir instabilities, J. Fluid Mech., 333 (1997), 327-373.  doi: 10.1017/S0022112096004284.  Google Scholar

[4]

T. B. Benjamin and P. J. Olver, Hamiltonian structure, symmetries and conservation laws for water waves, J. Fluid Mech., 125 (1982), 137-185.  doi: 10.1017/S0022112082003292.  Google Scholar

[5]

J. L. BonaD. Lannes and J.-C. Saut, Asymptotic models for internal waves, J. Math. Pures Appl., 89 (2008), 538-566.  doi: 10.1016/j.matpur.2008.02.003.  Google Scholar

[6]

A. Compelli, Hamiltonian formulation of 2 bounded immiscible media with constant non-zero vorticities and a common interface, Wave Motion, 54 (2015), 115-124.  doi: 10.1016/j.wavemoti.2014.11.015.  Google Scholar

[7]

A. Compelli, Hamiltonian approach to the modeling of internal geophysical waves with vorticity, Monatsh. Math., 179 (2016), 509-521.  doi: 10.1007/s00605-014-0724-1.  Google Scholar

[8]

A. Compelli and R. Ivanov, On the dynamics of internal waves interacting with the Equatorial Undercurrent, J. Nonlinear Math. Phys., 22 (2015), 531-539.  doi: 10.1080/14029251.2015.1113052.  Google Scholar

[9]

A. Compelli and R. Ivanov, The dynamics of flat surface internal geophysical waves with currents, J. Math. Fluid Mech., 19 (2017), 329-344.  doi: 10.1007/s00021-016-0283-4.  Google Scholar

[10]

A. Constantin, Nonlinear water waves with applications to wave-current interactions and tsunamis,, CBMS-NSF Regional Conference Series in Applied Mathematics, 81 Society for Industrial and Applied Mathematics, Philadelphia, 2011. doi: 10.1137/1.9781611971873.  Google Scholar

[11]

A. Constantin and J. Escher, Symmetry of steady periodic surface water waves with vorticity, J. Fluid Mech., 498 (2004), 171-181.  doi: 10.1017/S0022112003006773.  Google Scholar

[12]

A. Constantin and J. Escher, Analyticity of periodic traveling free surface water waves with vorticity, Ann.Math., 173 (2011), 559-568.  doi: 10.4007/annals.2011.173.1.12.  Google Scholar

[13]

A. Constantin and R. Ivanov, A Hamiltonian approach to wave-current interactions in two-layer fluids, Phys. Fluids, 27 (2015), 086603.  doi: 10.1063/1.4929457.  Google Scholar

[14]

A. ConstantinR. Ivanov and C. -I. Martin, Hamiltonian formulation for wave-current interactions in stratified rotational flows, Arch. Rational Mech. Anal., 221 (2016), 1417-1447.  doi: 10.1007/s00205-016-0990-2.  Google Scholar

[15]

A. ConstantinR. Ivanov and E. Prodanov, Nearly-Hamiltonian structure for water waves with constant vorticity, J. Math. Fluid Mech., 10 (2008), 224-237.  doi: 10.1007/s00021-006-0230-x.  Google Scholar

[16]

A. Constantin and R. S. Johnson, The dynamics of waves interacting with the Equatorial Undercurrent, Geophys. Astrophys. Fluid Dyn., 109 (2015), 311-358.  doi: 10.1080/03091929.2015.1066785.  Google Scholar

[17]

A. ConstantinD. Sattinger and W. Strauss, Variational formulations for steady water waves with vorticity, J. Fluid Mech., 548 (2006), 151-163.  doi: 10.1017/S0022112005007469.  Google Scholar

[18]

A. Constantin and W. Strauss, Exact steady periodic water waves with vorticity, Comm. Pure Appl. Math., 57 (2004), 481-527.  doi: 10.1002/cpa.3046.  Google Scholar

[19]

W. Craig and M. Groves, Hamiltonian long-wave approximations to the water-wave problem, Wave Motion, 19 (1994), 367-389.  doi: 10.1016/0165-2125(94)90003-5.  Google Scholar

[20]

W. CraigP. Guyenne and H. Kalisch, Hamiltonian long-wave expansions for free surfaces and interfaces, Comm. Pure Appl. Math., 58 (2005), 1587-1641.  doi: 10.1002/cpa.20098.  Google Scholar

[21]

W. CraigP. Guyenne and C. Sulem, Coupling between internal and surface waves, Nat. Hazards, 57 (2011), 617-642.  doi: 10.1007/s11069-010-9535-4.  Google Scholar

[22]

W. Craig and C. Sulem, Numerical simulation of gravity waves, J. Comp. Phys., 108 (1993), 73-83.  doi: 10.1006/jcph.1993.1164.  Google Scholar

[23]

A. V. Fedorov and J. N. Brown, Equatorial waves, in Encyclopedia of Ocean Sciences (ed. J. Steele), Academic, San Diego, Calif., 2009,271–287. doi: 10.1016/B978-012374473-9.00610-X.  Google Scholar

[24]

A. S. Fokas and M. J. Ablowitz, The inverse scattering transform for the Benjamin-Ono equation – a pivot to multidimensional problems, Stud. Appl. Math., 68 (1983), 1-10.  doi: 10.1002/sapm19836811.  Google Scholar

[25] R. S. Johnson, A Modern Introduction to the Mathematical Theory of Water Waves, Cambridge University Press, Cambridge, 1997.  doi: 10.1017/CBO9780511624056.  Google Scholar
[26]

I. G. Jonsson, Wave-current interactions, In: The sea 9 (1990), Wiley: New York, 65–120. Google Scholar

[27]

D. J. Kaup and Y. Matsuno, The inverse scattering transform for the Benjamin-Ono equation, Stud. Appl. Math., 101 (1998), 73-98.  doi: 10.1111/1467-9590.00086.  Google Scholar

[28]

D. Milder, A note regarding `On Hamilton's principle for water waves', J. Fluid Mech., 83 (1977), 159-161.   Google Scholar

[29]

J. Miles, On Hamilton's principle for water waves, J. Fluid Mech., 83 (1977), 153-158.  doi: 10.1017/S0022112077001104.  Google Scholar

[30]

F. Nansen, Farthest North: Volume I, Harper and Brothers, 1897. Google Scholar

[31]

H. Ono, Algebraic solitary waves in stratified fluids, J. Phys.Soc. Japan, 39 (1975), 1082-1091.  doi: 10.1143/JPSJ.39.1082.  Google Scholar

[32]

D. H. Peregrine, Interaction of water waves and currents, Adv. Appl. Mech., 16 (1976), 9-117.   Google Scholar

[33]

A. F. Teles da Silva and D. H. Peregrine, Steep, steady surface waves on water of finite depth with constant vorticity, J. Fluid Mech., 195 (1988), 281-302.  doi: 10.1017/S0022112088002423.  Google Scholar

[34]

G. P. Thomas and G. Klopman, Wave-current interactions in the near-shore region, In: Gravity Waves in Water of Finite Depth, Advances in Fluid Mechanics, Computational Mechanics Publications, 1997. Google Scholar

[35]

E. Wahlén, A Hamiltonian formulation of water waves with constant vorticity, Lett. Math. Phys., 79 (2007), 303-315.  doi: 10.1007/s11005-007-0143-5.  Google Scholar

[36]

V. Zakharov, Stability of periodic waves of finite amplitude on the surface of a deep fluid,, Zh. Prikl. Mekh. Tekh. Fiz., 9 (1968), 86–94 (in Russian); J. Appl. Mech. Tech. Phys., 9 (1968), 190–194 (English translation). doi: 10.1007/BF00913182.  Google Scholar

Figure 1.  The system under study
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