Was That Assumption Necessary? Reconsidering Boundary Conditions for Analytical Solutions to Estimate Streambed Fluxes

Abstract

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.

Monitoring Streambed Scour/Deposition Under Nonideal Temperature Signal and Flood Conditions

“…For completeness, we also report the equation to quantify the Darcian seepage flux, q : where γ = ( ρ m c m )/( ρ w c w ), and ρ refers to density and c to specific heat capacity, while the subscripts w and m refer to the water and the sediment pore water matrix, respectively. In this work, we use γ = 0.957 as reported by Luce et al () for our set of laboratory experiments.…”

“…The seepage velocity estimate from the process will reflect the average velocity over the sampling window for temperature, so errors would be most pronounced during rapid changes. The temperature signal can be nonstationary, which is generally the case in the field, and this does not affect the analytical solution (Luce et al, 2017).…”

“…where c 5 (q m c m )/(q w c w ), and q refers to density and c to specific heat capacity, while the subscripts w and m refer to the water and the sediment pore water matrix, respectively. In this work, we use c 5 0.957 as reported byLuce et al (2017) for our set of laboratory experiments.…”

Quantitative guidance for efficient vertical flow measurements at the sediment–water interface using temperature–depth profiles

“…The attributes of the diurnal temperature signal-based analytical solutions have been investigated broadly, including the influence of heterogeneity (Birkel et al, 2016;Irvine, Cranswick, Simmons, Shanafield, & Lautz, 2015), nonsinusoidal temperature signals (Luce, Tonina, Applebee, & DeWeese, 2017), nonconstant fluid fluxes Rau, Cuthbert, McCallum, Halloran, & Andersen, 2015), multidimensional flow (Cuthbert & Mackay, 2013;Lautz, 2010;Reeves & Hatch, 2016), and the uncertainty in flux estimates that results from uncertainties in thermal properties Shanafield, Hatch, & Pohll, 2011).…”

Limits on Groundwater‐Surface Water Fluxes Derived from Temperature Time Series: Defining Resolution‐Based Thresholds

“…These methods are not without their limitations, and numerous studies have been performed to analyze the conditions where the use of periodic heat signals to quantify Darcy fluxes break down as well as to help provide appropriate guidance when transitioning to a field setting. These studies include the investigation of nonideal field conditions that violate the assumptions established by Stallman () such as nonuniform flow fields (Cuthbert & Mackay, ; Reeves & Hatch, ), nonsinusoidal temperature signals (Lautz, ; Luce et al, ), transient fluid fluxes (Rau et al, ), heterogeneity (Birkel et al, ; Irvine, Cranswick, et al, ), and thermal property uncertainty (Irvine, Lautz, et al, ; Shanafield et al, ). Results from studies that are based on the violation of fundamental assumptions, while useful for exploring what errors are introduced when these assumptions are violated, do not provide a field‐based metric for determining if a Darcy flux estimate is accurate or not, nor do they provide a means for determining if a field‐derived Darcy flux is the result of an assumption violation.…”