Astronomy and Astrophysics – Astrophysics
Scientific paper
Jul 1999
adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1999phdt.........3c&link_type=abstract
Thesis (PhD). COLUMBIA UNIVERSITY, Source DAI-B 60/01, p. 209, Jul 1999, 144 pages.
Astronomy and Astrophysics
Astrophysics
Diffusive Relaxation, Negative Energy Modes
Scientific paper
This work explores two problems in hydrodynamic stability theory whose primary mathematical distinction is the manner in which unstable equilibria relax to a stable state. The first problem involves a dissipative fluid for which the diffusive relaxation of a linear instability is the primary saturation mechanism. The second problem arises from a conservative system where the relaxation is governed either by linear instabilities or nonlinear interactions among linearly stable modes. The dissipative problem considers thermohaline convection and the diffusive relaxation of two-dimensional, convectively unstable rolls. For this situation, the disparate diffusion rates of the heat and salt play a crucial role not only in triggering an instability, but also in dictating the nature of the saturation. We study these issues in two particular contexts. In the first case the effect of a nonlinear, basic state salt profile on the bifurcation to instability is examined, and some of the possible patterns in the weakly nonlinear regime are discussed. The second case explores the dynamics of a salt field diffusing so slowly that it does not have time to redistribute itself on the timescale over which the weakly unstable pattern develops. This necessitates a mathematical description that differs from the amplitude equations traditionally employed to capture the nonlinear saturation. The conservative system comes from astrophysics and cosmology. Motivated by the coexistence of dark and luminous matter in the cosmos, we investigate the instabilities of two interpenetrating fluids that interact only through the gravitational field. The result of primary importance is that negative energy modes may exist if the relative motion is substantial. This fact can have dramatic consequences on the nonlinear stability of the system if linearly stable waves of different energy signature conspire to resonate. In the absence of dissipative forces, the nature of the saturation must be a nonlinear one, and therefore fundamentally differs from the diffusive relaxation involved in the thermohaline problem. The possible nonlinear dynamics are elucidated within the context of the free-energy principle, the validity of which hinges on the existence of a conserved Hamiltonian functional whose definiteness or indefiniteness reveals the crucial stability information. When the Hamiltonian is indefinite, both positive and negative energy modes can coexist, and a triad of resonant waves may experience explosive instability. The implication of this for structure formation is briefly discussed. The influence of dissipation on the negative energy modes is also examined. Of primary concern is the condition under which dissipation may render a previously stable system unstable by feeding more negative energy to a negative energy wave. Criteria are presented that describe whether dissipation of various types projects onto the negative energy modes to promote instability, or whether the dissipation in some sense ``misses'' the negative energy modes and induces their decay by feeding them positive energy.
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