**Principal Investigator**
Douglas R Christie **Project** q51,s52

Research School of Earth Sciences, **Machine**
VP

**Co-Investigator** David
J Brown

Research School of Earth Sciences

**Propagation of Highly Nonlinear Solitary Waves
in Stratified Shear Flows**

This project is concerned with the theoretical description of the propagation and interaction of internal solitary waves of arbitrary amplitude in both incompressible and compressible fluids. Numerical solutions to the Dubreil-Jacotin-Long equation, the governing equation for stationary internal wave motions of finite amplitude in incompressible, inviscid fluids, have been found for a wide variety of physically realistic stability and shear profiles in both Boussinesq and non-Boussinesq fluids. These solutions have been compared with the results of a numerical study of solitary wave propagation in viscous, compressible fluids and also with the predictions of weakly nonlinear wave theory. In addition, an investigation of the properties of interacting highly nonlinear atmospheric solitary waves has been completed along with a study of fluid transport processes in waves of large amplitude.

**What are the basic questions addressed?**

How accurate is weakly nonlinear wave theory? How do the propagation properties of internal solitary waves depend on the waveguide stability and shear profiles? Can the Boussinesq approximation be used to describe internal solitary wave motions in geophysical fluids? Can compressibility and viscosity be ignored in the theoretical description of highly nonlinear waves in the lower atmosphere? Are large amplitude internal solitary waves with recirculating flow stable?

**What are the results to date and future of the
work?**

This investigation has led to the discovery of a number of new and interesting properties of internal solitary waves including waves which propagate with maximum speed and amplitude, waves in viscous fluids which periodically develop internal overturning instabilities and a new family of highly nonlinear waves in which recirculating flow is completely suppressed. It has been found that weakly nonlinear wave theory provides an accurate description of wave properties only at modest wave amplitudes. In addition, it has been shown that wave morphology is surprisingly sensitive to the detailed form of the waveguide stability and shear profiles. It has also been shown that the use of the Boussinesq approximation can have a dramatic influence on the properties of internal solitary waves. The results of these numerical calculations show that the properties of internal solitary wave motions are strongly influenced by viscosity. Even in viscous fluids, however, it has been found that internal solitary waves of large amplitude are surprisingly stable features which exhibit soliton-like properties during collision. The results of this detailed study provide a firm foundation for a comparison of fully nonlinear wave theory with geophysical field observations.

**What computational techniques are used and why
is a supercomputer required?**

Two essentially different computational techniques are used in the numerical studies described here. Solutions to the time-independent Dubreil-Jacotin-Long equation are obtained using a novel algorithm based on an efficient adaptive over-relaxation technique. The properties of time-dependent nonlinear wave phenomena are determined from a series of numerical simulations which are based on finite-difference solutions of the full nonhydrostatic primitive ensemble-averaged equations for a compressible fluid. The use of a supercomputer is essential because calculations of this type are very computationally intensive and require large memory.

**Publications**

*Interacting Morning Glories over Northern Australia*,
M. J. Reeder, D. R. Christie, R. K. Smith and R. Grimshaw, Bull.
Amer. Meteor. Soc.**, 76**, 1165-1171 (1995).