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

Weakly nonlinear waves in stratified shear flows

Anna Geyer 1, and  Ronald Quirchmayr 2,,

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.

Keywords: Weakly nonlinear waves, KdV equation, stratified flows, internal waves, vorticity, shear flow, Burns condition, dispersion relation.
Mathematics Subject Classification: 35R35, 76B55, 76B70.
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. Compelli, R. I. Ivanov, C. 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. Constantin, R. 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. Compelli, R. I. Ivanov, C. 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. Constantin, R. 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) $
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