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%\title{A sediment transport model for suspended and bed load including erosion and deposition effects}
\title{On a two-layers / two-phases / Exner model for sediment transport with erosion and deposition effects}
\authors{
L. Bonaventura$^1$, E.D. Fern\'andez-Nieto$^2$ \& \underline{G. Narbona-Reina}$^2$
}

\affiliations{
$^1$MOX -- Politecnico di Milano, Milano, Italy\\
%{\tt luca.bonaventura@polimi.it}
$^2$Departamento Matem\'atica Aplicada I, Sevilla, Spain
}

\maketitle

\begin{summary}
Sediment transport, suspended load, bed load, erosion rate, deposition rate, two phase, two layer.\\
\end{summary}

In this work we deal with the geophysical problem of  sediment transport in river. Our objective is to propose a complete model that 
describes both suspended and bed load sediment transport, as well as the interactions between them. 
\\\\
The whole physical system includes particles moving as bed load, suspended load and wash load, which is  seen indeed as a subset of the suspended load. 
%The bed load is the sediment transport that occurs near the bottom, by saltation and rolling due to the interaction between the particles and the water flow. If the flow is strong enough, some of the smaller particles become suspended.
%The suspended load is the amount of sediment particles that are carried by the water flow. They can be deposited depending on the flow rate and the size of the particles. 
%Finally, the wash load refers to the very small particles that are always suspended and can be seen as a subset of the suspended load. It causes the turbidity and can be also related to the salty character of the oceans, for example.
In the literature on this topic,  several proposals to tackle the modelling of the sediment transport have been advanced. In general, the basis are the two-layer and two-phase models. The two-layer models are based on the fact that the different sediment transport types can be seen as separate layers with some kind of interactions among them. The two-phase models keep the property of one fluid formed up of two different components that interacts intrinsically, water and sediment, so the total column is seen as a mixture. Of course, both kind of models are not incompatible and the suitability of each one depends on the physical situations that one would like to focus on.
\\
Furthermore, in this type of geophysical problems, the derivation of depth-integrated models $-$under the assumption of shallow flows$-$ are extendedly accepted in order to get models mathematically and numerically manageable together with a good review in results that captures the essential effects. %But, in other way, as it is well known, with this kind of approximation one lose the vertical nature of these flows. Indeed, the introduction of the multilayer models is aimed at resolving this lack inherent to the simple one-layer shallow models. That's why the multilayer models are more complete and reward the modelling from the physical point of view.
\\\\
In this work, we present the derivation of a two-layer sediment transport model for both suspended-load and bed-load phenomena, that includes the interaction between the different sediment loads. For this goal we use some established developments in order to obtain a new model that is  as complete as possible. 
We consider the fluid-solid mixture model proposed by Jackson \cite{jackson} as a basis and we follow different derivations to obtain the dynamics for the two layers. 
The top layer  is defined as a suspension layer made up of water and grains, so we derive a two-phase model based on \cite{luca_sve, GaRoBo}. For the layer below, we follow the derivation developed in \cite{sve_nos} to obtain an Exner type model for arbitrary sloping beds. Erosion, suspension and deposition are considered in the whole system through appropriate boundary conditions. In order to well define the exchange of mass between sediment layers, mass and momentum conservation at the interface are imposed. The rate of erosion/deposition/suspension are defined following \cite{zech09}. Thus, this model can be seen as a generalization of the existing models. Finally, several numerical tests will be presented. 


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%%%%%%%%%%%
  
\begin{thebibliography}{1}

%\bibitem{bib:Galilei1633} 
%G. Galilei,
%\textit{E pur si muove !},
%J. Planet. Sci. \textbf{1}, 007 (1633).

\bibitem{sve_nos} E.D. Fern\'andez-Nieto, T. Morales de Luna, G. Narbona-Reina, J.D. Zabsonr\'e. 
\textit{Formal deduction of the Saint-Venant-Exner model including arbitrarily sloping sediment beds and associated energy}, submitted to M2AN, (2015). 

\bibitem{GaRoBo} G. Garegnani, G. Rosatti, L. Bonaventura. 
\textit{Free surface flows over mobile bed: mathematical analysis and numerical modeling of coupled and decoupled approaches}, Communications In Applied And Industrial Mathematics, \textbf{2(1)},  (2011).

\bibitem{luca_sve} G. Garegnani , G. Rosatti, L. Bonaventura. 
\textit{On the range of validity of the Exner-based models for mobile-bed river flow simulations}, 
Journal of Hydraulic Research, \textbf{51:4}, 380-391, (2013).

\bibitem{jackson} R. Jackson, 
\textit{The Dynamics of Fluidized Particles}, Cambridge University Press, Cambridge, (2000).

\bibitem{zech09} Y. Zech, S. Soares-Fraz\~ao, B. Spinewine, C. Savary, L. Gouti\`ere. 
\textit{Inertia effects in bed-load transport models}, 
Can.J.Civ.Eng. \textbf{36}, 1587--1597, (2009).

 \end{thebibliography}

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