A Time-Dependent Numerical Model for Spherically Symmetric Hailstone Growth Thermodynamics under Constant Ambient Conditions

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  • 1 Division of Meteorology, Department of Geography, University of Alberta, Edmonton, Canada
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Abstract

A model of spherical hailstone growth thermodynamics is presented, and used to examine the validity of the continuous growth and heat balance assumptions frequently employed in the “classical” hail growth models. The model is similar to the spherically symmetric model formulated by Macklin and Payne (1969), but solutions to the model equations are obtained by means of finite-difference numerical methods. In the model, we do not try to simulate the discrete accretion process of individual drops. Instead, we attempt to identify the implications of the discrete, time-dependent nature of the icing process, by examining the accretion of a thin uniform layer of supercooled water over the entire surface of the sphere. The heat transfer equations both with the air and within the hailstone axe then solved assuming radial symmetry. By the addition of several such layers, the finite growth of a spherical hailstone can be simulated. In the present paper, only growth in constant ambient conditions is considered. It is shown that there are large internal heat fluxes during the interval between the accretion of successive layers (typically ≲1 s), which cause the temperatures near the surface to oscillate several degrees above and below their time-mean value. Nevertheless, the time-averaged temperature over an accretion cycle is almost uniform throughout the hailstone and, when the environmental conditions are constant, is approximately equal to the equilibrium surface temperature predicted by the “classical” models. As the hailstone grows under constant environmental conditions, it continually adapts to the classical equilibrium temperature, warming up almost uniformly throughout. The time scale for this adjustment to a quasi-equilibrium state is found to be of the order of the internal diffusive time scale R2/k. It is speculated therefore that if the environmental conditions change slowly (over time scales large compared with R2/k) the hailstone thermodynamics will be adequately described by the classics equilibrium theories. However, if conditions change rapidly, internal heat conduction and time-dependent (non-equilibrium) effects may have to be taken into account.

Abstract

A model of spherical hailstone growth thermodynamics is presented, and used to examine the validity of the continuous growth and heat balance assumptions frequently employed in the “classical” hail growth models. The model is similar to the spherically symmetric model formulated by Macklin and Payne (1969), but solutions to the model equations are obtained by means of finite-difference numerical methods. In the model, we do not try to simulate the discrete accretion process of individual drops. Instead, we attempt to identify the implications of the discrete, time-dependent nature of the icing process, by examining the accretion of a thin uniform layer of supercooled water over the entire surface of the sphere. The heat transfer equations both with the air and within the hailstone axe then solved assuming radial symmetry. By the addition of several such layers, the finite growth of a spherical hailstone can be simulated. In the present paper, only growth in constant ambient conditions is considered. It is shown that there are large internal heat fluxes during the interval between the accretion of successive layers (typically ≲1 s), which cause the temperatures near the surface to oscillate several degrees above and below their time-mean value. Nevertheless, the time-averaged temperature over an accretion cycle is almost uniform throughout the hailstone and, when the environmental conditions are constant, is approximately equal to the equilibrium surface temperature predicted by the “classical” models. As the hailstone grows under constant environmental conditions, it continually adapts to the classical equilibrium temperature, warming up almost uniformly throughout. The time scale for this adjustment to a quasi-equilibrium state is found to be of the order of the internal diffusive time scale R2/k. It is speculated therefore that if the environmental conditions change slowly (over time scales large compared with R2/k) the hailstone thermodynamics will be adequately described by the classics equilibrium theories. However, if conditions change rapidly, internal heat conduction and time-dependent (non-equilibrium) effects may have to be taken into account.

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