Z and satisfy the same eigenvalue problem so, if they are both scaled to have a maximum value of 1, they are identical. The theoretical approximations are now compared with fully nonlinear waves. The bottom to surface density difference is 0.01 in both cases. For both stratifications the wave half-widths are approximately 100 m for the largest waves. Comparisons of the isopycnal displacement along the vertical line through the centre of a number of waves are shown in figures 1 and 2.
Solid curves are profiles from the fully nonlinear waves.
Dashed curves are the theoretical profiles. In the Euler-Lagrange formulation the leading order isopycnal displacement is . For the Eulerian formulation the isopycnal displacement is obtained by expanding which gives at leading-order. Since both expression have the identical form however there is a crucial difference between the two. In the Eulerian expression is the vertical displacement of an isopycnal as a function of the displaced isopycnal height.