Calculation of Black Ice Thickness and Heat Fluxes inside the Ice and at the Water–Ice Boundary in a Boreal Lake

Abstract

This paper presents the results of the calculation of black ice thickness, as well as conductive heat fluxes inside the ice and at the water–ice boundary during the winter in the shallow boreal Lake Vendyurskoe (Russia). The calculation was carried out on the basis of experimental data obtained from a thermistor chain with nine sensors, five of which were successively frozen into the black ice during the winter of 1995–1996. Data processing was carried out by two methods, whose novelty lay in the simultaneous use of the temperature series of two sensors frozen into the ice and those that were in the water column directly under the lower ice boundary. The resulting estimates of black ice growth rates varied widely: maximum values (up to 8.5 mm/day) were observed in December during first month of ice period, with an average growth rate of 3.4 mm/day from December to the end of February. The heat flux in the black ice sheet varied significantly over synoptic time intervals; the highest values (up to 40 W/m2) were observed during the first two weeks of measurements, then a downward trend was noted, to values of ~10 W/m2. Black ice was isothermal from the end of February to the end of April due to the release of water on the ice surface after heavy snowfall. During this period the heat flux inside the black ice was zero, and there was no increase in black ice thickness. The calculation of the water–ice heat flux gives results that are very sensitive to both measurement limitations and the variability of external parameters. However, the estimates of this flux for moments in time when the sensors were frozen in the ice are values 1–2 W/m2, which are quite close to the previous estimates for Lake Vendyurskoe. The limitations of the presented method are related to the thermal inertia of black ice and make it possible to calculate of ice thickness with a time delay of several days. To quantify the effects of thermal inertia of ice, a model problem of heat propagation in the ice sheet is considered for the case of periodic temperature changes at its upper boundary. The attenuation of the amplitude and the delay of a heat wave during its propagation in the ice are estimated, and accordingly, the conditions, under which the temperature profile in the ice sheet is close to linear, are analyzed.

Keywords

• shallow boreal lake
• black ice
• ice growth rate
• water-to-ice heat fluxes
• temperature gradients
• Stefan’s boundary condition