资源环境科技发展态势分析平台

Two common refrains about using the one-dimensional advection diffusion equation to estimate fluid fluxes and thermal conductivity from temperature time series in streambeds are that the solution assumes that (1) the surface boundary condition is a sine wave or nearly so, and (2) there is no gradient in mean temperature with depth. Although the mathematical posing of the problem in the original solution to the problem might lead one to believe these constraints exist, the perception that they are a source of error is a fallacy. Here we develop a mathematical proof demonstrating the equivalence of the solution as developed based on an arbitrary (Fourier integral) surface temperature forcing when evaluated at a single given frequency versus that derived considering a single frequency from the beginning. The implication is that any single frequency can be used in the frequency-domain solutions to estimate thermal diffusivity and 1-D fluid flux in streambeds, even if the forcing has multiple frequencies. This means that diurnal variations with asymmetric shapes or gradients in the mean temperature with depth are not actually assumptions, and deviations from them should not cause errors in estimates. Given this clarification, we further explore the potential for using information at multiple frequencies to augment the information derived from time series of temperature.

Plain Language Summary Measuring temperature over time at different depths in streambeds has become a common practice for estimating infiltration into streambeds or upwelling water rates. These values are important for fish eggs, stream-bottom insects, filtering of pollutants in streams, and managing water resources. It has often been assumed that real-life temperature fluctuations, which are a bit noisy and only look approximately like sine waves, caused errors in these calculations. It was also thought that a warm stream flowing above cold groundwater (another common situation) could cause errors. The mathematical derivations in this paper and accompanying laboratory work show that neither of these assumed sources of error actually cause errors. This is good news, as it crosses two suspects off the list when we try to trace errors, and we can turn our attention toward more serious issues when thinking about how to improve these calculations.