July  2022, 21(7): 2309-2325. doi: 10.3934/cpaa.2022061

Weakly nonlinear waves in stratified shear flows

1. 

Delft University of Technology, Delft Institute of Applied Mathematics, Faculty of EEMCS, Mekelweg 4, 2628 CD Delft, The Netherlands

2. 

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

*Corresponding author

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

Fund Project: The second author is supported by the Austrian Science Fund (FWF), Erwin Schrödinger fellowship J 4339-N32

We develop a Korteweg–De Vries (KdV) theory for weakly nonlinear waves in discontinuously stratified two-layer fluids with a generally prescribed rotational steady current. With the help of a classical asymptotic power series approach, these models are directly derived from the divergence-free incompressible Euler equations for unidirectional free surface and internal waves over a flat bed. Moreover, we derive a Burns condition for the determination of wave propagation speeds. Several examples of currents are given; explicit calculations of the corresponding propagation speeds and KdV coefficients are provided as well.

Citation: Anna Geyer, Ronald Quirchmayr. Weakly nonlinear waves in stratified shear flows. Communications on Pure and Applied Analysis, 2022, 21 (7) : 2309-2325. doi: 10.3934/cpaa.2022061
References:
[1]

T. B. Benjamin, The solitary wave on a stream with an arbitrary distribution of vorticity, J. Fluid Mech., 12 (1962), 97-116.  doi: 10.1017/S0022112062000063.

[2]

J. Burns, Long waves in running water, Math. Proc. Cambridge Philos., 49 (1953), 695-706.  doi: 10.1017/S0305004100028899.

[3]

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.

[4]

A. Compelli and R. I. 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.

[5]

A. Compelli, R. I. Ivanov and M. Todorov, Hamiltonian models for the propagation of irrotational surface gravity waves over a variable bottom, Phil. Trans. R. Soc. A, 376 (2018), 15 pp. doi: 10.1098/rsta.2017.0091.

[6]

A. CompelliR. I. IvanovC. I. Martin and M. D. Todorov, Surface waves over currents and uneven bottom, Deep Sea Res. Part II, 160 (2019), 25-31.  doi: 10.1016/j.dsr2.2018.11.004.

[7]

A. Constantin and R. I. Ivanov, A Hamiltonian approach to wave-current interactions in two-layer fluids, Phy. Fluids, 27 (2015), 8 pp. doi: 10.1063/1.4929457.

[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. 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.

[10]

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.

[11]

J. Cullen and R. I. Ivanov, On the intermediate long wave propagation for internal waves in the presence of currents, Eur. J. Mech. B Fluids, 84 (2020), 325-333.  doi: 10.1016/j.euromechflu.2020.07.001.

[12]

N. C. Freeman and R. S. Johnson, Shallow water waves on shear flows, J. Fluid Mech., 42 (1970), 401-409.  doi: 10.1017/S0022112070001349.

[13]

A. Geyer and R. Quirchmayr, Shallow water equations for equatorial tsunami waves, Philos. Trans. Roy. Soc. London Ser. A, 376 (2018), 12 pp. doi: 10.1098/rsta.2017.0100.

[14]

A. Geyer and R. Quirchmayr, Shallow water models for stratified equatorial flows, Discrete Contin. Dyn. Syst., 39 (2019), 4533-4545.  doi: 10.3934/dcds.2019186.

[15]

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

[16]

R. I. Ivanov, Hamiltonian model for coupled surface and internal waves in the presence of currents, Nonlinear Anal. Real World Appl., 34 (2017), 316-334.  doi: 10.1016/j.nonrwa.2016.09.010.

[17]

R. S. Johnson, On the nonlinear critical layer below a nonlinear unsteady surface wave, J. Fluid Mech., 167 (1986), 327-351.  doi: 10.1017/S0022112086002847.

[18]

R. S. Johnson, On solutions of the Burns condition (which determines the speed of propagation of linear long waves on a shear flow with or without a critical layer), Geophys. Astrophys. Fluid Dyn., 57 (1991), 115-133.  doi: 10.1080/03091929108225231.

[19] R. S. Johnson, A Modern Introduction to the Mathematical Theory of Water Waves, Cambridge University Press, Cambridge, UK, 1997.  doi: 10.1017/CBO9780511624056.
[20]

R. S. Johnson, A problem in the classical theory of water waves: weakly nonlinear waves in the presence of vorticity, J. Nonlinear Math. Phys., 19 (2012), 137-160.  doi: 10.1142/S1402925112400128.

[21]

R. S. Johnson, An ocean undercurrent, a thermocline, a free surface, with waves: a problem in classical fluid mechanics, J. Nonlinear Math. Phys., 22 (2015), 475-493.  doi: 10.1080/14029251.2015.1113042.

[22]

C. I. Martin, Azimuthal equatorial flows in spherical coordinates with discontinuous stratification, Phys. Fluids, 33 (2021), 9 pp. doi: 10.1063/5.0035443.

[23]

C. I. Martin and R. Quirchmayr, Exact solutions and internal waves for the Antarctic Circumpolar Current in spherical coordinates, Stud. Appl. Math., 48 (2022), 1021-1039.  doi: 10.1111/sapm.12467.

show all references

References:
[1]

T. B. Benjamin, The solitary wave on a stream with an arbitrary distribution of vorticity, J. Fluid Mech., 12 (1962), 97-116.  doi: 10.1017/S0022112062000063.

[2]

J. Burns, Long waves in running water, Math. Proc. Cambridge Philos., 49 (1953), 695-706.  doi: 10.1017/S0305004100028899.

[3]

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.

[4]

A. Compelli and R. I. 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.

[5]

A. Compelli, R. I. Ivanov and M. Todorov, Hamiltonian models for the propagation of irrotational surface gravity waves over a variable bottom, Phil. Trans. R. Soc. A, 376 (2018), 15 pp. doi: 10.1098/rsta.2017.0091.

[6]

A. CompelliR. I. IvanovC. I. Martin and M. D. Todorov, Surface waves over currents and uneven bottom, Deep Sea Res. Part II, 160 (2019), 25-31.  doi: 10.1016/j.dsr2.2018.11.004.

[7]

A. Constantin and R. I. Ivanov, A Hamiltonian approach to wave-current interactions in two-layer fluids, Phy. Fluids, 27 (2015), 8 pp. doi: 10.1063/1.4929457.

[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. 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.

[10]

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.

[11]

J. Cullen and R. I. Ivanov, On the intermediate long wave propagation for internal waves in the presence of currents, Eur. J. Mech. B Fluids, 84 (2020), 325-333.  doi: 10.1016/j.euromechflu.2020.07.001.

[12]

N. C. Freeman and R. S. Johnson, Shallow water waves on shear flows, J. Fluid Mech., 42 (1970), 401-409.  doi: 10.1017/S0022112070001349.

[13]

A. Geyer and R. Quirchmayr, Shallow water equations for equatorial tsunami waves, Philos. Trans. Roy. Soc. London Ser. A, 376 (2018), 12 pp. doi: 10.1098/rsta.2017.0100.

[14]

A. Geyer and R. Quirchmayr, Shallow water models for stratified equatorial flows, Discrete Contin. Dyn. Syst., 39 (2019), 4533-4545.  doi: 10.3934/dcds.2019186.

[15]

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

[16]

R. I. Ivanov, Hamiltonian model for coupled surface and internal waves in the presence of currents, Nonlinear Anal. Real World Appl., 34 (2017), 316-334.  doi: 10.1016/j.nonrwa.2016.09.010.

[17]

R. S. Johnson, On the nonlinear critical layer below a nonlinear unsteady surface wave, J. Fluid Mech., 167 (1986), 327-351.  doi: 10.1017/S0022112086002847.

[18]

R. S. Johnson, On solutions of the Burns condition (which determines the speed of propagation of linear long waves on a shear flow with or without a critical layer), Geophys. Astrophys. Fluid Dyn., 57 (1991), 115-133.  doi: 10.1080/03091929108225231.

[19] R. S. Johnson, A Modern Introduction to the Mathematical Theory of Water Waves, Cambridge University Press, Cambridge, UK, 1997.  doi: 10.1017/CBO9780511624056.
[20]

R. S. Johnson, A problem in the classical theory of water waves: weakly nonlinear waves in the presence of vorticity, J. Nonlinear Math. Phys., 19 (2012), 137-160.  doi: 10.1142/S1402925112400128.

[21]

R. S. Johnson, An ocean undercurrent, a thermocline, a free surface, with waves: a problem in classical fluid mechanics, J. Nonlinear Math. Phys., 22 (2015), 475-493.  doi: 10.1080/14029251.2015.1113042.

[22]

C. I. Martin, Azimuthal equatorial flows in spherical coordinates with discontinuous stratification, Phys. Fluids, 33 (2021), 9 pp. doi: 10.1063/5.0035443.

[23]

C. I. Martin and R. Quirchmayr, Exact solutions and internal waves for the Antarctic Circumpolar Current in spherical coordinates, Stud. Appl. Math., 48 (2022), 1021-1039.  doi: 10.1111/sapm.12467.

Figure 1.  Fig. 1a shows a sketch of the stratified fluid domain bounded by a free surface at $ \bar z = \bar\eta(\bar x, \bar t) $ and a fixed bottom at $ \bar z = -\bar d $ with an interface at $ \bar z = -\bar h + \bar H(\bar x, \bar t) $ separating the upper fluid with density $ \bar\rho = \bar\rho_0 $ from the denser lower one, where $ \bar\rho = \bar\rho_0(1+r) $. Fig. 1b illustrates an example of a background current $ \bar U(\bar z) $
[1]

Adrian Constantin. Dispersion relations for periodic traveling water waves in flows with discontinuous vorticity. Communications on Pure and Applied Analysis, 2012, 11 (4) : 1397-1406. doi: 10.3934/cpaa.2012.11.1397

[2]

Calin Iulian Martin. Dispersion relations for periodic water waves with surface tension and discontinuous vorticity. Discrete and Continuous Dynamical Systems, 2014, 34 (8) : 3109-3123. doi: 10.3934/dcds.2014.34.3109

[3]

Calin Iulian Martin. A Hamiltonian approach for nonlinear rotational capillary-gravity water waves in stratified flows. Discrete and Continuous Dynamical Systems, 2017, 37 (1) : 387-404. doi: 10.3934/dcds.2017016

[4]

Delia Ionescu-Kruse, Anca-Voichita Matioc. Small-amplitude equatorial water waves with constant vorticity: Dispersion relations and particle trajectories. Discrete and Continuous Dynamical Systems, 2014, 34 (8) : 3045-3060. doi: 10.3934/dcds.2014.34.3045

[5]

Raphael Stuhlmeier. Effects of shear flow on KdV balance - applications to tsunami. Communications on Pure and Applied Analysis, 2012, 11 (4) : 1549-1561. doi: 10.3934/cpaa.2012.11.1549

[6]

Wenzhang Huang. Weakly coupled traveling waves for a model of growth and competition in a flow reactor. Mathematical Biosciences & Engineering, 2006, 3 (1) : 79-87. doi: 10.3934/mbe.2006.3.79

[7]

Walter A. Strauss. Vorticity jumps in steady water waves. Discrete and Continuous Dynamical Systems - B, 2012, 17 (4) : 1101-1112. doi: 10.3934/dcdsb.2012.17.1101

[8]

Alex Mahalov, Mohamed Moustaoui, Basil Nicolaenko. Three-dimensional instabilities in non-parallel shear stratified flows. Kinetic and Related Models, 2009, 2 (1) : 215-229. doi: 10.3934/krm.2009.2.215

[9]

Rui Huang, Ming Mei, Yong Wang. Planar traveling waves for nonlocal dispersion equation with monostable nonlinearity. Discrete and Continuous Dynamical Systems, 2012, 32 (10) : 3621-3649. doi: 10.3934/dcds.2012.32.3621

[10]

Yuqian Zhou, Qian Liu. Reduction and bifurcation of traveling waves of the KdV-Burgers-Kuramoto equation. Discrete and Continuous Dynamical Systems - B, 2016, 21 (6) : 2057-2071. doi: 10.3934/dcdsb.2016036

[11]

Jundong Wang, Lijun Zhang, Elena Shchepakina, Vladimir Sobolev. Solitary waves of singularly perturbed generalized KdV equation with high order nonlinearity. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022124

[12]

Patrick Martinez, Jean-Michel Roquejoffre. The rate of attraction of super-critical waves in a Fisher-KPP type model with shear flow. Communications on Pure and Applied Analysis, 2012, 11 (6) : 2445-2472. doi: 10.3934/cpaa.2012.11.2445

[13]

Minoru Murai, Kunimochi Sakamoto, Shoji Yotsutani. Representation formula for traveling waves to a derivative nonlinear Schrödinger equation with the periodic boundary condition. Conference Publications, 2015, 2015 (special) : 878-900. doi: 10.3934/proc.2015.0878

[14]

Eric S. Wright. Macrotransport in nonlinear, reactive, shear flows. Communications on Pure and Applied Analysis, 2012, 11 (5) : 2125-2146. doi: 10.3934/cpaa.2012.11.2125

[15]

David Henry, Rossen Ivanov. One-dimensional weakly nonlinear model equations for Rossby waves. Discrete and Continuous Dynamical Systems, 2014, 34 (8) : 3025-3034. doi: 10.3934/dcds.2014.34.3025

[16]

Nate Bottman, Bernard Deconinck. KdV cnoidal waves are spectrally stable. Discrete and Continuous Dynamical Systems, 2009, 25 (4) : 1163-1180. doi: 10.3934/dcds.2009.25.1163

[17]

H. Beirão da Veiga. Vorticity and regularity for flows under the Navier boundary condition. Communications on Pure and Applied Analysis, 2006, 5 (4) : 907-918. doi: 10.3934/cpaa.2006.5.907

[18]

Raphael Stuhlmeier. Internal Gerstner waves on a sloping bed. Discrete and Continuous Dynamical Systems, 2014, 34 (8) : 3183-3192. doi: 10.3934/dcds.2014.34.3183

[19]

José Raúl Quintero, Juan Carlos Muñoz Grajales. Solitary waves for an internal wave model. Discrete and Continuous Dynamical Systems, 2016, 36 (10) : 5721-5741. doi: 10.3934/dcds.2016051

[20]

Jerry L. Bona, Henrik Kalisch. Models for internal waves in deep water. Discrete and Continuous Dynamical Systems, 2000, 6 (1) : 1-20. doi: 10.3934/dcds.2000.6.1

2021 Impact Factor: 1.273

Metrics

  • PDF downloads (206)
  • HTML views (111)
  • Cited by (0)

Other articles
by authors

[Back to Top]