An enhanced depth-averaged tidal model for morphological studies in the presence of rotary currents

Abstract

A simple and efficient method to improve morphological predictions using depth-averaged tidal models is presented. The method includes the contribution of secondary flows in sediment transport using the computed flow field from a depth-averaged model. The method has been validated for a case study using the 3D POLCOMS model and ADCP data. The enhanced depth-averaged tidal model along with the SWAN wave model are applied to morphological prediction around the Lleyn Peninsula and Bardsey Island as a case study in the Irish Sea. Due to the presence of a headland in this area two asymmetrical tidal eddies are developed in which the cyclonic eddy is stronger as a result of Coriolis effects. The results show that the enhanced model can effectively predict formation of sand banks at the centre of cyclonic eddies, while the depth-averaged model, due to its inability to accommodate secondary flow, is inadequate in this respect.

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Continental Shelf Research 27 (2007) 82–102
An enhanced depth-averaged tidal model for morphological
studies in the presence of rotary currents
Simon P. Neilla,?, Mohammad R. Hashemib, Alan J. Elliotta
aCentre for Applied Marine Sciences, University of Wales (Bangor), Marine Science Laboratories, Menai Bridge, LL59 5AB, UK
bDepartment of Water Engineering, Shiraz University, Shiraz, Iran
Received 18 July 2005; received in revised form 23 June 2006; accepted 8 September 2006
Available online 2 November 2006
Abstract
A simple and efficient method to improve morphological predictions using depth-averaged tidal models is presented.
The method includes the contribution of secondary flows in sediment transport using the computed flow field from a
depth-averaged model. The method has been validated for a case study using the 3D POLCOMS model and ADCP data.
The enhanced depth-averaged tidal model along with the SWAN wave model are applied to morphological prediction
around the Lleyn Peninsula and Bardsey Island as a case study in the Irish Sea. Due to the presence of a headland in this
area two asymmetrical tidal eddies are developed in which the cyclonic eddy is stronger as a result of Coriolis effects. The
results show that the enhanced model can effectively predict formation of sand banks at the centre of cyclonic eddies, while
the depth-averaged model, due to its inability to accommodate secondary flow, is inadequate in this respect.
r 2006 Elsevier Ltd. All rights reserved.
Keywords: Coastal morphology; Sand banks; Coriolis force; Rotary currents; Secondary flow; Irish Sea
1. Introduction
Tidal stirring of the seas in the region of
headlands may concentrate sediments and result in
the formation of tidal banks (Pingree, 1978). Head-
lands produce characteristic eddy patterns which
are a result of vorticity transfer from the tide to the
residual flow. This vorticity generation is a result of
two separate mechanisms (Robinson, 1981): in-
creased flow near the headland tip creates a
proportionally larger frictional force inshore, while
a shallower depth inshore results in greater depth-
averaged friction. Both mechanisms create a similar
sense of vorticity, hence leading to eddy formation.
As a result of these eddies, bed material is either
lifted into suspension or transported as bed load
around the eddy during periods of high tidal stress
(Pingree, 1978). This may take place gradually
within each tidal cycle or suddenly under storm
conditions. The eddies are expected to generate
secondary flows, with convergence towards the
centre of the eddies at the bed and divergence at
the surface. For curved shallow water flow, the
circular motion of the eddy is maintained by an
inward pressure gradient matching the centripetal
force, and this pressure gradient forces a flow
towards the centre of the eddy near the bed, where
friction reduces the strength of the centripetal force
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doi:10.1016/j.csr.2006.09.002
?Corresponding author. Tel.: +441248713808;
fax: +441248716729.
E-mail address: s.p.neill@bangor.ac.uk (S.P. Neill).

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(Dyer and Huntley, 1999). For smaller flow speeds
or curvature, the Coriolis force becomes more impor-
tant. The original location for sand bank formation
is controlled hydrodynamically (Pattiaratchi and
Collins, 1987), but as the sand bank grows in
dimensions, interaction between the tidal currents
and the sand bank enhances the eddy, hence
encouraging further bank growth until a state of
equilibrium is reached.
Morphological models can be divided into three
main categories depending on the timescale of
simulation (de Vriend et al., 1993). Initial sedimen-
tation/erosion (ISE) models go through the se-
quence of constituent models (e.g. tidal, wave,
sediment transport and morphological) only once,
hence the assumption is made that the bed
topography is invariant for all models other than
the final morphological model (i.e. no feedback
mechanism exists between the evolving bedform and
the hydrodynamics). The other two classes of
model, where modelled bed evolution affects the
hydrodynamics are medium-term morphodynamic
models and long-term morphodynamic models,
suitable for weekly and yearly time scales, respec-
tively. In this paper, an ISE model is developed
since the time scales studied are of order 1–2 days
(i.e. a single storm event), therefore the expected
change in bed level due to sediment transport during
a storm will have a negligible effect on the
hydrodynamics. It is calculated (in this paper) that
the maximum change in bed level during a 48h
period (including a storm) is of order 0.1m in water
of depth 30m. Assuming that depth changes of 10%
have a significant effect on the hydrodynamics
(Soulsby, 1997), a conservative estimate of time-
scales when an ISE model is no longer applicable in
the chosen study area is of order 60 days. ISE
models are generally considered to be diagnostic
tools for morphological process analysis and not
quantitative predictors of morphological evolution.
In the current work, the aim is not to make accurate
quantitative predictions of changing bedforms, but
to examine the suitability of a method which
includes secondary flow characteristics in 2D
models, hence an ISE model is sufficient for the
study.
With an increasing trend in computational power,
there is a general shift occurring in morphological
prediction from the use of 2D to 3D hydrodynamic
simulations. However, it is not always necessary to
make use of the vertical resolution in morphological
models. In one example of the sedimentation of a
breakwater, Lesser et al. (2004) obtained approxi-
mately the same results using a 3D morphological
model as did Nicholson et al. (1997) using a simpler
2D model. Additionally, 3D models include com-
plex 3D effects automatically, i.e. they do not
necessarily increase understanding of a morpholo-
gical situation. The additional computing power
available may be of more use in many situations in
improving the horizontal rather than the vertical
resolution.
One of the advantages of 3D morphological
models over 2D morphological models is that they
include complex effects ‘‘by default’’. This can be a
major issue in 2D sediment transport models of
river meanders. In the real case, a helical secondary
flow is induced by the flow curvature which leads to
a bed load transport of material towards the inner
bank of the meander. Erosion occurs on the outer
banks and deposition occurs on the inner bank
(Kikkawa et al., 1976). 3D models account for this
effect by solving the vertical profile of secondary
velocity in the river cross-section. 2D models,
however, must be modified to account for this
secondary effect (e.g. Struiksma et al., 1985).
Assuming that bed load transport (compared to
suspended load transport) has a major contribution
to bed level change, a simplified method to be
applied to 2D sediment transport models need only
solve the near-bed contribution of secondary flow.
The transport direction can then be calculated as the
resultant of the main flow direction and this
secondary flow direction. Such a method has been
developed by several authors to study river mean-
ders based on flow curvature and water depth
(Kikkawa et al., 1976; Engelund, 1976). Attempts
have been made to quantify the secondary flow
velocity in tidal flows around headlands (Geyer,
1993; Alaee et al., 2004) using theoretical curves for
secondary flow based on Kalkwijk and Booij (1986).
Some authors have attempted to apply this simpli-
fied method to calculate the consequences for sand
transport (and morphological change) due to
secondary flow (e.g. Wang et al., 1995). However,
as far as the authors of the present paper are aware,
no-one has attempted to apply these methods to the
evolution of sand banks which are generated by
rotary currents (e.g. headland eddies). This paper
attempts to address the latter issue.
A morphological model is described in Section 2,
consisting of a tidal model, wave model, sediment
transport model and bed level change model. In
Section 3, the method for determining the secondary
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flow based on flow curvature and other 2D flow
field variables is discussed. The method is also
verified using a three-dimensional model and ADCP
data. In Section 4, the method is applied to a 2D
modelling case study of a headland/island system in
the Irish Sea. Finally, a discussion on the validity of
the results and further justification of the method
are given in Section 5.
2. Development of morphological model
The morphological model consists of an interac-
tion between several different models: a tidal model
for prediction of tidal currents, wind-generated
currents and wind surge, a wave model for
prediction of wave orbital velocity, a model to
predict total sediment transport under the action of
currents and waves, and finally a bed level change
model to predict the morphological change. In this
paper, 2D depth-averaged tidal modelling is con-
sidered for morphological studies although a 3D
tidal model of the study area is developed for
verification of the proposed method.
2.1. Hydrodynamic models
POLCOMS is the Proudman Oceanographic
Laboratory Coastal Ocean Modelling System (Holt
and James, 2001). POLCOMS is three-dimensional
(using s coordinates in the vertical) and is for-
mulated in spherical coordinates. For turbulence
closure, the Mellor–Yamada–Galperin level 2.5
scheme is used (Mellor and Yamada, 1974; Galperin
et al., 1988). The model can also be run in depth-
averaged mode, hence it is possible to examine both
2D and 3D current fields. Boundary conditions
required for POLCOMS are elevation and the
normal component of velocity. These are obtained
by running an outer coarse grid and extracting time
series of the boundary points for the inner nested
region. A harmonic analysis is calculated for each
time series using T_TIDE (Pawlowicz et al., 2002) to
create an independent high-resolution model with
no feedback to the outer nest. To include storm
surges and wind-generated currents in the model, all
of the model grids were run with a normal stress
(atmospheric pressure) and shear stress (wind speed
and direction) using 3-hourly spatially varying data
obtained from Met E´ireann. These additional surge
components were extracted as a time series for the
boundary points of the inner nested region and
added to the tidal component. This procedure was
repeated for all levels of nesting from the outer grid,
through intermediate grids and finally to the inner
nested region.
SWAN (Simulating WAves Nearshore) is an
Eulerian formulation of the discrete wave action
balance equation (Booij et al., 1999). The model is
discrete spectral in frequencies and directions and
the kinematic behaviour of waves (including the
effect of currents) is described with the linear theory
of gravity waves. The deep water physics of SWAN
are taken from the WAM model (Komen et al.,
1994). SWAN is run by applying data of wind speed
and direction firstly on a coarse outer grid. Tidal
currents and water level are included in the SWAN
simulation, using output from the POLCOMS
model. A series of inner nested grids is then used,
with boundary conditions of spectral
transferred from the outer to inner grids. Significant
wave height Hsand peak wave period Tpare output
for each grid point of the inner-most model, and
used to calculate the bottom orbital velocity using
the method described by Soulsby (1987).
density
2.2. Total sediment transport under action of waves
and currents
Numerous non-cohesive sediment transport mod-
els exist in the literature and these are often
compared against each another (e.g. Davies and
Villaret, 2002). In this study, sediment transport is
calculated as a total load transport by waves plus
currents using the Soulsby–Van Rijn formula. It is
based on the model of Van Rijn (1989) with curve
fitting over a range of wave and current conditions
by Soulsby (1997). This formulation contains a
large enhancement of transport rate due to wave
action. The wave action has an important contribu-
tion to the suspended load when considering total
transport. The formula is valid for non-cohesive
sediments in the range 0.1–2.0mm. Bed roughness is
parameterised using sediment size. Total sediment
transport rate is
"
?ð1 ? 1:6tanbbÞ,
where As¼ Asbþ Assand
Asb¼0:005hðd50=hÞ1:2
qt¼ AsUU2þ0:018
CD
U2
rms
??1=2
? Ucr
#2:4
ð1Þ
½ðs ? 1Þgd50?1:2,(2)
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Ass¼0:012d50D?0:6
½ðs ? 1Þgd50?1:2
and
?
(3)
Ucr¼ 0:19ðd50Þ0:1log4h
d90
for 100pd50p500mm,
(4)
or
Ucr¼ 8:50ðd50Þ0:6log4h
d90
for 500pd50p2000mm,
(5)
where U is the depth-averaged current velocity,
Urmsis the root-mean-square wave orbital velocity,
CDis the drag coefficient due to current alone, Ucr
is the threshold current velocity, bbis the bed slope,
h is the water depth, s is the relative density of
sediment and D?is the dimensionless grain size. The
Soulsby–Van Rijn formula produces a scalar
quantity for sediment transport. It is assumed that
the direction of sediment transport is determined by
the tidal flow and not the wave direction. Although
longshore transport can easily be included as a
source term in the sediment transport model, it was
neglected in this study to allow a focus on the effects
of rotary currents in the region of sand banks.
The method of calculating the total sediment
transport was implemented using the 2D tidal
model and the wave model (Section 2.1). This
provides a source term for the morphological model
(Section 2.3), hence bed evolution due to tidal and
wave effects can be studied.
2.3. Bed level change model
When considering long-term morphodynamics, it
is important to include the interaction between the
hydrodynamic and the morphodynamic compo-
nents of the scheme (Nicholson et al., 1997). In this
paper, the timescale is of order 48h, hence this
feedback has not been included. Generally, the
results of two-dimensional morphological models
are sufficient for practical applications, but three-
dimensional models have the advantage that com-
plex three-dimensional flow effects (despite a
possible lack of understanding) can be automati-
cally included in a simulation (Lesser et al., 2004).
However, for this study the use of a 3D morpho-
logical model is not helpful since a method is sought
to reduce the complexity (and similarly reduce
computational time) of a three-dimensional pro-
blem to two dimensions.
Assuming that the sediment content of the water
column does not change significantly over time,
morphological development can be modelled in
two-dimensions using (e.g. Van der Molen et al.,
2004)
?
where z is the bed level, p is the bed porosity and qi
is the transport of sediment in the i direction. This
equation, known as the Exner equation, was solved
using the Lax–Friedrichs finite differencing scheme
which has first order accuracy (Chung, 2003).
qz
qt¼ ?
1
1 ? p
qqx
qxþqqy
qy
?
,(6)
3. Secondary flow and enhanced 2D tidal model
3.1. Effect of rotary currents on morphodynamics
Secondary flow leads to an accumulation of tea
leaves at the centre of a tea cup after stirring with a
spoon. This tea cup phenomenon is described in
several papers (e.g. Pingree, 1978) but is usually
taken as an analogy for sediment dynamics in
practical situations. Taking into consideration the
curvature of a river meander rather than the
curvature of a tea cup, this phenomenon has been
applied to the movement of river sediments
(Kikkawa et al., 1976). A more detailed theory of
the secondary circulation due to flow curvature is
explained in Section 3.2. However, for the purposes
of this section it is sufficient to say that there is a
surface divergence of suspended material and a
convergence of bed material at the centre of an
eddy. In the case of a river meander, this will lead to
scour at the outer banks and deposition along the
inner banks since the movement of bed load
material is towards the centre of curvature. The
same phenomenon exists in tidal flows, particularly
in the case of eddies due to headlands and islands
(an island can be considered as a headland with a
line of symmetry). A headland eddy will cause
material to gather at the centre of the eddy, the
process being approximately symmetrical between
flood and ebb, resulting in the formation of two
sand banks. However, for increasing latitude head-
lands, the system becomes progressively less sym-
metricaluntilone of
considerably stronger than the other, hence material
tends to be gathered preferentially in the region
of one residual eddy and a single sand bank is
theresidual eddiesis
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generated (Pingree, 1978). This is a result of the
increasing importance of Coriolis force with latitude
and its importance is indicated by the Rossby
number. In Section 4, a case study is examined in
which there is a single sand bank due to tidal
asymmetry of a headland. Similar convergence of
bed material due to secondary flow will exist in
eddies created by tidal flow around groynes and
harbours as another engineering application.
3.2. Earlier theories of secondary flow based on river
hydraulics
The concept of secondary flow which causes a
helical or spiral motion of water particles in curved
current fields was first investigated for curved open
channel flow (e.g. Van Bendegom, 1947) and is
discussed in this section. As Fig. 1 shows, the
secondary flow component occurs in a plane
perpendicular to the main flow direction. Although
the magnitude of secondary flow (as with main flow)
is zero at the bed (no-slip), the maximum occurs
slightly above the bed and so is important for bed
load transport. After Falco ´ n (1984), consider the
Navier–Stokes equation in the radial direction for a
curved steady flow field in a very simplified case
U2
gr¼qh
m
qr?1
rg
qtzr
qz, (7)
where Umis main current velocity, tzris radial shear
stress, r is the density of water and r is the radial
coordinate. This equation has three basic terms:
pressure gradient term, centripetal force term and
the radial shear stress term. Assuming a hydrostatic
pressure distribution, the radial pressure gradient is
equal to the radial water surface slope. Hence, there
is no significant variation of this term in the vertical
direction. Due to bed resistance, however, a
boundary layer develops near the bed which causes
a vertical variation in the streamwise velocity (Um)
and hence a vertical variation in centripetal force.
The difference between centripetal and pressure
gradient is balanced by the radial shear stress
gradient. The radial shear stress is related to the
secondary component of velocity through the
Boussinesq equation (Boussinesq, 1872)
tzr¼ r?qUs
qz,(8)
where Us is secondary flow velocity and ? is eddy
viscosity. Substituting tzrinto Eq. (7), the simplified
relation between main and secondary flow velocity
may be written as
U2
gr¼qh
m
qr??
g
q2Us
qz2. (9)
By implementing the above discussion, some re-
searchers (mainly for estimation of secondary flow
in rivers) have attempted to quantify the secondary
flow based on the main flow characteristics (e.g.
Engelund, 1974; Kikkawa et al., 1976). Generally,
the magnitude of the secondary flow is shown to be
proportional to the main flow velocity, curvature of
the main flow field and water depth which can be
written as
w ¼Us
Um
¼ KhFðz;Cz?Þ, (10)
where w is the strength of secondary flow, z is
relative depth and Um is the depth-averaged main
velocity, Cz?is the dimensionless Chezy coefficient
(i.e., Cz?¼ Cz=
cient), K is curvature (i.e., 1=R, where R is radius of
curvature) and Fðz;Cz?Þ is a coefficient which
represents the effect of bed resistance on the vertical
distribution of main flow and varies with depth. For
example, Engelund (1974, 1976) derived the func-
tion
ffiffiffig
p
where Cz is the Chezy coeffi-
Fðz;Cz?Þ ¼ 13Cz?½1
2ða ? 1Þz2þ1
þ K0ða;bÞ?,
6bz4?1
30b2z6
ð11Þ
where a, b and K0are also functions of Cz?and are
constants.
Although the above concept is valid in certain
circumstances for oceanographic flows, there are
some other factors which should be considered in
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Um
Us
Fig. 1. Secondary flow generated due to flow curvature.
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