Quasi-steady approximations to initial value problems with application to sediment transport

Abstract

We discuss the issues associated with quasi-steady approximations to the heat equation on both finite and infinite domains. The long time behavior of unsteady, diffusive problems is often the primary interest of many “first order”, analytical modeling studies especially where precise initial value information is unavailable. Approximations to initial value problems (IVPs) that are independent of the specified initial condition are here described as quasi-steady solutions. To obtain a quasi-steady approximation, temporal behavior is necessarily modeled in terms of time independent, i.e. spatial information. By requiring that the same the functional basis underlying separation of variables for finite domain problems (eigenvalue problems) we can delimit a simple function class for the temporal terms. Similarly, a basis consistent with the introduction of a self-similar variable for the infinite domain problems (boundary value problems) can be reproduced by the quasi-steady approximation be selecting a derivative based steady closure. Given this closure we then discuss the approximation relationships in terms of a set of canonical attributes of solution character. Discussions include behavior of several simple eigenvalue and boundary value problems, approximations using (along with the convergence properties) successive substitution and, finally, behavior and physical interpretation of Lagrangian formulations of the quasi-steady ODE. Finally application of the quasi-steady approximation to the development of alluvial fans, a geological problem where initial condition information is not available, has been shown to provide a useful, elementary result.