Abstract for the 18th Physics of Estuaries and Coastal Seas Conference, 2016
Equilibrium Morphology and Tidal conditions in Estuaries.
Sierd de Vries1*, Marco Gatto1, Lodewijk de Vet1, Bram van Prooijen1,
Zheng Bing Wang 1,2
Keywords: equilibrium Morphology, estuaries, tidal propagation.
Abstract
This study presents a new theory that relates the propagation of the tidal wave to the equilibrium morphology of
tidal estuaries. The concept of equilibrium morphology has been long discussed within the scientific community.
Although some practical applications relying on this assumption have been developed, others criticize whether such an
equilibrium actually exists. Discussions on spatio-temporal scales of morphological equilibrium are relevant and
complicate matters, especially since collected data is often insufficient for a thorough analysis. However, theories
relating shape of the tidal wave to basin morphology have been developed based on the limited available data. Notable
examples are the studies by Dronkers (1986) and Friedrichs and Aubrey (1988).
Based on an analytical description of the tidal wave propagating into a basin, Dronkers et al. (1986) hypothesised
that the difference in flood period and ebb period is proportional to the difference in average water depth during the two
stages. Assuming morphological equilibrium is achieved when the ebb period is equal to the flood period, a
corresponding basin geometry can be derived. In order to support the theory, Dronkers used the Dutch Wadden Sea and
Western Scheldt as test cases. Although opportunities to predict morphological equilibrium are provided, the theory
needs a somewhat crude assumption on the representative channel cross-section.
Conversely, Friedrichs and Aubrey (1988) adopted a numerical approach. That is based on numerical simulations of
tidal wave propagation for different morphologies. They found ebb- and flood-dominance could be predicted based on
the basin geometry. From their results it can be argued that, when neither ebb nor flood dominance occurs, morphology
is in equilibrium. This theory also implies a crude assumption with respect to the shape of the channel’s cross-section.
In order to overcome the restrictions regarding assumptions on the channel shape, we hypothesize that morphology
is in equilibrium if the propagation speed (c) of the tide is independent of the water depth (H). Propagation speed, water
depth, channel width (b), gravitational acceleration (g) and cross sectional area (A) are related according to:
𝑐 = !𝑔𝐻 = !𝑔
𝐴
𝑏
If the (frictionless) celerity is constant, we propose:
𝑏 = 𝛼𝐴
and since:
then:
𝜕𝐴
=𝑏
𝜕𝑧
𝜕𝐴
= 𝛼𝐴
𝜕𝑧
The analytical solution to this equation is:
𝐴 = 𝐴! 𝑒
with
𝛼=
!
!! !
!
𝑏!
1
=
𝐴! 𝐻!
where z = bed level, A0= cumulative area at z=0 (~MSL), H0 = mean depth. The solution is in the form of a hypsometric
curve, in which the surface area A of the domain under a certain level (z) develops exponentially with the bed level
itself.
The theory is tested by using historical, measured bathymetries from the Wadden Sea and the Western Scheldt
estuary (The Netherlands). Especially the Western Scheldt can be considered close to morphological equilibrium as
compared to the Wadden Sea. The top panel of Figure 1 displays the measured bathymetry of the Western Scheldt and
the bottom panel the relationship between bed level and volume of water below. Results highlight that the hypsometry
of the Western Scheldt fits our analytical solution for morphological equilibrium. The adaptation to different sections of
1
Department of Hydraulic Engineering, Delft university of Technology
Marine and Coastal Ecosystems, Deltares
*Corresponding author - [email protected]
2
the Wadden Sea and Western Scheldt will be presented, giving us the chance to highlight morphodynamic differences
between estuaries and tidal basins.
Figure 1. Top: Western Scheld morphology measured in 2013. Bottom panel shows the hypsometric curve and the derived analytical
solution.
References
Friedrichs, C. T., Aubrey, D. G., 1988. Non-linear tidal distortion in shallow well-mixed estuaries: a synthesis. Estuarine, Coastal
and Shelf Science, 27, 521-545.
Dronkers, J., 1986. Tidal asymmetry and estuarine morphology. Netherlands Journal of Sea Research. 20, 117 – 131.