averaging theorems, and conservation equations

6 7 Abstract: The paper reports the double-averaged (in space and in time) hydrodynamic equations for 8 mobile-boundary conditions which are derived based on the refined double-averaging theorems, 9 modified Reynolds decomposition, and improved definitions of the spatial and time bed porosities. 10 The obtained double-averaged conservation equations provide a mathematical framework for 11 studying mobile-boundary flows such as gravel-bed rivers during flood events or flows over 12 vegetated beds. These equations will help in designing measurement campaigns for obtaining 13 mobile-bed data and their interpretation and parameterisation, eventually leading to improved and 14