In the Euler-Lagrange expression the vertical displacement of an isopycnal is given as a function of the rest height of the isopycnal. In figure 1 it can be seen that for density 1 the Eulerian description is much better than the Euler-Lagrange description while figure 2 shows that the opposite is true for density 2. Use of first-order theory for density 2 also reduces the error significantly but the agreement is not as good. For density 2 the largest wave is far from breaking. In figures 3 and 4 vertical profiles of the horizontal velocity through the centre of the waves are shown.
Solid curves are the profiles for the fully-nonlinear waves.
Dashed profiles are the theoretical results. For leading-order theory the terms are dropped. For the first-order expressions for u the value of B xx is needed in the centre of the wave. Here at the wave centre. For density 1 the theoretical estimate is very good for even the largest wave.