6.8.1: Memory effects
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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).