On the form and stability of seafloor stratigraphy and shelf profiles: A mathematical model and solution method

There are a variety of different marine sediment transport processes that affect the evolution of continental shelf profiles. Physically based numerical models make it possible to simulate the evolution of the continental shelf in response to these processes, which may be combined with different magnitudes, frequencies and durations. The basic model that is employed in most numerical simulations is based on conservation of mass and is a moving boundary problem since the river mouth progrades into the receiving basin. The general form of these stratigraphic or seafloor evolution models is a first-order, Exner-type differential equation that is forced by a sediment deposition/erosion function. This latter function depends on which sediment transport processes are modeled and how they are modeled, so it is desirable to have a solution method that works for any such function. This paper presents a general solution method to this type of evolution equation that works for any initial bathymetry and sediment deposition function given as 1D functions of seaward distance. It is based on the Laplace transform. Numerical results often exhibit ‘‘equilibrium profile’’ solutions for a wide variety of different model scenarios. These profiles may be viewed as travelling waves, such that the rate of progradation gives the speed of this wave as it advances into the receiving basin. When realistic initial and boundary conditions are used, profiles are often seen to evolve toward forms that are independent of the initial bathymetry. Although episodic perturbations such as turbidities disrupt these forms, they still serve as attractors for the system dynamics. An application of the solution method presented in this paper also allows the shape of equilibrium profiles and the progradation rate to be determined from a specification of the initial bathymetry and the sediment deposition function. Several illustrative examples are given in closed form. Results are also presented that make it possible to compute the amount of time required for an equilibrium profile to be re-established after a perturbation. r 2008 Elsevier Ltd. All rights reserved.