Multiple outflows, spatial components, and nonlinearities in age theory

Water age has become an important variable for the characterization of hydrologic systems. The goal of this paper is to analyze the role of multiple outflows, spatial components, and nonlinearities in age theory. We first extend the theory to linear systems with multiple outflows, including the relationship between age distribution at death and survival time distribution at birth. We further show that for each outflow there is a survival time distribution at birth, which normalized corresponds to the impulse-response function for the specific outflow. We also analyze how the impulse-response function affects both the amplitude gain and time delay of the outflow and the long-term average partitioning. With regard to linear spatially extended systems, we link the impulse-response function to the Green's function. This allows us to easily compute the loss function and the age distribution for the system. Finally, we focus on nonlinear systems to analyze the effects of storage-dependent and age distribution-dependent loss functions. By considering the Burgers' equation, we show how the relationships between spatial dynamics and the age distribution are complicated by nonlinearities.