Avalanche forecasting ?Avalanche forecasters measure the ?temperature gradient dT , dh which is the rate at which the temperature in a snowpack ?T? changes with respect to its depth ?h.? If the temperature gradient is large, it may lead to a weak layer of snow in the snowpack. When these weak layers collapse, avalanches occur. Avalanche forecasters dT use the following rule of thumb: If dh exceeds 10°C/m anywhere in the snowpack, conditions are favorable for weak layer formation and the risk of avalanche increases. Assume the temperature function is continuous and differentiable. a. An avalanche forecaster digs a snow pit and takes two temperature measurements. At the surface (?h? = 0) the temperature is ?12°C. At a depth of 1.1 m, the temperature is 2°C.Using the Mean Value Theorem, what can he conclude about the temperature gradient? Is the formation of a weak layer likely? b. One mile away, a skier finds that the temperature at a depth of 1.4 m is ?1 °C. and at the surface it is ? 12°C. What can be concluded about the temperature gradient? Is the formation of a weak layer in her location likely? c. Because shaw is an excellent insulator, the temperature of snow-covered ground is near 0°C Furthermore, the surface temperature of snow in a particular area does not vary much from one location to the next. Explain why a weak layer is more likely to form in places where the snowpack is not too deep. d. The term ?isothermal? is used to describe the situation where all layers of the snowpack are at the same temperature (typically near the freezing point). Is a weak layer likely to form in isothermal snow? Explain.
Solution 28E Step 1 In this problem we are given that the temperature in a sno wpack T in C changes with o respect toits depth h in meter. Thus the temperature gradient is dh in C/mand also given that the temperature function is continuous and differentiable. And if the gradient exceeds 10 C/mthen the condition is favorite for weak pack formation in snowpack and and the risk of avalanche increases. …. (1) dT We have to find the required dh for various given conditions. Step 2 a. An avalanche forecaster digs a snow pit and takes two temperature measurements. At the surface (h = 0) the temperature is 12°C. At a depth of 1.1 m, the temperature is 2°C.Using the Mean Value Theorem, what can he conclude about the temperature gradient Is the formation of a weak layer likely Given: At the surface (h = 0) the temperature is 12°C and at a depth of 1.1 m, the temperature is 2°C. Initially, at the surface h1= 0 mthe temperature is T = 12 C o o And at the depth of h = 2.1 mthe temperature is T = 2 C Let us use the mean value theorem to conclude about the temperature gradient. Mean Value theorem: If f is defined and continuous on the closed interval [a,b]and differentiable on the open interval (a,b)then there is at least one point cin (a,b)that is f(b) f(a) a < c < bsuch that f(c) = ba Now, according to the mean value theorem, the temperature gradient will be as: dT = T(h2) T(1 ) dh h2h1 Now substituting all the given values, we get o o dT 2 C (12 C) dh = 1.1 m 0 m 2 C +12 C = 1.1 m 14 C = 1.1 m o o = 12.7 C/m > 10 C/m o Here the temperature gradient dT = 12.7 C/m dh o Since this is greater than 10 C/m, by (1) we say that, It is favourable to form weak pack formation in snowpack.