Introduction
Estuaries develop as a result of dynamic interactions between hydrodynamic
conditions, sediment supply, underlying geology and ecological environment
. One model for the resulting morphology is that
of the “ideal estuary” that is hypothesised to have along-channel uniform
tidal range, constant depth and current velocity, and a channel width that
exponentially converges in the landward direction such that the loss of tidal
energy by friction is balanced by the gain in tidal energy by convergence
. One would expect
that in this case the along-channel variation in hypsometry is also
negligible. However, natural estuaries deviate from ideal ones as a result
of a varying degree of sediment supply, lack of time for adaptation and
sea-level rise , and locations wider than ideal
are filled with tidal bars
(Fig. ). Differences in bed-level profiles between
ideal and non-ideal estuaries are further enhanced by damming, dredging,
dumping, land reclamation and other human interference
e.g.. All
these natural deviations from the ideal estuary mean that there is no
straightforward relation between the planform geometry of the estuary and the
hypsometry or distribution of depths.
Bathymetry from (a) the Western
Scheldt (NL), (b) the Dovey (Wales),
(c) the Eems–Dollard (NL) and
(d) the Columbia River Estuary (USA). Source: (a, c)
Rijkswaterstaat (NL), (b) Natural Resources Wales,
(d) Lower Columbia Estuary Partnership.
Hypsometry captures key elements of geomorphological features
(Fig. ). The hypsometric method was
developed by and to relate the planform
area of a basin to elevation. Later the resulting functions were used to
predict the influence of basin morphology on the asymmetry of the horizontal
and vertical tides, to predict flood or ebb dominance and maturity of an
estuary , and to
characterise the trend of salt marsh development .
Furthermore, hypsometry was used as a data reduction method to characterise
entire reaches spanning bars and channels in estuaries
and shapes of individual tidal bar tops . Hypsometry has also
been used to describe the dimensions of channels and tidal flats in an idealised
model . In the latter, a parabolic shape was prescribed for the low
water channel, a linear profile for the intertidal low zone and a convex
profile for the intertidal high zone. While these profiles are valid for
perfectly converging channels, it is unknown to what extent they are
applicable to estuaries with irregular planforms and whether the currently
assumed profiles are valid to assess flood or ebb dominance. Hypsometric
profiles and derived inundation duration are also relevant indicators for
habitat composition and future transitions from mudflat to salt marsh
. To predict and characterise the morphology and assess
habitat area, we need along-channel and cross-channel bed-level predictions
for systems without measured bathymetry .
For only a few natural estuaries is bathymetry available, which leaves many
alluvial estuaries with irregular planforms from all around the world
underinvestigated. However, many estuaries are visible in detail on satellite
imagery, which raises the question of whether there is a relation between
planform geometry and depth distribution. Such a relation is known to exist
in rivers in the form of hydraulic geometry depending on bar pattern and
meander pool depths depending on planform channel curvature
e.g.. Therefore, it seems likely that
such a relation between the horizontal and vertical dimensions exists for
sandy estuaries as well, but this is not reported in the literature.
Morphological models can simulate 3-D bed levels with considerable accuracy
, but these models are
computationally intensive and need calibration and specification of initial
and boundary conditions. To study unmapped systems for which only aerial
photography is available, it would be useful to be able to estimate bed-level
distributions from planform geometry. Here we investigate this relation.
Previously, we showed in how locations and widths of tidal
bars can be predicted from the excess width, which is the local width of the
estuary minus the ideal estuary width. The summed width of bars in each
cross section was found to approximate the excess width. This theory
describes bars as discrete recognisable elements truncated at low water level
on what is essentially a continuous field of bed elevation that changes in
the along-channel direction . However, to predict morphology in
more detail, predictions of along-channel and cross-channel bed elevations
are required. While hypsometry can summarise bed elevation distribution as a
cumulative profile, it is unknown whether the shape of the profile is
predictable. Our hypothesis is that the along-channel variation in hypsometry
depends on the degree to which an estuary deviates from its ideal shape.
Therefore, we expect that locations with large excess width and thus a large
summed width of bars have a more convex hypsometry
(Fig. c, d). In the case of an ideal estuary
with (almost) no bars (Fig. a, b), we
expect concave hypsometry.
Hypsometric functions to
describe morphological systems; modified from . (a, b) Effect of z values with the r value kept constant. (c) Inverted version of the Strahler
function (see Eq. ).
Example cross sections and
hypsometry, suggesting that channel-dominated morphology (a, b) generally
results in concave hypsometric functions (high z value in
Fig. ), while bar complexes
(c, d) generally result in convex hypsometry (low z value in
Fig. ). In the case of narrow bars with a
flat top and relatively steep transition from bar top to channel, the
original hypsometry function by is less appropriate and
the inverted function fits better.
The aim of this paper is to investigate the relation between estuary
planform outline and along-channel variation in hypsometry. To do so, first
hypsometric curves are used to summarise the occurring bed elevations in a
cumulative profile. Then, we use the original function of
to fit the data obtained from the bathymetry of four
estuaries (Fig. ). In the results, we develop an
empirical function to predict hypsometry. The quality and applications of the
predictor are assessed in the discussion.
Methods
In this section, first, the definition of an ideal estuary is given with a
description of how we derived a geometric property that characterises
deviation from an ideal shape. Second, the general form of a hypsometric
curve is described. Then, the available datasets that were used for curve
fitting are given. Last, the methodology to fit a hypsometric function to
bathymetry in systems is presented.
Deviation from “ideal”
A useful model to describe the morphology of estuaries is that of the “ideal
estuary”, in which the energy per unit width remains constant along-channel.
The ideal state can be met when tidal range and tidal current are constant
along-channel such that the loss of tidal energy by friction is balanced by
the gain in tidal energy per unit width by channel convergence
. In the case that the depth is constant
along-channel, the ideal estuary conditions are approximately met when the
width is exponentially decreasing in the landward direction
,
which also implies an along-channel converging cross-sectional area. However,
when depth and friction are not constant along-channel, for example linearly
decreasing in the landward direction, less convergence in width is required to
maintain constant energy per unit of width.
Therefore, many natural estuaries are neither in equilibrium nor in a
condition of constant tidal energy per unit width. They deviate from the
ideal ones as a result of a varying degree of sediment supply, lack of time for
adaptation to changing upstream conditions and sea-level rise
. Whether continued sedimentation would reform
bar-built estuaries with irregular planforms into ideal estuaries remains an
open question. While we expect a somewhat different degree of convergence
such that the ideal state of constant energy per unit of width is
approximately maintained, we do not study the deviation of this convergence
length from that in ideal estuaries.
Ideally, we would assess the degree to which an estuary is in equilibrium
from an aerial photograph. However, the only indicator derivable is channel
width and thus deviation from a converging width profile. Therefore, in
, we defined the excess width, which is the local width of
the estuary minus our approximation of the potential ideal estuary width.
Here, the ideal estuary width is approximated as an exponential fit on the
width of the mouth and the width of the landward river. While the empirical
measure of “ideal width” should not be confused with the “ideal state” of
an estuary, it is the only practical way to estimate deviation from an ideal
estuary based on the estuary outline only. Moreover, it proved to be a good
indicator of occurring bar patterns and will therefore be
applied in this paper to study hypsometries.
General hypsometric curve
In the past, multiple authors have proposed empirical relations for the
hypsometric shape of terrestrial landscapes and
(partially) submerged bodies
seefor
a review. All equations, except for , predict a
fairly similar hypsometric curve based on the volume and height range of the
landform . While it is of interest to use these empirical
relations to predict the occurring altitude variation of a landform, the
framework here is different. Here, we apply the general hypsometric curve to
characterise the occurring cross-sectional hypsometry along-channel. This approach is
similar to the approach of , who fitted a power function
to 15 zones along the Western Scheldt. However, the zoned approach smooths
out all the differences between bar complex and channel-dominated zones,
which are of interest for this study. For this purpose, it is less relevant
for which environment the hypsometric relation was proposed, as long as it is
capable of describing the range of occurring hypsometries. For the case of the
estuarine environment (Fig. ), the
hypsometric curve should be able to describe variations in concavity and
variations in the slope of the curve at the inflection point. Here we use the
original formulation, which is capable of doing so, but in
principle any equation that fits well could be used.
formulated the general hypsometric curve as
hz=rr-1z1(1-r)y+r-1z,
in which hz is the value of the bed elevation, above which fraction y
of the width profile occurs. In other words, hz is the proportion of
total section height and y the proportion of section width. r sets the
slope of the curve at the inflection point in a range of 0.01–0.50, with
sharper curves for lower values of r
(Fig. a, b). z determines the
concavity of the function in a range of 0.03–2, with lower values giving a
more convex profile and higher values giving a more concave profile
(Fig. ). Our
approach changes the original definition of r and z to make them fitting
parameters. It is expected that z values depend on excess width because
the fraction of the width occupied by bars becomes larger with excess width,
resulting in a more convex hypsometric profile
(Fig. c, d), or the presence of bars
generates excess width.
Excess width is defined as the local width minus the ideal width, which is
given by
Wideal(x)=Wm⋅e-x/LW,
in which x is the distance from the mouth, Wm the width of the mouth
and LW is the width convergence length , which
can be obtained conservatively from a fit on the width of the mouth and the
landward river width :
LW=-s1lnWsWm,
in which Wm is the local width measured at the mouth of the
estuary, Ws is the width measured at the landward side of the
estuary and s is the distance between these locations measured along the
centreline. This practical method makes the convergence length somewhat
sensitive to the selected position of the seaward and landward limit.
The landward limit was selected at the location where the width ceases to converge on an
image at the resolution of the full estuary scale and the landward width was
measured between the vegetated banks. The seaward limit was selected as the
location with the minimum width in the case that bedrock geology, human engineering or
a higher elevated spit confined the mouth because in these cases the minimum
width limits the inflow of tidal prism. In other cases, the mouth was chosen
at the point at which the first tidal flats were observed in the estuary or
the sandy beach ends at the mouth of the estuary. However, when the
mouth is chosen at a location where sand bars are present, the ideal width
will be overestimated and the width of intertidal area underestimated. It is
therefore recommended to either choose the mouth at a location where bars are
absent or subtract the width of bars from the measured width at the mouth to
obtain the ideal width profile.
Data availability and classification
Detailed bathymetries were available for four systems: the Western Scheldt
esturary (NL), the Dovey estuary (Wales), the Eems–Dollard estuary (NL) and
the Columbia River Estuary (USA) (Fig. ,
Table ). Data for the Western Scheldt and
Eems were obtained from Rijkswaterstaat (NL), for the Dovey estuary from
Natural Resources Wales and for the Columbia River Estuary from the Lower
Columbia Estuary Partnership. Bed elevations were extracted from these
bathymetries as follows. First, the estuary outline was digitised, excluding
fully developed salt marshes, and subsequently a centreline was determined
within this polygon following the approach of. Bed
elevations were collected on equally spaced transects perpendicular to the
centreline of the estuary. The bed levels extracted at each transect were
subsequently sorted by bed-level value and made dimensionless to obtain
hypsometric profiles (see Fig. for
examples).
Characteristics of the estuaries used
in this study. hm is the depth at the mouth, hr is
the depth at the landward river, Wm is the width at the mouth, Wr is
the width at the landward river, a is the tidal amplitude, “Area” is
the surface area, “% intertidal” is the percentage of intertidal area and Qr is the river
discharge.
hm (m)
hr (m)
Wm (m)
Wr (m)
2a (m)
Area (km2)
% intertidal
Qr (m3 s-1)
Western Scheldt
25
15
4500
350
5
300
20
100
Columbia River
40
20
4000
800
2.5
900
30
7000
Dovey
10
2
450
50
3
12
75
30
Eems
25
8
3500
350
3.5
260
30
80
(a) Width along the
Western Scheldt, with the maximum ideal converging width profile indicated.
The green area is defined as the excess width cf. . For
each along-channel transect of the estuary, the optimal fit of z and r in
the function
(Fig. , Eq. 1) was determined.
(b, c) Results for the Western Scheldt when both z and r are
freely fitted (solid line) and the results when r is fixed to a constant
value of 0.5 (dashed line). (d) The quality of the fits remains
about the same when r is set to a fixed value of 0.5 as indicated by the
root mean square error (RMSE). (e) Fitted z values show similar
trends as the ideal width divided by local width.
We classified the transects by morphological characteristics and potential
susceptibility to errors. The following morphological classes were used:
mouth, bar junction, bar complex, narrow bar, point bar, channel, pioneer
marsh. The mouth is the location where the estuary transitions into the sea.
A bar junction is the most seaward or most landward tip of tidal bars. A bar
complex, also called a compound bar, is a location where a large
bar is dissected by barb channels or multiple smaller bars
are present. Narrow bar is used when the bars present were narrow along their
entire length and often also relatively flat on their top. Point bar is a bar
in the inner bend of a large meander. Channel was assigned when bars were
largely absent. Pioneer marsh was assigned when aerial photographs or
bathymetry gave visual indications of initial marsh formation, such as the
presence of small tidal creeks and pioneering vegetation. Fully developed
marsh is excluded from the outline.
The following classes were used to indicate possible errors: the presence of
harbours, major dredging locations, the presence of a sand spit, the presence
of drainage channels for agriculture, constraints by hard layers, human
engineering works. Either a locally deep channel or scour occurred at one of
the sides of these transects or they lacked a natural transition from channel
to estuary bank, thus ending in their deepest part on one side of the
transect. Major dredging locations have unnaturally deeper channels and
shallower bars, resulting in a hypsometric shape that is relatively flat in
the highest and lowest part and is steep in between
(Fig. e, f). Furthermore, in a few cases
side channels were perpendicular to the orientation of the main channel of
the estuary. This resulted in transects being along-channel of these side
channels, which biases transect data towards larger depth and creates a flat
hypsometric profile at the depth of the side channel.
Data processing
Least-squares fits resulted in optimal values of z and r in
Eq. () (Fig. ) for each
transect, using three different approaches. First, a regular least-squares
curve fitting was used for r and z, which resulted in along-channel
varying values for z, but an almost entirely constant along-channel value
for r of 0.5 (Fig. b, c, solid lines). In
the second approach we set r to a constant value of 0.5 and only fitted to
obtain z (Fig. c, dashed line). We found
that the quality of the fit was the same, as indicated by the
root mean square error (RMSE) (Fig. d), and
therefore apply this second approach in the remainder of this paper.
Locations where the RMSE was relatively large correspond to locations where
major dredging occurred in the past century. This possibly resulted in a
hypsometry characterised by a larger fraction of the width occupied by high
tidal flats, a larger fraction of the width occupied by deep channels and a
smaller fraction of the width occupied by the zone between channels and bars
(Fig. e, f). Because the hypsometric
curve at these locations deviated from the original Strahler function
(Eq. ), our third approach was to apply a modified function to
find optimal values for z and r. To do so, the original formulation of
was inverted to allow for hypsometries that describe
steep transitions from bar top to channel bottom because the original does
not fit nearly as well:
hz,inv=y1/z(1-r)r+1-1-r1-r.
Applying this modified function resulted in better fits, but only at
locations that were classified to be excluded because of possible errors.
Therefore, results from this approach are not shown here and it is suggested
to study the effect of dredging and dumping on hypsometry in more detail in
future studies.
Hypsometric curves as
extracted from bathymetry in the Western Scheldt, Dovey, Eems and Columbia
River. Cross-sectional profiles were extracted along the centreline
(Sect. ) and were subsequently classified as
channel when sand bars were (mostly) absent and as bar complex when one
larger bar dissected by barb channels or multiple smaller bars was present.
Channel-dominated morphology generally results in concave hypsometric
profiles (a) and bar complexes in convex profiles (b).
In principle, both the bed elevation (hz) and the width fraction y in
Eq. () are dimensionless. To compare the resulting predictions
with measured values, the prediction needs to be dimensionalised. Values for
y are scaled with the local estuary width. We test three options to scale
hz. The first option is to scale hz between the highest bed
elevation and lowest bed elevation in the given cross section, which is
sensitive to the precise cut-off of the bathymetry. The second option is to
scale hz between the local high water level (HWL) and the maximum
estuary depth in that cross section, which is sensitive to bathymetric
information that is usually not available in unmapped estuaries.
The third option requires a prediction of depth at the upstream or downstream
boundary. Width-averaged depth profiles along estuaries are often (near)
linear , which includes horizontal profiles
with constant depth. Therefore, only the channel depth at the mouth of the
estuary and at the upstream river have to be estimated, and subsequently a
linear regression can be made. Channel depth at an upstream river
(hr) is estimated with hydraulic geometry relations
e.g.:
hr=0.12Wr0.78. The depth at the mouth is estimated
from relations between tidal prism and cross-sectional area
e.g..
Here we used
hm=0.13×10-3PWm,
in which P is the tidal prism, which can be estimated by multiplying the
estuary surface area with the tidal range . This assumes a
flat water surface elevation along the estuary and neglects portions of the
estuary that might get dry during the tidal cycle . Locally,
the maximum depth may be deeper or lower than predicted due to the presence
of resistant layers in the subsurface or where banks are fixed or protected.
While this may affect the accuracy of the locally predicted maximum channel
depth, it has a minor effect on the calculations of subtidal and intertidal area.
Moreover, the upper limit for dimensionalisation is chosen as the high water
line, which implies that the supratidal area is not included in the
predictions. We will show results with all methods, but only the third can be
applied when information about depth is entirely lacking.
Statistical analyses in the remainder of the paper were approached as follows. In
linear regressions, we minimised residuals in both the x and
y directions. This results in regressions that are more robust than when
residuals in only one direction are minimised. In the case that regressions are
plotted, the legend will specify the multiplication factor that the
confidence limits plot above or below the trend. R2 values are given to
indicate the variance around the regression. In cases in which the quality of
the correlation between two along-channel profiles is assessed, we used the
Pearson product-moment correlation coefficient (r).
Results
We found a strong relation between along-channel variation in hypsometry and
the degree to which an estuary deviates from its ideal shape. Below, we will
first show how the hypsometry of typical channel morphology deviates from
that of bar complexes. Subsequently, the data are presented per system and
classified on their morphology and potential for errors due to human
interference, method and other causes. Then we combine all data to derive an
empirical relation to predict hypsometry. Last, we apply this relation to
predict the along-channel variation in intertidal and subtidal width and
validate the results with measurements from bathymetry.
Empirical relation between morphology and hypsometry
As hypothesised, it is indeed observed that channel-dominated morphology
results in more concave hypsometry profiles (high z value), while bar
complex morphology results in more convex hypsometry (low z value)
(Fig. ). Values for z in
Eq. () range from 0.83 to 1.14 for channels, with an average
value of 1.0. In contrast, z ranges from 0.36 to 0.41 for bar complexes,
with an average of 0.39.
Results of
hypsometry fitting in which clustering indicates a relation to planform
geometry. (a, c, e, g) z values were fitted for cross-channel
transects in the bathymetry of four estuaries and plotted by morphological
classification. (b, d, f, h) Regressions for z value as a function
of ideal width divided by local width. Data points that were influenced by
human interference, bedrock geology, or errors in methodology or data,
indicated in red (a, c, e, g), were excluded. Confidence limits
are plotted at 2 standard deviations above and below the regression and
their multiplication factor compared to the trend is given in the legend.
Clustering of morphological classes strongly suggests a relation between
hypsometry and planform estuary shape
(Fig. ). Mouth and
channel-dominated morphologies typically plot at the right-hand side of the
plots in Fig. a, c, e and g, thus
being locations close to ideal width. In the case of the Western Scheldt this
results in the highest values of z. In the case of the Dovey and Columbia
River, the mouth region was respectively influenced by a spit and human
engineering, which resulted in the formation of tidal flats on the side and
thus led to a lower z value.
Fitted z values as a function of
deviation from the ideal width. Colours indicate the location along the
estuary, with dark blue colours at the mouth transitioning into dark red
colours at the landward end. Some zones show scatter in fitted z values,
while some other zones (e.g. green to orange to red in c) show quasi-periodic
behaviour.
Fitted z values of filtered data
increase with the fraction of ideal width and local width, which indicates
that hypsometric shapes become progressively more concave when the local
width approaches the ideal width and become more convex when the local width
becomes larger than the ideal width (i.e. the excess width increases). The
data shown as asterisks are used for the regression. Confidence limits are
plotted at 2 standard deviations above and below the regression and their
multiplication factor compared to the trend is given in the legend.
Illustration of uncertainty in
the
predicted hypsometry from Eq. () with uncertainty
margins (Fig. ). Resulting prediction for
hypothetical location where (a) local width is equal to the ideal
width and (b) local width is 5 times larger than the ideal width.
Results are compared against a typical tidal range in order to show
the uncertainty of predicted intertidal high width, intertidal low width and
subtidal width as a fraction of the total estuary width.
Comparison of measured
and predicted values of intertidal high, intertidal low and subtidal width
for the Columbia River Estuary
(a, c, e) and the Western Scheldt (b, d, f). Predicted
nondimensional hypsometry was dimensionalised for each cross section using
three methods (explained in Methods) and uncertainty margins are given for
one of the predictions (solid black line). In the legend, r indicates the
Pearson product-moment correlation coefficient and “dev” the average factor
of deviation between the predicted (TP-HWL) and measured lines.
Comparison of measured and
predicted width of intertidal and subtidal width. The (solid) line of
equality indicates a perfect fit and dashed lines indicate a deviation of a
factor of 2. The percentage of measurements within these margins is indicated in
Table .
Bar complexes occur at the other end of the spectrum; these locations are
generally much wider than the ideal shape and are characterised by
hypsometries with a z value well below 1. Bar junctions, as well as narrow
bars, are generally found at the transition from channel-dominated morphology
to bar complex morphology and therefore also occur between these types in the
plots. The point bar in the Western Scheldt (the Plaat van
Ossenisse) shows hypsometry comparable to bar complexes
(Fig. a), which reflects the
complex history of formation by multiple bar amalgamations. Also, the
locations in the Columbia River where pioneer marsh is present show the same
trend as the locations where unvegetated bar complexes occur
(Fig. g).
In a few cases, the transects used to extract bathymetry were not
perpendicular to the main channel of the estuary. For example, landward and
seaward of the point bar in the Western Scheldt (Plaat van
Ossenisse) transects were inclined, covering a larger part of the channel
than perpendicular transects, resulting in higher z values as a consequence
of the apparent channel-dominated morphology. Immediately landward of the
spit in the Dovey, transects are almost parallel to the shallow side channel.
Fitting hypsometry at these locations resulted in relatively low z values
because it is a relatively shallow side channel.
For the Western Scheldt and Eems it is known at which locations major
dredging and dumping takes place
e.g.. Even
though the resulting z values at these locations do not cause major
outliers, the quality of the fits is typically lower and the inverted Strahler
function (Eq. ) fitted better. These points were
therefore excluded from further analysis.
The filtered data show quasi-cyclicity in along-channel hypsometry
(Fig. ). In general, the width at the mouth of the
estuary and at the upstream estuary is close to ideal and the hypsometry is
concave, except in systems with wide mouths and bars in the inlet. The part
in between is characterised by variations in the local width and therefore
gradual increases and decreases in the ratio between local width and ideal
width. In some cases, quasi-cyclic loops are visible (e.g.
Fig. c) caused by the asymmetry in bar complexes.
In other cases, the points show more zigzag or clustered patterns, which
indicate minor variation in the bar complexes or scatter in the fit applied
to the bathymetry.
Hypsometry predictor
The relations between excess width and hypsometric function are similar for
all estuaries, which suggests that a universal function is of value. Combining
all the filtered data resulted in a regression between the extent to which an
estuary deviates from the ideal shape and the predicted z value in the
hypsometry formulation (Fig. ). Data from
the Columbia River, Eems in 1985, the Western Scheldt in 2013 and Dovey were used to
obtain this relation. Other data are shown in
Fig. but not used in the regression.
These results mean that we found a predictive function for hypsometry, where
r is set to a constant value of 0.5 and the z(x) is calculated as
z(x)=1.4Wideal(x)W(x)1.2,
in which Wideal(x) is the ideal estuary width
(Eq. ) and W(x) is the measured local width. The
confidence limits of the regression plot a factor of 1.9 higher and lower than
the regression, which indicates that the z value can be predicted within a
factor of 2 (see Fig. for an example of
prediction with uncertainty). While not used in the regression, hypsometry
from bathymetry in other years shows similar trends and scatter as the data
used in the regression.
The predictor (Eq. ) was applied to the Columbia River
Estuary, Western Scheldt, Dovey and
Eems to check the quality of the resulting along-channel predictions of
intertidal high, intertidal low and subtidal width
(Figs. and
). These zones can be derived after
dimensionalising hypsometry and imposing a tidal range
(Fig. b). For almost the entire
along-channel profile, the predictions are within a factor of 2 of the
measured value (Fig. ) and the best
agreement was obtained when the hypsometry was dimensionalised between the
minimum and maximum measured bed level for each transect
(Fig. ).
Percentage of points predicted
within a factor of 2 from the measured value.
Estuary
% for subtidal
% for intertidal
Western Scheldt
100
84
Columbia River
90
79
Dovey
54
71
Eems
91
59
Discussion
Results from this study illustrate that the bed-level distributions of channel
and bar patterns in estuaries are topographically forced. The estuary outline
that is observable from the surface translates into the three-dimensional
patterns below the water surface. Bar-built estuaries typically have a
quasi-periodic planform, in which major channel
confluences occur at locations where the estuary is close to its ideal shape
. The parts between the confluences are typically filled
with intertidal bar complexes. These findings are consistent with hypsometry
zonations previously found for the Western Scheldt with more concave
hypsometries for channel-dominated morphology and more convex hypsometries
for bar complex morphology . Our cross-sectional approach
additionally revealed quasi-periodic behaviour within these zones.
In contrast to an empirical description, ideally, a physics-based
determination of the hypsometry would be favourable. However, with the
current state of the art of bar theory and relations for
intertidal area, tidal prism, cross-sectional area and flow velocities
it is not yet possible to derive a
theoretical prediction of hypsometry. For example, bar theory
could predict occurring bar patterns on
top of an (ideal) estuary shape, but current theories overpredict their
dimensions and it is still impossible to scale these to
bed-level variations because the theories are linear. In addition to that,
the resulting predictions would need to meet the requirement that the
predicted bed levels and the intertidal area together lead to hydrodynamic
conditions that fit the estuary as well.
Previously, hypsometry was used to summarise the geometry of entire tidal
basins or estuaries . The whole
system descriptions are consistent with the original
concept of a basin hypsometry based on plan area, which is a valid
description in a landform context. However, these descriptions oversimplify
the along-channel variability in estuaries that are relatively long. These
estuaries typically have a linear bed profile varying from an along-channel
constant depth to strongly linear sloping (e.g. the Mersey in the UK). In the
latter case, the elevation at which subtidal and intertidal area occurs varies
significantly along-channel . Additionally, friction and
convergence may cause the tidal range to either dampen or amplify, causing
variation in tidal elevation, subtidal area and tidal prism
. Consequently, the along-channel cross section
hypsometry should be assumed to be relative to an along-channel varying high
water level or mean sea level rather than an along-channel fixed vertical
datum. Interpreting these along-channel variations remains an open question
because of the reasons outlined above. Nevertheless, if desired,
along-channel varying hypsometry predictions can be converted into one single
summarising curve (Fig. ), which shows that
the basin hypsometry can also be predicted when limited data are available.
Hypsometry as summarised in a single
curve for the entire estuaries. Solid lines are measured from bathymetry,
and dashed lines are based on the predictions.
Our results show that hypsometry is not only a tool to predict morphology
when limited data are available, but that hypsometry can also be used to
reduce a large dataset of bathymetry and to study the evolution of bathymetry
over time. In the case of , hypsometry fits result in
along-channel profiles of z and r values, but in practice any function
or shape could be fitted. For example, the locations along the Western
Scheldt where major dredging and dumping took place showed a weaker
correlation with the original shape
(Fig. ). In these cases, fits with
higher quality (lower RMSE) were obtained when we used the inverted
hypsometric function (Eq. )
(Fig. b, e). So in practice, one
could fit a range of different hypsometry shapes and subsequently find out
which of these shapes fit best on the dataset used. It can indicate that
certain parts along the estuary require a separate hypsometric description.
The fitting parameters are a method to describe the along-channel variation
(Fig. b, d). Hypsometry can be fitted
to compare data from nature, physical experiments and numerical modelling and
subsequently study, for example, the effect of vegetation, cohesive mud and the
influence of management on these systems.
(a) Evolution of
a cross section in the Western Scheldt where it has been significantly
dredged and dumped (the Drempel van Hansweert). (c) Measured
hypsometric profiles of the same time steps at the same cross section.
(e) Best-fit hypsometries using the original Strahler equation
(solid) and inverted Strahler function (Eq. )
(dashed). (b) The quality of the two types of fits shows that the
shape of the best-fitting hypsometric curve changes from the original
Shrahler to the inverted equation in the 1970s–1980s. An additional transect
is added in the panels on the right. (d) Fitting coefficients for
z increase over time for both hypsometry types and both transects.
(f) Intertidal high area increased over this period, while
intertidal low area remained constant (transect 1) or decreased
(transect 2).
(a) Ecotope map of the Western
Scheldt (2012) obtained from Rijkswaterstaat. (b) Prediction of the
width in which Salicornia (black) and Spartina (blue) can
occur when assuming that Spartina occurs between MSL and 1.5 m
above and Salicornia occurs between 1.0 and 2.5 m, while ignoring
velocity, salinity and sediment type constraints. Red line indicates measured
width of vegetation based on ecotope map. The Drowned Land of Saeftinghe is
excluded in the predictions because the high water line was the boundary of
the analysed bathymetry, while it is included in the measured data.
Implications for management of estuaries
In many estuaries from around the world, subtidal channels are used as
shipping fairways, while the intertidal bars (or shoals) form valuable
ecological habitats e.g.. For example in the Western
Scheldt, the shipping fairway is now maintained at a depth of 14.5 m below
the lowest astronomical tide . With empirical
hypsometry predictions, we can estimate the width below a certain depth
required for shipping, which gives estimates of what volume to dredge and at
what locations along the estuary, which is relevant for the construction of
future shipping fairways in estuaries for which we may have limited data.
In contrast, low-dynamic intertidal areas are valuable ecological habitats;
for example, for the Western Scheldt there is an obligation to maintain a
certain amount of intertidal area . Previously,
showed that basin hypsometry can be a tool to design
breaches in managed realignment sites and can provide an indication of
habitat composition. Hypsometry analysis per cross section shows that estuary
outline translates into intertidal area, which implies that locations where
the estuary is relatively wide have a relatively wide intertidal area. The
ecological value is determined by the area of low-dynamical shallow water and
intertidal areas (for settling and feeding) . This means
that the edges should neither become steeper nor higher (leading to permanent
dry fall) or deeper. Hypsometry fits (in the case of available data) or
predictions (in the case of limited data) can indicate which locations along the
estuary have a risk to transform away from low-dynamic area or have the
potential to become low-dynamic area by the suppletion of dredged sediment.
The occurrence of vegetation species depends on bed elevation, salinity,
maximum flow velocity and sediment type . Even
though predicted hypsometry only gives bed elevations, a comparison of the
height interval in which Salicornia and Spartina can occur
showed similar trends and the same
order of magnitude as the measured vegetation from ecotope maps of the
Western Scheldt in 2012 (Fig. ). Some underpredictions
arise in parts along the estuary where bed elevations above the high water
level occur, such as at the Drowned Land of Saeftinghe. However, in general,
the vegetation width is overpredicted because (1) hypsometry is stretched
between the high water line and channel depth and (2) other constraining
biotic and abiotic factors were excluded.