SOME PROBLEMS OF HYDRODYNAMIC STABILITY ARISING IN GEOPHYSICAL FLUID DYNAMICS (ORDER & CHAOS, DOUBLE DIFFUSION, LANGMUIR CIRCULATIONS)

Two problems of hydrodynamic instability are considered. One concerns the linear and nonlinear dynamics of Langmuir circulations in the ocean arising from the destabilizing influence of current shear and wave action. The second considers instabilities of geophysical Ekman layers.; The theoretical model of Leibovich (1977a) is used to investigate the influence of the thermal environment on Langmuir circulations. Stable stratification is found to limit both the strength and the vertical penetration of the circulations: strong thermoclines act as a virtual bottom. Linear stability characteristics for finite layer problems with various combinations of mechanical and thermal boundary conditions are obtained numerically. Limit cases of interest are treated analytically, a case of particular interest establishes an analogy between Langmuir circulations and Marangoni convection.; The nonlinear physics of Langmuir circulations is studied with carefully chosen finite difference simulations. Even with weak stable stratification the onset of steady circulations is subcritical. As the stratification is increased two classes of steady states differing in spatial form arise, one of which is generally unstable. Increasing stratification results in additional steady states branching from the stable family. Time periodic states also arise with sufficient stratification; these lose stability to steady states as the destabilization increases. The chaotic dynamics, suggested by a study of the low order truncations, is not found. Quasi-periodic behavior arises, however, in the vicinity of parameter values for which two modes are simultaneously unstable. The present results may be interpreted in terms of double-diffusion analogs, establishing the importance of boundary and symmetry conditions on the nonlinear behavior of such systems, and suggesting new avenues yet to be explored in familiar double-diffusion problems.; Instabilities of atmospheric and oceanic Ekman layers are found to be affected by the horizontal component of Earth's angular velocity. This influence is generally destabilizing, it decreases the critical Reynolds number R, increases the growth rate of unstable modes, and widens the band of unstable wavenumbers. Several new features of the high Reynolds number limit are established, including the existence of multiple (possibly infinite) unstable modes, and the viscous behavior of the lower branch of the neutral stability curve.