August  2019, 39(8): 4747-4770. doi: 10.3934/dcds.2019193

On stratified water waves with critical layers and Coriolis forces

University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria

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

Received  October 2018 Revised  January 2019 Published  May 2019

We consider nonlinear traveling waves in a two-dimensional fluid subject to the effects of vorticity, stratification, and in-plane Coriolis forces. We first observe that the terms representing the Coriolis forces can be completely eliminated by a change of variables. This does not appear to be well-known, and helps to organize some of the existing literature.

Second we give a rigorous existence result for periodic waves in a two-layer system with a free surface and constant densities and vorticities in each layer, allowing for the presence of critical layers. We augment the problem with four physically-motivated constraints, and phrase our hypotheses directly in terms of the explicit dispersion relation for the problem. This approach smooths the way for further generalizations, some of which we briefly outline at the end of the paper.

Citation: Miles H. Wheeler. On stratified water waves with critical layers and Coriolis forces. Discrete & Continuous Dynamical Systems - A, 2019, 39 (8) : 4747-4770. doi: 10.3934/dcds.2019193
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. Akers and S. Walsh, Solitary water waves with discontinuous vorticity, Journal de Mathématiques Pures et Appliquées, 124 (2019), 220-272. doi: 10.1016/j.matpur.2018.06.008.

[3]

B. Buffoni and J. Toland, Analytic Theory of Global Bifurcation, Princeton Series in Applied Mathematics, Princeton University Press, Princeton, NJ, 2003, An introduction. doi: 10.1515/9781400884339.

[4]

R. M. ChenS. Walsh and M. H. Wheeler, Existence and qualitative theory for stratified solitary water waves, Ann. Inst. H. Poincaré Anal. Non Linéaire, 35 (2018), 517-576. doi: 10.1016/j.anihpc.2017.06.003.

[5]

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.

[6]

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.

[7]

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.

[8]

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.

[9]

A. Constantin, Nonlinear Water Waves with Applications to Wave-Current Interactions and Tsunamis, vol. 81 of CBMS-NSF Regional Conference Series in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2011. doi: 10.1137/1.9781611971873.

[10]

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

[11]

A. ConstantinW. Strauss and E. Vǎrvǎrucǎ, Global bifurcation of steady gravity water waves with critical layers, Acta Math., 217 (2016), 195-262. doi: 10.1007/s11511-017-0144-x.

[12]

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

[13]

A. Constantin and W. A. Strauss, Stability properties of steady water waves with vorticity, Comm. Pure Appl. Math., 60 (2007), 911-950. doi: 10.1002/cpa.20165.

[14]

M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Functional Analysis, 8 (1971), 321-340. doi: 10.1016/0022-1236(71)90015-2.

[15]

M. L. Dubreil-Jacotin, Sur la détermination rigoureuse des ondes permanentes périodiques d'amplitude finie, (French) 1934. 75 pp.

[16]

M. D. Groves, Steady water waves, J. Nonlinear Math. Phys., 11 (2004), 435-460. doi: 10.2991/jnmp.2004.11.4.2.

[17]

K. R. Helfrich and W. K. Melville, Long nonlinear internal waves, in Annual Review of Fluid Mechanics. Vol. 38, Annual Reviews, Palo Alto, CA, 38 (2006), 395–425. doi: 10.1146/annurev.fluid.38.050304.092129.

[18]

D. Henry, Large amplitude steady periodic waves for fixed-depth rotational flows, Comm. Partial Differential Equations, 38 (2013), 1015-1037. doi: 10.1080/03605302.2012.734889.

[19]

D. Henry, Steady periodic waves bifurcating for fixed-depth rotational flows, Quart. Appl. Math., 71 (2013), 455-487. doi: 10.1090/S0033-569X-2013-01293-8.

[20]

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

[21]

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.

[22]

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.

[23]

H.-C. Hsu, Exact nonlinear internal equatorial waves in the $f$-plane, J. Math. Fluid Mech., 19 (2017), 367-374. doi: 10.1007/s00021-016-0285-2.

[24]

D. Ionescu-Kruse and C. I. Martin, Periodic equatorial water flows from a Hamiltonian perspective, J. Differential Equations, 262 (2017), 4451-4474. doi: 10.1016/j.jde.2017.01.001.

[25]

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.

[26]

V. Kozlov, N. Kuznetsov and E. Lokharu, Steady water waves with vorticity: An analysis of the dispersion equation, J. Fluid Mech., 751 (2014), R3, 13pp. doi: 10.1017/jfm.2014.322.

[27]

V. Kozlov and N. Kuznetsov, Dispersion equation for water waves with vorticity and Stokes waves on flows with counter-currents, Arch. Ration. Mech. Anal., 214 (2014), 971-1018. doi: 10.1007/s00205-014-0787-0.

[28]

O. A. Ladyzhenskaya and N. N. Ural'tseva, Linear and Quasilinear Elliptic Equations, Translated from the Russian by Scripta Technica, Inc. Translation editor: Leon Ehrenpreis, Academic Press, New York-London, 1968.

[29]

H. Le, Elliptic equations with transmission and Wentzell boundary conditions and an application to steady water waves in the presence of wind, Discrete Contin. Dyn. Syst., 38 (2018), 3357-3385. doi: 10.3934/dcds.2018144.

[30]

T. Levi-Civita, Determinazione rigorosa delle onde irrotazionali periodiche in acqua profonda, Rend. Accad. Lincei, 33 (1924), 141-150.

[31]

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

[32]

A.-V. Matioc, Steady internal water waves with a critical layer bounded by the wave surface, J. Nonlinear Math. Phys., 19 (2012), 1250008, 21pp. doi: 10.1142/S1402925112500088.

[33]

A. Mielke, Reduction of quasilinear elliptic equations in cylindrical domains with applications, Math. Methods Appl. Sci., 10 (1988), 51-66. doi: 10.1002/mma.1670100105.

[34]

J. W. Miles, Solitary waves, in Annual Review of Fluid Mechanics, Vol. 12, Annual Reviews, Palo Alto, Calif., 43 (1980), 11–43.

[35]

A. I. Nekrasov, On steady waves, Izv. Ivanovo-Voznesensk. Politekhn. In-ta, 3.

[36]

D. V. Nilsson, Internal gravity-capillary solitary waves in finite depth, Math. Methods Appl. Sci., 40 (2017), 1053-1080. doi: 10.1002/mma.4036.

[37]

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.

[38]

B. Sandstede and A. Scheel, Relative Morse indices, Fredholm indices, and group velocities, Discrete Contin. Dyn. Syst., 20 (2008), 139-158. doi: 10.3934/dcds.2008.20.139.

[39]

W. A. Strauss, Steady water waves, Bull. Amer. Math. Soc. (N.S.), 47 (2010), 671-694. doi: 10.1090/S0273-0979-2010-01302-1.

[40]

J. F. Toland, Stokes waves, Topol. Methods Nonlinear Anal., 7 (1996), 1-48. doi: 10.12775/TMNA.1996.001.

[41]

E. Wahlén, Steady water waves with a critical layer, J. Differential Equations, 246 (2009), 2468-2483. doi: 10.1016/j.jde.2008.10.005.

[42]

S. Walsh, Stratified steady periodic water waves, SIAM J. Math. Anal., 41 (2009), 1054-1105. doi: 10.1137/080721583.

[43]

S. WalshO. Bühler and J. Shatah, Steady water waves in the presence of wind, SIAM J. Math. Anal., 45 (2013), 2182-2227. doi: 10.1137/120880124.

[44]

L.-J. Wang, Small-amplitude solitary and generalized solitary traveling waves in a gravity two-layer fluid with vorticity, Nonlinear Anal., 150 (2017), 159-193. doi: 10.1016/j.na.2016.11.012.

[45]

M. H. Wheeler, Simplified models for equatorial waves with vertical structure, Oceanography, 31 (2018), 36-41. doi: 10.5670/oceanog.2018.307.

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. Akers and S. Walsh, Solitary water waves with discontinuous vorticity, Journal de Mathématiques Pures et Appliquées, 124 (2019), 220-272. doi: 10.1016/j.matpur.2018.06.008.

[3]

B. Buffoni and J. Toland, Analytic Theory of Global Bifurcation, Princeton Series in Applied Mathematics, Princeton University Press, Princeton, NJ, 2003, An introduction. doi: 10.1515/9781400884339.

[4]

R. M. ChenS. Walsh and M. H. Wheeler, Existence and qualitative theory for stratified solitary water waves, Ann. Inst. H. Poincaré Anal. Non Linéaire, 35 (2018), 517-576. doi: 10.1016/j.anihpc.2017.06.003.

[5]

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.

[6]

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.

[7]

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.

[8]

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.

[9]

A. Constantin, Nonlinear Water Waves with Applications to Wave-Current Interactions and Tsunamis, vol. 81 of CBMS-NSF Regional Conference Series in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2011. doi: 10.1137/1.9781611971873.

[10]

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

[11]

A. ConstantinW. Strauss and E. Vǎrvǎrucǎ, Global bifurcation of steady gravity water waves with critical layers, Acta Math., 217 (2016), 195-262. doi: 10.1007/s11511-017-0144-x.

[12]

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

[13]

A. Constantin and W. A. Strauss, Stability properties of steady water waves with vorticity, Comm. Pure Appl. Math., 60 (2007), 911-950. doi: 10.1002/cpa.20165.

[14]

M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Functional Analysis, 8 (1971), 321-340. doi: 10.1016/0022-1236(71)90015-2.

[15]

M. L. Dubreil-Jacotin, Sur la détermination rigoureuse des ondes permanentes périodiques d'amplitude finie, (French) 1934. 75 pp.

[16]

M. D. Groves, Steady water waves, J. Nonlinear Math. Phys., 11 (2004), 435-460. doi: 10.2991/jnmp.2004.11.4.2.

[17]

K. R. Helfrich and W. K. Melville, Long nonlinear internal waves, in Annual Review of Fluid Mechanics. Vol. 38, Annual Reviews, Palo Alto, CA, 38 (2006), 395–425. doi: 10.1146/annurev.fluid.38.050304.092129.

[18]

D. Henry, Large amplitude steady periodic waves for fixed-depth rotational flows, Comm. Partial Differential Equations, 38 (2013), 1015-1037. doi: 10.1080/03605302.2012.734889.

[19]

D. Henry, Steady periodic waves bifurcating for fixed-depth rotational flows, Quart. Appl. Math., 71 (2013), 455-487. doi: 10.1090/S0033-569X-2013-01293-8.

[20]

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

[21]

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.

[22]

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.

[23]

H.-C. Hsu, Exact nonlinear internal equatorial waves in the $f$-plane, J. Math. Fluid Mech., 19 (2017), 367-374. doi: 10.1007/s00021-016-0285-2.

[24]

D. Ionescu-Kruse and C. I. Martin, Periodic equatorial water flows from a Hamiltonian perspective, J. Differential Equations, 262 (2017), 4451-4474. doi: 10.1016/j.jde.2017.01.001.

[25]

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.

[26]

V. Kozlov, N. Kuznetsov and E. Lokharu, Steady water waves with vorticity: An analysis of the dispersion equation, J. Fluid Mech., 751 (2014), R3, 13pp. doi: 10.1017/jfm.2014.322.

[27]

V. Kozlov and N. Kuznetsov, Dispersion equation for water waves with vorticity and Stokes waves on flows with counter-currents, Arch. Ration. Mech. Anal., 214 (2014), 971-1018. doi: 10.1007/s00205-014-0787-0.

[28]

O. A. Ladyzhenskaya and N. N. Ural'tseva, Linear and Quasilinear Elliptic Equations, Translated from the Russian by Scripta Technica, Inc. Translation editor: Leon Ehrenpreis, Academic Press, New York-London, 1968.

[29]

H. Le, Elliptic equations with transmission and Wentzell boundary conditions and an application to steady water waves in the presence of wind, Discrete Contin. Dyn. Syst., 38 (2018), 3357-3385. doi: 10.3934/dcds.2018144.

[30]

T. Levi-Civita, Determinazione rigorosa delle onde irrotazionali periodiche in acqua profonda, Rend. Accad. Lincei, 33 (1924), 141-150.

[31]

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

[32]

A.-V. Matioc, Steady internal water waves with a critical layer bounded by the wave surface, J. Nonlinear Math. Phys., 19 (2012), 1250008, 21pp. doi: 10.1142/S1402925112500088.

[33]

A. Mielke, Reduction of quasilinear elliptic equations in cylindrical domains with applications, Math. Methods Appl. Sci., 10 (1988), 51-66. doi: 10.1002/mma.1670100105.

[34]

J. W. Miles, Solitary waves, in Annual Review of Fluid Mechanics, Vol. 12, Annual Reviews, Palo Alto, Calif., 43 (1980), 11–43.

[35]

A. I. Nekrasov, On steady waves, Izv. Ivanovo-Voznesensk. Politekhn. In-ta, 3.

[36]

D. V. Nilsson, Internal gravity-capillary solitary waves in finite depth, Math. Methods Appl. Sci., 40 (2017), 1053-1080. doi: 10.1002/mma.4036.

[37]

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.

[38]

B. Sandstede and A. Scheel, Relative Morse indices, Fredholm indices, and group velocities, Discrete Contin. Dyn. Syst., 20 (2008), 139-158. doi: 10.3934/dcds.2008.20.139.

[39]

W. A. Strauss, Steady water waves, Bull. Amer. Math. Soc. (N.S.), 47 (2010), 671-694. doi: 10.1090/S0273-0979-2010-01302-1.

[40]

J. F. Toland, Stokes waves, Topol. Methods Nonlinear Anal., 7 (1996), 1-48. doi: 10.12775/TMNA.1996.001.

[41]

E. Wahlén, Steady water waves with a critical layer, J. Differential Equations, 246 (2009), 2468-2483. doi: 10.1016/j.jde.2008.10.005.

[42]

S. Walsh, Stratified steady periodic water waves, SIAM J. Math. Anal., 41 (2009), 1054-1105. doi: 10.1137/080721583.

[43]

S. WalshO. Bühler and J. Shatah, Steady water waves in the presence of wind, SIAM J. Math. Anal., 45 (2013), 2182-2227. doi: 10.1137/120880124.

[44]

L.-J. Wang, Small-amplitude solitary and generalized solitary traveling waves in a gravity two-layer fluid with vorticity, Nonlinear Anal., 150 (2017), 159-193. doi: 10.1016/j.na.2016.11.012.

[45]

M. H. Wheeler, Simplified models for equatorial waves with vertical structure, Oceanography, 31 (2018), 36-41. doi: 10.5670/oceanog.2018.307.

Figure 1.  Fluid configurations with multiple layers using the notation (1.1) and (1.2). (a) A configuration with N = 4 layers and a rigid lid. (b) A configuration with N = 2 layers and a free surface. This is the type of configuration which will be considered in Section 3
Figure 2.  Shear flows $\overline U(z)$ corresponding to the stream functions $\overline\Psi_1, \overline\Psi_2$ in (3.9). Both flows have $\omega_2 < 0 < \omega_1$ and $c > 0$. (a) A flow with a critical layer at the marked point in $D_1$ where $\overline U_1 = c$. (b) A flow without a critical layer
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