Temperature Profile in Lake Bottom Sediments: An Analytical Self-Similar Solution

The vertical temperature structure of bottom sediments in lakes is discussed. Observations indicate that the temperature profile in the bottom sediments has the form of a thermal wave. The wave starts at the water-sediment interface and propagates downward as the lake water and the bottom sediments are heated during spring and summer. When heating ceases and cooling sets in, a new wave starts at the water-sediment interface and propagates downward as the lake water and the sediments are cooled over autumn and winter. Although the temperature in the thermally active layer of bottom sediments varies considerably with time, a characteristic shape of the temperature-depth curve remains approximately the same. Based on this observational evidence, a two-layer parametric representation of the temperature profile in lake bottom sediments was proposed. In this note, a theoretical explanation for the observed self-similarity of the temperature profile is offered. Assuming a travelling wave-type behaviour of the temperature profile in bottom sediments, an analytical solution to the heat transfer equation is found. This solution is compared with data from measurements in a number of lakes and in a laboratory tank.