ARTICLE IN PRESS
Continental Shelf Research 27 (2007) 798–831
www.elsevier.com/locate/csr
Vertical circulation in shallow tidal inlets and
back-barrier basins
Emil V. Staneva,c,, Burghard W. Flemmingb, Alex Bartholomäb,
Joanna V. Stanevaa, Jörg-Olaf Wolffa
a
Institute for Chemistry and Biology of the Sea (ICBM), University of Oldenburg, Postfach 2503, D-26111 Oldenburg, Germany
b
Senckenberg Institute, Südstrand 40, 26382 Wilhelmshaven, Germany
c
School of Environmental Science, University of Ulster, Cromore Road, Coleraine BT52 1SA, UK1
Received 4 June 2006; received in revised form 27 November 2006; accepted 28 November 2006
Available online 19 December 2006
Abstract
In this paper, we analyse the contribution of tidally induced drift in the surface layer to the overall dynamics of wellmixed tidal basins undergoing drying and flooding. The study area covers the East Frisian Wadden Sea (German Bight,
Southern North Sea), which consists of seven tidal basins. The major interest is focused on the tidal basin behind the
islands of Langeoog and Spiekeroog and the inlet connecting it with the North Sea. The comparison between theoretical
concepts, results from direct observations, and simulations with a numerical model helps to understand the underlying
physics controlling the tidal response. The data were collected during the period 1995–1998 and consist of cross-channel
ADCP transects. The identification of the dominant spatial patterns and their temporal variability is facilitated by
applying an EOF analysis to the data. The numerical simulations are based on the 3-D primitive equation General
Estuarine Transport Model (GETM) with a horizontal resolution of 200 m and terrain-following vertical coordinates. We
find distinct differences between the temporal variability of the transports near the surface and those in deeper layers of the
tidal inlets. The near surface transport is dominated by the tidally induced drift (similar to the Stokes drift), whereas the
deeper layer transport is dominated by asymmetries caused by the hypsometric properties of the intertidal basins. These
transports, when averaged over a tidal period, have opposite directions and compensate each other. This explains the
establishment of a vertical overturning cell: landward motion in the upper layers and seaward motion in the deeper parts of
the tidal channels. This vertical circulation cell is also observable in our numerical simulations and shows a clear
dependency of the temporal asymmetry in the transport patterns on the local depth. In deep tidal channels, the overall
properties of the tidal signal show a clear ebb dominance, whereas in the shallow extensions of the channels the transports
during flood are larger than during ebb. Although, our research area can be characterized as a well mixed estuary,
baroclinicity associated with the fresh water flux from the coast can substantially affect vertical overturning.
r 2007 Elsevier Ltd. All rights reserved.
Keywords: Wadden Sea; Tidally induced Stokes transport; ADCP measurements; EOF analysis; Numerical modelling; Tidal asymmetry
Corresponding author. Institute for Chemistry and Biology of
the Sea (ICBM), University of Oldenburg, Postfach 2503,
D-26111 Oldenburg, Germany. Fax: +49 441 798 3404.
E-mail address: e.stanev@icbm.de (E.V. Stanev).
1
Present affiliation.
0278-4343/$ - see front matter r 2007 Elsevier Ltd. All rights reserved.
doi:10.1016/j.csr.2006.11.019
1. Introduction
The ratio d between tidal range and depth, which
is known as the external Froude number (Jay and
Smith, 1988), controls the shallow water dynamics.
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E.V. Stanev et al. / Continental Shelf Research 27 (2007) 798–831
This control becomes very pronounced when the
local depth is comparable to the tidal range
(Ianniello, 1977), i.e. when d tends to unity.
In the case of the tidal basins of the East Frisian
Wadden Sea (Fig. 1), the mean depth can be even
smaller than the tidal range and large areas of the
basins undergo drying during part of the tidal cycle.
This very specific dynamics necessitates a more
detailed analysis because earlier studies have addressed mostly weakly non-linear systems in which
the external Froude number is much smaller than
unity (e.g. Ianniello, 1977). Effects associated with
drying and flooding also need more attention.
In this paper, we will illustrate some important
effects resulting from the shallow depth of the
Wadden Sea using data from observations and
numerical modelling, and analyse the consistency of
observations and numerical simulations with theory. The considerations above are reminiscent of the
classical problem of surface gravity waves where
variations of sea level over time induce a Stokes
drift. This issue has been the subject of a number of
studies (Longuet-Higgins, 1969; Ianniello, 1977;
Ianniello, 1979; Jay and Smith, 1988; Jay, 1990).
We can present the transport, vertically integrated from the bottom H to the ocean surface
z and averaged over a full tidal cycle T, as
Z z
hUi ¼
u dz ,
(1)
H
where
1
hwi ¼
T
Z
T
w dt.
0
(2)
799
This transport can be decomposed into two parts
(see e.g. LeBlond and Mysak, 1978):
Z z
Z 0
hui dz þ
u dz .
(3)
hUi ¼
H
0
In the simplest case of linear waves
z ¼ a cosðkx otÞ
(4)
the velocity components ~
v ¼ ðu; v; wÞ being given by
u¼
gak cosh½kðz þ HÞ
cosðkx otÞ,
o
cosh kH
v ¼ 0,
(5)
(6)
gak sinh½kðz þ HÞ
sinðkx otÞ,
(7)
o
cosh kH
where a, k and o are the amplitude, wave number
and frequency, respectively, the time-averaged
velocity is zero, and the first integral in Eq. (3)
vanishes. However, the second integral which
measures the contribution of the interval between
the troughs and the crests of the waves to the total
transport of momentum is not zero, but proportional to a2 . This results in a forward (in the
direction of wave propagation) transport of mean
momentum, which is concentrated at the surface
(Fig. 2). Thus, at any level above z ¼ a there is
more transport forward than backward, which leads
to a non-zero second-order drift. Lagrangian and
Eulerian mean velocities can differ considerably
(Longuet-Higgins, 1969), the difference between
them is the Stokes drift measured by the correlation
between surface velocity and sea level. Furthermore,
velocities associated with the Stokes drift can exceed
w¼
Fig. 1. The East Frisian Wadden Sea. The plot displays the model topography (see also Section 4.1) and the locations of observations and
model samples discussed in the text. The depths are represented as negative numbers (m) below the mean sea level. The thin meridional
sections in the extension of Otzumer Balje in the tidal basin of Spiekeroog Island are sections sampled every 5 min from the model
simulations. ADCP measurements were taken along Section 1.
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E.V. Stanev et al. / Continental Shelf Research 27 (2007) 798–831
Ocean
Land
Bay
A C: inlet cross
sectional area
Inlet
LC
A B: surface area
Fig. 3. Schematic representation of the system ocean–inlet–
bay–land.
Fig. 2. Schematic representation of Stokes drift. (a) Orbital
motions in surface gravity waves, (b) Eulerian representation: the
Eulerian net transport is between apzpa, (c) Lagrangian
representation: the exponential decrease of the radius of the
orbits with increasing depth leads to an exponential decrease of
the Lagrangian drift. The vertical line could be interpreted as a
column of dye at t ¼ 0 and the tips of the arrows (forming an
exponential line) give the progression of the same column after
several wave periods.
the velocities associated with river discharge by an
order of magnitude (Pritchard, 1958).
One simple configuration of an open-ocean/tidal
basin system is shown in Fig. 3. The net transport
through the inlet is zero because the mass in the
tidal bay should be conserved (we assume that river
discharge is zero). However, the tidally induced
Stokes (hereinafter TIS) drift is always in the
direction of the wave motion. Mass conservation
dictates that a vertical shear of the currents is
needed so that the landward transport in the upper
layers (TIS drift) is compensated by a seaward
transport in the deeper layers. This possibility has
been revealed earlier by Jay (1990), who pointed out
that the net Lagrangian current across the channel
must be zero; thus a compensating Eulerian current
must be provided by a second-order surface slope to
balance the TIS drift. Furthermore, tidal non-linear
generation of residual flow is related to an ebb–
flood asymmetry, the latter being the primary
factor determining the profile of the residual
current (see also Jay, 1990). We stress here that,
unlike the classical Stokes drift, the TIS one is
strongly dependent on the friction in the shallow
water. The quantification of these transports for
the East Frisian Wadden Sea is focal to the
present study.
The first objective of the paper is to consider the
specific appearance of the TIS drift in the Wadden
Sea focusing on the spatial dependence. This is an
important issue because the external Froude number varies spatially. Furthermore, the back-barrier
basins include relatively deep channels and broad
tidal flats, which is a complicated case compared to
the known theoretical cases of channels with simple
topography. In such settings divergence (convergence) of horizontal flows becomes dominant
(Jay, 1990) and the tidal excursion scale takes the
control on the residual transport. Under such
conditions, the Eulerian residual currents will differ
substantially from Lagrangian ones (Signell and
Geyer, 1990), the latter being extremely complex
(Zimmerman, 1976a,b). Understanding dynamics in
such estuaries requires consideration of the 3-D
problem. Another strong argument to apply numerical modelling is that, to the best of the authors
knowledge, a theory of TIS transport for intertidal
flats (undergoing drying and flooding) has not yet
been developed. This seems not an easy theoretical
issue, and numerical simulations can be regarded as
a first step in this direction.
The second objective of this paper is to deepen the
understanding of the tidal response in the East
Frisian Wadden Sea to external forcing as outlined
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in our previous papers (Stanev et al., 2003b,
hereafter SWBBF and Stanev et al., 2003a, hereafter
SFW). For this purpose, Acoustic Doppler Current
Profiler (ADCP) data across channel sections are
analysed using empirical orthogonal functions
(EOFs). With this information, we can identify the
dominant temporal and spatial pattern in the tidal
response. An analysis of some fundamental parameters and non-dimensional numbers will also be
presented allowing to derive the basic dynamical
controls in the area. The tidal response is then
further analysed with the help of numerical simulations. We show in the paper that the dynamics are
not simply another manifestation of the well known
gravitationally dominated circulation in estuaries.
Open questions like the competition between TIS
drift and gravitational circulation and the interplay
between shallow depths and movable boundaries
(drying and flooding of tidal flats) are fundamental
to our area of interest and will be addressed in
this paper.
Dyer (1988) pointed out that the current velocities
produce a larger (landward) volume transport near
high water, whereas the transport near low water is
smaller due to the smaller cross-sectional area of the
inlet (see also Dyer, 1997). This concept is very
important in the present study because in the East
Frisian Wadden Sea the cross-sectional area of the
inlets below the low water level is comparable to the
cross-sectional area between low and high water
levels. The effects of changing cross-sectional areas
are discussed in Section 3.2. Furthermore, we will
illustrate that a specific vertical structure of the
transport is established in the Wadden Sea, which is
well documented both in observations and numerical simulations. The vertical transport cell could
play a major role in the processes responsible for
sorting and redistributing tracers and sediments in
the tidal basins. This important possibility is one of
the main motivations to put the emphasis in this
paper on the vertical overturning. The sediment
response to tidal forcing is addressed elsewhere
(Stanev et al., 2006, 2007). The paper is structured
as follows: the theory is presented in Section 2, the
results of observations are discussed in Section 3,
the numerical model and the simulated data are
briefly described in Section 4, and this is followed by
with theory is presented in Sections 2 and 3, the
results of observations are discussed in Section 4,
the numerical model and the simulated data are
described in Section 5, and this is followed by
general conclusions.
801
2. Oceanographic characteristics of the East Frisian
Wadden Sea
Most of the analyses in this paper, in particular
the ones based on observations, are for the tidal
basin of Spiekeroog Island. In this extremely
shallow area, which is representative for most of
the tidal basins in the East Frisian Wadden Sea, the
longitudinal momentum balance is between pressure
gradient and friction. The horizontal dimension of
the basin is 8 20 km. The red colours of the
near-coastal zone (Fig. 1) indicate areas which are
prone to drying and flooding.
The tidal prism DV ¼ V h V l , where V h and V l
are the volumes for high and low waters, respectively, amounts to 145 106 m3 at spring tide,
which substantially exceeds V l 40 106 m3 . In
similar settings, the export and import of waters
through the inlets may have different characteristic
times, depending on the ratio between the maximum
storage capacity of the basin and the volume of
water permanently stored in it.
Direct observations (Flemming and Ziegler,
1995; Davis and Flemming, 1995; Nyandwi and
Flemming, 1995) and numerical modelling
(SWBBF) contributed to a better understanding of
the dynamics of the East Frisian Wadden Sea. It has
been demonstrated that the major dynamical control is exerted by the narrow inlets where velocities
reach magnitudes of 1 m s1 . The velocity profiles
reveal a clear friction dependence (i.e. a logarithmic
profile in the bottom layer). In SFW is was shown
that the asymmetry of the tidal signal in the deep
channels is to a large extent governed by the
hypsometry of the respective tidal basin.
In a number of studies (among them Hansen and
Rattray, 1966; Fischer et al., 1979; Prandle, 1985;
Jay and Smith, 1988; Garvine, 1995) basic dynamical controls of estuaries have been studied and
classification schemes have been proposed based on
several important parameters such as: Coriolis
parameter, inlet width, depth, tidal amplitude,
vertical and horizontal salinity difference, and
velocity. In the following, we will estimate the basic
non-dimensional numbers for the back-barrier basin
of Spiekeroog Island and its main channel. The
geometrical characteristics are: depth hc 10 m,
width W c 3 km, length from the mouth to where
its depth remains larger than 4 m Lc 20 km (‘‘c’’
stands for ‘‘channel’’). The channel is considered as
narrow not only because W c 5Lc (here we have to
keep in mind that the width of the deep part of the
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E.V. Stanev et al. / Continental Shelf Research 27 (2007) 798–831
802
channel is about half of W c ), but also because W c is
much smaller than the tidal excursions of 20 km,
as shown by observations with shallow-depth
Lagrangian drifters in the back-barrier area of
Spiekeroog Island carried out by the University of
Oldenburg in the frame of the research project
WATT (Oliver Punken, personal communication).
The most important characteristic of the East
Frisian Wadden Sea is the large external Froude
number
a
d¼ ,
(8)
hc
where a is the sea-level amplitude. This number
ranges from 0.15 in the deep channels to values
larger than 5 in the areas undergoing drying and
flooding.
For the barotropic motion the Rossby radius
pffiffiffiffiffiffiffi
ghc
RE ¼
(9)
f
is 100 km (here, the index ‘‘E’’ stands for external
motion, g is the acceleration due to gravity, f is the
Coriolis parameter).
The ratio of the estuary width W c to the Rossby
radius is often called the Kelvin number
KE ¼
Wc
.
RE
(10)
For the Spiekeroog channel (Otzumer Balje, see Fig. 1)
K E ¼ 3 102 , proving that for the barotropic
motion the horizontal scale RE at which the earth’s
rotation starts to be important is much larger than the
geometric scale W c . In such systems, Coriolis acceleration does not have adequate space to act.
The horizontal aspect ratio (Ianniello, 1977)
dE ¼
sW c
sW c
¼ pffiffiffiffiffiffiffi
fRE
ghc
(11)
measures the importance of the cross-channel
velocity in the cross-channel momentum balance.
Taking for the frequency of the M2 tide s ¼ 1:4 104 s1 yields dE ¼ 4:2 102 , allowing to neglect
in the theoretical studies the y-momentum equation
and cross-channel velocity in the x-momentum
equation.
Observations in the Wadden Sea show that the
contribution of temperature to the vertical density
stratification is usually small, although situations
are also possible when extreme winter cooling or
summer warming might dominate the stratification.
For the estimates below, we will consider the
Wadden Sea as dominated by a ROFI regime
(Regions of Freshwater Influence, Simpson, 1997),
i.e. the fresh water buoyancy flux from the coast
exceeds the seasonal input of buoyancy over the
shelf. We use station data measured continuously in
the Otzumer Balje during the last few years (R.
Reuter, personal communication). For the example
given below, we take observations of mid-January
2004 when air temperature was relatively warm, but
at the same time the temperature difference between
surface and bottom waters was negligible. During
this period, the salinity difference measured by
sensors at 3.5 and 9 m above the ground was
0:521, thus we will take the value DS ¼ 1 as the
mean salinity difference between the surface and the
bottom. Using a linear equation of state with a
coefficient of salinity expansion b ¼ 0:78 103 (per mil)1 yields for the reduced gravity
(negative buoyancy)
g0 ¼ g
Dr
gbDS ¼ 7:8 103 m2 s1 .
r
(12)
The corresponding Brunt–Väisälä frequency is
rffiffiffiffiffi
g0
N¼
¼ 2:8 102 s1 ,
(13)
hc
which is much higher than the M2 and inertial
frequencies. With the above values, the baroclinic
Rossby radius
RD ¼
N
hc ¼ 2:5 103 m
f
(14)
approximately equals the width of the strait, and the
corresponding baroclinic Kelvin number K D ¼ W c =
R 1. Because the Otzumer Balje is only about one
internal Rossby radius wide it cannot fully accommodate features found in wide estuaries (substantially affected by the rotation of earth): shelf waves,
baroclinic instabilities and eddies. For comparison,
we remind that in Delaware Bay, giving one example
of a gravitationally driven circulation (RD 5 km and
W c 10245 km), K D is substantially larger than
unity, thereby explaining the larger contribution of
the Coriolis force and the more pronounced transversal circulation. Nevertheless, 3-D baroclinic models are needed in order to appropriately address
dynamics in the East Frisian Wadden Sea.
Continuous observations in the Otzumer Balje
indicate that salinity in the surface and deep layers is
almost in phase; moreover, the amplitudes at 3.5
and 9 m above the bottom are almost equal. This is
an illustration that the circulation is not typically
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E.V. Stanev et al. / Continental Shelf Research 27 (2007) 798–831
estuarine (in typical estuaries, water flows seaward
in the surface layer and landward in the deep layer).
Based on this evidence, one can assume that tidal
straining is not as pronounced as in some other
ROFI-s (Simpson, 1997), which gives an indirect
proof that the Wadden Sea is well mixed.
We can estimate the vertical eddy viscosity AM
V
using the formula of Csanady (1976) and Garrett
and Loder (1981)
AM
V ¼
u2
F ðRiÞ,
200f
(15)
where u is the average friction velocity. Expressing
the latter as
1=2
u ¼ C d U rms ,
(16)
where U rms ¼ 1 m s1 is the rms tidal current in the
channel of Spiekeroog and C d ¼ 2:5 103 is the
friction parameter, results in u ¼ 5 102 m s1 .
This number supports the estimates for friction
velocity produced by the numerical simulations of
SWBBF. The function F ðRiÞ is given by
F ðRiÞ ¼ ð1 þ 7RiÞ1=4 ,
where
2 N
RD =f 2
g0 hc
Ri ¼
¼
¼ 2
U rms =hc
U rms
U rms
AM
V
11,
fh2c
and the Ekman depth
pffiffiffiffiffiffi
D ¼ 2E hc 50 m,
Rie ¼ g0
(19)
(20)
i.e. even in the deepest channel ðhc 10 mÞ the entire
water column is dominated by friction.
As demonstrated by Wong (1994) and Kasai et al.
(2000), in cases dominated by gravitational circulation and large Ekman number, inflows are
observed in the deep channels, while outflow is over
the flats. This situation is in agreement with the
Qf
W c U 3rms
,
(21)
which gives the ratio of the gain of potential energy
due to the fresh water discharge to the mixing power
of tidal currents, and the densimetric Froude
number
Fre ¼
is the Richardson number. With the above parameters Ri ¼ 7:8 102 . Obviously, this is a very
small number, compared to what is known for other
estuaries (e.g. 1.2 for the Ise Bay, Kasai et al., 2000)
and the correction factor F ðRiÞ in Eq. (15)
approaches unity.
Estimates for AM
V from Eq. (15) of 1:25 101 m2 s1 support the values obtained by SWBBF
from numerical simulations in the same area. The
Ekman number
E¼
concept based on a number of observations (see
Valle-Levinson and Lwiza, 1995 and the citations
therein) that in most estuaries mean Eulerian
outflow is over shoals and inflow over the deep
areas. This is exactly opposite to what is observed in
the East Frisian Wadden Sea, thus revealing
different dynamical controls. Some candidates to
explain this difference are the vast areas of drying
and flooding, weak gravitational circulation, and
TIS transport opposing ‘‘classical’’ estuarine circulation (detailed estimates are given further in this
paper). We remind that Wong (1994) and Kasai et
al. (2000) neglected TIS transport, instead focusing
on the density-induced gravitational circulation.
The estuarine Richardson number
(17)
(18)
803
Qf
pffiffiffiffiffiffiffiffi
W c hc g0 hc
(22)
were used by Fischer et al. (1979) to classify
estuarine dynamics. With a fresh water flux
Qf ¼ 102 m3 s1 , which is a rather large value for
the area of the Spiekeroog Back-Barrier basin
(internal report of Niedersächsischer Landesbetrieb
für Wasserwirtschaft und Küstenschutz, Aurich,
Germany), the above non-dimensional numbers are
estimated as Rie 2:5 104 and Fre 1:3 103 ,
correspondingly. The small Richardson number is
far below the transition region (0.08, Fischer et al.,
1979) between mixed and stratified estuaries. We
thus deal with a well mixed estuary, which is
explained by the fact that the ratio of the tidal
prism of 140 106 m3 (SWBBF) and fluvial
discharge is at least of the order of 30. The
stratifying influence of the buoyancy flux is thus
relatively weak compared to the stirring effects of
mechanical forcing, and density effects are negligible (according to the classification of Fischer et al.,
1979). In such systems, one can expect gravitational
circulation to be too weak to compete sufficiently
with the TIS drift, i.e. the tidally induced mean
residual flow dominates over the gravitational
circulation.
As a measure of the importance of the horizontal
density gradient driving an along-channel current,
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804
Ianniello (1977) proposed the number ðh2c =a2 Þ
ðDx r=rÞ, which for a mean depth of a channel of
4.5 m, a1:5 m and an along-channel density
difference Dx r ¼ 4 kg m3 is 3:6 101 . This
small value implies that in our case the density
gradient term is small compared to the nonlinear terms.
Finally, we will provide an estimate for the
contribution of the terms in the momentum
equation. For the time evolution and the non-linear
terms, we take the M2 tidal period and the radius of
curvature of a channel as the characteristic temporal
and spatial scales. We also assume that the velocity
changes from zero at the bottom to U rms at the middepth of channel, which does not overestimate the
weight of the friction term. With the parameters
used above, the different acceleration terms are
aevol ¼ 0:2104 , anonlinear ¼ 0:5 104 , aCor ¼ 104 ,
and africt ¼ 50 104 . Obviously, like in many
other shallow areas, the main balance in the East
Frisian Wadden Sea is between pressure force and
friction, non-linear and Coriolis terms being of
secondary importance.
bay). Note that, unlike in most analyses based on
the above set of equations, we assume here that: (1)
the bay area Ab is a function of depth below the
mean sea level (z ¼ 0), and (2) the cross-sectional
area of the inlet Ac is a function of sea level height.
The first important generalization was addressed by
Green (1992) who showed that the inclusion of
sloping side walls changes the behaviour of the
Helmholtz oscillator which becomes non-linear.
Later Maas (1997) and Maas and Doelman (2002)
extended the theory of Green towards the response
of semi-enclosed bays driven by tidal oscillations.
The relevance of that generalization to the area
investigated in this paper was addressed by SFW. In
the present study, we will focus on the second term
on the right-hand side of Eq. (26), which cannot be
neglected if zhc (large external Froude number).
This means that we can expect effects which are
characteristic of finite-amplitude waves.
For the standard case of a Helmholtz oscillator
(i.e. basin area and cross-sectional area of the inlet
are constant and friction is neglected), the continuity Eq. (24) has the form
3. TIS transport in the Wadden Sea
u¼
3.1. Oscillations in tidal bays with shallow inlets.
Basic equations
and its substitution in Eq. (23) leads to a secondorder linear equation for z
The dominating balances in the tidal basins under
consideration allow to describe the oscillations as a
simple inlet–bay system (Fig. 3) by the following
system of equations which consists of a momentum
and a continuity equation:
d2 z
¼ o2H ðz0 zÞ,
dt2
du
g
¼ ðz zÞ C d juju,
dt Lc 0
0
(26)
is the cross-sectional area of the inlet, A0c ¼ W c hc
and Ab ðzÞ is the area of the bay (index ‘‘b’’ stays for
(27)
(28)
where
o2H ¼
(23)
dV
¼ Ac u,
(24)
dt
where u is the current velocity through the inlet, z is
the sea-level elevation in the bay, z0 is the sea level
in the open ocean,
Z z
Ab ðzÞ dz
(25)
V¼
is the excess volume,
z
0
Ac ¼ Ac 1 þ
hc
A0b dz
A0c dt
g A0c
Lc A0b
(29)
is the pumping frequency. The corresponding
characteristic length scale LH can be obtained from
o2H
¼ ghc
k2H
(30)
and
L2H ¼
1
A0b AL
¼
,
A0c
k2H
(31)
where AL ¼ Lc hc is the area of the along-inlet
section. With the parameters given in Section 2 and
taking the area of Spiekeroog basin at mean sea
level as A0b ¼ 42 km2 , this yields LH 11 kmLc =2.
Eq. (30) is an analogue to the dispersion equation
for shallow water waves.
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This analogy between oscillations in tidal basins
and shallow water waves is instructive for an
explanation of the basic physics of the TIS
transport. We recall that the classical example given
in the Introduction is not directly applicable
because the oscillations in a shallow sea are not
orbital but simply back and forth. For a simple
progressive wave
z ¼ A cosðkx otÞ þ B sinðkx otÞ
with
rffiffiffi
g
u¼
z,
h
(32)
(33)
the water transport averaged over a period T is
rffiffiffi
Z
1 T
1 g 2
ðz þ hÞu dt ¼
(34)
ðA þ B2 Þ.
U¼
T 0
2 h
Eq. (34) cannot be applied to the tidal basin shown
in Fig. 3 where the net transport through the inlet is
zero. The discrepancy can be removed either by
establishment of flood–ebb asymmetries or a
vertical circulation cell, or both (see also Jay, 1990).
3.2. Asymmetrical tidal response controlled by
inlet depth
Here we want to demonstrate the basic physics
caused by sea-level variability for the situation of a
shallow inlet and a tidal basin with a simple
topography. To achieve this, we simplify Eqs. (23)
and (24) under the assumption that the basin area
increases linearly with decreasing depth:
z
Ab ¼ A0b 1 þ
,
(35)
zs
where zs accounts for the slope of the bottom. In
this case, the excess volume is
z
V ¼ A0b 1 þ
z
(36)
2zs
and using Eq. (26) we can re-write Eq. (24) as
A0 1 þ z=zs dz
.
(37)
u ¼ b0
Ac 1 þ z=hc dt
It is clear that the non-linear effects of basin area
and cross-sectional area of the inlet oppose each
other. However, this effect is not symmetrical
because zs and hc might differ. Before analyzing
the consequences of variable basin areas and inlet
cross-sections we will assume here that zozs and
zohc . The first inequality implies that the basin
805
cannot fall completely dry at low water (the basin
area, Eq. (35) is always positive), the second
inequality implies that the connection between the
tidal basin and the open sea never falls dry either
(the cross-sectional area is also positive). These
inequalities correspond to the real case and reduce the variety of possible responses described in
Eq. (37).
In order to illustrate the effects of area control,
we suppose that the sea level in the tidal basin
approximately follows the sea level in the open
ocean. This assumption was analysed in more detail
by SFW who showed its relevance to the case of the
East Frisian Wadden Sea, which is a friction
dominated area. If we suppose that the forcing
signal (a semi-diurnal lunar tide M2 ) is almost
harmonic, i.e. z ¼ a sin ot t, we can easily compute
the tidal response from Eq. (37) as expressed by the
velocity. The results are shown in Fig. 4 for three
combinations of parameters. In all cases, we use
a ¼ 1:5 m. For parameters zs and hc we take either
2:5 m or 10 m. The first case (zs ¼ 2:5 m and
hc ¼ 10 m) almost repeats the simulations reported by SFW. The black curve in Fig. 4b demonstrates the asymmetry found by SFW in the case of
a linear dependence of the basin area with depth. It
was shown in this study that in the case Ac ¼ A0c ¼
const the non-linear hypsometric control tends
to create a tidal asymmetry in the transport
such that the time required for the transition
between maximum ebb current and maximum
flood current exceeds the remaining part of the
tidal period.
If we assume Ab const, which is ensured if zs ¼
10 m and take hc ¼ 2:5 m, i.e. a very shallow inlet
whose depth is close to the tidal amplitude, we
obtain the red curve in Fig. 4b. In this case, it takes
much less time for the transition between ebb and
flood than for the inverse process. This kind of
response is due to the fact that an equal change in
sea level in the open sea induces different velocities
in the inlet. For equal dz=dt the velocities at low
water are larger than at high water. Adding to these
(volumetrically induced) asymmetries, a phase lag
due to friction can create a residual transport
because of the non-zero correlation between velocity and sea level.
The results in the mixed case when both Ab and
Ac depend strongly on sea level ðzs ¼ hc ¼ 2:5 mÞ are
represented by the red line in Fig. 4a. Obviously, for
the chosen set of parameters the tidal response
becomes symmetrical.
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E.V. Stanev et al. / Continental Shelf Research 27 (2007) 798–831
a
b
frequency of 1:2 MHz set at a vertical resolution
of 25 cm. The length of the cross-section was about
800 m and it took less than 5 min for the boat to
cross the channel. Since this duration is much
shorter than the tidal period, every cross-section can
be considered as being quasi-instantaneous. The
along-track resolution of 15 m is quite sufficient to
allow the study of horizontal patterns of estuarine
transports. More details about these surveys are
reported in Santamarina Cuneo and Flemming
(2000).
To give a general idea of the data quality of the
ADCP observations during ebb and flood, two
representative cross-sections are presented in Fig. 5.
At the chosen bin size, not only the general patterns
of the currents but also the boundary layers are
resolved quite well. The gradients in the bottom
layer (not shown here) reveal a strong shear,
indicating the existence of a logarithmic layer
(Davis and Flemming, 1991). Unfortunately, not
all cross-sections showed the same good quality as
in Fig. 5, and lots of data were eliminated to avoid
Fig. 4. Temporal asymmetries of the tidal response. (a) Sea level
and inlet current in the case of compensation of topographic
controls, (b) currents in the case of hypsometric control (black
curve) and control by the section area of inlet (red curve). The
results are shown in non-dimensional form by scaling the sea
level by 1 m and velocities by aot A0b =A0c . Positive values mean
landward current.
4. Observations
4.1. Description of observations
In recent years, ADCP data have been widely
used in oceanography and contributed substantially
to the understanding of the basic dynamics of the
coastal ocean (Valle-Levinson and Lwiza, 1995;
Münchow, 1998). However, to the authors knowledge, in-depth analyses of such data are still lacking
for the East Frisian Wadden Sea, in particular
concerning temporal and spatial patterns. In the
following, we analyse measurements of velocity
profiles along a cross-section in the Otzumer Inlet
(Fig. 1), which have been carried out by the
Senckenberg Institute in Wilhelmshaven, Germany,
since 1995. The surveys were carried out several
times a year and covered time periods of about one
tidal period in each case using ADCP with a
Fig. 5. Along-channel velocities (positive landwards) between
Spiekeroog and Langeoog Islands on the 19th of August 1997,
(a) at 7:21 (b) 13:46. The position of cross-sections is given by the
section line 1 in Fig. 1. The x and y axes give the number of pixel
and depth, correspondingly.
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E.V. Stanev et al. / Continental Shelf Research 27 (2007) 798–831
problems with filling gaps, interpolation, etc. The
good quality data were further corrected for the
deviation of the course of the boat from the shortest
line between start and end of the survey profile. The
sections taken at time intervals shorter than 20 min
were averaged. The data were then rotated from the
earth’s coordinate frame to a frame representing
along and across channel directions, interpolated
onto a regular grid of 15 m 0:25 m horizontally
and vertically, and finally interpolated onto regular
time intervals.
In the following, only the data series obtained
on the 19th of August, 1997 and 21st of April, 1998
are discussed because they have a relatively good
time coverage and are characteristic for spring and
neap tide conditions, respectively. In Fig. 6, we
show a sequence of six cross-sections illustrating
the variability-pattern of the along-channel component of velocity in April 1998. The following
characteristics are noteworthy. The maximum ebb
velocity slightly exceeds the maximum flood velocity. Although the boundary layers are only
resolved by a few bins, their structure seems
hydrodynamically consistent (see e.g. Fig. 6e, f).
The core of the ebb current is displaced towards the
steep bank, whereas the flood currents (Fig. 6c, d)
also spread onto the shallow part of the channel.
From there the flood current constantly shifts
towards the deeper part of the channel to form a
sloping core (Fig. 6f) observed in all cases with
different tidal amplitudes.
During most of the tidal cycle, the net transport
in the shallow part of the channel is generally
landward. When this area is dominated by seaward transport during ebb, this transport is much
smaller than in the deep part. This is convincingly
demonstrated in Fig. 7 where the currents in the
deepest part of the channel udeep and on the shallow
part ushallow are shown. The depth of the two
locations is the same (15 m above the zero
depth in Fig. 6) and chosen such that the shallow
location never falls dry under spring tide. The two
curves in Fig. 7 are very different, particularly
during ebb. Obviously, the transport in the channel
is dominated not only by a pronounced vertical
shear, but also by lateral gradients and large
differences in the temporal variability (phase
lags or even different modal structures). We will
come to this peculiarity later when we demonstrate
that shallow areas are characterized by flood
dominated currents, whereas deep channels are
ebb dominated.
807
4.2. EOF analysis
The last result in the previous subsection motivated us to find the dominant temporal and spatial
patterns of variability of the transports. A statistical
decomposition of the time–space signal with the
help of the EOF analysis allows to find the most
important spatial patterns and their time evolution
(see e.g. von Storch and Zwiers, 1999).
Although the application of the EOF technique to
analyse tidal data is relatively new, there are already
a number of examples in the field of tidal dynamics
(Münchow, 1998) and sediment dynamics (McManus and Prandle, 1997). Münchow (1998) gives an
example to what extent EOF modes can be used to
predict tidal responses.
The statistically identified spatial patterns and
their time evolution (the so-called principle components), as shown in Figs. 8–11, sometimes allow to
identify a direct link between the time evolution of a
pattern with a particular physical process. In the
two cases analysed in this paper, the tidal forcing,
which is rather simple (approximately one mode
sinusoidal), leads to the establishment of a relatively
simple spatial structure of the along-inlet transport.
As a result of the clear and simple structure of the
signals, the first EOF describes almost 95% of the
variance.
EOF patterns are not necessarily (or not always)
clearly connected with specific physical characteristics of the analysed fields. However, in our case,
(see Fig. 8a,b) the dominant pattern shows the
landward/seaward current in the deep channel, the
boundary layers along the walls of the channel and
above the bottom, as well as the shallow western
part where the correlation with the currents in the
channel interior is weak (see also Fig. 7b).
The first principal component (PC-1) is a onemodal curve illustrating the well known time
evolution of a tidally driven transport in this area
(see e.g. SWBBF). The major asymmetry is revealed
by the longer time T e!f needed for the transition
between maximum ebb and maximum flood as
compared to the time needed for the inverse
transition T f !e (T e!f : T f !e ¼ 2: 1). The inflection
in the curve occurs approximately at the time of low
water and proves that PC-1 captures the major
signals of the tidal response, which is associated
with the asymmetries created by the hypsometric
control (see SFW).
Unlike the EOF-1, which almost repeats the
profile of the deep channel, the EOF-2 shows a
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808
a
b
c
d
e
f
Fig. 6. A sequence of along-channel velocity snapshots plotted with equidistant time interval of
section is given by the section line 1 in Fig. 1.
more complicated structure with a correlation
between areas where the ebb current (steep channel
wall) and flood current (shallow bank) appear first.
1
6
tidal period. The position of cross-
The comparison between the two patterns reveals
that EOF-2 does not show coastal boundary layertype structures, but rather a contrast between the
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a
b
Fig. 7. Temporal variability of along-channel velocity udeep (full
squares) and ushallow (empty squares) during spring (a) and neap
(b) tide. See also the notations in text. Positive values correspond
to landward currents.
shallow and deep parts of the channel (the
extremum on the tidal flats is in the surface layer).
Furthermore, no pronounced vertical structure is
observed, possibly indicating that the density does
not separate the water column into layers, which is
consistent with the classification of the East Frisian
Wadden Sea as a well mixed estuary.
The two-modal PC-2 curve suggests that this
variability is associated with the non-linear control.
This control depends on the square of the velocity.
It is important to note here that while the EOF-2
shows similar patterns (Fig. 8c, d), PC-2 shows a
large difference in the appearance of the two-modal
curves (Fig. 9). This could be due to the strong
dependence of the temporal variability on the
magnitude of the oscillations, a typical behaviour
in non-linear systems. One possible indication
supporting the above conclusion is that the PC-2
is almost identical in all neap (low amplitude) tide
809
situations (not shown), but is rather different at
spring tide. In the latter case, the deeper minimum
(observed in the neap-tide-curve) deepens even
more, while the second minimum almost flattens
(notice the shift in phases caused by the different
initial time with respect to the tide in the two
surveys).
Higher EOF-s show quite different patterns
during neap and spring tide conditions, their PC-s
are also being very different, and they are probably
due to noise in the data rather than to clear physical
processes. We will not discuss these patterns here
because their contribution to the total variance is
negligible.
The EOF analysis of across-channel velocity
(Figs. 10 and 11) gives a further understanding of
the dominating dynamics, in particular the secondary (transversal) circulation. Although its energy is
100 times smaller than the energy of alongchannel circulation, the signals are very clear.
EOF-1 shows characteristic patterns in the deep
part of the channel penetrating from the surface to
the bottom. In both coastal areas (left and right
bank) the current is much weaker. During ebb, the
sea level is higher along the right bank (looking in
the direction of the outflowing current), which is
due to the velocity convergence. In this case, the
vertical circulation is counter-clockwise. The opposite situation develops during flood when the
circulation cell changes the sense of rotation to
clockwise (looking from the coast to the open sea).
The second EOF, in particular in the more
energetic spring tide case, reveals two circulation
cells. They can be identified in Fig. 10c by the two
blue-coloured layers (at the surface and bottom)
separated by the red-coloured intermediate layer.
It becomes clear that there is a high level of
correlation between surface velocities along the
eastern (right) bank and at the end of the shallow
bottom along the western (left) bank. The PC curve
of the secondary circulation is again one-modal,
which is very asymmetric (Fig. 11c). During the
neap tide EOF-2 is very noisy.
4.3. Vertical shear of mean transport in tidal inlets
The up-and-down motion of the sea level and the
associated transport through the tidal inlets make
the observations presented in the preceding sections
a good source of data to test the relevance of the
TIS transport for the dynamics of tidal basins. For
the analyses below, we only used section data where
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810
a
b
c
d
Fig. 8. The first (a) and (b) and second (c) and (d) EOF of along-channel velocity corresponding to spring (left plots) and neap (plots on
the right) conditions. The analysis is done for locations below 14:5 m (see also Fig. 6) to ensure continuous observations throughout the
tidal cycle.
the ADCP profiles are of good quality for the entire
section. This reduces the amount of data, but avoids
interpolations which could produce misleading
results. Therefore, in the neap and spring tide cases
the entire tidal period is sampled by only 10–11
sections. In Fig. 12a, b we show the dominant
characteristics describing the dynamics of tidal
inlets. Although the vertical resolution of the
measurements with the ADCP is not better than
0:25 m, the distance between the boat and bottom
nevertheless captures the variability of the sea
surface (the upper left panel). The phase difference
between different curves in Fig. 12a and b results
from the fact that the spring and neap tide surveys
were initiated during different phases of the tide.
The tidal ranges calculated as the maximum
difference between the thickness of the water
column at high and low water are r ¼ 3:5 and 2 m,
respectively.
We can consider the tidal channel as being
composed of three parts: (1) a deep part, which
never falls dry, (2) a rectangular section extending
from the top of the deep part (15 m above the
deepest part of the channel) to the sea surface (the
slope of sea level can lead to a small deviation from
a rectangular shape) and (3) a shallow part, which is
the remaining triangular part of the channel.
Because the slope of the sea level along the channel
is relatively small, the area of the second (rectangular) section is a linear function of the mean sea
level. This is not the case for the shallow part of the
channel where the section area shows a clear nonlinear dependence on the sea level (Fig. 12a,b
second panels).
In the following, we analyse the transport
through the three sections defined above. The total
transport integrated over the entire channel (the
third panels in Fig. 12a,b) follow the course of PC-1
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a
b
c
d
811
Fig. 9. The first (a) and (b) and second (c) and (d) PC of along-channel velocity corresponding to spring (left plots) and neap (plots on the
right) conditions. See the comment in Fig. 8.
(Fig. 9a, b) and reveals the well known asymmetries
in tidal basins dominated by hypsometric control
(SFW). During neap tide, the area of the shallow
section is small and the transports there (fifth panels
in Fig. 12a,b) are much smaller than those during
spring tide. Not only are the transport maxima
higher there but in addition, the duration of the
flood currents is longer. Obviously, this area gives a
positive net contribution to the increasing volume of
the tidal basin. However, in the two cases displayed
in Fig. 12, the transport through the shallow section
is negligible compared to the one in the interior of
the channel (third panels in Fig. 12a,b). Because of
this, we will discuss the differences between transport patterns in the third and fourth panels of
Fig. 12a,b in greater detail below. The total
transport reaches 4 103 and 5:2 103 m3 s1 at
the considered neap and spring tide situations,
respectively. The contribution of the transport
through the upper section in the two cases to the
total transport is 10% and 30%, respectively.
One fundamental difference between the upper
and lower layer transport becomes clear from the
following example. The net water mass exchanged
between the ocean and the tidal basin in parts (1)
and (2) of the tidal channel during one spring tide
period is 0:8 106 m3 . This small transport in the
deep part of the channel compensates the small
positive contribution of the shallow area. The
transports in parts (1) and (2) of the tidal channel
are much larger (the net transport in part (1) reaches
9 106 m3 , i.e. about 6% of the tidal prism) but
oppose each other. This vertical asymmetry decreases strongly during neap tide in April 1988
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812
a
b
c
d
Fig. 10. The first (a) and (b) and second (c) and (d) EOF of across-channel velocity corresponding to spring (left plots) and neap (plots on
the right) conditions. The analysis is done for locations below 14:5 m (see also Fig. 6 to ensure continuous observations throughout the
tidal cycle).
( 1:61 106 m3 ) but is still in the same direction:
i.e. a landward drift in the upper layer and a
compensating outflow in the deeper part of the
channel (recall that the tidal prism amounts to
100 106 and 140 106 m3 during neap and
spring tide, respectively). The above transport
values indicate a vertical overturning in the tidal
basin. This issue is subject of the reminder of the
paper where we use numerical simulations to
generate a more complete data set in order to study
the vertical overturning.
Ending this observational part, we will refer to
Münchow et al. (1992) who compared tidal and
Stokes mean transport from observations in the
vicinity of Delaware Bay. They found a landward
transport of about 1:7 103 m3 s1 , which was
balanced by a seaward Eulerian mean flow. Like
in our case, both transports are much smaller
than the tidal transport of about 1:5 105 m3 s1 .
However, important is that both in the Wadden Sea
and in the Delaware Bay these transports contribute
to the vertical overturning.
5. The numerical model and some results of the
simulations
5.1. Description of the model
Theories addressing the tidal response either in
narrow channels or in shallow areas are only useful
for the general understanding of dominating processes, but are no longer appropriate for the East
Frisian Wadden Sea. Because the area of our
interest includes narrow channels and wide (mostly
neutrally stratified) tidal flats, the problem seems to
be fully 3-D and therefore requires realistic simulations with 3-D models. The present study uses the
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a
b
c
d
813
Fig. 11. The first (a) and (b) and second (c) and (d) PC of across-channel velocity corresponding to spring (left plots) and neap (plots on
the right) conditions. See the comment in Fig. 8.
GETM which is a 3-D primitive equation numerical
model (Burchard and Bolding, 2002).
The governing in Cartesian coordinates read:
2
qu
qðu Þ qðuvÞ
qðuwÞ
þg
þ
fv þ
qt
qx
qy
qz
1 qp q
qu
2
þ
AM
¼
ð38Þ
þ AM
V
H r u,
r0 qx qz
qz
qv
qðuvÞ qðv2 Þ
qðvwÞ
þg
þ
þ fu þ
qt
qx
qy
qz
1 qp q
qv
2
þ
AM
¼
þ AM
V
H r v,
r0 qy qz
qz
qu qv qw
þ þ
¼ 0,
qx qy qz
ð39Þ
(40)
qðT; SÞ
qðT; SÞ
qðT; SÞ
qðT; SÞ
þu
þv
þw
qt
qx
qy
qz
q
qðT;
SÞ
AðT;SÞ
¼
r2 ðT; SÞ,
þ AðT;SÞ
V
H
qz
qz
ð41Þ
where AM
V ðk; ; gÞ is a generalized form of the vertical
ðT;SÞ
ðT;SÞ
eddy viscosity coefficient, AV
and AH
are
vertical and horizontal eddy diffusivity coefficients,
correspondingly, k the turbulent kinetic energy
(TKE) per unit mass, and the eddy dissipation
rate (EDR) due to viscosity. The lateral eddy
ðT;SÞ
viscosity AM
¼ 10 m2 s1 .
H ðx; yÞ ¼ AH
In GETM, the process of drying and flooding is
incorporated in the hydrodynamical equations
through a parameter g which equals unity in
regions where a critical water depth Dcrit is exceeded
and which approaches zero when the thickness of
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814
a
b
Fig. 12. Temporal variability of the sea level (first panel), total transport (third panel), upper layer transport (fourth panel) and shallow
area transport (fifth panel). The correlation between sea level height and shallow area is shown in the second panel; (a) spring tide (on the
left), (b) neap tide (on the right). Positive values correspond to landward transport.
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the water column D ¼ H þ z tends to a minimum
value Dmin :
D Dmin
g ¼ min 1;
,
(42)
Dcrit Dmin
where H is the local depth (constant in time), taken
as the bottom depth below mean sea level in the
model area. Actually, the ‘‘drying corrector’’
reduces the influence of some terms in the momentum equations in situations of very thin fluid
coverage on the intertidal flats. The minimum
allowable thickness Dmin of the water column is
2 cm and the critical thickness Dcrit is 10 cm
(Burchard and Bolding, 2002, and SWBBF). For a
water depth greater than 10 cm ðDXDcrit Þ; g ¼ 1,
and the full physics are included. In the range
between critical and minimal thickness (between 10
and 2 cm), the model physics are gradually switched
towards friction domination, i.e. by reducing the
effects of horizontal advection and Coriolis acceleration in Eqs. (38) and (39) and varying the vertical
eddy viscosity coefficient AM
V according to
AM
V ¼ nt þ ð1 gÞng ,
(43)
2
where ng ¼ 10 m2 s2 is a constant background
viscosity. The eddy viscosity nt is obtained from the
relation
k2
,
(44)
where cm ¼ 0:56 (see, e.g. Rodi, 1980). The drying
and flooding algorithm is volume and mass conserving (Burchard et al., 2004).
In GETM, the momentum equations (38) and
(39) and the continuity equation (40) are supplemented by a pair of equations describing the time
evolution of the TKE and EDR (k2 turbulence
model).
Close to the bed, TKE and EDR are governed by
the law of the wall with
nt ¼ c4m
ðub Þ3
,
(45)
1=2
kðz0 þ z0 Þ
cm
pffiffiffiffiffiffiffiffiffiffi
where ub ¼ tb =r is the friction velocity at the sea
floor, tb ¼ rnt ðqu=qzÞ is the bed shear-stress, z0 is the
distance from the bed, z0 is the bottom roughness
length, k is the von Karman constant, and rw is the
water density. The parameter z0 , which gives a
general representation of the bottom roughness is
taken constant over the whole area (SWBBF).
Obviously, this simplification does not account for
complex bedforms (e.g. ripples), which are impork¼
ðub Þ2
;
e¼
815
tant elements of the local bedload transport.
Because in this study we do not address local
morphodynamics, but rather larger scale balances,
we avoid the introduction of additional parameterizations on small-scale topography. In the horizontal, we resolve the model domain with equidistant
steps of 200 m, the horizontal matrix including
324 88 grid-points in the zonal and meridional
directions, respectively. This horizontal resolution is
still too coarse to resolve currents in the minor
channels, which could motivate further research, in
particular, if combined with more sophisticated
parameterizations.
In the vertical, the model uses terrain-following
coordinates. The vertical discretization consists of
10 equidistant layers extending from the bottom H
to the sea surface z. Because z changes continuously
during the model integration the thickness of the
water column D becomes a function of the sea level,
the vertical discretization changes with time.
The model can be run in a 3-D barotropic mode,
as well as a fully baroclinic model. The arguments
given in Section 2 speak for using the former mode
because the East Frisian Wadden Sea is a well mixed
water body. Furthermore, working with ðT; SÞ ¼
const facilitates the understanding of processes
associated with the TIS drift only. Otherwise,
variable temperature and salinity fields would
largely ‘‘contaminate’’ the simulations. Thus, focusing on cases with no fresh water flux from the coast
enables us to illustrate more clearly the basic physics
addressed in this paper. Nevertheless, simulations
with more realistic forcing (including fresh water
flux from land) have been carried out, just to give an
idea about how different the responses to tidal
forcing in homogenous ocean are from the ones in a
more realistic (baroclinic) tidal system.
We carried out three types of simulations: (I) in
idealistic (‘‘I’’ stays for idealistic) basins, (RT) in
realistic (R) basins with realistic tidal (T) forcing,
and RTB in realistic basins with tidal and buoyancy
(B) forcing (see Table 1). Three simulations belong
to the I-class where the grid and dimensions of the
computational area are the same as in the realistic
simulations. In the first I-experiment (IS, ‘‘S’’ stays
for shallow), the depth changes linearly from 20 m
at the open boundary to 2 m at the coast (Fig. 13).
Because the prescribed tidal forcing at the open
boundary has an amplitude of only 1:5 m, there is
no flooding and drying in this simulation. In the
second simulation, the bottom is 20 m deeper (ID).
In the third I-simulation, the bottom is 2 m
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816
Table 1
Description of numerical simulations
Simulation
Topography
IS
Idealistic,
shallow
Idealistic,
deep
Idealistic,
flooding and
drying
Realistic
Realistic
ID
IFD
RT
RTB
Tidal forcing
Buoyancy
forcing
p
p
p
p
p
p
Fig. 13. Bottom topography in idealized (I-) experiments (see
legend and Table 1).
shallower than in IS, i.e. this simulation allows
flooding and drying of the near-coastal area (IFD).
The comparison between IS and ID gives an
estimate of the TIS drift, the comparison between IS
and IFD is supposed to explain the competition
between transports due to drying and flooding and
the TIS transport.
The simulation with realistic topography and
both tidal and buoyancy forcing (RTB) is used here
in order to (1) find an answer to the question about
the competing TIS and gravitational circulation and
(2) to produce improved estimates for the responses
of a realistic (baroclinic) coastal system. The details
of RTB are subject to a separate publication.
The forcing data in runs R (sea level and salinity
at the open boundaries) are generated by the
operational model of the BSH (Bundesanstalt für
Seeschifffahrt und Hydrographie). The BSH model
is a 3-D prognostic model (Dick and Sötje, 1990;
Dick et al., 2001), which operates in two versions:
(1) a coarse resolution model including the North
Sea and Baltic Sea (grid size is 10 km) and (2) a
higher resolution model of the German Bight where
the horizontal resolution is 1:8 km. The boundary
conditions at the open boundaries are formulated
using tidal values calculated from the tidal constituents of 14 partial tides. The model predicts
currents, water level, water temperature, salinity,
and ice coverage. At the sea surface, the model is
forced with meteorological and wave forecasts
(wind, atmospheric pressure, wave characteristics,
air temperature, specific humidity, and clouds),
which are provided by the German Weather Service
(Deutscher Wetterdienst, DWD).
The output of the BSH model incorporates the
main elements of the regional circulation, which is
the coastal wave associated with the well known
amphidromy at ð55:5 N; 5:5 EÞ. The tidal signal
crosses the model area from west to east in
50 min. The vertical motion of the sea level at
the open boundary and its slope provide the major
driving force for the model (more technical details
describing the forcing of our regional model are
given in SWBBF).
The simulations analysed by SWBBF focus on a
very short period, October 16–18, 2000, which is
representative for the general conditions during
spring tide and excellently illustrate the asymmetry
of transports in the vertical plane. For the aims of
the present paper, we reran these simulations for a
longer period (one month) overlapping the above
period and produced new model diagnostics (RT).
The extended runs now focus on the control of the
TIS drift on the exchange through the inlets and the
contribution of baroclinicity.
The simulations in RTB are carried out for the
same period as for RT. The observed fresh water
fluxes from the main tributaries in the region are
taken from an internal report of the Niedersächsischer Landesbetrieb für Wasserwirtschaft und
Küstenschutz, Aurich, Germany.
5.2. TIS transport in basins with movable boundaries
In all simulations with an idealistic topography
(IS, ID, IFD), the response to harmonic tidal
forcing ðM2 Þ reveals a simple structure of the
currents (Fig. 14). The surface velocity in ID is
about half of that in IS, which is purely a result of
the larger depth in ID. The comparison of the
results in IS and IFD reveals that the change of
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a
b
c
Fig. 14. Tidal response in I-experiments, (a) IS, (b) ID, (c) IFD. Meridional velocity in the middle of idealized basins (in cm s1 ) is plotted
as a function of distance from coast and time. Positive values point seawards.
topography, although relatively small, has a pronounced effect on the currents. Shallower depths
tend to increase the surface velocity. However, in
IFD an asymmetry is formed in the near-coastal
zone (compare Fig. 14a and c).
Because the major focus of this paper is to
illustrate the role of the TIS transport we interpret
our simulations as sensitivity studies aimed at
checking whether the model can resolve this
transport. Because it is difficult to define the TIS
transport in areas subject to flooding and drying, we
Table 2
TIS transport vz
Simulation
IS
ID
IFD
Transport per unit length
ð103 cm2 s1 Þ
5.2
1.4
3.0
show the results from the three simulations only
for the wet area. The overall result is presented in
Table 2 as an area integrated TIS transport vz.
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Consistent with the theory, the landward transport
decreases in the simulation with idealistic deep
topography (ID) compared to the shallow topography in IS. The effect of movable boundaries through
drying and flooding (IFD) introduces a more
complex response. If the decrease in depth provides
the major control one would expect the TIS
transport in IFD to be larger than in IS. However,
our simulations show just the opposite effect, which
demonstrates that flooding and drying competes
with the TIS transport. The differences in the
transports in these simulations with idealistic
topography are comparable to the mean transports,
indicating that the compensation of different effects
could be an important mechanism in the Wadden
Sea dynamics.
5.3. The circulation in the East Frisian Wadden Sea
and transports through the inlets
Because part of the simulations used in the
present study are the same as reported in the study
of SWBBF, the results below are presented very
briefly and with more focus on the transports
through the inlets. The circulation in the model
area is dominated by westward transport during ebb
and eastward transport during flood (Fig. 15). It has
been demonstrated by SWBBF that, while the
transport through the inlets is mainly governed by
the amplitude of the tidal oscillations (see Eq. (27)),
the along-shore circulation as well as the circulation
in the intertidal areas are governed by the spatial
properties of the forcing signal. Our simulations are
consistent with observations (e.g. Santamarina
Cuneo and Flemming, 2000) demonstrating that
maximum velocities in the Otzumer Balje exceed
1 m s1 . There is a pronounced similarity between
the simulated dynamics in the individual inlets,
particularly in the larger ones (from the Harle to the
Accumer Ee), which confirms that the dynamics in
the individual basins obey the same physical
balances.
5.4. Temporal variability of velocity profiles
The evolution of the tidal signals over time, as
well as along and across the tidal channels, is the
main subject discussed in the remainder of this
paper. We will first illustrate this process with the
help of time versus depth diagrams in Fig. 16,
plotted for the middle location in the first and last
sections in Fig. 1. The ratio between depth and tidal
amplitude for the two locations is about 10 and 3,
respectively. We can thus expect that the effects
resulting from a large external Froude number will
apply in this case.
The time versus depth plots of transport and
turbulence in Fig. 16 demonstrate that two velocity
maxima are simulated every tidal period. The first
maximum corresponds to the flood and the second
one to the ebb. During most of the time, the entire
water column shows relatively strong vertical gradients in velocity and therefore a high level of
turbulence. Only during slack water (duration of
1 h) the level of turbulence diminishes significantly.
Fig. 16a clearly shows the asymmetry of the
simulated tidal signals. This asymmetry is revealed
by the difference between the time intervals during
which the maximum flood and ebb are established
(SWBBF). The maximum ebb velocity is observed
shortly before the rate of sea-level fall reaches its
maximum. However, the maximum flood velocity is
delayed by 2 h with respect to the maximum rate
of sea-level rise. These general properties of transports through the inlets, as explained by SFW,
reflect the case of hypsometric control of the basin
area presented in Section 2 (Fig. 4b, the black
curve).
With increasing distance from the inlet (i.e. when
approaching the coast) the asymmetry in the tidal
response changes in such a way that the velocity
maxima, which are close to each other at the time of
high water, come close to each other at the time of
low water. This means that the relative length of
the periods during which maximum flood is
established change. The numerical simulations thus
enable to distinguish two types of asymmetries
corresponding to ones following from theoretical
considerations and analysis of observations: (1) an
asymmetry driven by hypsometric control (SFW,
see also Figs. 7a and 9a), and (2) an asymmetry
dominated by the shallow channel (Fig. 4b, the red
curve).
The analysis of tidal asymmetries by SWBBF
and SFW is extended in the present paper in order
to gain a better understanding of the spatial
characteristics of the signals. To this end, we show
in Fig. 17 the time versus north–south distance
diagram for the zonally averaged surface current in
the back-barrier basins of Langeoog and Spiekeroog. The patchy structures in the back-barrier area
are due to the fact that the channel direction
changes several times, the zonal average thus
depends on the ratio between channel and flat
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a
b
c
d
Fig. 15. Vertically integrated velocity during high water (a), ebb (b), low water (c) and flood (d) phases of spring tidal cycle.
areas. More important in the present context is that
the flood maximum, particularly in the basin of
Spiekeroog, is sharper and is situated closer to the
coast. This is additional proof that the tidal
response of the Wadden Sea is characterized by a
pronounced spatial variability.
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a
b
c
d
Fig. 16. Time (expressed by number of periods) versus depth diagrams of along-channel
pffiffiffiffiffiffiffiffi currents at the deepest part of sections 2 (a) and 7
(c). The corresponding level of turbulence (measured by friction velocity u ¼ t=r) for the same sections is shown in (b) and (d). The
data are remapped from a sigma coordinate system with time variable thickness of the water column onto an equidistant z coordinate
system. This results in the stepwise change of the sea level, which is just an artefact of the vertical discretization in the z coordinate system.
The temporal evolution of the level of turbulence
(Fig. 16b,d) follows the evolution of velocity (the
time of appearance of maximum velocity almost
coincides with the time of appearance of maximum
friction velocity). The important difference between
the two patterns (on the left and on the right of
Fig. 16) is that maximum velocities are at the sea
surface (the model is tidally driven), whereas the
maximum friction velocity is at the bottom where
the turbulence is generated.
5.5. Cross-channel velocity asymmetry
The asymmetry of the along-channel velocity
presented as a function of time and cross-channel
distance is illustrated in Fig. 18. The distance in
these diagrams is proportional to the difference
between the number of grid points on the section
line (see Fig. 1 for the position of section lines). The
general transport properties are manifested by an
almost symmetrical appearance of the current cores,
‘‘coming close to each other’’ at the time of high
water (the interval between times when maximum
current is reached is shorter) and ‘‘very distant’’
(longer time interval) at low water (Fig. 18a).
Superimposed on this major asymmetry in the
tidal response is the cross-channel asymmetry
manifested by the southward (looking in the
direction of the transport) displacement of the
current core during the flooding tide and northward
(this time toward the Island of Spiekeroog) displacement of the core during the ebbing tide. This is
a usual behaviour of geophysical fluids on a rotating
earth, and the simulations (also observations, see
Section 3) demonstrate that the Coriolis force is not
negligible, although in coastal systems such as the
East Frisian Wadden Sea the first-order balance is
between friction and pressure forces.
The analysis above is not definite because the
curvature of the channels can also result in a
displacement of the core of the current during ebb
and flood. Because changing the geometry of the
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a
b
Fig. 17. Time versus distance from the coast diagrams of the cross-shore (meridional) current zonally averaged for the tidal basins of
Langeoog (upper panel) and Spiekeroog (bottom panel). Positive values correspond to landward current.
channels would necessitate another set of idealized
experiments, and it would not be straightforward to
establish the relevance of theoretical experiments to
the real channel system, we carried out a simple
experiment with a zero Coriolis parameter and
compared it with simulations of RT.
Because the effects discussed below are well
traced only in the deep channels we take a zoomin look at the central region of our area of interest.
Fig. 19a illustrates that the sea level in the coastal
zone is slightly lower in the case when we account
for the earth’s rotation. More pronounced are the
differences between simulated transports (Fig. 19c),
indicating that in the deep channels there are two
well pronounced zones where the differences are
either positive or negative. It is expected that in the
case without earth’s rotation the flow in the channel
would have a maximum in the deepest parts. If the
Coriolis force is important (in the case when we
account for the earth’s rotation) the flow will tend
to be displaced to the right (looking in the direction
of the flow). Simulations show that the velocity is
larger along the banks in the case with earth’s
rotation, which could indicate that during ebb the
core of the current is displaced to the east whereas
during flood it is displaced to the west. However, the
differences (Fig. 19c) are much smaller than the
averaged (Eulerian mean) meridional velocity (Fig.
19b), indicating that the above mentioned effects are
relatively small.
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a
b
Fig. 18. Temporal variability of current magnitude along meridional sections, (a) Section 1, (b) Section 7, see Fig. 1 for the position of
section lines. The time versus distance along the section plots are for z ¼ 1:5 m below mean sea level. The distance is given as the interval
between grid points ð200 mÞ. During a short time at low water this depth gets dry, therefore the white strips in the figure. This
representation of the results (with separating strips) was chosen because white strips represent a clear timing indicator. The length of the
section in the figure is shorter than the sections in Fig. 1 because below 1:5 m the number of wet points is smaller.
5.6. Tidal response along tidal channels
The asymmetry in the tidal response in the deep
and shallow parts of the tidal channels (described in
the previous subsection) motivated us to analyse the
transition of the signal between these two extremes
(i.e. the changing interval between the time of
occurrence of high and low water) in greater detail.
These extremes are illustrated in Fig. 20a, b by the
temporal variability of transports in the surface and
deep layers (deeper than 2 m below mean sea level).
In the deep channel, the model qualitatively
simulates the same peculiarities as discussed on the
basis of observations: a quasi-periodic signal with a
sawtooth like form, characterized by a short
descend time and a much longer ascend time.
The appearance of minima and maxima in the
upper layer almost coincide with the appearance of
these extrema in the deep layers. This is just a
demonstration that the vertical structure of the
signal is rather simple, as already shown in the EOF
analysis of the available observations. The difference in the two signals (upper and deep layer) is (1)
the magnitude of the transport, and (2) the
pronounced inflection in the upper layer curve.
The former difference is due to the fact that the
upper layer is much thinner than the deeper one.
The latter difference is associated with the fact that
the deep layer curve is more representative for the
total transport. It depends on the hypsometry of the
entire tidal basin (see SFW). In basins with a linear
hypsometry, where the permanently submerged area
is substantial, the sawtooth-like curve shows only a
small inflection. In the case when the entire basin
falls dry at low water, the curve has a form
corresponding to the upper layer curve shown in
Fig. 12 (we remind that the phase lag in the
representation of observations and simulations is
due to the different initial phase).
Another important result here is that the integral
of the upper-layer curve is larger during flood than
during ebb, which results in a net landward
transport of 40 106 m3 per tidal period across
Section 1. This transport is larger than in the
observations, which may be due either to different
forcing conditions, to insufficient space resolution
of the strait, or to smaller friction in the model than
in nature. However, for the qualitative analysis it is
more important that the upper layer transport
(landward) is compensated by a net ebb transport
in the deep channels (in the same way as in the
observations). When approaching the shallow sections, the net upper layer transport decreases (from
30 106 m3 across Section 2, down to 106 m3
across Section 5). Possible consequences of this
vertical asymmetry are discussed below.
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I
a
b
c
Fig. 19. Sensitivity of dynamics to earth’s rotation. Time mean of: (a) difference between sea level simulated with and without earth’s
rotation, (b) meridional surface currents in the case with earth’s rotation, and (c) difference between meridional surface currents simulated
with and without earth’s rotation.
The situation in the shallow channel (Fig. 20b) is
quite different from the one in the deep channel.
Here, the upper layer signal has a larger amplitude
than the signal in the deep layer. This is explained
by the fact that the contribution of the channel area
to the total area of cross-section 7 is much smaller
in comparison to the case of Section 1. This
statement is supported by the following values.
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a
b
c
d
Fig. 20. Temporal variability of transport across Section 1 (a) and 7 (b), see legend. The phase diagrams (c and d) give an idea about the
different correlation between upper and deep layer transports in deep and shallow channels. The colours in these diagrams follow as:
black, red, green, blue, and yellow, which makes it possible to see the rotation in the phase plane. The abbreviations in the plots ‘‘D. L.’’
and ‘‘S. L.’’ are for deep layer and surface layer, correspondingly.
For the cross-sections in Fig. 1, which extend over
10 grid steps ð2000 mÞ, the mean section area in the
upper layer is 3000 m2 (we remind that the
amplitude of the sea-level variability is 1:5 m).
However, the deep Section 1 has an area of 6700 m2 ,
Section 4 of 2050 m2 , Section 6 of 1200 m2 , and
Section 7 of 500 m2 . Therefore, as we have seen in
Fig. 20b, the transport across the shallow sections is
mostly dominated by the surface layer transport, the
latter being controlled by a non-linear tidal
response.
The dynamics of tidal flats are characterized by a
pronounced time lag between surface and deep layer
transports on the flats. The deep layer transport
reaches its flood maximum earlier than the surface
maximum is reached. By contrast, the ebb minimum
appears much later than the minimum in the surface
layer transport. It is thus clear that the model
captures the major variability pattern in shallow
water and we see that the sawtooth-like pattern of
the transport curves in the deep sections reverses
(faster increase of flood current and much slower
establishment of ebb current maximum), as was the
case in the theoretical analysis (Fig. 4).
Even though the tidal response appears to follow
relatively simple variability patterns (see also Section 4), the analysis of our simulations in the phase
plane described by the surface and upper layer
transport provides a better explanation of the
variability. The two ‘‘leaves’’ in the phase plane
are due to the four different slopes in the curve of
the surface transport (before and after the inflection,
and before and after maximum flood velocity).
These leaves ‘‘rotate’’ in the shallow extension of the
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channel to orient themselves along the y-axis (note
the different scales of the two plots).
Two questions arise from the comparison of the
two extreme situations as exemplified at cross-section
1 and 7. First, how representative are the total
transports for the variability in the tidal channels (we
remind that the channel in the Section 7 has a small
contribution to the area of Section 7)? Second, how
different would the results be if we analysed velocity
instead of transport (we remind that the curves in
Fig. 4 show velocity)? The answers to these questions
follow from Fig. 21, where we show the velocity
across several sections at different depths. That some
of the curves disappear during part of the tidal
825
period is due to the fact that (at the corresponding
depth of the sample results) the channel section has
fallen dry.
Fig. 21 demonstrates several important points: (1)
the signal is almost coherent at all levels, (2) the
velocity shear is larger in the shallower part of the
channel (compare Sections 1 and 2), (3) the
amplitudes of the oscillations in Section 5 are
smaller compared to those in Section 7 (this is not
observed in the case of total transport), and (4) the
slopes of the curves clearly demonstrate that a
substantial change in the shape of the tidal signal
occurs mostly in the shallow parts of the channels
where the tidal range becomes comparable to the
a
b
c
d
Fig. 21. Temporal variability of along-channel velocity sampled from the model simulations at section 1 (a), 2 (b), 5 (c), and 7 (d). The
data correspond to the deepest profile. The depths corresponding to every curve are given in the legends. The numbers in the legend give
the number of level in z coordinate system used to plot the data. The distance between levels in this coordinate system is 0:25 m. The last
number corresponds to the last layer above the bottom. Some differences in sampling depth in different stations are due to the different
depth of channels. Positive values correspond to landward transport.
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depth, (5) the three types of velocity curves revealed
in the theoretical case (Fig. 4) are well developed
along the axis of the channel. We reiterate that no
exact analogy exists between enclosed tidal basins
and channels. However, we can roughly assume that
the channels play the role of the inlet and their
drainage area the role of the tidal basin. The
agreement between 3-D simulations and the simple
theory indicates that this assumption is valid.
The transition of the tidal response along the
channels is summarized in Fig. 22 where we show
the along-channel transport across the sections in
the upper and deep layers plotted against time and
distance (for the position of the sections see Fig. 1).
The transport at the deep level has almost a
coherent appearance in the first two sections. The
change in the oscillation pattern is the largest
between Section 3 and 4 where the channel changes
direction. In addition, the channel cross-sectional
area is about three times smaller there than along
Section 1. This trend continues down to Section 7
where the cross-sectional area is an order of
a
b
Fig. 22. Time versus distance (from section 1) diagram of transports in the deep layer (a) and surface layer (b). Positive values correspond
to eastward transport.
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Fig. 23. Velocity profiles in the deepest parts of sections in Fig. 1 averaged over one tidal period. Positive values correspond to landward
transport.
magnitude smaller than the area of the deep part of
Sections 1–3. For these reasons, the effects resulting
from the shallowness of the channels and tidal flats
begin to dominate and the transport maxima are
displaced towards low water.
Unlike the case of the deep layer transports, the
surface layer ones follow almost the same variability
pattern, and the slight slopes of the contours in Fig.
22 give a measure for the retardation of the signal
on the tidal flats. The fact that the maxima are not
in the section nearest to the inlet demonstrates that
recirculations occur between Sections 1 and 4. This
could indicate that the displacement of water does
not only follow the deep channels, but also occurs
outside them.
5.7. The vertical circulation cell
The TIS drift is considered below on the basis of
numerically simulated data. As in the case of the
observations, the velocity in the deep channels
averaged over the tidal period is directed towards
the open sea in the deep layers and towards the
coast in the surface layer (Fig. 23). The zero point is
close to the level of mean low water. With increasing
distance from the open sea, the velocity shear
Fig. 24. Stokes stream function. The plot seems ‘‘shallower’’ than
the profiles in Fig. 23 because, by plotting, the shallower of the
two neighbouring stations is taken (S.No. is section number, see
Fig. 1).
decreases (green line). However, up to the shallowest extension of the channel the velocity profile
reveals a two-layer structure of the currents. At all
locations, the transport in the surface layer is
landward. The decrease of the mean velocity above
the level of the surface maximum results of the fact
that the period of averaging is the same at all
depths. However, during part of that time the
shallow levels fall dry and the current is zero.
The two-layer transport along the channel is well
represented by the Stokes stream function (Fig. 24).
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The larger size fractions would become subject to
export by ebb currents in the deep channels.
This stream function is defined as
u¼
w¼
dc
,
dz
(46)
5.8. The role of buoyancy
dc
dx
(47)
and illustrates the landward transport in the surface
layer and the seaward transport in the deep layer.
Accounting for the vertical circulation cell, we can
anticipate that the finer sediments (dominating the
surface layer) are transported towards the coast.
Here, we compare results from simulations RT
and RTB with the aim to estimate the role of
baroclinicity and its importance for surface transports in a very shallow ocean. The difference
between surface currents (Fig. 25b) and TIS
transport (Fig. 25c) in the two experiments averaged
for one month proves that, although we deal with a
a
b
c
Fig. 25. Time averaged difference between baroclinic and barotropic simulations (‘‘baroclinic-barotropic’’). (a) Sea level, (b) meridional
surface currents, (c) meridional TIS transport.
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well mixed estuarine system, the baroclinicity has a
pronounced effect on the long term averages. The
major difference between surface velocities in the
two simulations is observed in the coastal zone (a
ROFI regime). In these areas, fresh water enhances
the seaward surface transport. While this result can
be expected, the origin of pronounced differences
between the two simulations north of the barrier
islands (Fig. 25b) is less clear. Presumably, it is due
to ‘‘retention’’ of fresh water in several areas (see
differences in sea level in Fig. 25) or the joint effect
of baroclinicity and relief.
In the context of the issues addressed in this
paper, it is more important to analyse the differences in the TIS transport simulated in a baroclinic
and a barotropic case (Fig. 25c). The general
conclusion from this plot is that in the deep areas
(north of the barrier islands and in the deep
channels) the relative difference of meridional
velocities is negative. Thus, in these areas, the TIS
transport in a baroclinic ocean is enhanced. On the
tidal flats, by contrary, baroclinicity works against
the TIS transport. It may be expected that the two
effects tend to compensate each other. The mean
transport due to the difference between the baroclinic and the barotropic case is comparable to the
mean transport shown in Fig. 19b, which reveals
that, even in the well mixed estuaries, baroclinicity
cannot be neglected when addressing long-term
processes.
6. Summary and conclusions
In this paper, we have analysed the tidal response
of a typical back-barrier tidal flat in the East Frisian
Wadden Sea. Simple theoretical concepts of a
shoaling and narrowing tidal inlet connecting the
open sea with the tidal flat allowed us to demonstrate the existence of quite distinct response
patterns in velocity and transports. Using ADCP
observations for spring and neap tide periods we
were able to extract the major spatial patterns and
their time variability (EOF-Analysis). Numerical
simulations based on a 3-D primitive equation
model were used to further quantify the horizontal
and vertical circulation in the tidal channel and on
the tidal flats. The major results are the following:
(i) near surface transports (tidally averaged) are
landward and are dominated by a TIS drift, whereas
deeper layer transports are seaward and are defined
by the hypsometric properties of the connected
intertidal basin, (ii) in the deeper parts of the tidal
829
channel, the tidal signal shows a clear ebb dominance, whereas on the shallower parts of the
channels and the intertidal flats we observe a
flood-dominance, (iii) dynamics associated with
flooding and drying tend to oppose the TIS
transport, (iv) baroclinicity due to the fresh water
flux from the coast also opposes the TIS transport,
as could be expected. However, the changes of
dynamics in the deep channels and north of the
barrier islands are something less trivial. In these
areas, effects of baroclinicity add to those of the TIS
transport.
Some authors assume that the landward transport
of fine sediment is a result of a transport asymmetry
dominated by the flood current (Groen, 1967;
Postma, 1982). Larger flood velocities or temporal
asymmetries would thus create a more efficient
mechanism for the resuspension of sediments
deposited at slack tide. This contributes to a longer
retention time of suspended particles in the water
column as a result of which they are gradually
transferred towards the coast. However, as demonstrated by observations and model simulations
(SWBBF and SFW), the tidal channels in the East
Frisian Wadden Sea are ebb dominated and the
channel networks are similar to a fluvial drainage
system on land (Flemming and Davis, 1994;
Flemming, 1998). This apparent contradiction has
been resolved in the present study by demonstrating
that the tidal flats of the Wadden Sea are in fact
flood dominated, and that the resulting asymmetry
evidently promotes a landward transport of progressively finer sediments. This is explained by the
vertical asymmetry in the transports, and the basic
process behind this asymmetry is the TIS transport.
Acknowledgements
We are indebted to H. Burchard and K. Bolding for
making GETM available to us. The help of N. Saleck
in processing ADCP data is also acknowledged.
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