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Was That Assumption Necessary? Reconsidering Boundary Conditions for Analytical Solutions to Estimate Streambed Fluxes
Author(s) -
Luce Charles H.,
Tonina Daniele,
Applebee Ralph,
DeWeese Timothy
Publication year - 2017
Publication title -
water resources research
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.863
H-Index - 217
eISSN - 1944-7973
pISSN - 0043-1397
DOI - 10.1002/2017wr020618
Subject(s) - thermal diffusivity , forcing (mathematics) , mathematics , series (stratigraphy) , mathematical analysis , boundary value problem , boundary (topology) , equivalence (formal languages) , thermal conduction , frequency domain , perturbation (astronomy) , temperature gradient , advection , curse of dimensionality , statistics , meteorology , physics , geology , thermodynamics , paleontology , discrete mathematics , quantum mechanics
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.