doi: 10.3934/jgm.2022004
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Nonlinear dispersion in wave-current interactions

Department of Mathematics, Imperial College London, SW7 2AZ, UK

*Corresponding author: Ruiao Hu

Received  August 2021 Early access March 2022

Fund Project: DH is partially supported by ERC Synergy Grant 856408 - STUOD (Stochastic Transport in Upper Ocean Dynamics). RH is supported by an EPSRC scholarship [grant number EP/R513052/1]

Via a sequence of approximations of the Lagrangian in Hamilton's principle for dispersive nonlinear gravity waves we derive a hierarchy of Hamiltonian models for describing wave-current interaction (WCI) in nonlinear dispersive wave dynamics on free surfaces. A subclass of these WCI Hamiltonians admits emergent singular solutions for certain initial conditions. These singular solutions are identified with a singular momentum map for left action of the diffeomorphisms on a semidirect-product Lie algebra. This semidirect-product Lie algebra comprises vector fields representing horizontal current velocity acting on scalar functions representing wave elevation. We use computational simulations to demonstrate the dynamical interactions of the emergent wavefront trains which are admitted by this special subclass of Hamiltonians for a variety of initial conditions.

In this paper, we investigate:

(1) A variety of localised initial current configurations in still water whose subsequent propagation generates surface-elevation dynamics on an initially flat surface; and

(2) The release of initially confined configurations of surface elevation in still water that generate dynamically interacting fronts of localised currents and wave trains. The results of these simulations show intricate wave-current interaction patterns whose structures are similar to those seen, for example, in Synthetic Aperture Radar (SAR) images taken from the space shuttle.

Citation: Darryl D. Holm, Ruiao Hu. Nonlinear dispersion in wave-current interactions. Journal of Geometric Mechanics, doi: 10.3934/jgm.2022004

References:
[1]

X. BaiX. LiK. G. Lamb and J. Hu, Internal solitary wave reflection near dongsha atoll, the south china sea, Journal of Geophysical Research: Oceans, 122 (2017), 7978-7991.  doi: 10.1002/2017JC012880.

[2]

R. Barros and W. Choi, Inhibiting shear instability induced by large amplitude internal solitary waves in two-layer flows with a free surface, Stud. Appl. Math., 122 (2009), 325-346.  doi: 10.1111/j.1467-9590.2009.00436.x.

[3]

J. Burbea and C. R. Rao, Entropy differential metric, distance and divergence measures in probability spaces: A unified approach, J. Multivariate Anal., 12 (1982), 575-596.  doi: 10.1016/0047-259X(82)90065-3.

[4]

H. CendraD. D. HolmJ. E. Marsden and T. S. Ratiu, Lagrangian reduction, the Euler-Poincaré equations, and semidirect products, Translations of the American Mathematical Society-Series 2, 186 (1998), 1-26.  doi: 10.1090/trans2/186/01.

[5]

S. Chakravarty and Y. Kodama, Soliton solutions of the KP equation and application to shallow water waves, Stud. Appl. Math., 123 (2009), 83-151.  doi: 10.1111/j.1467-9590.2009.00448.x.

[6]

S. ChenC. FoiasD. D. HolmE. OlsonE. S. Titi and S. Wynne, Camassa-Holm equations as a closure model for turbulent channel and pipe flow, Phys. Rev. Lett., 81 (1998), 5338-5341.  doi: 10.1103/PhysRevLett.81.5338.

[7]

S. ChenC. FoiasD. D. HolmE. OlsonE. S. Titi and S. Wynne, A connection between the Camassa-Holm equations and turbulent flows in channels and pipes, Phys. Fluids, 11 (1999), 2343-2353.  doi: 10.1063/1.870096.

[8]

M. ChenS.-Q. Liu and Y. Zhang, A two-component generalization of the Camassa-Holm equation and its solutions, Lett. Math. Phys., 75 (2006), 1-15.  doi: 10.1007/s11005-005-0041-7.

[9]

W. Choi and R. Camassa, Weakly nonlinear internal waves in a two-fluid system, J. Fluid Mech., 313 (1996), 83-103.  doi: 10.1017/S0022112096002133.

[10]

W. Choi and R. Camassa, Fully nonlinear internal waves in a two-fluid system, J. Fluid Mech., 396 (1999), 1-36.  doi: 10.1017/S0022112099005820.

[11]

A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations, Arch. Ration. Mech. Anal., 192 (2009), 165-186.  doi: 10.1007/s00205-008-0128-2.

[12]

C. J. CotterJ. ElderingD. D. HolmH. O. Jacobs and D. M. Meier, Weak dual pairs and jetlet methods for ideal incompressible fluid models in $n\ge 2$ dimensions, J. Nonlinear Sci., 26 (2016), 1723-1765.  doi: 10.1007/s00332-016-9317-6.

[13]

C. J. Cotter, D. D. Holm, H. O. Jacobs and D. M. Meier, A jetlet hierarchy for ideal fluid dynamics, J. Phys., 47 (2014), 352001, 8 pp. doi: 10.1088/1751-8113/47/35/352001.

[14]

C. J. CotterD. D. Holm and J. R. Percival, The square root depth wave equations, Proc. Roy. Soc. A Math. Phys., 466 (2010), 3621-3633.  doi: 10.1098/rspa.2010.0124.

[15]

V. Duchêne, Many Models for Water Waves, Doctoral dissertation, Université de Rennes, 2021, https://tel.archives-ouvertes.fr/tel-03282212

[16]

V. DuchêneS. Israwi and R. Talhouk, A new class of two-layer Green-Naghdi systems with improved frequency dispersion, Stud. Appl. Math., 137 (2016), 356-415.  doi: 10.1111/sapm.12125.

[17]

G. Falqui, On a Camassa-Holm type equation with two dependent variables, J. Phys. A, 39 (2006), 327-342.  doi: 10.1088/0305-4470/39/2/004.

[18]

D. M. FarmerL. ArmiL. Armi and D. M. Farmer, The flow of Atlantic water through the Strait of gibraltar, Progress in Oceanography, 21 (1988), 1-103.  doi: 10.1016/0079-6611(88)90055-9.

[19]

C. FoiasD. D. Holm and E. S. Titi, The Navier-Stokes-alpha model of fluid turbulence, Phys. D, 152/153 (2001), 505-519.  doi: 10.1016/S0167-2789(01)00191-9.

[20]

C. FoiasD. D. Holm and E. S. Titi, The three dimensional viscous Camassa-Holm equations, and their relation to the Navier-Stokes equations and turbulence theory, J. Dynam. Differential Equations, 14 (2002), 1-35.  doi: 10.1023/A:1012984210582.

[21]

F. Gay-Balmaz and C. Vizman, Dual pairs in fluid dynamics, Ann. Global Anal. Geom., 41 (2012), 1-24.  doi: 10.1007/s10455-011-9267-z.

[22]

D. D. Holm, Variational principles, geometry and topology of Lagrangian-averaged fluid dynamics, An Introduction to the Geometry and Topology of Fluid Flows, 47 (2001), 271-291.  doi: 10.1007/978-94-010-0446-6_14.

[23]

D. D. Holm, Variational principles for stochastic fluid dynamics, Proc. A., 471 (2015), 20140963, 19 pp. doi: 10.1098/rspa.2014.0963.

[24]

D. D. Holm and H. O. Jacobs, Multipole vortex blobs (MVB): Symplectic geometry and dynamics, J. Nonlinear Sci., 27 (2017), 973-1006.  doi: 10.1007/s00332-017-9367-4.

[25]

D. D. Holm and J. E. Marsden, Momentum maps and measure-valued solutions (peakons, filaments, and sheets) for the EPDiff equation, In Progr. Math., Birkhäuser Boston, 232 (2005), 203-235. doi: 10.1007/0-8176-4419-9_8.

[26]

D. D. HolmJ. E. Marsden and T. S. Ratiu, Euler-Poincaré models of ideal fluids with nonlinear dispersion, Physical Review Letters, 80 (1998), 4173-4176.  doi: 10.1103/PhysRevLett.80.4173.

[27]

D. D. HolmJ. E. Marsden and T. S. Ratiu, The Euler-Poincaré equations and semidirect products with applications to continuum theories, Adv. Math., 137 (1998), 1-81.  doi: 10.1006/aima.1998.1721.

[28]

D. D. Holm, L. O'Náraigh and C. Tronci, Singular solutions of a modified two-component Camassa-Holm equation, Phys. Rev. E, 79 (2009), 016601, 13 pp. doi: 10.1103/PhysRevE.79.016601.

[29]

D. D. Holm and M. F. Staley, Interaction dynamics of singular wave fronts, preprint, arXiv: 1301.1460.

[30]

D. D. Holm and C. Tronci, Geodesic flows on semidirect-product Lie groups: Geometry of singular measure-valued solutions, Proc. R. Soc. Lond. Ser. A, 465 (2009), 457-476.  doi: 10.1098/rspa.2008.0263.

[31]

D. Ionescu-Kruse, Mathematical Methods in Water Wave Problems, Habilitation Thesis, Romanian Academy, 2021.

[32]

T. C. Jo and W. Choi, Dynamics of strongly nonlinear internal solitary waves in shallow water, Stud. Appl. Math., 109 (2002), 205-227.  doi: 10.1111/1467-9590.00222.

[33]

H. P. Kruse, J. Scheurle and W. Du, A two-dimensional version of the Camassa-Holm equation, In Symmetry and Perturbation Theory, (2001), 120-127. doi: 10.1142/9789812794543_0017.

[34]

R. Liska and B. Wendroff, Analysis and computation with stratified fluid models, J. Comput. Phys., 137 (1997), 212-244.  doi: 10.1006/jcph.1997.5806.

[35]

A. K. LiuY. S. ChangM. K. Hsu and N. K. Liang, Evolution of nonlinear internal waves in the East and South China Seas, J. Geophys. Res., 103 (1998), 7995-8008.  doi: 10.1029/97JC01918.

[36]

J. E. Marsden and A. Weinstein, Coadjoint orbits, vortices, and Clebsch variables for incompressible fluids, Physica D: Nonlinear Phenomena, 7 (1983), 305-323.  doi: 10.1016/0167-2789(83)90134-3.

[37]

J. R. Percival, C. J. Cotter and D. D. Holm, A Euler-Poincaré framework for the multilayer Green-Naghdi equations, J. Phys. A: Math. Theoret., 41 (2008), 344018, 13 pp. doi: 10.1088/1751-8113/41/34/344018.

[38]

T. Soomere, Solitons interactions, in Mathematics of Complexity and Dynamical Systems, Springer, New York, NY, 2012. doi: 10.1007/978-1-4614-1806-1_101.

[39]

G. Verdoolaege, Information geometry, MDPI, 2019. doi: 10.3390/books978-3-03897-633-2.

[40]

A. Weinstein, The local structure of Poisson manifolds, J. Differential Geom., 18 (1983), 523-557.  doi: 10.4310/jdg/1214437787.

[41]

Z. ZhaoB. Liu and X. Li, Internal solitary waves in the China seas observed using satellite remote-sensing techniques: A review and perspectives, International Journal of Remote Sensing, 35 (2014), 3926-3946.  doi: 10.1080/01431161.2014.916442.

show all references


References:
[1]

X. BaiX. LiK. G. Lamb and J. Hu, Internal solitary wave reflection near dongsha atoll, the south china sea, Journal of Geophysical Research: Oceans, 122 (2017), 7978-7991.  doi: 10.1002/2017JC012880.

[2]

R. Barros and W. Choi, Inhibiting shear instability induced by large amplitude internal solitary waves in two-layer flows with a free surface, Stud. Appl. Math., 122 (2009), 325-346.  doi: 10.1111/j.1467-9590.2009.00436.x.

[3]

J. Burbea and C. R. Rao, Entropy differential metric, distance and divergence measures in probability spaces: A unified approach, J. Multivariate Anal., 12 (1982), 575-596.  doi: 10.1016/0047-259X(82)90065-3.

[4]

H. CendraD. D. HolmJ. E. Marsden and T. S. Ratiu, Lagrangian reduction, the Euler-Poincaré equations, and semidirect products, Translations of the American Mathematical Society-Series 2, 186 (1998), 1-26.  doi: 10.1090/trans2/186/01.

[5]

S. Chakravarty and Y. Kodama, Soliton solutions of the KP equation and application to shallow water waves, Stud. Appl. Math., 123 (2009), 83-151.  doi: 10.1111/j.1467-9590.2009.00448.x.

[6]

S. ChenC. FoiasD. D. HolmE. OlsonE. S. Titi and S. Wynne, Camassa-Holm equations as a closure model for turbulent channel and pipe flow, Phys. Rev. Lett., 81 (1998), 5338-5341.  doi: 10.1103/PhysRevLett.81.5338.

[7]

S. ChenC. FoiasD. D. HolmE. OlsonE. S. Titi and S. Wynne, A connection between the Camassa-Holm equations and turbulent flows in channels and pipes, Phys. Fluids, 11 (1999), 2343-2353.  doi: 10.1063/1.870096.

[8]

M. ChenS.-Q. Liu and Y. Zhang, A two-component generalization of the Camassa-Holm equation and its solutions, Lett. Math. Phys., 75 (2006), 1-15.  doi: 10.1007/s11005-005-0041-7.

[9]

W. Choi and R. Camassa, Weakly nonlinear internal waves in a two-fluid system, J. Fluid Mech., 313 (1996), 83-103.  doi: 10.1017/S0022112096002133.

[10]

W. Choi and R. Camassa, Fully nonlinear internal waves in a two-fluid system, J. Fluid Mech., 396 (1999), 1-36.  doi: 10.1017/S0022112099005820.

[11]

A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations, Arch. Ration. Mech. Anal., 192 (2009), 165-186.  doi: 10.1007/s00205-008-0128-2.

[12]

C. J. CotterJ. ElderingD. D. HolmH. O. Jacobs and D. M. Meier, Weak dual pairs and jetlet methods for ideal incompressible fluid models in $n\ge 2$ dimensions, J. Nonlinear Sci., 26 (2016), 1723-1765.  doi: 10.1007/s00332-016-9317-6.

[13]

C. J. Cotter, D. D. Holm, H. O. Jacobs and D. M. Meier, A jetlet hierarchy for ideal fluid dynamics, J. Phys., 47 (2014), 352001, 8 pp. doi: 10.1088/1751-8113/47/35/352001.

[14]

C. J. CotterD. D. Holm and J. R. Percival, The square root depth wave equations, Proc. Roy. Soc. A Math. Phys., 466 (2010), 3621-3633.  doi: 10.1098/rspa.2010.0124.

[15]

V. Duchêne, Many Models for Water Waves, Doctoral dissertation, Université de Rennes, 2021, https://tel.archives-ouvertes.fr/tel-03282212

[16]

V. DuchêneS. Israwi and R. Talhouk, A new class of two-layer Green-Naghdi systems with improved frequency dispersion, Stud. Appl. Math., 137 (2016), 356-415.  doi: 10.1111/sapm.12125.

[17]

G. Falqui, On a Camassa-Holm type equation with two dependent variables, J. Phys. A, 39 (2006), 327-342.  doi: 10.1088/0305-4470/39/2/004.

[18]

D. M. FarmerL. ArmiL. Armi and D. M. Farmer, The flow of Atlantic water through the Strait of gibraltar, Progress in Oceanography, 21 (1988), 1-103.  doi: 10.1016/0079-6611(88)90055-9.

[19]

C. FoiasD. D. Holm and E. S. Titi, The Navier-Stokes-alpha model of fluid turbulence, Phys. D, 152/153 (2001), 505-519.  doi: 10.1016/S0167-2789(01)00191-9.

[20]

C. FoiasD. D. Holm and E. S. Titi, The three dimensional viscous Camassa-Holm equations, and their relation to the Navier-Stokes equations and turbulence theory, J. Dynam. Differential Equations, 14 (2002), 1-35.  doi: 10.1023/A:1012984210582.

[21]

F. Gay-Balmaz and C. Vizman, Dual pairs in fluid dynamics, Ann. Global Anal. Geom., 41 (2012), 1-24.  doi: 10.1007/s10455-011-9267-z.

[22]

D. D. Holm, Variational principles, geometry and topology of Lagrangian-averaged fluid dynamics, An Introduction to the Geometry and Topology of Fluid Flows, 47 (2001), 271-291.  doi: 10.1007/978-94-010-0446-6_14.

[23]

D. D. Holm, Variational principles for stochastic fluid dynamics, Proc. A., 471 (2015), 20140963, 19 pp. doi: 10.1098/rspa.2014.0963.

[24]

D. D. Holm and H. O. Jacobs, Multipole vortex blobs (MVB): Symplectic geometry and dynamics, J. Nonlinear Sci., 27 (2017), 973-1006.  doi: 10.1007/s00332-017-9367-4.

[25]

D. D. Holm and J. E. Marsden, Momentum maps and measure-valued solutions (peakons, filaments, and sheets) for the EPDiff equation, In Progr. Math., Birkhäuser Boston, 232 (2005), 203-235. doi: 10.1007/0-8176-4419-9_8.

[26]

D. D. HolmJ. E. Marsden and T. S. Ratiu, Euler-Poincaré models of ideal fluids with nonlinear dispersion, Physical Review Letters, 80 (1998), 4173-4176.  doi: 10.1103/PhysRevLett.80.4173.

[27]

D. D. HolmJ. E. Marsden and T. S. Ratiu, The Euler-Poincaré equations and semidirect products with applications to continuum theories, Adv. Math., 137 (1998), 1-81.  doi: 10.1006/aima.1998.1721.

[28]

D. D. Holm, L. O'Náraigh and C. Tronci, Singular solutions of a modified two-component Camassa-Holm equation, Phys. Rev. E, 79 (2009), 016601, 13 pp. doi: 10.1103/PhysRevE.79.016601.

[29]

D. D. Holm and M. F. Staley, Interaction dynamics of singular wave fronts, preprint, arXiv: 1301.1460.

[30]

D. D. Holm and C. Tronci, Geodesic flows on semidirect-product Lie groups: Geometry of singular measure-valued solutions, Proc. R. Soc. Lond. Ser. A, 465 (2009), 457-476.  doi: 10.1098/rspa.2008.0263.

[31]

D. Ionescu-Kruse, Mathematical Methods in Water Wave Problems, Habilitation Thesis, Romanian Academy, 2021.

[32]

T. C. Jo and W. Choi, Dynamics of strongly nonlinear internal solitary waves in shallow water, Stud. Appl. Math., 109 (2002), 205-227.  doi: 10.1111/1467-9590.00222.

[33]

H. P. Kruse, J. Scheurle and W. Du, A two-dimensional version of the Camassa-Holm equation, In Symmetry and Perturbation Theory, (2001), 120-127. doi: 10.1142/9789812794543_0017.

[34]

R. Liska and B. Wendroff, Analysis and computation with stratified fluid models, J. Comput. Phys., 137 (1997), 212-244.  doi: 10.1006/jcph.1997.5806.

[35]

A. K. LiuY. S. ChangM. K. Hsu and N. K. Liang, Evolution of nonlinear internal waves in the East and South China Seas, J. Geophys. Res., 103 (1998), 7995-8008.  doi: 10.1029/97JC01918.

[36]

J. E. Marsden and A. Weinstein, Coadjoint orbits, vortices, and Clebsch variables for incompressible fluids, Physica D: Nonlinear Phenomena, 7 (1983), 305-323.  doi: 10.1016/0167-2789(83)90134-3.

[37]

J. R. Percival, C. J. Cotter and D. D. Holm, A Euler-Poincaré framework for the multilayer Green-Naghdi equations, J. Phys. A: Math. Theoret., 41 (2008), 344018, 13 pp. doi: 10.1088/1751-8113/41/34/344018.

[38]

T. Soomere, Solitons interactions, in Mathematics of Complexity and Dynamical Systems, Springer, New York, NY, 2012. doi: 10.1007/978-1-4614-1806-1_101.

[39]

G. Verdoolaege, Information geometry, MDPI, 2019. doi: 10.3390/books978-3-03897-633-2.

[40]

A. Weinstein, The local structure of Poisson manifolds, J. Differential Geom., 18 (1983), 523-557.  doi: 10.4310/jdg/1214437787.

[41]

Z. ZhaoB. Liu and X. Li, Internal solitary waves in the China seas observed using satellite remote-sensing techniques: A review and perspectives, International Journal of Remote Sensing, 35 (2014), 3926-3946.  doi: 10.1080/01431161.2014.916442.

Figure 2.  Synthetic Aperture Radar (SAR) Image of internal-wave signatures on the South China Sea
Figure 3.  Custom colour map for the upcoming $ |u|^2 $ plots
Figure 4.  Locations of the 1D profiles of $ |\boldsymbol{{u}}| $ and $ \overline{D} $ in the 2D numerical simulations
Figure 5.  Evolution of $ |u|^2 $ with the "plate" initial condition with $ \alpha = w_0 $
Figure 6.  Evolution of $ \overline{D}-\overline{b} $ with the "plate" initial condition with $ \alpha = w_0 $
Figure 7.  Evolution of $ |u|^2 $ with the "plate" initial condition with $ \alpha = w_0/8 $
Figure 8.  Evolution of $ \overline{D}-\overline{b} $ with the "plate" initial condition with $ \alpha = w_0/8 $
Figure 9.  Evolution of $|u|^2$ with the "skew" initial condition with $\alpha = w_0$
Figure 10.  Evolution of $\overline{D}-\overline{b}$ with the "skew" initial condition with $\alpha = w_0$
Figure 11.  Evolution of $|u|^2$ with the "skew" initial condition with $\alpha = w_0/8$
Figure 12.  Evolution of $\overline{D}-\overline{b}$ with the "skew" initial condition with $\alpha = w_0/8$
Figure 13.  Evolution of $ |u|^2 $ with the "wedge" initial condition with $ \alpha = w_0 $
Figure 14.  Evolution of $ \overline{D}-\overline{b} $ with the "wedge" initial condition with $ \alpha = w_0 $
Figure 15.  Evolution of |u|2 with the "wedge" initial condition with 8α = w0
Figure 16.  Evolution of $ \overline{D}-\overline{b} $ with the "wedge" initial condition with 8α = w0
Figure 17.  Evolution of $ |u|^2 $ with the "parallel" initial condition with $ \alpha = w_0 $
Figure 18.  Evolution of $ \overline{D}-\overline{b} $ with the "parallel" initial condition with $ \alpha = w_0 $
Figure 19.  Evolution of $ |u|^2 $ with the "parallel" initial condition with $ 8\alpha = w_0 $
Figure 20.  Evolution of $ \overline{D}-\overline{b} $ with the "parallel" initial condition with $ 8\alpha = w_0 $
Figure 21.  Evolution of $ \overline{D}-\overline{b} $ with the "dam break" initial condition with $ \alpha = w_0 $
Figure 22.  Evolution of $ |u|^2 $ with the "dam break" initial condition with $ \alpha = w_0 $
Figure 23.  Evolution of $\overline{D}-\overline{b}$ with the "dam break" initial condition with $8\alpha = w_0$
Figure 24.  Evolution of $|u|^2$ with the "dam break" initial condition with $8\alpha = w_0$
Figure 25.  Evolution of $ \overline{D}-\overline{b} $ with the "dam break" initial condition with $ \alpha = w_0 $ and zero bathymetry
Figure 26.  Evolution of $|u|^2$ with the "dam break" initial condition with $\alpha = w_0$ and zero bathymetry
Figure 27.  Evolution of $ \overline{D}-\overline{b} $ with the "dam break" initial condition with $ 8\alpha = w_0 $
Figure 28.  Evolution of $|u|^2$ with the "dam break" initial condition with $8\alpha = w_0$
Figure 29.  Evolution of |u|2 with the "dual dam break" initial condition with α = w0
Figure 30.  Evolution of $ \overline{D}-\overline{b} $ with the "dual dam break" initial condition with α = w0
Figure 31.  Evolution of |u|2 with the "dual dam break" initial condition with 8α = w0
Figure 32.  Evolution of $ \overline{D}-\overline{b} $ with the "dual dam break" initial condition with 8α = w0
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