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Solidification of binary aqueous solutions under periodic cooling. Part 2. Distribution of solid fraction

  • Guang-Yu Ding (a1) (a2), Andrew J. Wells (a3) and Jin-Qiang Zhong (a1)


We report an experimental study of the distributions of temperature and solid fraction of growing $\text{NH}_{4}\text{Cl}$ $\text{H}_{2}\text{O}$ mushy layers that are subjected to periodical cooling from below, focusing on late-time dynamics where the mushy layer oscillates about an approximate steady state. Temporal evolution of the local temperature $T(z,t)$ at various heights in the mush demonstrates that the temperature oscillations of the bottom cooling boundary propagate through the mushy layer with phase delays and substantial decay in the amplitude. As the initial concentration $C_{0}$ increases, we show that the decay rate of the thermal oscillation with height also decreases, and the propagation speed of the oscillation phase increases. We interpret this as a result of the solid fraction increasing with $C_{0}$ , which enhances the thermal conductivity but reduces the specific heat of the mushy layer. We present a new methodology to determine the distribution of solid fraction $\unicode[STIX]{x1D719}(z)$ in mushy layers for various $C_{0}$ , using only measurements of the temperature $T(z,t)$ . The method is based on the phase behaviour during thermal modulation, and opens up a new approach for inferring mushy-layer properties in geophysical and engineering settings, where direct measurements are challenging. In our experiments, profiles of the solid fraction $\unicode[STIX]{x1D719}(z)$ exhibit a cliff–ramp–cliff structure with large vertical gradients of $\unicode[STIX]{x1D719}$ near the mush–liquid interface and also near the bottom boundary, but much more gradual variation in the interior of the mushy layer. Such a profile structure is more pronounced for higher initial concentration $C_{0}$ . For very low concentration, the solid fraction appears to be linearly dependent on the height within the mush. The volume-average of the solid fraction, and the local fluctuations in $\unicode[STIX]{x1D719}(z)$ both increase as $C_{0}$ increases. We suggest that the fast increase of $\unicode[STIX]{x1D719}(z)$ near the bottom boundary is possibly due to diffusive transport of solute away from the bottom boundary and the depletion of solute content near the basal region.


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Aussillous, P., Sederman, A. J., Gladden, L. F., Huppert, H. E. & Worster, M. G. 2006 Magnetic resonance imaging of structure and convection in solidifying mushy layers. J. Fluid Mech. 552, 99125.
Backstrom, L. G. E. & Eicken, H. 2006 Capacitance probe measurements of brine volume and bulk salinity in first-year sea ice. Cold Reg. Sci. Technol. 46 (3), 167180.
Beckermann, C. & Viskanta, R. 1988 Double-diffusive convection due to melting. Intl J. Heat Mass Transfer 31, 20772089.
Carslaw, H. S. & Jaeger, J. C. 1959 Conduction of Heat in Solids, 2nd edn. Oxford University Press.
Chalmers, B. 1964 Principles of Solidification. Wiley.
Chen, C. F. 1995 Experimental study of convection in a mushy layer during directional solidification. J. Fluid Mech. 293, 8198.
Chen, C. F. & Chen, F. 1991 Experimental study of directional solidification of aqueous ammonium chloride solution. J. Fluid Mech. 227, 567586.
Chen, F. 1997 Formation of double-diffusive layers in the directional solidification of binary solution. J. Cryst. Growth 179, 277286.10.1016/S0022-0248(97)00098-5
Chiareli, A. O. P. & Worster, M. G. 1992 On measurement and prediction of the solid fraction within mushy layers. J. Cryst. Growth 125 (3–4), 487494.
Copley, S., Giamei, A., Johnson, S. & Hornbecker, M. 1970 The origin of freckles in unidirectionally solidified castings. Metall. Mater. Trans. B 1 (8), 21932204.
Ding, G.-Y., Wells, A. J. & Zhong, J.-Q. 2019 Solidification of binary aqueous solutions under periodic cooling. Part 1. Dynamics of mushy-layer growth. J. Fluid Mech. 870, 121146.
Eicken, H., Bock, C., Wittig, R., Miller, H. & Poertner, H. O. 2000 Magnetic resonance imaging of sea-ice pore fluids: methods and thermal evolution of pore microstructure. Cold Reg. Sci. Technol. 31 (3), 207225.
Feltham, D. L., Untersteiner, N., Wettlaufer, J. S. & Worster, M. G. 2006 Sea ice is a mushy layer. Geophys. Res. Lett. 33, L14501.
Golden, K. M., Eicken, H., Heaton, A. L., Miner, J., Pringle, D. J. & Zhu, J. 2007 Thermal evolution of permeability and microstructure in sea ice. Geophys. Res. Lett. 34 (16), L16501.
Hallworth, M. A. & Huppert, H. E. 2004 Crystallization and layering induced by heating a reactive porous medium. Geophys. Res. Lett. 31, L13605.
Hallworth, M. A., Huppert, H. E. & Woods, A. W. 2005 Dissolution-driven convection in a reactive porous medium. J. Fluid Mech. 535, 255285.
Head, M. J. 1983 The Use of Miniature Four-electrode Conductivity Probes for High Resolution Measurement of Turbulent Density or Temperature Variations in Salt-stratified Water Flows. University of California, San Diego.
Hobbs, P. V. 2010 Ice Physics. Oxford University Press.
Huguet, L., Alboussiere, T., Bergman, M. I., Deguen, R., Labrosse, S. & Lesceur, G. 2016 Structure of a mushy layer under hypergravity with implications for earth’s inner core. Geophys. J. Intl 204, 17291755.
Hunke, E. C., Notz, D., Turner, A. K. & Vancoppenolle, M. 2011 The multiphase physics of sea ice: a review for model developers. Cryosphere 5 (4), 9891009.
Hunkeler, P. A., Hendricks, S., Hoppmann, M., Farquharson, C. G., Kalscheuer, T., Grab, M., Kaufmann, M. S., Rabenstein, L. & Gerdes, R. 2015 Improved 1D inversions for sea ice thickness and conductivity from electromagnetic induction data: inclusion of nonlinearities caused by passive bucking multifrequency EM sea ice inversions. Geophysics 81 (1), WA45.
Huppert, H. E. 1990 The fluid mechanics of solidification. J. Fluid Mech. 212, 209240.
Huppert, H. E. & Worster, M. G. 1985 Dynamic solidification of a binary melt. Nature 314, 703707.
Jackson, K., Wilkinson, J., Maksym, T., Meldrum, D., Beckers, J., Haas, C. & Mackenzie, D. 2013 A novel and low-cost sea ice mass balance buoy. J. Atmos. Ocean. Technol. 30 (11), 26762688.10.1175/JTECH-D-13-00058.1
Jeevaraj, C. G. & Imberger, J. 1991 Experimental study of double-diffusive instability in sidewall heating. J. Fluid Mech. 222, 565586.
Lieb-Lappen, R. M., Golden, E. J. & Obbard, R. W. 2017 Metrics for interpreting the microstructure of sea ice using x-ray micro-computed tomography. Cold Reg. Sci. Technol. 138, 2435.
Loper, D. E. & Roberts, P. H. 1983 Compositional convection and the gravitationally powered dynamo. Stellar and Planetary Magnetism (ed. Soward, A. M.), pp. 297327. Gordon and Breach Science Publishers.
Neufeld, J. A. & Wettlaufer, J. S. 2008 An experimental study of shear-enhanced convection in a mushy layer. J. Fluid Mech. 612, 363385.
Notz, D., Wettlaufer, J. S. & Worster, M. G. 2005 A non-destructive method for measuring the salinity and solid fraction of growing sea ice in-situ. J. Glaciol. 51, 159166.
Notz, D. & Worster, M. G. 2008 In situ measurements of the evolution of young sea ice. J. Geophys. Res. Oceans 113 (C3), C03001.
Notz, D. & Worster, M. G. 2009 Desalination processes of sea ice revisited. J. Geophys. Res. Oceans 114 (C5), C05006.10.1029/2008JC004885
Peppin, S. S. L., Huppert, H. E. & Worster, M. G. 2008 Steady-state solidification of aqueous ammonium chloride. J. Fluid Mech. 599, 465476.
Pesci, A. I., Porter, M. A. & Goldstein, R. E. 2003 Inertially driven buckling and overturning of jets in a Hele-Shaw cell. Phys. Rev. E 68, 056305.
Pringle, D. J., Eicken, H., Trodahl, H. J. & Backstrom, L. G. E. 2007 Thermal conductivity of landfast Antarctic and Arctic sea ice. J. Geophys. Res. Oceans 112 (C4), C04017.10.1029/2006JC003641
Rees Jones, D. & Worster, M. 2013 Fluxes through steady chimneys in a mushy layer during binary alloy solidification. J. Fluid Mech. 714, 127151.
Richter-Menge, J. A., Perovich, D. K., Elder, B. C., Claffey, K., Rigor, I. & Ortmeyer, M. 2006 Ice mass-balance buoys: a tool for measuring and attributing changes in the thickness of the Arctic sea-ice cover. Ann. Glaciol. 44, 205210.
Rosenberger, F. E. 1979 Fundamentals of Crystal Growth I: Macroscopic Equilibrium and Transport Concepts. Springer.
Sampson, C., Golden, K. M., Gully, A. & Worby, A. P. 2011 Surface impedance tomography for Antarctic sea ice. Deep-Sea Res. II 58 (9), 11491157.10.1016/j.dsr2.2010.12.003
Shirtcliffe, T. G. L., Huppert, H. E. & Worster, M. G. 1991 Measurement of the solid fraction in the crystallization of a binary melt. J. Cryst. Growth 113 (3–4), 566574.
Tait, S. & Jaupart, C. 1989 Compositional convection in viscous melts. Nature 338 (6216), 571574.
Thorpe, S. A., Hutt, P. K. & Soulsby, R. 1969 The effect of horizontal gradients on thermohaline convection. J. Fluid Mech. 38, 375400.
Wells, A. J., Wettlaufer, J. S. & Orszag, S. A. 2010 Maximal potential energy transport: a variational principle for solidification problems. Phys. Rev. Lett. 105 (25), 254502.
Wettlaufer, J. S., Worster, M. G. & Huppert, H. E. 1997 Natural convection during solidification of an alloy from above with application to the evolution of sea ice. J. Fluid Mech. 344, 291316.
Worster, M. G. 1986 Solidification of an alloy from a cooled boundary. J. Fluid Mech. 167, 481501.
Worster, M. G. 1991 Natural convection in a mushy layer. J. Fluid Mech. 224, 335359.
Worster, M. G. 1992 On measurement and prediction of the solid fraction within mushy layers. J. Cryst. Growth 125, 487494.
Worster, M. G. 1997 Convection in mushy layers. Annu. Rev. Fluid Mech. 29, 91122.
Worster, M. G. 2000 Perspectives in Fluid Dynamics: A Collective Introduction to Current Research, pp. 393446. Cambridge University Press.
Yu, J., Bergman, M. I., Huguet, L. & Alboussiere, T. 2015 Partial melting of a Pb-Sn mushy layer due to heating from above, and implications for regional melting of Earth’s directionally solidified inner core. Geophys. Res. Lett. 42, 70467053.
Zhong, J.-Q., Fragoso, A. T., Wells, A. J. & Wettlaufer, J. S. 2012 Finite-sample-size effects on convection in mushy layers. J. Fluid Mech. 704 (2), 89108.10.1017/jfm.2012.219
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