July  2022, 21(7): 2337-2356. doi: 10.3934/cpaa.2022057

Energy considerations for nonlinear equatorial water waves

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

Received  December 2021 Revised  February 2022 Published  July 2022 Early access  March 2022

In this article we consider the excess kinetic and potential energies for exact nonlinear equatorial water waves. An investigation of linear waves establishes that the excess kinetic energy density is always negative, whereas the excess potential energy density is always positive, for periodic travelling irrotational water waves in the steady reference frame. For negative wavespeeds, we prove that similar inequalities must also hold for nonlinear wave solutions. Characterisations of the various excess energy densities as integrals along the wave surface profile are also derived.

Citation: David Henry. Energy considerations for nonlinear equatorial water waves. Communications on Pure and Applied Analysis, 2022, 21 (7) : 2337-2356. doi: 10.3934/cpaa.2022057
References:
[1]

A. Aivaliotis, On the symmetry of equatorial travelling water waves with constant vorticity and stagnation points, Nonlinear Anal. Real World Appl., 34 (2017), 159-171.  doi: 10.1016/j.nonrwa.2016.08.010.

[2]

A. Aleman and A. Constantin, On the decrease of kinetic energy with depth in wave-current interactions, Math. Ann., 378 (2020), 853-872.  doi: 10.1007/s00208-019-01910-8.

[3]

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

[4]

P. Bonneton and D. Lannes, Recovering water wave elevation from pressure measurements, J. Fluid Mech., 833 (2017), 399-429.  doi: 10.1017/jfm.2017.666.

[5]

D. Clamond, Note on the velocity and related fields of steady irrotational two-dimensional surface gravity waves, Philos. Trans. R. Soc. Lond. Ser. A, 370 (2012), 1572-1586.  doi: 10.1098/rsta.2011.0470.

[6]

D. Clamond, New exact relations for easy recovery of steady wave profiles from bottom pressure measurements, J. Fluid Mech., 726 (2013), 547-558.  doi: 10.1017/jfm.2013.253.

[7]

D. Clamond and A. Constantin, Recovery of steady periodic wave profiles from pressure measurements at the bed, J. Fluid Mech., 714 (2013), 463-475.  doi: 10.1017/jfm.2012.490.

[8]

D. Clamond and D. Henry, Extreme water-wave profile recovery from pressure measurements at the seabed, J. Fluid Mech., 903 (2020), 12 pp. doi: 10.1017/jfm.2020.729.

[9]

A. Constantin, Nonlinear Water Waves with Applications to Wave-Current Interactions and Tsunamis, CBMS-NSF Conference Series in Applied Mathematics, SIAM, Philadelphia, 2011. doi: 10.1137/1.9781611971873.

[10]

A. Constantin, An exact solution for equatorially trapped waves, J. Geophys. Res.: Oceans, 117 (2012), 8 pp. doi: 10.1029/2012jc007879.

[11]

A. Constantin, On the recovery of solitary wave profiles from pressure measurements, J. Fluid Mech., 699 (2012), 376-384.  doi: 10.1017/jfm.2012.114.

[12]

A. Constantin, On equatorial wind waves, Differ. Integral Equ., 26 (2013), 237-252. 

[13]

A. Constantin, Mean velocities in a Stokes wave, Arch. Ration. Mech. Anal., 207 (2013), 907-917.  doi: 10.1007/s00205-012-0584-6.

[14]

A. Constantin and R. I. Ivanov, Equatorial wave-current interactions, Commun. Math. Phys., 370 (2019), 1-48.  doi: 10.1007/s00220-019-03483-8.

[15]

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

[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.

[17]

A. Constantin and R. S. Johnson, An exact, steady, purely azimuthal equatorial flow with a free surface, J. Phys. Oceanogr., 46 (2016), 1935-1945. 

[18]

A. Constantin and R. S. Johnson, Current and future prospects for the application of systematic theoretical methods to the study of problems in physical oceanography, Phys. Lett. A, 380 (2016), 3007-3012. 

[19]

A. Constantin and R. S. Johnson, A nonlinear, three-dimensional model for ocean flows, motivated by some observations of the Pacific Equatorial Undercurrent and thermocline, Phys. Fluids, 29 (2017), 21 pp. doi: 10.1063/1.4984001.

[20]

G. D. Crapper, Introduction to Water Waves, Ellis Horwood, Chichester, 1984.

[21]

B. Cushman-Roisin and J. M. Beckers, Introduction to Geophysical Fluid Dynamics: Physical and Numerical Aspects, Academic, Waltham, Mass., 2011.

[22]

R. G. Dean and R. A. Dalrymple, Water Wave Mechanics for Engineers and Scientists, World Scientific Publishing, 1991.

[23]

L. Fan, Mean velocities in an irrotational equatorial wind wave, Appl. Numer. Math., 141 (2019), 158-166.  doi: 10.1016/j.apnum.2019.03.001.

[24]

A. V. Fedorov and J. N. Brown, Equatorial waves, in Encyclopedia of Ocean Sciences, Academic, San Diego, Calif., 2009.

[25] L. E. Fraenkel, An Introduction to Maximum Principles and Symmetry in Elliptic Problems, Cambridge University Press, Cambridge, 2000.  doi: 10.1017/CBO9780511569203.
[26] A. Gill, Atmosphere-Ocean Dynamics, Academic Press, New York, 1982. 
[27]

D. Henry, On three-dimensional Gerstner-like equatorial water waves, Philos. Trans. Roy. Soc. A, 376 (2018), 20170088, 16 pp. doi: 10.1098/rsta.2017.0088.

[28]

D. Henry, On nonlinearity in three-dimensional equatorial flows, J. Nonlinear Math. Phys., 25 (2018), 351–357. doi: 10.1080/14029251.2018.1494780.

[29]

D. Henry, On the energy of nonlinear water waves, Proc. R. Soc. A., 477 (2021), 12 pp. doi: 10.1098/rspa.2021.0544.

[30]

D. Henry and C. I. Martin, Exact, free-surface equatorial flows with general stratification in spherical coordinates, Arch. Ration. Mech. Anal., 233 (2019), 497–512. doi: 10.1007/s00205-019-01362-z.

[31]

D. Henry and C. I. Martin, Stratified equatorial flows in cylindrical coordinates, Nonlinearity, 33 (2020), 3889–3904. doi: 10.1088/1361-6544/ab801f.

[32]

D. Henry and A.-V. Matioc, On the existence of equatorial wind waves, Nonlinear Anal., 101 (2014), 113–123. doi: 10.1016/j.na.2014.01.018.

[33]

D. Henry and A.-V. Matioc, On the symmetry of steady equatorial wind waves, Nonlinear Anal. Real World Appl., 18 (2014), 50–56. doi: 10.1016/j.nonrwa.2014.01.009.

[34]

D. Henry and G. P. Thomas, Prediction of the free-surface elevation for rotational water waves using the recovery of pressure at the bed, Philos. Trans. Roy. Soc. A, 376 (2018), 20170102, 21 pp. doi: 10.1098/rsta.2017.0102.

[35]

D. Ionescu-Kruse and A.-V. Matioc, Small-amplitude equatorial water waves with constant vorticity: dispersion relations and particle trajectories, Discrete Contin. Dyn. Syst., 34 (2014), 3045–3060. doi: 10.3934/dcds.2014.34.3045.

[36]

R. S. Johnson, Application of the ideas and techniques of classical fluid mechanics to some problems in physical oceanography, Phil. Trans. R. Soc. A, 376 (2018), 20170092, 19 pp. doi: 10.1098/rsta.2017.0092.

[37] J. Lighthill, Waves in Fluids, Cambridge University Press, Cambridge-New York, 1978. 
[38]

M. S. Longuet-Higgins, On the decrease of velocity with depth in an irrotational water wave, Math. Proc. Camb. Philos. Soc., 49 (1953), 552–560.

[39]

M. S. Longuet-Higgins, Integral properties of periodic gravity waves of finite amplitude, Proc. R. Soc. Lond. A, 342 (1975), 157–174. doi: 10.1098/rspa.1975.0018.

[40]

M. S. Longuet-Higgins, New integral relations for gravity waves of finite amplitude, J. Fluid Mech., 149 (1984), 205–215. doi: 10.1017/S0022112084002615.

[41]

C. I. Martin, Equatorial wind waves with capillary effects and stagnation points, Nonlinear Anal. TMA, 96 (2014), 1–17. doi: 10.1016/j.na.2013.10.025.

[42]

C. I. Martin, On periodic geophysical water flows with discontinuous vorticity in the equatorial $f-$plane approximation, Phil. Trans. Roy. Soc. A, 376 (2018), 20170096, 23 pp. doi: 10.1098/rsta.2017.0096.

[43]

R. Quirchmayr, On irrotational flows beneath periodic traveling equatorial waves, J. Math. Fluid Mech., 19 (2017), 283–304 doi: 10.1007/s00021-016-0280-7.

[44]

L. Roberti, On the decrease of velocity with depth in irrotational periodic water waves, Monatsh. Math., 193 (2020), 671–682. doi: 10.1007/s00605-020-01451-2.

[45]

V. T. Starr, Momentum and energy integrals for gravity waves of finite height, J. Mar. Res. 6 (1947), 175–193.

[46]

G. P. Thomas, The theory behind the conversion of ocean wave energy: a review, in Ocean Wave Energy: Current status and Future Perspectives, 41–91, Springer, Berlin-Heidelberg, 2008.

[47]

G. Thomas and G. Klopman, Wave-current interactions in the nearshore region, in Gravity Waves in Water of Finite Depth, 215–319, Advances in Fluid Mechanics, Southhampton, United Kingdom, 1997.

[48]

G. K. Vallis, Atmospheric and Oceanic Fluid Dynamics, Cambridge University Press, 2006.

show all references

References:
[1]

A. Aivaliotis, On the symmetry of equatorial travelling water waves with constant vorticity and stagnation points, Nonlinear Anal. Real World Appl., 34 (2017), 159-171.  doi: 10.1016/j.nonrwa.2016.08.010.

[2]

A. Aleman and A. Constantin, On the decrease of kinetic energy with depth in wave-current interactions, Math. Ann., 378 (2020), 853-872.  doi: 10.1007/s00208-019-01910-8.

[3]

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

[4]

P. Bonneton and D. Lannes, Recovering water wave elevation from pressure measurements, J. Fluid Mech., 833 (2017), 399-429.  doi: 10.1017/jfm.2017.666.

[5]

D. Clamond, Note on the velocity and related fields of steady irrotational two-dimensional surface gravity waves, Philos. Trans. R. Soc. Lond. Ser. A, 370 (2012), 1572-1586.  doi: 10.1098/rsta.2011.0470.

[6]

D. Clamond, New exact relations for easy recovery of steady wave profiles from bottom pressure measurements, J. Fluid Mech., 726 (2013), 547-558.  doi: 10.1017/jfm.2013.253.

[7]

D. Clamond and A. Constantin, Recovery of steady periodic wave profiles from pressure measurements at the bed, J. Fluid Mech., 714 (2013), 463-475.  doi: 10.1017/jfm.2012.490.

[8]

D. Clamond and D. Henry, Extreme water-wave profile recovery from pressure measurements at the seabed, J. Fluid Mech., 903 (2020), 12 pp. doi: 10.1017/jfm.2020.729.

[9]

A. Constantin, Nonlinear Water Waves with Applications to Wave-Current Interactions and Tsunamis, CBMS-NSF Conference Series in Applied Mathematics, SIAM, Philadelphia, 2011. doi: 10.1137/1.9781611971873.

[10]

A. Constantin, An exact solution for equatorially trapped waves, J. Geophys. Res.: Oceans, 117 (2012), 8 pp. doi: 10.1029/2012jc007879.

[11]

A. Constantin, On the recovery of solitary wave profiles from pressure measurements, J. Fluid Mech., 699 (2012), 376-384.  doi: 10.1017/jfm.2012.114.

[12]

A. Constantin, On equatorial wind waves, Differ. Integral Equ., 26 (2013), 237-252. 

[13]

A. Constantin, Mean velocities in a Stokes wave, Arch. Ration. Mech. Anal., 207 (2013), 907-917.  doi: 10.1007/s00205-012-0584-6.

[14]

A. Constantin and R. I. Ivanov, Equatorial wave-current interactions, Commun. Math. Phys., 370 (2019), 1-48.  doi: 10.1007/s00220-019-03483-8.

[15]

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

[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.

[17]

A. Constantin and R. S. Johnson, An exact, steady, purely azimuthal equatorial flow with a free surface, J. Phys. Oceanogr., 46 (2016), 1935-1945. 

[18]

A. Constantin and R. S. Johnson, Current and future prospects for the application of systematic theoretical methods to the study of problems in physical oceanography, Phys. Lett. A, 380 (2016), 3007-3012. 

[19]

A. Constantin and R. S. Johnson, A nonlinear, three-dimensional model for ocean flows, motivated by some observations of the Pacific Equatorial Undercurrent and thermocline, Phys. Fluids, 29 (2017), 21 pp. doi: 10.1063/1.4984001.

[20]

G. D. Crapper, Introduction to Water Waves, Ellis Horwood, Chichester, 1984.

[21]

B. Cushman-Roisin and J. M. Beckers, Introduction to Geophysical Fluid Dynamics: Physical and Numerical Aspects, Academic, Waltham, Mass., 2011.

[22]

R. G. Dean and R. A. Dalrymple, Water Wave Mechanics for Engineers and Scientists, World Scientific Publishing, 1991.

[23]

L. Fan, Mean velocities in an irrotational equatorial wind wave, Appl. Numer. Math., 141 (2019), 158-166.  doi: 10.1016/j.apnum.2019.03.001.

[24]

A. V. Fedorov and J. N. Brown, Equatorial waves, in Encyclopedia of Ocean Sciences, Academic, San Diego, Calif., 2009.

[25] L. E. Fraenkel, An Introduction to Maximum Principles and Symmetry in Elliptic Problems, Cambridge University Press, Cambridge, 2000.  doi: 10.1017/CBO9780511569203.
[26] A. Gill, Atmosphere-Ocean Dynamics, Academic Press, New York, 1982. 
[27]

D. Henry, On three-dimensional Gerstner-like equatorial water waves, Philos. Trans. Roy. Soc. A, 376 (2018), 20170088, 16 pp. doi: 10.1098/rsta.2017.0088.

[28]

D. Henry, On nonlinearity in three-dimensional equatorial flows, J. Nonlinear Math. Phys., 25 (2018), 351–357. doi: 10.1080/14029251.2018.1494780.

[29]

D. Henry, On the energy of nonlinear water waves, Proc. R. Soc. A., 477 (2021), 12 pp. doi: 10.1098/rspa.2021.0544.

[30]

D. Henry and C. I. Martin, Exact, free-surface equatorial flows with general stratification in spherical coordinates, Arch. Ration. Mech. Anal., 233 (2019), 497–512. doi: 10.1007/s00205-019-01362-z.

[31]

D. Henry and C. I. Martin, Stratified equatorial flows in cylindrical coordinates, Nonlinearity, 33 (2020), 3889–3904. doi: 10.1088/1361-6544/ab801f.

[32]

D. Henry and A.-V. Matioc, On the existence of equatorial wind waves, Nonlinear Anal., 101 (2014), 113–123. doi: 10.1016/j.na.2014.01.018.

[33]

D. Henry and A.-V. Matioc, On the symmetry of steady equatorial wind waves, Nonlinear Anal. Real World Appl., 18 (2014), 50–56. doi: 10.1016/j.nonrwa.2014.01.009.

[34]

D. Henry and G. P. Thomas, Prediction of the free-surface elevation for rotational water waves using the recovery of pressure at the bed, Philos. Trans. Roy. Soc. A, 376 (2018), 20170102, 21 pp. doi: 10.1098/rsta.2017.0102.

[35]

D. Ionescu-Kruse and A.-V. Matioc, Small-amplitude equatorial water waves with constant vorticity: dispersion relations and particle trajectories, Discrete Contin. Dyn. Syst., 34 (2014), 3045–3060. doi: 10.3934/dcds.2014.34.3045.

[36]

R. S. Johnson, Application of the ideas and techniques of classical fluid mechanics to some problems in physical oceanography, Phil. Trans. R. Soc. A, 376 (2018), 20170092, 19 pp. doi: 10.1098/rsta.2017.0092.

[37] J. Lighthill, Waves in Fluids, Cambridge University Press, Cambridge-New York, 1978. 
[38]

M. S. Longuet-Higgins, On the decrease of velocity with depth in an irrotational water wave, Math. Proc. Camb. Philos. Soc., 49 (1953), 552–560.

[39]

M. S. Longuet-Higgins, Integral properties of periodic gravity waves of finite amplitude, Proc. R. Soc. Lond. A, 342 (1975), 157–174. doi: 10.1098/rspa.1975.0018.

[40]

M. S. Longuet-Higgins, New integral relations for gravity waves of finite amplitude, J. Fluid Mech., 149 (1984), 205–215. doi: 10.1017/S0022112084002615.

[41]

C. I. Martin, Equatorial wind waves with capillary effects and stagnation points, Nonlinear Anal. TMA, 96 (2014), 1–17. doi: 10.1016/j.na.2013.10.025.

[42]

C. I. Martin, On periodic geophysical water flows with discontinuous vorticity in the equatorial $f-$plane approximation, Phil. Trans. Roy. Soc. A, 376 (2018), 20170096, 23 pp. doi: 10.1098/rsta.2017.0096.

[43]

R. Quirchmayr, On irrotational flows beneath periodic traveling equatorial waves, J. Math. Fluid Mech., 19 (2017), 283–304 doi: 10.1007/s00021-016-0280-7.

[44]

L. Roberti, On the decrease of velocity with depth in irrotational periodic water waves, Monatsh. Math., 193 (2020), 671–682. doi: 10.1007/s00605-020-01451-2.

[45]

V. T. Starr, Momentum and energy integrals for gravity waves of finite height, J. Mar. Res. 6 (1947), 175–193.

[46]

G. P. Thomas, The theory behind the conversion of ocean wave energy: a review, in Ocean Wave Energy: Current status and Future Perspectives, 41–91, Springer, Berlin-Heidelberg, 2008.

[47]

G. Thomas and G. Klopman, Wave-current interactions in the nearshore region, in Gravity Waves in Water of Finite Depth, 215–319, Advances in Fluid Mechanics, Southhampton, United Kingdom, 1997.

[48]

G. K. Vallis, Atmospheric and Oceanic Fluid Dynamics, Cambridge University Press, 2006.

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