# 6.8.1: Memory effects

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Suspended load transport is a quite common condition in the coastal environment, maybe even more common than pure bed load. From a morphological modelling point of view, we must distinguish two types of suspended load: one which is determined entirely by the hydrodynamic conditions and the sediment properties at the point of consideration, and one which includes a ‘memory effect’ and responds to the condi- tions in all points it has come through in the past. The former type can be modelled with a sediment transport formula (e.g. the Bijker or the Bailard formula), or with an intra-wave model which describes the suspension process during a wave cycle (see Sect. 6.6).

The sediment concentration associated with the second type of suspended load trans- port is described by an advection/diffusion equation of the type (e.g. Katopodi and Ribberink, 1992 for tidal currents, and Wang and Ribberink, 1986 for nearshore applications):

\[T_A \dfrac{\partial c}{\partial t} + L_A \left [\dfrac{u}{u_{\text{tot}}} \dfrac{\partial c}{\partial x} + \dfrac{v}{u_{\text{tot}}} \dfrac{\partial c}{\partial y} \right ] = c_e - c \label{eq6.8.1.1}\]

in which \(T_A\) is a timescale of the order of magnitude \(h/w_s\) and \(L_A\) is a length scale of the order of magnitude . The adaptation time- and length scales increase for finer sediment (smaller settling velocity). The equilibrium concentration, \(c_e\), corresponds with the spatially uniform situation and is usually derived from a sediment transport formula, or from a simpler model which applies to uniform situations.

Note that Eq. \(\ref{eq6.8.1.1}\) can also be considered as a decay equation of the type:

\[\dfrac{Dc}{Dt} + \dfrac{c}{T_A} = \dfrac{c_e}{T_A}\]

in which the \(D/Dt\) stands for the material derivative, i.e. moving along with the sediment flow. If the concentration at \(t = 0\) is given, the general solution can be written as:

\[c(t) = c(0) e^{-t/T_A} + \int_{0}^{t} c_e (\tau) e^{-(t - \tau)/T_A} d\tau\]

in which \(\tau\) is a formal time variable which runs from 0 to the actual time \(t\). Apparently, all values of the equilibrium condition encountered in the time interval between 0 and \(t\) contribute to the forcing term (i.e. the second term in the right-hand part of the equation), but their contributions become smaller as they occurred longer ago. This shows that the memory of the system is not infinite, but has a certain timescale \(T_A\).

The memory effect of the suspended load concentration can be rather important in a tidal inlet, with its strong spatial variations of the bed topography and the hydro-dynamic conditions (see Sect. 9.7.3).