A Lagrangian study of internal Gerstner- and Stokes-type gravity waves

Abstract

By using a direct Lagrangian formulation, two different solutions for the nonlinear motion in inviscid internal gravity waves between horizontal planes in a non-rotating fluid are discussed. The first solution assumes a constant Brunt–Väisälä frequency N. The main findings are that the vertical mean displacement of the isopycnals and the horizontal mean drift are both zero to second order (Sanderson, 1985). We here point out that this is characteristic for a Gerstner-type wave. Attention is drawn to a second solution for arbitrary stable stratification which yields a non-zero vertical mean isopycnal displacement. Although internal waves are rotational, this solution characterizes a Stokes-type wave. The vertical mean displacement, which can be positive or negative depending on the spatial location and the wave mode in question, is due to the divergence effect in Lagrangian terms. It is shown that the vertical mean displacement at a certain level is proportional to the depth-dependent, or partial Stokes flux. Finally, it is demonstrated that the two wave types have different mean vorticities to second order. For constant , they are equal in magnitude, but have opposite signs. In the Appendix the existence of a mean drift in internal waves of the Stokes-type is verified by a direct Lagrangian calculation for a slightly viscous fluid with constant N.