Mathematics – Logic
Scientific paper
Feb 1991
adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1991phdt........81w&link_type=abstract
Thesis (PH.D.)--UNIVERSITY OF WASHINGTON, 1991.Source: Dissertation Abstracts International, Volume: 52-12, Section: B, page: 64
Mathematics
Logic
Scientific paper
A sodium chloride solution is used as a model for the natural solidification of seawater. A linear perturbation theory is used to show that the one-dimensional steady state describing the unidirectional solidification of a dilute H_2O-NaCl solution is morphologically unstable. This instability breaks the translational symmetry of the steady state, resulting in the transition from a planar to a cellular solid-liquid interface. The cellular interface represents a state with a different translational symmetry; parallel arrays of ice platelets. Consistent with other studies, for fixed far-field solute concentration C_infty, it is found that there is a range of solidification velocity V_ {c} < V < Va for which the system is linearly unstable to a range perturbations. Naturally occurring sea ice grows at rates between theses limits, so it will always have a cellular solid-liquid interface. The system exhibits weak wavelength selection near critical, that is, the wavelength of the instability is only weakly dependent on V near Vc, At low values of the segregation coefficient k, the predicted value of Vc is so small that it invalidates a continuum theory. The fluid layer adjacent to the solid-liquid interface is found to be hydrodynamically unstable for mean growth velocities between V_ {c} and V ~ 10^{-4} cm s^ {-1}. A weakly non-linear Landau analysis reveals that, during natural solidification, the transition to a cellular interface occurs via a subcritical bifurcation. That is, there is a 'jump' transition to cells for V close to V c. In three cases we derive scaling laws that relate the wavelength to the mean growth velocity of the interface: In the long wavelength limit (a) for V close to V c (b) for V close to V a, and in the short wavelength limit for velocities between Vc and Va. The linear theory provides a characteristic equation relating the disturbance growth rate to wavelength, mean growth velocity and other control variables. We view the characteristic equation as a one-parameter unfolding f of a cuspoid normal form N = xm where m = 3. This approach allows us to study the solidification system in terms of its basic geometric structure and to distinguish it from other formulations using geometric criteria. It is recommended that the critical point of instability be investigated experimentally for temperature gradients one or two orders of magnitude greater than the typical geophysical value. Interferometry can be used to detail the solute field adjacent to the interface and to assess the role of convection.
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