Flood propagation modeling with the Local Inertia Approximation: Theoretical and numerical analysis of its physical limitations

Attention of the researchers has increased towards a simplification of the complete Shallow water Equations called the Local Inertia Approximation (LInA), which is obtained by neglecting the advection term in the momentum conservation equation. This model, whose physical basis is discussed here, is commonly used for the simulation of slow flooding phenomena characterized by small velocities and absence of flow discontinuities. In the present paper it is demonstrated that a shock is always developed at moving wetting-drying frontiers, and this justifies the study of the Riemann problem on even and uneven beds. In particular, the general exact solution for the Riemann problem on horizontal frictionless bed is given, together with the exact solution of the non-breaking wave propagating on horizontal bed with friction, while some example solution is given for the Riemann problem on discontinuous bed. From this analysis, it follows that drying of the wet bed is forbidden in the LInA model, and that there are initial conditions for which the Riemann problem has no solution on smoothly varying bed. In addition, propagation of the flood on discontinuous sloping bed is impossible if the bed drops height has the same order of magnitude of the moving-frontier shock height. Finally, it is found that the conservation of the mechanical energy is violated. It is evident that all these findings pose a severe limit to the application of the model. The numerical analysis has confirmed the existence of the frontal shock in advancing flows, but has also demonstrated that LInA numerical models may produce numerical solutions, which are unreliable because of mere algorithmic nature, also in the case that the LInA mathematical solutions do not exist. Following the preceding results, two criteria for the definition of the applicability limits of the LInA model have been considered. These criteria, which are valid for the very restrictive case of continuously varying bed elevation, are based on the limitation of the wetting front velocity and the limitation of spurious total head variations, respectively. Based on these criteria, the applicability limits of the LInA model are discouragingly severe, even if the bed elevation varies continuously. More important, the non-existence of the LInA solution in the case of discontinuous topography and the non-existence of receding fronts radically question the viability of the LInA model in realistic cases. It is evident that classic SWE models should be preferred in the majority of the practical applications.