Avalanche forecasters measure the temperature gradient dT/dh, which is the rate

Question

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 use the following rule of thumb: If dT/dH exceeds 10 degrees 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 degees C . At a depth of 1.1 m the temperature is 2 degees C. Using the mean value Theorem, what can he conclude about the temperature gradient? Is the information of a weak layer likely? b. One mile away, a skier finds that the temperature at a depth of 1.4m is -1 degrees C, and at the surface it is -12 degees C. What can be concluded about the temperature gradient? Is the information of a weak layer in her location likely?

Explanation / Answer

a) dT/dh = (2+12)/1.1 = 12.72. So temperature gradient is greater than 10 degree C/m. Hence weak layer is likely in that location. b) dT/dh = (-1+12)/1.4 = 7.857. So temperature gradient is less than 10 degree C/m. Hence weak layer is not likely in her location.

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