Extending Active Network Length Versus Catchment Discharge Relations to Temporarily Dry Outlets

River networks are not steady blue lines drawn in a map, since they continuously change their shape and extent in response to climatic drivers. Therefore, the flowing length of rivers (L) and the corresponding catchment-scale streamflow (Qsur) co-evolve dynamically. This paper analyzes the relationship between the wet channel length and the streamflow of a river basin, formulating a general analytical model that includes the case of temporarily dry outlets. In particular, the framework relaxes the common assumption that when the discharge at the outlet tends to zero the upstream flowing length approaches zero. Different analytical expressions for the L(Qsur) law are derived for the cases of (a) a perennial outlet; (b) a non-perennial outlet that dries out only when the whole network is dry; and (c) a temporarily dry outlet, that experiences surface flow for less time than other network nodes. In all cases, the shape of the L(Qsur) relationship is controlled by the distribution of the specific subsurface discharge capacity along the network. For temporarily dry outlets, however, the relation between L and Qsur might depend on an unknown shifting factor. Three real-world examples are presented to demonstrate the flexibility and the robustness of the theory. Our results indicate that the whole shape of the L(Qsur) relation might not be empirically observable if a significant fraction of the network is perennial or some reaches in the network experience surface flow for longer than the discharge gauging station. The study provides a basis for integrating empirical L(Qsur) data gathered in diverse sites.