The following is inspired on well-known active layer approaches in river bed sediment transport. All quantities with subscript $f$ are facies dependent. Sediment is measured in meters of deposited material. $P_f$ is the production of sediment per facies in $m/s$. Further unit calculations would be more readable if we consider the unit of sediment as separate, so for instance it doesn't cancel against $m^2$ in the units of sediment flux. TBD
We suppose that loose sediment, either fresh production or disintegrated older sediment, is being transported in a layer on top of the sea bed. The flux in this layer is assumed to be directly proportional to the local slope of the sea bed $| \nablax \eta |$, where $\eta_ = \sum_f \eta_f$, the sum over all facies contributions.
The active layer now contains a concentration $C_f$ particles of different grain size (for each facies $f$). If needed, $C_f = \alpha_f P_f$ where $\alpha_f$ is some facies parameter determining the fraction of production that is available for transport. The sediment flux is given as,
In our modelling we keep track of individual contributions per facies over time. Note that in other approaches to active layer transport there would be a factor $1/C_f$. Here we have a different interpretation to what the concentration means: the sediment settles down after transport, such that the concentration has no impact on the change in sediment surface elevation.
Combining these equations, and ignoring subsidence for the moment (which is a global effect and can't be expressed on a per-facies basis), we get a component-wise diffusion equation
In our model we need to solve this equation one time-step each iteration. If we solve this using forward methods, we should be reminded of the CFL limit for diffusion equations (depending on the diffusion constants and grid size we shouldn't pick the time steps too large). Alternatively, for these two-dimensional situations, an implicit approach is feasible. Also we should take care that somehow $\nabla(\nu\alpha P \nabla \eta) + P > 0$. The interpretation being that we can't transport more than we produce, even if there is capacity to do so.
Active Layer Transport
The following is inspired on well-known active layer approaches in river bed sediment transport. All quantities with subscript $f$ are facies dependent. Sediment is measured in meters of deposited material. $P_f$ is the production of sediment per facies in $m/s$. Further unit calculations would be more readable if we consider the unit of sediment as separate, so for instance it doesn't cancel against $m^2$ in the units of sediment flux. TBD
In a model without transport, we could write
$$\sigma + \sum_f {{\partial \eta_f} \over {\partial t}} = \sum_f P_f,$$
where $\sigma$ is the subsidence rate in $m/s$.
We suppose that loose sediment, either fresh production or disintegrated older sediment, is being transported in a layer on top of the sea bed. The flux in this layer is assumed to be directly proportional to the local slope of the sea bed $| \nablax \eta |$, where $\eta_ = \sum_f \eta_f$, the sum over all facies contributions.
The active layer now contains a concentration $C_f$ particles of different grain size (for each facies $f$). If needed, $C_f = \alpha_f P_f$ where $\alpha_f$ is some facies parameter determining the fraction of production that is available for transport. The sediment flux is given as,
$${\bf q_f} = -\nu_f C_f {\bf \nablax} \eta*.$$
The following is the mass balance:
$$\sigma + \sum_f {{\partial \eta_f} \over {\partial t}} = -\sum_f {\bf \nabla_x} \cdot {\bf q_f} + \sum_f P_f,$$
In our modelling we keep track of individual contributions per facies over time. Note that in other approaches to active layer transport there would be a factor $1/C_f$. Here we have a different interpretation to what the concentration means: the sediment settles down after transport, such that the concentration has no impact on the change in sediment surface elevation.
Combining these equations, and ignoring subsidence for the moment (which is a global effect and can't be expressed on a per-facies basis), we get a component-wise diffusion equation
$${{\partial \eta_f(x)}\over{\partial t}} = {\bf \nabla_x} \cdot \bigg[ \nu_f \alpha_f\ P_f(x)\ {\bf \nablax} \eta{*}(x) \bigg] + P_f(x),$$
In our model we need to solve this equation one time-step each iteration. If we solve this using forward methods, we should be reminded of the CFL limit for diffusion equations (depending on the diffusion constants and grid size we shouldn't pick the time steps too large). Alternatively, for these two-dimensional situations, an implicit approach is feasible. Also we should take care that somehow $\nabla(\nu\alpha P \nabla \eta) + P > 0$. The interpretation being that we can't transport more than we produce, even if there is capacity to do so.