1.5.4: Equilibrium concept

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Judith Bosboom & Marcel J.F. Stive
Delft University of Technology via TU Delft Open

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If a morphological system is not in equilibrium with the forcing (waves, tides, currents), morphological adjustments start to take place immediately; the morphological system reacts to disturbances. The rate of morphological adjustment has been observed to depend on the magnitude of the still existing disruption (the difference between the actual situation and the equilibrium situation).

Sometimes, morphological changes are induced very abrupt, such as the closure of parts of a tidal basin. Other changes take place more slowly, such as the response of the shape of a cross-shore profile to sea-level rise. Usually, the morphological response to (sudden) changes shows a certain variation with time. The process of morphological response will be fast at first and decelerates when the new equilibrium situation is approached. Often such a morphological adjustment process can be approximated exponentially. In terms of for instance sediment volume content of a certain morphological unit we would then have:

\[V(t) = V_{old} + (V_{new} - V_{old})(1 - e^{-t/\tau})\label{eq1.5.4.1}\]

where

\(V(t)\)	characteristic volume in morphological unit at time \(t\)	\(m^3\)
\(t\)	time after the distortion	\(yr\)
\(V_{old}\)	the (equilibrium) volume before distortion	\(m^3\)
\(V_{new}\)	the new equilibrium volume	\(m^3\)
\(r\)	morphological timescale	\(yr\)

E.g. Stive and De Vriend (1995) and Eysink (1991) use this type of approximation to describe adaptation processes of units with large scales. Within a time \(t\) equal to the morphological timescale \(\tau\) a good 63 % (namely: \(1 - 1/e\)) of the changes required to reach new equilibrium (namely: \(V_{new} - V_{old}\)) have taken place. The morphological timescale \(\tau\) is the time that would be required to reach equilibrium, if the rate of morphological adjustment \(dV/dt\) would remain equal to the rate at \(t = 0\):

\[r = \dfrac{(V_{new} - V_{old})}{(dV/dt)_{t = 0}}\label{eq1.5.4.2}\]

Equation \(\ref{eq1.5.4.2}\) can be found by differentiation of Eq. \(\ref{eq1.5.4.1}\) and evaluating the resulting expression at \(t = 0\). Sediment transports are supposed to drive the morphological unit to its new equilibrium and are therefore implicitly known. Such an implicit approach to determine sediment transport rates based on the deviation from a predefined equilibrium is very different from a more process-based approach to describe sediment transport. The latter approach will be taken in Ch. 6.