Directional Solidification of Salt Water: Deep and Shallow Cells

The common observation of cellular substructure ((1 + at the seaice-ocean interface is explained by modelling the natural solidification of seawater as that of a dilute H20-NaCl solution. Linear and nonlinear perturbation theories reveal that the onedimensional planar steady state is morphologically unstable, and a bifurcation to cells occurs in geophysically relevant growth regimes. We compute the range of solidification velocity V, < V < V, in which the system is unstable for fxed far-field solute concentration C, , and bound the geophysical observations. The system exhibits weak wavelength selection near V, and the nonlinear theory shows that the bifurcation to cells is subcritical. Measurements reveal that during growth, mature Arctic Sea ice has a cellular solid-liquid interface [ 13. The substructure consists of horizontal c-axes, with random intergrain orientation, and vertical pure ice platelets extending downward, separating regions of concentrated seawater (fig. 1). This two-phase structure influences the bulk mechanical and thermophysical properties of sea ice in much the same way as in binary alloys. A review of the growth and structure of naturally occurring sea ice can be found elsewhere [l]. Other relevant topics include freezing potentials induced by preferential ion incorporation [2], and the influence of bulk solution on acoustic and electromagnetic propagation through the solidified material [3]. Previously, horizontal frezing of ice cylinders from aqueous solutions of concentrations much less than seawater has been investigated[4]. To study the dependence of sea ice platelet spacing on growth parameters, the only approach has been to assume that the stable interfacial growth form is cellular [5]. Qualitatively this approach is useful in studying the transition from cells to dendrites [6], or for long-time simulations of bulk properties in growth from an impure melt [7]. We are interested in the physics responsible for the substructure in this system. The directional solidification system [8-101 is used in a variety of forms to examine the uni-directional solidification of dilute binary alloys. The alloy is pulled with a constant speed V through an imposed thermal field, and a mean position is established at which the planar solid-liquid interface is located. This state can be maintained for growth velocities V less than a critical velocity V, , or above an absolute velocity V, . The influence of the thermal fields and surface tension must overcome the destabilizing effects of solute rejection at the interface to maintain this state [%lo]. Mullins and Sekerka [9] showed that for an alloy solidifying within V, V < V, , a bifurcation to cells occurs, because solute diffusion is too slow to remove local 338 EUROPHYSICS LETTERS Fig. 1. Horizontal thin section of Arctic Sea ice exhibiting a substructure of pure ice platelets with uniform intergrain spacing. The crystallographic c-axis (parallel to arrow) is in the plane of the section and oriented perpendicular to the platelet structure of individual crystals. Bulk seawater is trapped between the platelets. The come scale is in mm. (Photograph is courtesy of Dr. k J. Gow.) impurity-induced undercooling. The purpose of this letter is to show that the ice platelet substructure results from a diffusive instability, and to call for careful experiments on this transparent system, in this geometry. As mature sea ice cools, an adverse temperature gradient will develop, so that V approaches V, from below (V+ V , ) . For the salt-water system we find that V,, for the onset of morphological instability, depends weakly on the disturbance wavelength, and that the entire instability range is a one of subcritical bihrcation, meaning that cells appear via a jump transition as V+.V; , e.g. [8,10-121. The solidification of seawater is approximated as that of a dilute H2 O-NaCl solution, since NaCl is the major impurity species. We analyze a thermally nonsymmetric (thermal conductivities of both phases differ), chemically one-sided model (solute diffusion in the solid is negligible). Because the intergrain platelet spacing is uniform (fig. l), we focus on a single crystal grain, and ignore grain boundary energy effects. Convective transport of heat or solute is prescribed, because the critical wavelength for convective instability is two orders of magnitude greater than that of morphological instability. The reference frame is attached to the solidification front which is moving with a speed V into the liquid. The planar interface is disturbed as h(x, t), along the direction x, perpendicular to the direction of V. The lengths and times in the problem are scaled on the solute diffusion scales x , x = ( x ’ , x ’) V/D and t = = t ‘ V 2 /D , where the primed quantities have dimensions, and D is the diffusivity of solute in the liquid. The temperature and solute scales are TL, s = (TL, s Td)/G * D/V, and C = = [C’k C , ] / ( k 1) C, , where the far-field solute concentration is C, and k is the segregation coefficient, giving the ratio of the solute in the solid to that in the liquid. The low solubility of salt in ice is represented by k < 1. These are related to the scales from a thermally symmetric zero-latent-heat analysis [13], but here we include latent heat. The subscripts L, S denote the liquid and solid phases, and Ti is the reference temperature of the planar interface. The dimensional temperatures Ti,s = TL*,s T, are measured relative to J. S. WETTLAUFER: DIRECTIONAL SOLIDIFICATION OF SALT WATER ETC. 339 the bulk melting temperature of ice T , . The quantity G * = (2Gi + LV/kL) / ( l + n) is the average temperature gradient at the interface. The dimensional liquid temperature gradient is GL', the latent-heat per unit volume is L, and the ratio of solid-to-liquid thermal conductivity is n = ks /kL . Using the above scales, the dimensionless diffusion fields obey the following two-dimensional equations and boundary conditions: for the liquid, x > h(x,t)