Dispersion relations for fixed mean-depth flows with two discontinuities in vorticity

Dispersion relations for steady periodic water waves of fixed mean-depth with two rotational layers

“…A further piecewise constant vorticity distribution in the fixed mean-depth setting was considered in [40]. We would also like to note that dispersion relations were analyzed in several rotational flows of continuous non-constant vorticity [39].…”

“…Analysis of the dispersion relation. The unique positive solution of (40) gives the relative speed at the surface of the laminar flows in terms of several parameters. In our case, those parameters are the fixed mean depth of the flow, the location of the jump in vorticity, the vorticity distribution itself and the wavelength (which is not explicit but that can be recovered by means of the dimension variables prior to the scaling (3)).…”

Dispersion Relations for Fixed Mean-Depth Steady Periodic Water Waves with Affine Vorticities

“…The work [16] is a great breakthrough for rotational water waves, and has been extended to more general settings. See [2][3][4][5]26,34] for rotational water waves, [9,17,25,27,[29][30][31]36] for water waves with discontinuous vorticity distributions. We also refer the reader to the monograph [8] for more results on rotational water waves.…”

“…To fix the mean-depth of the water wave is more natural and desirable from both intuitive and physical viewpoints. Following Henry's idea, the problem of water waves with a fixed mean-depth has been studied in [3,6,20,21,25,29,[34][35][36] in different settings. As we will show in Sect.…”

“…A particular character is that the pronounced density stratification of the equatorial regions is greater than anywhere else in the ocean [19]. See [1,4,6,7,[10][11][12][13][14]24,26,[29][30][31][32][33][34][35]37] for recent results on equatorial flows. We will focus on the dispersion relation for equatorial water waves when the vorticity functions admit piecewise affine distributions.…”