Solutions for the diurnally forced advection‐diffusion equation to estimate bulk fluid velocity and diffusivity in streambeds from temperature time series

[1] Work over the last decade has documented methods for estimating fluxes between streams and streambeds from time series of temperature at two depths in the streambed. We present substantial extension to the existing theory and practice of using temperature time series to estimate streambed water fluxes and thermal properties, including (1) a new explicit analytical solution to predict one-dimensional fluid velocity from amplitude and phase information; (2) an inverse function, also with explicit formulation; (3) methods to estimate fluid velocity from temperature measurements with unknown depths; (4) methods to estimate thermal diffusivity from the temperature time series when measurement depths are known; (5) methods to track streambed elevation between two sensors, given knowledge of the thermal diffusivity from (4) above; (6) methods to directly calculate the potential error in velocity estimates based on the measurement error characteristics ; and (7) methods for validation of parameter estimates. We also provide discussion and theoretical insights developed from the solutions to better understand the physics and scaling of the propagation of the diurnal temperature variation through the streambed. In particular, we note that the equations developed do not replace existing equations applied to the analysis, rather they are new equations representing new aspects of the process, and, as a consequence, they increase the amount of information that can be derived from a particular set of thermal measurements.