July  2022, 21(7): 2399-2414. doi: 10.3934/cpaa.2022041

Particle paths in equatorial flows

Department of Computing & Mathematics, Waterford Institute of Technology, Waterford, Ireland

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

We investigate particle trajectories in equatorial flows with geophysical corrections caused by the earth's rotation. Particle trajectories in the flows are constructed using pairs of analytic functions defined over the labelling space used in the Lagrangian formalism. Several classes of flow are investigated, and the physical regime in which each is valid is determined using the pressure distribution function of the flow, while the vorticity distribution of each flow is also calculated and found to be effected by earth's rotation.

Citation: Tony Lyons. Particle paths in equatorial flows. Communications on Pure and Applied Analysis, 2022, 21 (7) : 2399-2414. doi: 10.3934/cpaa.2022041
References:
[1]

A. Aleman and A. Constantin, Harmonic maps and ideal fluid flows, Arch. Rat. Mech. Anal., 204 (2012), 479-513.  doi: 10.1007/s00205-011-0483-2.

[2] A. Bennett, Lagrangian Fluid Dynamics, Cambridge Monographs on Mechanics, Cambridge University Press, 2006.  doi: 10.1017/CBO9780511734939.
[3]

A. Constantin, On the deep water wave motion, J. Phys. A, 34 (2001), 1405-1417.  doi: 10.1088/0305-4470/34/7/313.

[4]

A. Constantin, The trajectories of particles in Stokes waves, Invent. Math, 166 (2006), 523-535.  doi: 10.1007/s00222-006-0002-5.

[5]

A. Constantin, Nonlinear Water Waves with Applications to Wave-Current Interactions and Tsunamis, SIAM, 2011. doi: 10.1137/1.9781611971873.

[6]

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

[7]

A. Constantin, Particle trajectories in extreme Stokes waves, IMA Jour. Appl. Math., 77 (2012), 293-307.  doi: 10.1093/imamat/hxs033.

[8]

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

[9]

A. Constantin and R. S. Johnson, Large gyres as a shallow-water asymptotic solution of euler's equation in spherical coordinates, Proc. Royal Soc. A: Math. Phys. Eng. Sci., 473 (2017), 18 pp. doi: 10.1098/rspa. 2017.0063.

[10]

A. Constantin and W. Strauss, Pressure beneath a Stokes wave, Commun. Pure Appl. Math., 63 (2010), 533-557.  doi: 10.1002/cpa.20299.

[11]

A. Constantin and W. A. Strauss, Trochoidal solutions to the incompressible two-dimensional Euler equations, J. Math. Fluid Mech., 12 (2010), 181-201.  doi: 10.1007/s00021-008-0281-2.

[12]

A. Constantin and G. Villari, Particle trajectories in linear water waves, J. Math Fluid Mech., 10 (2008), 1-18.  doi: 10.1007/s00021-005-0214-2.

[13]

A. ConstantinM. Ehrnström and G. Villari, Particle trajectories in linear deep-water waves, Nonlin. Anal. B, 9 (2008), 1336-1344.  doi: 10.1016/j.nonrwa.2007.03.003.

[14]

O. Constantin and M. J. Martín, A harmonic maps approach to fluid flows, Math. Ann., 369 (2017), 1-16.  doi: 10.1007/s00208-016-1435-9.

[15] B. Cushman-Roisin and J. M. Beckers, Introduction to Geophysical Fluid Dynamics: Physical and Numerical Aspects, Academic Press, 2011. 
[16]

F. Gerstner, Theorie der Wellen, Anal. Phys., 32 (1809), 412-445. 

[17]

D. Henry, The trajectories of particles in deep-water Stokes waves, Internl. Math. Res. Not., 2006 (2006), 23405, 1–13. doi: 10.1155/IMRN/2006/23405.

[18]

D. Henry, On Gerstner's water wave, J. Nonlin. Math. Phys., 15 (2008), 87-95.  doi: 10.2991/jnmp.2008.15.s2.7.

[19]

D. Henry, On the deep-water Stokes wave flow, Int. Math. Res. Not., 2008 (2008), rnn071. doi: 10.1093/imrn/rnn071.

[20]

D. Henry, Internal equatorial water waves in the $f$–plane, J. Nonlin. Math. Phys., 22 (2015), 499-506.  doi: 10.1080/14029251.2015.1113046.

[21]

D. Henry, Exact equatorial water waves in the $f$-plane, Nonlin. Anal. B, 28 (2016), 284-289.  doi: 10.1016/j.nonrwa.2015.10.003.

[22]

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.

[23]

B. Kinsman, Wind Waves: Their Generation and Propagation on the Ocean Surface, Prentice Hall Inc., Englewood Cliffs, N. J., 1965.

[24]

M. Kluczek, Equatorial water waves with underlying currents in the $f$–plane approximation, Appl. Anal., 97 (2018), 1867-1880.  doi: 10.1080/00036811.2017.1343466.

[25]

M. Kluczek and R. Stuhlmeier, Mass transport for Pollard waves, Appl. Anal., 1–10.

[26]

T. Lyons, Particle trajectories in extreme Stokes waves over infinite depth, Disc. Contin. Dynam. Sys., 34 (2014), 3095-3107.  doi: 10.3934/dcds.2014.34.3095.

[27]

T. Lyons, Geophysical internal equatorial waves of extreme form, Disc. Contin. Dyn. Syst., 39 (2019), 4471-4486.  doi: 10.3934/dcds.2019183.

[28]

M. J. Martin and J. Tuomela, 2D incompressible Euler equations: new explicit solutions, Discrete Contin. Dyn. Syst., 39 (2019), 4547-4563.  doi: 10.3934/dcds.2019187.

[29]

A. V. Matioc, On particle trajectories in linear deep-water waves, Commun. Pure Appl. Anal., 11 (2012), 1537-1547.  doi: 10.3934/cpaa.2012.11.1537.

[30]

L. M. Milne-Thomson, Theoretical Hydrodynamics, Courier Corporation, 2013.

[31] F. W. J. OlverD. W. LozierR. F. Boisvert and C. W. Clark, NIST Handbook of Mathematical Functions, Cambridge University Press, 2010. 
[32]

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

[33]

W. J. M. Rankine, On the exact form of waves near the surface of deep water, Phil. Trans. R. Soc. Lond., 127–138.

[34]

A. Rodríguez-Sanjurjo, Internal equatorial water waves and wave–current interactions in the $f$–plane, Monats. Math., 168 (2018), 685-701.  doi: 10.1007/s00605-017-1052-z.

[35]

S. R. ValluriD. J. Jeffrey and R. M. Corless, Some applications of the Lambert W function to physics, Canadian J. Phys., 78 (2000), 823-831. 

show all references

References:
[1]

A. Aleman and A. Constantin, Harmonic maps and ideal fluid flows, Arch. Rat. Mech. Anal., 204 (2012), 479-513.  doi: 10.1007/s00205-011-0483-2.

[2] A. Bennett, Lagrangian Fluid Dynamics, Cambridge Monographs on Mechanics, Cambridge University Press, 2006.  doi: 10.1017/CBO9780511734939.
[3]

A. Constantin, On the deep water wave motion, J. Phys. A, 34 (2001), 1405-1417.  doi: 10.1088/0305-4470/34/7/313.

[4]

A. Constantin, The trajectories of particles in Stokes waves, Invent. Math, 166 (2006), 523-535.  doi: 10.1007/s00222-006-0002-5.

[5]

A. Constantin, Nonlinear Water Waves with Applications to Wave-Current Interactions and Tsunamis, SIAM, 2011. doi: 10.1137/1.9781611971873.

[6]

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

[7]

A. Constantin, Particle trajectories in extreme Stokes waves, IMA Jour. Appl. Math., 77 (2012), 293-307.  doi: 10.1093/imamat/hxs033.

[8]

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

[9]

A. Constantin and R. S. Johnson, Large gyres as a shallow-water asymptotic solution of euler's equation in spherical coordinates, Proc. Royal Soc. A: Math. Phys. Eng. Sci., 473 (2017), 18 pp. doi: 10.1098/rspa. 2017.0063.

[10]

A. Constantin and W. Strauss, Pressure beneath a Stokes wave, Commun. Pure Appl. Math., 63 (2010), 533-557.  doi: 10.1002/cpa.20299.

[11]

A. Constantin and W. A. Strauss, Trochoidal solutions to the incompressible two-dimensional Euler equations, J. Math. Fluid Mech., 12 (2010), 181-201.  doi: 10.1007/s00021-008-0281-2.

[12]

A. Constantin and G. Villari, Particle trajectories in linear water waves, J. Math Fluid Mech., 10 (2008), 1-18.  doi: 10.1007/s00021-005-0214-2.

[13]

A. ConstantinM. Ehrnström and G. Villari, Particle trajectories in linear deep-water waves, Nonlin. Anal. B, 9 (2008), 1336-1344.  doi: 10.1016/j.nonrwa.2007.03.003.

[14]

O. Constantin and M. J. Martín, A harmonic maps approach to fluid flows, Math. Ann., 369 (2017), 1-16.  doi: 10.1007/s00208-016-1435-9.

[15] B. Cushman-Roisin and J. M. Beckers, Introduction to Geophysical Fluid Dynamics: Physical and Numerical Aspects, Academic Press, 2011. 
[16]

F. Gerstner, Theorie der Wellen, Anal. Phys., 32 (1809), 412-445. 

[17]

D. Henry, The trajectories of particles in deep-water Stokes waves, Internl. Math. Res. Not., 2006 (2006), 23405, 1–13. doi: 10.1155/IMRN/2006/23405.

[18]

D. Henry, On Gerstner's water wave, J. Nonlin. Math. Phys., 15 (2008), 87-95.  doi: 10.2991/jnmp.2008.15.s2.7.

[19]

D. Henry, On the deep-water Stokes wave flow, Int. Math. Res. Not., 2008 (2008), rnn071. doi: 10.1093/imrn/rnn071.

[20]

D. Henry, Internal equatorial water waves in the $f$–plane, J. Nonlin. Math. Phys., 22 (2015), 499-506.  doi: 10.1080/14029251.2015.1113046.

[21]

D. Henry, Exact equatorial water waves in the $f$-plane, Nonlin. Anal. B, 28 (2016), 284-289.  doi: 10.1016/j.nonrwa.2015.10.003.

[22]

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.

[23]

B. Kinsman, Wind Waves: Their Generation and Propagation on the Ocean Surface, Prentice Hall Inc., Englewood Cliffs, N. J., 1965.

[24]

M. Kluczek, Equatorial water waves with underlying currents in the $f$–plane approximation, Appl. Anal., 97 (2018), 1867-1880.  doi: 10.1080/00036811.2017.1343466.

[25]

M. Kluczek and R. Stuhlmeier, Mass transport for Pollard waves, Appl. Anal., 1–10.

[26]

T. Lyons, Particle trajectories in extreme Stokes waves over infinite depth, Disc. Contin. Dynam. Sys., 34 (2014), 3095-3107.  doi: 10.3934/dcds.2014.34.3095.

[27]

T. Lyons, Geophysical internal equatorial waves of extreme form, Disc. Contin. Dyn. Syst., 39 (2019), 4471-4486.  doi: 10.3934/dcds.2019183.

[28]

M. J. Martin and J. Tuomela, 2D incompressible Euler equations: new explicit solutions, Discrete Contin. Dyn. Syst., 39 (2019), 4547-4563.  doi: 10.3934/dcds.2019187.

[29]

A. V. Matioc, On particle trajectories in linear deep-water waves, Commun. Pure Appl. Anal., 11 (2012), 1537-1547.  doi: 10.3934/cpaa.2012.11.1537.

[30]

L. M. Milne-Thomson, Theoretical Hydrodynamics, Courier Corporation, 2013.

[31] F. W. J. OlverD. W. LozierR. F. Boisvert and C. W. Clark, NIST Handbook of Mathematical Functions, Cambridge University Press, 2010. 
[32]

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

[33]

W. J. M. Rankine, On the exact form of waves near the surface of deep water, Phil. Trans. R. Soc. Lond., 127–138.

[34]

A. Rodríguez-Sanjurjo, Internal equatorial water waves and wave–current interactions in the $f$–plane, Monats. Math., 168 (2018), 685-701.  doi: 10.1007/s00605-017-1052-z.

[35]

S. R. ValluriD. J. Jeffrey and R. M. Corless, Some applications of the Lambert W function to physics, Canadian J. Phys., 78 (2000), 823-831. 

Figure 1.  The particle trajectories given by (4.21)--(5.8) with $ \omega_{1}=1+2i $, $ \omega_{2}=3+i $, $ \zeta_{1}=4+2i $ and $ \zeta_{2}=2.5+1.94i $
Figure 2.  Elliptical particle trajectories obtained from exponential profiles with complex parameters $ \omega_{1}=1-2i $, $ \omega_{2}=2+i $, $ \zeta_{1}=1+i $ while the remaining parameter is $ \kappa=\frac{\pi}{4} $ for $ b\in\{-2.0,-1.0,-0.15\} $ with $ b_{0}=-0.15 $
Figure 3.  An elliptical particle trajectory with $ \omega_{1}=1+2i=\bar{\omega}_{2} $ and $ \zeta_{1}=2-4i $ (solid curve) and a second trajectory with $ \omega_{1}=1-3i=\bar{\omega}_{2} $ and $ \zeta_{1}=2+5i $ (dashed curve). The additional parameter is and $ \kappa=\frac{\pi}{2} $
Figure 4.  A cycloid (solid curve) and trochoids (dashed curve) described by the map (5.27a)–(5.27b) wave with $ \omega_{1}=1 $ and $ \zeta_{2}=5i $ and $ \kappa=\frac{\pi}{10} $
Figure 5.  The free surface b = b0 = −12.93 m (solid curve) and the cycloid b = bcrit = 74.88 m when $ l_{1}=1 $ and $ M_{2}=9.95 $ with Patm = 101.325 m2 s−2 and wavelength 300 m
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