Two-dimensionality of gravity water flows governed by the equatorial f-plane approximation

Calin Iulian Martin

doi:10.1007/s10231-017-0663-2

Abstract

We show that gravity wave trains governed by the equatorial f-plane approximation propagate at the free surface of a rotational water flow of constant vorticity vector (Omega _1, Omega _2, Omega _3) over a flat bed only if the flow is two-dimensional. Owing to the presence of Coriolis effects, our result is also true even if the vorticity vector vanishes. This represents a striking difference when compared with the cases without geophysical effects discussed in Constantin (Europhys Lett 86:29001, 2009, Eur J Mech 30:12–16; 2011) and Martin (J Math Fluid Mech 2016. doi:10.1007/s00021-016-0306-1), where the conclusion about the two-dimensionality of the flow was possible under the assumption of constant nonvanishing vorticity vector. Another upshot is that the only nonzero component of the vorticity that may not vanish is Omega _2, that is, the one pointing in the horizontal direction orthogonal to the direction of wave propagation.

Highlights

There is an increased interest nowadays in the study of geophysical water flows
We show that gravity wave trains governed by the equatorial f -plane approximation propagate at the free surface of a rotational water flow of constant vorticity vector ( 1, 2, 3) over a flat bed only if the flow is two-dimensional
Owing to the presence of Coriolis effects, our result is true even if the vorticity vector vanishes. This represents a striking difference when compared with the cases without geophysical effects discussed in Constantin (Europhys Lett 86:29001, 2009, Eur J Mech 30:12–16; 2011) and Martin (J Math Fluid Mech 2016. doi:10.1007/s00021-016-0306-1), where the conclusion about the two-dimensionality of the flow was possible under the assumption of constant nonvanishing vorticity vector

Summary

Introduction

There is an increased interest nowadays in the study of geophysical water flows. These are flows whose motion is influenced by the Earth’s rotation reflected through the manifestation of the Coriolis force. Abstract We show that gravity wave trains governed by the equatorial f -plane approximation propagate at the free surface of a rotational water flow of constant vorticity vector ( 1, 2, 3) over a flat bed only if the flow is two-dimensional.

Results

Conclusion