On fully nonlinear, vertically trapped wave packets in a stratified fluid on the f-plane

Abstract

The ubiquity of solitary and solitarylike internal waves in the coastal ocean has been recognized for some time. Recent theoretical studies of a strongly nonlinear, weakly nonhydrostatic set of layer-averaged model equations have predicted that rotation, for example, on the f-plane, can lead to the decay and subsequent reemergence of internal solitary waves. We reconsider this problem using high resolution numerical simulations of the rotating stratified Euler equations. We find that in certain cases the initial disturbances indeed fission into nonlinear wave packets, with the constituent waves making up the wave packet being, in themselves, nonlinear. However, for typical coastal ocean parameters this only occurs at rotation rates higher than those on Earth on the time scales we are able to simulate. We confirm, using the Dubreil–Jacotin–Long equation, that the vertical structure of the wave-induced currents is well predicted by the fully nonlinear theory of nonrotating internal solitary waves and that weakly nonlinear Korteweg–de Vries equation-based theory fails to describe this structure accurately. Subsequently, we consider flat-crested solitary waves that allow us to fix the wave amplitude while varying the horizontal wavelength. We find that as the waves’ horizontal extent nears the baroclinic Rossby radius more energy is deposited into the wave tail. However, no wave overtaking is observed, and an explanation for this fact is proposed. Finally, we discuss the effects of the horizontal component of the rotation vector and derive an exact equation for rotation modified waves near the equator. This equation demonstrates that in this situation, rotation modifies the structure of the fully nonlinear waves but does not lead to solitary wave decay.

Experimental study of the effect of rotation on nonlinear internal waves

“…Several numerical experiments of model equations, including the Ostrovsky equation (see equation (6) below), and of fully nonlinear equation systems have shown that background rotation tends to inhibit the production of internal solitary waves, see Helfrich 9 , Grimshaw and Helfrich 6,7 , and Stastna et al 16 , and the references therein. In particular, Helfrich 9 found that an initial solitary wave disturbance decayed through the radiation of inertial gravity waves, and eventually an envelope wave packet formed.…”

The fully nonlinear stratified geostrophic adjustment problem

“…Motivated by the results presented in Stastna et al (2009), a domain 4 times longer than the physical tank (Lx = 52 m) was used, as 13 m is an insufficient length when considering lower (closer to physical) rotation rates. In addition to this it was decided to change the tank depth to a more evenly divisible 0.4 m (from a laboratory value of 0.36 m).…”

“…Additionally, they also performed collision experiments, finding that the packets that emerge from the initial solitary waves can merge during collisions and hence do not interact as classical solitons. Finally, Stastna et al (2009) also found that by increasing the width of a flat-crested wave, more energy was deposited into the tail. It remains to reconcile the two sets of results in detail, likely by systematically reducing the solitary wave amplitude used as an initial condition.…”

“…Helfrich suggested that the initial solitary wave breaks down into a coherent leading wave packet with a trailing tail of waves. Stastna et al (2009) suggested that for large amplitude, exact internal solitary waves that are solutions to the Dubreil-Jacotin-Long (DJL) equation, Helfrich's result was observed for artificially high rotation rates, while rotation rates typical of mid-latitudes led to a disturbance that never fully separated from the trailing tail. Despite differences in details, the qualitative features observed in both studies were quite similar.…”

“…Work has also been performed using models with higherorder nonlinearity (Helfrich, 2007) but weak nonhydrostatic effects, as well as with the full set of stratified Euler equations (Stastna et al, 2009). Both of these studies consid-ered the breakdown of an initial solitary wave in the presence of rotation.…”